Formal Objects and Propositions in Megalodon as of February 2021

Chad E. Brown

Draft of February 11, 2021

2

Contents

1 Basic Logic  11
2 Equality  13
2.1 Eq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Notation for Equality . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Functional Extensionality . . . . . . . . . . . . . . . . . . . . 14
3 Existential Quantifiers  15
3.1 Ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Notation for Existential Quantifiers . . . . . . . . . . . . . . . 15
4 Further Primitives and Axioms  17
5 Some Basic Results  21
5.1 PropN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Exactly 1 of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.4 Exactly 1 of 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5 More Basic Results . . . . . . . . . . . . . . . . . . . . . . . . 41
5.6 If-then-else on Sets . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Basic Set Theory  45
7 Natural Numbers I  65
8 Ordinals  83
9 Comparing the Sizes of Sets  89

10 Misc  95
11 Description and If-then-else  99
11.1 Descr_ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.2 Descr_iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.3 Descr_iio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.4 Descr_Vo1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.5 Descr_Vo2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.6 If_ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.7 If_iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.8 If_Vo1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11.9 If_iio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11.10If_Vo2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12 Recursion on Sets  107
12.1 EpsilonRec_i . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
12.2 EpsilonRec_ii . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
12.3 EpsilonRec_iii . . . . . . . . . . . . . . . . . . . . . . . . . . 108
12.4 EpsilonRec_iio . . . . . . . . . . . . . . . . . . . . . . . . . . 109
12.5 EpsilonRec_Vo1 . . . . . . . . . . . . . . . . . . . . . . . . . 109
12.6 EpsilonRec_Vo2 . . . . . . . . . . . . . . . . . . . . . . . . . 110
13 If-then-else, Description and Recursion again  111
13.1 If_Vo3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
13.2 Descr_Vo3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
13.3 EpsilonRec_Vo3 . . . . . . . . . . . . . . . . . . . . . . . . . 112
13.4 If_Vo4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
13.5 Descr_Vo4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
13.6 EpsilonRec_Vo4 . . . . . . . . . . . . . . . . . . . . . . . . . 114
14 Predicates and Relations  115
15 Zermelo’s Well-Ordering Theorem  121
15.1 Zermelo1908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
16 More Logical Properties  125

17 Natural Numbers II  127
17.1 NatRec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
17.2 NatArith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
18 Natural Numbers III and Ordinals  143
19 Disjoint Unions  155
20 Pairs, Sums, Functions and Products  163
20.1 pair_setsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
20.2 Dependent Sums . . . . . . . . . . . . . . . . . . . . . . . . . 168
20.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
20.4 Tuples as Functions on Naturals . . . . . . . . . . . . . . . . . 177
20.5 Dependent Products . . . . . . . . . . . . . . . . . . . . . . . 178
20.6 Pairs as Tuples . . . . . . . . . . . . . . . . . . . . . . . . . . 181
20.6.1 Abstracting with Two Variables . . . . . . . . . . . . . 182
20.7 Encodings of Functions and Predicates as Sets . . . . . . . . . 186
20.7.1 Tuples with More Than 2 Components . . . . . . . . . 197
20.8 Notation for Sums and Products . . . . . . . . . . . . . . . . . 206
21 Surreal Numbers I  211
21.1 Surreal Numbers as Predicates on Ordinals . . . . . . . . . . . 211
21.2 Surreal Numbers as Sets . . . . . . . . . . . . . . . . . . . . . 236
21.3 TaggedSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
21.4 Surreal Numbers as Sets II . . . . . . . . . . . . . . . . . . . . 245
21.5 TaggedSets2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
21.6 Ordinals are Surreal Numbers . . . . . . . . . . . . . . . . . . 269
21.7 SurrealRecI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.8 SurrealRecII . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
21.9 SurrealRec2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
21.10More Results about Surreal Numbers . . . . . . . . . . . . . . 273
22 Explicit Number Structures  303
22.1 explicit_Nats . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
22.2 Omega gives Explicit Naturals . . . . . . . . . . . . . . . . . . 305
22.3 explicit_Nats_zero . . . . . . . . . . . . . . . . . . . . . . . . 306
22.4 explicit_Nats_one . . . . . . . . . . . . . . . . . . . . . . . . 307
22.5 explicit_Nats_transfer . . . . . . . . . . . . . . . . . . . . . . 311

23 Groups  313
23.1 AssocComm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
23.2 Group1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
23.2.1 Group1Explicit . . . . . . . . . . . . . . . . . . . . . . 314
23.2.2 Group1Explicit2 . . . . . . . . . . . . . . . . . . . . . 318
Group1Explicit2RepIndep . . . . . . . . . . . . . . . . 318
23.2.3 Group1Explicit3RepIndep . . . . . . . . . . . . . . . . 319
23.3 Groups Encoded as a Set . . . . . . . . . . . . . . . . . . . . . 320
23.4 Group2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
23.5 Group3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
23.6 Subgroup as a Relation on Encoded Groups . . . . . . . . . . 324
23.7 Group4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
23.8 Groups of Permutations . . . . . . . . . . . . . . . . . . . . . 327
23.9 Group2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
24 Rings  331
24.1 explicit_Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
24.2 explicit_Ring_with_id . . . . . . . . . . . . . . . . . . . . . . 334
24.3 explicit_Ring_with_id_RepIndep2 . . . . . . . . . . . . . . . 339
24.4 explicit_CRing_with_id . . . . . . . . . . . . . . . . . . . . . 340
24.5 explicit_CRing_with_id_RepIndep2 . . . . . . . . . . . . . . 345
24.6 Packing two Binary Operations and a Constant into a Set . . 346
25 Explicit Reals  351
25.1 explicit_Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
26 Rings II  365
26.1 CRing_with_id . . . . . . . . . . . . . . . . . . . . . . . . . . 366
27 Explicit Reals II and Fields  371
27.1 explicit_Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
27.1.1 explicit_Reals_Q_min_props . . . . . . . . . . . . . . 372
27.1.2 Q is Minimal as a Subfield of R . . . . . . . . . . . . . 373
27.2 explicit_Field_transfer . . . . . . . . . . . . . . . . . . . . . . 373
27.3 explicit_Field_RepIndep2 . . . . . . . . . . . . . . . . . . . . 374
27.4 Fields Encoded as Sets . . . . . . . . . . . . . . . . . . . . . . 375
27.5 explicit_OrderedField_transfer . . . . . . . . . . . . . . . . . 376
27.6 Selectors for Fields . . . . . . . . . . . . . . . . . . . . . . . . 376

27.7 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
27.8 Field2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
27.9 Basic Results about Fields . . . . . . . . . . . . . . . . . . . . 390
27.10explicit_Reals_transfer . . . . . . . . . . . . . . . . . . . . . 396
28 Surreal Numbers II  397
28.1 SurrealArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 397
28.2 Complex Surreals . . . . . . . . . . . . . . . . . . . . . . . . . 434
28.3 Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
28.4 Complex II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
28.5 Int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
28.6 Packing Two Operations, a Relation and Two Constants … 445
28.7 explicit_OrderedField_RepIndep2 . . . . . . . . . . . . . . . 453
28.8 Unpacking Ordered Fields . . . . . . . . . . . . . . . . . . . . 454
28.9 explicit_Reals_RepIndep2 . . . . . . . . . . . . . . . . . . . . 455
28.10Unpacking Explicit Reals . . . . . . . . . . . . . . . . . . . . . 455
28.11RealsStruct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

8 CONTENTS

Preface

This document lists most of the objects defined and propositions proven in the Egal theory of Proofgold (using Megalodon) as of February 2021. When relevant, corresponding 256-bit identifiers and 162-bit addresses are given for objects and propositions so that users can find them in the Proofgold blockchain. The objects and propositions here are those distributed with

Megalodon in the file PfgEFeb2021Preamble.mgs.

The document is a draft and much of it was automatically generated (using the -latex option in Megalodon), so please overlook any formatting problems.

10 CONTENTS

Chapter 1

Basic Logic

Primitive. The name Eps_i is a primitive term of type (ι → o) → ι. Specifically it is “Primitive 0” in the Proofgold Egal theory.

Axiom 1.1 ∀P : ι → o.∀x : ι.P x → P (Eps_i P). The proposition is identified by the following information:

Pure Prop Id: c3f0de4cb966012957ca752938aa96a32c594389e7aea45227d571c0506618ba
Pure Prop Address: TMaqNdrw1xUUqfmGqr4T5HNsS6X1kThiGJp
Theory Prop Id: 756d6e520a540dee983a2bc983f030fca1e15bc5c01cd10daa59ba9846775a89
Theory Prop Address: TMZeDQEPZjtPy34irdW6qoDmhfGuT9rLoo3

Definition 1.1 We define True to be ∀p : o.p → p of type o identified by the following information:

Pure Object Id: f81b3934a73154a8595135f10d1564b0719278d3976cc83da7fda60151d770da
Pure Object Address: TMXh1PeTNoRdeQMkipsnTZ1NBoLMCSQGszg
Theory Object Id: 94e6a74d7fb4010e37ca58528b5eefb6d84e85f0fcd9598afcfdb3000334a38a
Theory Object Address: TMdZywkjk5BP16zSdFbfK4CcoVzmBS54DHA

Definition 1.2 We define False to be ∀p : o.p of type o identified by the following information:

Pure Object Id: 1db7057b60c1ceb81172de1c3ba2207a1a8928e814b31ea13b9525be180f05af
Pure Object Address: TMVcBGjSsw48XJ4L45VWUk51f7pywF6kvVX
Theory Object Id: 266ad502104b0eadbdf70b5ce22ec14cce255c0b312cfc3719813f538ed6de1b
Theory Object Address: TMZx4Q2psRTfsQjwWdVEVQnyn3ZRxXosKPq

Definition 1.3 We define not to be λA : o.A → False of type o → o identified by the following information:

Pure Object Id: f30435b6080d786f27e1adaca219d7927ddce994708aacaf4856c5b701fe9fa1
Pure Object Address: TMWAvR96ZU4RCj94CSoZhrtMR5rdaTaLB5R
Theory Object Id: 0dc3c779c7cc720c63503b5ae618908388fe9f8ef7b20ce33a5baa0de6389c1b
Theory Object Address: TMJJZawymNdUcMFLBwRqSW8DSQsHg76ans7

12 CHAPTER 1. BASIC LOGIC

Notation. We use ¬− as a prefix operator corresponding to applying term not.

Definition 1.4 We define and to be λAB : o.∀p : o.(A → B → p) → p of type o → o → o identified by the following information:

Pure Object Id: 2ba7d093d496c0f2909a6e2ea3b2e4a943a97740d27d15003a815d25340b379a
Pure Object Address: TMHVPoyLuccMxozW2JkzTcKaFNorhQvcqVz
Theory Object Id: 37da83639bcf9d8c8a37734df867cfc15d7d22573b6c8ccfc52e5c2e7a5abc81
Theory Object Address: TMMEUAXx5wQ61b6khohvEMcR5eUbjjJSYAw

Notation. We use ∧ as a left associative infix operator corresponding to applying term and.

Definition 1.5 We define or to be λAB : o.∀p : o.(A → p) → (B → p) → p of type o → o → o identified by the following information:

Pure Object Id: 9577468199161470abc0815b6a25e03706a4e61d5e628c203bf1f793606b1153
Pure Object Address: TMRrSAiwhsGG8gNN8EukN1zLVibyVKQXaFq
Theory Object Id: 86ae6491fbd8f20f1f7e4986f09463cdcc489bdedfda845c840df66cb8f76edb
Theory Object Address: TMbacaYWYcRwAFxG4ALNMm3EyDKWEtRrh42

Notation. We use ∨ as a left associative infix operator corresponding to applying term or.

Definition 1.6 We define iff to be λAB : o.and (A → B) (B → A) of type o → o → o identified by the following information:

Pure Object Id: 98aaaf225067eca7b3f9af03e4905bbbf48fc0ccbe2b4777422caed3e8d4dfb9
Pure Object Address: TMNPkfzmoJmg4Qsbe28fcbyaXL1XTynPhik
Theory Object Id: 6588e5a76078e7919b50307e23503d854af1b869a2eedd6b091f6610609777ba
Theory Object Address: TMc6fJsUiuiG3KCWP4EAREq7NWM56fQqJHi

Notation. We use ⇔ as an infix operator corresponding to applying term iff.

Equality

2.1 Eq

Let α be a type.

Definition 2.1 We define eq to be λxy : α.∀Q : α → α → o.Q x y → Q y x of type α → α → o.

Since Proofgold has no polymorphism, there is no single object corresponding to eq. Instead there is a constructor Eq that takes a type argument and is always expanded away using the definition above as part of normalization.

Definition 2.2 We define neq to be λxy : α.¬eq x y of type α → α → o.

Again, there is no single object in Proofgold corresponding to neq. Instead it is represented by its definition which is then expanded away as part of normalization.

2.2 Notation for Equality

Notation. We use = as an infix operator corresponding to applying term eq. Notation. We use ≠ as an infix operator corresponding to applying term neq.

14 CHAPTER 2. EQUALITY

2.3 Functional Extensionality

Let αβ be types. Axiom 2.1 ∀fg : α → β.(∀x : α.f x = g x) → f = g. Functional extensionality is built in as part of the proof term data structure in Proofgold, so this proposition has no counterpart as a proposition published in the chain.

Existential Quantifiers

3.1 Ex

Let α be a type.

Definition 3.1 We define ex to be λQ : α → o.∀P : o.(∀x : α.Q x → P) → P of type (α → o) → o.

Since Proofgold has no polymorphism there is no single object corresponding to existential quantification. Instead there is a constructor Ex that takes a type argument and is always expanded away using the definition above as part of normalization.

3.2 Notation for Existential Quantifiers

Notation. We use ∃x, …, y. as a binder notation corresponding to a term constructed using ex.

16 CHAPTER 3. EXISTENTIAL QUANTIFIERS

Further Primitives and Axioms

Axiom 4.1 ∀pq : o.iff p q → p = q. The proposition is identified by the following information:

Pure Prop Id: 00ac81fe17c156a4ac6f8a4b0f730686a7acf0e26a713a5d0e7f9aab342ef273
Pure Prop Address: TMT4ViLziW9cx8676T7siQNUsc4pLGajCAh
Theory Prop Id: 1d1e565be0b2225dbdd84a1b9f843374545e89ad6f25ebf6856e801170f8d60b
Theory Prop Address: TMK6YdqYUKB5vJfwfTMJNgJJGj1TmDiA7rj

Primitive. The name In is a primitive term of type ι → ι → o. In particular it is “Primitive 1” in the Proofgold Egal theory.

Definition 4.1 Notation. We use ∈ in infix notation for In.

We define Subq to be λAB.∀x ∈ A.x ∈ B of type ι → ι → o identified by the following information:

Pure Object Id: 81c0efe6636cef7bc8041820a3ebc8dc5fa3bc816854d8b7f507e30425fc1cef
Pure Object Address: TMbQvibiSdPYTcyHirAiT7dojaZd9QVHnnx
Theory Object Id: 317007aa687cfc3fa6d33a4d45b72e2b84878595f194108dee11852a04950639
Theory Object Address: TMJciJ4WVfdZbE7XT2SUcAkkoSYiF2gabRX

Notation. We use ⊆ in infix notation for Subq.

Axiom 4.2 ∀XY : ι.X ⊆ Y → Y ⊆ X → X = Y. The proposition is identified by the following information:

Pure Prop Id: b3b6981b6de9a71189167e6208f1fa2be38ade4bd2a77084c6878323e6d97ab9
Pure Prop Address: TMP6dV9kMfpLcfVgkH1rTbsurTnQKfFkFUf
Theory Prop Id: 21238ac2a61503233d86f0ab5bbd1cfebd49fecaa7aa3a131715c5e08003430e
Theory Prop Address: TMQ3CJdXpjT9Vyjdd8x58bzAfVa2uHKMmnZ

Axiom 4.3 ∀P : ι → o.(∀X : ι.(∀x ∈ X.P x) → P X) → ∀X : ι.P X. The proposition is identified by the following information:

18 CHAPTER 4. FURTHER PRIMITIVES AND AXIOMS

Pure Prop Id: 04521121bc85ca3c89ee96767237629584fc96b34de1cf1a3f790ea4af4b0047
Pure Prop Address: TMZDd5XWnW8qoDTqVGuvgh8AQGEhcN5haVH
Theory Prop Id: 20faa51007fd5aff9dc54e343b9c7b57fab2eb829c71a0daa2e90c858707d2b7
Theory Prop Address: TMMzAhfU4QKJhywTdGXVuR4P8u6KoKixxYw

Notation. We use ∃x, …, y. as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and. Primitive. The name Empty is a primitive term of type ι. In particular it is “Primitive 2” in the Proofgold Egal theory.

Axiom 4.4 ¬∃x : ι.x ∈ Empty. The proposition is identified by the following information:

Pure Prop Id: 4da93150e41630a98ea3fc30e9bc894dee0da4364b020ac8ee0a79d04508434f
Pure Prop Address: TMQMLLe8Cz5vwfXB4uUFfsKFVfvwBMf4VPJ
Theory Prop Id: c7b0d7f466fd9528f6264e7a3a9eaeff8c18159e8679b3d6b840d2689bda8486
Theory Prop Address: TMaPKWsC7HDHhrC2zYBmYcX19RqLy2VefbW

Primitive. The name Union is a primitive term of type ι → ι. In particular it is “Primitive 3” in the Proofgold Egal theory.

Axiom 4.5 ∀Xx.x ∈ Union X ⇔ ∃Y.x ∈ Y ∧ Y ∈ X. The proposition is identified by the following information:

Pure Prop Id: bf4c263ac4776c12ed6de40b226c687445f1e634eddf121ad52b2d095224ec74
Pure Prop Address: TMHRWvQi72bggXNdrukFL23tqhkHk6iri2y
Theory Prop Id: 40660fae1988a12c0dd433fed65b5971371ce274cfa7acb0152f6c8b11856323
Theory Prop Address: TMdiuFh9LBKb2nXjbQi7GzQCn57wX2izbEV

Primitive. The name Power is a primitive term of type ι → ι. Specifically it is “Primitive 4” in the Proofgold Egal theory.

Axiom 4.6 ∀XY : ι.Y ∈ Power X ⇔ Y ⊆ X. The proposition is identified by the following information:

Pure Prop Id: 746e5a9564518580f673371a7891c1db2e38c91762155f302694c6b4cbf7c649
Pure Prop Address: TMTzabXXxRmuTxrdG42VtpMDWCUmgK2UMgY
Theory Prop Id: d67d718c245e91b738c984751e77f84e6db2b59131a6dd3dbf033e5c5b151dc0
Theory Prop Address: TMbRJTDNmcR6KZC6t4TGXakKw4XSAYS3VN8

Primitive. The name Repl is a primitive term of type ι → (ι → ι) → ι. Specifically it is “Primitive 5” in the Proofgold Egal theory.

Notation. We write Repl A (λx.s) as {s|x ∈ A}.

Axiom 4.7 ∀A : ι.∀F : ι → ι.∀y : ι.y ∈ {F x|x ∈ A} ⇔ ∃x ∈ A.y = F x. The proposition is identified by the following information:

Pure Prop Id: 4f2357276dc7f58ae12c278fb7f133f1de0a4e899d0a1b4df22db544ef8fbfea
Pure Prop Address: TMHvGDzbjKwgKUfmhz4YstLTy6FK6VEf8VX
Theory Prop Id: 471122ee9aecdb9185d0ed855c4fd074596525d6ade7150ea945d05f732c1d91
Theory Prop Address: TMQwYTqizk3ouisSfLQUbv5JP258W4VhxxV

Definition 4.2 We define TransSet to be λU : ι.∀x ∈ U.x ⊆ U of type ι → o identified by the following information:

Pure Object Id: 538bb76a522dc0736106c2b620bfc2d628d5ec8a27fe62fc505e3a0cdb60a5a2
Pure Object Address: TMFFkFhggUWb8iaKbBVFF2NXQtBDp9ocvey
Theory Object Id: e69f92130adf04e6fb850fe93c130a82c6a84226e2914139b5b46a4c8b46dd59
Theory Object Address: TMaE1Wfu3Ekit1RP2q2rs1eExhVT4eYQUWZ

Definition 4.3 We define Union_closed to be λU : ι.∀X : ι.X ∈ U → Union X ∈ U of type ι → o identified by the following information:

Pure Object Id: 57561da78767379e0c78b7935a6858f63c7c4be20fe81fe487471b6f2b30c085
Pure Object Address: TMGQhGvaoAR8Ac54JYsvH3PiWjPoz8NS8J6
Theory Object Id: e45ed946b4201c2ac3bce08c594bcc5b37e16935b9de8d06f9515256bfcb6c5d
Theory Object Address: TML288688fkgDpJdibigbTFnrScrqv2iKt8

Definition 4.4 We define Power_closed to be λU : ι.∀X : ι.X ∈ U → Power X ∈ U of type ι → o identified by the following information:

Pure Object Id: 8b9cc0777a4e6a47a5191b7e6b041943fe822daffc2eb14f3a7baec83fa16020
Pure Object Address: TMGEQQpyASUMyKRHqWbnFXopjtYsATE5AVk
Theory Object Id: d2954ea0f8a59a40ddc3fc5be9d473a03e8b92e4a279ea7e2bf10debed186ffd
Theory Object Address: TMNbYYR8P3gh4AeJAfTdNEMMkiD5HvA97jx

Definition 4.5 We define Repl_closed to be

λU : ι.∀X : ι.X ∈ U → ∀F : ι → ι.(∀x : ι.x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U

of type ι → o identified by the following information:

Pure Object Id: 5574bbcac2e27d8e3345e1dc66374aa209740ff86c1db14104bc6c6129aee558
Pure Object Address: TMHsmhDUjBaGfXeB9XmeViaPhpFM3GPGFEK
Theory Object Id: 5135346e3c2b9f78fd2ca0f8dc77f2075a100d43dfdecf64bc2792a54493f446
Theory Object Address: TMFer2uiF5iECeDKX8Yjf4jvbfj2RQYen9J

Definition 4.6 We define ZF_closed to be λU : ι.Union_closed U ∧ Power_closed U ∧ Repl_closed U of type ι → o identified by the following information:

Pure Object Id: 1bd4aa0ec0b5e627dce9a8a1ddae929e58109ccbb6f4bb3d08614abf740280c0
Pure Object Address: TMK8id4rjoGzRQvGv2RVDkaCpE2gvidq2jS
Theory Object Id: e1eccf55c627a83d82a0c5fbb2d8effe908afbba6259c3a3a9f7af4839904407
Theory Object Address: TMKM8HgiAgaT2ShpWewk4PuTK8LpKuESdvs

Primitive. The name UnivOf is a primitive term of type ι → ι. In particular it is “Primitive 6” in the Proofgold Egal theory.

20 CHAPTER 4. FURTHER PRIMITIVES AND AXIOMS

Axiom 4.8 ∀N : ι.N ∈ UnivOf N. The proposition is identified by the following information:

Pure Prop Id: efcc1652fa565b11ad9b7e6b61992946e170a56766124011245e25a95ef533d8
Pure Prop Address: TMWDMXfHXNT3hXXgPN8FdFXa6TtH5VEC8L2
Theory Prop Id: e805d44e6e3b129108fe3a5d7b86456ab1ded7fbabdb1e013db339c19e74fc3c
Theory Prop Address: TMXQAwXZaXUD3yTSAm1fLH7g2HqyDMYutzR

Axiom 4.9 ∀N : ι.TransSet (UnivOf N). The proposition is identified by the following information:

Pure Prop Id: 05357566d69c1e942fba918ba630fe1befcd3b76747edaa23bf9e22c07e0bf80
Pure Prop Address: TMPFnHy6wH7xHdqPH1LVnFGKXNXBVDCGhcc
Theory Prop Id: cf4f5cdd03fb3fcd9b880d27f1bb10dfa697dda82fb891d4f01e8901f99e0ddd
Theory Prop Address: TMQDPgN5NEyK5uGnLoLPsCtuYVhgM8BFwd8

Axiom 4.10 ∀N : ι.ZF_closed (UnivOf N). The proposition is identified by the following information:

Pure Prop Id: 770e356024df94789dc767ad29154fb817b6339b6526f6b4826cc678ae6937ab
Pure Prop Address: TMUaRKj3xqMnJ3tpKZ46beaVE82Vps7qYVG
Theory Prop Id: 0e736334ea6ea7d6454eede353e88ba0fa7a5e4c3a17f6c90921a2414dab6d08
Theory Prop Address: TMK8JSDxgUygbLA49dTtQCjnRY2ijj3pG6u

Axiom 4.11 ∀NU : ι.N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U. The proposition is identified by the following information:

Pure Prop Id: 042dda2b34a2ebfa193a104324faf5219666788a71551ab9dd079136a7c6281d
Pure Prop Address: TMSbghKuyRptdLDe7mXjA1e8YRJUc3PRgkx
Theory Prop Id: 1f793348811ae94e9d72c764a3e780c39a9a2f45f55507133ecc732e15133ce6
Theory Prop Address: TMGeTmqMVx11mKzRV46CRSj6xUcaf13udQk

Some Basic Results

FalseE

Theorem 5.1 False → ∀p : o.p. The proposition is identified by the following information:

Pure Prop Id: 0dcfe33e27797b7b436d2f4d048a94573a53d6753db4a3cc25933a3dd6f89015
Pure Prop Address: TMKsmG8CTue9Uq3xFgtUwj8xZhW6Vsv7LaE
Theory Prop Id: f9a230def2f4412d999116d577818e7c13149b3ffd6ec8c0f216fb2570056eab
Theory Prop Address: TMJrbzrCp7MPJvNNanPW79aqzPDUV4gcM4o

TrueI

Theorem 5.2 True. The proposition is identified by the following information:

Pure Prop Id: f81b3934a73154a8595135f10d1564b0719278d3976cc83da7fda60151d770da
Pure Prop Address: TMXh1PeTNoRdeQMkipsnTZ1NBoLMCSQGszg
Theory Prop Id: eb5cbf9e00c1dc6fbac87f0e7dd91901a69a5afc430de9f265def7009d06523d
Theory Prop Address: TMdaMQ9TGYHTUYWtAiezq6j2m9Lh8nTGaJX

notI

Theorem 5.3 ∀A : o.(A → False) → ¬A. The proposition is identified by the following information:

Pure Prop Id: 67ebfbb7241f6a92dbb75a0571d5396f165877e75a006c85956aeff1c756032a
Pure Prop Address: TMMfBXPbPGW7MmkNMsTQVtkyifxwbPw9VeV
Theory Prop Id: 12f948374053b3cf6da14578610534d466feec847f6e53f73e2bb362ee644abb
Theory Prop Address: TMUAmMSt4Wx5Red5pwuUbb3SEBepWAW1UwN

notE

22 CHAPTER 5. SOME BASIC RESULTS

Theorem 5.4 ∀A : o.¬A → A → False. The proposition is identified by the following information:

Pure Prop Id: 23bb78c7c174d96fa3a932c82b14d7e88bfb71751aaaf84541df8ff1a4ceaf2e
Pure Prop Address: TMVPdAHpYAjBMsM3in5F2sssApoxWNtPMfR
Theory Prop Id: 856e325af2a33a3529471e559a4573ad6338bfc260d1c5073519e9b223161c81
Theory Prop Address: TMe11m2tYgGjWUJtGEX5X1tvrSDZWn8w2j1

andI

Theorem 5.5 ∀AB : o.A → B → A ∧ B. The proposition is identified by the following information:

Pure Prop Id: 3c4f59ac38ec13161d275eff419ecea194e7a962d51e721d2c93249a9130e97c
Pure Prop Address: TMdtgzJ3v8tqbJfC6ZLrrzp79xerKpJKvkB
Theory Prop Id: 3bf1985f2198e37de5686038da57a16773e504ab6257e9686d49bfa2edd1762f
Theory Prop Address: TMFJPpp7jnkJTJUfrH6ULPhVR9zhYrkB16u

andEL

Theorem 5.6 ∀AB : o.A ∧ B → A. The proposition is identified by the following information:

Pure Prop Id: c2f006baa588b23089bf32c5f1ecbbf59d67cec883a690fc7a3dfd4f92b52e28
Pure Prop Address: TMNrPqMjXhvQDz1y7fMcUrWvqz8EvpMAuGq
Theory Prop Id: 718be46f06f0025b5dbef8d8ac0138d307c9b79d1e220278818af7e7137afaf1
Theory Prop Address: TMP2MCwM6msNpa6CvfXzjby7fxBAXQCafdT

andER

Theorem 5.7 ∀AB : o.A ∧ B → B. The proposition is identified by the following information:

Pure Prop Id: f66b82a6432b84ea5d3544eec5c9459bd603e2c74b7dc20fdb37c6c0278a30d2
Pure Prop Address: TML2fqTH52b5jHHUqj4UWsE6pCJqVKiMDrP
Theory Prop Id: 5caea9b570346a0a4827977bc317350a63bfb51530b8489373b29970868e822a
Theory Prop Address: TMRHXWMGqfM8cUBGJnz16hksWXqTpSxL6AN

orIL

Theorem 5.8 ∀AB : o.A → A ∨ B. The proposition is identified by the following information:

Pure Prop Id: 55be8c084d3d69aaadcdfedd53b12314b3f00b7aa0fe62bba95d8e20a6913eef
Pure Prop Address: TMcR7JViW1Pt5HqJG1ZrAu2LnZ6BoYWBfiw
Theory Prop Id: 38267393635d01946334424cfe8e37c8739b8cefeb185f7310c8efc17536d3b6
Theory Prop Address: TMWwPCgtJ9GA5byYB2oN3wyknGgpGw6sABs

orIR

5.1. PROPN 23

Theorem 5.9 ∀AB : o.B → A ∨ B. The proposition is identified by the following information:

Pure Prop Id: 3c63fc5739c09fc9b12fb6d378e9de249f610fbbf2df1cce28f85000280f7bfb
Pure Prop Address: TMTfYZuPbL4xJdZTnLMEYGVXQTCZrDeAcef
Theory Prop Id: 8d8e400885f8d52e5c13e42b0127916d2d04fa1a6277be7ce7b01afd1b7e90bc
Theory Prop Address: TMShT5hWbYuTfR7XBLwEnzy4XoJLhwhTWfs

orE

Theorem 5.10 ∀ABC : o.(A → C) → (B → C) → A ∨ B → C. The proposition is identified by the following information:

Pure Prop Id: 5af08e7b4b57e0dbdc28c2fff18dfe38e04e61787986420a7b92d8802bd7ebef
Pure Prop Address: TMRxb9x7hsQcQdArwJPp9wuFPEjCqtSpjvD
Theory Prop Id: 27f42ce3892a9e7634f07197c79efa6d1892500cb92de0ba160f103333bae4e5
Theory Prop Address: TMXA3QBiXM4MyCoYLxr1N5ZWgF5hLhEUP3r

5.1 PropN

Let P 1P 2P 3 : o be given.

and3I

Theorem 5.11 P 1 → P 2 → P 3 → P 1 ∧ P 2 ∧ P 3. The proposition is identified by the following information:

Pure Prop Id: 23ac9ca708caa1dd38b9d722781b028bb2d1c60d27249f613f0f58a46f811df6
Pure Prop Address: TMTCLZz943DGXmGagi5cxC1VhkU7Xh7gmxj
Theory Prop Id: cb999bf5d18c324be0801b18f227a943efbdaa196adbe869b96f43be18f1790d
Theory Prop Address: TMHU8VN1xhmJmu7CW62fpKC6YQYCHXWhdNt

and3E

Theorem 5.12 P 1 ∧ P 2 ∧ P 3 → (∀p : o.(P 1 → P 2 → P 3 → p) → p). The proposition is identified by the following information:

Pure Prop Id: 445f7614b5a460ac2ee2c65f60676f43814b16323277b97c9e03b7e2d5ef1bca
Pure Prop Address: TMHw71AzVXN296b5iiE9yrMayr1gPK6Dmgn
Theory Prop Id: 50fabe4d504ce92226b4ca92d7a1ab747f919e6c3d6c87a33765f260310d73ef
Theory Prop Address: TMRpTuB7NQrKHNzC82mBhAzGZdBLPpwn8YZ

or3I1

Theorem 5.13 P 1 → P 1 ∨ P 2 ∨ P 3. The proposition is identified by the following information:

24 CHAPTER 5. SOME BASIC RESULTS

Pure Prop Id: e44f34f452d8efd0440da0051ba3966fc604cdf0f1a7d5a43a24ebca3f6477e6
Pure Prop Address: TMY8tfGHE5SpXjBtYW9GnGTRRRYZjTkVC1h
Theory Prop Id: ef9071fe3a6dd6111bf8b4b88913d688c289a81f6af948df2b99c0f80d2ac6c1
Theory Prop Address: TMKCha5JJm2rxV2CyYeXrokWFDR2ryrF4rE

or3I2

Theorem 5.14 P 2 → P 1 ∨ P 2 ∨ P 3. The proposition is identified by the following information:

Pure Prop Id: 8df6e153401808c8d1877008f68f33f35f4bb6901c96d857f1e0f13e1519cce6
Pure Prop Address: TMcANDxLDCL4xKQuqhsiQV2MPpGnTrcX2tb
Theory Prop Id: 827a91555bc9a5f9fbc87e42c706559e78927705587b35109c356ece2a13307d
Theory Prop Address: TMJjiRdEq1FHszQpepkDPdNLaNYqcVQvb3g

or3I3

Theorem 5.15 P 3 → P 1 ∨ P 2 ∨ P 3. The proposition is identified by the following information:

Pure Prop Id: 017a00202c1447a85eee3f717fabbb57b1194539474715a7958654c3eeff6dba
Pure Prop Address: TMMKVc34VAUfiURqFmHXpwwfDmcdTEMi8p9
Theory Prop Id: 3b7f5abd745db6a80ff6dd817b2bc916034448d1d8cca1a8e2958f0b3257c71c
Theory Prop Address: TMc5vyzo3ArE8GAUf3UUvpSQdgk1mgJAYae

or3E

Theorem 5.16 P 1 ∨ P 2 ∨ P 3 → (∀p : o.(P 1 → p) → (P 2 → p) → (P 3 → p) → p). The proposition is identified by the following information:

Pure Prop Id: 5d069d8e90aea128ccbc62404954d9fa5e38854f6dea8d51e13e68ff413e0ab1
Pure Prop Address: TML79YQmK3xkoSvhsjHcVf5xiAdsKJpEJBe
Theory Prop Id: 2120c9770c0268721584c931577292b619113ab7eeb88e02ffbf738291ce46f1
Theory Prop Address: TMTxTUaNh3Y288kongaMx7qkfusZvuByRau

Let P 4 : o be given.

and4I

Theorem 5.17 P 1 → P 2 → P 3 → P 4 → P 1 ∧ P 2 ∧ P 3 ∧ P 4. The proposition is identified by the following information:

Pure Prop Id: ed32cbf9688b8877bdd20c59f473bb9c69432ffd24de548cfc4c59616dd11c5a
Pure Prop Address: TMdAoQLQ3RM117hTZoE6J4XtvGzkLFHKGQm
Theory Prop Id: 906aaf2cf039cec16718707b69fda8d19357ca96611472679c8c38ac5552d505
Theory Prop Address: TMXRatgrgSzbMm1bvkukK5JVrsGh2z8rQ6u

and4E

Theorem 5.18 P 1 ∧ P 2 ∧ P 3 ∧ P 4 → (∀p : o.(P 1 → P 2 → P 3 → P 4 → p) → p). The proposition is identified by the following information:

Pure Prop Id: 7f6e6c52947184716f0a59a54ee2c3fd4562ee529d8e109ad2527c908611d3be
Pure Prop Address: TMTwtsu5L9Ezi887h92X1qCCtTSgg6Z5Fxr
Theory Prop Id: b41cb764baa71ac5c454b1c9bff373c0bc8ceac5c10a56f18ba2e4a30db98bd1
Theory Prop Address: TMT4yM4NV4sxuvWZUkXew6X7hJ9zigrwUfZ

or4I1

Theorem 5.19 P 1 → P 1 ∨ P 2 ∨ P 3 ∨ P 4. The proposition is identified by the following information:

Pure Prop Id: d61309b1a6dfcdb5b57f074782c88d8e6004fb5bd8c217f4ef2e88a3376de181
Pure Prop Address: TMNvs8HY31Qn4Ju5WNUWKVjbaoTkg7MDXDG
Theory Prop Id: d10b2eb23a5fa505396dff509fa1c442d6870e72d887a41bf455239abe6dff54
Theory Prop Address: TMaTMcRZ9E1Rb4Nuq4C1Ho7bwKAyCZhJbn1

or4I2

Theorem 5.20 P 2 → P 1 ∨ P 2 ∨ P 3 ∨ P 4. The proposition is identified by the following information:

Pure Prop Id: 17c763a32537244fd99d77c2973672e33ccdcb0195284690a5b20a50d917f1bc
Pure Prop Address: TMMFmRtJXqQaL2pJyCq2NXmUoQo6iEcBUeF
Theory Prop Id: f9e4cf5b7be5972585b09b77e8fe2de549ffcd217050934f83ca2a0269d7c22e
Theory Prop Address: TMXYRWPJ6eBJ2veji2CB4sPXSweJfbMW9NC

or4I3

Theorem 5.21 P 3 → P 1 ∨ P 2 ∨ P 3 ∨ P 4. The proposition is identified by the following information:

Pure Prop Id: 7500b33566aaf93de84ea48148bfb3bb21b30ec225170d993339f8bb0cbbaaf1
Pure Prop Address: TMZiwfSWdAkAZ7WSEnmx8EfCq26LV9zrvDT
Theory Prop Id: 1a5881c849fb86fd77a8a4f11ef6783644b06e8c3f74fb02af0c13b7a816c3a7
Theory Prop Address: TMHnh1oKCafRDqpe4JxXfg2qfSkei64vT35

or4I4

Theorem 5.22 P 4 → P 1 ∨ P 2 ∨ P 3 ∨ P 4. The proposition is identified by the following information:

Pure Prop Id: 7a20061f67b705b52da52d1619188bf47b727e3bf41cab7aab5e3b9c6c9e1e9f
Pure Prop Address: TMVBHMZoCoxkQBLGMKjpPfCyWzaL2nkaZx4
Theory Prop Id: 1ecd5bc77d1ec7549783816ee07b8441f43a51ff5ab63ec30ca559c442813060
Theory Prop Address: TMFhHjtCoxKWhG8QRGUQ92dvizsEVBwWvYQ

or4E

26 CHAPTER 5. SOME BASIC RESULTS Theorem 5.23 P 1 ∨ P 2 ∨ P 3 ∨ P 4 → (∀p : o.(P 1 → p) → (P 2 → p) → (P 3 → p) → (P 4 → p) → p). The proposition is identified by the following information:

Pure Prop Id: d05b31b0be8cd483fa07a3f0f19264b60e47f6c21659902935462906bdf6b76d
Pure Prop Address: TMQwRKYH5ei57PtG8MGuyMNQBKe8cuszmF4
Theory Prop Id: 1fdc2a30b796702db155c2eb5a9c6e71c48a99beca4fc05f20102b22e9cdce47
Theory Prop Address: TMWhPKKnyM7dRKMXmnAxawenezrpwrV9cgz

Let P 5 : o be given.

and5I

Theorem 5.24 P 1 → P 2 → P 3 → P 4 → P 5 → P 1 ∧ P 2 ∧ P 3 ∧ P 4 ∧ P 5. The proposition is identified by the following information:

Pure Prop Id: 041ced6dc277faece2d470de6befa0e97006998af0c9cb1989fc9901b3ec866f
Pure Prop Address: TMVd1Ghg1sxR7zcgk4GPNy72pYJ9AQdBzd6
Theory Prop Id: 11fc478665305a3b2e27aa913e8dcb933d84e00ee923c3e503f0ed51d4ed7d8f
Theory Prop Address: TMFEndoHPA1ez9Y8dc2y2YAf6QxYxCFg4BB

and5E

Theorem 5.25 P 1 ∧ P 2 ∧ P 3 ∧ P 4 ∧ P 5 → (∀p : o.(P 1 → P 2 → P 3 → P 4 → P 5 → p) → p). The proposition is identified by the following information:

Pure Prop Id: b9e36eb0772059378de6e2bae57aec129036e5378cd37e203e1791e7b44f92cf
Pure Prop Address: TMJBWvgmbPvfKX7ymNvDkg1M46iQe7Ny1z7
Theory Prop Id: a02e0e4e7793f2f93af907335f12098d37abc74e46d2eeb4465324ba9d8046db
Theory Prop Address: TMMmsE2QkKEBNxNaoo9zCfmr5o7QBZgTPGQ

or5I1

Theorem 5.26 P 1 → P 1 ∨ P 2 ∨ P 3 ∨ P 4 ∨ P 5. The proposition is identified by the following information:

Pure Prop Id: 39ea457dead1717f0a108c7304916bfd9cab247cf4cf2d3b67f5065814a8e0b4
Pure Prop Address: TMUs12JtV2vEaiiahyRUXpuovbU11fq7zLB
Theory Prop Id: 4e8f4064dae12df31e7df3da1e5c8535b89046a7b5319d8f2305e5de1b24f829
Theory Prop Address: TMTPcpiEwfkrghjEEposf5yS1v869ke7Rrj

or5I2

Theorem 5.27 P 2 → P 1 ∨ P 2 ∨ P 3 ∨ P 4 ∨ P 5. The proposition is identified by the following information:

Pure Prop Id: 8494de963a2912613d2bfb1d747fc055d6eb362d422be1896eee6d866c358af9
Pure Prop Address: TMFZtXVUhAJgMrzbc8SGQr8PvkqfircW7cL
Theory Prop Id: 32035af2da626c1acd3d71cbf3197f52b67159384f068ebb3b5f4c50a08ce3db
Theory Prop Address: TMR8LzrcpDPkw2uf7T8WHukPntFRQWVakmR

or5I3

Theorem 5.28 P 3 → P 1 ∨ P 2 ∨ P 3 ∨ P 4 ∨ P 5. The proposition is identified by the following information:

Pure Prop Id: 09182b1c056c3e6ec773e5fb3c476f7e026643a15998bb79deb35e131fc516a4
Pure Prop Address: TMV1nEkVMZR4hjji67oiYpLCFwcPavF6bSp
Theory Prop Id: 4aca859beb67b87c07448642e2f80b02e346c431898060774f6806896b8bc060
Theory Prop Address: TMVdimQJwC5PcNHpaQeD8vVev4mkSnqKUDh

or5I4

Theorem 5.29 P 4 → P 1 ∨ P 2 ∨ P 3 ∨ P 4 ∨ P 5. The proposition is identified by the following information:

Pure Prop Id: 51cd1ca65c14f555453c13f820eac600735e6825f73e4e668138659cb752ce82
Pure Prop Address: TMRxkMs8kLmmDi8zdv7a2mgmiiwm8gtbeQC
Theory Prop Id: 8e109e9f1c205c91c5b79a556c11e2b2fb94d2d77c537f904faaf5e7cdf2fce4
Theory Prop Address: TMTZe6z6z3GDdCznmKAm5d5ZAas3rSPduhh

or5I5

Theorem 5.30 P 5 → P 1 ∨ P 2 ∨ P 3 ∨ P 4 ∨ P 5. The proposition is identified by the following information:

Pure Prop Id: 1d6362c9b2c69a1b4b1360eef0b6f9f9763f1089e1bbde2904b3db2c6544b5b9
Pure Prop Address: TMHM6WDtzUk5yH5hFkLWwTSXzie4fHKJkeR
Theory Prop Id: 5947729b14c31f7f5dfb495f3dc278156afc20b954ecaa24e6839650aef497bf
Theory Prop Address: TMWqVbPVCB7MunfAfT8ReFmNyoB4QbkEzA2

or5E

Theorem 5.31

P 1 ∨ P 2 ∨ P 3 ∨ P 4 ∨ P 5 → (∀p : o.(P 1 → p) → (P 2 → p) → (P 3 → p) → (P 4 → p) → (P 5 → p) → p).

The proposition is identified by the following information:

Pure Prop Id: b956eb2b7204f5a6e780e90bc1bd534e682ab77b2ad4cb7c5d03d8acacfc9a2e
Pure Prop Address: TMURL8fhzF9KpcCx1BootRK4CFuzooNhaUw
Theory Prop Id: 780a9f1fdc60b8ddc55d630d2b78a74557585751e69471682b41605c6a27eef9
Theory Prop Address: TMVMNDhbZznQnqiXkLo51F83SmRwT6czcWC

28 CHAPTER 5. SOME BASIC RESULTS Let P 6 : o be given.

and6I

Theorem 5.32 P 1 → P 2 → P 3 → P 4 → P 5 → P 6 → P 1 ∧ P 2 ∧ P 3 ∧ P 4 ∧ P 5 ∧ P 6. The proposition is identified by the following information:

Pure Prop Id: 9817c7a535d81bed18efaa95b110653ebba78a7ad991eaca8054c41020b664da
Pure Prop Address: TMM8hfvyGGBPHT3XKYSQ5WjPZBdQu8CHFmh
Theory Prop Id: a5dc2935b019c23338edba6ccdea3a654b6a51efe6170f8887e55dc4e5d6e4ff
Theory Prop Address: TMNPT8qknaGSp8UzvMyiNy5UD7HRGHNMvZ9

and6E

Theorem 5.33 P 1 ∧ P 2 ∧ P 3 ∧ P 4 ∧ P 5 ∧ P 6 → (∀p : o.(P 1 → P 2 → P 3 → P 4 → P 5 → P 6 → p) → p). The proposition is identified by the following information:

Pure Prop Id: 3668da80c73d5b9a85e7b685ab4ed93a92bdbb2a7cacf0c764a2353bffd98bd5
Pure Prop Address: TMQ5JF8b5hxhEB7utWT6iBHdgtNte14WFKZ
Theory Prop Id: 1c4fc650221a6c544c529f279d39c9effe360f4bbb342bf30ac2ef1bb0cae7bd
Theory Prop Address: TMXE3S7PKbVYmocgEe4jSS2d628TPGWZpTj

Let P 7 : o be given.

and7I

Theorem 5.34 P 1 → P 2 → P 3 → P 4 → P 5 → P 6 → P 7 → P 1 ∧ P 2 ∧ P 3 ∧ P 4 ∧ P 5 ∧ P 6 ∧ P 7. The proposition is identified by the following information:

Pure Prop Id: 6a4d8ce6f0dd012e01148d6cfcd0336a7bdfd1f6e0a108700387aa6534f21e3a
Pure Prop Address: TMcFEeC2wp56VXpGCwrrXAy9KX7iPCggM9d
Theory Prop Id: c89388f0b732abd74a2fb4d6fef23202a20e588058361e4617d162de86cd6b7d
Theory Prop Address: TMFwsYdcsUc2UYbNFsd4RzP6m2K8FrCgFrq

and7E

Theorem 5.35 P 1 ∧ P 2 ∧ P 3 ∧ P 4 ∧ P 5 ∧ P 6 ∧ P 7 → (∀p : o.(P 1 → P 2 → P 3 → P 4 → P 5 → P 6 → P 7 → p) → p). The proposition is identified by the following information:

Pure Prop Id: 2ea6ee206fc0e0bac89e94ecdeb5cc8218a37b0e2ad93881ee4b947fe261aca5
Pure Prop Address: TMarbzMgS52Zmt6DZPQwD7NrgfFnGmniZBg
Theory Prop Id: 02714a2b656b0d600f52d924edb42df9d13fd49d79bacc042df6ff567767a5a5
Theory Prop Address: TMJ3TDaD9hfZwexV6a4jq3FXE9KhjambBpi

5.2. FURTHER RESULTS 29

5.2 Further Results

iffI

Theorem 5.36 ∀AB : o.(A → B) → (B → A) → (A ⇔ B). The proposition is identified by the following information:

Pure Prop Id: 7516b9382e323d500a2b396c81bf98c3859ca15e7e7b359310c7cea2aeed5c8a
Pure Prop Address: TMMjuJFnsPng1UMvQzKS18LFFsgWsi9d8xU
Theory Prop Id: 4d966a5f5da0679b05fd8ce1619ac4165c9654be7a87ee2cceaaec3ceac10633
Theory Prop Address: TMddTKwGeWwMHruZq1JPdHyJmpeTrFg89Xi

iffEL

Theorem 5.37 ∀AB : o.(A ⇔ B) → A → B. The proposition is identified by the following information:

Pure Prop Id: 0e94c69dbdcf6a1baca24d804ad190121b2de563e5642fa8c452be1c6595979b
Pure Prop Address: TMPAKwBu2btn7LfWCbudNDBAqu86wzbzmEY
Theory Prop Id: 9a07f60730cc4e5044b6b2af10fd8467b0bedf6f66364ca33c0818495467c2ed
Theory Prop Address: TMTL1Jyg3GdY9swgWWQPXN6ceqVnBuFt1Dn

iffER

Theorem 5.38 ∀AB : o.(A ⇔ B) → B → A. The proposition is identified by the following information:

Pure Prop Id: 54702aec0e7fabe49df0fa4b3a00910bf49c4c03c646ddc286485e7af2899fe8
Pure Prop Address: TMLxQTofs8NSTdPqLWVf9T5mVck9C5B8m8s
Theory Prop Id: f5dfc0631a7727945b67f112d323c8a1558a724a0d6b5c261d4a11649d925f64
Theory Prop Address: TMGZN34A8TcN1Z21rxqNAMYJRjG67dAotvb

iff_ref

Theorem 5.39 ∀A : o.A ⇔ A. The proposition is identified by the following information:

Pure Prop Id: 01338dd4902fcd80dd89af75713f451fe1b1e4804229f05955b962fe6def6250
Pure Prop Address: TMSKz5gXPFakcgFRBJ5RJYfDQC2q8ocrFi1
Theory Prop Id: 7b71b52f0f8ece18ad34af695ce60729d9853085b816941378947b88e44cbf0a
Theory Prop Address: TML3Gcc3WbCE559uqmKxvpDahXs8iDJjURL

neq_i_sym

Theorem 5.40 ∀xy.x ≠ y → y ≠ x. The proposition is identified by the following information:

Pure Prop Id: 5c0ec9ea191d0f6e263cc182b469ec839768d633284966a6a6a2058c1778ed0f
Pure Prop Address: TMXXdxoTHLJ7BfG4hU2CwgemEXjQ8FmtQ1C
Theory Prop Id: 1a98981725a2236f37b1c465a0e41336ebef451702e4d9da44b1fab316a2e844
Theory Prop Address: TMdSA41VnJVw63YmxPtgsvson8sPkVFMoKc

Definition 5.1 We define nIn to be λxX.¬In x X of type ι → ι → o identified by the following information:

Pure Object Id: 36808cca33f2b3819497d3a1f10fcc77486a0cddbcb304e44cbf2d818e181c3e
Pure Object Address: TMSqgddy21cUnLoRkMPcBAxXhAmCfaQeLQj
Theory Object Id: 253451c995b7e8af5baae1333a990b55d0dea9af34d44b09b418d3f09fe72de4
Theory Object Address: TMReNhKc6t82pWW6stKo9YNy1BuLWXDCyeC

Notation. We use ∉ as an infix operator corresponding to applying term nIn.

Eps_i_ex

Theorem 5.41 ∀P : ι → o.(∃x.P x) → P (Eps_i P). The proposition is identified by the following information:

Pure Prop Id: be36f0573e3745e47e74354bb117860bd387b3f6c26ba3a752a7388e6b66d647
Pure Prop Address: TMFoJytvVebF2Sg18ua9AU1WgCbeqEijvbZ
Theory Prop Id: 6a28328c813ae82b72e327564626a89c19306d9b1f89653ada2b22e954e90881
Theory Prop Address: TMXYa9wuYkRrAmfCgqhH6hY5uK5GFw8gzHB

pred_ext

Theorem 5.42 ∀P Q : ι → o.(∀x.P x ⇔ Q x) → P = Q. The proposition is identified by the following information:

Pure Prop Id: e9e64b253f257e12e8cc34a6936324678cd05c9763263d2d12e48da4bd7198e7
Pure Prop Address: TMYnqc7V192uGaHR6Z4JQYWcpukZNrMq6cy
Theory Prop Id: d4b2d3f3eabb3ff09d4d1a4e7e1663d78c31373aac18adcfbef2e2c081bb44fa
Theory Prop Address: TMFdQW2ooCy3wDncDrzFGtMSsTwAzHRg5kG

prop_ext_2

Theorem 5.43 ∀pq : o.(p → q) → (q → p) → p = q. The proposition is identified by the following information:

Pure Prop Id: a9ede2c9e3cf1d70dfbf9f57d775f61afa29105059e99ada89c0de099e2c1a26
Pure Prop Address: TMNH7NvYuoPP9nYGA3ZMWdycLUsdiXP5REb
Theory Prop Id: a56fc7bc69f2eed7db4340adfe025ace2f0f83cdfc470f09b018498a097775ab
Theory Prop Address: TMYdTtZfAJ1wEsDSWCBMpXk181wbLRKQ2tT

pred_ext_2

Theorem 5.44 ∀P Q : ι → o.P ⊆ Q → Q ⊆ P → P = Q. The proposition is identified by the following information:

5.2. FURTHER RESULTS 31

Pure Prop Id: c558ffab38d9b7b5cc1866022b11c5f544a9b798f1745926e15f3391ff6cc87d
Pure Prop Address: TML1d3NrCUcMsEZrsdaVLuyf7VGJq9zKYD8
Theory Prop Id: c5df8491488c0d5344f2a38c75caa07e6cfe3d5e89feb23ebc08798a9fdccfd4
Theory Prop Address: TMW8biyFW9jcdzzGjDz5sXDkojWntfdbQ1t

Subq_ref

Theorem 5.45 ∀X : ι.X ⊆ X. The proposition is identified by the following information:

Pure Prop Id: 4c53952cc6ecb22968534a4d43adab78af1f0b047ed070da10ea521d270d1f2a
Pure Prop Address: TMWfjRz3PusvtrSBv2umMamuPFQ3vpfqq5R
Theory Prop Id: 2efbacece8d60c274641e6775400fab460246bacd8ac2d18f883d059a7ac8358
Theory Prop Address: TMLkYwQzhxE5mhkLWGyedCPNtZ5dCRp8qmB

Subq_tra

Theorem 5.46 ∀XY Z : ι.X ⊆ Y → Y ⊆ Z → X ⊆ Z. The proposition is identified by the following information:

Pure Prop Id: 79accb4060735cc396e0e3ea36014c3e94c87d8e52a22ed4c18d55d6af348d9e
Pure Prop Address: TMXhwEo6mEGCoaDosK1LZDEK8xXZr8kji3H
Theory Prop Id: fed662887c1b855fd3ff74e7894edbaebabacb12dcae138964b49804bddac9a6
Theory Prop Address: TMaozUoYZxHxHHTvoggGVukYwGVgw9VLyP8

Subq_contra

Theorem 5.47 ∀XY z : ι.X ⊆ Y → z ∉ Y → z ∉ X. The proposition is identified by the following information:

Pure Prop Id: d15e87bdbc1fd0a307b0f0b273414e5fca8c31ec8256241a668bcab8d25dc157
Pure Prop Address: TMZtpMMhjtkFakD25QWYgBYDJcz4naaYj5w
Theory Prop Id: bd5a2f337d80e9eca8abab55ca08c824896fb8274ddacef26a0d1e1bf8b8f0a3
Theory Prop Address: TMFK5BFNrPEtMNeyxT8Acey4o4GybQWYQ8i

EmptyE

Theorem 5.48 ∀x : ι.x ∉ Empty. The proposition is identified by the following information:

Pure Prop Id: 93a77a0c78f3c49660c44dd07c723ad23e5e55e64c0dcd527a22c0278b829b8e
Pure Prop Address: TMaSWKbJHDuPvqXN5ddrdRfghAejYyXef7K
Theory Prop Id: 8349abf2aeba6cf6d62d407fc6062a629c409a81df869ba5e46a2adde87b67a5
Theory Prop Address: TMUoYxyibdoLdU1WGAmWKr2UQqgAiyrf8sb

Subq_Empty

Theorem 5.49 ∀X : ι.Empty ⊆ X. The proposition is identified by the following information:

32 CHAPTER 5. SOME BASIC RESULTS

Pure Prop Id: a517cdaf3478a733b128a31cce59618dd840c79c17df99f29410ff3e9ba455e4
Pure Prop Address: TMWWMiBhRYozNqky8uYhTyxKxDXXVBjHmqH
Theory Prop Id: 2faec377f13468a82bbfcb310b43ad7a193b5d26e72c513a4b6fbb139dd1f852
Theory Prop Address: TMJxeX1k56XRyrVDH9k6F9DZp2n6xFpPAck

Empty_Subq_eq

Theorem 5.50 ∀X : ι.X ⊆ Empty → X = Empty. The proposition is identified by the following information:

Pure Prop Id: 2dd95966ce09dd2345607d594b51d1640857501d96dea096bacb21e292697f31
Pure Prop Address: TMbzyCseCE8NavdH4tvfvqjNSXHHLkNaTk4
Theory Prop Id: 988691971197671404c5f64923b3e6f170a71f5f8f48e44c06089873913e08c9
Theory Prop Address: TMQENPUZ2wP5s7vyaLEtBCdaJErFFSUpJSV

Empty_eq

Theorem 5.51 ∀X : ι.(∀x.x ∉ X) → X = Empty. The proposition is identified by the following information:

Pure Prop Id: 3de0fdba2afc8aa47ba046f6d57a7edaabc38eebe8925bdc1b3803ca0a447373
Pure Prop Address: TMM1agcdRVrGxbz8kV5AmjKvcKDfDHWAxHb
Theory Prop Id: 8b915bb396e264c2d82d33dc8dfd93f4374585f0ab37c01fc964b73f4e28b7a3
Theory Prop Address: TMcosVQ8txADpELPSRpWSvATFdw3vWrW8H2

UnionI

Theorem 5.52 ∀XxY : ι.x ∈ Y → Y ∈ X → x ∈ Union X. The proposition is identified by the following information:

Pure Prop Id: 51b5499aea657de4516b761629ee7a8bf5292810ac70c13d0745541a55023b02
Pure Prop Address: TMQCYZzADyCKcXXBekJX5crLwK9MZzh7paV
Theory Prop Id: bade054056415b264e8e0ad162a917d91a5399de193fc12ede490c8bf786acf3
Theory Prop Address: TMHDQ27UKqnJLKFeQDx27tH4XqzefSAZp1w

UnionE

Theorem 5.53 ∀Xx : ι.x ∈ Union X → ∃Y : ι.x ∈ Y ∧ Y ∈ X. The proposition is identified by the following information:

Pure Prop Id: db4bc1188573af6ee726c91f30668d0d047de9916b17056407fb50b728e30efe
Pure Prop Address: TMF2PbBRuwVtGUAm2dBcakxEFts1b3gfeHi
Theory Prop Id: ab781b05690ff7302616c39024e38d17990dfb88f8dc15950b3d5f47ab0a4324
Theory Prop Address: TMPGvCnxEDNcLzGEpegCEggB87LUw7GvgQ9

UnionE_impred

Theorem 5.54 ∀Xx : ι.x ∈ Union X → ∀p : o.(∀Y : ι.x ∈ Y → Y ∈ X → p) → p. The proposition is identified by the following information:

5.2. FURTHER RESULTS 33

Pure Prop Id: 8efd578687dc029da349c22ca087953113491e4794018f04c73b2c7ae5084bf0
Pure Prop Address: TMH5ng38evRBjZnMvcn9FJLMLXbDC1Bz9aY
Theory Prop Id: d621b2d16dc72ef52c1fe29162e3508201e053e42f163701b3713472780e9a5e
Theory Prop Address: TMHeeXhX7g1GDga7C7LQdPcR4t9R2vstffp

Union_Empty

Theorem 5.55 Union Empty = Empty. The proposition is identified by the following information:

Pure Prop Id: 2c511ebf784d1ac4263638dde61f03ebe06b666994224125d685445a9af596e3
Pure Prop Address: TMLm3kKCuXxhm2w4H3avGFTRVMtJZ4TZQPW
Theory Prop Id: 72b1c4f60fbbe561816842cf8d137a012a23b49ea41913fed5ee67e33bbf6c52
Theory Prop Address: TMMwTUayoXbEd6iFoK2PNm3DAtTEe5cdhtf

PowerI

Theorem 5.56 ∀XY : ι.Y ⊆ X → Y ∈ Power X. The proposition is identified by the following information:

Pure Prop Id: 7143d71ecf102401ab2636034e61d377bec399d276d42b9e418431501853f4c6
Pure Prop Address: TMYd9dR9ghFpJALxMSh1bifCcoHmd6XFuvA
Theory Prop Id: 3c48d3c62b94e7a41c69deefa90b2e8c771cbc0aa279fcba058fefb98579331d
Theory Prop Address: TMXdAMd1HZUyw6mE1ZFaV71nZ63jBYMLiUj

PowerE

Theorem 5.57 ∀XY : ι.Y ∈ Power X → Y ⊆ X. The proposition is identified by the following information:

Pure Prop Id: 149030a919d6402241e63172077fe438016089bfde0e7da4cbc536dc04901bb1
Pure Prop Address: TMWSJ1X7N37P2Pemk8UyDMfYjkY3zie7nYZ
Theory Prop Id: 6470703fa38f09b9dbdffe7bcefe88a4d4c94d929d7e93c64556a9f00ad14694
Theory Prop Address: TMTYtQq2xbReLwQoZqj15EvuksZjaG3A9fA

Power_Subq

Theorem 5.58 ∀XY : ι.X ⊆ Y → Power X ⊆ Power Y. The proposition is identified by the following information:

Pure Prop Id: 6d8f173a14a5bbfef48f6ce86bee8512556a0deed405c0bd072d5141d2299af9
Pure Prop Address: TMUvnczMnpH9X3CybY6TzCw5SCHqkt7ir7u
Theory Prop Id: 9dd5f3b3f3808a5118155ae75d2e1c08c28ea7ba30690f84aae27623efd601dd
Theory Prop Address: TMLB1eXPkLwvDDtkFvqtauiz4FckV2UwQPk

Empty_In_Power

Theorem 5.59 ∀X : ι.Empty ∈ Power X. The proposition is identified by the following information:

34 CHAPTER 5. SOME BASIC RESULTS

Pure Prop Id: d2656b52d4bf2bab93fe5255f13b797a5512a056ca12c4fb73ccdc10829f6f17
Pure Prop Address: TMaK4ZHB63mPURsGQ3iXFc6dERVy8jXzgG5
Theory Prop Id: 487855dcfff4149b1efb5bd9ea0572b18fe9005d660d86fcfc5ff90110ba6df9
Theory Prop Address: TMa8XHamyspUjhkeiNgAYuj8aecCrFsHqjg

Self_In_Power

Theorem 5.60 ∀X : ι.X ∈ Power X. The proposition is identified by the following information:

Pure Prop Id: f81ef2a25f07c69ddd40acce90e3caf25609ce25f8183e2509b613cb0092508e
Pure Prop Address: TMQycQfNrWABmzjhJNUDx6UCqmAXeAy1qsv
Theory Prop Id: 8336ff563585b5335f5d748d44c68d6e04e2e671537cfd5537a4b37acde6a25a
Theory Prop Address: TMKY4E4GGAWwCfDftsZmANC17noKbnuCrbo

Union_Power_Subq

Theorem 5.61 ∀X : ι.Union (Power X) ⊆ X. The proposition is identified by the following information:

Pure Prop Id: 7f4d2bc5abb24988ada92aecad1eeb9b7e0902f2bf91f6062f31024af17a8bed
Pure Prop Address: TMTgvEtxJEiGcbTTuKLumZaRiNe3WkRbVBG
Theory Prop Id: 1b4311aee3fe38a1eb6d381638c72194c02e1025b68cb8fd7f24d84b4aa8ff10
Theory Prop Address: TMQdwUwhhyCEESAhoqgTeHBAY85kLo5KAYP

xm

Theorem 5.62 ∀P : o.P ∨ ¬P. The proposition is identified by the following information:

Pure Prop Id: 62e4c3fec424181797ab4ef1d385902e9cdc35c7558a8f861b90f48a494c0e0e
Pure Prop Address: TMKtBvuSXX3aDsty6aeAaXbywwMJa4KhJpC
Theory Prop Id: 7b6737622a32781568aa9679a7eb1928791917e8dbf1a5fea21e2f0cbccee791
Theory Prop Address: TMMPzhDBTSXgihVRhAwaiREjdQ5Jeotv34y

dneg

Theorem 5.63 ∀P : o.¬¬P → P. The proposition is identified by the following information:

Pure Prop Id: 852f346d61a871c2db340ac4fd7de308d10ae313849575cd39cfac1469b8995e
Pure Prop Address: TMHNYQKxvrS4ZDs2FctrNBCPfYUgZB6RFrx
Theory Prop Id: b7cc631d047502f5a0c47618dc8c76a4947e2224be400faf040b7ef95f7994da
Theory Prop Address: TMRYv1F8fFeWtceTAuu9zbNET7tH5xtFHQn

imp_not_or

Theorem 5.64 ∀pq : o.(p → q) → ¬p ∨ q. The proposition is identified by the following information:

5.3. EXACTLY 1 OF 2 35

Pure Prop Id: a8655639be6ebff69772dc3e37a7cf1414005132611dfae586f7e43fa02f82f9
Pure Prop Address: TMcvyPAhaTi7kXdf8YYSBsEH4XcVThUkX9L
Theory Prop Id: 30c98c972cfd8712dbdce74f465991aad8aa72159a4511fc01e975a23d063a5c
Theory Prop Address: TMH4ymv2c9JczjQGEQqzDcP9xj2yX1JHHfx

not_and_or_demorgan

Theorem 5.65 ∀pq : o.¬(p ∧ q) → ¬p ∨ ¬q. The proposition is identified by the following information:

Pure Prop Id: 29877a368f94842edeff462e056244efa22376525930141a2d2926e5f3d0ef3b
Pure Prop Address: TMQfs5TGwUrjea616pdTxbLXGNvhBxY4UzL
Theory Prop Id: 63ddfe069e38d4763ee4f112c78b39e12a120ccaa496fd47e6a4646fc9391093
Theory Prop Address: TMaCC7RwYa5bssu9qewCUzTz5xikj6JohLz

5.3 Exactly 1 of 2

Definition 5.2 exactly1of2 is the opaque object of type o → o → o identified by the following information:

Pure Object Id: 163602f90de012a7426ee39176523ca58bc964ccde619b652cb448bd678f7e21
Pure Object Address: TMS9Bz8WFNT8hbFokWDmzwFmNr2ZD7CfP2e
Theory Object Id: a0b94402d8edc93e77f0dd488bd6709a5ab850b3f07e42bedc0d1e40163aec06
Theory Object Address: TMEqbBVsvC394VsHqRdcJB6GggoMgLnuS6A

exactly1of2_I1

Theorem 5.66 ∀AB : o.A → ¬B → exactly1of2 A B. The proposition is identified by the following information:

Pure Prop Id: 8e1e4515e8d2e1265d837e207a0eaa000d82c55e2fa708c4ebd9ad229b5a5d38
Pure Prop Address: TMQnjH6acMFmZdz797stCauUdrSebKX7Mev
Theory Prop Id: 4f94fafdd5f7f8d4dc0f3e1b7277b17a0c7c19add3a6af231254dcedab30fe39
Theory Prop Address: TMJL9YXWiTdUUzhvi5WWHy9wvvoebH1tQzv

exactly1of2_I2

Theorem 5.67 ∀AB : o.¬A → B → exactly1of2 A B. The proposition is identified by the following information:

Pure Prop Id: c9fd90a0917801b2b7fae79f83bdc259f931342830e8158acece4f3c1a283659
Pure Prop Address: TMMAKooscST9mjX4iuDPzg3FY58bpufyfVU
Theory Prop Id: 6c1ad919f0b4255c719d573f707feb239d05426a99ece78b2f738ef43e3e1a51
Theory Prop Address: TML5oMK64kBLn2JLS7ATshVeqCgaCsuNK6e

exactly1of2_impI1

Theorem 5.68 ∀AB : o.(A → ¬B) → (¬A → B) → exactly1of2 A B. The proposition is identified by the following information:

36 CHAPTER 5. SOME BASIC RESULTS

Pure Prop Id: 51838fa63f26d38944c5e35f2413ef312130a04668affc103545649e6845b841
Pure Prop Address: TMRhwQ9VVpjkPMMkpzHArgyLqFJ2sLxakNb
Theory Prop Id: 517be2b6e298510b19ae2a4576b522c9cc8fe9bf89dfa1630bc8b79219852955
Theory Prop Address: TMcoNtefvy7eJpxYPtwBmyQuGKfMJ9PMpkn

exactly1of2_impI2

Theorem 5.69 ∀AB : o.(B → ¬A) → (¬B → A) → exactly1of2 A B. The proposition is identified by the following information:

Pure Prop Id: f32aab556c283c02cca82ae1401040f6003bf6a1ac60788c12ad6329d832a293
Pure Prop Address: TMSY4dBSVErbAWxEf4c743PqBZ9EABV3BJY
Theory Prop Id: 811f013a6ac0abc6e0c8f752b4c6fbd661919f540a84f0f8b6fc99dd0d22aa07
Theory Prop Address: TMGJHhrbg2Ea8nrQBjPv3SdM2aoAj4dxq8y

exactly1of2_E

Theorem 5.70 ∀AB : o.exactly1of2 A B → ∀p : o.(A → ¬B → p) → (¬A → B → p) → p. The proposition is identified by the following information:

Pure Prop Id: 29b901ba4db12851de74e31368539f6d1cc25bbc1c301edaa62335d7e6c17545
Pure Prop Address: TMRCeD4ey2b4rXZn7HkMnX7YburjvMK1Eyg
Theory Prop Id: c674348923fa7045ff80c55a4bb504730f6f8ee0dcdf90e8c71d35291ea85112
Theory Prop Address: TMJHeaLcDFsg71BZKr8vRobyqEDxdcim8fW

exactly1of2_or

Theorem 5.71 ∀AB : o.exactly1of2 A B → A ∨ B. The proposition is identified by the following information:

Pure Prop Id: 18af7395e3694a283ae114ccec2d06247033786f17700e124d0fd73c83572675
Pure Prop Address: TMLi8xLyPBxTfc12noiHmSmqavSb4tpNQeQ
Theory Prop Id: e092f672ff0d584cbaf46120347c86b03fbcec240a9d5100380e8240373fd2be
Theory Prop Address: TMXLeWxQ5A2VGtDdKgtMuBn5HCa3VzxF1pb

exactly1of2_impn12

Theorem 5.72 ∀AB : o.exactly1of2 A B → A → ¬B. The proposition is identified by the following information:

Pure Prop Id: d3009840b3d85b74b9e0a9620f4308a8e1436d22cdd2a69e83dd3c46c1487323
Pure Prop Address: TMWzgfM5wbPGyJhNsxsijwiYzmX4TthmdKH
Theory Prop Id: ffdec3681a2e06d83ddcbdc151520cfe5b3b623d2db529d7a849209f1d6fe7cc
Theory Prop Address: TMG66wx9tzCkMhQc9NRiUxq74peErmdEviC

exactly1of2_impn21

Theorem 5.73 ∀AB : o.exactly1of2 A B → B → ¬A. The proposition is identified by the following information:

5.4. EXACTLY 1 OF 3 37

Pure Prop Id: 29929fd4030494b67c9eeeaa7dc35578573efb328bcafaf20f5efb6ad854e87b
Pure Prop Address: TMJz2j2BW2BvWFzAm664pFqbVtvTqWXzGq7
Theory Prop Id: 7499d10110c69f4c2d84763d224e28a23a5525fa48cfc823789b09051b07db66
Theory Prop Address: TMPjFxrRCRMeS8z17aDuupoUhLQX4Kodx6A

exactly1of2_nimp12

Theorem 5.74 ∀AB : o.exactly1of2 A B → ¬A → B. The proposition is identified by the following information:

Pure Prop Id: 0fb71ab4a166db8b8a5493187a1623cb2573bdafdb80c215e1df61d68b8fa34b
Pure Prop Address: TMYo2ThWofRLhEMmrLih7wWsVmnmor928bi
Theory Prop Id: d000b6d72c072278302ade061ba0e7a3735507b341fa9f3cb4b8ea13a683c623
Theory Prop Address: TMdXoSSvPrdQxAK2M2kCdSMvu74LozCHPWp

exactly1of2_nimp21

Theorem 5.75 ∀AB : o.exactly1of2 A B → ¬B → A. The proposition is identified by the following information:

Pure Prop Id: 0ef7cfe1ed1435414acfd763fb0f561715bba0ebfee6abe5163770fb6289d861
Pure Prop Address: TMdxUZ8ab8VgywY6cY6h1Q7GH65rTvgRbZM
Theory Prop Id: bced87455f1dbd71dca4499cef836fc13af9c4cd9d900b0e299d0289a36ec08d
Theory Prop Address: TMXMKQXQCPoboD5XPkTZaaJc8c6rPbjnF7n

5.4 Exactly 1 of 3

Definition 5.3 exactly1of3 is the opaque object of type o → o → o → o identified by the following information:

Pure Object Id: aa4bcd059b9a4c99635877362627f7d5998ee755c58679934cc62913f8ef06e0
Pure Object Address: TMKyEfBd2FYAx6zajz2sxvYYYXLUo86oiwP
Theory Object Id: c2eb06c534871c653d9f25994d6e8a39d134bed3c7ab7fe19a4f2ecc1eaa1c9c
Theory Object Address: TMVfRiAA5KRi6JvZP4W3VJUw9mtsQVKMdhf

exactly1of3_I1

Theorem 5.76 ∀ABC : o.A → ¬B → ¬C → exactly1of3 A B C. The proposition is identified by the following information:

Pure Prop Id: 1310b63ae562af7333aee9625e0aa6979394ff104503c204c828d7b627b814d9
Pure Prop Address: TMcMEtNxcD358TggS5kHahcWnA74uxTqbqr
Theory Prop Id: b7468ae93994040f6662cbfbfdfebbcd1b25bb59711010960cc711c9538dfac3
Theory Prop Address: TMYCtwze3JZKavf143XCtrbzsmDzM8qK318

exactly1of3_I2

Theorem 5.77 ∀ABC : o.¬A → B → ¬C → exactly1of3 A B C. The proposition is identified by the following information:

38 CHAPTER 5. SOME BASIC RESULTS

Pure Prop Id: 00b5b6f76b51f729be0782aed25cf933b24c8784fde8d05bdc69f9b47b28a170
Pure Prop Address: TMbGNDSr1GGAFnxH4SmSg37nzwWa3ctnxUP
Theory Prop Id: 9660f910a33194ba73b4fbc257bf1c9c423ef83597391b5889dfe8f6a38be5cb
Theory Prop Address: TMXqa8cMXntrgJc53wodmauXTvXn3M1bkAY

exactly1of3_I3

Theorem 5.78 ∀ABC : o.¬A → ¬B → C → exactly1of3 A B C. The proposition is identified by the following information:

Pure Prop Id: fb6319ee090e002b279304bb4d2ec36b0740f58ea070dd36a44eed28f3b20035
Pure Prop Address: TMUH4ZTnmgc4sLH4v3sQNX3n4SQUyCZmnRE
Theory Prop Id: 98b137b027f483863b18f8083c21acbebd5fb61bcf6f96cf6dbd18c4e987f7fe
Theory Prop Address: TMR4EdHrWcDi4aPaboYPdGxLPTj5h4zc5Jb

exactly1of3_impI1

Theorem 5.79 ∀ABC : o.(A → ¬B) → (A → ¬C) → (B → ¬C) → (¬A → B ∨ C) → exactly1of3 A B C. The proposition is identified by the following information:

Pure Prop Id: ad228087f6c62536f8e84aa14c71244af2a1324cab874e74e539fce731299438
Pure Prop Address: TMTheuMKMYKhVdkDXgyaffeAMHxELJkUktD
Theory Prop Id: 83079842b76616f048df954d49c762e5ac75af2a19adeb1da721b43ed57eb0bf
Theory Prop Address: TMREFPJ1oU8sqtHxNQZq9p3RdA4ch42V8qu

exactly1of3_impI2

Theorem 5.80 ∀ABC : o.(B → ¬A) → (B → ¬C) → (A → ¬C) → (¬B → A ∨ C) → exactly1of3 A B C. The proposition is identified by the following information:

Pure Prop Id: 92d78d63c2f16429dfa172220b50c32bf70c813aefa4c1fbfc82982a1db5b261
Pure Prop Address: TMPq35SCVMNKL4cFiGZvRwpmRBR7XyHk3FM
Theory Prop Id: e9f574b7e9b23c6e5e976545161fe61fed9af69c6425e8992913d0188d809d95
Theory Prop Address: TMFXVhkyu6LNHsoHcGv18AE41FHiuVDFocM

exactly1of3_impI3

Theorem 5.81 ∀ABC : o.(C → ¬A) → (C → ¬B) → (A → ¬B) → (¬A → B) → exactly1of3 A B C. The proposition is identified by the following information:

5.4. EXACTLY 1 OF 3 39

Pure Prop Id: 80acac2dac94c4ea11d970424ef6748ca187d4a2d65bb70dd89cbb140688d23c
Pure Prop Address: TMXWj2i7Ck9NwC1rdBpeVZCnufshBHrocD1
Theory Prop Id: 78f24ba81b345b2c6ab858eba1eb4f9a5b836a0669931ce7c85b99e2a517410e
Theory Prop Address: TMPZhiykV6QMjsj1rdDoZ3FEXBoTtzSGsW1

exactly1of3_E

Theorem 5.82

∀ABC : o.exactly1of3 A B C →

∀p : o.(A → ¬B → ¬C → p) → (¬A → B → ¬C → p) → (¬A → ¬B → C → p) → p. The proposition is identified by the following information:

Pure Prop Id: 9abcef9e10449a86443719e4eccde7cc0122aea0cdd87559e72070b5056c68a3
Pure Prop Address: TMSs9kYqUVNU4be4UnVpebwjefQ88pwnfPh
Theory Prop Id: f4999a084ffdbda508b13e281a016002d1099fdaf2727b624c9a352ff505061e
Theory Prop Address: TMRGu2Bkor4inX6t54rtKgCqaA85yMnm4Fn

exactly1of3_or

Theorem 5.83 ∀ABC : o.exactly1of3 A B C → A ∨ B ∨ C. The proposition is identified by the following information:

Pure Prop Id: b9e1330f726a56d262ecb418f9a9f2892f85275d460094e7c8f3b8a38518f227
Pure Prop Address: TMLoZZfbmUJwXoBoU6BXeji1WYei2vbwgjy
Theory Prop Id: 46638186a29363d865020e0cf0a5e9f76ca956ed7c669bee7f98f4d7fd242530
Theory Prop Address: TMLUgggjEPAVsx7wBkaGiLQy8AdUeXJZWjM

exactly1of3_impn12

Theorem 5.84 ∀ABC : o.exactly1of3 A B C → A → ¬B. The proposition is identified by the following information:

Pure Prop Id: aa5009ff9a11492e840f32443e73a771ca267d22293893156f1d4deb4aaab2be
Pure Prop Address: TMYv5ButtzZjAKGh8ZS12xVuGTD6YJQcH2b
Theory Prop Id: ddde3fa5accd00c57075aa8e16e5ddd6096787e97651a7d00fe4c06313bd25e5
Theory Prop Address: TMEsRotLbeZwM4Vb8CEFrHUrjwZobtVbQmC

exactly1of3_impn13

Theorem 5.85 ∀ABC : o.exactly1of3 A B C → A → ¬C. The proposition is identified by the following information:

Pure Prop Id: 610826600590b2f8df19bcab4fe7ab0aaf66cf995d6f68f1d9ef9fd3139ef3b3
Pure Prop Address: TMadDers8VBnavfoVVcRpPTxDYWgEMaqYaA
Theory Prop Id: 829d4d5e39a13c5555db82570159451aaf7dedfe2b643ebc2c21720e8128ade4
Theory Prop Address: TMKWE5kXYcypJ3VwhHAfwnUZ524Pbhidn2h

exactly1of3_impn21

Theorem 5.86 ∀ABC : o.exactly1of3 A B C → B → ¬A. The proposition is identified by the following information:

Pure Prop Id: 1d346d501d1ba647921c3025a1335977b63c27a8a61bd46533fcfc6ba14b2d81
Pure Prop Address: TMMzqWU52BqgHAJK8WaH7Z9CCq7JwxqJcP4
Theory Prop Id: 397117cbef66df36202b151aefd2ddef44318f936aabf85d3a81c5b6185a89a7
Theory Prop Address: TMRitWJaC1PAk8gRhTEJE7Eoz2VanS86r9f

exactly1of3_impn23

Theorem 5.87 ∀ABC : o.exactly1of3 A B C → B → ¬C. The proposition is identified by the following information:

Pure Prop Id: e61d04bf5bbf41805809956d45bdde949bb6e8b00f4defdce071649f26b9bd42
Pure Prop Address: TMXB6Z3uL2sQmzDCqC3abT4uqUmke5iLtk6
Theory Prop Id: a5f7c26af5306214a6ff6a38a92231f9ad04f3095b211b4147ed6e6fa1272d23
Theory Prop Address: TMSaFCqhniFsaMDK85YeQmyVxVZbpNzbFHt

exactly1of3_impn31

Theorem 5.88 ∀ABC : o.exactly1of3 A B C → C → ¬A. The proposition is identified by the following information:

Pure Prop Id: fca5376e31e8f8fc2e5c6f166569ff186415ff18941d23f3f6f029f569148dd7
Pure Prop Address: TMFM62rQpPoQyXj5ji7YpQmUU3nG3SpUV6h
Theory Prop Id: 35250014509cb9f52a6cd6ac464fff7cd4df30035a9ea5b3c48aaedd2b42946b
Theory Prop Address: TMTbHqaT1XHTxp9kCkBgrphFzfDCg9TtuTi

exactly1of3_impn32

Theorem 5.89 ∀ABC : o.exactly1of3 A B C → C → ¬B. The proposition is identified by the following information:

Pure Prop Id: b27fbaab4bdf4912ad94528a90059e4fe9b1aa06f9c87ceed32c02a2c49b5dc5
Pure Prop Address: TMN18VWDLkauUPx1tHnX2hvXJFtMx1JohBA
Theory Prop Id: fd95f6b42e5f234c90b8218b2e2ebe7858c531af928f420deb984c1f39e731d6
Theory Prop Address: TMY9goAmFbnGXqWRMRHhrFiULnAUs9A1FYY

exactly1of3_nimp1

Theorem 5.90 ∀ABC : o.exactly1of3 A B C → ¬A → B ∨ C. The proposition is identified by the following information:

Pure Prop Id: 6f09b4a396feb709ceb93d0a17d24519b1d958895abedbaba2715b65b7e7584e
Pure Prop Address: TMJM8ao3vrQPrWM5rgtqDZ9GARGusrQ5wm3
Theory Prop Id: 21dd5ba0fc3af8a67d61c902bc798489c825a4490a81e54581755b00b24c8b73
Theory Prop Address: TMQ1J7pdfn7NmxfAzvuJMKcSZ7sdKVx5yff

exactly1of3_nimp2

5.5. MORE BASIC RESULTS 41

Theorem 5.91 ∀ABC : o.exactly1of3 A B C → ¬B → A ∨ C. The proposition is identified by the following information:

Pure Prop Id: f93f16a2acab0e2c70aaa6ba0f16c5f8fae767769ebf1191ccb689aae347f908
Pure Prop Address: TMan3tfnxxCb6QyS1xebfjHrUsCXBfkbrta
Theory Prop Id: 0a20f6a1f5ec9ea9dc84d10e81affaaeca954a6fb32ae14805896ac1d06bfb30
Theory Prop Address: TMTyG7NfHu1i8TzsieyTpERQxhdGXmw7o4P

exactly1of3_nimp3

Theorem 5.92 ∀ABC : o.exactly1of3 A B C → ¬C → A ∨ B. The proposition is identified by the following information:

Pure Prop Id: 6e599c436e1c33f48b986f575992a80e57b99534e289fc843916f91e1d1a347d
Pure Prop Address: TMbF9MFUNWCLgbMpnwTvAAwrKRm9QfXyEba
Theory Prop Id: d19918a97b442e1a1f214592d4b8db4cdda02c3fe87acf67f5a1951ba2cab40a
Theory Prop Address: TMaKsz23V9zHBq8Nc44i1QWCrQiM7XnB9N3

5.5 More Basic Results

ReplI

Theorem 5.93 ∀A : ι.∀F : ι → ι.∀x : ι.x ∈ A → F x ∈ {F x|x ∈ A}. The proposition is identified by the following information:

Pure Prop Id: b146c5f4439d8676c0dccf44c83e811ebb81f9038983f510c8607ddc1eb34420
Pure Prop Address: TMP2e3huiS7VJFLqC2ppXhrjDJ2bUGLJV5e
Theory Prop Id: f84982ee3b076ecba0e4ff60abdc2f37c622ad1afac9429103f323e4cdfe3e19
Theory Prop Address: TMHGmtrvwBUQsPubrpbb6tLvjthQVG8iwCD

ReplE

Theorem 5.94 ∀A : ι.∀F : ι → ι.∀y : ι.y ∈ {F x|x ∈ A} → ∃x ∈ A.y = F x. The proposition is identified by the following information:

Pure Prop Id: 311cdfe6686cf4d3ef41da3e91d25bb9180eb27a0f3a38739e52ece413817da1
Pure Prop Address: TMU6c8sDinZc5CK9bdAsnrw7LPnQPhim3R3
Theory Prop Id: d07e78df51e6c40a1fb15182455062bac21a7cd73536d47d7e4aeba8426d8f70
Theory Prop Address: TMdxK4ZQijyUW6813F1x5brcPHDKGCWdQpm

ReplE_impred

Theorem 5.95 ∀A : ι.∀F : ι → ι.∀y : ι.y ∈ {F x|x ∈ A} → ∀p : o.(∀x : ι.x ∈ A → y = F x → p) → p. The proposition is identified by the following information:

Pure Prop Id: 31e90a1cfe1339b3fdd723ef9c177dda85e4f3f565d6cd3cfb4f371ac4c304f8
Pure Prop Address: TMFAa4xtcBS2jBu91NQ7stWLB3d3RE3qzYR
Theory Prop Id: 34fefdfa28cb805cd6e029487d3c32cc301dc5cfe9635fa1b9dded4506aca32e
Theory Prop Address: TMJVpXF9QNoP2mDr6CyNe7oNaDmpm2R1zMb

42 CHAPTER 5. SOME BASIC RESULTS

Repl_Empty

Theorem 5.96 ∀F : ι → ι.{F x|x ∈ Empty} = Empty. The proposition is identified by the following information:

Pure Prop Id: 2af2ac513652a1800583c65d812613479314c35762cfa051dd456a4370470c27
Pure Prop Address: TMNTKJzUTMuevhemTED7aYZL43yxKSfLyaw
Theory Prop Id: 0835bc06e82fa4c2e4d9eb7e519d643457109db965d61841af4879944ee45344
Theory Prop Address: TMdRuwDbtmFquZoGEj4pWkc2saYwQmcLziT

ReplEq_ext_sub

Theorem 5.97 ∀X.∀F G : ι → ι.(∀x ∈ X.F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}. The proposition is identified by the following information:

Pure Prop Id: eb4c1d386f80c2ee581f2c1069f67f723e405c7993ee30a79f8bbd10d84d9ca3
Pure Prop Address: TMcq6sFfV1su5tjWvGTgkj2EP4oNFAVRcup
Theory Prop Id: 56aad0710a7b30d465d2dbe0bd8d3c290b9f836ccb5a6fa4519911a80b05626a
Theory Prop Address: TMZZYGkpW7fMLbAob3MNpFsZFKRvPdWXvXH

ReplEq_ext

Theorem 5.98 ∀X.∀F G : ι → ι.(∀x ∈ X.F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}. The proposition is identified by the following information:

Pure Prop Id: 81eb906670f9d334d181dc7e4b9de214b26ca6dcc378c218bba5f27088c69e41
Pure Prop Address: TMNrQK5e5dPgQCprQNuNJ3RRcsb98VaqbXM
Theory Prop Id: 7818462d4aac5c1290215c65d85d4a4cae2160757b04afdf041748738daa94de
Theory Prop Address: TMHUULG8EX1BGFc9jYacJPvM5JWuzmqSJoh

5.6 If-then-else on Sets

Definition 5.4 If_i is the opaque object of type o → ι → ι → ι identified by the following information:

Pure Object Id: b8ff52f838d0ff97beb955ee0b26fad79602e1529f8a2854bda0ecd4193a8a3c
Pure Object Address: TMFXposu1uQxEse8ZSjwvaRFgnCw5z9YTK6
Theory Object Id: 76c4c0ead9836400beb62a9590f920012eeb5b529d51bc098f0a4540dd3492bf
Theory Object Address: TMTZcwmiXZdp3zV8ehxEX8eEg2jwH6B4gXM

Notation. We write if p then x else y for If_i p x y.

If_i_correct

Theorem 5.99 ∀p : o.∀xy : ι.p ∧ (if p then x else y) = x ∨ ¬p ∧ (if p then x else y) = y. The proposition is identified by the following information:

5.6. IF-THEN-ELSE ON SETS 43

Pure Prop Id: 2a17ae8615c55763933cbf62cfdea60aab440ec5370b1a166e5a90b2dbbdaa33
Pure Prop Address: TMYAPcTTzBuQuezPwJWXPnQTtHjtuAmrh3a
Theory Prop Id: b5c26bc5c00a1fae5b9eb6e6e2dddacab303e5c17f31736c985646e26d160188
Theory Prop Address: TMSSvVBAsGTpqvzGHpFj5YS4oCq5oKaLmoJ

If_i_0

Theorem 5.100 ∀p : o.∀xy : ι.¬p → (if p then x else y) = y. The proposition is identified by the following information:

Pure Prop Id: fde1a8047aea787315fd3c48c3bf4b8dcf677b9f13975abdc0cf25a46bd38dbe
Pure Prop Address: TMS3vZha9dDF2eKGvMvTFmh9tT8CjUJYmHN
Theory Prop Id: a9abd320bb96a93c0bfd6f8cd02d5a4b36f1c06bc80d8a0d61d8f7f21e978b60
Theory Prop Address: TMXimAMUQhCm2DKHwaiTu61r3YFoxo2t7aF

If_i_1

Theorem 5.101 ∀p : o.∀xy : ι.p → (if p then x else y) = x. The proposition is identified by the following information:

Pure Prop Id: b1aa3ff40ccd3b5e52ea574b197c5921dcbf1d479f4fc972636f9a536dc4c6c7
Pure Prop Address: TMG6G29dgsYC3Bvq1bUdKCsiS18QXwMmERv
Theory Prop Id: a93c2c61afa1f7d589c4078389126876aa054a9fb356be52dbf9a1ee994afe29
Theory Prop Address: TMLX1L6KQv9yGQm2cqdp146LHmN8xmhBiSp

If_i_or

Theorem 5.102 ∀p : o.∀xy : ι.(if p then x else y) = x ∨ (if p then x else y) = y. The proposition is identified by the following information:

Pure Prop Id: 3e14398be827f1c6e32094ea4dc2734502882cbca27eddafe5dce31ef4356336
Pure Prop Address: TMVLUMbzPPFhysrpDbLFxMCLeFJ1VVL1jZC
Theory Prop Id: 4de742c8e4700f7927be7ee9ab705ba6fd76b46b20d1ba83576b15b076f51a7e
Theory Prop Address: TMRFxe5VNckUFijgfvt4yKBVqJJFGDGD9Un

If_i_eta

Theorem 5.103 ∀p : o.∀x : ι.(if p then x else x) = x. The proposition is identified by the following information:

Pure Prop Id: c790ef8c8cbc1d8b8baccef44b4431f462ec3f875068c16063b3f8e4f9606c96
Pure Prop Address: TMEsYnnoa7TBfEDReC6dhETQtLYNdEGfVEm
Theory Prop Id: e413bd92fbd8dffea8c40f40fa4c82bd8732a0d4a82572f9264e5767a06a8ade
Theory Prop Address: TMTiR3dSMwWpEh2ReAKGGJZjp5K2AjJ4xrH

44 CHAPTER 5. SOME BASIC RESULTS

Basic Set Theory

Definition 6.1 UPair is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 74243828e4e6c9c0b467551f19c2ddaebf843f72e2437cc2dea41d079a31107f
Pure Object Address: TMVRUmJTotqxWfVB8wE3prBP8W149nntsis
Theory Object Id: dc313483331e27588bf1a1b69cd5f5c7e098db82519bb4f97f7410379392841d
Theory Object Address: TMX4EvisNRDE5LYNME7fAsXyPiC8WnAhWDf

Notation. We write {x, y} for UPair x y.

UPairE

Theorem 6.1 ∀xyz : ι.x ∈ {y, z} → x = y ∨ x = z. The proposition is identified by the following information:

Pure Prop Id: 5009cc2fb54e181e885e6ba3ebb81d1e7e8a680ac7694595d03d2bae261e7b96
Pure Prop Address: TMGwnYYsQikhZ63aRncU1cXAkHjKyj9bmC5
Theory Prop Id: e67fc59933588258cbf12faa14d08a2171e75c4be3c6375f1c244269e8f0a620
Theory Prop Address: TMZokDUxWLSRz89jtEJyPSvuwJgZmLVRkEK

UPairI1

Theorem 6.2 ∀yz : ι.y ∈ {y, z}. The proposition is identified by the following information:

Pure Prop Id: b1eea2f239b14cf1d5593f79c76232b611aa52ae4809066c3fff5897fe4a9d56
Pure Prop Address: TMPfeeVFA54LBD1V3Ejp5GiM71yeFdjdpEG
Theory Prop Id: bf36495fbbdc72eb575c69fdfc681c506ae9eebfb94538f10549bc8f191ce07b
Theory Prop Address: TMd3cJnuukmmggqVuKQroUQUyw9xFNmdytb

UPairI2

46 CHAPTER 6. BASIC SET THEORY

Theorem 6.3 ∀yz : ι.z ∈ {y, z}. The proposition is identified by the following information:

Pure Prop Id: 27455b29a050cd7d072a6fecd331e4463360653ad07926e47c46ee56e5b66f75
Pure Prop Address: TMdg8pGYbTqgx8xPL9PwPu2pJJbSoxCnRHy
Theory Prop Id: a0a5b852859049d097662ba32a64381a0c4251956d0027ceea65617e12ec7e68
Theory Prop Address: TMTrsMdWZgjAwNgkT5uqy2DqzBnBXnxWupZ

UPair_com

Theorem 6.4 ∀xy : ι.{x, y} = {y, x}. The proposition is identified by the following information:

Pure Prop Id: b0847098a703dd9518520ce0538ef2bb61b423ef3bfce3e056d6ddebc96a5b10
Pure Prop Address: TMTDNchhQ5726g1za89cNkp72gXjpPYs7Nm
Theory Prop Id: 792d7d18b9ffdad302a08ed667c940da3d4f02f0bfbaf07fef81aa2961d74531
Theory Prop Address: TMVjMcbkMeWtsqfqTEQnrRhtUFG4TMoXw5s

Definition 6.2 Sing is the opaque object of type ι → ι identified by the following information:

Pure Object Id: bd01a809e97149be7e091bf7cbb44e0c2084c018911c24e159f585455d8e6bd0
Pure Object Address: TMRoFJh3XPuDqcMDh3fauvfQL55yG4ZW8xi
Theory Object Id: b7dfc1c317fc5a717f457d8782ced05f886230b550688a8ad8b1a60364455741
Theory Object Address: TMYwdTEJpbYhdjjsshw7LP6uC2s7dSrhjMJ

Notation. We write {x} for Sing x.

SingI

Theorem 6.5 ∀x : ι.x ∈ {x}. The proposition is identified by the following information:

Pure Prop Id: 6591bd63306a8b20a34c3c62753b21a3972cf8174ef23e9454980a5b6b82618d
Pure Prop Address: TMWZhsiyoAnszzrXM7jCkxtmgh4ETFtiEfk
Theory Prop Id: 2f93be9a22d244304c3bc2f3253e0b4a512c5f66bb0f005cb881e61f6eb26071
Theory Prop Address: TMbi5x8pysc7XDzhg16jDTJ4KrkuJzZS7BK

SingE

Theorem 6.6 ∀xy : ι.y ∈ {x} → y = x. The proposition is identified by the following information:

Pure Prop Id: 9341fe885f477660bd3dfc0b46ef487b4623351d45911bddb267d44cdf6d0821
Pure Prop Address: TMNoMEGQFgAVj7maHozhYTCSXBfUaNXtuAN
Theory Prop Id: 3b0dd829f31946c18010e2f35a58c321cf20e3464f3c10d63efcab64a4f018fb
Theory Prop Address: TMGwutpkdmmtAbHgjLsWPgK3gW6vqLx2s43

Definition 6.3 binunion is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 5e1ac4ac93257583d0e9e17d6d048ff7c0d6ccc1a69875b2a505a2d4da305784
Pure Object Address: TMKadjsG1yH8Ra5dXxSA9jstbEvqFo88bxo
Theory Object Id: e14989b9e54d0036d328f81b7c2ca9bf3b1c6d7d6dfd5cbc1d8421fd6cc0159e
Theory Object Address: TMWh5oU44xPmpsrB4CwWV2e49m2HBPkrW2D

Notation. We use ∪ as a left associative infix operator corresponding to applying term binunion.

binunionI1

Theorem 6.7 ∀XY z : ι.z ∈ X → z ∈ X ∪ Y. The proposition is identified by the following information:

Pure Prop Id: 753b6ef046ed0483fcc02f203ffb7e80213207fe9ca6b95995de9c4eaea00411
Pure Prop Address: TMRRF1KJnf4XF9rqDvQv3UaJJUWLeXK8ZP9
Theory Prop Id: e43c8db7c2b8d05d548b5a0c56a8d23f40736402d22a35cb8a212ee0ab4be72c
Theory Prop Address: TMMufFv1jkCn3ZdkSG7Y7viPMpw7eZgnmeh

binunionI2

Theorem 6.8 ∀XY z : ι.z ∈ Y → z ∈ X ∪ Y. The proposition is identified by the following information:

Pure Prop Id: b8821c633c5f298b2822c44b4eecc8fdef6356f33c6b8bdd93a53e96d7729690
Pure Prop Address: TMY2zmhR5aQU4DsXTEmDmSgQYhYPh1Py1VW
Theory Prop Id: 959535d02d0caf6be5dc6d781bb10a45795b705bd32a8de6d41ef5ccdca130ce
Theory Prop Address: TMLpWxJ4RXSb2TWfpnMMgvDQbHM1A4XSH9X

binunionE

Theorem 6.9 ∀XY z : ι.z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y. The proposition is identified by the following information:

Pure Prop Id: 5a319544d131da120ce102ea37cee203a594453cf4350bac27592a462809271b
Pure Prop Address: TMHtDhNz67RzwWxi6YMkqKVbqx8c9XaiDyX
Theory Prop Id: 1ceeb045243b9d5012dc124e31a841805b4e6059c2e99973ef3d2663b5039471
Theory Prop Address: TMMe9w3Qz1uA5B6MSbrBAwnM2Ydf6fFAksa

Definition 6.4 We define SetAdjoin to be λXy.X ∪ {y} of type ι → ι → ι identified by the following information:

Pure Object Id: 1f3a09356e470bff37ef2408148f440357b92f92f9a27c828b37d777eb41ddc6
Pure Object Address: TMXya8p7m5AB5g1dVJ1MSDKpnXtcu2N87Mz
Theory Object Id: 5bdda84d481c907ad5ac9225d0210c23eca52ec933bd26ac51fc7debfa5cd394
Theory Object Address: TMLYeA5y1CUhy319tM7uM8NUTufJJyM6eya

Notation. We now write {x_{1}, …, x_{n}} in general with the following conventions. If n = 0, the notation means Empty. If n = 1, the notation means Sing x_{1}. If n = 2, the notation means UPair x_{1} x_{2}. If n > 2, then SetAdjoin is used to reduce to the case with one fewer elements.

Power_0_Sing_0

Theorem 6.10 Power Empty = {Empty}. The proposition is identified by the following information:

Pure Prop Id: 6526a1550f9c7084aa6631fc5bd8b4d8d339fd0ce60e7d548edc293a22206f43
Pure Prop Address: TMFtrGrnuqpdT6zKi1x2QYqakvTT82UnEGx
Theory Prop Id: a39fd0f7312cd15f6f2e488850e84694dd268ddebb9a0a84a4dec2e6cce4b2e2
Theory Prop Address: TMZJjQSh37Giw23bdQW2e8z3PhLqdGbYhbC

Repl_UPair

Theorem 6.11

∀F : ι → ι.∀xy : ι.{F z|z ∈ {x, y}} = {F x, F y}.

The proposition is identified by the following information:

Pure Prop Id: cacbdd41cb5a9699e9d7e46d5d2d536a174a15a2dad0e76a3885d739f3bf9618
Pure Prop Address: TMavxam5T7BVKzszyZpmyDGWmzwq22rrVr1
Theory Prop Id: 412745a32a74658e5f4747ff82265690389455bc948a7dc4ae5ff1cc468be3bd
Theory Prop Address: TMSeQz641QN3jhU5ykuM6QagGpUJisdc52E

Repl_Sing

Theorem 6.12 ∀F : ι → ι.∀x : ι.{F z|z ∈ {x}} = {F x}. The proposition is identified by the following information:

Pure Prop Id: 1b6f04154ad08fd4e5ac8a4a7e0666ebb6990dd65162ddc49c4c2aec615a9338
Pure Prop Address: TMXmXBaEMHiXuMsDum7NZFR84rqzoKxHCpU
Theory Prop Id: bd9e64fedf4189ba490aa0c11b9fbd79890c422133c09d3193c333247af88b1b
Theory Prop Address: TMV9HYczMJjbDeoJuBm27QfyM9yXxF6FN3Y

Repl_restr

Theorem 6.13 ∀X : ι.∀F G : ι → ι.(∀x : ι.x ∈ X → F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}. The proposition is identified by the following information:

Pure Prop Id: 81eb906670f9d334d181dc7e4b9de214b26ca6dcc378c218bba5f27088c69e41
Pure Prop Address: TMNrQK5e5dPgQCprQNuNJ3RRcsb98VaqbXM
Theory Prop Id: 7818462d4aac5c1290215c65d85d4a4cae2160757b04afdf041748738daa94de
Theory Prop Address: TMHUULG8EX1BGFc9jYacJPvM5JWuzmqSJoh

Definition 6.5 famunion is the opaque object of type ι → (ι → ι) → ι identified by the following information:

Pure Object Id: b3e3bf86a58af5d468d398d3acad61ccc50261f43c856a68f8594967a06ec07a
Pure Object Address: TMWQ76Uj1RnGeogmDxhYaxbaaPGB1LBoCiP
Theory Object Id: 241349d21f55f878b9cd5590cf66f9dc362c98552772a88eccdedc7910500fd0
Theory Object Address: TMaTpVSSy7hrg8WXTVmsiohBmjmfrDfMMJG

Notation. We use ⋃_{x∈−} .− as a binder notation corresponding to a term constructed using famunion.

famunionI

Theorem 6.14 ∀X : ι.∀F : (ι → ι).∀xy : ι.x ∈ X → y ∈ F x → y ∈ ⋃_{x∈X} F x. The proposition is identified by the following information:

Pure Prop Id: 0c23d0ec3aeca92359022d4236f139bbce8322597bc7aa615dfd5957df202628
Pure Prop Address: TMc9t7j1HpKhoSwH9rdLbnytztKsuCGA1n3
Theory Prop Id: e4f490a807c7166931ad2baca93cfaa01c42669f58e0e3fa9d5a4e910da403f1
Theory Prop Address: TMSPh7dPrn6WaLbJUMvCgyCRAXKpqw7tSxW

famunionE

Theorem 6.15 ∀X : ι.∀F : (ι → ι).∀y : ι.y ∈ (⋃_{x∈X} F x) → ∃x ∈ X.y ∈ F x. The proposition is identified by the following information:

Pure Prop Id: 48683a19e0b9cac431e35763caf26ab25d3b68d0eec6fb8193bb111ac8d2a57a
Pure Prop Address: TMS1BrnhPG5Rb3BqURjxG5UF4FXEg1CWJ6h
Theory Prop Id: ede6e5d506bad2c8dd1acc9526f74416f2e8ed084472c376ae92dd12e47b1b36
Theory Prop Address: TMN9kEqZptjbB8TVx3pwjNpZJsSZxFeje71

famunionE_impred

Theorem 6.16

∀X : ι.∀F : (ι → ι).∀y : ι.y ∈ ( ⋃_{x∈X} F x) → ∀p : o.(∀x.x ∈ X → y ∈ F x → p) → p.

The proposition is identified by the following information:

Pure Prop Id: ab22661d1f7c130e5684c80d6642dea2ccbe8caea270129cb32e218a75dc3b14
Pure Prop Address: TMM2k1dfrqcpKhy5sotaNPvjvVMC2zbAF94
Theory Prop Id: 562023a05cb52f7958f9f74add2503404a86b5d121085bea792c6ae40d578143
Theory Prop Address: TMbvpfPvzznj6iG7UVhksbQDUQpsJeGR69p

UnionEq_famunionId

Theorem 6.17 ∀X : ι.Union X = ⋃_{x∈X} x. The proposition is identified by the following information:

50 CHAPTER 6. BASIC SET THEORY

Pure Prop Id: c0407ac29149de6b96007edfd11708f9cad613bf44851fb9a688ec0b72823abc
Pure Prop Address: TMJq16zmpETvRMnCrNiWd4hPxiCF9wLQGvM
Theory Prop Id: 5b5383939b1feed9224b810975652ed8da9eef8545e5df1b9311c75e151ca976
Theory Prop Address: TMazYyBoEesPvNn5L5cyG4oKnoYJWf1MTXS

ReplEq_famunion_Sing

Theorem 6.18 ∀X : ι.∀F : (ι → ι).{F x|x ∈ X} = ⋃_{x∈X} {F x}. The proposition is identified by the following information:

Pure Prop Id: dfa22e97a0f977492075ad87b7ec5f1e874e8cc726cad73016122eff2e495382
Pure Prop Address: TMcpQ6xMXyb8neoFWcHWRhPuHkeDjU7r5Sc
Theory Prop Id: 9829d1ccabddee314bee7238491144e99da007ceb90bcc3b1aade28e6bc30824
Theory Prop Address: TMbPFYZ4Eaf5LC8DmM7MPALZ4kWFxfV1npd

Power_Sing

Theorem 6.19 ∀x : ι.Power {x} = {Empty, {x}}. The proposition is identified by the following information:

Pure Prop Id: f735ed6a9bad818e002e12cc5ef0819b362cbc929180991433e3e3237282e883
Pure Prop Address: TMYYws5iekM1u7qxpPo65kGRFcdPoaPVCa9
Theory Prop Id: e5fb8fe5f5f2baf36fccd9d1f35398ed4a86da3345f90b8e9d2336d609a38a7b
Theory Prop Address: TMYxTSeb1gFn8ZUnNyFfjYHJv1cLqspoYGo

Power_Sing_0

Theorem 6.20 Power {Empty} = {Empty, {Empty}}. The proposition is identified by the following information:

Pure Prop Id: 24373cc848ed23a872ca0879dfdbcf7a451233726845c72b4cfccef56ea08e80
Pure Prop Address: TMVoMhEC9xC2sTc7tiuSozHc8YaTWHcTdNa
Theory Prop Id: 173fc92aca3f0c56bd3327162bd62d8aa42795423e3bd3a7692afa1bcbead4cc
Theory Prop Address: TMTeCtmtdQ9KwHpHJU1W4m7ytbUgCGJRUV5

Definition 6.6 Sep is the opaque object of type ι → (ι → o) → ι identified by the following information:

Pure Object Id: f336a4ec8d55185095e45a638507748bac5384e04e0c48d008e4f6a9653e9c44
Pure Object Address: TMUtfXxDDLj9Tk47vMVXxn7X4YsuGxPH9iG
Theory Object Id: 7008ec0ee19d11cc67c5f11635c82852ccb27360b8bd8431676b0c7e03219428
Theory Object Address: TMU5xEoVJ4JK7LsxiB8DMm5ouqBu3Rgy3CL

Notation. We write {x ∈ A|varphi} for Sep A (λx.φ).

SepI

Theorem 6.21 ∀X : ι.∀P : (ι → o).∀x : ι.x ∈ X → P x → x ∈ {x ∈ X|P x}. The proposition is identified by the following information:

Pure Prop Id: ea56bb93dea4f3624b90c68591a17a4b0e370a444cd3e405fd7fcdb52632635b
Pure Prop Address: TMR97AbUhdgTucfxJ7ujRoLGzK6qJ2E1jyn
Theory Prop Id: 147a6ae90a60bb6d7a3135bb216628e19a94a3f9a5fb6e915e96bbbf26d60ba0
Theory Prop Address: TMUrJfQgr3jYT2UBYmVX9Mv2z2Jgr6upg8L

SepE

Theorem 6.22 ∀X : ι.∀P : (ι → o).∀x : ι.x ∈ {x ∈ X|P x} → x ∈ X ∧ P x. The proposition is identified by the following information:

Pure Prop Id: 102830994311842627e3ed866fe29eb1f07e8f9431d5cb028658a623df943412
Pure Prop Address: TMGr4JhwYm65vLL1JdgtV7Z4udWfVUCyR7C
Theory Prop Id: 00899a751488a4862ed01d1ae8f2ff0dbfbfe3fe46a1349c813bec079a6f9ede
Theory Prop Address: TMSaSd5PfDFDpCcrVNF6RUq6GkDHu9ks3pQ

SepE1

Theorem 6.23 ∀X : ι.∀P : (ι → o).∀x : ι.x ∈ {x ∈ X|P x} → x ∈ X. The proposition is identified by the following information:

Pure Prop Id: d83445946f710ccd1847efc45d29fcc99bebd491806d37ec544f5350bcfa0f86
Pure Prop Address: TMWSj1Q1gssEN1FRYzrkAPVjkpNyJhfDv8F
Theory Prop Id: 6399b0a98035d8a70e8306896b0f872730b3288dcdbb66b851b1f84cc75089c8
Theory Prop Address: TMSjwkTcc2asZAUTrPPnJyXprfdL48BnKdS

SepE2

Theorem 6.24 ∀X : ι.∀P : (ι → o).∀x : ι.x ∈ {x ∈ X|P x} → P x. The proposition is identified by the following information:

Pure Prop Id: aa61ca17580417f6d32cdf9fba918e207307848ff899382c5acbd1e2027eebd5
Pure Prop Address: TMV6AB23w8THH3JMfPMPg1fJahEHtfBWk9A
Theory Prop Id: 104227af0e258de974a06dacd30d9996af2cc290588f50a9c669aaf9868024f6
Theory Prop Address: TMUZTejdmRUG8mdTCdynTZZpHdZx8LNmpbi

Sep_Subq

Theorem 6.25 ∀X : ι.∀P : ι → o.{x ∈ X|P x} ⊆ X. The proposition is identified by the following information:

Pure Prop Id: 97d0809464d57d8f031c3968a64d1d7a35cc9bf2076517a32bfada2a891489f0
Pure Prop Address: TMKXby5zNJ4KVAvdU1gcKmDEYMkBvsYGdtv
Theory Prop Id: c17d45b17052435d24c1e2d23e7e0c0589d735640358f24c5893103ac9cd487f
Theory Prop Address: TMYQD6DWoPBDiMim118Sqg7B5nz4JVyqYui

Sep_In_Power

Theorem 6.26 ∀X : ι.∀P : ι → o.{x ∈ X|P x} ∈ Power X. The proposition is identified by the following information:

52 CHAPTER 6. BASIC SET THEORY

Pure Prop Id: 612c94a6047cff99b0373d060e43552169d05dddccd4ea06be3c8d6a86650764
Pure Prop Address: TMHfdLZNWAmDjm1ZXXcZtP8QxYVC7ugYRTA
Theory Prop Id: fba53ed8ea5e05c876d2ead08cbb41c2104737958be70e5c48d509f88ed420a6
Theory Prop Address: TMd59svET4t7PBsSUNEFGENePU74pbhVRyk

Definition 6.7 ReplSep is the opaque object of type ι → (ι → o) → (ι → ι) → ι identified by the following information:

Pure Object Id: ec807b205da3293041239ff9552e2912636525180ddecb3a2b285b91b53f70d8
Pure Object Address: TMLdtkt2SAyu6uRx3VwN4EmguAaBBbX15ky
Theory Object Id: 9bdb562173d8d14c707097187f2b9ec994d4d550da69fd27247ea7a1734dc0ea
Theory Object Address: TMRDkBndFptKLHv3ZdFcEM1zsnPucRbrZsD

Notation. We write {s|x ∈ A, φ} for ReplSep A (λx.φ) (λx.s).

ReplSepI

Theorem 6.27 ∀X : ι.∀P : ι → o.∀F : ι → ι.∀x : ι.x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}. The proposition is identified by the following information:

Pure Prop Id: 35b8ecafee8ec7ce07622bb50b3833b0aab40abc8e7a7253fef934747ce888e9
Pure Prop Address: TMWGYkRQTsDoJuJbmJZ1z6DVf1M5n7LNCgC
Theory Prop Id: aa7b2b992fe9f4cd5864693019fc63fcdd7e92856cb452371ce4409e421bf1e7
Theory Prop Address: TMGFrvHwbGaUn5voQsk3ha4eAC8RLvBtTSB

ReplSepE

Theorem 6.28 ∀X : ι.∀P : ι → o.∀F : ι → ι.∀y : ι.y ∈ {F x|x ∈ X, P x} → ∃x : ι.x ∈ X ∧ P x ∧ y = F x. The proposition is identified by the following information:

Pure Prop Id: 415ff1723537afd685cb0fed04bae8d9d3e3c9fc8dd81f94e3e5ef00af2fcd62
Pure Prop Address: TMErPUtuvpjwR2ReNB9nGcmnK8AYV664v6C
Theory Prop Id: b1bc76c44d5f7b7b23a5be6966a678cb5d36886ab4685b00d3d5157e8daf69e3
Theory Prop Address: TMQMpL9gjiXL2MUteH6hiemmLJG4AhbLTXr

ReplSepE_impred

Theorem 6.29 ∀X : ι.∀P : ι → o.∀F : ι → ι.∀y : ι.y ∈ {F x|x ∈ X, P x} → ∀p : o.(∀x ∈ X.P x → y = F x → p) → p. The proposition is identified by the following information:

Pure Prop Id: 5d26e1469193d5bd2136b195439a688359b4bf58fb7a17905abb3c0cf02e93af
Pure Prop Address: TMMk3dLDDxdwMUP7GQkv6ep261PQbXFJKi5
Theory Prop Id: 063cead7e54785ba03db6bfc07a0999226785a5ad4252309bcbbf4d94eb92422
Theory Prop Address: TMM2umBqoRLF5bftij5PkXw6MCfbhi2G26n

Definition 6.8 ReplSep2 is the opaque object of type ι → (ι → ι) → (ι → ι → o) → (ι → ι → ι) → ι identified by the following information:

Pure Object Id: da098a2dd3a59275101fdd49b6d2258642997171eac15c6b60570c638743e785
Pure Object Address: TMNVwxznYhNZZoYoQL3ESJeSeewae4Z5ubB
Theory Object Id: bd46a4454a608867c1da526c8a8d052f33ebc39c49a21d7e84f4d329218203b0
Theory Object Address: TMdeKFiZBi4BgUrRypBtmVFrLRWnZ8EbVUP

ReplSep2I

Theorem 6.30 ∀A.∀B : ι → ι.∀P : ι → ι → o.∀F : ι → ι → ι.∀x ∈ A.∀y ∈ B x.P x y → F x y ∈ ReplSep2 A B P F . The proposition is identified by the following information:

Pure Prop Id: c012d962b28e32216b7c116bb9b93b568f22cdbd01c2ad49eb0973e193d89ca5
Pure Prop Address: TMbDYFXkGgf7ZjkUpxAt2c8KfCPQnzBVHXo
Theory Prop Id: d75c053450bb002a32e5862bcdf41f63858ceb9bca07b0b78edc448c3cd382e5
Theory Prop Address: TMTZAhgLueBybAHKpFbownqVFw34ydwJv3A

ReplSep2E_impred

Theorem 6.31

\begin{matrix}∀A.∀B : ι → ι.∀P : ι → ι → o.∀F : ι → ι → ι. \\ ∀r ∈ ReplSep2 A B P F .∀p : o.(∀x ∈ A.∀y ∈ B x.P x y → r = F x y → p) → p. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 9cf007f5a509413458fb2202520eeadc0e1282d8f5e08ea4e998973e0c49f6eb
Pure Prop Address: TMH6JGjNiuhPqpFuX9oizzHFk5Le2zk5WAH
Theory Prop Id: fc01e896457b69f97929e28b9ca1348b7bec5c0f73942fd10cd6424efd4ff8da
Theory Prop Address: TMP5H56SXZ4FHy2J2NYCGC9QGApNByq6Fuv

ReplSep2E

Theorem 6.32

\begin{matrix}∀A.∀B : ι → ι.∀P : ι → ι → o.∀F : ι → ι → ι.∀r ∈ ReplSep2 A B P F . \\ ∃x ∈ A.∃y ∈ B x.P x y ∧ r = F x y. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 7c43a59f9be76561c287a01d1503cef13020f9d5467c7fe07a42e5ca37944da3
Pure Prop Address: TMWPJMkxvgzqUKqtFU66YkSgt15bbJMX2GN
Theory Prop Id: fb571ab298cdb2355f09c7b4be2d8e161ca5df97f2f94f5907375ecfd6c02700
Theory Prop Address: TMR6JYS9aq1RbPLrzpqs7eQpgMEUJFbekS2

54 CHAPTER 6. BASIC SET THEORY

binunion_asso

Theorem 6.33 ∀XY Z : ι.X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z. The proposition is identified by the following information:

Pure Prop Id: e0f701721ae7f584ad335901c46f861f37994ae2c421659e2fc7d24d52c4ad58
Pure Prop Address: TMSkYkrEs3aZ5sGQH7NK9w6LMvc7ihPrtiY
Theory Prop Id: 61dfbad9a60ca916e2a74941dde462165a35addadb67dd1aa0e14bf13e972036
Theory Prop Address: TMay4ygEWLRWTtSwLgj2avwuNTtFqy4VPaD

binunion_com

Theorem 6.34 ∀XY : ι.X ∪ Y = Y ∪ X. The proposition is identified by the following information:

Pure Prop Id: e20f4f80912d4279be9396097afd9e80c497a2f339edd6413343e5d23c789f0e
Pure Prop Address: TMaLPUJawPt39v249vAR1DqP1n84oA3fd8g
Theory Prop Id: bed33ed76997bd9546985ac652f8933bf140216a164aa11462375a79b95e7a08
Theory Prop Address: TMY4TyspYtVRBA6PfYNL89qxpZvncatAvsk

binunion_idl

Theorem 6.35 ∀X : ι.Empty ∪ X = X. The proposition is identified by the following information:

Pure Prop Id: 8dc61661d4d5e1f88a3162d142c0a3e25f64eacc329dfd0068a1dc72be818a05
Pure Prop Address: TMHvo75eYy3DzHQyXDhy7nSXmaRpD1QL8Xv
Theory Prop Id: 52cd24837ea762ce63b82815496f8b7cf911a666c277332e8ecf1679702408c2
Theory Prop Address: TMM6roc6rAT3Nk8tdS3jEzf3rjCYtEtKppo

binunion_idr

Theorem 6.36 ∀X : ι.X ∪ Empty = X. The proposition is identified by the following information:

Pure Prop Id: 67e82fb8d1d026e5488f27f866055b8ad7b6ae3107546e436d50455676cb7db6
Pure Prop Address: TMYUCbeWsYbEVYGjC69S1voGpXmnhiyPS1j
Theory Prop Id: d5a0059cf51bb85c8614882cfb5aa46e5986047c17f59943f59e9afdca587577
Theory Prop Address: TMK8JAPKwtcTnWyvdtnBF1r93mW75pcUzMM

binunion_idem

Theorem 6.37 ∀X : ι.X ∪ X = X. The proposition is identified by the following information:

Pure Prop Id: 810a82013ba606cd5918b3f6b58976e1a987615eaad568a3e32930a3239a3e37
Pure Prop Address: TMbQmiBEMSBKRMaesDwRUx5iHKc86Jpvahe
Theory Prop Id: 10e6968d996355460bf1486862c7a957c7b64f3eaa21e7c5530de09d0acc76bd
Theory Prop Address: TMRyZSEzQ6SeCXSBXN2HWBcfU7GgBsUCzp7

binunion_Subq_1

Theorem 6.38 ∀XY : ι.X ⊆ X ∪ Y. The proposition is identified by the following information:

Pure Prop Id: b2c65821a8c09a0b77af768d4af071f6f52e54e3ab9f874132e22d9b780fec03
Pure Prop Address: TMNMiyzoBmmTgby8dHSUT9GNJXBRATKB1Je
Theory Prop Id: a9b3a8f498f868b3e5f0ad0bf8f135aeabf5748b92ad79003393964533a788f6
Theory Prop Address: TMKuARFTAuYGC1dr9qRt4msXcsg294CE5fx

binunion_Subq_2

Theorem 6.39 ∀XY : ι.Y ⊆ X ∪ Y. The proposition is identified by the following information:

Pure Prop Id: 3cf2292964400c071542529f75f809fb1cccc1b48658e084626f9f60981211a2
Pure Prop Address: TMWp6qtFQ6sNjLamnX9ZFCwWRY29tAeEj4n
Theory Prop Id: 1507ce49772284307aa14b5f6312cbe2050013469059484c4e6af749d27296c7
Theory Prop Address: TMHECM3VikFfrZ52FBVEeJcMZtdHizHKgfk

binunion_Subq_min

Theorem 6.40 ∀XY Z : ι.X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z. The proposition is identified by the following information:

Pure Prop Id: cc32a5be9e847d0d6b3321ae657ec25e33e4efa0fa1a8b187388477416c1b8a3
Pure Prop Address: TMZ3UZw7snyBWjGXoD2PLSgQ7Z2tViLeZHE
Theory Prop Id: d64da44c02293ec947fdce4b6a1334307011456947961dce8950499be075b837
Theory Prop Address: TMEmr475JNDN72vef9HbrMFEnw3HeFp8ckR

Subq_binunion_eq

Theorem 6.41 ∀XY.(X ⊆ Y) = (X ∪ Y = Y). The proposition is identified by the following information:

Pure Prop Id: f8ee53d9e4447bc52c14da3fcc90d5e8615f20617571fe5255ea2f6eb381e06d
Pure Prop Address: TMbuiip99HVRqBDM9hnt3UzPZe6NvLXEw43
Theory Prop Id: c2d39365f127c2c7b045442887fe24ea64a512918a28eaa5c725c8592f45e775
Theory Prop Address: TMZAQ7e8J964d8E63qrdsixRgixW7twAP2T

binunion_nIn_I

Theorem 6.42 ∀XY z : ι.z ∉ X → z ∉ Y → z ∉ X ∪ Y. The proposition is identified by the following information:

Pure Prop Id: a815887d2483bf0106a08715fcac061cae3e6bcbb8e465a668937ec74c34a252
Pure Prop Address: TMHmep2oeAYUaJwZC3hiCXb2vG3pYtH1QiF
Theory Prop Id: b26c78a45180d7dd20f8b0147d272da614cde185efd40b05c72c389284a5bdfd
Theory Prop Address: TMNdwhoFsncxh4zt41JCJjMiky1UTdLveND

binunion_nIn_E

Theorem 6.43 ∀XY z : ι.z ∉ X ∪ Y → z ∉ X ∧ z ∉ Y. The proposition is identified by the following information:

Pure Prop Id: eb90dbeba507f6184d9a4519639032fd25e13e8a5665b55e204384457add4651
Pure Prop Address: TMLKugoiqVxQHU6VmQg13s94YEdBHkpk3c6
Theory Prop Id: 64c8be6bcd58b6ff3005c6e68cea0923081c0c08b375ed3440f58e2ae4590507
Theory Prop Address: TMMzSapwNEj6zYnr4sqrPXz964RVJfZhXvJ

Definition 6.9 binintersect is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: b2abd2e5215c0170efe42d2fa0fb8a62cdafe2c8fbd0d37ca14e3497e54ba729
Pure Object Address: TMLEtM84AAELgYDKCd7u4BvMP8XhNj6rBEU
Theory Object Id: 2c68d89742466253865fe4ac454f660dc5ef3aabe295adca6101b1cfe29a9b4b
Theory Object Address: TMTxtnFNepi91FFerdTCcnaeQ6ZJWf8pc76

Notation. We use ∩ as a left associative infix operator corresponding to applying term binintersect.

binintersectI

Theorem 6.44 ∀XY z.z ∈ X → z ∈ Y → z ∈ X ∩ Y. The proposition is identified by the following information:

Pure Prop Id: fd656d06038bf1e8213bf796c99d2730d0934ac5207ea1a324a38e8ffb7f1d72
Pure Prop Address: TMavDaUgQY4Yc7jvGaUAZLBxvza2KYE7CTa
Theory Prop Id: 9081a09947c1d3d1991f93fd8066a329cc8dd137d5232ac2ec7d204ac6f93e3c
Theory Prop Address: TMW6j5eRPzucAp43ZDsETbD83EUAMzqu3gS

binintersectE

Theorem 6.45 ∀XY z.z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y. The proposition is identified by the following information:

Pure Prop Id: 463b5a7d659e015ae77f37224a2da56f16082db6bae0d607624285937805d0ac
Pure Prop Address: TMGgPwxMGdV2DoStuTEK7hiAAwBA496E6VG
Theory Prop Id: ca06365460a4a4310d69039b45771ed26e9f64b8553c1685d7d26b38bc88fce3
Theory Prop Address: TMT3UicyLqut6GdZ4E3a5HpLsx7pQRGYv4o

binintersectE1

Theorem 6.46 ∀XY z.z ∈ X ∩ Y → z ∈ X. The proposition is identified by the following information:

Pure Prop Id: b12123fa4efe378e34ca200da469f86ccfc917210d536de59d52c5e9de77116c
Pure Prop Address: TMVXNYZCzRuA7LZtTrfmGXLqpJKtWXchcg3
Theory Prop Id: 51005ef0a8ef018c9807d92cb93099aee35ab19d9b57fee1f06b6d001123d66e
Theory Prop Address: TMGso6ks75EXfsrAgvoTysUxjJQVJBRAjXQ

binintersectE2

Theorem 6.47 ∀XY z.z ∈ X ∩ Y → z ∈ Y. The proposition is identified by the following information:

Pure Prop Id: 39656fd10aea14619f5a39d872360e27a8a2adbf407ac9f383cc565db9601f30
Pure Prop Address: TMFtaVMiexDH341MBZktuvoMPVCTRcMZ5E6
Theory Prop Id: 599de44995ff57c083eb2f45a7abd687d47df353ae136fb3a5621a3444c2edaf
Theory Prop Address: TMckjJj9KRQHyQMWWeL4Y7CKxNta6jpAGYB

binintersect_Subq_1

Theorem 6.48 ∀XY : ι.X ∩ Y ⊆ X. The proposition is identified by the following information:

Pure Prop Id: f296ad2186681350539ee33fda247d0b2db3407ef4c9b9878863d410f9f6c174
Pure Prop Address: TMHYoEPaHzR6jsg5oRyMdnfEQqNRzrVv5fj
Theory Prop Id: 303833566de1cc91ab34cfc55ff0aa1141fcb51e6ffe3026695258fa36b408ef
Theory Prop Address: TMbU54waGS53UwVgzDZ1rm2yMPzD36vBhkj

binintersect_Subq_2

Theorem 6.49 ∀XY : ι.X ∩ Y ⊆ Y. The proposition is identified by the following information:

Pure Prop Id: fcd5f54826312c964c01eee203b1512d720124e0ee6cda87c6500675af973a10
Pure Prop Address: TMdHZ7X2QicJLMked82RYwwjaLTcFLaALcQ
Theory Prop Id: 8b9fd275589ddd59b577f6592484143e175ec81d47f79fecbc691c373af53bf6
Theory Prop Address: TMdhk1ZqZYfFHLHXb5kzMk3FTGGmYXyd6pv

binintersect_Subq_eq_1

Theorem 6.50 ∀XY.X ⊆ Y → X ∩ Y = X. The proposition is identified by the following information:

Pure Prop Id: 277eecb6dd017782dd48738677bd8a5982157e4c5cd5cc05918e2124c974cd07
Pure Prop Address: TMGURzfSeKMD3UYDfmxJ2DFVhggDoZwZnUy
Theory Prop Id: ec1cc3fd8120069fd440f892bb823f090399876a574880710582ba7186a73de5
Theory Prop Address: TMcUHjKDeFZpkMnMczJ2SUAQuA1B6GVgFMp

binintersect_Subq_max

Theorem 6.51 ∀XY Z : ι.Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y. The proposition is identified by the following information:

Pure Prop Id: 95624ddfbe331265c4ca2f86b896a19a583eb8540109d5bf381eca2fd9c5a2ce
Pure Prop Address: TMZEyDaHk9qJtwS7tYcTVD3TPZbywzpMvLN
Theory Prop Id: 64636bafad5e1f11c7c5b8d776a60352971b7c848370b9d8c312eba0ba7c8a7d
Theory Prop Address: TMYQdf7jp47hrnGmxiraKcX3aZTPo8rVu7B

58 CHAPTER 6. BASIC SET THEORY

binintersect_asso

Theorem 6.52 ∀XY Z : ι.X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z. The proposition is identified by the following information:

Pure Prop Id: 888f8c948c07fff1208257cfdcb30e80aba97a0ea19895a0ddf3f2a7ccae2cdb
Pure Prop Address: TMSNBUzV3aQxYYeZPY2TT8DCJH4y5uZtPS8
Theory Prop Id: e32fe808b50d4a7d26feb7c47fba0677ea7248260462759537117a8ded2fb5dc
Theory Prop Address: TMYMBJhEnU2xzWURsT3VUeTKQ6932cZ99sf

binintersect_com

Theorem 6.53 ∀XY : ι.X ∩ Y = Y ∩ X. The proposition is identified by the following information:

Pure Prop Id: 1a2c6f83dfc7eef1f0f5d8a6ef71433168e521d79874af4f4612a7cae5c5b035
Pure Prop Address: TMb2URdQekPfF7sFmwNvzkDv5BMnydvoeSe
Theory Prop Id: ea8a1cf7428c5c48bad97c047bde68748dd123f1e3e2886bb9723b9685f19f52
Theory Prop Address: TMPHGC6FLeDyJUBBy5raiX5Q2fa4jN5brmt

binintersect_annil

Theorem 6.54 ∀X : ι.Empty ∩ X = Empty. The proposition is identified by the following information:

Pure Prop Id: 19837ac3a4cc72e88c8423496b51c2617beb1ae51cfbefea7f1962c0b5f50a16
Pure Prop Address: TMHFgk1LgJ2LLMW8CoCeuxb6mvWZeaznuN8
Theory Prop Id: 66dfcafe7e8daa11a68daa9116d60646fe5aa8d58830a016d31ad2426013468c
Theory Prop Address: TMUGqogYhT1MpqtsHd9QtihJT2pr5trMXwD

binintersect_annir

Theorem 6.55 ∀X : ι.X ∩ Empty = Empty. The proposition is identified by the following information:

Pure Prop Id: 0e7e2f4e6df513f1e6b5ccaea6a1621ebbca518c2b68b695a894a4ce698c4e0c
Pure Prop Address: TMS8kC9HsHPYsso9Ze468n6eVHK6b88et1P
Theory Prop Id: 347af54bba6e5ad1b72402cecf209ec4580f847c599c14715314311d43e1dcef
Theory Prop Address: TMKWuV4D5gCFenDioXWGtrYFYLc52e3UzGG

binintersect_idem

Theorem 6.56 ∀X : ι.X ∩ X = X. The proposition is identified by the following information:

Pure Prop Id: dbddbcb4a8c0a37d70f0630df520746c472b6a1a846e6011ecc825783815db34
Pure Prop Address: TMHBW8eKU6ryBGgk4HWYuA3dnXogpEKcixW
Theory Prop Id: d2a963ba345439222b37255ea718d0d0044869fef2d1dbf88d33f1c74f8181af
Theory Prop Address: TMVpfMftXnY2WNrX82PsUg6Ffz8dhHZCCbY

binintersect_binunion_distr

Theorem 6.57 ∀XY Z : ι.X ∩ (Y ∪ Z) = X ∩ Y ∪ X ∩ Z. The proposition is identified by the following information:

Pure Prop Id: 36e5834dc2dad9ecc45b407b7436248dd124ef4599aa9404e9c6e2d285fdf8ce
Pure Prop Address: TMJTm4UUo6gbzs6LVeQouXfnfj1XXyMHF1A
Theory Prop Id: 8127d38d4909f520e4a84c6845262a06189700ca1e234d7cbdaf3d2ce716f3ca
Theory Prop Address: TMbjiZjgzVpkQ3KA7NniPjhJJdwASKPh6M6

binunion_binintersect_distr

Theorem 6.58 ∀XY Z : ι.X ∪ Y ∩ Z = (X ∪ Y) ∩ (X ∪ Z). The proposition is identified by the following information:

Pure Prop Id: a9a58a72e353790019b05940c85c859102eea0d847382f289d58e8bd468aacc5
Pure Prop Address: TMUsf49mgzPX91G3rqfz1ovatYjDfYdvK98
Theory Prop Id: 4a9b047f4f17cdf61b7b7dd40930b1c044fa82525a4c6116a7e11a025b94040f
Theory Prop Address: TMSNTa34aJse1X5BoYYckuaeBfJRNGqKuDG

Subq_binintersection_eq

Theorem 6.59 ∀XY : ι.(X ⊆ Y) = (X ∩ Y = X). The proposition is identified by the following information:

Pure Prop Id: 86d529c3fd349b6b3715efdec80a7ca160419af4d312f017c9b6429831a8b6fa
Pure Prop Address: TMS6VwxbSL2tMcsJ33Gzb3sYguz8uL14aJW
Theory Prop Id: 33cf2fc02a709c5c13da3cccb6bb79ee7947a9d4791092613388daca4cfe3096
Theory Prop Address: TMM8Sp6PDTpeVhKhLLCUpaWCnnBiqynotsR

binintersect_nIn_I1

Theorem 6.60 ∀XY z : ι.z ∉ X → z ∉ X ∩ Y. The proposition is identified by the following information:

Pure Prop Id: 073b4038a060c4b1bde2ede64117c357d3bdce362ac825ec6a720138f563c8c5
Pure Prop Address: TMVoJYkpJgivDmuJexxTuap5J3uDDg1wMek
Theory Prop Id: 9c3d131fc1d769080781e93eb8ec589b7a2c2d6c60bcbc11c1c14a15a604d93a
Theory Prop Address: TMQojPTSE5pppDmrTox3ZJojc7GqQ5pR8nk

binintersect_nIn_I2

Theorem 6.61 ∀XY z : ι.z ∉ Y → z ∉ X ∩ Y. The proposition is identified by the following information:

Pure Prop Id: 4cd97a3920660cc1baab84a02cde2f0752cc800bdbf6948734149b840f916d31
Pure Prop Address: TMHeuLBfC3AwL6jnzuUSDhrdaQmCoHoj5D3
Theory Prop Id: 095f40265118c2d28d3547f12e68d5c28ea77112d6ccc6b086c324fc60b2794e
Theory Prop Address: TMMM6TarDYx9DpZCUpsD2z8vW1PAAJQ5vPa

binintersect_nIn_E

Theorem 6.62 ∀XY z : ι.z ∉ X ∩ Y → z ∉ X ∨ z ∉ Y. The proposition is identified by the following information:

Pure Prop Id: f6260f7735d4122a10a80856c4f9049d1519687f57b7b19edb745ee2cf5cfd3e
Pure Prop Address: TMdweHzbtRnrtxQVKunco97o2a12LP8roAQ
Theory Prop Id: 84390c4eb6fdb43452a191bfde74ee0a790cd0285b2a874167fa19e4f893bf40
Theory Prop Address: TML22Gpy4Bf79Bv1gy1Ndk2iWQUPF7ZTQMZ

Definition 6.10 setminus is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: c68e5a1f5f57bc5b6e12b423f8c24b51b48bcc32149a86fc2c30a969a15d8881
Pure Object Address: TMHxJ13rHG66sG7Z5d8qoQ2Wf71i3cMXRPW
Theory Object Id: 913da1489106a5031ed6923d7aac74c19bb471c7cb6d0b317001452fc6f29b30
Theory Object Address: TMNfftowRQHpY83NVqg5iVfi5uncZiEp6pF

Notation. We use \ as an infix operator corresponding to applying term setminus.

setminusI

Theorem 6.63 ∀XY z.(z ∈ X) → (z ∉ Y) → z ∈ X \ Y. The proposition is identified by the following information:

Pure Prop Id: 0feeb8d8a9b7ef65218a930ce4e11ecfc7dbb1545c5ef07add1f4b81afb9c3e5
Pure Prop Address: TMQPezxvx5iDwGozY2n44p1QSmzLW9c33gh
Theory Prop Id: 80db6afb2b95204c3816c5c17e3193ee6ef8bd580a5c82b1d40de15b0dd67e5c
Theory Prop Address: TMdgvunw2TsmUp4GCmh9xtJANKkfVaELTtY

setminusE

Theorem 6.64 ∀XY z.(z ∈ X \ Y) → z ∈ X ∧ z ∉ Y. The proposition is identified by the following information:

Pure Prop Id: 6774f4f653e1e92356f11506e8157d8abe7dbb9be9e6bcb14db1f0bceab51ef3
Pure Prop Address: TMXy1QEyBGVDaiEg8aD2Cam9qThxoomLi6f
Theory Prop Id: 71763af047667a6ca6327c62400dccb91551bc8f9f136c9e479e28698216380b
Theory Prop Address: TMdogwAC88ANifD7adTtE8r86D6W3bUPM9S

setminusE1

Theorem 6.65 ∀XY z.(z ∈ X \ Y) → z ∈ X. The proposition is identified by the following information:

Pure Prop Id: 064bd5d663dc259ca997e99bb2bd4b8152814550c0aa05c43ce0df3746079700
Pure Prop Address: TMTjEjKNmtdB3PuULs3bKw7WREdMMYnLNNi
Theory Prop Id: d74bafc908140cfff43333693f1863756a00f38b654f98d03ff010b973d027f6
Theory Prop Address: TMMW2tdJZPbYz3RFp8VEL3uz9UEjkov951Y

setminusE2

Theorem 6.66 ∀XY z.(z ∈ X \ Y) → z ∉ Y. The proposition is identified by the following information:

Pure Prop Id: 3e696906861b398caf75e058badb44e8df22002a1dbcebbd1e432ae75ebf73e8
Pure Prop Address: TMbw77MdhQRu71U2YgwKpdioxpTzvqPeTph
Theory Prop Id: 36b2737dca3d3ac20725828013e922045b7f000d58c5bd39fbdbde5ba29e8f17
Theory Prop Address: TMK2S85jwBL3f1YkbqgioAc3U9L13yLZ4dq

setminus_Subq

Theorem 6.67 ∀XY : ι.X \ Y ⊆ X. The proposition is identified by the following information:

Pure Prop Id: 5a497a676d0c43a84c28067a8d230220dfff42d4ecda3069868bd45d6864a3a1
Pure Prop Address: TMZpkAydiSX6PF3eKuUhdGAjSwYNgM8nANN
Theory Prop Id: 66f982019f8b0f9fb9cede258e617e735d7ee6afe6be70204a016fa60645b2ad
Theory Prop Address: TMXKBY1x4AJNkQvcmzLSA5oKynMfanH8j8g

setminus_Subq_contra

Theorem 6.68 ∀XY Z : ι.Z ⊆ Y → X \ Y ⊆ X \ Z. The proposition is identified by the following information:

Pure Prop Id: b1f440be52ba3f0081ae25bcbcaeb0ae2996726d1652a9b43221ba081ebb4046
Pure Prop Address: TML7YKpcn2xkVnB8jzzhMFivx3UtkkeTwEf
Theory Prop Id: 9a06b276a8a3609d482bb051c783d938e2b4332e33f0c788b5b1eaa70620ea52
Theory Prop Address: TMYGQFvSYDuhgZUsfQwKDf3uo8fipNdRzPL

setminus_nIn_I1

Theorem 6.69 ∀XY z.z ∉ X → z ∉ X \ Y. The proposition is identified by the following information:

Pure Prop Id: a34d147d253b8631f71a8ad885d6b07b51174f6adb06f33ff18f2e17cc3a05d8
Pure Prop Address: TMP4Br6DGx2zPHwukCi4NHAb1J6UCTGEUTH
Theory Prop Id: 0cf01b8566a440309bf989695975a3eb8c7b5b2b72839f4c5258a27e06a47268
Theory Prop Address: TMKc4mWQxd5nPWm3Twn6HUoZZ3TpTLTY6kk

setminus_nIn_I2

Theorem 6.70 ∀XY z.z ∈ Y → z ∉ X \ Y. The proposition is identified by the following information:

Pure Prop Id: bcb7ba48c815e20d7bee194dfed0cc862dd0e3d301f8c9b1d0280812f816ad6c
Pure Prop Address: TMYK5VS87wrMj21qU9jtK2xRsBEsBv9CAaa
Theory Prop Id: 358a53764f4ed435a531d19ce7ac02af9dfb2c98400d9d931a04c2593b2bee95
Theory Prop Address: TMRc5nmsKibAuSoPbYJ7nMiCh26fBzFmE26

62 CHAPTER 6. BASIC SET THEORY

setminus_nIn_E

Theorem 6.71 ∀XY z.z ∉ X \ Y → z ∉ X ∨ z ∈ Y. The proposition is identified by the following information:

Pure Prop Id: 9bc7c60b2c9e62f8b2a9af59f911de6c5d68761f4a2fe77a94c69a7d3eaf26e2
Pure Prop Address: TMW5CoiZXunGqAiTQBjRhmUmvjXoJz3mude
Theory Prop Id: 56f952bfe0bd75d292717ecaa4b14fda05ca0cf14fadaaa475598d568bff0ad2
Theory Prop Address: TMLi6vhY79EBfSSmntPk9snhH4VNXepc9wJ

setminus_selfannih

Theorem 6.72 ∀X : ι.(X \ X) = Empty. The proposition is identified by the following information:

Pure Prop Id: d3b191afa78022d380048239a2968a37051b62e5c02fa9ac36ee4db31e1083aa
Pure Prop Address: TManQQz76f8CsK8ZetcsbHQhCjjZe659QGn
Theory Prop Id: 478fe86072ac5772bd0bc55b172cc272698845d92efc4458128557ac479044a8
Theory Prop Address: TMd6YxJcNSPF58sbPVXKMa7uGjNh2yUe8vf

setminus_binintersect

Theorem 6.73 ∀XY Z : ι.X \ Y ∩ Z = (X \ Y) ∪ (X \ Z). The proposition is identified by the following information:

Pure Prop Id: 2625c6089792c3161e4afb2ac98c688356a04763067fa6803e414d5dcedf8d4f
Pure Prop Address: TMdyYr2Jj41ATEHaGEpR1dQgkGsyZP4FuYL
Theory Prop Id: f764127a273ebffdd46d2090e345b6124d00e4e10a76d5047d199152fb751f21
Theory Prop Address: TMUUoPBVNSzcS2ijLW9PqkQhqPjPTs1Z7MA

setminus_binunion

Theorem 6.74 ∀XY Z : ι.X \ Y ∪ Z = (X \ Y) \ Z. The proposition is identified by the following information:

Pure Prop Id: 0b9317714989035a7cbcbdd8eba497db661ca19eccedc9faeb0ae391b2c8987d
Pure Prop Address: TMFwTVRbH9sP3ic4Jj5K77JHuuDaHidkTTo
Theory Prop Id: 651326888d51f7ae0c489bc08dde18dffd32270d616f68516db81047548f94b9
Theory Prop Address: TMT4EsAWT7qGdwNHgiPcC19uNg3gnsu5RfF

binintersect_setminus

Theorem 6.75 ∀XY Z : ι.(X ∩ Y) \ Z = X ∩ (Y \ Z). The proposition is identified by the following information:

Pure Prop Id: d1bd304c20aac47c64bd3668e5ee0c534b4dbff3de4e7a34ab27ea65f5a5e53e
Pure Prop Address: TMYXNZDQUqfnGsDS2NpMohAoZPUMeuh4Puw
Theory Prop Id: f50067170c57c9d167d2bbce86fb7bba1d5f359a694a3e8d87c42ea80fc8f4a9
Theory Prop Address: TMXQGHghJkaMcQb53tZ6WfydPdKPLw4Fbx3

binunion_setminus

Theorem 6.76 ∀XY Z : ι.X ∪ Y \ Z = (X \ Z) ∪ (Y \ Z). The proposition is identified by the following information:

Pure Prop Id: 99c8981445a26f2a8f3c67db827e6580c563359a4f20df37e41dca73f6393a97
Pure Prop Address: TMbfg9u25oTvirxRNPHRobxdagcb2y9Y1Ez
Theory Prop Id: 1c1cfdea82cec4035b3de28d39481169301aa46b0d0912a7f83e9228797f5a48
Theory Prop Address: TMQW2SjLqSWSKT8J2EQPCpxjys4GYkHvDX2

setminus_setminus

Theorem 6.77 ∀XY Z : ι.X \ (Y \ Z) = (X \ Y) ∪ (X ∩ Z). The proposition is identified by the following information:

Pure Prop Id: 60e720f6c807ccdae50a934ea56981944603f9d6f18983c678c322715f66a587
Pure Prop Address: TMLXRcQtVsZ2gokZw2jjxkAgJhArzeWbwhF
Theory Prop Id: cab095f0a1f43e4dc50e5489f05911c74f5fd42b4a8dced8ecea3a53e69f1d71
Theory Prop Address: TMEiQhFR4FhRQESPDAccJrCE98eg1tzdAEE

setminus_annil

Theorem 6.78 ∀X : ι.Empty \ X = Empty. The proposition is identified by the following information:

Pure Prop Id: fd99d2feff8bf081dd1e98ac3ced977726230ad5af0c70364ed17032b322e958
Pure Prop Address: TMPjszawGe4kxyHBsc9D4AVrRBAh4nHf7YB
Theory Prop Id: 3d2f5b9dfe25bba40fc9e8e784f45fff9a4a929712ef764617c8b137da6f2cdb
Theory Prop Address: TMPrnLS8t41sPUjW6B8bNjh2dhMBwTbU9S8

setminus_idr

Theorem 6.79 ∀X : ι.X \ Empty = X. The proposition is identified by the following information:

Pure Prop Id: 20e1086759c8557ab7f7e06736e90912f8684d26fb4e569f3e23fc4960e2a173
Pure Prop Address: TMQi4nBawCq2rp7HipQymKHj8UhkFm1gCeM
Theory Prop Id: ebb8ec6864341bd40d215d011ec32f77f22a7bfdb6d5d8332a0487fa1efbbaf9
Theory Prop Address: TMYkPV9ze3vB5CvjACMPmYU1b8ea1z2jSPg

In_irref

Theorem 6.80 ∀x.x ∉ x. The proposition is identified by the following information:

Pure Prop Id: 12629a025dcc8a596e030daeffbf6322b3d51631eeebf85bcca0a44b6a7964cd
Pure Prop Address: TMKwkB6s3TXZJG8hL6bLbEGFMysQgpweKpz
Theory Prop Id: 7bfd46e967631be2a9098717487f5ae8f88e2867117fd98cce921a47bed1bdbf
Theory Prop Address: TMZagTqZqCeDpgAoeuYZejsuagxzALpQvWL

64 CHAPTER 6. BASIC SET THEORY

In_no2cycle

Theorem 6.81 ∀xy.x ∈ y → y ∈ x → False. The proposition is identified by the following information:

Pure Prop Id: 6d0cb3f4b527344ea21bf482f99b38b72d358ba458097b2ed205e1de6bd2c82f
Pure Prop Address: TMcBgknu3MmmLrKct3NGTcziaJv71unmDbY
Theory Prop Id: 344f3082291501cfe856ef475802aea283ae4928899f0eba4e2c3a6556451b82
Theory Prop Address: TMNd1Mv58X5hcYFChdQ93tpgtHdhyahABAg

In_no3cycle

Theorem 6.82 ∀xyz.x ∈ y → y ∈ z → z ∈ x → False. The proposition is identified by the following information:

Pure Prop Id: 660f202c6752b9347ac4d5b7b55e28507e6fdf71c23bf56ce2eae287f04ec6f3
Pure Prop Address: TMQcimvCarrH1MmuAYoNABafQguio3NLZeY
Theory Prop Id: 90080b22e6d3c8f5944bbe4d048a7d9fdffe58eac2e3d918ff8ad0d16d572a8e
Theory Prop Address: TMZPDFZYJchoNx1RFvKeMUbtyTaSeQj9LrH

Natural Numbers I

Definition 7.1 ordsucc is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 65d8837d7b0172ae830bed36c8407fcd41b7d875033d2284eb2df245b42295a6
Pure Object Address: TMavRsMUqs8nxad2sig41TibDMTMy1H1mNn
Theory Object Id: 1452f7d6b185ccc8dd0cf665789667547457cc1e37f37eeb4542ea13bb0f3b7c
Theory Object Address: TMKDT5PUJNLA2gdidecqgknUTRNKvDTHhkr

ordsuccI1

Theorem 7.1 ∀x : ι.x ⊆ ordsucc x. The proposition is identified by the following information:

Pure Prop Id: 5a2f15e1e0256bce0e0a36c10194ffcbfe641dbb51c699626b909df04caf5bc8
Pure Prop Address: TMPc16mnjiUyzqEV7bqCxpSd6RmykST31RK
Theory Prop Id: 367d807338b46735c105597b4d7b718fc7262d8a19b606bca8b2fcba8f5b1e15
Theory Prop Address: TMPyyF13kHh14HmE9nFU4oA64FKkfCootMK

ordsuccI2

Theorem 7.2 ∀x : ι.x ∈ ordsucc x. The proposition is identified by the following information:

Pure Prop Id: 2f6efdd09ddf2d88f93f929612974d444c064325e7c941dfb1462b672c87b2af
Pure Prop Address: TMVqzvRwWJ3SEK7WcdSkCGWDSdUaejyiBbZ
Theory Prop Id: 2be6be024332289d2aa440c03da38fb1d593f63adf8e2fd3e60d61ed384710cb
Theory Prop Address: TMMZ4GfqtzWS5jhaGuWH9wJJD8HBcxQMpbD

ordsuccE

Theorem 7.3 ∀xy : ι.y ∈ ordsucc x → y ∈ x ∨ y = x. The proposition is identified by the following information:

66 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: 01c4eee053a9b09e095c4f3572245e6c0eab90795b95854b85d38a0087607c4c
Pure Prop Address: TMUfsQNiuSFqHup4XjacKRWBsfxf8BGBqRc
Theory Prop Id: 3849f5cb870c46683ed2e33b5b86c0c8d2d347b9f9ef20b2d6c3591e603a785b
Theory Prop Address: TMHz6H3wiAaYh9iPsBhq4bonjGvMxx7bRgr

Notation. We now use natural numbers 0, 1, 2, … as terms where 0 means Empty and n + 1 means ordsucc n.

neq_0_ordsucc

Theorem 7.4 ∀a : ι.0 ≠ ordsucc a. The proposition is identified by the following information:

Pure Prop Id: 1b20cba417f676cb6dbc9672ee2b03ded8ab6cfed86cd29335c5ffc3734e019f
Pure Prop Address: TMQj2ULKJiSRkfDiw5djhEYAiKFnD8V8ChY
Theory Prop Id: 04c4104e681b528cc961597d733a565b395e183dad8c992b88138202d9471b6c
Theory Prop Address: TMSZGeeBC5QqAdFny2AFc64sYMBqNSG7WsX

neq_ordsucc_0

Theorem 7.5 ∀a : ι.ordsucc a ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 9b391ea2bb9a775053e83e0216e5088a30439bc1052e445f7e2b049a5e87f59e
Pure Prop Address: TMHpzvw5s6p11s2PhBAUH5tA63dzykXSMV4
Theory Prop Id: 346ee0e70829c6afd0125feef2af949dbac67cd5e69584a5150bc6084f37e8e7
Theory Prop Address: TMG4GdZg9J7SSN69pwxvT4GRe1Qx3Dvxqof

ordsucc_inj

Theorem 7.6 ∀ab : ι.ordsucc a = ordsucc b → a = b. The proposition is identified by the following information:

Pure Prop Id: 40cad67f9c954d4e382390b2aba8222e29db73bba21874edcdab5f1e65bcae9e
Pure Prop Address: TMJhxcvNP3aGv2vSzykSRZBpaD8KiY3FWPU
Theory Prop Id: 4869a11c62e30e2d41c7f4524c576be90f3c64f919fc7dfe1442add6e7c4b685
Theory Prop Address: TMNg9uVctRmhnFUUaeZ81Gj1Y7kuPbhrwyc

ordsucc_inj_contra

Theorem 7.7 ∀ab : ι.a ≠ b → ordsucc a ≠ ordsucc b. The proposition is identified by the following information:

Pure Prop Id: 79130458153e5721466856a630c586b16f2738bbfc6e0259d8313cb5ade1bfa0
Pure Prop Address: TMa7r3NCaBWkq3wxWzUgF3xxk3pTHF1U4qs
Theory Prop Id: 66f4df16ff59481c9056308d94b75bc6bc55adb8b8e510c166184cb6cc83d26c
Theory Prop Address: TMKVLdh2rykBFF1YXiKx1pwSwX7cpuoEp9t

Theorem 7.8 0 ∈ 1. The proposition is identified by the following information:

Pure Prop Id: e984b0b4b8bd4560157f8d6d3577f303bcbfad9f5202acbc6aafe41c46f5a1e2
Pure Prop Address: TMSTYDAFnhcsxKw7rE1h1KyMkzowN5nMHPH
Theory Prop Id: 3be80a55870dc121ca054e5dacdfa1de52caa7939a81bb69beaf6bdbc1246eaf
Theory Prop Address: TMQTUaJ1s8JSg8LytU2CzSFNeMJJfqKmUPn

In_0_2

Theorem 7.9 0 ∈ 2. The proposition is identified by the following information:

Pure Prop Id: 56dc2cbeb7b04a87557e31dd8850f675eda2fcb01d7fe2d95714032d7726ec34
Pure Prop Address: TMXDJP7PuoXtQw49z9dst2KviKHpUhQEeng
Theory Prop Id: 5a202e3c8e0f6cac6103c72106cd4d8b5fafa891dddb8c40e68ea9b5e1e4d088
Theory Prop Address: TMVCcMZv24dUjhcz1MQJEf7nxq1pjQVJLEk

In_1_2

Theorem 7.10 1 ∈ 2. The proposition is identified by the following information:

Pure Prop Id: 8704b3e5f3972c8160904d3c95ef653a8d86fd991a0e3e89d6c4ba3a0989cffa
Pure Prop Address: TMcojLEL6YY4Y3YxV2dkpKy8Vbt3vCHJed1
Theory Prop Id: 431e9e50b33a518fbfbcb0617273d0749a4068ee4d20068f34f568c8ca0ca98d
Theory Prop Address: TMPAkzgdLgKfuKVB27MKBPmY5KQWhbL2dWE

Definition 7.2 We define nat_p to be λn : ι.∀p : ι → o.p 0 → (∀x : ι.p x → p (ordsucc x)) → p n of type ι → o identified by the following information:

Pure Object Id: 458be3a74fef41541068991d6ed4034dc3b5e637179369954ba21f6dff4448e4
Pure Object Address: TMdNqBmjo7vJ5wFYe6UamcKdgnn9WXg7rEJ
Theory Object Id: cfd8eb3ef07509602aa3910d783c30847ee19a1fa94457706e8bcc91da1a0685
Theory Object Address: TMQwsyTWGfzEce8GjWygs11T1pzxgCk15vk

nat_0

Theorem 7.11 nat_p 0. The proposition is identified by the following information:

Pure Prop Id: 1617bca54b26e5228d76be996df23b2acf38a1e8ccb04bf94ec2a951660e6a42
Pure Prop Address: TMUBAZcQKUGLBQhSYCTVsFgkCVP66bx3XMG
Theory Prop Id: 457a72a6740767a11460a92c05bf74c2f85ec3d9d68b112eda7a6ccb7585d72a
Theory Prop Address: TMMYbseVvbEE5mCqsfJbV6FkYHcx1cF2uL8

nat_ordsucc

Theorem 7.12 ∀n : ι.nat_p n → nat_p (ordsucc n). The proposition is identified by the following information:

68 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: 87f001f9ea63dd0ca64b98fb5d48ee465b362dbdb42cd8a8108ac9dde660ade1
Pure Prop Address: TMaWAUtmodzvGsvp6q6xHpk4Yuham8jZXcm
Theory Prop Id: 57e940f024fda91a7159ab98b6fcc0219d72fb6e7bb86ae7f0b6c4d781dfd58c
Theory Prop Address: TMKZQMyAvFRX9iJG3iXqmXhhgXw3WPru1ww

nat_1

Theorem 7.13 nat_p 1. The proposition is identified by the following information:

Pure Prop Id: 2f83faeb354b73c788d02a2f05af45d694cfea901b6a7b10dfa230c853b96453
Pure Prop Address: TMKVKW3snRuquEuPq79vsPPwjabNMfZWF9F
Theory Prop Id: 672d0d5e7bbed7f2c3c492dd058d40c6f76f54880250fc30060ac1efe4135adc
Theory Prop Address: TMUjC2bRLBWtBtHMGwSpPZY777xJhfnQBRo

nat_2

Theorem 7.14 nat_p 2. The proposition is identified by the following information:

Pure Prop Id: a8be1568ff355447f7db7c89da6ae23d1686bd32ec45719d79659417af6f447d
Pure Prop Address: TMWqNeynfSDURZP8Wzc3iZasb4wYMvcrwSj
Theory Prop Id: 9112e975c2b7883a953c49c1d03e3a50209fee33f81345ec21f36647283d3373
Theory Prop Address: TMTTG4UrkKbYb6ukW8afwvToLqvgXuokzbh

nat_3

Theorem 7.15 nat_p 3. The proposition is identified by the following information:

Pure Prop Id: c0849ff3f73eb686bb761cb6944a4784c9c2d96f7206b9a8024656f2b406317d
Pure Prop Address: TMNxJRiy5Z5ySkwD4bVtViWKE7vD4jrAD15
Theory Prop Id: 133bb750b1d962e5d82a3fae62787479ad909ed8e960846d440b4c68d71ae6da
Theory Prop Address: TMQXcPujARi3ryfse2NkWbvZ4uGumLKcEWS

nat_4

Theorem 7.16 nat_p 4. The proposition is identified by the following information:

Pure Prop Id: 2c8e1aae548c1357c58dde606cc0aec5ffac22c5e66251a8ce0805caf0f28922
Pure Prop Address: TMH3HZH2XtjuKpCAF7SNvq8oWwWq1gMGf9g
Theory Prop Id: 69ffc33641fda2b620e65b8c7a42ab36d1575ef4f6cdffca4419166392954387
Theory Prop Address: TMcTTobfSQSnJ7AbXsvRGCQ7y3HaQnEeeFs

nat_5

Theorem 7.17 nat_p 5. The proposition is identified by the following information:

Pure Prop Id: 1a2477a480bd21b24c7f26877221e1b8a5a0977ebc3409b63d8b81adc052b208
Pure Prop Address: TMaLjbE49i38tGGYWFzVVSxUtcuU82qvhmk
Theory Prop Id: b84b25e58b844e2ddd85bea48fe48f329f13a25c4c0456681e28158f15b3a5df
Theory Prop Address: TMFj5aVyECSdNXwpDgCbbGDuCSC7gTYNSn4

nat_6

Theorem 7.18 nat_p 6. The proposition is identified by the following information:

Pure Prop Id: 25de88744534dcfb82c0d0379635001cdffce949810f8ad348a12948d32b0af7
Pure Prop Address: TMS5YEXhfPvtcWyPkWtRChML7geomnrXHCd
Theory Prop Id: b4ec6745b4fa46eb645b20d7578dbd15a3c3bd98922e7daf874d00b8ec50fb0c
Theory Prop Address: TMStihyrXSQ2eoCRa882VokNRU7nbYWQD8F

nat_0_in_ordsucc

Theorem 7.19 ∀n.nat_p n → 0 ∈ ordsucc n. The proposition is identified by the following information:

Pure Prop Id: 9b50b9ff20796d5e40a96b518b4c6306f3f1fbcdac11d2c3f6be7a95ea734347
Pure Prop Address: TMNGcWtDYoP2Bidne3KeT6KVqrriDbdzHjr
Theory Prop Id: 251f3d891ace960a597b3f004c9759fedf5c6738929d3028dba9a61a95fbf120
Theory Prop Address: TMHA18C6v8mXaGqFjk7jf2v9egdfvKKqWuW

nat_ordsucc_in_ordsucc

Theorem 7.20 ∀n.nat_p n → ∀m ∈ n.ordsucc m ∈ ordsucc n. The proposition is identified by the following information:

Pure Prop Id: 819399e48921ef23f6b1409081964e191ac9faf077bcea0b877deb52ebc19088
Pure Prop Address: TMRPbaUhLFQA5eMs4qgHq2eHkAum5RDSzqx
Theory Prop Id: 4c2c9521cdcb93eeae5ab0b3bddd72a6ac20699773fda70d1c40ad8369603f79
Theory Prop Address: TMWSu4yCbzbJPdvkEV8EErMS2USSHmDvFoQ

nat_ind

Theorem 7.21 ∀p : ι → o.p 0 → (∀n.nat_p n → p n → p (ordsucc n)) → ∀n.nat_p n → p n. The proposition is identified by the following information:

Pure Prop Id: 5d06f5ffa757285d834b9b984ab55693e9c64caee1e15c187462d6b08a91cfb6
Pure Prop Address: TMHLu8uvpDa5T7p3FX9tm1QcSzRDu8EgeoS
Theory Prop Id: ecdce8bd5bb09bae03975edfeb1c1d944e5b0fb2eace3d673428b70400e410f2
Theory Prop Address: TMLKPWPGi2sVCW27Hu2rRaW71Je8eYbfMK9

nat_inv

Theorem 7.22 ∀n.nat_p n → n = 0 ∨ ∃x.nat_p x ∧ n = ordsucc x. The proposition is identified by the following information:

70 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: 5d4296fccf80fc82c53e9d0330ce9f3d5e1106daedee7d4d25d13d7d4b63ba57
Pure Prop Address: TMa76r1GZgLGs96CTRuUt7foxrtZPG4axsF
Theory Prop Id: 4f4167d99f52765043e392f09f9c226ca035e3f1640818079090d6f7ab744d14
Theory Prop Address: TMR4i7cHgevq2znVdMhJMiqjT6wu9k8kJN8

nat_complete_ind

Theorem 7.23 ∀p : ι → o.(∀n.nat_p n → (∀m ∈ n.p m) → p n) → ∀n.nat_p n → p n. The proposition is identified by the following information:

Pure Prop Id: d32e5b24fa29af5ae41e8ad1bdabc16e354331cd6f147252b6051abcd45747b4
Pure Prop Address: TMSAj2j7jwL81oXRAhtY9BFvgyXQUt34jN9
Theory Prop Id: 1e16087110e02042e56a9e5c1063925cc2f7ec418bc4e813ba1663102359d49f
Theory Prop Address: TMSj9PFLh5Qr9YAvzYmFtQ9ahr3yMf1DzhX

nat_p_trans

Theorem 7.24 ∀n.nat_p n → ∀m ∈ n.nat_p m. The proposition is identified by the following information:

Pure Prop Id: e884494b492bcc0edee39dcf5b1043dc2f35d009618bfad2e00c24a9c042215a
Pure Prop Address: TMHs2RYjZjBVk1upHj4ggqMDExPU6aYRTU5
Theory Prop Id: 03b874ce14b10c68c8b003d5521fedd9dbde8f272d9947fada3cf5bee55416fc
Theory Prop Address: TMMK6MMXkYaSfwkm9GkAhh9Lo4bkVBtYWdV

nat_trans

Theorem 7.25 ∀n.nat_p n → ∀m ∈ n.m ⊆ n. The proposition is identified by the following information:

Pure Prop Id: 6bc6e12c22a5d14f7294be0cdc44e66abf13bd33013e1c985c60b88343f6a7e9
Pure Prop Address: TMFwnmbaMpakWwrrmKjohDndW8KqnGgrE2R
Theory Prop Id: cad6fe1daa30ce548334ddc0217dd046070f343d4279370eed3639bb6552015e
Theory Prop Address: TMQmjjPV3zeSxpELyxYbuTXa9bAc7s9QGaw

nat_ordsucc_trans

Theorem 7.26 ∀n.nat_p n → ∀m ∈ ordsucc n.m ⊆ n. The proposition is identified by the following information:

Pure Prop Id: e42e581777e9b184d838738cd0f2e42b37dbddeaedc030fc6bcbb1017c2c4acf
Pure Prop Address: TMWNTUPr5UHTy5qBphSAPE6eNxNxTmDMM3c
Theory Prop Id: bc868dd13fb819f4f89016d584069c5811ebf3ca5efcc8c1c4d079e962cf85db
Theory Prop Address: TMV35265NGTDZcH9QrCs5zZVseRcm6nuMzu

Union_ordsucc_eq

Theorem 7.27 ∀n.nat_p n → Union (ordsucc n) = n. The proposition is identified by the following information:

Pure Prop Id: 87308c20573b24369c99ff4fe9bc8822508ac2939b6f6d5756b6098dda4b527f
Pure Prop Address: TMHVBTxQoQ4YadqkTHbe8MXRCPm3xY51g35
Theory Prop Id: e8cdeba740894b73be89b96729f0c00cde7781b068388e485ec8a7ef82de11cf
Theory Prop Address: TMWEc3LBttyZMJdbQThEZMNmW5T1bNLMb8k

In_0_3

Theorem 7.28 0 ∈ 3. The proposition is identified by the following information:

Pure Prop Id: 5d5d5f868b630108f6d0ef72c022b1a1fdcb3bba9e9ac36ffa44fbd60e4334ae
Pure Prop Address: TMYMGQ2bVW21zWN5CbbTLaasyax3hxdLvqK
Theory Prop Id: 313d97e81a38fa191ac39cd36a8d7be9e4da2ad48aa4710d01a9a001638e022b
Theory Prop Address: TMU4Cdfau2fRjQ3n9SbuJ1p1TcVEvxWfhqp

In_1_3

Theorem 7.29 1 ∈ 3. The proposition is identified by the following information:

Pure Prop Id: 85201c4239d9221474ddfc6342817247634f124875f424ae9f7175ce6df17f51
Pure Prop Address: TMGfU5npaRR11H1ymdoVZKTfX7BrhZKDwjf
Theory Prop Id: 8f3cec3de25f5229ffbd8a9526581652ba38c8b4ace6f65a6c6b0c34e257b18d
Theory Prop Address: TMZAip8oPziRqgUtYsjaNbkMfq6G8pgm3NH

In_2_3

Theorem 7.30 2 ∈ 3. The proposition is identified by the following information:

Pure Prop Id: 4d67cef397f17c6f96d94a96fb406d790704d539c9dcd26f464a7ede52fd09ea
Pure Prop Address: TMPo1MLQ944guurv3GfbEq47E776WbpEpzr
Theory Prop Id: ebdb5d69cf555fc8c7f6126302fc59f1945db62dc77e65788bb147ab62800823
Theory Prop Address: TMFx1bn63w8vypVgVbQs3uqihXU9xXDRXcb

In_0_4

Theorem 7.31 0 ∈ 4. The proposition is identified by the following information:

Pure Prop Id: 0a2665a34864dd30cb973fd074b11e77208b4a6c95f91a17bc48472090308db8
Pure Prop Address: TMbWh6fxUCtxFfTwjF1FBTop3nZpchLnWC7
Theory Prop Id: 2d4164fa05829f0fb01d9e4ae31ab49773fd9055499b85021468859ab970f821
Theory Prop Address: TMG6uSmoWvJMEa12QT4E89xXUiPGnfsLvJb

In_1_4

Theorem 7.32 1 ∈ 4. The proposition is identified by the following information:

72 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: 5cabbbcb5f4d3c4366e68c4306207f64e478077557ddbd21b737a9ed99b1bde8
Pure Prop Address: TMFSB8wCuJCgms892RboQ8XaFFohyZ5VoXi
Theory Prop Id: 1e2d5bc32b050352877615a028897b5966602f62b83077de259c38244906a955
Theory Prop Address: TMb5w175AXMmikqn3jnFTEuWyxRWW1q9VR1

In_2_4

Theorem 7.33 2 ∈ 4. The proposition is identified by the following information:

Pure Prop Id: 33c2ff88221f741e21ca67de81295fbf2c5a50f85469edc4c578006445422047
Pure Prop Address: TMSmCUEe8UeCDsTpkgeVsJafexbUrbDtBJT
Theory Prop Id: 2cbb94cec93497c80bc1c990383ab0d819918ebba46428b8d07b50c2a8dd3873
Theory Prop Address: TMZ4rUSfGohxvov39V3nn9XfgUBuVUnDP81

In_3_4

Theorem 7.34 3 ∈ 4. The proposition is identified by the following information:

Pure Prop Id: f25e446ff25ccb26c9eaf5000bbb167cdc2eee3efcb7a6c5d97b91ed9f586259
Pure Prop Address: TMHhdWYgy6pNRNN33esNvfW6grxMUZbxmQy
Theory Prop Id: 22a0af713d24716cae60d0f24b75fa2f7a4806d9e04268f23e037d5bfda1d9ab
Theory Prop Address: TMY5nCeLnTtDAbqPEDMuJXoU1synsTZUbX2

In_0_5

Theorem 7.35 0 ∈ 5. The proposition is identified by the following information:

Pure Prop Id: d496b37c7578aa3dd3d6688724973b63a69ab8843dcdd9a9cbf404cee4a9e756
Pure Prop Address: TMWXaxNKNEtuRGm8q1B3GYkjFHYAnyqhofA
Theory Prop Id: 965bc2aab360b39c6dcd958d77ad07534cbbe30f14358d204ad4e6b7b7b46d6c
Theory Prop Address: TMK6cfwkMoqpWBimhTSkWz16DVZMkJZvErn

In_1_5

Theorem 7.36 1 ∈ 5. The proposition is identified by the following information:

Pure Prop Id: be96a32ec1836ba1630ad994edab9d7936fc2693d7809d868797766ea4190943
Pure Prop Address: TMXHZTLBSjY6b6uscunc7vkkgtZToJhnJx3
Theory Prop Id: 34c02d4ab3de796ce3192599f046a83544b80ab964c752f5812cab439ef73dac
Theory Prop Address: TMJSvdJuHEbLX569a7rARyuiAc4TBR5z7jq

In_2_5

Theorem 7.37 2 ∈ 5. The proposition is identified by the following information:

Pure Prop Id: ec68b2a5af6caf3d108a9db5a8c18bdba7461c7ae67c3f170c1567638841250e
Pure Prop Address: TMcknqzryhsoUsEHJDCubHX54MtQJ7bamVZ
Theory Prop Id: 7ed8e4dcf1f9fa87265224b605b86bd3741e69943d43b50b1e1184851339bc69
Theory Prop Address: TMKyPhdwyzSyueTwxxHt94eVxfRmv1aYr8Z

In_3_5

Theorem 7.38 3 ∈ 5. The proposition is identified by the following information:

Pure Prop Id: eddd369a108e071365a07eeb0102084ea704a8d55dd011c4fa3a11984c9d03a6
Pure Prop Address: TMFAtiv8XgWv9eW46mFVpYtzxFPKr6Zdm9r
Theory Prop Id: df92fe69df5632294637d1b16ca10e692f827160f76ad0dea688f1cb47248e6d
Theory Prop Address: TMEsH4FBsT68Y7qAu35cdgeMxfncBpZDQGB

In_4_5

Theorem 7.39 4 ∈ 5. The proposition is identified by the following information:

Pure Prop Id: e72ad7d7edf5bc6c656af02486d73733da6dab698e0c8324a4ef1729c1350b9c
Pure Prop Address: TMM44jRqVsjAWBEjN5MPxHU2crXNCqdxJt9
Theory Prop Id: 03b9ff65b44d816e3599756cc6e5dc3b6b50ca8f07749f7adfd9e48f19469c45
Theory Prop Address: TMbtsFeGDXqrFQb2Xe3LnUKy3TXKV8WPAoQ

In_0_6

Theorem 7.40 0 ∈ 6. The proposition is identified by the following information:

Pure Prop Id: 4b3417730a51ee94caf04c6f4de510c62f9f4925400122c58b21177776e3b787
Pure Prop Address: TMLu5nuHNV3AzNorNqbEsf2hWrAxPF519QM
Theory Prop Id: 9f78a575d1b7281fbcfa8c95d8e8c97c9bff83b15cc425c99c1cebf91540f0c5
Theory Prop Address: TMVHzfqGiL4fJjTzThd6tjGntRMTKW4S98D

In_1_6

Theorem 7.41 1 ∈ 6. The proposition is identified by the following information:

Pure Prop Id: 4c8428c2d2820e6ddd02f33412f82a456fbb92337a7c24bd2949838775230efa
Pure Prop Address: TMc8kjaqr7JWG5gtEKyEAZGMhS9xsyCNsQu
Theory Prop Id: 327244a71fe9738bca8eeac7c0c9b9ec3a4bb249e97ea6275ba6ab2dfd446afe
Theory Prop Address: TMRoR6jxpANVwezSACwWKEdCd4NxVcABGsa

In_2_6

Theorem 7.42 2 ∈ 6. The proposition is identified by the following information:

74 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: a8317784c20b4dc83403cf330adc750c702b00a3cd65c663bc5c901a0367573d
Pure Prop Address: TMXd4ThRkPLZAiyGBMqRWy2aiMvbxmyP3bv
Theory Prop Id: a8f5a1b9820e7d62e90597d5e69f41e69bad70f2d1c754a0ca922ddeed9a53f7
Theory Prop Address: TMGReynZEJe34xYHBotustobc5dLWKRgJyQ

In_3_6

Theorem 7.43 3 ∈ 6. The proposition is identified by the following information:

Pure Prop Id: a261e5ada36e036a105a0af5df2c1a6bf280d14dceadb5fb843b1c5b93e6eca9
Pure Prop Address: TMGXG2p1QMD6aexgGew972chtNsckboghZa
Theory Prop Id: 9f4635e903ae4d1ec8343f5c6b3c43eb6b2f022214d0c7c82673da42ba715ac7
Theory Prop Address: TMPdrRy79F8aNf9CVxzNbnL4JAAP286vxFk

In_4_6

Theorem 7.44 4 ∈ 6. The proposition is identified by the following information:

Pure Prop Id: 1b41da8a9246684b8db369a60d5fe5356821065ca71fb292dcb7f664a7828e8c
Pure Prop Address: TMMLyZwuwf8JFrgJFBNjioGMCZyYvyzDqAn
Theory Prop Id: 5d38f6e045507f4a93cad11fae8c1540a1eae90b3f52dbbefdcc9ed45b1d3c8e
Theory Prop Address: TMLPcbeA4bjw9wrq33aFxyRQLGSz8g4hz1u

In_5_6

Theorem 7.45 5 ∈ 6. The proposition is identified by the following information:

Pure Prop Id: 2b492691b7ed3865b84636fa15db3d5054a753686f97633cc4ede78f686ef8d1
Pure Prop Address: TMPEvT5ScMWzFTfTs4L9HbhU3MydujRMzZs
Theory Prop Id: 6d1dd13e78e58b97900513a8b85b88143e8f8b02809c1190db86c9dd2790e39f
Theory Prop Address: TMW8GXcL8fcN4SrFvpTeB4P4JkZjrbQxnBM

cases_1

Theorem 7.46 ∀i ∈ 1.∀p : ι → o.p 0 → p i. The proposition is identified by the following information:

Pure Prop Id: 7066058004d5c0b851f77ebf2a55649c2a748e523d3f217177dfd9510fd577ca
Pure Prop Address: TMModSu1YSRUz9bXeP9MgEyHwZQ9newZ6CC
Theory Prop Id: c6f0c0649935f65fab6b1d1d427a6aac062e1ba5d1610667c31ecaa47cb0c54e
Theory Prop Address: TMcp6xnjxks7qVeybXpyew3hEbkn2Y1Yov6

cases_2

Theorem 7.47 ∀i ∈ 2.∀p : ι → o.p 0 → p 1 → p i. The proposition is identified by the following information:

Pure Prop Id: 4693b27abb85322341f1944c565b525391a1ae698d8535be845fd1293fbc3624
Pure Prop Address: TMKetHxomnrco2Vqg88PFWW5AconfvozyEs
Theory Prop Id: 74563af5af19787ebf01165e4b1ceb3b061086f68c594ad4c780e78738af0c68
Theory Prop Address: TMZquspohj8HSbBs2iZWfi9rM4tD9PXhYCf

cases_3

Theorem 7.48 ∀i ∈ 3.∀p : ι → o.p 0 → p 1 → p 2 → p i. The proposition is identified by the following information:

Pure Prop Id: 2a0ca2e4c94eb26ab5c19412a6da10600e180d42150f6e6127f0717ef3a8c112
Pure Prop Address: TMUXV8fVuZa8oY4GuQa6GtAfrazqQgpJXZJ
Theory Prop Id: 1a8a209d9754777b2d79de7e41a0d52b1cb29112b497f6494c04278866122964
Theory Prop Address: TMJNiwboDvoErEPCNUWNtB2URCDsvp59127

cases_4

Theorem 7.49 ∀i ∈ 4.∀p : ι → o.p 0 → p 1 → p 2 → p 3 → p i. The proposition is identified by the following information:

Pure Prop Id: d983698cedfbfb0d50150666cddd3b822dedfadb18caeb579770c029f207cd45
Pure Prop Address: TMa3a2ZsjVQYDT6yBAp8gzu9pDMZmvxMhFX
Theory Prop Id: 58ec175e1fd7d83946fa2cbdc271e4619d08449b2fd4b9aa206d25c5c0250722
Theory Prop Address: TMSUJZnbtQsbxnZdQiLbj8KeXys9eLdtKmD

cases_5

Theorem 7.50 ∀i ∈ 5.∀p : ι → o.p 0 → p 1 → p 2 → p 3 → p 4 → p i. The proposition is identified by the following information:

Pure Prop Id: f09c249f31e8eeee27f856932cdd5b636b782b2b3b2e32f82647235920772138
Pure Prop Address: TMaNUYXuGQLR3vViJceHm29PgX2B1DvZRQi
Theory Prop Id: f380534e00e1acdd7956770233333b4d2c419506f8ed1664966fbd5a79325ea4
Theory Prop Address: TMbKWQtmy4GP7grvnKR87i4GqU3qf8y7RHj

cases_6

Theorem 7.51 ∀i ∈ 6.∀p : ι → o.p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p i. The proposition is identified by the following information:

Pure Prop Id: 4dc95970bb9227a3f22f3850c0b8ae6a1fd851594d05dbd5ea64f678a4c05b04
Pure Prop Address: TMTngRfM6nYyAGMJnk3QjkHDpxpz3pM2qJM
Theory Prop Id: ed34a795973972d067788ff2e988cafe8a031b1f5916062b65c812cd42b40e21
Theory Prop Address: TMJ5Xh7NbKFHd1vwDJmsHKLnMiEE6Lffx2L

neq_0_1

Theorem 7.52 0 ≠ 1. The proposition is identified by the following information:

76 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: 9630fda1b35eebeb16a6330ac81eba7219b48d89e756733cef5f05db380572c9
Pure Prop Address: TMby1mMrnJnG3eebgn4AYd1GyHbrhqpL1fH
Theory Prop Id: 74867a84ff3fe03d477cec3700c3c0504745b30ba4082bcf6867d3bfce7e06fd
Theory Prop Address: TMW8jqbkDUF7dxtE4b7h7yVNVnwFchv7r4T

neq_0_2

Theorem 7.53 0 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: 05558045c651b5291665c412a39d06de2be8367ab59466b82289dd5533d9320e
Pure Prop Address: TMZu44mT4xzQgx5223eLUCX58xAjU5Jf5QA
Theory Prop Id: 9c9a3f601515a2d7da3b8b00ab6b88f05b63c5fb44c173cd5e868be872f2892a
Theory Prop Address: TMYNJEaA1HL5UDfSf43JCRXZeNbHuYw2YXp

neq_1_2

Theorem 7.54 1 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: 1d088a05070bdb1abb11db258e0e1a91308bf33b4bfd90fac2b4027e776ab0d7
Pure Prop Address: TMGhpsGUJvMfvoyyDMhyLQnsaoSjoUHHex1
Theory Prop Id: e5542eddbbedca6158ac40fec1d023b24b164c16667074aa41d1e2aa21991e23
Theory Prop Address: TMWMpuvExT7y6Sv9W8iFya7qZnGC8mnKqbz

neq_1_0

Theorem 7.55 1 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 3a047842d8ee288fb6cf5794c28a9d57967ac47fed1474610f3d9471590139d4
Pure Prop Address: TMMwBjpCmpUHnTqc64ZdF7on5Eq6WCgDoy8
Theory Prop Id: 952fed722f255a691a61f7c092a9f63980b2e4e12b60313b9cdac6dbfd0d38d4
Theory Prop Address: TMVVgmiuwcrrYrrRVLykRqhZy2oz2EkZq5W

neq_2_0

Theorem 7.56 2 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 4d54d80a48e261479b57582099a83eaa414e72e5021d582b40a631d6440dcbd7
Pure Prop Address: TMSj4zVM1S7jpaZ1AVkAMacHh7z2a7w1mih
Theory Prop Id: 00b357093a27461d60280a55739ba21a89b42f257242ae1a9fb792ab4984b5be
Theory Prop Address: TMSyCWK4mwhWDmvJL9T59TwDeVDzmo5GCZW

neq_2_1

Theorem 7.57 2 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: 6dd4c5f69b6b5a30c50cc8870ff9c94a49eb2257fc738c81c43eebd10b0b7fc8
Pure Prop Address: TMHNLPWx722xRdb6Lf4hip5kXBPKr3JD1kp
Theory Prop Id: 2949ced3d070583d25d9e92478b3bb34672bfefe1c3018361515b55a230cbf19
Theory Prop Address: TMJ4wBoghYfjCp1GwwmbywTP6TCGue5eUGD

neq_3_0

Theorem 7.58 3 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: f27effee2c2c94b3616a5e05a191639ff0e04b481a37195eb4c6a1044f82ac79
Pure Prop Address: TMP1HRK27b3SUVqnCvdZEHQAsyFfrJj3ZFT
Theory Prop Id: 799b0593a2ee6a2b5ac8d4d9a2bc8d8251c90a17927eb218b54fa15559ab2a26
Theory Prop Address: TMHR2rge7NMrthvfUFGcDrapFUGSLd3GMoR

neq_3_1

Theorem 7.59 3 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: 7fa352ff4f2f05596173373889b1ff9973ab441dd35cbebda9739720ebb7f5be
Pure Prop Address: TMVVi7vv3wSKb2C8z3cEZ4a9C551bwwE95s
Theory Prop Id: 8e4c7a0b39ad3b859373543800077739eb387a05d6213b9129540303af3968aa
Theory Prop Address: TMdwMoiJi93xr8T9C61J9zFJibuUWHxu1oa

neq_3_2

Theorem 7.60 3 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: 90afb7a36efe45370d8283989fbac866a0e120e2b883d38ea2fe6fad1e2bbb4f
Pure Prop Address: TMRJB22T27bFhKS9ThQj3eqAGQXbT11BEvn
Theory Prop Id: 4672768d6a3c4e4a1c04351ed841430bbf66bed9a742ca18ddb5c4ccfef3f93b
Theory Prop Address: TMG3u3ZqBrijhNAwNj1kYTN99h2HYB63pTL

neq_4_0

Theorem 7.61 4 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 9bd122b0da951d1af4b4d4c32e237bc4422ce033e491090ee9806587dc9cafb3
Pure Prop Address: TMKZ8uqsY7szUTPxqEmAn695ZoVVu1V891e
Theory Prop Id: dbfe9c5bd8206b487aa4c1e0b115d0543cf6674122a0517ff19c410ec0cd3a44
Theory Prop Address: TMJvPDtd46ULWYjgiP8gUuZNywq5ExcmHBc

neq_4_1

Theorem 7.62 4 ≠ 1. The proposition is identified by the following information:

78 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: 021d04962f43e9033ed2121c6eeb4e5d5e7ebd24e7e42ee06cfef6191229d177
Pure Prop Address: TMQffDs5XzPDL8shLUBqNhq3buem1wiUybw
Theory Prop Id: bbd62921beb6f45383aa837f460867836347d2cde92052a3b2f54987f227e5e8
Theory Prop Address: TMHipYaxxuowEA9gh3W7Mg9siWtCc1uwmCj

neq_4_2

Theorem 7.63 4 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: 7a3354868dab1b0643c846ebb21f339ebbb411d9da95783a5cd4b5d559f44204
Pure Prop Address: TMRxE4YdWypoUuxdDA1hJAxrZPPnK1xbzyA
Theory Prop Id: 0a09eff12034956f5d3fb29bf25815b76d297d68220b9f03eeb417bd726f5609
Theory Prop Address: TMJcWeMVmo9TXWzmFKFkPMhtzFbGJGJ6b8o

neq_4_3

Theorem 7.64 4 ≠ 3. The proposition is identified by the following information:

Pure Prop Id: fdb8798e8cacfe9c22e7a5300ab6ad312c527880e2d6114e93ee6909092b939e
Pure Prop Address: TMWoPuE96Hes4hFCCiWYiKQFfnv8kusTH4Q
Theory Prop Id: 2a56945cd215c22b3db025b3a1115bea3f8be9f8cf7e86f135c9d62fe1b0d0ee
Theory Prop Address: TMTJo37vnbVGCAZzhJgVrDjVt3X2aricxbz

neq_5_0

Theorem 7.65 5 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: bf9251e9bb7b6c3fcadb2f0da4a63d6b36fd8576e49f09a3abe2d7b832c72c36
Pure Prop Address: TMEqk6oZYNwmG5wB1ZGKrZ4T3yWfbqHjRqV
Theory Prop Id: 29ce2701ae30ad2dbf565d3b4a8bf0d77c130da112d208cdb3edf1b1daf72161
Theory Prop Address: TMFJq3wfqDg8x2vPXgbwmSzud3MUPwdSgcc

neq_5_1

Theorem 7.66 5 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: f97fb195b34610dd92dc1c9a46a8fe29d0b99faddd0ee0cb789d7ff153041aa4
Pure Prop Address: TMLhHuF4WrVXhN5KWBBaWXPp9K7YsCP7XNf
Theory Prop Id: cdb672d8c8dfd8a0236d3cfb650c10ae9a5ce01b958ebc60d0bba75bcd57ab5d
Theory Prop Address: TMahxTsu3gQAQUaNifvwRtxDuvarKz9wQP3

neq_5_2

Theorem 7.67 5 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: 25bd00866c407af3c2b8f45d94058aae95d0bd9f06352dd9963408e6745e7f2a
Pure Prop Address: TMLxeQNww6bks2wbWReMUo5Q5HCFEpkctEq
Theory Prop Id: 72066ec74ade8ba81c3aefe23451a21df4aca05a08491cc0af4ff4b1131a38f3
Theory Prop Address: TMLSxacJKrd3N21R6mQxy5fSfYVEUaG9E8y

neq_5_3

Theorem 7.68 5 ≠ 3. The proposition is identified by the following information:

Pure Prop Id: 3372df01e49c94e55fd317e47ba515344be9ed24947b63ae3b7f4648de435eda
Pure Prop Address: TMPpVH6dFjx3FEaJ9sSmDs9wSvDKLg7DmMq
Theory Prop Id: c26bb5b01832d8111735852c7f606f729b39acf9552a1df641404300808a241c
Theory Prop Address: TML4qfcmrd4raBqNGGB7QyhUxV5XtrDK1Q4

neq_5_4

Theorem 7.69 5 ≠ 4. The proposition is identified by the following information:

Pure Prop Id: 909c7c9d7cea9bcfbcb09a16340b2474d31cd36261d27ffef43b7102b4a4eb71
Pure Prop Address: TMJHtW82A2WHP7ViQ391KAdZ1w1wRKs8HYL
Theory Prop Id: 20c3f2d381518f33179709cb5e92c3242267b3069264b9539368291bdf5c269b
Theory Prop Address: TMX5VEFqEbmZM1jXnm6B1dHJJ3UZE1gT5QR

ZF_closed_I

Theorem 7.70 ∀U.Union_closed U → Power_closed U → Repl_closed U → ZF_closed U. The proposition is identified by the following information:

Pure Prop Id: f643b48ed401d2a1c3f6e0b2127cb23b125acf15c4cb21f77a1c45bb01c85847
Pure Prop Address: TMJPXWCGGsgp2RauzJmcXDkupZ2Bwq8zM8g
Theory Prop Id: 6d69881b387db1470aab7b9513ee8d85714e5e34821b8dcdfbe77a49b4958281
Theory Prop Address: TMbg2Nr5QH7Ui64ri9oF1hQu8esX9wKCtSG

ZF_closed_E

Theorem 7.71

∀U.ZF_closed U → ∀p : o.

(Union_closed U → Power_closed U → Repl_closed U → p) → p. The proposition is identified by the following information:

Pure Prop Id: d795d416d63c658dd17353b60159cec522fd5799091016c00532a86a98184a9c
Pure Prop Address: TMFJaZKQXV2BpbnNf3V5PpeaeGyXeZd9p8t
Theory Prop Id: c78c790145214f0f70a0200b6fba47786c8f97e50408def3352d6b836a599b37
Theory Prop Address: TMZSTib6nnUwYgZqRxuRTmGJX2JoBnPeAGv

ZF_Union_closed

80 CHAPTER 7. NATURAL NUMBERS I

Theorem 7.72 ∀U.ZF_closed U → ∀X ∈ U.Union X ∈ U. The proposition is identified by the following information:

Pure Prop Id: a7cdf13e39c8049cf56acd30a1e889d54edc18ff38a1bbc2e1ed347d9682a5ad
Pure Prop Address: TMUofvcpBARSdnFetm8dQrjhiWg4kWAGFB4
Theory Prop Id: 250496cf2e1875d7ec9219017e8d4508f9ca7718cbfd9c968c7bade397f00ea9
Theory Prop Address: TMcGjahmQUapH4bqbAdGBQ6EgJgQDSjPLwm

ZF_Power_closed

Theorem 7.73 ∀U.ZF_closed U → ∀X ∈ U.Power X ∈ U. The proposition is identified by the following information:

Pure Prop Id: 5d781fd8e7019faff9a70275545ea6d69a0eb9adbe24e83a2e06143bffd06746
Pure Prop Address: TMHz75Wp3yULtSkKHqZV4euzQkSQyKevgdT
Theory Prop Id: 7411dd8f49dfca8e20b3d4524ebb50b5fb3351ca8588eb34afde921cd3727004
Theory Prop Address: TMFeLtgqxJCBDZ3U5JYcbg9M6iiuFekPu6M

ZF_Repl_closed

Theorem 7.74

∀U.ZF_closed U → ∀X ∈ U.∀F : ι → ι.
(∀x ∈ X.F x ∈ U) → {F x|x ∈ X} ∈ U.

The proposition is identified by the following information:

Pure Prop Id: 312268feb1dd5835c81ed94141bfe83ca9dda2b26ecf8263cee15af85f88b8b2
Pure Prop Address: TMPP71S8C6hLepoQ5MX4wJnJZ5iqRp8wjHj
Theory Prop Id: 18f8acc434076d8590e9958a09b3287e80e677748bfa68846c9246019587ec5c
Theory Prop Address: TMQuAnjb2usoLHA9bv3uUoCs1dSHiL1SFdi

ZF_UPair_closed

Theorem 7.75 ∀U.ZF_closed U → ∀xy ∈ U.{x, y} ∈ U. The proposition is identified by the following information:

Pure Prop Id: 54256a260e0aeb219ec7b9c1f5470e1c879e31656131aab6f98d5191437dc819
Pure Prop Address: TMJ8PnAXtGvJBQ2w7xRg2sQuiEx27bk436v
Theory Prop Id: ef42d3cc50e9cba4432b33a79bbd1192bd239f4d284470ad502781b5fb35ca8e
Theory Prop Address: TMcjrdqKzeo7g2gDjiEsVum3CAuDTSFiFuB

ZF_Sing_closed

Theorem 7.76 ∀U.ZF_closed U → ∀x ∈ U.{x} ∈ U. The proposition is identified by the following information:

Pure Prop Id: 5d8a834f1cf303ccfa19193124cf05f94ba5eadb60ba9505bc698fae5beefe9f
Pure Prop Address: TMRHMv7QRuD1cNMZ1DKFBfgHLajcAvhMXXa
Theory Prop Id: d02d66f005b189993512ef90ced927d55db86cbffc2242c0a868958712bdead1
Theory Prop Address: TMQJBe5GNPMz6xkyZp1New6CprvjNNWbVQB

ZF_binunion_closed

Theorem 7.77 ∀U.ZF_closed U → ∀XY ∈ U.(X ∪ Y) ∈ U. The proposition is identified by the following information:

Pure Prop Id: 8a62402878aa34b70e4993833d0aac7cf60d7b31747f91d9f6bbb966b91e20e1
Pure Prop Address: TMEqKpBN15S7iTeSq7zbuWutkNifE2ingyu
Theory Prop Id: 1bacfdb236f4aeb5ec63cb4b7c4d4bb0744883de7265fff6b8b706d8b410cece
Theory Prop Address: TMdzxaBW52KHaW3S2NvEVtisq8WoKg6MZvp

ZF_ordsucc_closed

Theorem 7.78 ∀U.ZF_closed U → ∀x ∈ U.ordsucc x ∈ U. The proposition is identified by the following information:

Pure Prop Id: 68a6ba11b0ad56c445b6e8bc81041f5c2cb0db9142f6b147cfcfa7ef1d2f27f3
Pure Prop Address: TMH2dwKRQudFN9m3qx3RNdaBLvqypKizZv5
Theory Prop Id: 946e6124c0e8be96a15baa61fedb45d36338fafc05a2f31ce54a95174700bf8f
Theory Prop Address: TMPqDp5W4hUggbHVsbPDuU2ytBMzDWrbM2M

nat_p_UnivOf_Empty

Theorem 7.79 ∀n : ι.nat_p n → n ∈ UnivOf Empty. The proposition is identified by the following information:

Pure Prop Id: c885b475dae13164eaf466c05904664c48aa5e96b6499036d6bbd7ab2b3b01e3
Pure Prop Address: TMTM67QGgM77h8RbmGoR4QhuHU14bxuur8y
Theory Prop Id: 645aee297f02d0a7caa06a5efea238cec06af7df69b2643d2adc7c5d9b418911
Theory Prop Address: TMZEUzvGsZE4EoguW8R1ovkAWzhDE4znAr3

Definition 7.3 omega is the opaque object of type ι identified by the following information:

Pure Object Id: 6fc30ac8f2153537e397b9ff2d9c981f80c151a73f96cf9d56ae2ee27dfd1eb2
Pure Object Address: TMb4KgMPXEQBvZWN5yMtrjHJCTxAvdStvip
Theory Object Id: ad9ad81c42d92fd52433645cf2466f4f0179ab10ee124be71bb12991deb2ddf0
Theory Object Address: TMNkXuKe4mYMmqP6e4nkiPeWyEzHKwZy2zC

omega_nat_p

Theorem 7.80 ∀n ∈ omega.nat_p n. The proposition is identified by the following information:

82 CHAPTER 7. NATURAL NUMBERS I

Pure Prop Id: f44a309f3fadd69adfefbc11bb8cd5a0e2d204f0f0ccde446b0cf927293e46dd
Pure Prop Address: TMQLBjRkNsbEFRVrBkuh1bmZQqCeUscTWR1
Theory Prop Id: c2c61b35088d7afd198e0d398dabb21f1be9e52d6c73d798f0fa57761801eaa0
Theory Prop Address: TMcBbUsnDN99QLS1dM6hhdDj1W8pvPaMBBH

nat_p_omega

Theorem 7.81 ∀n : ι.nat_p n → n ∈ omega. The proposition is identified by the following information:

Pure Prop Id: 6a1b445f977fd3fae4e1e1cc46011455e1f0531f1a15c5c25278556928dcee83
Pure Prop Address: TMd8YkcuN2F2evUjAbU6Ts5vNLwKCBES7rW
Theory Prop Id: d5080f911ac6cf73280f9bbcf0edc2218ca24806bdc57695210e6115b9406c38
Theory Prop Address: TMWv1SjyTw4FBL65LJ4Dydky2jqrjzFo4WF

omega_ordsucc

Theorem 7.82 ∀n ∈ omega.ordsucc n ∈ omega. The proposition is identified by the following information:

Pure Prop Id: 246d636d93c78b4e3365422a679abe57556849f5546b6f8e8fbaeeab0e834905
Pure Prop Address: TMYTp6SmJ6Xb74ZB58T5drE74MrfSMJ56Xd
Theory Prop Id: 17fb646a22b11762cb7ae21fb49306beac81729c7c3e89992f09e10eff2d8f9e
Theory Prop Address: TMcKRUvCqbX8WtX4XYQZ7VoDqsewDsQmCWB

Ordinals

Definition 8.1 We define ordinal to be λα : ι.TransSet α ∧ ∀β ∈ α.TransSet β of type ι → o identified by the following information:

Pure Object Id: ee28d96500ca4c012f9306ed26fad3b20524e7a89f9ac3210c88be4e6a7aed23
Pure Object Address: TMMUuqYqPdrVu6cDtXgARhaXmQ84ZrMXG6P
Theory Object Id: 53ee8f616e489e1965c33226035b0df42058ce50fbf01e6172cdf3b427f5a693
Theory Object Address: TMJ4CnbKugQJ97k3pfeF56M78Ug4AubAWjZ

ordinal_TransSet

Theorem 8.1 ∀α : ι.ordinal α → TransSet α. The proposition is identified by the following information:

Pure Prop Id: 0dbaa3d497ea2fe357d7cdf6de2639b3f4cf815ee1d613eaddc6fb305b46de56
Pure Prop Address: TMdzYsAgm9hgqSggzyEMmMw5w6aRH1fmS1D
Theory Prop Id: c9e708b22b5dc60c895eed22a527c202e52f14066b2f95ce8df1cb630eb1df7e
Theory Prop Address: TMXHSD7yBJJjNhye2R2DtR5uPTVLTRGPMVo

ordinal_In_TransSet

Theorem 8.2 ∀α : ι.ordinal α → ∀β ∈ α.TransSet β. The proposition is identified by the following information:

Pure Prop Id: f4e77fe3f77ebff8ff08239d5f4175451624803d11b0018e85f316c34806264c
Pure Prop Address: TMYKCKb2mgJ361o53ZDmr4bqCotaiAa9UMT
Theory Prop Id: cc1f55fe1800e237a0d36d0f295ac28bf748b236473d1e1f9caf5515be62a995
Theory Prop Address: TMXH96JbZ5RRa3chjsLwakm9MEVpiN5ygZV

ordinal_Empty

Theorem 8.3 ordinal Empty. The proposition is identified by the following information:

84 CHAPTER 8. ORDINALS

Pure Prop Id: ce6bea11043bbd03d9af274f7b824c53be0840dd9f324cd7c9afc0d4f811b316
Pure Prop Address: TMHjHTkWmLerJqJCyfCzQ2x3bAbjxupX5rN
Theory Prop Id: 7d95d14b163c311c6050bb25900b2f0e05a1fb83c01aacb93a1b6cf03a658823
Theory Prop Address: TMcRLhPoJiCW9i4Gg3Z8GVhyHi3DGQmpBpx

ordinal_Hered

Theorem 8.4 ∀α : ι.ordinal α → ∀β ∈ α.ordinal β. The proposition is identified by the following information:

Pure Prop Id: a7e78b55af56fd38debe61d4cf2ccd6512ff509a33690ff7bcb062d67233a307
Pure Prop Address: TMYATDhQjyWwu2denmfYctz9jzc5T6pvxDY
Theory Prop Id: 43b0f3ddbec2adab900eae39d5c70d5f8e32937abf8669f79ab503a2cba3c688
Theory Prop Address: TMNMsaVbc4QJAjitEJmAT3A6577bnGUMenX

TransSet_ordsucc

Theorem 8.5 ∀X : ι.TransSet X → TransSet (ordsucc X). The proposition is identified by the following information:

Pure Prop Id: fc215771da05b7091bde93e8fd1dbf94168ddb5e1c68f7934a03aeb58f1ebd07
Pure Prop Address: TMds6RBqX4EBdbVb2jzpQF4BGu62TbmQW6S
Theory Prop Id: ebf9e34c8261f406aaf9188564b534bc4eee97dba2e5fafb75b1f972cbc9d365
Theory Prop Address: TMTZpuHkdymxGmLoep6QY3LRhQf6Nr8Myhy

ordinal_ordsucc

Theorem 8.6 ∀α : ι.ordinal α → ordinal (ordsucc α). The proposition is identified by the following information:

Pure Prop Id: 0ba97fa745a108db180034344887cefa8fa42aa10e94eb93ee5f6ba8a383902a
Pure Prop Address: TMNpPkbjLapq9BtN9phPiFFNbRzuB8jfTRk
Theory Prop Id: ab0f7f2be575c7979819df5a27b6d7276069f61fd5ec603b15b317dca63b3e76
Theory Prop Address: TMbVvz6WT788KucEJn9rQE17f9WVYDdtwv7

nat_p_ordinal

Theorem 8.7 ∀n : ι.nat_p n → ordinal n. The proposition is identified by the following information:

Pure Prop Id: 274fc409055bdbb382e1defb45f54a49e7f797ab883d97e76722ddad8971f35d
Pure Prop Address: TMKFEu7wFJqZoMviwrZLpLaCP1QjLcwtCwk
Theory Prop Id: 47cd49a599f17ace658f9f1d538502f51cde62c8cd2d997a51ce938eecd15d70
Theory Prop Address: TMR9b86LuWgjhZL4m5yEvQhHdYU1YcYRkn3

ordinal_1

Theorem 8.8 ordinal 1. The proposition is identified by the following information:

Pure Prop Id: c94d41f958c74e02ac9b04ef1997708d28df66c7d56e3a98177bfa3ef526a409
Pure Prop Address: TMT3Q25U1Ux1eVUUm1s1FeitpAvwbXTk1dm
Theory Prop Id: e325a502344d5146d6b51df0408cd431c99b978bd785e7aebd69d0cbd1d3630a
Theory Prop Address: TMLMkFbUXyMVHafQwFnHAFJwHJiUL8uizu8

ordinal_2

Theorem 8.9 ordinal 2. The proposition is identified by the following information:

Pure Prop Id: a570756c9f8d8069da7f0920bf81bd27397ddff102f6b7444e080f7bed3390ff
Pure Prop Address: TMFLbGodmA7UJer5TihfdxoFBgHSxZjK3cY
Theory Prop Id: e64a0df795b030c5c40d39ceceb101d04488a3dbf090fae286e8f8e284d9727b
Theory Prop Address: TMEjrJXdnsbDexBJZRJysy6hDoZye5ffvEX

omega_TransSet

Theorem 8.10 TransSet omega. The proposition is identified by the following information:

Pure Prop Id: 8e89e66d9788201a152dc0d2e564bf2e428e165c4c240da4dd210f8f102b0b9b
Pure Prop Address: TMNtK2hop4HkuAGDS35C3HQG1M44NKe3USF
Theory Prop Id: c79e752f422afbb22e542dfbd0d6ee1f417aa6fe06a04b74efbecce1e0b036e0
Theory Prop Address: TMKTjT9CEwmv6ZQwdPQWnn1MRx5wxzNZP5R

omega_ordinal

Theorem 8.11 ordinal omega. The proposition is identified by the following information:

Pure Prop Id: 3d621a0d65590ab57c57a1848257b5fa064c0e3a4847bb476e17a549b2bb1179
Pure Prop Address: TMEhcpuAFpHXHFSKrZU9N7ytoAc1KYSKfwJ
Theory Prop Id: 177484ec90afd915b52f7e111ce39d29114b2d0f97fd75908b5de1e8b5a405b4
Theory Prop Address: TMdVAnxzLqF7mrAcYCABmsfnfihincw4N1F

ordsucc_omega_ordinal

Theorem 8.12 ordinal (ordsucc omega). The proposition is identified by the following information:

Pure Prop Id: 66659c006c982a0ac99ea4ff01a85f6f4ff5c6737431c0032276450048e05cdf
Pure Prop Address: TMPkd9dTQz5vTvQaDf2GPn1vNYBQR6FCxTM
Theory Prop Id: ecfad7b774257cdf70613958fe3477637a520ac37c9c970c486229116833e51e
Theory Prop Address: TMF6YnnSCpzaZf1K8mSGfwFbrQh9ZW2n7Y9

TransSet_ordsucc_In_Subq

Theorem 8.13 ∀X : ι.TransSet X → ∀x ∈ X.ordsucc x ⊆ X. The proposition is identified by the following information:

86 CHAPTER 8. ORDINALS

Pure Prop Id: d5d88203bb9e4f58e19b53c1e5a62f15aedc32de8c5bc76e2580367c4174de4d
Pure Prop Address: TMLQBckdoSFHRuVFwGbGbuRrp1fdK4R87ov
Theory Prop Id: 3c2e4593e300f67b32ba028952543ba0dd7e523521087467b0174ba873c292af
Theory Prop Address: TMTRBiP8XXTzUuthK3uDUFLV6ZLsiJDuXee

ordinal_ordsucc_In_Subq

Theorem 8.14 ∀α.ordinal α → ∀β ∈ α.ordsucc β ⊆ α. The proposition is identified by the following information:

Pure Prop Id: 57345e9c93f2215aa2812d32af6e5102ec8faa4224750fe0e5a65c9f1afcd5ff
Pure Prop Address: TMQpZgVSt9a4G3EQn5TNVzeb9wssdHKNW7u
Theory Prop Id: b00304a76db0c730b04afb1cf7ab4e339ad984c2397a5aa9dfcbe8b7ff5fbb77
Theory Prop Address: TMaxmykHWjhaXzn1rWbPuqhxTLTTER9wNAp

ordinal_trichotomy_or

Theorem 8.15 ∀αβ : ι.ordinal α → ordinal β → α ∈ β ∨ α = β ∨ β ∈ α. The proposition is identified by the following information:

Pure Prop Id: 078a34fdf9fecc3740d32b870c8d09ccc0128f7642e5b75098ade0d1089ece30
Pure Prop Address: TMFhw4k9MkF4Xj6fGtXNVWovjCfNEDFgiCV
Theory Prop Id: 21f2d68bdab52d51a33741771dbbe589a47a82c6a747d3f27f2c6e7e7fb46323
Theory Prop Address: TMQiHm6s7fQpsV7SnBFtLZFvEeXFV9nDbmm

ordinal_In_Or_Subq

Theorem 8.16 ∀αβ.ordinal α → ordinal β → α ∈ β ∨ β ⊆ α. The proposition is identified by the following information:

Pure Prop Id: 70fe0df6b00a9a1cdc3d97c6a8ed1db39034d341a16f8a30bddebf8a0a8b8e4f
Pure Prop Address: TMU41PmuGiADNxp6RQqwepi4ZBaznh8t8Dy
Theory Prop Id: 7b99b0def95577474ca6a6371ba472b8ef499bf320c511f3b539db9524a87727
Theory Prop Address: TMUy7ee37ThdPDzbRspVTZUkthgdjtEyQYb

ordinal_linear

Theorem 8.17 ∀αβ.ordinal α → ordinal β → α ⊆ β ∨ β ⊆ α. The proposition is identified by the following information:

Pure Prop Id: 3f30a55f5c77831d749f6727a93c5a8bce525eb93957884747af3b16fd2817dc
Pure Prop Address: TMPY4A7R7aYy2invvHTTAywc1usxqJAm5u8
Theory Prop Id: 3b1fce49dbbdca0b438f6709b0e0a85233faa824682dcf5ed3f74df3c3bb6ea0
Theory Prop Address: TMM2dfXndDyhGP7QGpLd2hvEgv48peHsWcx

ordinal_ordsucc_In_eq

Theorem 8.18 ∀αβ.ordinal α → β ∈ α → ordsucc β ∈ α ∨ α = ordsucc β. The proposition is identified by the following information:

Pure Prop Id: 500378eb85429ae7b33a80121ead8cda752628caffbde51467158317d0e18d74
Pure Prop Address: TMHGn9G1ExDEjiH1MPhwk1Vyodkz876yMRi
Theory Prop Id: 0c19e771defd6f69e7d104285ac72cf1f7b2cf9c9100d4849916879cb6681814
Theory Prop Address: TMVF297QZUVVpWRs4i96ncxvbVNynLTkL7q

ordinal_lim_or_succ

Theorem 8.19 ∀α.ordinal α → (∀β ∈ α.ordsucc β ∈ α) ∨ (∃β ∈ α.α = ordsucc β). The proposition is identified by the following information:

Pure Prop Id: 4450c013030397143dd5b0c0a5fc68ccac288751c8cd230b4f4abcdeec1c5c6d
Pure Prop Address: TMankcEMnZXFAP8cvcfDqwi1HBKu5b8txfW
Theory Prop Id: a12697d237d0a645eb8cbf1e70cd0b6d46292c225e8a36a42a59dbe9350e27f6
Theory Prop Address: TMLRGTTc9BqSSZNBMbxF34StFhkCfsPU8B7

ordinal_ordsucc_In

Theorem 8.20 ∀α.ordinal α → ∀β ∈ α.ordsucc β ∈ ordsucc α. The proposition is identified by the following information:

Pure Prop Id: f4b1ed473dc1d1af402cb8a1a56b16920553d37b8b365cf4e72a74e2467668f7
Pure Prop Address: TMZQdYrrW7LPMeUbwQJGaSaWYZJ342j6TLR
Theory Prop Id: 8ab1dc0a15cde7e11ff2fda7e5194042b72dc495121dbe98a3cb9d5f23df0650
Theory Prop Address: TMF2GA2skuyQZS3EhFZYEwjYb1i5V7CiuYY

ordinal_Union

Theorem 8.21 ∀X.(∀x ∈ X.ordinal x) → ordinal (Union X). The proposition is identified by the following information:

Pure Prop Id: c733ee1a1d9fd2449e70663578b0382c3bccae896c32e0ec362f1494fda06c3e
Pure Prop Address: TMW1Z5hkuor8TPuLNVR1zbiXXaoiAEs63WD
Theory Prop Id: 8f2519b0a9c7cdd7d42c73a6733e14191e924dccd043f97339e7ea377d746f2b
Theory Prop Address: TMJhDobuvDdYqLTuiL9wE7gzst4xAaHHwqt

ordinal_famunion

Theorem 8.22 ∀X.∀F : ι → ι.(∀x ∈ X.ordinal (F x)) → ordinal (⋃_{x∈X} F x). The proposition is identified by the following information:

Pure Prop Id: fc035982e0d1f528e70c7a7b191727c9d3afc3eb8cde8f6cf06880f52db2e3d4
Pure Prop Address: TMU1RzbEJzVnikGUeRHDEeJKZdU2gv4Lu9U
Theory Prop Id: b3f1e1ecfee50fc6903bc622a5d3170a7e8f834e8b8e4cd173ce0c699d33d141
Theory Prop Address: TMNBvDL5kWic9wxjPXEkTh7WAf3f2AnJ1NW

ordinal_binintersect

Theorem 8.23 ∀αβ.ordinal α → ordinal β → ordinal (α ∩ β). The proposition is identified by the following information:

88 CHAPTER 8. ORDINALS

Pure Prop Id: 43d507cd3ab7cd23861b8705272c6f88b7930ceecde12dab179ead328e6a2cd6
Pure Prop Address: TMGAFLa8mQYJ6L1kp1WKCNwaY3YrMwPqad2
Theory Prop Id: 0253282876aaa86e7cea3ea89459e20513307e9d2b9ffe1cc2ba50b357bbb4a9
Theory Prop Address: TMQLySje1TXDy7YnSUR855qqzweodR4BBbP

ordinal_binunion

Theorem 8.24 ∀αβ.ordinal α → ordinal β → ordinal (α ∪ β). The proposition is identified by the following information:

Pure Prop Id: 861ae3bbc499b574608e38df76236c43f8ff5f552de1d70b9002feb31d7edc87
Pure Prop Address: TMWj1DiEDHWbFoRK4tceGziMDSzNMjXvATC
Theory Prop Id: b24989c4f74b8369768af389e0867d9ae88d987aae3abe7414441666572cf91a
Theory Prop Address: TMb8JjtU8NrQYjHW2dN4KBHmc6dZLD8tX2H

ordinal_Sep

Theorem 8.25 ∀α.ordinal α → ∀p : ι → o.(∀β ∈ α.∀γ ∈ β.p β → p γ) → ordinal {β ∈ α|p β}. The proposition is identified by the following information:

Pure Prop Id: 9c0d4265c1c0802cb20047122fd9d5c097bbab37a60f64d593b6a44e1bd26859
Pure Prop Address: TMMc3fJbFMyUS6zszTmv9rSMmYJS2mMJVFh
Theory Prop Id: b5f69f338e4258431a3a6159a38976e853cda2445044d06e4ed0a660100c1a65
Theory Prop Address: TMHv1MoXTBWSqnddBfw56ZCoB3HkEABQP52

Comparing the Sizes of Sets

Definition 9.1 We define inj to be λXY f.(∀u ∈ X.f u ∈ Y) ∧ (∀uv ∈ X.f u = f v → u = v) of type ι → ι → (ι → ι) → o identified by the following information:

Pure Object Id: 264b04a8ece7fd74ec4abb7dd2111104a2e6cde7f392363afe4acca8d5faa416
Pure Object Address: TMQsS5cocdJLxvSrfwFrtyVb2HtH4sHnoSq
Theory Object Id: 442958a234e56cda2843c516d8ba0be921c19f4486af20ea18cd78e0b5587a8e
Theory Object Address: TMEtNCh6NfJv35kUGcgFB52zooWEchr4mq3

Definition 9.2 We define surj to be λXY f.(∀u ∈ X.f u ∈ Y) ∧ (∀w ∈ Y.∃u ∈ X.f u = w) of type ι → ι → (ι → ι) → o identified by the following information:

Pure Object Id: 4b68e472ec71f7c8cd275af0606dbbc2f78778d7cbcc9491f000ebf01bcd894c
Pure Object Address: TMcR6n2BpcwoiDwRhTtfhBS2wRexeumPoBF
Theory Object Id: f022f92791e019fecb8ae163b387436a5af8f1fc86e8403db0d2add2d326d2cc
Theory Object Address: TMSbXiP7EjGYNy1Jeo9r2YG6YXA8qunsPXV

Definition 9.3 We define bij to be

λXY f.(∀u ∈ X.f u ∈ Y)
∧(∀uv ∈ X.f u = f v → u = v)
∧(∀w ∈ Y.∃u ∈ X.f u = w)

of type ι → ι → (ι → ι) → o identified by the following information:

Pure Object Id: 76bef6a46d22f680befbd3f5ca179f517fb6d2798abc5cd2ebbbc8753d137948
Pure Object Address: TMWBv3fDiJHG6PbQ5p9tmBynBsCvJmYFGXN
Theory Object Id: f7d93b7e704e3215ae66ae7c330f283bc8b4b71850013429da3d75d7ab6ee432
Theory Object Address: TMWitYCyjoeiXGg1451hhTG6x1DmSshpEsw

bijI

90 CHAPTER 9. COMPARING THE SIZES OF SETS Theorem 9.1

\begin{matrix}∀XY.∀f : ι → ι.(∀u ∈ X.f u ∈ Y) → (∀uv ∈ X.f u = f v → u = v) \\ → (∀w ∈ Y.∃u ∈ X.f u = w) → bij X Y f. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: a9064fdc226bed9225585cb4f2610b131c0a14aad760bfeb55112ab19f50c269
Pure Prop Address: TMVtA5kA7XycdNhnjcWoD7YH6yuuhkFVjMt
Theory Prop Id: 3229f986ac1acb74b9c25702305e679c33fee187075ba5e7e5521402a44407f4
Theory Prop Address: TMHZ6dKwiqgD43Q9CKokQdMMtyUWTqNe73b

bijE

Theorem 9.2

∀XY.∀f : ι → ι.bij X Y f → ∀p : o.

((∀u ∈ X.f u ∈ Y) → (∀uv ∈ X.f u = f v → u = v) → (∀w ∈ Y.∃u ∈ X.f u = w) → p)

→ p.

The proposition is identified by the following information:

Pure Prop Id: 2cf2785657b7836bb8cf1988ba517c500841f4597e31341232a8e4ae9a2ac11b
Pure Prop Address: TMPhihHJHHFvCn9A4dYL5TQ6pqHpkEvceX8
Theory Prop Id: cf075922dfef18d341a5a232bb48285fa39e11a63b966fd9557f7bbc707557bc
Theory Prop Address: TMVhokQU6h9PfVvoAJ7Se6F6yDdFRx38gX3

Definition 9.4 inv is the opaque object of type ι → (ι → ι) → ι → ι identified by the following information:

Pure Object Id: 896fa967e973688effc566a01c68f328848acd8b37e857ad23e133fdd4a50463
Pure Object Address: TMYsHRgQQXjN5d3YZ2sLdp9j73Hd4HiYRei
Theory Object Id: f3a0156f444b33934c3c28ed4b3f22eae35078c37c1e0c9f4f931d3c1e1c9266
Theory Object Address: TMNXEs7Lfg3PCmc11RhANqjcTXio2UJSNhw

surj_rinv

Theorem 9.3 ∀XY.∀f : ι → ι.(∀w ∈ Y.∃u ∈ X.f u = w) → ∀y ∈ Y.inv X f y ∈ X ∧ f (inv X f y) = y. The proposition is identified by the following information:

Pure Prop Id: ebf3718d5c90ac32637e5ff7de771588bd899f20728aac1c034cc4575a9f2bd5
Pure Prop Address: TMdfPpzpdtB7VwgaPueTNcMshHsVs117kuo
Theory Prop Id: 6489d2e57f0dee884152f4a33ad50d245f1c35dc285ca99ba09b67a783f89c93
Theory Prop Address: TMVFndmdYppMYsHoVKPEwfE1yfNTP6HYsK6

inj_linv

Theorem 9.4 ∀XY.∀f : ι → ι.(∀uv ∈ X.f u = f v → u = v) → ∀x ∈ X.inv X f (f x) = x. The proposition is identified by the following information:

Pure Prop Id: f9fc2fb6cfb4eb217cfd90f255fb35b64bd1729aa515ae6c662014a80909baf4
Pure Prop Address: TMEuKQvRXDUoH8sLbTA8MpzpoVNt2Uo41RH
Theory Prop Id: 74870e1df65b0cc2ae2c71c585c9735524f602fc82ed42ac4a046496a20db8bd
Theory Prop Address: TMTFjNxnxSQzmggpMN48iCzGFGMZ6u5HUUU

bij_inv

Theorem 9.5 ∀XY.∀f : ι → ι.bij X Y f → bij Y X (inv X f). The proposition is identified by the following information:

Pure Prop Id: ff1332269c5993981bc337566181eb6a5ac760b35f3e8f90eed0abf70ebc7c17
Pure Prop Address: TMRvnWitWbimHQFYgNtor4w8xmFZq6WVJT5
Theory Prop Id: 69c53767bc4248c32eca1d3c840bc6ee0513e90488a4e3d38dba20033986cce4
Theory Prop Address: TMYU31Q8vC6ff6UyuWLEMF43Pms8CHUmb2x

bij_comp

Theorem 9.6 ∀XY Z : ι.∀fg : ι → ι.bij X Y f → bij Y Z g → bij X Z (λx.g (f x)). The proposition is identified by the following information:

Pure Prop Id: abf85117d4cc5830f8c4e2302319f45ffb293b1c89044d95312722edcc59ebda
Pure Prop Address: TMbJ4RHULgz1EDJb4kgryxvfAijajbPaBWZ
Theory Prop Id: 4c90458f330b4d7cdc780aefbd2a209237607b7b0ce813a26728e290eaf8ae8e
Theory Prop Address: TMMeMmBNYFCGY7zJTLg49As2Zm5US2hziHm

bij_id

Theorem 9.7 ∀X.bij X X (λx.x). The proposition is identified by the following information:

Pure Prop Id: a8cefb4071a07310e8c514fb13ecd4fb64481818a12472770c065383dc391c08
Pure Prop Address: TMKDAnvGLshjhnT7DcuJYqVH8EphuBsUUcs
Theory Prop Id: 707f370beec6bb62870cc94c9fa8f45ced6eaaf202ba0af81dab213981d7b6a9
Theory Prop Address: TMFbyfnRF5QWxotLuNUgJzCGP2XV6QZ4y1c

bij_inj

Theorem 9.8 ∀XY.∀f : ι → ι.bij X Y f → inj X Y f. The proposition is identified by the following information:

Pure Prop Id: e3635e28871703e7f8247d1db9057e8a348d2ccd04747dd154febb02a37f9401
Pure Prop Address: TMMrgAbiSR6cnvcK7CVfGRP9kNu67vN98ZP
Theory Prop Id: 00097cfdcf04769f3acdf746b6155be23b2782a6c12e002b03fecfc4ed998644
Theory Prop Address: TMRRNL1K8TWBDtgrjqzbQBsxYHZsqGCVu11

92 CHAPTER 9. COMPARING THE SIZES OF SETS

bij_surj

Theorem 9.9 ∀XY.∀f : ι → ι.bij X Y f → surj X Y f. The proposition is identified by the following information:

Pure Prop Id: e7a049991e8ce73cb9d2cde3714035b3bfd567ffd29a6588380d6afe28d0d422
Pure Prop Address: TMZ1Df2XJMAY4VQtRKyVeYTrjgp9uNrkat1
Theory Prop Id: 7fa9a5a19214ed9b46b7c2672858b7198b12a117652c2884ba242b0d3023bb78
Theory Prop Address: TMK69KkYqvqjbC3pgQQ4xrkBTDfGpX9dhLN

inj_surj_bij

Theorem 9.10 ∀XY.∀f : ι → ι.inj X Y f → surj X Y f → bij X Y f. The proposition is identified by the following information:

Pure Prop Id: ccff0258f3d7f02dbb9c92e1c2c588c500749a613923b0ece6497a2266072218
Pure Prop Address: TMZSLNi3XSR9xpWQPVMJxKdbpuqMGcXGcTa
Theory Prop Id: de895fd8c8d353a1a3e58226dea93e0bcab39207eaea6f1bd63ca76728da5443
Theory Prop Address: TMKfqwemEJSEU94d4EWk9R5Z9o4KFevVF79

surj_inv_inj

Theorem 9.11 ∀XY.∀f : ι → ι.(∀y ∈ Y.∃x ∈ X.f x = y) → inj Y X (inv X f). The proposition is identified by the following information:

Pure Prop Id: ea4775ce7b91dca8fcab0b476c6da12b3812e7c3bb1c40ac217ef8ec4f7e2421
Pure Prop Address: TMMfpEoFj9egg8ymZiXFBa7mqr6V1aAyR52
Theory Prop Id: dde3b90d9180ac2126f48d3f539c000ff8f515f94358c02d782de14599dfd8e8
Theory Prop Address: TMYyJkfwSTRDidng9k7W8zBoQ7M964PoQu8

Definition 9.5 We define atleastp to be λXY : ι.∃f : ι → ι.inj X Y f of type ι → ι → o identified by the following information:

Pure Object Id: 9bb9c2b375cb29534fc7413011613fb9ae940d1603e3c4bebd1b23c164c0c6f7
Pure Object Address: TMdn3jbiRnktYr3yHQk9h3fe2hNUWwmbigq
Theory Object Id: 13545c645a339da15781ed2b2b387176e9241a5497e95448eb000efd67f60842
Theory Object Address: TMVULsv3UtPX7zQrujvLPr8j5YcesgYNBcQ

Definition 9.6 We define equip to be λXY : ι.∃f : ι → ι.bij X Y f of type ι → ι → o identified by the following information:

Pure Object Id: eb44199255e899126f4cd0bbf8cf2f5222a90aa4da547822cd6d81c671587877
Pure Object Address: TMQwX3qjReYJVk5GFFSX3Mc3FXLfVZVK9kz
Theory Object Id: 7f2bf6cf81f51f17f6ce5c2dbf188c600cd671933bce98176ab19c48014cea47
Theory Object Address: TMLBEa2WvRd6EvHsbywWEHTnGKb6kqvBVtd

equip_ref

Theorem 9.12 ∀X.equip X X. The proposition is identified by the following information:

Pure Prop Id: cc7e804a09044eda808398af78c405e10d681cf2d3ebd1e427405bb3a0f82f4b
Pure Prop Address: TMYk6cm3eVPFH29r4U5oDKnwrrdfovk75q8
Theory Prop Id: c41fb1856e522372cc47cfb80303fc14cb2d9b7b560c6fbb7b1e6aa94caf89c4
Theory Prop Address: TMXscfQQ6JtEo6wQ2rvbuRae4u8ghd3TmK3

equip_sym

Theorem 9.13 ∀XY.equip X Y → equip Y X. The proposition is identified by the following information:

Pure Prop Id: f289541b211153f721f213a25c59969aa4873f985ed3dadb265f68134c24c94e
Pure Prop Address: TMY2oL6NVAFGfyVKFHZtMwxnebWdEANu4RE
Theory Prop Id: 272f04ef14a36c414e4557b20a0d2296a32dc853b3ed7c4ee2a194386feabcb3
Theory Prop Address: TMP5Vsiypad2yczNmG9eU243SqUyrcsGC6w

equip_tra

Theorem 9.14 ∀XY Z.equip X Y → equip Y Z → equip X Z. The proposition is identified by the following information:

Pure Prop Id: 322bfbeb779343f7873d6478ed537f6d26e801c3b1dbd29367976be6a6a8cd1e
Pure Prop Address: TMP3Pyh4cMndHz7KEMSUcuThjZ8NUECDxnz
Theory Prop Id: d616c1609dd61f5c209c6cb476fcba79031210d81c19459ab45828203b68f961
Theory Prop Address: TMMPNzj7ZfCkfZFdbzZMRPAohztBSpmcrY9

Definition 9.7 We define finite to be λX.∃n ∈ omega.equip X n of type ι → o identified by the following information:

Pure Object Id: 04632dc85b4cfa9259234453a2464a187d6dfd480455b14e1e0793500164a7e3
Pure Object Address: TMTx9uDMYWFe7F4kG1CvRU871EPE4S3LaMV
Theory Object Id: 4c54a587d9d6957aacb640a264f675f93a1254e92f4c6f7aad082ab5a4350c68
Theory Object Address: TMFKGVLZA5xtkoBs7Z7ZUYc2RvcSeaNRv3v

Definition 9.8 We define infinite to be λX.¬finite X of type ι → o identified by the following information:

Pure Object Id: 313e7b5980d710cf22f2429583d0c3b8404586d75758ffcab45219ac373fbc58
Pure Object Address: TMasuVNeN4ZSGDYVEGFs8RGYS2DpDeugNiN
Theory Object Id: fad8eaad083b98f773fe61e7e4c9c8e42ee4b7b5b96a8a831500e0f2097c3ccf
Theory Object Address: TMUDie5EKa9EaiBtjwiXwQDYd6oF2q9cCbx

KnasterTarski_set

Theorem 9.15

∀A.∀F : ι → ι.(∀U ∈ Power A.F U ∈ Power A) →

(∀UV ∈ Power A.U ⊆ V → F U ⊆ F V) → ∃Y ∈ Power A.F Y = Y .

94 CHAPTER 9. COMPARING THE SIZES OF SETS

Pure Prop Id: 2ffb9f17db9abba2e99cdc9e9ea64f507ea8b974c232ad295e4047e204a3c2df
Pure Prop Address: TMGYc2cyt576aoAHoJnf5fmecWDttyfoAEA
Theory Prop Id: 31ef9df03eb42ad3c6e310449b514b042b286ecbe91fd6d663ff0cb8c38f9bcc
Theory Prop Address: TMVA6dF8zyCJrTc6aSsUANd7jk3viaeNZ4W

image_In_Power

Theorem 9.16 ∀AB.∀f : ι → ι.(∀x ∈ A.f x ∈ B) → ∀U ∈ Power A.{f x|x ∈ U} ∈ Power B. The proposition is identified by the following information:

Pure Prop Id: 6799e88e9e80af6a6d035ad42a185bfe8562bd1a6140b5c7d8848156518b59f1
Pure Prop Address: TMbqTvjYLewp4gH2vMecz4sa5m85ihte7rH
Theory Prop Id: 43ed06ac181dbddeb1e0701aeafd6b2b48bdf7da7d513e9c557e99bbc7534b05
Theory Prop Address: TMXRAfJAyZSv2buTnY3cuenqyFKCq8KC47T

image_monotone

Theorem 9.17 ∀f : ι → ι.∀UV.U ⊆ V → {f x|x ∈ U} ⊆ {f x|x ∈ V }. The proposition is identified by the following information:

Pure Prop Id: ac6fe8101f40a432ad82f07e1e2665c922b286ef3145bb1445a03166259e236f
Pure Prop Address: TMP3hoEqEfoP9GJwJnrLJ876qGy57htpeFQ
Theory Prop Id: f923c6e98034954cfdec182ed28c5f254354e531c01861ce39d38a6191a03629
Theory Prop Address: TMHKKD8maX4Ek8ApzVty5uJ6NuWSzXkpFib

setminus_In_Power

Theorem 9.18 ∀AU.A \ U ∈ Power A. The proposition is identified by the following information:

Pure Prop Id: 062275a787c47ab2b1e0a2473d1970002748ce7a53f286abe400bbb934df521c
Pure Prop Address: TMS4pSRxLA1Dbq2igDLXVVHw7nhdRCiv3XJ
Theory Prop Id: f503e1c5ae49a7f07828b518f3acefa7e8740424fa3207093249cc6306769e3f
Theory Prop Address: TMPhrNAZaW4G1fJTosqoVHur2BY9wXz7Crc

setminus_antimonotone

Theorem 9.19 ∀AUV.U ⊆ V → A \ V ⊆ A \ U. The proposition is identified by the following information:

Pure Prop Id: 67bd0e5333f721c6c4ebed67eb48fac3aaa4d7c9e31ecd29eebad8127a29bc85
Pure Prop Address: TMKx1CgcNCpQuzLmEaWfsvEuHkFTPgSbL1D
Theory Prop Id: f9b735ec697563f76b4bf8db58fac77a909244fe1f67ee2c99fdd5259caf0064
Theory Prop Address: TMYUDtXjxfZhhgoCHvxoyj1yVnZ2NwE1zoN

SchroederBernstein

Theorem 9.20 ∀AB.∀fg : ι → ι.inj A B f → inj B A g → equip A B. The proposition is identified by the following information:

Pure Prop Id: e247a892fd3a3a3c6729106e07939389d6cba1af2c7dab631e4a1a2ad18cba2b
Pure Prop Address: TMNErsk8ZdnQDMxX11dVnP6drZGGGUKRnT7
Theory Prop Id: 03a2cd1c8a567976a696d45296506a96202d6ef88ed050e5bd0781df51d7b9b4
Theory Prop Address: TMajcAjnmAvQnwV59Qj54uAcCt6LNS6kuqZ

Misc

f_eq_i

Theorem 10.1 ∀f : ι → ι.∀xy.x = y → f x = f y. The proposition is identified by the following information:

Pure Prop Id: b0871b523a7b57f1eb5aaeab0fc7f0a48159bfb735912ea5ae95f631f41659b1
Pure Prop Address: TMPVNZYtUqHTp8RYLtZcxXu3DU3ivfR6rAr
Theory Prop Id: 2a60fd880e3c5cee57da00ac972ed9116312115366d80d946af04bdedf52ba41
Theory Prop Address: TMcdcpfCVRxKR6sYN1s4kVKEHiMfK814Qv1

f_eq_i_i

Theorem 10.2 ∀f : ι → ι → ι.∀xyzw.x = y → z = w → f x z = f y w. The proposition is identified by the following information:

Pure Prop Id: 99d67d79970e96836845a0ba205f2ae8cf38fbbe7c66edb2e3dd47679e886175
Pure Prop Address: TMVyNZC57hm2JCbUEkm54EFRxyjV7uKHLYV
Theory Prop Id: 47e5770eb29a6575bfe0da11c0c722b53eaa9c4f54a6338e54fda222161f1310
Theory Prop Address: TMJfWQ8qLkS2wi6eLaJhtLGmdrxxJvCx3w1

eq_i_tra

Theorem 10.3 ∀xyz.x = y → y = z → x = z. The proposition is identified by the following information:

Pure Prop Id: 193bbfd16b1f36c82abc18861eb9920fd3ebdab3fa714704e5d9ee842a572cc0
Pure Prop Address: TMMRUBbS4C1QBM5KZBX4P26LywnZaH7iKjP
Theory Prop Id: 7ecbc48d1e90e90e475c12e4250d6ed1b153fe65d79663458f1516c1b19ef29e
Theory Prop Address: TMMtqtU6uJfRVq7S9aPMj7up363DpEmoc8j

Definition 10.1 We define nSubq to be λXY.¬Subq X Y of type ι → ι → o identified by the following information:

96 CHAPTER 10. MISC

Pure Object Id: ea71e0a1be5437cf5b03cea5fe0e2b3f947897def4697ae7599a73137f27d0ee
Pure Object Address: TMZtJ57VpuWkqSBd372tXDYbQV1ru9RQiP6
Theory Object Id: 1b32290aed0c0437eee0d631ba16b62c8142cad8d60330e8a2c72a430b7453ae
Theory Object Address: TMUnveDfk5UPe9zq9bRZ9NGMx9j7NA22YVb

Notation. We use ⊈ as an infix operator corresponding to applying term nSubq.

Sing_inv

Theorem 10.4 ∀xY.{x} = Y → x ∈ Y ∧ ∀y ∈ Y.y = x. The proposition is identified by the following information:

Pure Prop Id: 57ae4c95e879681858c5ff28d750d4d70e50b072a6d15b2feb91ea9fcef02da8
Pure Prop Address: TMGqotQdzdFCuZC8q4vqVRkFBMW4qjArQ3h
Theory Prop Id: 199c6ccf6b8a2d5af1b102b165811324c6387df31287b0a86242c6611b629fed
Theory Prop Address: TMMav2V1grSCKmckxQqmeTmvNxWh6DUx8uL

TransSet_In_ordsucc_Subq

Theorem 10.5 ∀xy.TransSet y → x ∈ ordsucc y → x ⊆ y. The proposition is identified by the following information:

Pure Prop Id: f80c6ef5328d1aebbabe1261cb2a17127aad514383c2c09bbf3354725a9afe82
Pure Prop Address: TMPqbNun9yFToi13JVKcoYQrF59XATjGUYz
Theory Prop Id: 34199340a5562195ace40af331c0a1da26d29faac3847d7b38f9e829ab506bd5
Theory Prop Address: TMH3pW2NLQvrgTnYbQVg1dcwdjLcHFcyHCU

inv_Repl_eq

Theorem 10.6 ∀X.∀fg : ι → ι.(∀x ∈ X.f (g x) = x) → {f y|y ∈ {g x|x ∈ X}} = X. The proposition is identified by the following information:

Pure Prop Id: de1759e07c1622dd0cf2ac0a92f57c8d0d379b3d4ee285440b6de212666d9eae
Pure Prop Address: TMSStFLLm81AFEdPaJMUoYdRHkYa5ucHRD2
Theory Prop Id: 5a8fc2fd0b67b701a32a2d1b39179d4975ca830fffb1452d564b9af4e10a99b9
Theory Prop Address: TMTeQxUhBU7C2YeyWfVA7p37xMSPpxyofrK

invol_Repl_eq

Theorem 10.7 ∀X.∀f : ι → ι.(∀x ∈ X.f (f x) = x) → {f y|y ∈ {f x|x ∈ X}} = X. The proposition is identified by the following information:

Pure Prop Id: 87b0604e91d11a8b4c06ac87cda6148fe5e60301aa91d79609febf82042a8ce3
Pure Prop Address: TMQznwJJoHjcdkGoE3rvi8W9R6zhQ68ZwCc
Theory Prop Id: 393cc6aeb2c74236699aca97e243d3db17d6149941639ffde4f7d58dc26b2e6c
Theory Prop Address: TMW5YcT8N18dkwdFihvpKYK9P8scfdxJAhs

Eps_i_set_R

Theorem 10.8

∀X : ι.∀P : ι → o.∀x ∈ X.P x → Eps_i (λx.x ∈ X ∧ P x) ∈ X

∧P (Eps_i (λx.x ∈ X ∧ P x)).

The proposition is identified by the following information:

Pure Prop Id: b4f328ffb79aabb0df4389dfe10d26a1a90d7ed192a2b7e062aae20ce483c959
Pure Prop Address: TMFghbYQCmAiS83QkiJceQ1ocqagGS8fdof
Theory Prop Id: a78d0d896dcf99f0838ebc9fbab4b2d7723c3dc6238c649009cf14a169e85421
Theory Prop Address: TMF6oMG8oYpvhQhrGCFXfDFrJYvD2UUnWF3

exandE_i

Theorem 10.9 ∀P Q : ι → o.(∃x.P x ∧ Q x) → ∀r : o.(∀x.P x → Q x → r) → r. The proposition is identified by the following information:

Pure Prop Id: 083f822f238e73865a8e53f1b501a3274298103440996953b5dbe5b9f1c64a49
Pure Prop Address: TMcSPtKUzzFxn3xkTT3qXkbEzgxdcFKD64W
Theory Prop Id: b1b26e0e1db0dccbd32126585f43f1f764c8cd693eeb9898836853ddea6300a1
Theory Prop Address: TMYHN9RveUaKq4szKucPeZQCRGkDxZMkhgC

exandE_ii

Theorem 10.10

\begin{matrix}∀P Q : (ι → ι) → o.(∃x : ι → ι.P x ∧ Q x) \\ → ∀p : o.(∀x : ι → ι.P x → Q x → p) → p. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: c3e995324746e718ab713bbf6a225f342b11f85e44af87041c25e67c26eea74b
Pure Prop Address: TMRys3KSWGA2LC5vfb7h5HsWi6UReHYFA4A
Theory Prop Id: ed9d65f783165c1844de5ef6e565738ab285fb760e976e5b2a4a0aaea6db676a
Theory Prop Address: TMJjeW58a6DARdGUxoyLGC7KDSF7e5NyVkz

exandE_iii

Theorem 10.11

\begin{matrix}∀P Q : (ι → ι → ι) → o.(∃x : ι → ι → ι.P x ∧ Q x) \\ → ∀p : o.(∀x : ι → ι → ι.P x → Q x → p) → p. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 028a8f7f6f33f8d9071fed86dccba76e2db66b07c2f6b7330daccd1792bb2382
Pure Prop Address: TMNuTg4eH32x4XhwU5Q9dAJWP8zDQ3b8ZXC
Theory Prop Id: 80b3b0d1dac86077631ca90e19efac3b726c7697bfe6eeb63b5d74a056a3d3ee
Theory Prop Address: TMKrUpgWDX1Lip64jWzHGPhKdYPQ9L4XevF

98 CHAPTER 10. MISC

exandE_iiii

Theorem 10.12

\begin{matrix}∀P Q : (ι → ι → ι → ι) → o.(∃x : ι → ι → ι → ι.P x ∧ Q x) \\ → ∀p : o.(∀x : ι → ι → ι → ι.P x → Q x → p) → p. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 7807d4121dc1d8d66fac47e62c634706453ae9f2acfb88ce1adf46eedfce6b57
Pure Prop Address: TMPCN5CQwhpCx9SEJcLXFz7VTi2HpDCvNie
Theory Prop Id: d1536a49224a1dfb03981a17133a78f7a7384d730d0fea9ee123ba6b7a9eee92
Theory Prop Address: TMNR9cr6cmoDmVuobFG1uP3pM5DqVkVGcfe

exandE_iio

Theorem 10.13

\begin{matrix}∀P Q : (ι → ι → o) → o.(∃x : ι → ι → o.P x ∧ Q x) \\ → ∀p : o.(∀x : ι → ι → o.P x → Q x → p) → p. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 86782c60904be2e5e01ea093b9564a62c5d7b5c5248e75598e84cda4eaff6db1
Pure Prop Address: TMU81hY5npesCwKcB8HT2xq4ju8aURKMfbk
Theory Prop Id: 9e40e24a95644e50b58d83354186c23bf455543eef7af35f3dcfdf02d9c6ef87
Theory Prop Address: TMZrs5GwEKYE1pY5Ctt2Fz9smjaQEZ5RQyx

exandE_iiio

Theorem 10.14

\begin{matrix}∀P Q : (ι → ι → ι → o) → o.(∃x : ι → ι → ι → o.P x ∧ Q x) \\ → ∀p : o.(∀x : ι → ι → ι → o.P x → Q x → p) → p. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: bd6e2d3ff943629d1de39214b2120452cec619ef78043f7c33b48dfd70ae31e3
Pure Prop Address: TMLJQJfu5Yt7gyzMd54gxCSHZYxdRbKTTPQ
Theory Prop Id: ad03be44e266e20551e0c398f53a8d6eb2497ca6b16f6b686368373774c0e421
Theory Prop Address: TMQyqYULvp8WJD8eRf3khDrng9qgVuiu34m

Description and If-then-else

11.1 Descr_ii

Let P : (ι → ι) → o be given.

Definition 11.1 Descr_ii is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 3bae39e9880bbf5d70538d82bbb05caf44c2c11484d80d06dee0589d6f75178c
Pure Object Address: TMYQgJAnkDBZVfPWVEikmdFCVufuxzpsMuR
Theory Object Id: 3a4ad2bbdc8560996c770f069c5b0861aa6f677de8c35f7e6825c37a062870ea
Theory Object Address: TMPXygXB3Vatp7ehxXW1SzR7YDxpTfC383X

Assume the following.

∃f : ι → ι.P f  (11.1)

Assume the following.

∀fg : ι → ι.P f → P g → f = g  (11.2)

Descr_ii_prop

Theorem 11.1 P Descr_ii. The proposition is identified by the following information:

Pure Prop Id: 45bfc0f7e5a41baec35381c982dec98e3138394009ea6479b4734e2b51ee3015
Pure Prop Address: TMRUGkPteutefnmzwHhFudjsxFRU5QyH8LU
Theory Prop Id: e705d1e90646b5936b4d3f399b1ba2f451f5e0bfaa6d4ce06bde1703c2ac6afb
Theory Prop Address: TMdRU9JrS4rFdHpHAP14V6T8N7yZhtAKeUt

100 CHAPTER 11. DESCRIPTION AND IF-THEN-ELSE

11.2 Descr_iii

Let P : (ι → ι → ι) → o be given. Definition 11.2 Descr_iii is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: ca5fc17a582fdd4350456951ccbb37275637ba6c06694213083ed9434fe3d545
Pure Object Address: TMXQVeWJ4sPR355qf1XJWKLuqBobxgZ1f4j
Theory Object Id: 877e40e99321711a3798c510f4a5784236e411ebab77a7914b276da43ccba50c
Theory Object Address: TMUoa925kPFT8WA5VCxeG6JDVutAYyuK4A6

Assume the following.

∃f : ι → ι → ι.P f  (11.3)

Assume the following.

∀fg : ι → ι → ι.P f → P g → f = g  (11.4)

Descr_iii_prop

Theorem 11.2 P Descr_iii. The proposition is identified by the following information:

Pure Prop Id: 261c3dac795dc9d12a850c45e12570f947cf3e73e7aac1d8b97f4fd8dfcaab74
Pure Prop Address: TMJbcddb8cyABNJEUp19YzaZ62uVYxZcxzw
Theory Prop Id: 1604215f70483ff4639dd10934f543d9f25d48553a1e9dbbeae40186c6cf6c06
Theory Prop Address: TMLfZU6SKM2VC18naP9r39PVsCucjtdzWB9

11.3 Descr_iio

Let P : (ι → ι → o) → o be given. Definition 11.3 Descr_iio is the opaque object of type ι → ι → o identified by the following information:

Pure Object Id: e8e5113bb5208434f24bf352985586094222b59a435d2d632e542c768fb9c029
Pure Object Address: TMc7TWyjFWcXBop91LuVowEJbeVve25nK5s
Theory Object Id: 0423e66d8558444d27eb1068b080fe14267e388a7434254d29d1c826ce64ac94
Theory Object Address: TMJ6ajRQrfdbCkA3bH4AFazLF71vXbSokSz

Assume the following.

∃f : ι → ι → o.P f  (11.5)

Assume the following.

∀fg : ι → ι → o.P f → P g → f = g  (11.6)

11.4. DESCR_VO1 101

Descr_iio_prop

Theorem 11.3 P Descr_iio. The proposition is identified by the following information:

Pure Prop Id: 5dd3c7f22282c081da40fb112d8f2d8ffa44620b63f3136e2b179ca8f521ddc5
Pure Prop Address: TMVeGSPy1KHq36b4uHCMZWc5ghcexfzeT12
Theory Prop Id: 44dcac73e42284ed3d29fe0e058e46dcdf185b04829a0f7b5929c19617f0f40b
Theory Prop Address: TMK1Bc1JaKqvjwRNdnxP1qds6eXGi9r8UaD

11.4 Descr_Vo1

From now on we write V o (n + 1) to mean the type V o n → o where V o 0 is ι.

Let P : V o 1 → o be given.

Definition 11.4 Descr_Vo1 is the opaque object of type V o 1 identified by the following information:

Pure Object Id: 615c0ac7fca2b62596ed34285f29a77b39225d1deed75d43b7ae87c33ba37083
Pure Object Address: TMFQ3dBkT4GNkBxqPW23PKfo5eca5LbGLnM
Theory Object Id: a4ecddac25cdabd614bf297524a2acb05724493b89ab7e7e636cf452a8887b01
Theory Object Address: TMS1y5bGeSZGkV7sivczRgNAQUcrNTaGYAX

Assume the following.

∃f : V o 1.P f  (11.7)

Assume the following.

∀fg : V o 1.P f → P g → f = g  (11.8)

Descr_Vo1_prop

Theorem 11.4 P Descr_Vo1. The proposition is identified by the following information:

Pure Prop Id: f550ef6a927da855d9705aafeb7607fe3931aa8ec0f5f675da6011e035f22458
Pure Prop Address: TMMwzBC9pSvmYAJivGEMP4qt8Mt7WyJqduS
Theory Prop Id: 2213e2f6bf896b176ce81e3f28c06d4eda10bd33020cb296513b4b938fcd9056
Theory Prop Address: TMLLpH2SQbA7WJA4grh8eUgfwMWFcLKc8kH

102 CHAPTER 11. DESCRIPTION AND IF-THEN-ELSE

11.5 Descr_Vo2

Let P : V o 2 → o be given.

Definition 11.5 Descr_Vo2 is the opaque object of type V o 2 identified by the following information:

Pure Object Id: a64b5b4306387d52672cb5cdbc1cb423709703e6c04fdd94fa6ffca008f7e1ab
Pure Object Address: TMFZfGbPgS43AdHK51F7w5aNLJzHmKsdsJN
Theory Object Id: 546d66811c244b38780425746c3230f23c0a0c78c5d74a314c6ec8eca20e5571
Theory Object Address: TMQQUC9cTGz9T8416CfsGFpzXHbvVhpxAu2

Assume the following.

∃f : V o 2.P f  (11.9)

Assume the following.

∀fg : V o 2.P f → P g → f = g  (11.10)

Descr_Vo2_prop

Theorem 11.5 P Descr_Vo2. The proposition is identified by the following information:

Pure Prop Id: 95fc538f5d5098f54c64e376e95b1284227552f9e403c9a68ca16035b2a69c37
Pure Prop Address: TMLM1fK9jLGpTqnThj3QLyRf4RLDCjDXKzv
Theory Prop Id: 763c84ad5f1253ac14ad30d1f2765feaf5cc0cdc449dab033ce038bcc35d452c
Theory Prop Address: TMaBE8ZNQAt1zVAJ1G6cs2yjF99pYEBGHwz

11.6 If_ii

Let p : o be given. Let f, g : ι → ι be given.

Definition 11.6 If_ii is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 17057f3db7be61b4e6bd237e7b37125096af29c45cb784bb9cc29b1d52333779
Pure Object Address: TMJ5CGNSX9XjHHwnu1BSQsVHCKzSFJcwu4b
Theory Object Id: e290152395c5a355ecc6c5012e4cd4507f1895df9d71192a4e860e914c8db43e
Theory Object Address: TMPHWP247Vz8BvmmXSfTeVHxvDsaU8cpPU8

If_ii_1

Theorem 11.6 p → If_ii = f. The proposition is identified by the following information:

11.7. IF_III 103

Pure Prop Id: 31521ec60d1fb026a6fc18736822f20291fbb3d33597bb3efce99a03f03e11d9
Pure Prop Address: TMKosiG2dup6SxL5QSDUpucfR51n9gCyDgV
Theory Prop Id: 342a1f3e82c1abf29e708798f7feb58587435d6e872e0fad569a7aa4aff40c55
Theory Prop Address: TMc5keGi4Y146iSqaxLiaYsAZYq1XuEVqNB

If_ii_0

Theorem 11.7 ¬p → If_ii = g. The proposition is identified by the following information:

Pure Prop Id: 6acab05b8c4b66e4263e5b31aaa005d9f082668a8bbace0e6d83fd1b4f2a4e7c
Pure Prop Address: TMNxHhMohQMfeSQ46TzksCAdPGGHded4fQn
Theory Prop Id: 36418b196a3dc55851db392b475c08d495d7976847a1bfe1ffa0839c921a9904
Theory Prop Address: TMJAFTnXbqvZwsoNakQckwUkzUMcWByPVDq

11.7 If_iii

Let p : o be given. Let f, g : ι → ι → ι be given.

Definition 11.7 If_iii is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 3314762dce5a2e94b7e9e468173b047dd4a9fac6ee2c5f553c6ea15e9c8b7542
Pure Object Address: TMafEMcrDN1G9aE3ciBtqxzAhjPGbGo9wGT
Theory Object Id: dd7e2b57dd8bff886e536f76317306bce1d1c82c8d0c4990736c5435d5194472
Theory Object Address: TMXbifVhYnu2zzxcRpGBEz7yE8Zgdxct2bF

If_iii_1

Theorem 11.8 p → If_iii = f. The proposition is identified by the following information:

Pure Prop Id: 2d62ee998f0e93e8531c4fa56ae24814f441c745adbbf870b6533f8348e4105d
Pure Prop Address: TMXoAktu9YHuqADS1FQhmqPQo88e8Wfe9Tc
Theory Prop Id: 48bc10484ec823e29188216c5145c832ea01049091770c743d54872c69c3738d
Theory Prop Address: TMP1NQtm8tugHGMUGtnKEr16wjn5P3HapKj

If_iii_0

Theorem 11.9 ¬p → If_iii = g. The proposition is identified by the following information:

Pure Prop Id: 9b77d90fae20e665d930e198836d49c2f016c9359f15a9036520651d80559a50
Pure Prop Address: TMMxCCtSQ76TqwLa7cLQxLYRL1uvAhxf6G5
Theory Prop Id: d11201a0fa74b1965c04633e2fbf7c186e42c0fbf53635dd329a7bb860c6b930
Theory Prop Address: TMG9gLyxAgRsyCSfKKidGTT9tPyeptZY75x

104 CHAPTER 11. DESCRIPTION AND IF-THEN-ELSE

11.8 If_Vo1

Let p : o be given. Let f, g : V o 1 be given.

Definition 11.8 If_Vo1 is the opaque object of type V o 1 identified by the following information:

Pure Object Id: 2bb1d80de996e76ceb61fc1636c53ea4dc6f7ce534bd5caee16a3ba4c8794a58
Pure Object Address: TMJRubZpJkuKGf6tLSqj98TBUniTgSkcx2k
Theory Object Id: c5705984e5c61b59990fd2a143acd737bb2dc7c7db647a62e398dac074967687
Theory Object Address: TMaRPXSn2Fjb29o31JvamGmuMJnp68JPVyZ

If_Vo1_1

Theorem 11.10 p → If_Vo1 = f. The proposition is identified by the following information:

Pure Prop Id: bb1012256bb9266025d71663756a5ebfd5e16baf8e93a915bfdd7fcb42d706f3
Pure Prop Address: TMXYuJm2QUHmNybRpkXiJxzCLG25czuz5LQ
Theory Prop Id: 62b2b86d0737e9e18346982d5767e5ef03ab4118b70f45a9ff56d1c9e5216feb
Theory Prop Address: TMFtdqVK8VYTrQ6cgnnamxbg6KeHhpZgPcd

If_Vo1_0

Theorem 11.11 ¬p → If_Vo1 = g. The proposition is identified by the following information:

Pure Prop Id: 59bda37f5c7398ead6c4b3243129fbad85f4f9f3636487f1c51cce4e0e6c8cfd
Pure Prop Address: TMWqGQPgK8uAv1B3geaEm1MSYAnLtEgFcb9
Theory Prop Id: 78529c61edfcd02459bc4ec507c773553a628c9c32db5afd8b9669bdeb5624be
Theory Prop Address: TMFdPp5SLYgJtZvzVSfP9YVDKe598dst2rw

11.9 If_iio

Let p : o be given. Let f, g : ι → ι → o be given.

Definition 11.9 If_iio is the opaque object of type ι → ι → o identified by the following information:

Pure Object Id: bf2fb7b3431387bbd1ede0aa0b590233207130df523e71e36aaebd675479e880
Pure Object Address: TMSVGinFUSHBKXoj7UW42oxwqfA45nW9uYC
Theory Object Id: f0254ba33a025d14997148eaf8b6489b75cbcd0e552813782f2fee4dac52a1ad
Theory Object Address: TMVgQ7ZnQFc9zUqLQEygvRvocriNyw8Urgz

If_iio_1

Theorem 11.12 p → If_iio = f. The proposition is identified by the following information:

11.10. IF_VO2 105

Pure Prop Id: 6e5b2f9d60ee2b95f684c6c89fac222c87c115edd65ea7faa36260e77030c34b
Pure Prop Address: TMbdPHYFQQVTEPDFuKsr2RTPtLJWVQCkgmo
Theory Prop Id: 5c40c9c2a99e6214c0a2e1d4e4d8ffe6b45d1a1be5958c0a0e817a962bcb2e40
Theory Prop Address: TMYBc7MrgDRDwERk16x3XtsYSbpzRT4JfeS

If_iio_0

Theorem 11.13 ¬p → If_iio = g. The proposition is identified by the following information:

Pure Prop Id: 9eab66913f2b7839396a2642931e0c4f2b5b96d45aa947c433210d1358ad7ce9
Pure Prop Address: TMLjBZs3dbH9f9eUHYnJdu3t2Pqbht8aDBr
Theory Prop Id: 09748b1d13dc8a2265025a9ce3f481495c8a7bccd4d956b5d4fc36dd28dfbd2b
Theory Prop Address: TMbk4ju7Bfvef6aQhpURVB4h7ERGCJXtpm5

11.10 If_Vo2

Let p : o be given. Let f, g : V o 2 be given.

Definition 11.10 If_Vo2 is the opaque object of type V o 2 identified by the following information:

Pure Object Id: 6cf28b2480e4fa77008de59ed21788efe58b7d6926c3a8b72ec889b0c27b2f2e
Pure Object Address: TMH4op1eDosdCj4CskKLdVdQMoEMPyaKmKz
Theory Object Id: f6760c1a5f26147ef91098a2d633b2f19a2f38b9c1c811244491274cf056b285
Theory Object Address: TMZuq1Mx5McSmUfVdPQ43iubWhx4f9CctV4

If_Vo2_1

Theorem 11.14 p → If_Vo2 = f. The proposition is identified by the following information:

Pure Prop Id: 0f655bcb4e8ea858e1e47b901188715bcb9443ff29ca227273b5eeab5ccce4fa
Pure Prop Address: TMLf5qJmSHrs6Umik46niap6g6Puq8u38PZ
Theory Prop Id: 530c03e594647527928784ed02365579d201bd63d29f76cf2679ec77f9c3a348
Theory Prop Address: TMYAJeyWgmE86YRTxka8iZ3VDvWDz4KFVak

If_Vo2_0

Theorem 11.15 ¬p → If_Vo2 = g. The proposition is identified by the following information:

Pure Prop Id: ba20c1835570284aac2a734187c79818c51b3e6b4ef50125ec114563e95c56a3
Pure Prop Address: TMFhCsW5TbyvNPeCPg5q8UZQCNrh7ba7mc7
Theory Prop Id: 49a8cfcdec6a427975b6c3b9ac4c13c6b5dbfa0624fe913a360d44ecd3993241
Theory Prop Address: TMRyczcoeSo63tGt7RMgBM4C9d1EHxcvBZi

106 CHAPTER 11. DESCRIPTION AND IF-THEN-ELSE

Recursion on Sets

12.1 EpsilonRec_i

Let F : ι → (ι → ι) → ι be given. Definition 12.1 In_rec_i is the opaque object of type ι → ι identified by the following information:

Pure Object Id: fac413e747a57408ad38b3855d3cde8673f86206e95ccdadff2b5babaf0c219e
Pure Object Address: TMNdra6FF88BaWZhLQWoMVGDVkGgU5xZ1Px
Theory Object Id: 93aac4fb93be5cc13a42b363fb830917e096fe0d5b269d4cae1a2675fc6fe442
Theory Object Address: TMKBpYHnteFb5wpr6cJ5KfTg3HrzRoSBByu

Assume the following.

∀X : ι.∀gh : ι → ι.(∀x ∈ X.g x = h x) → F X g = F X h  (12.1)

In_rec_i_eq

Theorem 12.1 ∀X : ι.In_rec_i X = F X In_rec_i. The proposition is identified by the following information:

Pure Prop Id: 1106536fa01b17bf5ad01765e95f74fb2d59af6f6540159757750d84c1f699ae
Pure Prop Address: TMS4niVmsvQx9jqComxMJ15nwcNzZqmdhRN
Theory Prop Id: e8653b0bcd75b2d7d06c38f2ac4c66b817c99e8dc96d2647bd2daaee681a632e
Theory Prop Address: TMbKwV8D2cMfE3R9MZ8SdtrUWVk2FfWMq91

12.2 EpsilonRec_ii

Let F : ι → (ι → (ι → ι)) → (ι → ι) be given.

108 CHAPTER 12. RECURSION ON SETS

Definition 12.2 In_rec_ii is the opaque object of type ι → (ι → ι) identified by the following information:

Pure Object Id: f3c9abbc5074c0d68e744893a170de526469426a5e95400ae7fc81f74f412f7e
Pure Object Address: TMKaBL68miPaU3awFWxavJ9vpffruAuJG6m
Theory Object Id: 3841bd9652360e3358d466fb36d771d47d765b56ce11b4a446e1c8ee7792319a
Theory Object Address: TMVfeCfjKozKnhDvd1g6ayuJytZPUtNbSQT

Assume the following.

∀X : ι.∀gh : ι → (ι → ι).(∀x ∈ X.g x = h x) → F X g = F X h (12.2)

In_rec_ii_eq

Theorem 12.2 ∀X : ι.In_rec_ii X = F X In_rec_ii. The proposition is identified by the following information:

Pure Prop Id: f7aff626beae77fbb87e16d4b62b6dbf1d4e1fc8b4151ab95e33ff0a7eece1fd
Pure Prop Address: TMc1r8kDpq8yTVrxstWVdBTkGu4bAHa2KqZ
Theory Prop Id: 7bb1e907bd4a595013dd5253579a59ded78c5fc86a4aba5c0e6af261615e64e4
Theory Prop Address: TMF2cwdAE8DvbpFHHjfMdf7jfwBbgauH5K1

12.3 EpsilonRec_iii

Let F : ι → (ι → (ι → ι → ι)) → (ι → ι → ι) be given.

Definition 12.3 In_rec_iii is the opaque object of type ι → (ι → ι → ι) identified by the following information:

Pure Object Id: 9b3a85b85e8269209d0ca8bf18ef658e56f967161bf5b7da5e193d24d345dd06
Pure Object Address: TMGS3HxzKZpy47SjRNptZvfW3hGeTR87kw8
Theory Object Id: 651077708696fee3d23e186d6ab9c1805fc9c46eb287174c7c7726c14df5cd25
Theory Object Address: TMNNG5ehc1kjbSr79hs1JoXieVRroM54amr

Assume the following.

∀X : ι.∀gh : ι → (ι → ι → ι).(∀x ∈ X.g x = h x) → F X g = F X h (12.3)

In_rec_iii_eq

Theorem 12.3 ∀X : ι.In_rec_iii X = F X In_rec_iii. The proposition is identified by the following information:

Pure Prop Id: 5f0574ebbeaebb60edd0cf70b9e3649d00e8391b36dc79f8abc40b36f0236f5d
Pure Prop Address: TMLw55kNTBuMX9fEZ9ucafMGa8X9q9Vg5AS
Theory Prop Id: a0709d7e1ee7e89fa5e406d74c01159f7bc4a43f2d6eb3aecf054a5c987acced
Theory Prop Address: TMe1LiqcZFAP9bdmQLTeizcNBRHbFYkJKz8

12.4. EPSILONREC_IIO 109

12.4 EpsilonRec_iio

Let F : ι → (ι → (ι → ι → o)) → (ι → ι → o) be given.

Definition 12.4 In_rec_iio is the opaque object of type ι → (ι → ι → o) identified by the following information:

Pure Object Id: 8465061e06db87ff5ec9bf4bd2245a29d557f6b265478036bee39419282a5e28
Pure Object Address: TMKaaHZBwQLEGRN8PYDKcEfwZubgSevskVz
Theory Object Id: ebf7a42862e5ae14b0b042ad0dd80452de3ad1c5e8e168f4b1ab3d2fb945f61e
Theory Object Address: TMLjGqbKQQthLSzLhyZ5N8iPmocUeVoLr3N

Assume the following.

∀X : ι.∀gh : ι → (ι → ι → o).(∀x ∈ X.g x = h x) → F X g = F X h (12.4)

In_rec_iio_eq

Theorem 12.4 ∀X : ι.In_rec_iio X = F X In_rec_iio. The proposition is identified by the following information:

Pure Prop Id: a6bb8bb1ff02720df380edef1d6e1599715ed907c141a8134c85b26a0a245f9b
Pure Prop Address: TMG9Ak4owSMXtiyYkh2BpkesWY7gR3RoFCu
Theory Prop Id: bff321513dbb1d418dd4c5d50b267215cb2c98cc7e1db5de1c7148e57ce1044d
Theory Prop Address: TMZyTR9yto8MPdDZTwmCg5y3krsPUWxxW6u

12.5 EpsilonRec_Vo1

Let F : ι → (ι → V o 1) → V o 1 be given.

Definition 12.5 In_rec_Vo1 is the opaque object of type ι → V o 1 identified by the following information:

Pure Object Id: e9c5f22f769cd18d0d29090a943f66f6006f0d132fafe90f542ee2ee8a3f7b59
Pure Object Address: TMF7xrRrqMjz5Lk6579gDvuGzGVm7S7Xbm3
Theory Object Id: b1f6ea77e17286f0717ab97435a1b23d6c4cd8a5a8d1718c6bbe022c543fe9b5
Theory Object Address: TMRg5yF9XnKxQKRqe5XeJ2UENC7Gurp2pax

Assume the following.

∀X : ι.∀gh : ι → V o 1.(∀x ∈ X.g x = h x) → F X g = F X h (12.5)

In_rec_Vo1_eq

Theorem 12.5 ∀X : ι.In_rec_Vo1 X = F X In_rec_Vo1. The proposition is identified by the following information:

Pure Prop Id: fa315696609285c33f1dd6bb13f004c898139dd9c5a7f4d65239606647d55821
Pure Prop Address: TMKzEyM3ak62ud6SUpQnDC1eNXkY9NiztW3
Theory Prop Id: 69f9607db0c116ec22508e0bbe3620d5741374de361719968ce5a7a509dfca8c
Theory Prop Address: TMUP4D2yi5VENpkASUcnEs47UeEw4yGYhCw

110 CHAPTER 12. RECURSION ON SETS

12.6 EpsilonRec_Vo2

Let F : ι → (ι → V o 2) → V o 2 be given.

Definition 12.6 In_rec_Vo2 is the opaque object of type ι → V o 2 identified by the following information:

Pure Object Id: 8bc8d37461c7653ced731399d140c0d164fb9231e77b2824088e534889c31596
Pure Object Address: TMJV2odr76mGWeyBRTv7gyjmWSSMGzGy4vj
Theory Object Id: b67f4be4f70efe3a08c8984e82e3f0ace6bc31b42132285fbf626d533ad4a46e
Theory Object Address: TMMQWnTTmxXXX8n9NmHW2iCY3E2PWon4Gh3

Assume the following.

∀X : ι.∀gh : ι → V o 2.(∀x ∈ X.g x = h x) → F X g = F X h (12.6)

In_rec_Vo2_eq

Theorem 12.6 ∀X : ι.In_rec_Vo2 X = F X In_rec_Vo2. The proposition is identified by the following information:

Pure Prop Id: 1386f49a7aeb74994677ac4425fbfe5622be81d9fd2ba5678dc74f1e60cbc108
Pure Prop Address: TMZt6xCkE5e2YHuk9kBiQSM7r1AC1Cp8HNJ
Theory Prop Id: 573f9327980a8c0f52e474888757973c754fdd9f151a1b366673a160168ba0cb
Theory Prop Address: TMGqV65crtJCaDQBUgBqHQbtxYEWxrkKgh3

If-then-else, Description and Recursion again

13.1 If_Vo3

Let p : o be given. Let f, g : V o 3 be given.

Definition 13.1 If_Vo3 is the opaque object of type V o 3 identified by the following information:

Pure Object Id: 73dd2d0fb9a3283dfd7b1d719404da0bf605e7b4c7b714a2b4e2cb1a38d98c6f
Pure Object Address: TMcasq19pTUcLufiu26HYpBfxPmeYk6E8mT
Theory Object Id: 333c92c590bdda1d300a193515c5804607be8cb804a38180a59610b36281676f
Theory Object Address: TMKVqTVDwSwbDy1BLiFYKZbD6QhkcDSVJhc

If_Vo3_1

Theorem 13.1 p → If_Vo3 = f. The proposition is identified by the following information:

Pure Prop Id: 9cf80c77cf895a06b2027489d75da25a65f13d09dbb89106276596f1621e9190
Pure Prop Address: TMJ2NGWhftpmGTYwAy7YTG5gSV15BG4yDC4
Theory Prop Id: da37c739b9dd4956b241bec65be35f2c5b0640a3526959d750cfa4bcd78d0621
Theory Prop Address: TMNrJsMGCaKdKHP7L2e4Nm4EZXFs2uCAAw3

If_Vo3_0

Theorem 13.2 ¬p → If_Vo3 = g. The proposition is identified by the following information:

Pure Prop Id: 26938a02bd0536ce534ff744512cc88eb86cd2ee1226183b145cae855a142032
Pure Prop Address: TMUNMynXzuArTTrK7p5jFXqihZJb43NBe7x
Theory Prop Id: df49a9a1ae039c4a570658bf925b0179e4b0a63bbabd6021b74d66c7c709f80f
Theory Prop Address: TMb4qEYPVKmSarAqVUz7NwbCdaKVAkyNYPm

112CHAPTER 13. IF-THEN-ELSE, DESCRIPTION AND RECURSION AGAIN

13.2 Descr_Vo3

Let P : V o 3 → o be given.

Definition 13.2 Descr_Vo3 is the opaque object of type V o 3 identified by the following information:

Pure Object Id: f25ee4a03f8810e3e5a11b184db6c8f282acaa7ef4bfd93c4b2de131431b977c
Pure Object Address: TMQ2iYFqmoYhwP7TbZqwfTHcnxkgx715jsM
Theory Object Id: d479f5126402f52d9a58899c5731677dad9533674526ab2b066def43094eec28
Theory Object Address: TMSADgxCz6JBunRMJpa7zmWr51aLHrZXZwg

Assume the following.

∃f : V o 3.P f  (13.1)

Assume the following.

∀fg : V o 3.P f → P g → f = g  (13.2)

Descr_Vo3_prop

Theorem 13.3 P Descr_Vo3. The proposition is identified by the following information:

Pure Prop Id: 20174ae5a109b906a46115d971515a729f9249f4f20fa5da310e2c4cdeab6249
Pure Prop Address: TMLwRSBien4w3CzFgCJ48yAfqXd7PCssRGE
Theory Prop Id: 23c590cb67150ebe43754899c3f2b5d5f0927084676267173928ecff860e70b5
Theory Prop Address: TMWShaCPNhN2XvW2e4gefY2XgoDY4dWexFY

13.3 EpsilonRec_Vo3

Let F : ι → (ι → V o 3) → V o 3 be given.

Definition 13.3 In_rec_Vo3 is the opaque object of type ι → V o 3 identified by the following information:

Pure Object Id: 80f7da89cc801b8279f42f9e1ed519f72d50d76e88aba5efdb67c4ae1e59aee0
Pure Object Address: TMTrS1bWLxXMW5qTfHXLxP8rqsqDfcSj5it
Theory Object Id: d1066f5fb07b44eb37af262f1b8cbfa415ea64116ce40aa20ad290bbdbba780e
Theory Object Address: TMdJy3eq8ZYfdDZguQUCdxZ6rfJuYZZntCT

Assume the following.

∀X : ι.∀gh : ι → V o 3.(∀x ∈ X.g x = h x) → F X g = F X h (13.3)

In_rec_Vo3_eq

Theorem 13.4 ∀X : ι.In_rec_Vo3 X = F X In_rec_Vo3. The proposition is identified by the following information:

Pure Prop Id: 2306eb8b31f5ee276553420c5e1f5c8c0651e1258c44c87a9c42e96bd3e79503
Pure Prop Address: TMaEn9SrA4LtLzZLVV3UpDHnxpFiFAqzKBz
Theory Prop Id: 4d2789ab957df70c09911021b3cc17f5578b5a09007513d609947d6805b3ca93
Theory Prop Address: TMaSVdip28DZxPPo4ByXDcTwLWpi4sfc89Q

13.4. IF_VO4 113

13.4 If_Vo4

Let p : o be given. Let f, g : V o 4 be given.

Definition 13.4 If_Vo4 is the opaque object of type V o 4 identified by the following information:

Pure Object Id: 1a8f92ceed76bef818d85515ce73c347dd0e2c0bcfd3fdfc1fcaf4ec26faed04
Pure Object Address: TMFG5MpDVE3nFZV3NJZdM6wd42HBKVgpUkN
Theory Object Id: 6f23a6aeaffecd3215c88a1ad6970a5926ecd481bf402e6873321d936dc9a49b
Theory Object Address: TMPj7rvJRkP19dsCJAFArEnBpzMnvjtktEW

If_Vo4_1

Theorem 13.5 p → If_Vo4 = f. The proposition is identified by the following information:

Pure Prop Id: 06ad5c864f232538e5dcbad7b8a51389f2b7630bb8f0576bcc523d1811805afa
Pure Prop Address: TMLaXf8kJ7EdCg7P6GR8txEnTFCZNLrNGCx
Theory Prop Id: 3d1a5d619766f7ca661d7cf070c1c9e481cb4a26a2603723df79b27fbc512fc5
Theory Prop Address: TMTdC39nPta5qDs73eUJZFWgNT3PKvCQPkp

If_Vo4_0

Theorem 13.6 ¬p → If_Vo4 = g. The proposition is identified by the following information:

Pure Prop Id: bab37df6d4aabf30ec549e549b8fff3336e2c7eebd629e65365508a0a69be04b
Pure Prop Address: TMTttbgudyrZKJvKhjQTui2Nr8qLhE3y4SX
Theory Prop Id: 24dc8a58d4de615eeb9b719733472bdc39bddf577984e0488dbf60ad72aa5b3e
Theory Prop Address: TMdiwt98qwWvCbCfyBMTBrpGjo53TgoPrZM

13.5 Descr_Vo4

Let P : V o 4 → o be given.

Definition 13.5 Descr_Vo4 is the opaque object of type V o 4 identified by the following information:

Pure Object Id: 8b81abb8b64cec9ea874d5c4dd619a9733a734933a713ef54ed7e7273510b0c3
Pure Object Address: TMLVDACU3jLYPMh9iPh1uhucMrsTaATPeZN
Theory Object Id: 85f7ca0492b5dac0db0a41ea793b13fcebe7b788606ad42263186b19feb60364
Theory Object Address: TMd1yKi67f99nixn38zFHJA2mC8ypLfDMrV

Assume the following.

∃f : V o 4.P f  (13.4)

Assume the following.

∀fg : V o 4.P f → P g → f = g  (13.5)

114CHAPTER 13. IF-THEN-ELSE, DESCRIPTION AND RECURSION AGAIN

Descr_Vo4_prop

Theorem 13.7 P Descr_Vo4. The proposition is identified by the following information:

Pure Prop Id: a6473b3334fd689271114e94dd8501ed316b996fca678cfd212ee147617f3b63
Pure Prop Address: TMFnuCwCTVMsFiXKonkd3YBQ9HMaKew1LfE
Theory Prop Id: 3e862cb55935309dafceacc6dca361c6a70271794a80e68a76ec04b8d5b78104
Theory Prop Address: TMXUspN6Sx6CYJpANQyhMG8CHovhesMrXwZ

13.6 EpsilonRec_Vo4

Let F : ι → (ι → V o 4) → V o 4 be given.

Definition 13.6 In_rec_Vo4 is the opaque object of type ι → V o 4 identified by the following information:

Pure Object Id: d82c5791815ca8155da516354e8f8024d8b9d43a14ce62e2526e4563ff64e67f
Pure Object Address: TMFrL2oqNRAHoTkwnxvXz88hCQoB8Uoidfq
Theory Object Id: 1e5061b04de4514b3a4c4ea251fd26dd589083b04c1276fec71c858ef714657d
Theory Object Address: TMc9WkccuALwugHXmUhucdGnwNEmZuusP9z

Assume the following.

∀X : ι.∀gh : ι → V o 4.(∀x ∈ X.g x = h x) → F X g = F X h (13.6)

In_rec_Vo4_eq

Theorem 13.8 ∀X : ι.In_rec_Vo4 X = F X In_rec_Vo4. The proposition is identified by the following information:

Pure Prop Id: 222bdbe9dc303a7fe1b2a2e6e60547f51606b31268e865e7f48ccdfc78fab6dc
Pure Prop Address: TMLZ4TCFwAU7GVexuLeFpen6YpZfwwTmUrz
Theory Prop Id: ab3561ec7f79f59905a72739474d784abea229696e9c83807694ef845230da1b
Theory Prop Address: TMTTyAbVG9aCnaLjwPoFSBiNHFM6ZrztqWx

Chapter 14 Predicates and Relations

Definition 14.1 We define bigintersect to be

λ(D : (ι → o) → o)(x : ι).∀P : ι → o.D P → P x

identified by the following information:

Pure Object Id: 615c0ac7fca2b62596ed34285f29a77b39225d1deed75d43b7ae87c33ba37083
Pure Object Address: TMFQ3dBkT4GNkBxqPW23PKfo5eca5LbGLnM
Theory Object Id: a4ecddac25cdabd614bf297524a2acb05724493b89ab7e7e636cf452a8887b01
Theory Object Address: TMS1y5bGeSZGkV7sivczRgNAQUcrNTaGYAX

Definition 14.2 We define reflexive to be λR.∀x : ι.R x x of type

(ι → ι → o) → o

identified by the following information:

Pure Object Id: 6a6dea288859e3f6bb58a32cace37ed96c35bdf6b0275b74982b243122972933
Pure Object Address: TMT7Ad5B4WrSQkBPTK8ixJeatPMtbGf5cdF
Theory Object Id: a8e476299e8d33df6c6811c287e1b9fabc2185a0e461a2c739e7e02c9344377c
Theory Object Address: TMUbRcXexrARoyLrg2fK8SYuYLT1wF4tnjt

Definition 14.3 We define irreflexive to be λR.∀x : ι.¬R x x of type

(ι → ι → o) → o

identified by the following information:

Pure Object Id: e2ea25bb1daaa6f56c9574c322ff417600093f1fea156f5efdcbe4a9b87e9dcf
Pure Object Address: TMYbVtA5sMrdMd5HyhU7dkL61akED8Yzd4T
Theory Object Id: e3b5275d1d8ea65a249223038cf102aa00ce1f82a9a8ca9a4bde7a257092a89e
Theory Object Address: TMFSJ2T4un5uVsKW7djiQAvN1DAwTnNJUiC

116 CHAPTER 14. PREDICATES AND RELATIONS

Definition 14.4 We define symmetric to be λR.∀xy : ι.R x y → R y x of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 309408a91949b0fa15f2c3f34cdd5ff57cd98bb5f49b64e42eb97b4bf1ab623b
Pure Object Address: TMGcxUiVbTw5fsQKQMYB8qsMsgLdrMBxa7j
Theory Object Id: 120a5ebbb3c5e5ef6629806c0a1e8303ae93de2f42212d21a9ae33042777373c
Theory Object Address: TMQbC5GACt2K5MgypgrbgKeZqE3KMVpjpim

Definition 14.5 We define antisymmetric to be

λR.∀xy : ι.R x y → R y x → x = y

of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 9a8a2beac3ecd759aebd5e91d3859dec6be35a41ec07f9aba583fb9c33b40fbe
Pure Object Address: TMVbTEZn5mU4ZX9WkJ7WJ2yxJDVpXU5YAUm
Theory Object Id: 82374c2974eaa40576881231e8c1e031292e0279528cee22769ae90c0d1ca574
Theory Object Address: TMNn7rfiqnuh8jbUTzq7HQitSyHk6xsoUfm

Definition 14.6 We define transitive to be

λR.∀xyz : ι.R x y → R y z → R x z

of type (ι → ι → o) → o identified by the following information:

Pure Object Id: b145249bb4b36907159289e3a9066e31e94dfa5bb943fc63ff6d6ca9abab9e02
Pure Object Address: TMFM3TNU7g1EZuVHc24wspMS7Y75XqC8hvN
Theory Object Id: d9f0d1487b5ca94d78a7fe0eb19e9c8bad4aad24dc543be4879ce610d5bd5c53
Theory Object Address: TMW5XFJc1g5eT257Q4ZaCqtxLhMtDWiqqty

Definition 14.7 We define eqreln to be

λR.reflexive R ∧ symmetric R ∧ transitive R

of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 25d55803878b7ce4c199607568d5250ff00ee63629e243e2a1fd20966a3ee92c
Pure Object Address: TMSuiPhKNmJUrhs9cRrAHLcqoskTttXj1fT
Theory Object Id: c457d55160cd9e8319ea47367df7263cd5e391e71be5fe7a86c761faf2ab6f26
Theory Object Address: TMM8uEsESwkqXBF3DURiTCeRydaN9MdPa2n

Definition 14.8 We define per to be λR.symmetric R ∧ transitive R of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 5754c55c5ad43b5eaeb1fb67971026c82f41fd52267a380d6f2bb8487e4e959b
Pure Object Address: TMJFMWPiZz3oFCEc6qbqK6ewA1Vd1h48BRr
Theory Object Id: 2688d14197e49ace4c9134d31466ccf7c821723681dd3c0a8e19f30828b6e69b
Theory Object Address: TMZ1qY6E5fsXqNMWYcwbbpUKqq24VbUFKZN

Definition 14.9 We define linear to be λR.∀xy : ι.R x y ∨ R y x of type

(ι → ι → o) → o

identified by the following information:

Pure Object Id: 9db35ff1491b528184e71402ab4646334daef44c462a5c04ab2c952037a84f3f
Pure Object Address: TMaYw74DvBqB1nULY5YzEwuZZL4wkaUyyxg
Theory Object Id: 04b95b0ad1550900eb8cea4a13f9a204f7ee4ade8f078352bfea612b8b08bc4b
Theory Object Address: TMYics8vs8zP2a85EMxEbQ7v2qa88dpRPxV

Definition 14.10 We define trichotomous_or to be

λR.∀xy : ι.R x y ∨ x = y ∨ R y x

of type (ι → ι → o) → o identified by the following information:

Pure Object Id: cc2c175bc9d6b3633a1f1084c35556110b83d269ebee07a3df3f71e3af4f8338
Pure Object Address: TMWfJY8TrCuqdBbXjXtyyPgdMaJFFhSYPxa
Theory Object Id: 61a1215cd148d58b01ffff5d1f427041f8684c948585f39446780cb2f09332f5
Theory Object Address: TMShCrFwwqitm33uX4Zfg5LP3nxx3wiWms6

Definition 14.11 We define partialorder to be

λR.reflexive R ∧ antisymmetric R ∧ transitive R

of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 406b89b059127ed670c6985bd5f9a204a2d3bc3bdc624694c06119811f439421
Pure Object Address: TMW2bk6NEbDRdxmx9iWTp8eEaq5Bqs8wPgv
Theory Object Id: c6fcf7ac241b0cf0da35ccb0b15e5793c002ba2e047a608000d69fd9887f713f
Theory Object Address: TMbQ7jLDSWGqbR63ibKa1g9z7w1VxQwMmpi

Definition 14.12 We define totalorder to be λR.partialorder R ∧ linear R of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 0e5ab00c10f017772d0d2cc2208e71a537fa4599590204be721685b72b5d213f
Pure Object Address: TMKx8GCXERU42pBKUXaDWCBbgA3pQMRP8tN
Theory Object Id: f82fe6af52f2eb7dba22be27f833973c1fc595caf2ed9e8d39761e844f892c7f
Theory Object Address: TMdPhbDwqcJQrVhsNYionDKzTcsQAEnNsVD

Definition 14.13 We define strictpartialorder to be

λR.irreflexive R ∧ transitive R

of type (ι → ι → o) → o identified by the following information:

118 CHAPTER 14. PREDICATES AND RELATIONS

Pure Object Id: ee5d8d94a19e756967ad78f3921cd249bb4aefc510ede4b798f8aa02c5087dfa
Pure Object Address: TMKCFjFsuHbvaNUnpivHQWRNoRicsnmAMxK
Theory Object Id: c5200681a8c27017c79f5fdf3f12ab2bccac1fae4bdbf8e1fda11162db118256
Theory Object Address: TMUM3ZcpFdPNXn5zGKLbrqdF1jPboSGWNuB

Definition 14.14 We define stricttotalorder to be

λR.strictpartialorder R ∧ trichotomous_or R

of type (ι → ι → o) → o identified by the following information:

Pure Object Id: 42a3bcd22900658b0ec7c4bbc765649ccf2b1b5641760fc62cf8a6a6220cbc47
Pure Object Address: TMUQBjSjcsVDhSC7wkbaDUwYx5frU3mzZYL
Theory Object Id: fc46b3ff515f15dc7d235be5f2ed71d758b31a1c27552cfb9c6a33f42d1a3a4f
Theory Object Address: TMJjLnVKzQjT4WNmmFgDZ2Wjjd3Y3FEBCBr

per_sym

Theorem 14.1 ∀R : ι → ι → o.per R → symmetric R. The proposition is identified by the following information:

Pure Prop Id: 79f95ef7f3d535d6717f7ddc757df9b5e552d3ce9b4bdfb2e6a7fe0937c98439
Pure Prop Address: TMHJHazkkUfb6aCs84p71oZHrkAA9SrRF6B
Theory Prop Id: 9c47256c1ffdf0d6543ef86bbda771b7500ac846168ef7cd276fefdf6ae54a1f
Theory Prop Address: TMGvYGQrs8TGK53ZwpRhhXKcriX4siG4aWK

per_tra

Theorem 14.2 ∀R : ι → ι → o.per R → transitive R. The proposition is identified by the following information:

Pure Prop Id: 607d155b87eb1945d8bd159974a5bd7062c2dea047295c027068baf6d81c8994
Pure Prop Address: TMPq6w6Ar2UhxZJQGGLkL7M9a8YzYHyjaEw
Theory Prop Id: 8ce50de71d645606be80665db71c58b8813cd0d19d60d8b763ffc406b2efe6e4
Theory Prop Address: TMXqAeANVNhbDDSZf3ThqJh57d5d8akrVuU

per_stra1

Theorem 14.3 ∀R : ι → ι → o.per R → ∀xyz : ι.R y x → R y z → R x z. The proposition is identified by the following information:

Pure Prop Id: 25c6c513d28204a6d67aec12f103a90010140f5c065ee6e81a2a235388669203
Pure Prop Address: TMH8vH9JbJk1oRxfLTaWkXDsdp1tBajpRuU
Theory Prop Id: 7450d4de2a0dc28afe52b779285309d8272e4ecc9d18b29694db6d0e88c2c70e
Theory Prop Address: TMWwhnqMzeuBU4txn9M7KbX32G2MYUsYXWX

per_stra2

Theorem 14.4 ∀R : ι → ι → o.per R → ∀xyz : ι.R x y → R z y → R x z. The proposition is identified by the following information:

Pure Prop Id: 42041dce9407f5e451a83275d4ed3562d50e979b371ac9cfc9ce2bbf65ffb11b
Pure Prop Address: TMczKpKZoBa6ULrfgZpdRBj5NNSBdUXVrmD
Theory Prop Id: c1c6a191a79810f500d658f41a2be9542a0355f4071cdcddc0e1aca07f3db296
Theory Prop Address: TMYhQJwgjHAdPXsFBVaSDYV1rJZ3eh8ERrF

per_stra3

Theorem 14.5 ∀R : ι → ι → o.per R → ∀xyz : ι.R y x → R z y → R x z. The proposition is identified by the following information:

Pure Prop Id: d7b95260b2573ec1f7c5e60368e6752db7fa181ff7cfcb71c4f345b0bb109dd7
Pure Prop Address: TMGmGZb54Qqh3fp8xP84i6ikrSsAZHtQrK2
Theory Prop Id: 21217c9c11e8e6fc8b65cd0e9e3a540900e413d26333d881e5c72b28ed682b59
Theory Prop Address: TMR85C3Y3ZcsxRT7X96TVb5FfvijohyYrhB

per_ref1

Theorem 14.6 ∀R : ι → ι → o.per R → ∀xy : ι.R x y → R x x. The proposition is identified by the following information:

Pure Prop Id: c9ce8823077b6bac6fe963cb5c95ed848be8455934bbdab468e7fd51a1b26027
Pure Prop Address: TMdPMQFgkvEnLZgXHL1Y1PTzuAK9GLNXFGi
Theory Prop Id: 45e4c5e058614a4b8b7d87bd59727e65f215a29fc3061feeb56cde8a17916335
Theory Prop Address: TMTZp74GpF3FnWhQACszYMqVy96d8NVbFXr

per_ref2

Theorem 14.7 ∀R : ι → ι → o.per R → ∀xy : ι.R x y → R y y. The proposition is identified by the following information:

Pure Prop Id: bbc879269d0205d16beaa6eee6fd30afbcdffa8cec1ac7f6b3b8d26e715d2345
Pure Prop Address: TMbvg6t3f8pBbTLcBciLrre2aCKoFw7zfno
Theory Prop Id: 5755e01c047ecd7f625b9e98dc3a6c0651bc16c7c536503e8732f0ecc467f80d
Theory Prop Address: TMLeJkqhNb5gsQPB2bsXFJnDEF5uSEmMFUv

partialorder_strictpartialorder

Theorem 14.8

∀R : ι → ι → o.partialorder R → strictpartialorder (λxy.R x y ∧ x ≠ y). The proposition is identified by the following information:

Pure Prop Id: 6d855f41855d31ad7129a125faa7f0b81aed14cfe4b95537164fbea0352b7169
Pure Prop Address: TMGDbQFv1xTKnLt1axDcGD9skkrk1NDjguy
Theory Prop Id: f2be7026be53a9cc86a061447a9845aea99c3af1fd7b0e0a1be19f93eac40826
Theory Prop Address: TMW6QAQXciNiq23bCcarz6nxhYqvHh8imu8

120 CHAPTER 14. PREDICATES AND RELATIONS Definition 14.15 We define reflclos to be λRxy.R x y ∨ x = y of type

(ι → ι → o) → (ι → ι → o)

identified by the following information:

Pure Object Id: 71bff9a6ef1b13bbb1d24f9665cde6df41325f634491be19e3910bc2e308fa29
Pure Object Address: TMT2htU1EHitc1tMVfJYMWKQyBmwmD6rtWn
Theory Object Id: 231a7bacb9df61407922150b01fd5a8408982ba28cc53a7b1116373732b564e5
Theory Object Address: TMUaCQBAHyUetsLgqr9moZkwAZ6DHmvevzv

reflclos_refl

Theorem 14.9 ∀R : ι → ι → o.reflexive (reflclos R). The proposition is identified by the following information:

Pure Prop Id: 04de27b34b8ee3720731f1aa3b07e08c79a5f5fe35b989e0e31c85df7d76188b
Pure Prop Address: TMRXSNELTTP8r956vacXYEerEnidnHV3YDx
Theory Prop Id: 073a437b4c2e1d6de20d186d3a62ccd4fbe468541fc10b5301ffbd1460aebad2
Theory Prop Address: TMcjRsdk3p8hhg3NaZJTez4aiFcKytNSMzj

reflclos_min

Theorem 14.10 ∀RS : ι → ι → o.R ⊆ S → reflexive S → reflclos R ⊆ S. The proposition is identified by the following information:

Pure Prop Id: 3c9bfea39a96499adf86a25b70a01d86dda56919bef77987d48bd540eeb47105
Pure Prop Address: TMdFsDfTaJ75o8jviDeot8gUFggEbgrSQk9
Theory Prop Id: c6c85f19b2d660c314da672cf7ec92b47ac92205119f6d34f6f8d78bae7eb2aa
Theory Prop Address: TMbsP4tFMHwzMtNmcrDFHkzc6XuNP7FvzeF

strictpartialorder_partialorder_reflclos

Theorem 14.11 ∀R : ι → ι → o.strictpartialorder R → partialorder (reflclos R). The proposition is identified by the following information:

Pure Prop Id: aae20a4db679340a3b3489212833014ae04ff05454150367d01c07193f0b3f6e
Pure Prop Address: TMRH8GAJvU7QCQVeMCvrdQxYzWTt23ioYvb
Theory Prop Id: 8909d872000d95634b270e22a50ab08882a2efd39862ee4dc9372226770395f5
Theory Prop Address: TMMc6BxYynTzGhkvf3wwkqL4XNB328wz9Kc

stricttotalorder_totalorder_reflclos

Theorem 14.12 ∀R : ι → ι → o.stricttotalorder R → totalorder (reflclos R). The proposition is identified by the following information:

Pure Prop Id: 76031124f76b48f04bc4f4bef1b143e3db985fab98172116e9eb3b62c7a65a54
Pure Prop Address: TMTpzN3QGmt1LjMzKNCWXA5CrTNPPXdsGom
Theory Prop Id: 6057db1f3e57d31b555a933f5f17681a6a725f7108faaef6d41f2d8e944fd1ca
Theory Prop Address: TMLi2krY1WNGGUsbxEr2fxHoJC6YZNrmfvS

Zermelo’s Well-Ordering Theorem

15.1 Zermelo1908

Definition 15.1 ZermeloWO is the opaque object of type ι → ι → o identified by the following information:

Pure Object Id: 36a362f5d7e56550e98a38468fb4fc4d70ea17f4707cfdd2f69fc2cce37a9643
Pure Object Address: TMYCW3jauecphYdjePotuDSXP7ZXkfAzLaT
Theory Object Id: e62050256ab27fb78004baab27aff9e7e88265df5714b6aeafcd573b2e7015f0
Theory Object Address: TMZVEYfCFAjoTjAywDY2bSVjv6eXC6FdX5Z

ZermeloWO_Eps

Theorem 15.1 ∀a : ι.(Eps_i (ZermeloWO a)) = a. The proposition is identified by the following information:

Pure Prop Id: 967f2b509c968b55983160582bb89f4b3b392cdb74bd3014ddadec13cba49a90
Pure Prop Address: TMaCPTLDmtonLabJKS8VNg2KpQHWXLBoNXU
Theory Prop Id: 56530eac409ade83907328e28fb11e3478cc0f4637d0fbe97f076524758f297c
Theory Prop Address: TMTxzfNoxPLisqEQk9tTQ8t7JzkFz1bXCit

ZermeloWO_ref

Theorem 15.2 reflexive ZermeloWO. The proposition is identified by the following information:

Pure Prop Id: 0debc67241d3f1b73ddcdb9242748e16c0a306dd4c5bd3be38285ce4c810b221
Pure Prop Address: TMZLFAPmHBxLwpqpQcE1qZeY2pPcfX3QVi5
Theory Prop Id: 3087155d06daaf3f9264297022e0ed58f4b93c8b8bfbbeb282039734efef904e
Theory Prop Address: TMUkjTCw2R4W9Puur76M31DHpHBxtkcez6n

ZermeloWO_lin

122 CHAPTER 15. ZERMELO’S WELL-ORDERING THEOREM

Theorem 15.3 linear ZermeloWO. The proposition is identified by the following information:

Pure Prop Id: 79880f17e3e9a931262c694b18e962541b3131893ba53d44229166cd1edc06ee
Pure Prop Address: TMJkDnfbxikrHuEoUB2KH59QL83E8yZkNdP
Theory Prop Id: feb13fbc27d9aae829703d3a69e71d745f6bbb8008d16029b663316db4a0f0ca
Theory Prop Address: TMMBkNPnE1Xevjyn43F624QPHNDG2eYWp52

ZermeloWO_tra

Theorem 15.4 transitive ZermeloWO. The proposition is identified by the following information:

Pure Prop Id: 9206e3397ca8b9ed09881d5cb76cc80df1ee726499d1f8f1b5c1780d5e9f59ba
Pure Prop Address: TMSnBoLW5VjgRKasiFgj55RonswXagBwUxF
Theory Prop Id: 0cbc46f560bfdfcf23991ad12021c892678d58d3e631d71ff5bab43ce776b5a2
Theory Prop Address: TMSyNvgsZEkSPxH6fbpbuyr29yGLQk1uE2K

ZermeloWO_antisym

Theorem 15.5 antisymmetric ZermeloWO. The proposition is identified by the following information:

Pure Prop Id: db152b33b6ab9a13d674d4d926a9099a24633405bb2ac4113622a3bc701b6e86
Pure Prop Address: TMVoZux2uwtDAmhKY3JNavcVyuWpiLWkQ53
Theory Prop Id: 6e0efdc62d566eec4426eae46854f865abce0b3abf2038f5022a0c3129893eee
Theory Prop Address: TMdx1hkuD9K2Mfu17Z1gRygw4fnFG42CyZi

ZermeloWO_partialorder

Theorem 15.6 partialorder ZermeloWO. The proposition is identified by the following information:

Pure Prop Id: 0656f9a90963b7a76e123603eb83d8b92e386a909153edb2d3e6bfed1e801d65
Pure Prop Address: TMYe2evtB7V2D59AgeiBdZxBgksb6og2GHV
Theory Prop Id: 3499fde0183fea1cebe20732bc7ad2cb75fe284d9e4d45f292274921f510f343
Theory Prop Address: TMPx6RFHGz7VtFGVUwqZbmtdXG9SYpYHcfG

ZermeloWO_totalorder

Theorem 15.7 totalorder ZermeloWO. The proposition is identified by the following information:

Pure Prop Id: 88ef7c493983ed7becf7f9e7c5dd717b8b1ad11361f29a5250e8aea67ebb6d56
Pure Prop Address: TMQi6BX2vtRo2UJyxGGrvkYy62BsS4n19XS
Theory Prop Id: d20df0c3bde7031c1b562bdbbd689604af6728a5b9c1130cdd7fd5a07e1cef0d
Theory Prop Address: TMcqRgmAfxGTPiCTYL27ftCtLTjGab61vAE

ZermeloWO_wo

15.1. ZERMELO1908 123

Theorem 15.8 ∀p : ι → o.(∃x : ι.p x) → ∃x : ι.p x ∧ ∀y : ι.p y → ZermeloWO x y. The proposition is identified by the following information:

Pure Prop Id: 7e38ffca0702338d27fa602350391a1c15520286ef30a841a14806771c076a95
Pure Prop Address: TMSRd1JSF4kWdSSvoi4w2eq61nrH7PDF95s
Theory Prop Id: 321d3a35eac90f0ec080d2b70a9efa74fb4cb4002111c116f5cd2d55be9418e2
Theory Prop Address: TMLKb3jsZ1LNt1giMEWqCcJ48ttSzxEektg

Definition 15.2 We define ZermeloWOstrict to be λab : ι.ZermeloWO a b ∧ a ≠ b identified by the following information:

Pure Object Id: e5853602d8dd42d777476c1c7de48c7bfc9c98bfa80a1e39744021305a32f462
Pure Object Address: TMNLYaDifZ1BU7nW96xprhzT7ZrQZLkjzMb
Theory Object Id: cef2db95ff08afb8db1745dfd6898d6e39064f4098ca0acb0061eafac129c1b4
Theory Object Address: TMTjf9r8y9UHsnCWkcHJQd5ZgraPrRuvcnA

ZermeloWOstrict_trich

Theorem 15.9 trichotomous_or ZermeloWOstrict. The proposition is identified by the following information:

Pure Prop Id: 157b48f20feb46068f0f2aca7b452a507ead7d12b83628b031a51c5f14ce0e37
Pure Prop Address: TMGSAsnBLMuRws587dr7obynuEWa3KPPuiL
Theory Prop Id: ee31d14ab8f6580d4d2449c878828d85530f8e2efa920c10cbcd5cc992f53766
Theory Prop Address: TMJCGmPXs5syPbHaK8NokgLMNiDtwNKeZYH

ZermeloWOstrict_stricttotalorder

Theorem 15.10 stricttotalorder ZermeloWOstrict. The proposition is identified by the following information:

Pure Prop Id: 56f948bc4585b5f5c01ab14c1def7e85a8b8a620235ac296d35a8d94e34c7b1e
Pure Prop Address: TMYR5UnFMSJLwv3HU9FtTvsT9LpzdoKVbZ3
Theory Prop Id: c2537c16e167c192e2f11d7811816cd1fa923138f2630e88388223d36fa49e70
Theory Prop Address: TMLePQqfgDYUVPqeJYmRyjMFNhntRXwDCMy

ZermeloWOstrict_wo

Theorem 15.11

∀p : ι → o.(∃x : ι.p x) →

∃x : ι.p x ∧ ∀y : ι.p y ∧ y ≠ x → ZermeloWOstrict x y.

The proposition is identified by the following information:

Pure Prop Id: d7160045949b81cdd599a4103fed7b65d41988baaee1332de79e879bbbfb0dcf
Pure Prop Address: TMVwVvUZQVTnP3hnKczqoHkD8Zxt76L7hq4
Theory Prop Id: 458f09e17af7e183e42d050a345d94c93e451920cc3aa0f15d74c38ff9645ca4
Theory Prop Address: TMEvCCPnMvJRotk1TMESMYoU6fL8ARNf8cG

124 CHAPTER 15. ZERMELO’S WELL-ORDERING THEOREM

Zermelo_WO

Theorem 15.12

∃r : ι → ι → o.totalorder r∧

(∀p : ι → o.(∃x : ι.p x) → ∃x : ι.p x ∧ ∀y : ι.p y → r x y). The proposition is identified by the following information:

Pure Prop Id: d37c8533f52315f30dedfc2fe79f9a6c647286dec8db8cfc0917eb94b01c80f2
Pure Prop Address: TMTrdxnBmgaKrgsRSyEtgvNwTWRhUzG7QEW
Theory Prop Id: 3b56530ad7e26682eefc8e7eaf7c2e18d87597abef3ce1d4dd28adb08a1bab16
Theory Prop Address: TMXrj7GDDD1CizwhmhqkHEbXjtQ236xfNTd

Zermelo_WO_strict

Theorem 15.13

∃r : ι → ι → o.stricttotalorder r∧

(∀p : ι → o.(∃x : ι.p x) → ∃x : ι.p x ∧ ∀y : ι.p y ∧ y ≠ x → r x y). The proposition is identified by the following information:

Pure Prop Id: 77cb5b1b3db7a63607bc9d07653bc63fb19f0a5a2226c66935ac1668becd94cf
Pure Prop Address: TMGPaaZebUwDYiYk1ANGBmfqMyyCg7WMmg5
Theory Prop Id: 57f61ff6fb48481c18b75895db9d64e93f2f58e90b52ec3ec4e59d1efa68cd83
Theory Prop Address: TMSPtM5ooeGZYBnjiKvH9mKzYtMVNv3LDSW

More Logical Properties

eq_imp_or

Theorem 16.1 (λxy : o.(x → y)) = (λxy : o.(¬x ∨ y)). The proposition is identified by the following information:

Pure Prop Id: 06c3093aa740f38323ae4d7f0dbad8d508f40a0ca90fa41e9f01516651c3e3c1
Pure Prop Address: TMFQ6hVxLyhKxQEQs23Xv9LC6mgW4iwwgrX
Theory Prop Id: 4c9dc0935ef80cbfc0fc55cf8abb066203b42c66e60c4bb6af8b939a9f3dba5a
Theory Prop Address: TMRjkjWby7U8Aw7kefWq3i3PMiD7ahDD43F

famunion_Empty

Theorem 16.2 ∀F : ι → ι.(⋃_{x∈0} F x) = 0. The proposition is identified by the following information:

Pure Prop Id: fc1bdd286f0d898d898c29f18fbe095090efb65893028ab68d5f98d8caa05ce9
Pure Prop Address: TMZjHxMN3FYwok47JsaP7dDQjKXS6jyzcQx
Theory Prop Id: 22bd10cff6495d6e03e6e76619e1426110cc714a9d78d666213a9e4d3b7897b2
Theory Prop Address: TMTAVVZK8aANakE2V9u1NoVueYVDuUeiPWu

Empty_or_ex

Theorem 16.3 ∀X : ι.X = Empty ∨ ∃x : ι.x ∈ X. The proposition is identified by the following information:

Pure Prop Id: ea40ae60ad1226481a72c72991900ab1d75eaab2f2603b30f4021ec9bfa53243
Pure Prop Address: TMN7F3M7BEyv9u1wysCYB4eCnjR17p8fFPJ
Theory Prop Id: 16587d445d1c65fd759cdc12b7fec9089c4b514284913be85bb2bfbaa6fe4f57
Theory Prop Address: TMGoeYYeEjatMCpA5EmBZSLRbtsqeYFpDcB

126 CHAPTER 16. MORE LOGICAL PROPERTIES

Natural Numbers II

nIn_0_0

Theorem 17.1 0 ∉ 0. The proposition is identified by the following information:

Pure Prop Id: 09a099717c75645aead366c3b2036df70c9289907d2755e106f75199e5674813
Pure Prop Address: TMMk6er7mxLyAi5T6t3WpKRXJCMS7xBHwqu
Theory Prop Id: 865a398e739b1785dc276dbe8cd4ed7fb4f1d8782c59e0b8e3b09c70fd4034c3
Theory Prop Address: TMSwDSAjn1RkA5QxqBbEUmy3yKyr57g6mWj

nIn_1_0

Theorem 17.2 1 ∉ 0. The proposition is identified by the following information:

Pure Prop Id: b197676023d2efaa19e70e356beb7444d5240c7348ab140126a015307c650d50
Pure Prop Address: TMavi1ZRdgBBbDLvwBQchihXtPKTjnNdhsk
Theory Prop Id: 2fd3e19cdd9862468dc112216fa972acca941b4de769d35cc59d7ec1577cf0e0
Theory Prop Address: TMdqjwWDUF4ntu7oqJ4gg8mEYbpw158A4BJ

nIn_2_0

Theorem 17.3 2 ∉ 0. The proposition is identified by the following information:

Pure Prop Id: 135dfc520ebfeb7876e0ee1477daff54f352c678350712bc7747990b981da558
Pure Prop Address: TMXsBVAtddQ17BGoDpSMPvxAsXgE6R4Qm3B
Theory Prop Id: 556f07538b44668c0a147b4fc3e95e1df80a4f96ee895c6106b88cb0beddabcf
Theory Prop Address: TMZ7o2p95PaehpRzBsCsB83NrmH4JcRBHJN

nIn_1_1

128 CHAPTER 17. NATURAL NUMBERS II

Theorem 17.4 1 ∉ 1. The proposition is identified by the following information:

Pure Prop Id: e668f88d6e9498b6de122df37d7c3f2dc02798e0ebb033a625b6e3b40d880a0d
Pure Prop Address: TMLzAqBPo85Hx5GYyr7xuu18NVjreNUyRSj
Theory Prop Id: 7774094463b5d0735559a4bab3714251e4442b5e3727856df52e3ae54b23c64e
Theory Prop Address: TMUrdNWeTXSbGVqA84DoNZn9BGLr6upydqx

nIn_2_2

Theorem 17.5 2 ∉ 2. The proposition is identified by the following information:

Pure Prop Id: 31c298db536bab1cc362a2d791118533b2cff78953407ece1ed3125867a3beb4
Pure Prop Address: TMXFjntzEU8n1QS25SjFvEdpbj6sq7qbV7d
Theory Prop Id: 94398399c4d60eec85399c9cc75054c87b16f94488166ce079063afe25b9d45a
Theory Prop Address: TML9u9hUZHGL9ztWF6S5eMPgi1WMRevw1sY

Subq_0_0

Theorem 17.6 0 ⊆ 0. The proposition is identified by the following information:

Pure Prop Id: e9749df5528d6c773cdda6fe3121311c07ae4197f8e3b399882f384c8541d735
Pure Prop Address: TMLX5CPCFVK66pFehnmXXgUmMn3P7noB8s2
Theory Prop Id: 6e655c00e783ba0a41d0029eaa2f71c798229b93c15f02483aafd5566f0a81f7
Theory Prop Address: TMLxnx5doMjojfharrvSwqG5tst7XcCPdHE

Subq_0_1

Theorem 17.7 0 ⊆ 1. The proposition is identified by the following information:

Pure Prop Id: cad7c1cae1ff938a082e5d73d82de4e76f9337136b94ed225e559e21d73e498a
Pure Prop Address: TMbAbdRq1jz52CySGPWWJzvEMuCMw9PQWf2
Theory Prop Id: f75cd0a8e8a447c357f931be8b2e1f23b89f6b92f2086dc3ff43276f2e88d35d
Theory Prop Address: TMH3GTPuv5BQPyQShNEpPrnhYzZUwtmu5ib

Subq_0_2

Theorem 17.8 0 ⊆ 2. The proposition is identified by the following information:

Pure Prop Id: 4147baf022e55cf2dd6aacf5e6a00ad38638a106bc08411c05387011d04bb7e3
Pure Prop Address: TMT9Kq6Qfozmik7TG8YKt8eBb884fWgrGUV
Theory Prop Id: e55065b32674ffef3065ed52eb5884c80c49f9fade95b3f25de91a675649b8fe
Theory Prop Address: TMNVFjtQ2Uu5aRqUQUGEuZaFCmJzys2dTcX

nSubq_1_0

Theorem 17.9 1 ⊈ 0. The proposition is identified by the following information:

Pure Prop Id: cd48c3a937c072b881fc81c095acf2d3d19992c467934cbf8c623a3e0b31c7b8
Pure Prop Address: TMMWurweEZoERkEgMW94uw2qJFHhYfSdhr1
Theory Prop Id: f57d867ae9fd7ef331bb12d8bbc87507a39c321aa63dccd26566dd794deb29aa
Theory Prop Address: TMJfLY5kzU7711oJKtQgwAjc5L8YVWwNjDY

Subq_1_1

Theorem 17.10 1 ⊆ 1. The proposition is identified by the following information:

Pure Prop Id: 6810b4ba8d863b656650594ed3d6b824b91deebd03aaa139e90a032fdc3f761e
Pure Prop Address: TMVoM4uwCGDUnRZZHZKXUcXzt6mXEkzgVcY
Theory Prop Id: 786cb1e61bb2545b3c1fa96512a920199016566e4b90be78896f3912039ebdd6
Theory Prop Address: TMcdzMHtEBTXABdHGrdDC4Ly4fpUtxsqxBG

Subq_1_2

Theorem 17.11 1 ⊆ 2. The proposition is identified by the following information:

Pure Prop Id: 83e0b593f193b7c5aaa42ebec3fe80fdfec8811d24ad5d1ca621e5e328669b5f
Pure Prop Address: TMKdW2ZiFjAjtmtoqqe1oUa34818gAT8iNr
Theory Prop Id: eb94537ff458732f1b6a217159a18b54524b675d0382a95296b651d3ba8f33c9
Theory Prop Address: TMYt6f8BQGRNnMa17NCKpryL5i39M9NdrbM

nSubq_2_0

Theorem 17.12 2 ⊈ 0. The proposition is identified by the following information:

Pure Prop Id: 5810f58237fa5ea1a72b1b1a19c59174248b204a444cd813a87d5c24b492efed
Pure Prop Address: TMapUXLAzmMk7EFCn8wdLE9wngohhRx64bJ
Theory Prop Id: 3a7512da2ec785465bad179179151f9a8406012ca9652a22c5ee00b1c1e8d549
Theory Prop Address: TMc22TospZpLCRqLYeAzy9Aw2a8eBegKoHu

nSubq_2_1

Theorem 17.13 2 ⊈ 1. The proposition is identified by the following information:

Pure Prop Id: 63bae94df530a1ce94daf68c68462bbef84a4139da4811189d52312898d4d441
Pure Prop Address: TMSu6yS8YSy2wRfQSYGzR3kr6TiHARnPaqe
Theory Prop Id: 0a8588177b5ee4d07cdd286638afa3c070f40cd1a82c6cc63bee5d057c8b5211
Theory Prop Address: TMdVb69DrHqVTcNAtqfbK4id4jxAiXFYfQd

Subq_2_2

Theorem 17.14 2 ⊆ 2. The proposition is identified by the following information:

Pure Prop Id: 7b90ad5115eabb6dc9245caeb7cbed2927cbb691b26ae5dcecad88d6115f09fa
Pure Prop Address: TMPuu9Nr9pfr2rJDgk5gScNW6sYP8p19FRM
Theory Prop Id: 0a1db4164bb30172507271531b8defd31e8ab3f2d787c5b272ee3bebeb2966f3
Theory Prop Address: TMUA9eA6KznSVoDTCQaVPvd6cuC6M5SRgKy

In_0_7

Theorem 17.15 0 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: a41b2c54f4d65391d6b32fb603ebffcff201103f2878a4a55f687b12bbe8b91a
Pure Prop Address: TMdFjNSCWXW8Vpvt73BWqM4Zj86eN5S2KNu
Theory Prop Id: 4cb36dedf7517e301f2945346202891f39d34107c01ef2e3b86d9e51b259df54
Theory Prop Address: TMUvTmAX6RDWKPVpacNDGZQ9p9sYmuoibQG

In_1_7

Theorem 17.16 1 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: aa5cdecdaec91fe3a6884b0f09c06631327c30886010ae16f21daddf6ba9c94d
Pure Prop Address: TMJSrd3NEP4iMdJBKZ7F1cS2TZynGhaxyQZ
Theory Prop Id: 392ce8e7c3bb5ff3aa4fa822b0258c7f80e76dedcd760af5717a6e916fa61cae
Theory Prop Address: TMTHHZrGZFsevPcmaS6RDEwByfki1EHr2Fo

In_2_7

Theorem 17.17 2 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: b26cf008f7aa0e7ffcb65a467f1e4f72dd8227f1089ee19c2b649ac8f1118454
Pure Prop Address: TMZoShAAJg9Vr9iuc4kJeMjUubQz3hbFXXd
Theory Prop Id: b47fba676b401c0467d0573bc3d555b8474735646a191dadb139c79d11b2e544
Theory Prop Address: TMUSma8kwGkSuHzU6fZivqqsFZseHSZ6rHq

In_3_7

Theorem 17.18 3 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: e6779a7f3f395ebcac3bd6e88ecd8088dc9671cd31a2b7cd8763b2d1967a07e0
Pure Prop Address: TMZirYDzJAMpfJGxrd52NTyUfVEWH2SZGzA
Theory Prop Id: 38ad3433d50b88a23443fb181777702afa2fd1d0aef7d62ea69949525c1e36c0
Theory Prop Address: TMPbKx6jYxRGDnKufFfM4TW7RyjNLYW4g4p

Theorem 17.19 4 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: 2d470f9c2b92bd61f616a640439618f469a0d784cb0b08a1a90625b00c1d5ba6
Pure Prop Address: TMFfR8LTg8YNKJkWgWwamN1Lgup4j48Whny
Theory Prop Id: 07ae6986b58c8e30c6133f6e2016bfeae67ec868d1227fa18a921392adfd8d2a
Theory Prop Address: TMPy9kPwbcVRmzis4rmmQajdnJnWBPWoMWA

In_5_7

Theorem 17.20 5 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: 685d80ba7d066c28aa1d043f7db6d4b5114f616d882a3bfaf71cd542a148e4ba
Pure Prop Address: TMMku2XXJ9wEw7xtWmoDtdpUqHzpip5x126
Theory Prop Id: 81137f00c9dd3d97ce30878bfcd5a44b55d559d1dd83f2f941053dd60b37883f
Theory Prop Address: TMZb8pTSFybxEM3VZWF9cs1v9kTeYLx2Mn3

In_6_7

Theorem 17.21 6 ∈ 7. The proposition is identified by the following information:

Pure Prop Id: cb4ca2976fd69ef1a079994c73071be6b55cea8db7238fccc9916cc3dd669ed7
Pure Prop Address: TMMcWExWsKZNzZxK6mn8N5UozNgsLWG22jv
Theory Prop Id: fe9b561a0b2c561827d50aaf6b01fbc0a1fe14ef472c16cc21b1738fdf7693ce
Theory Prop Address: TMa39PQbzrwe6rPaWAdYhM6GqbCbvwB5GFE

In_0_8

Theorem 17.22 0 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: 0f58323098cf1c7b76cfc3eeea9678f53a42cf23af1f553934572eb5b3fedb4b
Pure Prop Address: TMbVc3Gz8q2ncEs1dxxpVYDJt5maYv7XRYR
Theory Prop Id: 7797f1bcb9aa6b1d1d07565f754d8639d16574c1a6b38967e7432403cfbda877
Theory Prop Address: TMQfdecCs6mDoNMAna5TMVPYcgtewWsADLX

In_1_8

Theorem 17.23 1 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: fbef0f51f12e7c2ae21b16ba602e7f68892a1dac9c238bf6b4c1f2bed4eb34d0
Pure Prop Address: TMXRwsKNE9viw546hC36BY9U9SuJziTMUDB
Theory Prop Id: 0d7bd5f8b87d778142f6aef2fce3ecfd64b37623004ce8979fac7bb6dc71ecc1
Theory Prop Address: TMXL6Tdr6bmwMFVqYqUkidhjWc9dqLm4zuV

Theorem 17.24 2 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: a07c8ddbe590e66a92f9257de8f0fd799ef1483c2b0db58b970ebd947d68ede1
Pure Prop Address: TMLXEF3MDk7aJtSHea6Ts5kyRqVvzMTLVMx
Theory Prop Id: a625dbe848c33627c83019ef18fb78feb77869dc01561231f770a2c4534e4fb6
Theory Prop Address: TMbknbBMXvxmm4ESoH75pgyt1UuzokTDX8U

In_3_8

Theorem 17.25 3 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: 7ecdad34b9fda66b48c56dc6bf6ef24f1dd82edaf5b10a676dcb5b1c71d69d4d
Pure Prop Address: TMKDDRJXEjpy9C7xEE4XcgqVTYixinVGGZq
Theory Prop Id: 9b36a6f466e575fb7e2a48f68b7361e970bb6edc56671aeea80a5bed424770b2
Theory Prop Address: TMN2SjRR2yH8LkDiZpxREFxmhDWA1tRrd8y

In_4_8

Theorem 17.26 4 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: 61791c2e5bf306497941d536e18c90c32dc8db7afd6c98cd73d9bbbf773d2493
Pure Prop Address: TMKmzqKEHSZcjMZkbLYvTdKBAypDeC2vkWw
Theory Prop Id: 219ccf028539e884cc6ed244a4235bffac83e454680efdcbe9746634113b96b9
Theory Prop Address: TMapcHhFiw8K5Eyguh2XbYVss7iatNsa3XA

In_5_8

Theorem 17.27 5 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: d79eb5b8bab2d0ecce7a5262efe191d5cfa47f1546b64fca19ed3bbb64807bca
Pure Prop Address: TMHdg4GnYxDRPSidyUUEWzVnoASUqBVUVB4
Theory Prop Id: 5625eb82bb63b6fa37d0b870f9d8240cc59102ee45b5c516d25f9c467415b0d9
Theory Prop Address: TMZyKWFWemTmDJxiJkpygV8twB2Rtp15Roo

In_6_8

Theorem 17.28 6 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: bb5a775918672f1f9754e58316f4d97366beae541ee45c784b2caa7cf74b3d95
Pure Prop Address: TMNL3L2gEnenuX7Xv2E3UpeGxkaNNyZdubb
Theory Prop Id: 3c5f2d23bf893b5c57c706c4203ef7618979c66711b505a3e143cd52883986c1
Theory Prop Address: TMGi4xjiJrNb1ynt5cGWncJKwUJnCmsHNBs

Theorem 17.29 7 ∈ 8. The proposition is identified by the following information:

Pure Prop Id: 126eeac20dfdb85457404f82039e474f2a8eb9e482703a36265ea720baec0a3c
Pure Prop Address: TMEjueeewmbW3RUuC5wTG9v3ube5BAnUEGQ
Theory Prop Id: cc5985ce2772e22df62ec877dd1108f076a8d4ad53bb1b7aa440ceb21c281908
Theory Prop Address: TMGFU5u8dWSnsHseshspGDHDJEYkpW2VFo7

In_0_9

Theorem 17.30 0 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: f12d8e4addf556e8ac897b31bd7774ba5f016a114488aa7da47a4982ecf730e0
Pure Prop Address: TMXfAUPidgroRBVsfRFdmxVG3LXaAGCDfc3
Theory Prop Id: 5cb21ec1e1dd5e4c549e48ad0c126c468ccd82b46d5eea40b232ccf7c035b032
Theory Prop Address: TMPTV8Q2AeyehU2pLq7PNkwxbmhsacvmwyR

In_1_9

Theorem 17.31 1 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: 5922261855dc9c35d7a19be6bbe7337de066e405197aa8c3f331b75cb8f166d2
Pure Prop Address: TMFXiJ1JnXuvAUNxeQTQDnCcXESe6geABqf
Theory Prop Id: 0d30f02a5d23823a40baed594ac7548d8af020600e5b2a30d8d190892e99ac7f
Theory Prop Address: TMLQmTKRPrhZ6kBwPoMiaYrb5eJ5CspjmFe

In_2_9

Theorem 17.32 2 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: 024448eff2c81d63335a32519066934b16f60b301c909ac62e1aa2496f034b12
Pure Prop Address: TMSxqdqe4vyyMqhgByQKNDjzpe7hLiYJShy
Theory Prop Id: 330a457d080fe85bbe9e6674a86a4ae3f9e863e5ef5d34f3d5fbd1229256ac61
Theory Prop Address: TMGa1rMZX1W4ZGdPLqcVb6bsiTpmjbxmANL

In_3_9

Theorem 17.33 3 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: afa2b48908b69981c7ad81b71c4bb9e93fb53c3c337307ba4e9d394be14b4eae
Pure Prop Address: TMa3VWUSB3PNWdYhRYTBKk4sraF2pPhodut
Theory Prop Id: 6e40119249c9eff81c846d321343f9181f0fbbae5206bde40c6383b8c0ec20bc
Theory Prop Address: TMP2oP3jaVQRn8x97tPUZDxRbcZQLEPeJ9x

Theorem 17.34 4 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: fdadb4647802a0e507d7b6593df90699056f72e47e5888a3135537fb646a3dbc
Pure Prop Address: TMSFuF88nVp1p3DHdC3pEkjfaYLf6goH5vi
Theory Prop Id: caa1f8ed95c772063de3aa6a8b2cbfb5e9d219c3229bb9c5ce31f5eca2f9de0b
Theory Prop Address: TMbbYX4XLfjzv5dH8d76TFx5HmsrMRJ4tm3

In_5_9

Theorem 17.35 5 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: 318ef9258894a50147815073ae5c4f15b2e60e8d262fe4baa2fd4ac0643491bc
Pure Prop Address: TMQXwEo9W8vVJvGC5AHtnroNxcEsfWyG8Kd
Theory Prop Id: f8aae6c774ede7b089d55ed0f0b30d8fa10e7089bcc1eb1db56f40c182c816c0
Theory Prop Address: TMWDNdKoeVzUChM4ATtv7DwzjNjWouiN5u7

In_6_9

Theorem 17.36 6 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: dc3ff488e971d760fe164604c2ee2de729dd5be38a0bc74d93cedb851d661ab9
Pure Prop Address: TMTGMooeB9YBX55ezejbgrg6SxmcrHwz22C
Theory Prop Id: 0c8dd335320b67f918a737c3818ade882eb57d2cecbad2d32baa17a47518ed1d
Theory Prop Address: TMXsYFAkpquNtmfFQAYjQstpLXT7uYETZbU

In_7_9

Theorem 17.37 7 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: 95d3ebc9e30fe58f8ee1d32b8b051ab32276f7f30609b2fdcd7cec1cf2c66cda
Pure Prop Address: TMHUzLX2Z8CF8zE5QGfoumbM8j4vavy7ff5
Theory Prop Id: 02d55222d7d63900ce597b6f79802629c35164ffc62e82ce1b5c9351ae90842b
Theory Prop Address: TMZYe552F7mK6w5rhKoUE4iCEyFHkzTaJsC

In_8_9

Theorem 17.38 8 ∈ 9. The proposition is identified by the following information:

Pure Prop Id: aa9c6bc99164b52c64952a5e4c3f8ab0450b9a32b7a19a40e692e6458352bbdc
Pure Prop Address: TMMVC6pC2fVWYrSnkueRZ7f3obgNuwZK5aW
Theory Prop Id: 981f43991092c5fd5e23973a29612ea078b2458ee398f5f4e27b1c669761e9ca
Theory Prop Address: TMdurdeoZjzzmm64ypv6qGYNuFSjkTQnJtF

17.1. NATREC 135

17.1 NatRec

Let z : ι be given. Let f : ι → ι → ι be given. Let F : ι → (ι → ι) → ι be λng.if Union n ∈ n then f (Union n) (g (Union n)) else z.

Definition 17.1 We define nat_primrec to be In_rec_i F of type ι → ι identified by the following information:

Pure Object Id: 3be1c5f3e403e02caebeaf6a2b30d019be74b996581a62908316c01702a459df
Pure Object Address: TMMruBGibDS8rkGTgYt2TbCgEshTqpc8T9L
Theory Object Id: 1a64d5d130b189ae403562ffd2a8a2fa63feaf78ed6abca2742cea1af4e1547a
Theory Object Address: TMLxfukbsKf284nSh5ByiFX1hp1cggVTAUF

nat_primrec_r

Theorem 17.39 ∀X : ι.∀gh : ι → ι.(∀x ∈ X.g x = h x) → F X g = F X h. The proposition is identified by the following information:

Pure Prop Id: c955cd8ed0958d2ed269c30204a3cdf992cbaa25660ad46a00085023b9a73746
Pure Prop Address: TMV6dvzCjCU8eRTowJftzbC63sPXHppC6uR
Theory Prop Id: 9ba88707241ec8607e5cec9c0325d12d528f63b0670344631f540a848092b465
Theory Prop Address: TMFCUCvpQLtDtMF2PaNT39UWn3VGzEMY9Cz

nat_primrec_0

Theorem 17.40 nat_primrec 0 = z. The proposition is identified by the following information:

Pure Prop Id: 416eafbf1b348bac1c33766729696046e8de834bf29675a0bcbf8e9e2f4a2dc9
Pure Prop Address: TMVRM6N2YRqKYHAeVikSGWqmkhKWLLqo61v
Theory Prop Id: be11a277d9d20c3312b9592199a4932dcad62cb9aee7679da0a117c5016e5477
Theory Prop Address: TMZ6oFEPsAfDfMvyeTRLHoi489W3WuPogFb

nat_primrec_S

Theorem 17.41 ∀n : ι.nat_p n → nat_primrec (ordsucc n) = f n (nat_primrec n). The proposition is identified by the following information:

Pure Prop Id: 79718550912e5445000ec2075df6854a99c68fbe7dc1bea4ca955eb813ce143d
Pure Prop Address: TMQkJomVG82rUFW7nbsE295nn7dZqExNfUV
Theory Prop Id: fe4c88c37a0678c8dd8db722e13c22b2d116aee8f889a70007d0ce07747f3f05
Theory Prop Address: TMXGF76gsDqDFEkGMFYsmNF4aDipvSuu3Xj

17.2 NatArith

Definition 17.2 We define add_nat to be λnm : ι.nat_primrec n (λ_r.ordsucc r) m of type ι → ι → ι identified by the following information:

Pure Object Id: afa8ae66d018824f39cfa44fb10fe583334a7b9375ac09f92d622a4833578d1a
Pure Object Address: TMQX4KdWNNKEeAgNGf2XNUoYDVDzvGnc31d
Theory Object Id: 29768b585374d1e88373f200fac530e11d07d4aec939607f546fda4fae32538d
Theory Object Address: TMQ7riv5frn6wUQus29DXdY5DVorPU5wZwh

Notation. We use + as a right associative infix operator corresponding to applying term add_nat.

add_nat_0R

Theorem 17.42 ∀n : ι.n + 0 = n. The proposition is identified by the following information:

Pure Prop Id: 402bcbdce23d41df6234b683ebd9d480b649637a52dbe21deab650c8ceded543
Pure Prop Address: TMYQQZGZaNiuYCPSimRia2Xfn2Q6QWbtqhS
Theory Prop Id: 23277f785fd4b6fdf8ebe586a27b9fd4a47086eec22dc99b0d691329fba31a4b
Theory Prop Address: TMXQW56dPBW8mAHqGUEmsjLRYKBrxAAonqY

add_nat_SR

Theorem 17.43 ∀nm : ι.nat_p m → n + ordsucc m = ordsucc (n + m). The proposition is identified by the following information:

Pure Prop Id: a53523a8fbb46795673b15c005b9c1c1d054e111ec64c524c3fa3b20ef02e932
Pure Prop Address: TMTGvLHefupPU9uKtSQRk1QWQzFMwgJrrZU
Theory Prop Id: e61e18415fcbe09a2e909f97e3a2d3fe2abd965f61f8aab6a960f324295f0c82
Theory Prop Address: TMZUDpxQFwyy7YPJopzHZxFeJpa5xzNgnSM

add_nat_p

Theorem 17.44 ∀n : ι.nat_p n → ∀m : ι.nat_p m → nat_p (n + m). The proposition is identified by the following information:

Pure Prop Id: a68eaa385d85714b752c87f2a2eee1c568587a1fffda0e97906a41c26003cf88
Pure Prop Address: TMc24JjrtUUsp2EU37UmE4mu7iJ7DKkmhBJ
Theory Prop Id: 4c1d5d018a9f1d2b9453690cae8404fa9eb620f367b5eea25a0cb86d97cbe223
Theory Prop Address: TMFY44nBuvZeLpSeXoD777seJQ3FwNH5bHN

add_nat_asso

Theorem 17.45

∀n : ι.nat_p n → ∀m : ι.nat_p m → ∀k : ι.nat_p k → (n + m) + k = n + (m + k).

17.2. NATARITH 137

Pure Prop Id: c81c09e2824f3ae69080689a9918f8a8786896c891f60deb4ed02d6f67f3cb62
Pure Prop Address: TMKhPKqB8txYNiBHfmLH1ruc7muR6qocZmM
Theory Prop Id: eec29406f31cd07c1e41a8cfa7dd645a14748dd76d405c653593376354c1801c
Theory Prop Address: TMJ8UyHV8GUb8Vws1rSbV1qG6FtayeCkDeZ

add_nat_0L

Theorem 17.46 ∀m : ι.nat_p m → 0 + m = m. The proposition is identified by the following information:

Pure Prop Id: 8ceeb9658771d534867c4e7fb857c194b87b3c269b78a59e437c53340f7a0786
Pure Prop Address: TMLoqyDa4T219PBE9qaSVziSgcNXAuyYZSy
Theory Prop Id: 84fbe4456a772afa70e2e251ea34167365bee559c7a5aeb17e73e850db6f7c53
Theory Prop Address: TMYQwEtyoLvmW7jPiZze4QVGPj5DZuaTf5Q

add_nat_SL

Theorem 17.47

∀n : ι.nat_p n → ∀m : ι.nat_p m → ordsucc n + m = ordsucc (n + m).

The proposition is identified by the following information:

Pure Prop Id: 9c442aaec4cf7410aad415dc7f4e04405898410e390d79880d0375c76066e5fa
Pure Prop Address: TMaAYzqo6wqjw8Cyowaj3GfJxJxvbhxYDju
Theory Prop Id: e6efb75f65615ef88f5bde274557d9a8df5606c47b340bdff64b75fde270e0f5
Theory Prop Address: TMHs3GjtfJJMSExofB5jjekwQXt9vQ9JVEp

add_nat_com

Theorem 17.48 ∀n : ι.nat_p n → ∀m : ι.nat_p m → n + m = m + n. The proposition is identified by the following information:

Pure Prop Id: 40e9527c8d2ba8c9684c597d041b44dd398a84708d4958ed949bd806aa07a045
Pure Prop Address: TMUKhdGt15R67GHJenCm4tPyhkXRshPnejp
Theory Prop Id: 4ab1793481bb6270a606171fd587949f75d9e5d9f76c6cabe92d879d2029efbb
Theory Prop Address: TMTvQoshXSimoWoX7FGXYgBcckAvFxKAzcM

Definition 17.3 We define mul_nat to be λnm : ι.nat_primrec 0 (λ_r.n + r) m of type ι → ι → ι identified by the following information:

Pure Object Id: 35a1ef539d3e67ef8257da2bd992638cf1225c34d637bc7d8b45cf51e0445d80
Pure Object Address: TMQ9VRpXGSTRvGimsfh4aDJ7zMynq4DQkpu
Theory Object Id: 80ccc668a68b08f05dbefe4777f8a2954ee878fde2459ea00e9c3f844abba676
Theory Object Address: TMSPdCUvrmtPueTXMGGcZnyt6hALJ9ctyFj

Notation. We use ∗ as a right associative infix operator corresponding to applying term mul_nat.

138 CHAPTER 17. NATURAL NUMBERS II

mul_nat_0R

Theorem 17.49 ∀n : ι.n ∗ 0 = 0. The proposition is identified by the following information:

Pure Prop Id: b2f56c05dc69c0b009e6ef48273e48c299236dba09369d6470f062d8d4b9f4e0
Pure Prop Address: TMb9Vpd2YbcAwzGuAijMPac1ZZd1JTyv4Xi
Theory Prop Id: 1e5b2cd8b1404db3fe789241877d2db676126a3ee22aef833bce50e59717f3b5
Theory Prop Address: TMYKPBEGEsiA6kAm2D5gNMaaoQznS2XzHfH

mul_nat_SR

Theorem 17.50 ∀nm : ι.nat_p m → n ∗ ordsucc m = n + n ∗ m. The proposition is identified by the following information:

Pure Prop Id: 37df54fff3ede09b5f27087075ed84f0c3c205aeb237fe98e376929562343633
Pure Prop Address: TMGeZ3mYtjU3qrjXWbppcFZ1rqeDWrfiSqC
Theory Prop Id: 6dd6fc3c946630902a37292306a5d4628d0df49b90a16a0b4bde060e327d01ba
Theory Prop Address: TMWCj8uVzNKMcnEEoUyEKaLXqDFHWBQkrus

mul_nat_p

Theorem 17.51 ∀n : ι.nat_p n → ∀m : ι.nat_p m → nat_p (n ∗ m). The proposition is identified by the following information:

Pure Prop Id: d63438e3958a2dc77d897cf86bd67bfea42d951dc7cc262804802907e9d2702b
Pure Prop Address: TMG1LZCLEQXdxeijnJMFK24mEb2ZENAEMJe
Theory Prop Id: e910cb7dbed9f807bb2b492fd880072b0fa60ff97bf29504de618c1bb02b9b7b
Theory Prop Address: TMHuRaAVff8dq9SXtxuPNkua68MeuTc6xgX

mul_nat_0L

Theorem 17.52 ∀m : ι.nat_p m → 0 ∗ m = 0. The proposition is identified by the following information:

Pure Prop Id: db8a357ff4bbc85955dad0c6561e2f2e46517b67140c8b29f3410a166806d8c9
Pure Prop Address: TMat1oD79SXX8wbvMUfsE7vpuGWs6TV2tvt
Theory Prop Id: 6a2dd6eb8d82e1ebdc8d29656a314ba5f0823446cfd1b7e3e023a868e5c576a8
Theory Prop Address: TMKees76uQpkLU55adTSh33A5D4a1pWMJ6o

mul_nat_SL

Theorem 17.53

∀n : ι.nat_p n → ∀m : ι.nat_p m → ordsucc n ∗ m = n ∗ m + m. The proposition is identified by the following information:

17.2. NATARITH 139

Pure Prop Id: 5cf7ece86a139e3814ee6c6c7d1036b5efd2cb2a7569655a5785267e6b0744c7
Pure Prop Address: TMPvhe9Rvam418EBDU6GUny2MjoZwaH5XpP
Theory Prop Id: 24e4ebc5814e4a359b7a34331c431f191c16ea51e3acd3df89dfc39cbb4f5310
Theory Prop Address: TMK29Vjye4s7vokoPSyBUEGUMF7out1xEKo

mul_nat_com

Theorem 17.54 ∀n : ι.nat_p n → ∀m : ι.nat_p m → n ∗ m = m ∗ n. The proposition is identified by the following information:

Pure Prop Id: 4525ef58273c984cb807830a9315b6c92ea0e51493bee52c9fdb11b2c032debb
Pure Prop Address: TMR99zvsWSdWrZH1xvD1ZPyyRe2sN3tw9DJ
Theory Prop Id: 5d6ce40bd822f0043fa5e91314acb4c38d4657e00036f8292d1c646d61867911
Theory Prop Address: TMK6YYFmjjLN7XjWZngbXtSHGuvLhCNW5sK

mul_add_nat_distrL

Theorem 17.55 ∀n : ι.nat_p n → ∀m : ι.nat_p m → ∀k : ι.nat_p k → n ∗ (m + k) = n ∗ m + n ∗ k. The proposition is identified by the following information:

Pure Prop Id: 7758f10f23cd5bde5f62e6e19176c07070d03250e7ee2f5445ddd169a94ac9cc
Pure Prop Address: TMEg6jVocmXvkpGecshFe76iHzxRPb9JEDU
Theory Prop Id: d57b0261b87e3402ab865d1448a120cf5a64d7df32d00662a23546d0192c5f7f
Theory Prop Address: TMRv9ASbnJeJHuEAVEnAXZtBdyUqKHy7Num

mul_add_nat_distrR

Theorem 17.56 ∀n : ι.nat_p n → ∀m : ι.nat_p m → ∀k : ι.nat_p k → (n + m) ∗ k = n ∗ k + m ∗ k. The proposition is identified by the following information:

Pure Prop Id: c2db2e2dc62062cd1a587ac9ba967b9afcf30e142bf9bd6067b24d6b2f537912
Pure Prop Address: TManfhaHUANQVLqQTLdBVmdTonLkBBLbBAf
Theory Prop Id: a456bc1368534ecb0130b6b056bce984b8cae31fe686bfe5fa19371ee7bc295d
Theory Prop Address: TMTE1UAeY9QMbiDArVw7LV9YCaXoyHNsWTu

mul_nat_asso

Theorem 17.57 ∀n : ι.nat_p n → ∀m : ι.nat_p m → ∀k : ι.nat_p k → (n ∗ m) ∗ k = n ∗ (m ∗ k). The proposition is identified by the following information:

140 CHAPTER 17. NATURAL NUMBERS II

Pure Prop Id: 564886c3dec6ac8d225ffa511e508fd66f08a374ef9f3441d381346c5fd9da75
Pure Prop Address: TMTj2mzQJp3MhibenRc1qV4e9VBJHGB5LmM
Theory Prop Id: 229d2cf885153a28edbff336b15f01f9a69ebb36151922686ca00a59c571593d
Theory Prop Address: TMRAaHnsfviFEtPvtoFHTUV4fNfwKjkAV61

add_nat_1_1_2

Theorem 17.58 1 + 1 = 2. The proposition is identified by the following information:

Pure Prop Id: 174b87c6660c5b9369cec9ae0bb6fbfa629fb619d7c5b2abc06dbe96f057f730
Pure Prop Address: TMaJcaiD1zqTNzV6GxQBzYBvpnhKKCh8BgZ
Theory Prop Id: 77f23f1fdaa42b143fa225c79a56a1ea865e80541dbf361ba18f855e404b3575
Theory Prop Address: TMWAt7x6EE1xxXg5JkjkfpLoXMkjM4S7GM9

Definition 17.4 We define divides_nat to be

λmn.m ∈ omega ∧ n ∈ omega ∧ ∃k ∈ omega.m ∗ k = n

of type ι → ι → o identified by the following information:

Pure Object Id: 1b17d5d9c9e2b97e6f95ec4a022074e74b8cde3e5a8d49b62225063d44e67f4f
Pure Object Address: TMFX8yTrN53rRgSLEZohcsRA6EaD3CSANo1
Theory Object Id: 355928b746d784c55c7688d88105ecefae6bd0215b4f742280a8381aabba7f6f
Theory Object Address: TMc53M2iKqdhRPHnQpz3E7ib2G4x4BAu8NK

Definition 17.5 We define prime_nat to be

λn.n ∈ omega ∧ 1 ∈ n ∧ ∀k ∈ omega.divides_nat k n → k = 1 ∨ k = n of type ι → o identified by the following information:

Pure Object Id: 729ee87b361f57aed9acd8e4cdffcb3b80c01378d2410a0dabcf2c08b759e074
Pure Object Address: TMdvXdQkmLZGboSGJXGXRKmLGB5pqxCZpoJ
Theory Object Id: a54ca44017bc4b34632c41638b465b39b14dd8cd1733ed77712e0d245baffed5
Theory Object Address: TMPsbUQYdAiibkkLrzYgRtSGEqJ5zvim61J

Definition 17.6 We define coprime_nat to be

λab.a ∈ omega ∧ b ∈ omega∧

∀x ∈ omega \ 1.divides_nat x a → divides_nat x b → x = 1 of type ι → ι → o identified by the following information:

Pure Object Id: 73e637446290a3810d64d64adc8380f0b3951e89a2162b4055f67b396b230371
Pure Object Address: TMFmmCt86ZmVJJyYkyLgqnb1fJ56frE48w6
Theory Object Id: 5e7a659c94f73f93fcc4101b5280fa38c54cf4b3836afc4acb20cdafe1a91f6c
Theory Object Address: TMHvTwzJu9htnnfJFGzTSVgNXc69bPtJJRj

17.2. NATARITH 141 Definition 17.7 We define equiv_nat_mod to be

λmkn.m ∈ omega ∧ k ∈ omega ∧ n ∈ omega∧ ((∃q ∈ omega.m + q ∗ n = k) ∨ (∃q ∈ omega.k + q ∗ n = m))

of type ι → ι → ι → o identified by the following information:

Pure Object Id: 23f216cd4975d7a2b1380d30e7b48227ee30bb436b18413678721f3abebc9c6a
Pure Object Address: TMbQxhb4L79WyCM6RdiNH8CwjnACqvJCArJ
Theory Object Id: b3e6279627f6e69ce3e9cae910a6faa1ed2221482dbf9b8cbfff595f2802cb5b
Theory Object Address: TMTkKJpV4fxDM1kVGWKTUL74oaK3t8akX8c

Definition 17.8 We define exp_nat to be λnm : ι.nat_primrec 1 (λ_r.n ∗ r) m of type ι → ι → ι identified by the following information:

Pure Object Id: 37c5310c8da5c9f9db9152285b742d487f1a5b1bd7c73a4207d40bcd4f4d13fb
Pure Object Address: TMQbaN33eGzJDYuzCJatBYA9orZ4FEpcRPM
Theory Object Id: 8e35a198b6dbafe8b2e95eb0500523f42f6367ca1c5bcf349ca278f24a3a07d7
Theory Object Address: TMEwKZ5cpkx2mJ4qE8BkCSWkbVNAA7Wvn6U

Definition 17.9 We define even_nat to be

λn.n ∈ omega ∧ ∃m ∈ omega.n = 2 ∗ m

of type ι → o identified by the following information:

Pure Object Id: 1725096a59f3309a3f55cd24aaa2e28224213db0f2d4e34a9d3e5f9741529a68
Pure Object Address: TMJGQEurbTgjiciPtBTutT9u7rTkMUN4MWJ
Theory Object Id: c28a39bb677f0bc8b9d085f62ff72d310153afc73e004a5ac360c1cd5397836a
Theory Object Address: TMWJQjHTr9nF6QRD4b8ffhEed2Z8ZMsa6nB

Definition 17.10 We define odd_nat to be

λn.n ∈ omega ∧ ∀m ∈ omega.n ≠ 2 ∗ m

of type ι → o identified by the following information:

Pure Object Id: 8358baafa9517c0278f00333f8801296b6383390ea4814caaadcff0dec35238d
Pure Object Address: TMGKNXPQbMNiuCqZnwUEjEUha8fcnFt3Jk2
Theory Object Id: e16741f9a0da9b00388d2994378dc8d14febef4c577e38c02cea2b3600c7b124
Theory Object Address: TMPkkurDzwQcwtKKmG9c9urhgvFf7SduuxD

Definition 17.11 We define nat_factorial to be

λn.nat_primrec 1 (λkr.ordsucc k ∗ r) n

of type ι → ι identified by the following information:

Pure Object Id: ff333a6e581c2606ed46db99bd40bdd3a1bab9a80526a0741eba5bddd37e1bba
Pure Object Address: TMZjHdCL77axNr9RebheBmYNoJLbcGbbnfG
Theory Object Id: 43798425ce2e239eb427ae260e9a7b313bf8bd7df1dab1d082f026f6c879692c
Theory Object Address: TMYeeBw9Gn9bU5FejUUExj87nTNnbgW5gbX

142 CHAPTER 17. NATURAL NUMBERS II

Natural Numbers III and Ordinals

PigeonHole_nat

Theorem 18.1

∀n.nat_p n → ∀f : ι → ι.(∀i ∈ ordsucc n.f i ∈ n) →

¬(∀ij ∈ ordsucc n.f i = f j → i = j).

The proposition is identified by the following information:

Pure Prop Id: d756b7f9b65b31513512c63b940fdfc4ecb49540116753f6c2d340626609ee82
Pure Prop Address: TMLBavCBsSBws7ahi4AVPeESXViLDv31Ak8
Theory Prop Id: 604a84ad91cecfb6b42f1126bfec73fd706444dd748ec6c15b12ce0bfb69053a
Theory Prop Address: TMVN7QY12B9fLrGqiNtTGBesGf96NCkiHRC

PigeonHole_nat_bij

Theorem 18.2

∀n.nat_p n → ∀f : ι → ι.(∀i ∈ n.f i ∈ n)

→ (∀ij ∈ n.f i = f j → i = j)
→ bij n n f.

The proposition is identified by the following information:

Pure Prop Id: cd3b8dc5389196c2477610f742fb1ee7dba1dcdf01fea50de9a063945a56cffb
Pure Prop Address: TMV5QRAwtRsRGashmQVHiReREqRjLYeRTHc
Theory Prop Id: 44a08f5c7754241cf51e71665e77acf0e4b00fd74a503742d7594661c4883eac
Theory Prop Address: TMQn8ujQbMM2W4yRjHGhzmzMYVZfwKNdhFW

144 CHAPTER 18. NATURAL NUMBERS III AND ORDINALS

cases_7

Theorem 18.3

∀i ∈ 7.∀p : ι → o.p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p i. The proposition is identified by the following information:

Pure Prop Id: 9d65eaceb8856524e55245becb873ac99520db85c1aa6190004c31f201ae565a
Pure Prop Address: TMMGJXuShsSGWBi3CaHZeVWjtSHGMJv1cqn
Theory Prop Id: 0b70c477b139fb0134ba4733415ed1d40b67071992567e7df1d98b3fa3ef4f9b
Theory Prop Address: TMXFRbLTTN6ihayKReZVwptm9miwKifcmVm

cases_8

Theorem 18.4 ∀i ∈ 8.∀p : ι → o.p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p 7 → p i. The proposition is identified by the following information:

Pure Prop Id: fb834c80b903a21792ab31c32ca79011f1ed7fc1ee605431cab0377c113ce789
Pure Prop Address: TMJ9pan6QGbicHY78ZhcRV5sjSxUpJ9hjAe
Theory Prop Id: 148753c82fc2d19e4d0ccfdd34fc8ac73852e40d98c8740e4bbf13f37f1469bc
Theory Prop Address: TMaE8ED9F5UkowTrWVxJoWwkbViqo9kPjoK

cases_9

Theorem 18.5 ∀i ∈ 9.∀p : ι → o.p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p 7 → p 8 → p i. The proposition is identified by the following information:

Pure Prop Id: 0d0c6a6d27f3197a59eed06051dc8e722f3c73606b083f56fa41970d63f24e12
Pure Prop Address: TMKuRMEvwYcoSa8Gzg1hrtZoBaBSPH2xcsW
Theory Prop Id: cc8c42c160b06cd70b0400e6d0cd711ceb06d2dd8fbbf1468400cdf70a509df5
Theory Prop Address: TMLp2MUsd6cFR8gFchdhfihDsX7oRanJWxR

nIn_2_1

Theorem 18.6 2 ∉ 1. The proposition is identified by the following information:

Pure Prop Id: d4d954c8170695a2939bec3c446c37edaeb27568ad48b1986c6ecaba24070132
Pure Prop Address: TMMJE7KaXfpvgsT5JWqzaMeStdMLFTu4s3R
Theory Prop Id: 9f9cbd90053ad69e8dc078d6b9aafbd1e870ff598f884002a52555d6cf96db04
Theory Prop Address: TMSytpFZDKijwdEXRuEoRS26emWEHwyyJn9

neq_6_0

Theorem 18.7 6 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: d94311e2671c987b8b42941206d260e5fad22da17796f0e0af8fd9718168e741
Pure Prop Address: TMJPPVAEoouTBu7DFSenpbm1vf43asWn9A9
Theory Prop Id: f94806573dcb9fa2ec502b75f210c999a07ae981958bf4172d246de9cef66017
Theory Prop Address: TMUnexmnNzMTxwUotBtEVHCSB7LQYjDXGCA

neq_6_1

Theorem 18.8 6 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: c950efa1242fe81c8f031eb108e16f8792592823a5d93bd4f60a0f648062bc05
Pure Prop Address: TMG4GcsPH6pN4zKNadStY83A7KTwr1YmB43
Theory Prop Id: 799bdf447bc496543bc3e105fee018f133d24d15ca3c6eb8fa5bb71688379c06
Theory Prop Address: TMNpiWkUsYako2ZQqsmQ8aMpiCajvjtw6WY

neq_6_2

Theorem 18.9 6 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: d5b69e228a453716ca00492b40a6fb313f2f582f72946915bc5cba6815ebd5b4
Pure Prop Address: TMady4g8qST1QcVR9VA5N56ED9kVDm8NbA8
Theory Prop Id: 2865cfc1f9a346744761d36b1b8ffe1def60d3f92967f64caf2bdd6621484595
Theory Prop Address: TMYMRVL2d8FLeABGAFvksAq5e8LTs37M9U5

neq_6_3

Theorem 18.10 6 ≠ 3. The proposition is identified by the following information:

Pure Prop Id: 03fcc4b85dfade6c906248f8f17bb4e90da92e6bc4bd7dae07dc87b1c793cafc
Pure Prop Address: TMZhLzSJKhPJ6LW7S8RESbikoubnNwygd8E
Theory Prop Id: 1ffa21cb3a06b988a3518b1986f09cebcda356923bb17b459190071df400c1fd
Theory Prop Address: TMdSxgcJr8YTuZy9uTgG9ajdHdm6pAfiETZ

neq_6_4

Theorem 18.11 6 ≠ 4. The proposition is identified by the following information:

Pure Prop Id: 0ebd5de52c3bf03fa06ad9a52177c94376ac6d6d21e4f7eec2ae524c20eb865a
Pure Prop Address: TMYRPQnnTxR6J8VzzoFmHbqoXkGa3GgD775
Theory Prop Id: b3e48dff2905f986f3b5caced5a64dcd7234bfbc66ef0306f80725cd1ece61c7
Theory Prop Address: TMbKF48GnbjUBGJE9sw73TPmMcB8d19vTHv

146 CHAPTER 18. NATURAL NUMBERS III AND ORDINALS

neq_6_5

Theorem 18.12 6 ≠ 5. The proposition is identified by the following information:

Pure Prop Id: e3c546719f53e598755a05b44ff1592522513e88eab553a3118e5486a81ed067
Pure Prop Address: TMYY56rp3pu1Md2tx7cYLemwdM1mR5gcmwn
Theory Prop Id: f44366e0c31269b36c404d0c744fe0d1b9ff6021d8c4b58406adb81f5b1c2657
Theory Prop Address: TMdM2SjU46csQY7mCoKywbHX9m8H2pRDGvc

neq_7_0

Theorem 18.13 7 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: f1aca13486b782dc00a51c22a046ddf10b9e5b6024aa2d8487df4668d134ad21
Pure Prop Address: TMTpLW3ApNNyU9zRQ4a5vLvH6GvaG3hDZsw
Theory Prop Id: 04777e1121acde4051c1076405353442ecc7c841d8f5b1e282a7bbc610a446dd
Theory Prop Address: TMakKfA3Sv9UTsXeCRpShgJQ4dUnFkNenjx

neq_7_1

Theorem 18.14 7 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: 7f68ac2aa17e9e283bde82aaa6848151f8aae61ca586307fee893ae0535713a5
Pure Prop Address: TMQ5xCGH582yDx55yDN51RKxK9oZ7i3pmXr
Theory Prop Id: c517bd071cddfea2ce1c10f9a4334e27e7bb4cf04db904a41ccb36e001fe703d
Theory Prop Address: TMMMZYiwPgt5qwiuqAdeUwY2qzRX4uwcTBy

neq_7_2

Theorem 18.15 7 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: e915b85aabc6ff55b599e8b765d9d05c01d9dda3408fd4fa0a9833c8f62024b5
Pure Prop Address: TMbBFKirdwXXTLhAyJKum1V8XdNj5BBozBE
Theory Prop Id: 719b90b534d821a74d146a517ec28b80f23cdcc8ebf4522af7dfd48863cb4fa8
Theory Prop Address: TMFfMwPwBr11dsiDFnzdKdva5FgSwooFhKZ

neq_7_3

Theorem 18.16 7 ≠ 3. The proposition is identified by the following information:

Pure Prop Id: bc468ab5a647c84bfe1cbf2177bddf3cd9dcbed388c41930e8429925a62165e5
Pure Prop Address: TMLWnYbD25mqPFYupL6Lc7hitrjdJcfniuS
Theory Prop Id: 7c0147f7e0c5d264d821868156432da25efcb90102666d5c3d25e4e52d14e374
Theory Prop Address: TMLUE8BbNrawrU1x36jKypY1Hc7Dh9sTeBx

neq_7_4

Theorem 18.17 7 ≠ 4. The proposition is identified by the following information:

Pure Prop Id: 0241dc3875c580942490118be9d0c699ec0514fe9d772ea7a904789477d54238
Pure Prop Address: TMGN51cm5R9iSyp1WjfR5ejM1X6VgXfxqwX
Theory Prop Id: b83dd2a8b6844c39918736d4bb65dae6cac7b407b3e1d7f856124a1b8c8b1bbb
Theory Prop Address: TMczss8yk3mckgdGRz59sXdSgkJJAmtAG5j

neq_7_5

Theorem 18.18 7 ≠ 5. The proposition is identified by the following information:

Pure Prop Id: 0d6c1682fbca7a87f757e77ddd09c5bb907565625c9bb3bd4d4106f080d57fc6
Pure Prop Address: TMRiUm9T8u2HUSVqbxBbzUpwb8M5kpBY8y9
Theory Prop Id: 2c37cdd7efc305b90d6607a132b9b9dd842317e03383654b67242cd93c5a9abe
Theory Prop Address: TMNyRaG3YG1Vk3iDDYuv3o2c1VCAw8DviHL

neq_7_6

Theorem 18.19 7 ≠ 6. The proposition is identified by the following information:

Pure Prop Id: 0d1fd098069d41630b89611eb93f1f08bce1c36d9c813d078d3d146904cdd42f
Pure Prop Address: TMdEyrS1cfBLmsvCe812P4oyjUa3LvHoT5m
Theory Prop Id: 7b8e224d3ff812c0454afa567fe7c3949fb0d82d46543a26c9e7ca9d51dd332e
Theory Prop Address: TMdRvdFMDC1LW6RXkFfr25AGkBNkYkv7Mt1

neq_8_0

Theorem 18.20 8 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: e3931d2d8a8848d0d9ee64433a011a9aa4a51737fbdb5b7ff75eda148be9a459
Pure Prop Address: TMMdAqero4SC35AeBiqqtuQEkyo6rgyk4ti
Theory Prop Id: 30657128da01d2a236a2d6ddcc73e1f8049084468c98dd69a2605d082c526bb1
Theory Prop Address: TMSNjV8GGDiVXMRQBxvQkSYEkGBVdZjVjFm

neq_8_1

Theorem 18.21 8 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: 6a16eeb33a53075e488dca90100eb48cca90e3c02804aed2943df9da34c76b07
Pure Prop Address: TMU2yPgAvkUoBCYpDgKoFG8xJAzksaqUjrV
Theory Prop Id: 3fab0e9aef535966602b629d8b325e7d4fcc22fddae36170ebb822bab8fe78f2
Theory Prop Address: TMPHMdwsKTGixUr9ovcL92TPpNn2DZ59xR3

148 CHAPTER 18. NATURAL NUMBERS III AND ORDINALS

neq_8_2

Theorem 18.22 8 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: 9f7e3d7052052f039493deaa72664346f98b44644d5ce8e1d6b29f702756dc87
Pure Prop Address: TMHiEy2g7ytqLkCShdFDkQrHwAxrHLJ4ewz
Theory Prop Id: e34cd1607418d341f87f72d6c64acebcadb8c4c22064d081a9a312fa6ea07e26
Theory Prop Address: TMKa84jucjzdciZ6ZrGg7jbnwmESXpiuuyX

neq_8_3

Theorem 18.23 8 ≠ 3. The proposition is identified by the following information:

Pure Prop Id: 338f0e75ea4c6be85d54aac86da1fd615b97b42931170925c246bdb4ead8b0c7
Pure Prop Address: TMXDqEySQWNgPDXaxytYNRxHDU8nnVJBQmE
Theory Prop Id: c0304effdb21f61184bffd799d5816aa2c736176a0f60054faf868b3e0667412
Theory Prop Address: TMcMdW4UMw76zq4HEXEdz38xzdN3uEfNvfU

neq_8_4

Theorem 18.24 8 ≠ 4. The proposition is identified by the following information:

Pure Prop Id: d95ab6bf4885a62645e050411d3dc149b0c30553a1b11777fb9626b5056dd31a
Pure Prop Address: TMZ1C39y5oktaYiujD5mMcSng5NPSLJeEQz
Theory Prop Id: 626b967685fe040cf728abce59c8526d52b9320642a11332b64fb176654bcba1
Theory Prop Address: TMdUEjMQYbenfRooZ94bsLXZekKQw7AwjpY

neq_8_5

Theorem 18.25 8 ≠ 5. The proposition is identified by the following information:

Pure Prop Id: 3582f2a69391c993ecc9217f73540497ab0af13613233acff03d6d05999a68b9
Pure Prop Address: TMNQpkxEPyJb6YgZiPPBGZQ9GbC7g7hb3TA
Theory Prop Id: f8515d4c2c16dac9f1318ebd500262d8f6382c1822da217b3713dd322b505844
Theory Prop Address: TMK22UzEck1xp8U1jNqX9wkEh8eNg2kiLgv

neq_8_6

Theorem 18.26 8 ≠ 6. The proposition is identified by the following information:

Pure Prop Id: 15c47f6efea919c6469a2b32df126c3a593df8a81ab4ebc5df17a98087ebf8e6
Pure Prop Address: TMcBgaEvvTaTaTDaifgkFsjxZQsqU4WzYoY
Theory Prop Id: f5b7bf37bd7435e07a0b28d0a188937c24f605630a543a4e882d91a080499e2a
Theory Prop Address: TMM574yv9Ufaw1F3N1jvDPpcg9GNzrTpYD6

neq_8_7

Theorem 18.27 8 ≠ 7. The proposition is identified by the following information:

Pure Prop Id: 8d3267526667c8bc389f97552b72f511aec9f5f758b6ffb5677e384888680a6f
Pure Prop Address: TMSmpMULYdUBC4xMvaX48vBU8PW3G6mmegc
Theory Prop Id: 4f0658fb5c5f798e5218782be81c0fc926d764645d937e30bf2af1ae4d64d8f2
Theory Prop Address: TMLSfV5KwoLtB3AyRu1AQrfTfXfUQMcVAmW

neq_9_0

Theorem 18.28 9 ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 7fad6a4021cfaad3cd966f9db7cbc322601a15b27f7bebd190db92d9227d7d9d
Pure Prop Address: TMYqJFCaupzUjNDmU955uhVqnu4ij9Mkuyn
Theory Prop Id: 6e3ac6edcc976b5d4163e1e746d46f422f0f0916bd84e45ef0045db9d55d4c19
Theory Prop Address: TMKrgJnjKmPUDz3Ei1FeB8qAqPZq38H7jTb

neq_9_1

Theorem 18.29 9 ≠ 1. The proposition is identified by the following information:

Pure Prop Id: d87ea84c9c5771eec45a1fd14ad9153c2ddcdc39dae8e258585d43f60d51fc0d
Pure Prop Address: TMJggtzHrkHQuUoShHgUHjqXcPh3YDT7pN1
Theory Prop Id: 80c63319a6d1e1304573edb4dae25f062780f4a278368ab2d11ffeb00f49b7c6
Theory Prop Address: TMcbAZhdMU2jbnR5osTzLG3UcaHdNK1HoSu

neq_9_2

Theorem 18.30 9 ≠ 2. The proposition is identified by the following information:

Pure Prop Id: e2f4a18541452ebc1ff6a767768646cff4511a43dacb70dc53eb524adbacf071
Pure Prop Address: TMW6fNB5sUi4nGQoEwhzSpZyo1Y1LXDN1Ax
Theory Prop Id: 906ccd2680d9bcd663a9b598d0e0b69b8c88fbeaa778ec48beaa22ea36b6f5b6
Theory Prop Address: TMWS25pQtSvRUgzYu1ZiDeYaCbk35nSzVQb

neq_9_3

Theorem 18.31 9 ≠ 3. The proposition is identified by the following information:

Pure Prop Id: c17c1de4ec497af0fe4297a258d7a7c3d135e22ec81c834e7c9a96add5cccf55
Pure Prop Address: TMcK89BgzJhDbNna2LzUzWNiyoYx6z5zftc
Theory Prop Id: d46f51d9a1c08f9230e9eb18a81e934945d4012ad2dbe5e8e979e80e9c2d0a15
Theory Prop Address: TML3NJwKxkRE2ZwqumY1ptX5orP6hMULsaL

150 CHAPTER 18. NATURAL NUMBERS III AND ORDINALS

neq_9_4

Theorem 18.32 9 ≠ 4. The proposition is identified by the following information:

Pure Prop Id: b2be6a2b14079f391b61f0f2c994e6c1f83f55f33e48888f5714fe2885e624f7
Pure Prop Address: TMG2u9oAHq7WMiN9fonJks9eiAQRK26GM6C
Theory Prop Id: b136fba6a8b8fe9273ffb6386834de99d159f7c0ebb273ae030c99bdb951d513
Theory Prop Address: TMUecGRuCAeSZwXfRoh3G38RVvj1529JShy

neq_9_5

Theorem 18.33 9 ≠ 5. The proposition is identified by the following information:

Pure Prop Id: b1aaed7edfc0b04b990d933a7c324285c4e528683b5444ae62b5f6a6a1fea631
Pure Prop Address: TMWqKkpLGp13Tno4pYWFNtmRzmAH8WgKrKw
Theory Prop Id: 0ae3b202241c8b6489ff2afc286f400f7db9f53d731d7c5c96fbc5157b8f2546
Theory Prop Address: TMUn176msQ6tprqPwqSfnayaBTxnWVJfjxL

neq_9_6

Theorem 18.34 9 ≠ 6. The proposition is identified by the following information:

Pure Prop Id: b0286fd9ca576d16fe9d5933787a03791678ce168ecfef6360c216512ecbe419
Pure Prop Address: TMYwbNuQhDPUxShjx7H2boyojGDg8rAw8MU
Theory Prop Id: 19a02fe5d8602e59495deff9213cd8d10a5cc30204dadfbad1f34d1f0fad400f
Theory Prop Address: TMbZ1StXWmE3GzbtLteGPhs8Bh7HEHWxABd

neq_9_7

Theorem 18.35 9 ≠ 7. The proposition is identified by the following information:

Pure Prop Id: aa22ad31d3f91e04f56851635a9322d13c23e3854cf45c973dea5a3cd0990b88
Pure Prop Address: TMXRgW5zUyKotMqQa3YaGwUZ6vFLUNEzUV9
Theory Prop Id: ff267fe87e45aca713d50f6e46c11d032357b90dc7acb206e3eefd8d01d36e82
Theory Prop Address: TMRwAzgMNqVuBgS6WGFdfKaAwYzx2HWCZgu

neq_9_8

Theorem 18.36 9 ≠ 8. The proposition is identified by the following information:

Pure Prop Id: 552eeb72bea999741bbc3cfc4807d1e73d2669cb5a06966cd70231206647a992
Pure Prop Address: TMRjxiyZgHvxgNtxTc3rKqKfBHHQ3fpUnty
Theory Prop Id: e47272afbed4b7d1b148ec0b93566130bd83c0c213facce0dfa529c59ee5d182
Theory Prop Address: TMXoTBB5rYLbYrboKc4G2FzdBcPPXa5C9uF

Subq_1_Sing0

Theorem 18.37 1 ⊆ {0}. The proposition is identified by the following information:

Pure Prop Id: aeae9a39918acd52e99e4648ef2ce22082b14cdb7f056833b3c9b6c7b08071bb
Pure Prop Address: TMd1acsG4T47vanN7AeZN9GBKVvieBkvmww
Theory Prop Id: abab2d4437158699c98b6dc7b4a1d462e539047454145838ea1260d68fc49944
Theory Prop Address: TMbUJrpN4q3Hv8rt9wMEJobfB2VW2EKtT8g

Subq_Sing0_1

Theorem 18.38 {0} ⊆ 1. The proposition is identified by the following information:

Pure Prop Id: cd786497c06a79d7f8391817754ef67063326d18ddbf160ab5ed1c22c0a322eb
Pure Prop Address: TMZDF4AA5AbafCq1V1cfLd6D4iDMcgeXjr5
Theory Prop Id: 1806bf9a09183656922b1ceb29490f6cc9898d3b353ad377a7e13edd309c7791
Theory Prop Address: TMHq3pPKYpd4KhjT9PUsaM9T7pKh9PDKCDK

eq_1_Sing0

Theorem 18.39 1 = {0}. The proposition is identified by the following information:

Pure Prop Id: 5e59965f8dea2376b44a4645087aa7b0fb447c5e0b32fdfa0a419f8c6730821f
Pure Prop Address: TMM87nnYFzft3WcRjLweo3mTQXoJfb1o2yT
Theory Prop Id: 264f38ba2201bb27cd34ae54a896eda9cfb3191811b417511fd59a42a1aad583
Theory Prop Address: TMJZ4wN5mJDvtTGG8T4dkZwXYestYUa7nSE

Subq_2_UPair01

Theorem 18.40 2 ⊆ {0, 1}. The proposition is identified by the following information:

Pure Prop Id: d908841bebc314441f7259cb1ae1e06c95ce753fde5f00fc59b6dde11d4d93c1
Pure Prop Address: TMHVo8N2YAFhqLdCesUKneKkycAMCjufnxX
Theory Prop Id: 94203e1d2b2c0faaa1b53b57d6e02377ba6909bbad4345113cca6c3b30ffc6c2
Theory Prop Address: TMdUkY7ub8vSPAiJMtCVLRMpJmEeZvULcKe

Subq_UPair01_2

Theorem 18.41 {0, 1} ⊆ 2. The proposition is identified by the following information:

Pure Prop Id: a855482b832c8ea776aa01316555b8b83b59d1577d45ef4b5d123674420d0e01
Pure Prop Address: TMGfNJyPsWS2KxUk5KVgkcMQB4XboikFQcH
Theory Prop Id: 0095605b59f5d2bbd6d2ac9dc6c80902a1f7530a558319eac4bd5e1225e3059a
Theory Prop Address: TMYm6neMj3cdSovA76Sn3Rzr22vfiLnJrMu

152 CHAPTER 18. NATURAL NUMBERS III AND ORDINALS

eq_2_UPair01

Theorem 18.42 2 = {0, 1}. The proposition is identified by the following information:

Pure Prop Id: da80075027dc7a68c335dc91d0d1e2008733d49f594a4654f4808a72f790fe18
Pure Prop Address: TMTAkWeRWfntW6KcrRUzppdf93RSZBsj2kq
Theory Prop Id: 7ae5a1ab159f51b92a97a85e8382349a9ddca06dc5c2f387f58ab6cada7ddc8d
Theory Prop Address: TMd33UnmSrxCctoLEbFMBCqkF1i4pvM11uq

ordinal_ind

Theorem 18.43

∀p : ι → o.(∀α.ordinal α → (∀β ∈ α.p β) → p α) → ∀α.ordinal α → p α. The proposition is identified by the following information:

Pure Prop Id: bb557547876580348d1b27707e8329c129045f75384f6cfee3b6b3660d679f76
Pure Prop Address: TMMjF9N5BddMz7dZjRy3jb2edCyq6JxUnjh
Theory Prop Id: 973cd8ebf285141bb206385be0589deb9b0d2b22f4291d4f8935767e2c8d72f7
Theory Prop Address: TMHyu4a19t1LKmHrnT3eDCrBexiXHK9SKmm

least_ordinal_ex

Theorem 18.44

∀p : ι → o.(∃α.ordinal α ∧ p α) → ∃α.ordinal α ∧ p α ∧ ∀β ∈ α.¬p β. The proposition is identified by the following information:

Pure Prop Id: df6c7763ab1d598ac956e39f705674757a597eb43301218c4a0cf210679e8487
Pure Prop Address: TMR2LGsWB47Dd2V32aJMtRmhkcuHkb32B5c
Theory Prop Id: ab6fe147dcaf15113c3482aa2b52e2418ef973d6f5f9ad0df1e81a91ac34fd0f
Theory Prop Address: TMWujpqPswYN98tJhWJajsNjSmNit3C1wxB

ordinal_trichotomy_or_impred

Theorem 18.45

∀αβ.ordinal α → ordinal β →

∀p : o.(α ∈ β → p) → (α = β → p) → (β ∈ α → p) → p.

The proposition is identified by the following information:

Pure Prop Id: 7c0f0bd46025b48ff67a7143c3e4a076e5e080c3c720416d50d3fda07959571a
Pure Prop Address: TMbyFsACnhcNuinkiWbTRsyq9Xw7dG3yxen
Theory Prop Id: bb6d2d91d4ebfae062a8c9b008250ab6b380ac9e09ebf6e8ec9dfd01afcfc71b
Theory Prop Address: TMX8gEh9epaBjfv8Rid5iBE5GnLWmimvYwE

ordinal_trichotomy

Theorem 18.46

∀αβ : ι.ordinal α → ordinal β → exactly1of3 (α ∈ β) (α = β) (β ∈ α). The proposition is identified by the following information:

Pure Prop Id: 2230cfc37c714b9145baabb918487c4c4cb9582cb14de24bf562f395dbffef9e
Pure Prop Address: TMUJTrRxNbofUZ7cPQahpGQK3QFoP7UEpr5
Theory Prop Id: 72667378ef8d960e362a49147b9f82d2da940398bc2f4f035524457ddae3b011
Theory Prop Address: TMadvrwG2zGWUTUNaw1gN8KJ6gPMX3d1LQJ

154 CHAPTER 18. NATURAL NUMBERS III AND ORDINALS

Disjoint Unions

Definition 19.1 We define Inj1 to be In_rec_i (λXf.{0} ∪ {f x|x ∈ X}) of type ι → ι identified by the following information:

Pure Object Id: 8f0026627bca968c807e91fff0fdc318bc60691e5ae497542f92c8e85be9fd7d
Pure Object Address: TMHfHa5vujJQpf428B9wNXuDE5s6mwdPL2X
Theory Object Id: 7736ca8991b680139f61982df3af1ac650f0f26acdc0d8f0eee357ccb025fa14
Theory Object Address: TMG73kjDALbhSPSDyKjUdTNFaHJFKmvRPR3

Inj1_eq

Theorem 19.1 ∀X : ι.Inj1 X = {0} ∪ {Inj1 x|x ∈ X}. The proposition is identified by the following information:

Pure Prop Id: 73d189eaaf79f7e70a5868626e8599a06905bffe21fe665a8b1af2a7532ef363
Pure Prop Address: TMVWdgZGFsyibGkrWpy2DrjXjAj4ZHnCcnv
Theory Prop Id: b804575f2f48fe85cc3e16f46d28aedf5c9e1bd2db40240cdc402976b31678dc
Theory Prop Address: TMc2FdiN6hAsJo3duwEKeNeMQK18PyUEFsh

Inj1I1

Theorem 19.2 ∀X : ι.0 ∈ Inj1 X. The proposition is identified by the following information:

Pure Prop Id: 3c2cc4eebcc7df71902a867c8e10780084c122974b276dfc2e42b3636fcb7f48
Pure Prop Address: TMUKupGQMJTgBLEF8gUr11TxXdBgWPczFDF
Theory Prop Id: 4ccac3180e2819292f9b0007f9583007576538ef5094469c5b9112c6324d5ae5
Theory Prop Address: TMFoqv2T2HTXaw58WBHGfN3Yg1LLmHDXTZY

Inj1I2

Theorem 19.3 ∀Xx : ι.x ∈ X → Inj1 x ∈ Inj1 X. The proposition is identified by the following information:

156 CHAPTER 19. DISJOINT UNIONS

Pure Prop Id: b6f23b38b854e4764eb026a61fb5de29c88c81746a47fcf866ef0b179e8cceb7
Pure Prop Address: TMZjKduCZEmbUN9ZBjsaBnKgkfw7LN2UDHT
Theory Prop Id: 086475da641652d7d705bb3a628b3be4360161ede19a126c9e4a25189e1af347
Theory Prop Address: TMcYPGZrNfjZ1uwVn5pPjvCKQnbnj8Fb9jY

Inj1E

Theorem 19.4 ∀Xy : ι.y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X.y = Inj1 x. The proposition is identified by the following information:

Pure Prop Id: d4177b1a2defe4e7a1dc9a2f0cdd24121f05df2353248c4fbce54a7813e386aa
Pure Prop Address: TMRaNkUGFqMNGWjPSkyqhVQ1u7CuZYnE44m
Theory Prop Id: 259217a7917e6f724d1d01957b59b045b53d464a75ffd225751fab998674d1eb
Theory Prop Address: TMKgb5K1zBg9yUsASeSHAXS5dXzMAp6g4mC

Inj1NE1

Theorem 19.5 ∀x : ι.Inj1 x ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 5e3197f060ee011e661df22a9f6fa83fd21ce6975b1e8fffe8bf822020036e6e
Pure Prop Address: TMZWFtRyxiZnniPvM2zyYdwdjemD8ju9G9q
Theory Prop Id: 09bd10c733e446cd49301e4db4b5b278f8d986e3543f97f02967ca0c21b264ad
Theory Prop Address: TMSEKm63FLbyd37zrxFC3ESBmnwSacXbajE

Inj1NE2

Theorem 19.6 ∀x : ι.Inj1 x ∉ {0}. The proposition is identified by the following information:

Pure Prop Id: 9e94b180d7b2cb52448d59e85f15103666357701440f2f18d5af02589fcb09ff
Pure Prop Address: TMaoFDrWfevDLaLmviNwM8qU5VfCbfQNyxm
Theory Prop Id: 3c4dd0ee065b1e59cfafc01a250811bc2c5b00c5d2b523db2b6fb82b36158b54
Theory Prop Address: TMdfQzzFY8EHNJ3UScdjLaKoU7T5kaYk9Gg

Definition 19.2 We define Inj0 to be λX.{Inj1 x|x ∈ X} of type ι → ι identified by the following information:

Pure Object Id: 8143218ffde429ff34b20ee5c938914c75e40d59cd52cc5db4114971d231ca73
Pure Object Address: TMNyr2ri5LN86eqRzW8u1M6rvw78xBZoUjT
Theory Object Id: 7d54bea1f00d9595006e5596af5681c32a7d5291e967d47e9c1f618685b8067e
Theory Object Address: TMM4pcVYzZz9VCeFoN6ah22KyBbZKKNTDAc

Inj0I

Theorem 19.7 ∀Xx : ι.x ∈ X → Inj1 x ∈ Inj0 X. The proposition is identified by the following information:

Pure Prop Id: 7fc2e938ecb986314c840c9ebc715b8faa04b3a118675b2f97af4d5a6475aa9f
Pure Prop Address: TMPfdYfdnemz7bK4pdryqUiJ88tNURoednk
Theory Prop Id: 5653780aa993662d9f130c83c987c41047896e81076539a071e059bef75c3a85
Theory Prop Address: TMWQMDFSDmKyL2MMoQEMXjXRGQkvWtCHUuQ

Inj0E

Theorem 19.8 ∀Xy : ι.y ∈ Inj0 X → ∃x : ι.x ∈ X ∧ y = Inj1 x. The proposition is identified by the following information:

Pure Prop Id: 41668460a96da96670eda3373b14c44145fecc8b7543d1eb48cc87d1a03b5873
Pure Prop Address: TMR3n3H2ExAyH57HwSFYymEuxUDc7FLsyoo
Theory Prop Id: 855213768c213323d014713167da61d832ba2c1f0e4bac38154d846f4887467d
Theory Prop Address: TMGnC4VJWDhLev8VPHUK3NMMFBEpWB7UKzh

Definition 19.3 We define Unj to be In_rec_i (λXf.{f x|x ∈ X \ {0}}) of type ι → ι identified by the following information:

Pure Object Id: f202c1f30355001f3665c854acb4fdae117f5ac555d2616236548c8309e59026
Pure Object Address: TMWjsgx6D3647vX6CZ7FACDq76sirVgNudL
Theory Object Id: 2e16753b23b9f054a644cd2c352a33cf3885b3e69791a426da64d5a3ff83d6ee
Theory Object Address: TMUQznLHs487u6czgXR1tuqBrKovEsSFuwB

Unj_eq

Theorem 19.9 ∀X : ι.Unj X = {Unj x|x ∈ X \ {0}}. The proposition is identified by the following information:

Pure Prop Id: 3b1aa65e273ab9e8836c2bc74542816dbc66db6da358e79281baa68c1eac4c51
Pure Prop Address: TMUzG5DSxEUo14WoVVEZHRUadq2y9fEMq9D
Theory Prop Id: b4c8525077a8bd9a2f4b5fe9b1db873fe4209bd38a98f300f580351d5d18d7d3
Theory Prop Address: TMLCXNWg3jeqjg6spuSQMWMiiuoNrDLSqsX

Unj_Inj1_eq

Theorem 19.10 ∀X : ι.Unj (Inj1 X) = X. The proposition is identified by the following information:

Pure Prop Id: e2cdfc6f275aeef110524d267044d56c4438d2db1293e2867fb46d60760a0f45
Pure Prop Address: TMRKw89oG3aDYwZKbJ2YCSPZoxR6oQrXVAM
Theory Prop Id: ae00e1714271d7aa2be5da6b28ba3d58bf1dcb1e86e45bd075db78fb8e268271
Theory Prop Address: TMMx1iggqKMgEC5fpswYPb7SMn6mXdV3Gd6

Inj1_inj

Theorem 19.11 ∀XY : ι.Inj1 X = Inj1 Y → X = Y. The proposition is identified by the following information:

Pure Prop Id: ce8b6635f1785f75c7b5b16d1f61111f3895e4072cc06a71b7f2548024ab9fcb
Pure Prop Address: TMPRTGLAeftK68qCgSftNHQAt5ZoPyRmGsh
Theory Prop Id: 3e918a17810ff35edf26ed66220d1d7a2ce50e44383669fb3e82f49186cdee06
Theory Prop Address: TMaXPPDHYrLmw3vrDF8cSrSzcALUJ1fr2vQ

Unj_Inj0_eq

Theorem 19.12 ∀X : ι.Unj (Inj0 X) = X. The proposition is identified by the following information:

Pure Prop Id: f3e39c79aeba9f53d5600670c3217bc096389fccc95b8a119918ee61bbd2ba1f
Pure Prop Address: TMZUKGCMAbV3pSZqA94UEu6kCWXx8F95cVU
Theory Prop Id: 3808ad5876773f87048df2f1763c5b4749fa9feb633aabecd3c5bc64e8bea8ba
Theory Prop Address: TMWJtkfszm1k5JmP9LnmZB5GUL6tyPJE95H

Inj0_inj

Theorem 19.13 ∀XY : ι.Inj0 X = Inj0 Y → X = Y. The proposition is identified by the following information:

Pure Prop Id: 662f53ab21221e41310d9ac6c9389a4c91b5f8a8c92bc0e7c16500877a0ee8d3
Pure Prop Address: TMc3veqimj5NLmacT7fJhju75JwAWeR5tRC
Theory Prop Id: 29f9450d88499de2d9169079cdcb356767832e57ed9a0111ca120c9eb731b8d1
Theory Prop Address: TMV8xsSpddijAdXaoJ58C6Ds3FGDXzmngXz

Inj0_0

Theorem 19.14 Inj0 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 171f471b1ee40b5e9e23d33ae47d474ea49798216a6286e99f7505fd596e0b1d
Pure Prop Address: TML5YcJHULJ6j353CwJ1qM8742qPffm73wi
Theory Prop Id: 57f5af3069e244db75ffa5bfd8014bb81b0c95ace6e267ef286f6f4f11003544
Theory Prop Address: TMbQhddJBnbci3edhLUCUA3LAWeGzxZF6vQ

Inj0_Inj1_neq

Theorem 19.15 ∀XY : ι.Inj0 X ≠ Inj1 Y. The proposition is identified by the following information:

Pure Prop Id: 21bb03ac63365d37ed4ef00915506a97c5a5b79e468a1019f8db539f0520ead3
Pure Prop Address: TMNaUd4a4oqrV7GxRGw17zvy3w3U9tCHd4W
Theory Prop Id: 2b14d886759a5ff3af07d29a955feeeebe2511955316846bdd6b6a0516947b7e
Theory Prop Address: TMNgqTkR1PJCufRmpkGQmktjoL6ruuRLg22

Definition 19.4 We define setsum to be λXY.{Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y } of type ι → ι → ι identified by the following information:

Pure Object Id: afe37e3e9df1ab0b3a1c5daa589ec6a68c18bf14b3d81364ac41e1812672537a
Pure Object Address: TMLfysEtzMCWeSWegZfR794mvNom3FuyKrm
Theory Object Id: 5b7e09a6e30fb5cf9e9ae6e183db0d299ed352490f06c4381c56c42c8e75cced
Theory Object Address: TMctw4uoy129ZSr7Njfx1xj2Bd38ipPNaKu

Notation. We use ⊕ as a left associative infix operator corresponding to applying term setsum.

Inj0_setsum

Theorem 19.16 ∀XY x : ι.x ∈ X → Inj0 x ∈ X ⊕ Y. The proposition is identified by the following information:

Pure Prop Id: 1c640838de3fb39c7e9ba79c5409e04f12c37645baf67043e9f7f4278de39039
Pure Prop Address: TMKTYzxjjWdrWetApF9Sj67Z6edBVD8EBTo
Theory Prop Id: 246ac5b8e5010b12d563e2f045e2abdb51d794611ea8dd590bc0fe11e5a7ad33
Theory Prop Address: TMTNL3J3wEnAzxucGw8kUWBscSaUn4BLgiu

Inj1_setsum

Theorem 19.17 ∀XY y : ι.y ∈ Y → Inj1 y ∈ X ⊕ Y. The proposition is identified by the following information:

Pure Prop Id: d47a0f2a013d89032cb64812b92012dc27b1d5fcc204fbc9a9c86638237204e2
Pure Prop Address: TMVgD1dS7zmrgJcgocmjP99ka8Dpg62ZCtL
Theory Prop Id: 2b4f9491906d1ef49131201b69a9b8bd7d2f1ebf95fe8d5c0244a559215334ac
Theory Prop Address: TMVk4zcLngHpsDrazL9rzQjzJ6jxFvj2f6c

setsum_Inj_inv

Theorem 19.18 ∀XY z : ι.z ∈ X ⊕ Y → (∃x ∈ X.z = Inj0 x) ∨ (∃y ∈ Y.z = Inj1 y). The proposition is identified by the following information:

Pure Prop Id: 645a6ce0a1cd5ad2ce3ac83a7b4d74f5954bae61db680341db619c0b10612a16
Pure Prop Address: TMGJozk41a81Kp4SygTYFZeKcuHrdPrC32q
Theory Prop Id: a36400c28b0f8902008e9de3432e7367f347381ef074614db8802940be43cbc6
Theory Prop Address: TMHiPKm9hQiccXFfb4skC57UHenaqznn87b

Inj0_setsum_0L

Theorem 19.19 ∀X : ι.0 ⊕ X = Inj0 X. The proposition is identified by the following information:

Pure Prop Id: 0be1e755f689c575a9eda08885f30343eeee939fbebeab4f3fa42931ed3bf82f
Pure Prop Address: TMK8zdDs3kr6c2kZDhfjxDumvkoP5SgQ9RW
Theory Prop Id: 36fabd108b18f5565ca33eddd506b2b62395333c16b3399474bae3abea3b0ada
Theory Prop Address: TMbUYG9UEmYDqfxYZUKHNBKT4Y75njU2mbq

Inj1_setsum_1L

Theorem 19.20 ∀X : ι.1 ⊕ X = Inj1 X. The proposition is identified by the following information:

Pure Prop Id: f8d72132080ef3e20ece3ff94da26f46176e6715a7f569d277aefa88c73c3774
Pure Prop Address: TMVZEiZNntaDr4daY1DmuXvonFq9uNTdmig
Theory Prop Id: 45b1c8dba7f1e6bb6910722f4bbf2ac5cec44dc5e8ebd74f245640488f2aa566
Theory Prop Address: TMME8xmRrBtE5QcayY5Ek96RbNEE8zSo5C2

nat_setsum1_ordsucc

Theorem 19.21 ∀n : ι.nat_p n → 1 ⊕ n = ordsucc n. The proposition is identified by the following information:

Pure Prop Id: 3be3cce904aadda436daf1b9679086129be4811093362608100e6f068bdd4039
Pure Prop Address: TMHYS9eCHQRks9avhfkoeHGSrbHGL8uaBv1
Theory Prop Id: 8fbdda58ab70c8d36db764f1f86d169ce58ba6cc946bd4cb21cc5936808f1183
Theory Prop Address: TMUdCytedFWrC4iKskiF2SDYbA8djETMgpn

setsum_0_0

Theorem 19.22 0 ⊕ 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 5dcaebebb574103d0edf890d85e23a5fa97b59db661795af9daeb7aa65f336a4
Pure Prop Address: TMaVr4aCWi6koabmAQVJ2wQdpL7mC7foUu9
Theory Prop Id: 4f3651b8f623daa8b048a8735ce8791d53d3c1d1b0f2aa681c537176ab51a619
Theory Prop Address: TMHtDbLm7fFXE2PBw9a9TaKFqUT1oKq4DFj

setsum_1_0_1

Theorem 19.23 1 ⊕ 0 = 1. The proposition is identified by the following information:

Pure Prop Id: c2107f17a63765673ccaf223da15ffd6fe6fcb31499a0a5fc5415779192120e5
Pure Prop Address: TMXomoBYE3zuVe9e7kyBAEnPduKtPVYL79A
Theory Prop Id: e8609a742bc15d0823294b218d9d7e9f9d926e9cb69df0e589c81a5f60f5366e
Theory Prop Address: TMGKXxfahnZuNyLnFqGHv1d446PYGT5n9MX

setsum_1_1_2

Theorem 19.24 1 ⊕ 1 = 2. The proposition is identified by the following information:

Pure Prop Id: 9c89c04ece2b66ca83b8a9b210a7f94ab137834db0b093883a5ab955eb6ef743
Pure Prop Address: TMSMi29gtadSxAPeBb5CCz27pw11Gt1r5Md
Theory Prop Id: ef3f41b81000348e190c1253edc936248dee43fc698ee536fa6d9f1c6e97c1aa
Theory Prop Address: TMH43YDo7oYg8zrgf9FjkpTTpmMyXLKMTTt

setsum_mon

Theorem 19.25 ∀XY W Z.X ⊆ W → Y ⊆ Z → X ⊕ Y ⊆ W ⊕ Z. The proposition is identified by the following information:

Pure Prop Id: 6e30186eb9483aaffe4c265cfe38dc213a2cf9faf3f5aa66b43cdfe7c9a5d2ca
Pure Prop Address: TMUfRBKC5kCVDQmt7f2Q83pWENdCggpCtEM
Theory Prop Id: 838d561df06bc2aee2d92713cb2b2d1c726613db9da9186614c6563f51c2bf86
Theory Prop Address: TMW7Tpt1vt99hraFFFRzz5pjENcaXQ9VV51

Definition 19.5 We define combine_funcs to be

λXY fgz.if z = Inj0 (Unj z) then f (Unj z) else g (Unj z)

of type ι → ι → (ι → ι) → (ι → ι) → ι → ι identified by the following information:

Pure Object Id: ccac4354446ce449bb1c967fa332cdf48b070fc032d9733e4c1305fb864cb72a
Pure Object Address: TMKSa978XJEMycuWYBhLHN2QGXRqqHaF4Bb
Theory Object Id: aa7bb3bbd2ec8e8fd8ba27aa97a6fc35a6ab1e3003dbbf6136a6110f1823bb93
Theory Object Address: TMJtkvSshA8XZqXDCbMyvjzjiyzZfNwnBom

combine_funcs_eq1

Theorem 19.26

∀XY.∀fg : ι → ι.∀x.combine_funcs X Y f g (Inj0 x) = f x.

The proposition is identified by the following information:

Pure Prop Id: 19f032786d18528530a8e52c17dfc6a4417805e17b1a6ee6ed1364c06b749913
Pure Prop Address: TMXtLgamyXaY4YgLpaWoWvf8a28MSYTmXZ7
Theory Prop Id: a0730ef167ec6039eaf3084667d145e2e26662596942c8973bd0711b71f18ad3
Theory Prop Address: TMa8SMHnUqoCuZw2Kk78KMtZVfqopa2somA

combine_funcs_eq2

Theorem 19.27

∀XY.∀fg : ι → ι.∀y.combine_funcs X Y f g (Inj1 y) = g y.

The proposition is identified by the following information:

Pure Prop Id: 709f34af1e90fda80a83b8965675935d92fb44648d48274e89b23dc20e8ebff5
Pure Prop Address: TMLVrnvc3Z1Pjpg39KChTrX1Jjknz3bF3ri
Theory Prop Id: 3933870bb4bccaad7d50c4972cba92adb92a7d7ff41c00d8ef4c83b8c60fad83
Theory Prop Address: TMTpr4M3N39vULCLJKgNMnuiESzd162fkqo

162 CHAPTER 19. DISJOINT UNIONS

Pairs, Sums, Functions and Products

20.1 pair_setsum

Let pair be setsum.

pair_0_0

Theorem 20.1 pair 0 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 5dcaebebb574103d0edf890d85e23a5fa97b59db661795af9daeb7aa65f336a4
Pure Prop Address: TMaVr4aCWi6koabmAQVJ2wQdpL7mC7foUu9
Theory Prop Id: 4f3651b8f623daa8b048a8735ce8791d53d3c1d1b0f2aa681c537176ab51a619
Theory Prop Address: TMHtDbLm7fFXE2PBw9a9TaKFqUT1oKq4DFj

pair_1_0_1

Theorem 20.2 pair 1 0 = 1. The proposition is identified by the following information:

Pure Prop Id: c2107f17a63765673ccaf223da15ffd6fe6fcb31499a0a5fc5415779192120e5
Pure Prop Address: TMXomoBYE3zuVe9e7kyBAEnPduKtPVYL79A
Theory Prop Id: e8609a742bc15d0823294b218d9d7e9f9d926e9cb69df0e589c81a5f60f5366e
Theory Prop Address: TMGKXxfahnZuNyLnFqGHv1d446PYGT5n9MX

pair_1_1_2

Theorem 20.3 pair 1 1 = 2. The proposition is identified by the following information:

164 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 9c89c04ece2b66ca83b8a9b210a7f94ab137834db0b093883a5ab955eb6ef743
Pure Prop Address: TMSMi29gtadSxAPeBb5CCz27pw11Gt1r5Md
Theory Prop Id: ef3f41b81000348e190c1253edc936248dee43fc698ee536fa6d9f1c6e97c1aa
Theory Prop Address: TMH43YDo7oYg8zrgf9FjkpTTpmMyXLKMTTt

nat_pair1_ordsucc

Theorem 20.4 ∀n : ι.nat_p n → pair 1 n = ordsucc n. The proposition is identified by the following information:

Pure Prop Id: 3be3cce904aadda436daf1b9679086129be4811093362608100e6f068bdd4039
Pure Prop Address: TMHYS9eCHQRks9avhfkoeHGSrbHGL8uaBv1
Theory Prop Id: 8fbdda58ab70c8d36db764f1f86d169ce58ba6cc946bd4cb21cc5936808f1183
Theory Prop Address: TMUdCytedFWrC4iKskiF2SDYbA8djETMgpn

Definition 20.1 We define proj0 to be λZ.{Unj z|z ∈ Z, ∃x : ι.Inj0 x = z} of type ι → ι identified by the following information:

Pure Object Id: 21a4888a1d196a26c9a88401c9f2b73d991cc569a069532cb5b119c4538a99d7
Pure Object Address: TMJAVsjow6vGiHhjkrvQ3hFGbgbvS9xcWyV
Theory Object Id: 9c5e4e6be64531b8ea143c51d4eae82a384c81ad95f7a944e1c670a8a30da3b9
Theory Object Address: TMaHY5KSgmVdGVqgzTcQgf4sHUsLLKmvAny

Definition 20.2 We define proj1 to be λZ.{Unj z|z ∈ Z, ∃y : ι.Inj1 y = z} of type ι → ι identified by the following information:

Pure Object Id: 7aba21bd28932bfdd0b1a640f09b1b836264d10ccbba50d568dea8389d2f8c9d
Pure Object Address: TMaSYWTJtnhHMswCbUGaheomcukAYucQua4
Theory Object Id: 7fd81e3c066cdbca377abd70b13843fd894f77ad69e67cf26842b8bc147db710
Theory Object Address: TMPJv5Do7FeQVBbPTL9ZY9H9YGMGpRV652o

Inj0_pair_0_eq

Theorem 20.5 Inj0 = pair 0. The proposition is identified by the following information:

Pure Prop Id: 0fe60101f9b90226a3183f7eee66d492a07dd05cb531e7baf46d3ab4302ca05f
Pure Prop Address: TMVm4GBpHVCYV2pJB5HL2QNkHnuSGWZ9qAh
Theory Prop Id: 490c2a25a7dca847f40f638f694a872a3f6f47d42aea7867c086f2c4cf3f6988
Theory Prop Address: TMNdMHTR5f8Y73jbhAeJgaesKTmGkru7yzB

Inj1_pair_1_eq

Theorem 20.6 Inj1 = pair 1. The proposition is identified by the following information:

Pure Prop Id: 0bbde85d4f9a32c3b689bb6a75b3954d57abb296cbd0aa2745de678af1df9a59
Pure Prop Address: TMPgPyPUzWL2JW9oss4msqgSCZ6zCy2Ztc1
Theory Prop Id: 8272d79377fa78cbfc8589d848b8f5e3862c8d19e10dd47a58c6bea9d16d8447
Theory Prop Address: TMNjuSVZSzPL5H7zv2hTWCGkBLDksUvykJN

20.1. PAIR_SETSUM 165

pairI0

Theorem 20.7 ∀XY x.x ∈ X → pair 0 x ∈ pair X Y. The proposition is identified by the following information:

Pure Prop Id: 21a4141f1e8cac366335c6139b8e6f6c143e493b4b78e1106e78e213967caf29
Pure Prop Address: TMPWuPuKr4fNYduR2p8k1ib9xWvBLuRs4wX
Theory Prop Id: 34813b26053e164b7ed43057b6f8ee6ff0cd96d01f5b6401b42ceae658468e01
Theory Prop Address: TMNQzaX8ZiddhSn6QTSqc1McYEwNG1Fe8wT

pairI1

Theorem 20.8 ∀XY y.y ∈ Y → pair 1 y ∈ pair X Y. The proposition is identified by the following information:

Pure Prop Id: bdaab525a50b9c2d070e4f67b9e4b3b931b06622695aa3c0f0b51683e1905bc7
Pure Prop Address: TMbhSedKYd5RyPq2mELrEkTMb7JkqLqrGWh
Theory Prop Id: b5248b96fbd2dbc9c65f96e27bfae42798a39bc110034aa9dcc82afc9fe0056a
Theory Prop Address: TMM3oDhF32usrf43sjdNDwEZbSftUuY8cjh

pairE

Theorem 20.9

∀XY z.z ∈ pair X Y → (∃x ∈ X.z = pair 0 x) ∨ (∃y ∈ Y.z = pair 1 y). The proposition is identified by the following information:

Pure Prop Id: aac6108d214baf821b35d72218cb0b4f9e63fc00aecbbb7a4ce9bcbb8a21b941
Pure Prop Address: TMUqfs3rToFZZEUcGSfnRKkNmLL7jyk7GqJ
Theory Prop Id: 0afdeb4f80887eb047d9b590fad88a119cbb419efa9edfb46ab666e1282c8fd4
Theory Prop Address: TMJLCz7nFpUma4iUHHjfnc85LCDmaCUqtN2

pairE0

Theorem 20.10 ∀XY x.pair 0 x ∈ pair X Y → x ∈ X. The proposition is identified by the following information:

Pure Prop Id: 042f308329163b02cbbc55bb0027f0352880a98c06f8ab271eadf3ec5e9f04e3
Pure Prop Address: TMJeqsKVgs3VYVeCG9rGS4ZNKAdR8nTs38m
Theory Prop Id: f50b255d63e6dbed7324927c7b1afe6317eea22dbd269f2a5b27d6f6565c5445
Theory Prop Address: TMLQ2VeBqCmuyodmpqK5dmQM5s98wqPSfk2

pairE1

Theorem 20.11 ∀XY y.pair 1 y ∈ pair X Y → y ∈ Y. The proposition is identified by the following information:

166 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 505cfcb3d7610169f4bfa8ba33596724c69138cb7d875cad9a309a91e2402eb6
Pure Prop Address: TMcH1uWQFgoCUsZbTGyxzwk17PtdC9cvCA6
Theory Prop Id: 30a181bf14bf2566fdcf937c9b42f3512a8a40f569e90dc2aef4aa7ae0db9e24
Theory Prop Address: TMQgeZWRp9TKUwYNKVDSo4jYfniQaLPJzRY

pairEq

Theorem 20.12

∀XY z.z ∈ pair X Y ⇔ (∃x ∈ X.z = pair 0 x) ∨ (∃y ∈ Y.z = pair 1 y). The proposition is identified by the following information:

Pure Prop Id: 98d332f11ba6fc7f438c81b60a59777921748d69f214c8a1788cc4374ca7f53b
Pure Prop Address: TMVrjFeVrNRrLEwuGQPfHQJgGH3nN34D2yd
Theory Prop Id: 65c2d317407da943881beedf59fac794f29f668a2676e85d7c1d17f0f0f1c0ca
Theory Prop Address: TMRyemtQaJ4VgBEH4b2NuAukq6NEzdWuaof

pairSubq

Theorem 20.13 ∀XY W Z.X ⊆ W → Y ⊆ Z → pair X Y ⊆ pair W Z. The proposition is identified by the following information:

Pure Prop Id: 6e30186eb9483aaffe4c265cfe38dc213a2cf9faf3f5aa66b43cdfe7c9a5d2ca
Pure Prop Address: TMUfRBKC5kCVDQmt7f2Q83pWENdCggpCtEM
Theory Prop Id: 838d561df06bc2aee2d92713cb2b2d1c726613db9da9186614c6563f51c2bf86
Theory Prop Address: TMW7Tpt1vt99hraFFFRzz5pjENcaXQ9VV51

proj0I

Theorem 20.14 ∀wu : ι.pair 0 u ∈ w → u ∈ proj0 w. The proposition is identified by the following information:

Pure Prop Id: be55b9a07966119a7062e99cbd0ce90614937b1321ad2f7982157399ef7cd51b
Pure Prop Address: TMMvCe8JzDEJMj9cXvrha66rpRYMz5CKGJ1
Theory Prop Id: 840c8df1f99f4da23ec51cac3d8dc3d97931d9d67121464862c6504eec69245a
Theory Prop Address: TMGtD6G4wVv2RV84KASZp4iaX1TDhDn2QMG

proj0E

Theorem 20.15 ∀wu : ι.u ∈ proj0 w → pair 0 u ∈ w. The proposition is identified by the following information:

Pure Prop Id: b9afa9f543fc43d88abece7f6e50c4eb30c889faceb03f4deb53230d9fb540ba
Pure Prop Address: TMQYzQmm1NFjQJzmtbAGu6ymzj9LF6Gx53A
Theory Prop Id: 5b7f586d6ff9e7dda36e45b26ab54d9324d38494897913660be264154c751908
Theory Prop Address: TMKRt7H9e4QrANP7FuuuVFh5G963n269NGA

proj1I

20.1. PAIR_SETSUM 167

Theorem 20.16 ∀wu : ι.pair 1 u ∈ w → u ∈ proj1 w. The proposition is identified by the following information:

Pure Prop Id: fbddb75c7d6c8d452a3849e8b97ab824ad9f6385bfaee6fab47eb8e903c72ff0
Pure Prop Address: TMMhHn9rASEBErsJ8RZqUMcdtgkQZHiTtjS
Theory Prop Id: 595b3cca1eb4a87a06e1fac1e30ea2a8d161b87216b075cfab0ff0066f3025c6
Theory Prop Address: TMPGXMep9UXBiUhuNpo5MxyTNz4GnkUrTMz

proj1E

Theorem 20.17 ∀wu : ι.u ∈ proj1 w → pair 1 u ∈ w. The proposition is identified by the following information:

Pure Prop Id: 06e32e59aafc4113d0d25461bbdd08d7d93a870dec775e19a9e48a763362b709
Pure Prop Address: TMTjqJpi2rtJW9qijp4JKmBpdTppjv1rP6f
Theory Prop Id: c3c3e6c3e94f9e03aeb2b4f93e33744decc142ead482a1fa50a8767e7fc99638
Theory Prop Address: TMcN2J3tSopfuEz9NRL1jfvCCu5kxE2v2No

proj0_pair_eq

Theorem 20.18 ∀XY : ι.proj0 (pair X Y) = X. The proposition is identified by the following information:

Pure Prop Id: d546f32a1c10eae38a9af6397e8d2af2759b1d0646a5f3d4868c51100c9e2395
Pure Prop Address: TMbPLyxLXKPkMnkdo72LQeKTjebPKPLSJmk
Theory Prop Id: 0c12fbce603161ada7fcfdceb916aa8184711c821cd1e1b2326d5152dfe84499
Theory Prop Address: TMHWPP4mLNCpHEx7SiPwPaQi8HiEHdVrm9b

proj1_pair_eq

Theorem 20.19 ∀XY : ι.proj1 (pair X Y) = Y. The proposition is identified by the following information:

Pure Prop Id: 9f98a8d2797cf5bb49a476c118564c127ff5edcaea373b1ca114e096c594679f
Pure Prop Address: TMLxPAuDd6Zdjkv16vcwxEZoxAdBz8adaQM
Theory Prop Id: e0cb0f860b00c8c1dea60e21e3bab80b7213e7c2da3750ce479bd67ef1eee60d
Theory Prop Address: TMM7LPbGJRfCrTrhXupSUxTM5PSBW4xxwqf

pair_inj

Theorem 20.20 ∀xywz : ι.pair x y = pair w z → x = w ∧ y = z. The proposition is identified by the following information:

Pure Prop Id: c7ef9fe33c7c1572d541f409e082fd4867a3959389d1246ddb8c6e790550ad23
Pure Prop Address: TMEvDEchwftSTVQExszQAystHF4dFnXXiiV
Theory Prop Id: 972ddf22ac0ad35ef54fbe2f14d3eb51f3932d2eb7c90de81cea04fd839a85b6
Theory Prop Address: TMTqqAmny9TT4MKV73TgCn5hxdYCZV3qB4j

pair_eta_Subq_proj

Theorem 20.21 ∀w.pair (proj0 w) (proj1 w) ⊆ w. The proposition is identified by the following information:

Pure Prop Id: a868f0131174256073ca2e69f0629c02909a5e8cddbddc8bf755c9d11a4fc8df
Pure Prop Address: TMSF6AUrFFqfp7WqPsWnmZrPyBmHXMFBof5
Theory Prop Id: 9afd8b29c6f5307394a6a98ed81326ea8d0a1e58965e3886e0fcd87155aba943
Theory Prop Address: TMNSTW7wqZuPxSL1kuFRRghsCCu3eRTUUC1

20.2 Dependent Sums

Definition 20.3 We define Sigma to be λXY.⋃_{x∈X} {pair x y|y ∈ Y x} of type ι → (ι → ι) → ι identified by the following information:

Pure Object Id: 8acbb2b5de4ab265344d713b111d82b0048c8a4bf732a67ad35f1054a4eb4642
Pure Object Address: TMMDLyvQS3eoiP32KBXjSQyuvc4keN5yMmX
Theory Object Id: d64bcbbefe13f9fc0035d2317f017d9ee3aeb3de90b836ed7cbe11d46f478799
Theory Object Address: TMS2X4GvB3DZiVdQjy5ZeDZQVVCDuJBZvgW

Notation. We use Σx, …, y. as a binder notation corresponding to a term constructed using Sigma.

pair_Sigma

Theorem 20.22 ∀X : ι.∀Y : ι → ι.∀x ∈ X.∀y ∈ Y x.pair x y ∈ Σx ∈ X.Y x. The proposition is identified by the following information:

Pure Prop Id: 0d8645a51ab956ff15d92813d2cab140e73529474636367747958b8e18bab905
Pure Prop Address: TML8ffdR3C51f2E9BMf7H2BS9S1rFurh827
Theory Prop Id: 1456cb5b576a7c97a2a8bc57c2896584b0d55dd30ca4212f1733b4c7ebe4f919
Theory Prop Address: TMaVj4zKhXkoLhyH11SzjuosBXYALhD49xN

Sigma_eta_proj0_proj1

Theorem 20.23

∀X : ι.∀Y : ι → ι.∀z ∈ (Σx ∈ X.Y x).

pair (proj0 z) (proj1 z) = z ∧ proj0 z ∈ X ∧ proj1 z ∈ Y (proj0 z).

The proposition is identified by the following information:

Pure Prop Id: afe96271dba741ef6022853ac10aa26abecb9b32d05c44f71e7c5089f782d778
Pure Prop Address: TMWpztGVWdAFQ1ovvNdLFx6cps617NMUj8f
Theory Prop Id: 77ba8e9a2110bd246cbd0efc93c8437da7906392890ad5f4156a4e526a76a950
Theory Prop Address: TMY1GRBXj483wdHtvtfTD9uZNgUp8CAiaRD

proj_Sigma_eta

Theorem 20.24

∀X : ι.∀Y : ι → ι.∀z ∈ (Σx ∈ X.Y x).pair (proj0 z) (proj1 z) = z.

The proposition is identified by the following information:

Pure Prop Id: 8b31cdfbbf34fe7c3b0423c153108eed30fc19b3ff82132f32fc21939416a2ef
Pure Prop Address: TMJH9UNCBubGXwZc46iPvjZm3iBB1y6mfHu
Theory Prop Id: f30f76cb2aef08be1838ea268e49d550fefb9694a95470f51dd35304cd46da86
Theory Prop Address: TMX4ChFaKVytLhSzjfKyAapz8j6LrbpcHM5

20.2. DEPENDENT SUMS 169

proj0_Sigma

Theorem 20.25

∀X : ι.∀Y : ι → ι.∀z : ι.z ∈ (Σx ∈ X.Y x) → proj0 z ∈ X. The proposition is identified by the following information:

Pure Prop Id: ae12a7b526c0a13cce84f529e93ced0f678ce1c9b63337256f2952f31af9eb47
Pure Prop Address: TMGDaF1fWHGqSd5NBBSVmU2WGERM8Yb3k9o
Theory Prop Id: 50686996f47a43fc8b07a9886de2c8d44fb1f23e5ebe1c95ea71b5d7a06d274a
Theory Prop Address: TMWkPywgVwGwhQpBQMdXhZTDgiXWekNhrCd

proj1_Sigma

Theorem 20.26

∀X : ι.∀Y : ι → ι.∀z : ι.z ∈ (Σx ∈ X.Y x) → proj1 z ∈ Y (proj0 z). The proposition is identified by the following information:

Pure Prop Id: 553db6938c6f87c354d6c9209f682b09c73f8cbc2eb21ab6513fee7dd2092d35
Pure Prop Address: TMaYkGoHBN1iZiAYUVzJRZLn3EgRSQWrx1H
Theory Prop Id: d79917b7b626dcc0902b44780a05f036fcf9f4445434650e2ea129fa0f955938
Theory Prop Address: TMFMACi6zbuVxKdSyevDC9cG2fhuphqdLvD

pair_Sigma_E0

Theorem 20.27 ∀X : ι.∀Y : ι → ι.∀xy : ι.pair x y ∈ (Σx ∈ X.Y x) → x ∈ X. The proposition is identified by the following information:

Pure Prop Id: c4b4172cb06b68610331df0bff40a80e131c5fe7a0f37012432d5896d5e2c4a8
Pure Prop Address: TMa5xp6pY52DfG6pfd7VzRGhjHUWXTk87nR
Theory Prop Id: 481fddcf7b10f918784064dcf83451596b1a8fec3423f6e7215608a7dccefb43
Theory Prop Address: TMRJQf5xNbV6uehrGdTPq6X4eyNfy3irJAQ

pair_Sigma_E1

Theorem 20.28 ∀X : ι.∀Y : ι → ι.∀xy : ι.pair x y ∈ (Σx ∈ X.Y x) → y ∈ Y x. The proposition is identified by the following information:

Pure Prop Id: c294a7d73a371c2ac828dbda009fa9e74a8b85adfe24c4731eb12086f8913b12
Pure Prop Address: TMMukS2p563YcLJwvCEj2kjfqr8z23JmFUE
Theory Prop Id: 81d8b0d6136a1b697d4b5753d87fb42789104c2db7a89e6023100fcb32516c3f
Theory Prop Address: TMWdjWXycEquBp15ycxwyeRHKb8YEFjrntf

Sigma_E

170 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS Theorem 20.29 ∀X : ι.∀Y : ι → ι.∀z : ι.z ∈ (Σx ∈ X.Y x) → ∃x ∈ X.∃y ∈ Y x.z = pair x y. The proposition is identified by the following information:

Pure Prop Id: 5b1bd2e14ea3dbf36ed03b2bca51d8d6329ffb7924a6198502c84e472877864e
Pure Prop Address: TMKZhCQwWmLxNmtRGpKgTrmnpUbXzpUqcC6
Theory Prop Id: 8ecfabfede2e6f041d7ca7aa6d8ddc7d93e1d4a8f77164e302775838b170ef2f
Theory Prop Address: TMHvzRUPwTDz6sNjr1Xu75jPe97kiKyiFRf

Sigma_Eq

Theorem 20.30 ∀X : ι.∀Y : ι → ι.∀z : ι.z ∈ (Σx ∈ X.Y x) ⇔ ∃x ∈ X.∃y ∈ Y x.z = pair x y. The proposition is identified by the following information:

Pure Prop Id: 49886b5979c9429dfa4b9590d2e5c308bccd95f87e7cc32f24dc682e35e497f8
Pure Prop Address: TMXKo25CKqXa5jWu42vnAgmVtNvhqv9e29Q
Theory Prop Id: e7bf483e07224467cd16dcad77572201131ce4f5471414967c469948e6ef38dd
Theory Prop Address: TMUV6V5qcBof6dwE3ebfhwBLiWuk2pD9W8w

Sigma_mon

Theorem 20.31 ∀XY : ι.X ⊆ Y → ∀ZW : ι → ι.(∀x ∈ X.Z x ⊆ W x) → (Σx ∈ X.Z x) ⊆ Σy ∈ Y.W y. The proposition is identified by the following information:

Pure Prop Id: 425a20091aa7362e407182389fe24965433da14fc0de3e08e13b5905910b2208
Pure Prop Address: TMYSaxNB5bpwjKeSA3Sf2F7RCAM2gnN5zrb
Theory Prop Id: a8addf8ad27e68150e96f0d2dbd7dd38f2d552c5eb34a1bd96b1975a2139fa44
Theory Prop Address: TMS4jg3iXSAoEt25M7crKBhYaShmP982iYR

Sigma_mon0

Theorem 20.32

∀XY : ι.X ⊆ Y → ∀Z : ι → ι.(Σx ∈ X.Z x) ⊆ Σy ∈ Y.Z y. The proposition is identified by the following information:

Pure Prop Id: 9b3fbfc10d8cb198135b15ef9120823b40c83a7048721a2a8f5bd592dccd207e
Pure Prop Address: TMHvdHmzNtjNYsphs5AwquVMExTDwRKzC6i
Theory Prop Id: 0c39e5d90a03c8b53d1f91b86500a8426061994e891ba8d4f0abf6ed55874388
Theory Prop Address: TMG1irPWX7hSKZCJSoz5U9WvofwpYiLcPrD

20.2. DEPENDENT SUMS 171

Sigma_mon1

Theorem 20.33 ∀X : ι.∀ZW : ι → ι.(∀x.x ∈ X → Z x ⊆ W x) → (Σx ∈ X.Z x) ⊆ Σx ∈ X.W x. The proposition is identified by the following information:

Pure Prop Id: 8f6c670d9cf58964547a2c8f5ecb2655c51fed679a89fcbaf042a9a98576b22c
Pure Prop Address: TMHxEXAifEfqBZatTVRBvvzdAKBzqA8xJC8
Theory Prop Id: 8a20d4273c3bd449f68d0446ea954be5016f26a33d1b32bc491ace72b24bfce8
Theory Prop Address: TMVezD2kkz73gqZqGBDvxeirmP7cryxcrP2

Sigma_Power_1

Theorem 20.34 ∀X : ι.X ∈ Power 1 → ∀Y : ι → ι.(∀x ∈ X.Y x ∈ Power 1) → (Σx ∈ X.Y x) ∈ Power 1. The proposition is identified by the following information:

Pure Prop Id: 7a3ae673c65b20d28f4b80b127c87fe8effc6abadb514348178f01786f8f687a
Pure Prop Address: TMTUPmfM2Mi7ztuVrQGCeVRgWXipKh9vVXb
Theory Prop Id: 889ba9237de623f9140e05e0574c4f3e91b2c3c0df705ac3109723a3a09dda78
Theory Prop Address: TMZYULyhhi7GEPAnwEARVDsEJgpr2MpVTbG

Definition 20.4 We define setprod to be λXY : ι.Σx ∈ X.Y of type ι → ι → ι identified by the following information:

Pure Object Id: fc0b600a21afd76820f1fb74092d9aa81687f3b0199e9b493dafd3e283c6e8ca
Pure Object Address: TMX5iqj6cvWjd5eoMqqTEF1367VHDQkj6Yq
Theory Object Id: 107b6bb74ab2c1972d7258ba373ae94a01269f3d05e17f604d541e405e9ab93f
Theory Object Address: TMFyVvPCFYsHfsACwhMzeFYF8jZqyyWje7X

Notation. We use × as a left associative infix operator corresponding to applying term setprod.

pair_setprod

Theorem 20.35 ∀XY : ι.∀(x ∈ X)(y ∈ Y).pair x y ∈ X × Y. The proposition is identified by the following information:

Pure Prop Id: f55ea3828e02273317f0b400d2105e5f6809d31a9ac1326fe0fea71657f5048d
Pure Prop Address: TMP1cFf4SMJvCYXWGuPAbdAT3rrBv9oFfUd
Theory Prop Id: 1dd4994dec4372a29bd9c00f15aa916beb53d6a66b15c1d25e72219e6a2d006c
Theory Prop Address: TMJCXgkH2PkoAWeAmeW98Fvbhtn48QfWyEd

proj0_setprod

Theorem 20.36 ∀XY : ι.∀z ∈ X × Y .proj0 z ∈ X. The proposition is identified by the following information:

Pure Prop Id: 510623d6d9fee58fe36195a65b68d9e88774298fd789c76bb947e43ba0e9ddd3
Pure Prop Address: TMbvsu8j8H6CMFRSSMgZzBoz1STTTdmJCQv
Theory Prop Id: f7a519336a9ebe4633840fe128d350ab70461bc5396fe75659857496a4df6c8e
Theory Prop Address: TMQGsuKAnuZCk7jn3UYPfyZCtmqNjDLEXm9

proj1_setprod

Theorem 20.37 ∀XY : ι.∀z ∈ X × Y .proj1 z ∈ Y. The proposition is identified by the following information:

Pure Prop Id: f6765c051b33e8891d8407887b98ac6204877cdb39b1c5336a548c49859ce225
Pure Prop Address: TMMHAkHerqWaT2pFe3dt7C1yXqqaPT3Lotj
Theory Prop Id: afe85e979f6864648f01f74dec8776c350283f9b19d37ad3b00f2558dc5da050
Theory Prop Address: TMVQEexDvYgcxvr3ptPV72JSsdGheT6ge7G

pair_setprod_E0

Theorem 20.38 ∀XY xy : ι.pair x y ∈ X × Y → x ∈ X. The proposition is identified by the following information:

Pure Prop Id: 107e92452bde569e548509981b8ad6b3760e221a2995bd8b8b11748c06428f6e
Pure Prop Address: TMKaLir23aqUncRog77yHWNUteSZpVWVbf5
Theory Prop Id: 92b4c14b744c79aa7e9d294133d7983d9ba8f30c9cc011c9d079a5f451447047
Theory Prop Address: TMSWuEEd13oCxUA81Augy7MtxEyUeafaiQQ

pair_setprod_E1

Theorem 20.39 ∀XY xy : ι.pair x y ∈ X × Y → y ∈ Y. The proposition is identified by the following information:

Pure Prop Id: 50179b781626f48b09be181df9f551b18fbc72192d119884122eea0b4427b916
Pure Prop Address: TMTFx9j8XQY7mMx8nXnVLkS1ReCnJvoc5VR
Theory Prop Id: 4e83a82fa29d4046c8ca4bd9154e06e2992eba3ed652c3aa954d141cccef49be
Theory Prop Address: TMYMex5ZPQm8KKBnWBPkZD51i9QL5Ao4tHE

20.3 Functions

Let lam : ι → (ι → ι) → ι be Sigma. We write λx ∈ X.t for lam X (λx.t). Note that Σ_{x∈X}.t and λx ∈ X.t are two notations for the same term.

Definition 20.5 We define ap to be λfx.{proj1 z|z ∈ f, ∃y : ι.z = pair x y} of type ι → ι → ι identified by the following information:

20.3. FUNCTIONS 173

Pure Object Id: c7aaa670ef9b6f678b8cf10b13b2571e4dc1e6fde061d1f641a5fa6c30c09d46
Pure Object Address: TMRkKHvzv8Tta9AhRXsVE4k7dQ68XWymCoD
Theory Object Id: a4d82c5a3643a7e174676c313ef995e0307b3b56f57e8ae761b7c6033bbc40db
Theory Object Address: TMULGKR83zvLzsaT5wtaLqEme8A8TV9uVcG

Notation. When x and y are of type ι, then xy means ap x y.

lamI

Theorem 20.40 ∀X : ι.∀F : ι → ι.∀x ∈ X.∀y ∈ F x.pair x y ∈ λx ∈ X.F x. The proposition is identified by the following information:

Pure Prop Id: 0d8645a51ab956ff15d92813d2cab140e73529474636367747958b8e18bab905
Pure Prop Address: TML8ffdR3C51f2E9BMf7H2BS9S1rFurh827
Theory Prop Id: 1456cb5b576a7c97a2a8bc57c2896584b0d55dd30ca4212f1733b4c7ebe4f919
Theory Prop Address: TMaVj4zKhXkoLhyH11SzjuosBXYALhD49xN

lamE

Theorem 20.41 ∀X : ι.∀F : ι → ι.∀z : ι.z ∈ (λx ∈ X.F x) → ∃x ∈ X.∃y ∈ F x.z = pair x y. The proposition is identified by the following information:

Pure Prop Id: 5b1bd2e14ea3dbf36ed03b2bca51d8d6329ffb7924a6198502c84e472877864e
Pure Prop Address: TMKZhCQwWmLxNmtRGpKgTrmnpUbXzpUqcC6
Theory Prop Id: 8ecfabfede2e6f041d7ca7aa6d8ddc7d93e1d4a8f77164e302775838b170ef2f
Theory Prop Address: TMHvzRUPwTDz6sNjr1Xu75jPe97kiKyiFRf

lamEq

Theorem 20.42 ∀X : ι.∀F : ι → ι.∀z.z ∈ (λx ∈ X.F x) ⇔ ∃x ∈ X.∃y ∈ F x.z = pair x y. The proposition is identified by the following information:

Pure Prop Id: 49886b5979c9429dfa4b9590d2e5c308bccd95f87e7cc32f24dc682e35e497f8
Pure Prop Address: TMXKo25CKqXa5jWu42vnAgmVtNvhqv9e29Q
Theory Prop Id: e7bf483e07224467cd16dcad77572201131ce4f5471414967c469948e6ef38dd
Theory Prop Address: TMUV6V5qcBof6dwE3ebfhwBLiWuk2pD9W8w

apI

Theorem 20.43 ∀fxy.pair x y ∈ f → y ∈ f x. The proposition is identified by the following information:

174 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 64667c3e12b9f3ae61ec356833747e367523a8c0ce6c2a9d93f4c38b0783bde5
Pure Prop Address: TMZLPpY9KNgV39N2yHL65R4C438SdAntCJF
Theory Prop Id: 7708163781ef5b94401f65bdb2a5b080ff257a5930faf05dd1bbe0b4789e9537
Theory Prop Address: TMPak8MUV4CmmTFj6jaPQ3fYN19JukFbBVX

apE

Theorem 20.44 ∀fxy.y ∈ f x → pair x y ∈ f. The proposition is identified by the following information:

Pure Prop Id: 85693a617f5d7231598e57242e17d01ba19426804680415415d83c36ca14dcf3
Pure Prop Address: TMM471VfyY9L9ihJZoGVZy3MTJucpLkPz71
Theory Prop Id: c08565940adcaebbb0bd8ea7dca3ebc52f096c6a861e7e3eb1dfcc522c057448
Theory Prop Address: TMNq9VhD4hU2LQs2wbRmvWrApYxACcfdNyd

apEq

Theorem 20.45 ∀fxy.y ∈ f x ⇔ pair x y ∈ f. The proposition is identified by the following information:

Pure Prop Id: 4c9d67e97abd0fc158c7647a05ca09ff2992ddd126c6b8d0b452e70e8cbcc406
Pure Prop Address: TMZQ7PrqUw2JPS363w3buW9bffJsBTNttxy
Theory Prop Id: bd396ee22cc900202a9db5ef15b2b6dc151ca2285723ce93f3f245def051fbe2
Theory Prop Address: TMSiUYhpZzUqv5JfXdvkMd9Vyn2Xj1gRY8z

beta

Theorem 20.46 ∀X : ι.∀F : ι → ι.∀x : ι.x ∈ X → (λx ∈ X.F x) x = F x. The proposition is identified by the following information:

Pure Prop Id: 95991de564f22218049f5c01a2054c9f531e2a0e7f89348f4cd3cce56269ee6a
Pure Prop Address: TMW5SkUgFpNvxZtwNs5tigJKy4uVF4aLEX4
Theory Prop Id: b5182dff18aa9200707f42f128e18ba010c3c5b0a3a598682cfd9fe3e36def5d
Theory Prop Address: TMUK7trdHToYkqBA9FA7nyBfnGCJVnzMrdt

beta0

Theorem 20.47 ∀X : ι.∀F : ι → ι.∀x : ι.x ∉ X → (λx ∈ X.F x) x = 0. The proposition is identified by the following information:

Pure Prop Id: 3bb4348d3945f138f08291a2d198487854d211b0b272222fe4f9e932e577522b
Pure Prop Address: TMc9Mv7ukGPvjkLta2ci5R66AEtkDgNuCSM
Theory Prop Id: 475c212b14b3b0ca0e09cf957e6ba095fef5db59bc5b0dcdbe6adbb59a281cce
Theory Prop Address: TMFBwjzjE5yzpGy66RMUx1PdJ9mHVfc6dHb

proj0_ap_0

Theorem 20.48 ∀u.proj0 u = u 0. The proposition is identified by the following information:

20.3. FUNCTIONS 175

Pure Prop Id: 588f379b68a9458a540fcdf77dc5d2f4783c00f4cce95dc51c95471cc943eb03
Pure Prop Address: TMYCXMFZ4HzrBKRUDMYgmo4eHSvQVMQ7p31
Theory Prop Id: d4f3513ee04ff36d85b2db23be3080385b47ffa013ac715ce8658accd5d66b38
Theory Prop Address: TMSgBry8vyDWLoFEMZhop4n2xDSn5k839xF

proj1_ap_1

Theorem 20.49 ∀u.proj1 u = u 1. The proposition is identified by the following information:

Pure Prop Id: d116b3ec53d3345a265357a353929b0e25b62670c0a9b2a3d51a8c73ef732f19
Pure Prop Address: TMNoHiTHocDKAmUg9hNSv1FxD2Amd3S3tUP
Theory Prop Id: 2328b0e51e8e97743e5baf0897eb65244f631c1e1667f87ea13b28f83b8939b1
Theory Prop Address: TMRaYAR6WpaEp6hPTm43iVAHVqRrJwCb9di

pair_ap_0

Theorem 20.50 ∀xy : ι.(pair x y) 0 = x. The proposition is identified by the following information:

Pure Prop Id: ac34b9cde0ee2208ea8e422cba572e3fb7982b46ebfdd714251b2dfcb897193e
Pure Prop Address: TMVVcCiRogXNQekjrhetLEc1AwKsjAoaCUb
Theory Prop Id: bb047796b50fbb719f3e14fe10b517fd118f2911a01dac2dc1972ece4948275e
Theory Prop Address: TMPNWnANeRPomJ9tENqayKNjmwfidUynXLZ

pair_ap_1

Theorem 20.51 ∀xy : ι.(pair x y) 1 = y. The proposition is identified by the following information:

Pure Prop Id: 2274fc6211c808c3b1604a439df9779ba0f1ddad055df119366dc394c2fb97f6
Pure Prop Address: TMSs4jYzqSwfDX666rdcAD6CeBDVaoMRoyc
Theory Prop Id: cce3d21c0d689ca2584d628eb0fdd682ec793195238268afe769ce51f75bb164
Theory Prop Address: TMFmHmqub8mZKhdbCxJVhYYkdtHahTZ7UAh

pair_ap_n2

Theorem 20.52 ∀xyi : ι.i ∉ 2 → (pair x y) i = 0. The proposition is identified by the following information:

Pure Prop Id: aa4c5b4688eb6b4f059f6d4ecc867417bf44703f3f25ae2c2a4bb84351e4918a
Pure Prop Address: TMJvhJVjoT8tymCLYAEd8KMpMSd7c5dzvrs
Theory Prop Id: 6a36da546596b4c9b0cab2739118ca96148377b2acb8a70b37d0a4aa0d404b22
Theory Prop Address: TMapLv3NXF9X9vWbexn1gRz4mSdux6SZs6w

ap0_Sigma

Theorem 20.53 ∀X : ι.∀Y : ι → ι.∀z : ι.z ∈ (Σx ∈ X.Y x) → (z 0) ∈ X. The proposition is identified by the following information:

176 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 861a795017a942802f4443c95b5a36332a33f167ce1441dfc6a24503f6268b3e
Pure Prop Address: TMKRf5LGKqydnVhB8vpAzDvodiXpqNvrMxB
Theory Prop Id: 1affcc338ae841e0e000cf9be970918006435331f595cedc951a44209748026c
Theory Prop Address: TMUBP4wFdxgzoA73DJqdvnWDxq8D4ZPGmSV

ap1_Sigma

Theorem 20.54

∀X : ι.∀Y : ι → ι.∀z : ι.z ∈ (Σx ∈ X.Y x) → (z 1) ∈ (Y (z 0)). The proposition is identified by the following information:

Pure Prop Id: ad3bfe036fa895ae325d64498611356bb49df8ebd34b91be258534581bb43fb7
Pure Prop Address: TMQ3NDTgxp9a1VuLPqoNmEGA5MjKwt6QJLH
Theory Prop Id: 520af5f2f80e3d8411b1460e91384a058099174706d3029c6f03f1ab389a8f32
Theory Prop Address: TMKB8YXKTKjDb1T1UTPbMwrfxTLy3KBSoJv

Definition 20.6 We define pair_p to be λu : ι.pair (u 0) (u 1) = u of type ι → o identified by the following information:

Pure Object Id: ade2bb455de320054265fc5dccac77731c4aa2464b054286194d634707d2e120
Pure Object Address: TMSvXNQFUtikJWHkJ8EmjpLPVE6eVHHPye3
Theory Object Id: 5c8a35346b94416c8921f9496e68b956c0e80aa2f5b966805dd30a7cc9bdb03e
Theory Object Address: TMbuqZXCg75fnX39GezV6NmULFhCWCyVQLX

pair_p_I

Theorem 20.55 ∀xy.pair_p (pair x y). The proposition is identified by the following information:

Pure Prop Id: 06d9a28b2c025648a5c4c367e18ca07d131c2483cbec5ef5ed9b8c412a2959e3
Pure Prop Address: TMJoDcS8DRqyUSNGb82nnjwSRKn9hqLmFdN
Theory Prop Id: 8f2fdee4c2a4b4668940264d97e6a4d34887ed1cb2b9462e5a68ea86f3267c92
Theory Prop Address: TMPcM8MemhXia2UuEVstbB2UDdYPEQwK1R1

pair_p_I2

Theorem 20.56 ∀w.(∀u ∈ w.pair_p u ∧ u 0 ∈ 2) → pair_p w. The proposition is identified by the following information:

Pure Prop Id: e4b35e6cad38e54d2a79aadbe2af0245099589e5c7fade68162ef8871c460979
Pure Prop Address: TMJUphhHJCGQn6H3JsbDsYQ2vhJYQpGSSq6
Theory Prop Id: f66937108389bce047e36e45cd53689abfbd69e81d439e978b5d6f03a84804fd
Theory Prop Address: TMUhxxoG6nPHQGfqpNzk4M5VX4pepkB3W7b

pair_p_In_ap

Theorem 20.57 ∀wf.pair_p w → w ∈ f → w 1 ∈ ap f (w 0). The proposition is identified by the following information:

Pure Prop Id: e93eb76f4ff4537fc952c7e2663dd348eba0a1eda133a280fa73a98aaed1afd7
Pure Prop Address: TMMpbcE9Nf47Qr4V5rA9WJeQ3beYck2RvoP
Theory Prop Id: cda5c674d061ba7e438ebe2318a287bc128928b90a50c0bbd5f76eb976eda17a
Theory Prop Address: TMVJ3KJ5wJShTfj2JebzJWxoqicyYhHmzhf

20.4. TUPLES AS FUNCTIONS ON NATURALS 177

20.4 Tuples as Functions on Naturals

Definition 20.7 We define tuple_p to be

λnu.∀z ∈ u.∃i ∈ n.∃x : ι.z = pair i x

of type ι → ι → o identified by the following information:

Pure Object Id: 83d48a514448966e1b5b259c5f356357f149bf0184a37feae52fc80d5e8083e5
Pure Object Address: TMPQRFB8kv9Mi3hJcbAVVFkKaCjpTiKn5DP
Theory Object Id: 7b913e689b741f5e3d1a5c19e7580fd46ffd026760b86242807b35879571cbe3
Theory Object Address: TMSwuuEUBwMbLNhY2oHRChGyRGNSMgRmKA1

pair_p_tuple2

Theorem 20.58 pair_p = tuple_p 2. The proposition is identified by the following information:

Pure Prop Id: edb4f2daac705fc5fde5fb625306e4ce3b70b5c324811dcfe9286a3352609130
Pure Prop Address: TMa7Ph4F792pWejWQxMqTYL67K6y3Atz3GB
Theory Prop Id: 4ab8c71265b072dffa2255ec379c857d2480a9f4cc0b5722bf710c7b2e3aaa21
Theory Prop Address: TMPDLkLax6xryGqZWfjiAzsWiueQwkeoq77

Notation. When x_{0}, …, x_{n−1} are of type ι, we write (x_{0}, …, x_{n−1}) for λi ∈ n.if i = 0 then x_{0} else ⋯ if i = n − 1 then x_{n−1} else 0.

tuple_p_2_tuple

Theorem 20.59 ∀xy : ι.tuple_p 2 (x, y). The proposition is identified by the following information:

Pure Prop Id: 982145ec24f8189cb7ff562fa9a82c89405822de093f290a7c78ff1b6560c79b
Pure Prop Address: TMRXKTaQZ2mMDiGAZrvuMg3ozPJcAZAgWGP
Theory Prop Id: f8f972a715455be65b85075f8f7f738798be5894513ebfc46853439a15150d79
Theory Prop Address: TMX5VxQzTL4dEJu18K3FBfoBgfDf59XiNpV

tuple_pair

Theorem 20.60 ∀xy : ι.pair x y = (x, y). The proposition is identified by the following information:

Pure Prop Id: c122f6c45d6ae2b29699ba19714006844818fe6aed0b78c730fa61d541f8f655
Pure Prop Address: TMNGGwosAVCCgzqRpMz6fsnenJVoBPWXYUc
Theory Prop Id: 18198e977ce2f209097f879f8fb82081371242788607f8a3fa8da991d443ee60
Theory Prop Address: TMUTEozs7LjRCKnvt77suPHQfR7gNvzEi7v

178 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

20.5 Dependent Products

Definition 20.8 We define Pi to be

λXY.{f ∈ Power (Σx ∈ X.Union (Y x))|∀x ∈ X.f x ∈ Y x} of type ι → (ι → ι) → ι identified by the following information:

Pure Object Id: 40f1c38f975e69e2ae664b3395551df5820b0ba6a93a228eba771fc2a4551293
Pure Object Address: TMMmsr8jojnNQnfEu32oXr8oHW6b2JucDYn
Theory Object Id: a7b1b09cb152a467eba4c7154e043aede9f10dd364de00ce2145c5178d3ad12b
Theory Object Address: TMb8zzbN5gj4sGhAcwYC7Re7qCi97Rtgo7h

Notation. We use Πx, …, y. as a binder notation corresponding to a term constructed using Pi.

PiI

Theorem 20.61

\begin{matrix}∀X : ι.∀Y : ι → ι.∀f : ι.(∀u ∈ f.pair_p u ∧ u 0 ∈ X) → \\ (∀x ∈ X.f x ∈ Y x) → f ∈ Πx ∈ X.Y x. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 2b7ef42e911f5c9d245f0e1650bdf7a017dc07c61643b180beacd902afa937ac
Pure Prop Address: TMHGz8t2EnwQDiuH6KDQpg8EnyxGgR8dS7e
Theory Prop Id: 1f8f6f437998cacea340ec1ea9cc9798e39298aa0d85498c0c20643140f46942
Theory Prop Address: TMWx4ti6JwTdu8xygg15tK2YVvzLMBHLWpP

PiE

Theorem 20.62

\begin{matrix}∀X : ι.∀Y : ι → ι.∀f : ι.f ∈ (Πx ∈ X.Y x) → \\ (∀u ∈ f.pair_p u ∧ u 0 ∈ X) ∧ (∀x ∈ X.f x ∈ Y x). \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: e900d056f38f163f77287b64052d5d9e602edf6c698bce849b83205e4fecd08a
Pure Prop Address: TMQswaTQQ8v2NGGPZrcYHSjahdLxdVqq8Bc
Theory Prop Id: c6a19abb37631cc1754b1e349cb516006e88cf3a4a1da2d0f9de9154571fabdc
Theory Prop Address: TMNmPoYTgXJ1oQ5bHuJoN3MvGSuMrhGxKb8

PiEq

20.5. DEPENDENT PRODUCTS 179 Theorem 20.63

\begin{matrix}∀X : ι.∀Y : ι → ι.∀f : ι.f ∈ Pi X Y ⇔ \\ (∀u ∈ f.pair_p u ∧ u 0 ∈ X) ∧ (∀x ∈ X.f x ∈ Y x). \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: aee68f00e9f9b4dfa4f93208b50b84f39e34d3143871795718db72cf160eceba
Pure Prop Address: TMLoctKVTPvcs1L1sQJmpzmMB7ZGefxweHF
Theory Prop Id: e80f95c3f8e7d257d04588c36cf7c2cbcef829f184e76311e443137e0d9bf708
Theory Prop Address: TMJq3SLpMrtKWAEFsc6WRLhigo1buRia2SA

lam_Pi

Theorem 20.64

\begin{matrix}∀X : ι.∀Y : ι → ι.∀F : ι → ι.(∀x ∈ X.F x ∈ Y x) → \\ (λx ∈ X.F x) ∈ (Πx ∈ X.Y x). \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: c0b88f5343d69ece7a5c1776f75f883b1ea441d9f54b23c0f2f9bfd3d63291ce
Pure Prop Address: TMK1tiobRV7Rk8oLJDiqRt9emhAiARd65fd
Theory Prop Id: 3733f8d8b8cfb744a472ecadfd3dd8ec6a02c94d9f4ba9cb8fe6fa05f899b65e
Theory Prop Address: TMR7xMjyWg42N9C62nDT2fP29rt63NXuwrb

ap_Pi

Theorem 20.65 ∀X : ι.∀Y : ι → ι.∀f : ι.∀x : ι.f ∈ (Πx ∈ X.Y x) → x ∈ X → f x ∈ Y x. The proposition is identified by the following information:

Pure Prop Id: 7c36179a82d8644940659a1f02b440b62c39cf8ece8fc0d2538fccfd980eaa68
Pure Prop Address: TMXNoSqzT5Lu24RbFYcKt5N8JRNvmf1MDSm
Theory Prop Id: cb962ef0bbe77996198527ec24108318b6142c9b6d3b1cacdc185efa044e94f6
Theory Prop Address: TMFXJjxwrkNdrbgK4hwdYsRNhanGr8DEELP

Pi_ext_Subq

Theorem 20.66

∀X : ι.∀Y : ι → ι.∀fg ∈ (Πx ∈ X.Y x).(∀x ∈ X.f x ⊆ g x) → f ⊆ g. The proposition is identified by the following information:

180 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 895d76cfaedb48089543791c77f82e5ff8402576798d94cc12c928b42720b3c7
Pure Prop Address: TMK7g7rTpdWeWhZFQAosbFDbAB6K9oH1vVL
Theory Prop Id: fcbe2211dda51b2c7c056b9eb958563051102a56a57c22caef81a1ef81d5955c
Theory Prop Address: TMQb6BZWimWiUUD2TKu7WGhQkSQpf98JfKA

Pi_ext

Theorem 20.67

∀X : ι.∀Y : ι → ι.∀fg ∈ (Πx ∈ X.Y x).(∀x ∈ X.f x = g x) → f = g. The proposition is identified by the following information:

Pure Prop Id: afabf60d2f0c584af57bc85ddf915b13c2173fba3ae62a67a3862a472418d979
Pure Prop Address: TMZQyqEpgM1nAy5SpShxXSu4QC8bRbftnW5
Theory Prop Id: 08bc8829905fa1c9330919448051750a09ed9872e5fe47a6c8f242ef8905a6e1
Theory Prop Address: TMFnU1vnCxgdDuPv6P9CoEak4jkUTjULRYT

Pi_eta

Theorem 20.68

∀X : ι.∀Y : ι → ι.∀f : ι.f ∈ (Πx ∈ X.Y x) → (λx ∈ X.f x) = f. The proposition is identified by the following information:

Pure Prop Id: 9ca3eb4ba8300315de6cc6aa4bc0569bf6750d0a55b1e27929bafce0326878c2
Pure Prop Address: TMZ1A4m5suKjS6q8ujjemybzpezfVbf3Ja5
Theory Prop Id: 9bd10ac78e1b9dd36b301ed8e06904d4553c0bd53f28e2dd52662fee5e5fd680
Theory Prop Address: TMXWQh8ieRCGD99r9bmA7JmXyAU4rDWeass

Definition 20.9 We define setexp to be λXY : ι.Πy ∈ Y.X of type ι → ι → ι identified by the following information:

Pure Object Id: 1de7fcfb8d27df15536f5567084f73d70d2b8526ee5d05012e7c9722ec76a8a6
Pure Object Address: TMdMKJiLvRmU1utdH9i5YSFhSMeAkFKRH1a
Theory Object Id: 021ae079aee80b7f24175c08d98ddd174bcabd841750953bdefaa681a83714e7
Theory Object Address: TMMDiZSCEZ1dDZ4PWaQHcXQkH4ocT26aDjJ

Notation. We use superscripts as notation for applying the term setexp.

pair_tuple_fun

Theorem 20.69 pair = (λxy.(x, y)). The proposition is identified by the following information:

Pure Prop Id: b02dfae335ec8db395520722941b8dc3a0d6514527797822554c3261ab3f1cb8
Pure Prop Address: TMQLyvPjAxztmV8UGpQ822nUAH8ziqxafSL
Theory Prop Id: ed37d0c53ab39329962382f39457e2117aa8fbbe349e0cb167d2c2d76af2a96e
Theory Prop Address: TMQyWvvFdsJND6mF8ATSUatvvo9p2RAUqY3

20.6. PAIRS AS TUPLES 181

lamI2

Theorem 20.70 ∀X.∀F : ι → ι.∀x ∈ X.∀y ∈ F x.(x, y) ∈ λx ∈ X.F x. The proposition is identified by the following information:

Pure Prop Id: d4a9d90bcc14a1607b933d2b6324f74a12e560f1ee23d66bbef65f0ab2300245
Pure Prop Address: TMTxDywb49ok9weqR5PA1YkH5HYVVQgfmLa
Theory Prop Id: de381fee5dd742f3e056c75fdc7abdebc389a9d213f325ce23fff6b0562da398
Theory Prop Address: TMYB1uFbN7KQg4EMgzCmiEUJxJcauCrH6gi

lamE2

Theorem 20.71

∀X.∀F : ι → ι.∀z : ι.z ∈ (λx ∈ X.F x) → ∃x ∈ X.∃y ∈ F x.z = (x, y). The proposition is identified by the following information:

Pure Prop Id: 2907d7e677f8d705f7bf5bfe75cf312a139a28537d4ee131f50b8e62731177d7
Pure Prop Address: TMMNkCJJRFygiBFaXLzPkZ9eYzm8gs5REwb
Theory Prop Id: c69df21f0c5d7582f84f032104dcbf0cdde6b5a1e91fbe54831893e9ab547722
Theory Prop Address: TMZ49ENvtTqA2Tv7yJYdvGtiYB5BtFdTeVV

tuple_2_inj

Theorem 20.72 ∀xywz : ι.(x, y) = (w, z) → x = w ∧ y = z. The proposition is identified by the following information:

Pure Prop Id: 5a5be1cedd8243deb6c455ce395e463e404fca87c52b0a3a277a9b3855cb78e9
Pure Prop Address: TMFbqwXKr8H1bgcdquem9YDZJMmsUJmHeKC
Theory Prop Id: 8feaceee2c95bc779292a107468e26dcc3aeefca4f4cec066bd23102d06d83ad
Theory Prop Address: TMNPCfqHk5t1KqeT14ns1V9EhdHeBU9wLyc

20.6 Pairs as Tuples

Let x0, x1 : ι be given.

tuple_2_0_eq

Theorem 20.73 (x0, x1) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: 42d4527df30bd5915ecca74999921e236618128fee94e9018ab8452c7d825590
Pure Prop Address: TMQ8nhtfV6W3JDU3mscgLR5QMXHfDZgZYK4
Theory Prop Id: ad903742c17c3fe99f8b82cdfafdc7c72f5443fcb7032eb137490fe2367b8749
Theory Prop Address: TMMmXMGjkWqC3h5qQxyBwLzymNx8CZr1RZA

tuple_2_1_eq

Theorem 20.74 (x0, x1) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 596569b2cc94a7411cd00ed3dcc85f68340782a2dea77d8454b30f81c7eb6661
Pure Prop Address: TMJ9iszZSDm4Q8D744Pr5ok8cAppzRJapau
Theory Prop Id: 2c0ad17788686eabbd5f29d00be65516a9112c392a4d4466c10b9df595bb100e
Theory Prop Address: TMNHajqbPTVi7TYeCTT1YBrSfZg2Srt6YhV

182 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

20.6.1 Abstracting with Two Variables

Definition 20.10 We define Sep2 to be

λXY R.{u ∈ Σx ∈ X.Y x|R (u 0) (u 1)}

of type ι → (ι → ι) → (ι → ι → o) → ι identified by the following information:

Pure Object Id: b86d403403efb36530cd2adfbcacc86b2c6986a54e80eb4305bfdd0e0b457810
Pure Object Address: TMJBzUyWurQtsdPSJTbqaZih7trGyeYoC3m
Theory Object Id: fc03e85944df87e6a24b2bdf2798d99cf3faa3ada8ae8ebef483367bb485a195
Theory Object Address: TMYXWf8P7ybU2LjpjrRcGd9ZaYEfsgynmzt

Sep2I

Theorem 20.75

\begin{matrix}∀X.∀Y : ι → ι.∀R : ι → ι → o.∀x ∈ X.∀y ∈ Y x. \\ R x y → (x, y) ∈ Sep2 X Y R. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: f2dd7d7d0de46ec460fe30813842b2da107da11fbc9608d96eea869888811cca
Pure Prop Address: TMZdgPdsrcEjsnAmQm7foGQpRLeMQ4R8i64
Theory Prop Id: 35493133c295677bdfd5c4e9ab9236a9a50b7d9c1c379734567bf32375fc9508
Theory Prop Address: TMYNAkcLqkkyU7u7qUwEAZxTvqQnKYD6FxK

Sep2E

Theorem 20.76

\begin{matrix}∀X.∀Y : ι → ι.∀R : ι → ι → o.∀u ∈ Sep2 X Y R. \\ ∃x ∈ X.∃y ∈ Y x.u = (x, y) ∧ R x y. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: aca0dd1d7527fc7cb76890b26390b9351092342fe8f0f530c801862f12edf753
Pure Prop Address: TMMMFQtAKedKiQ38BC5pn49TBQDJSeKoTYo
Theory Prop Id: f094fb5ccc800bcb94dec37b2867363bf66db0ad9e299e07454e901889561056
Theory Prop Address: TMYo6UtpLGR595GYoBTpBoawMbFG3BTJ6aH

Sep2E'

Theorem 20.77

\begin{matrix}∀X.∀Y : ι → ι.∀R : ι → ι → o. \\ ∀xy.(x, y) ∈ Sep2 X Y R → x ∈ X ∧ y ∈ Y x ∧ R x y. \end{matrix}

The proposition is identified by the following information:

20.6. PAIRS AS TUPLES 183

Pure Prop Id: c5ad5457a7903751d537c31ccfbac1f4b84637cc2029ba26617d81ccecc36758
Pure Prop Address: TMbEYwrs85BumegRS4Nue1VQK9pgGsjNjk6
Theory Prop Id: f89326bba190a5f12b5139594f24a9ffae64a5b1b99d9ad4ab4e87f02434d466
Theory Prop Address: TMEw2C6gFcC7Dv9bdaWQaJELJTrEwgTQ5p6

Sep2E'1

Theorem 20.78

∀X.∀Y : ι → ι.∀R : ι → ι → o.∀xy.(x, y) ∈ Sep2 X Y R → x ∈ X. The proposition is identified by the following information:

Pure Prop Id: d38eaf92e4d64f247f4a780f3acacb7e2b72c2c3161acddce4ff56153a152346
Pure Prop Address: TMKEJf9zJKiTAqodB3HUgFWJktNYut6txng
Theory Prop Id: 0df0e0988ad65ec90d05105e87f7530d402ff762879f9f56fd2cea42ba0e3868
Theory Prop Address: TMQamUmivkZALac37sX35Qy2VnbAqqLtDUR

Sep2E'2

Theorem 20.79

∀X.∀Y : ι → ι.∀R : ι → ι → o.∀xy.(x, y) ∈ Sep2 X Y R → y ∈ Y x. The proposition is identified by the following information:

Pure Prop Id: aa8d9b9df491ed7a1a582cbb2433626cfd9178aed409e00cfb68e55d7be32deb
Pure Prop Address: TMTb1KVCtVPHStwUTCiwdLPEPhorfx9R1DH
Theory Prop Id: b637cd9c32289ddb25bdd962484739bdb1e70ff50694648a27ce6bcbbd48aadf
Theory Prop Address: TMRNNqqH7CLCLvFwcFthUaz6sHcYuKw3TTR

Sep2E'3

Theorem 20.80

∀X.∀Y : ι → ι.∀R : ι → ι → o.∀xy.(x, y) ∈ Sep2 X Y R → R x y. The proposition is identified by the following information:

Pure Prop Id: 5bc2e18996dfa5f943f7598be61adad327423e248744c541671fbe4a34dbcdb9
Pure Prop Address: TMJnp3Ji7AtjsMbmW8cAVsi53FTqYVAGgdL
Theory Prop Id: 379abcf237632ff104c4aa456b6ade54b471ea851766aabd28e15e516d1cbc2d
Theory Prop Address: TMS5WEGW72CGY6U9AD8kjGtvBWaHnyN25En

Definition 20.11 We define set_of_pairs to be λX.∀x ∈ X.∃yz.x = (y, z) of type ι → o identified by the following information:

184 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Object Id: f0b1c15b686d1aa6492d7e76874c7afc5cd026a88ebdb352e9d7f0867f39d916
Pure Object Address: TMUjSWaE3xf1N5QcPC5311QNAmCHwv9P5Wh
Theory Object Id: fd6f81f636454aafecfc9ce1aff42ca28607f59856bf0029f61a1c9a3db299fd
Theory Object Address: TMGXX5uaVkMUzQ1LmgBVrBHf1qxYoxXsvza

set_of_pairs_ext

Theorem 20.81 ∀XY.set_of_pairs X → set_of_pairs Y → (∀vw.(v, w) ∈ X ⇔ (v, w) ∈ Y) → X = Y .

The proposition is identified by the following information:

Pure Prop Id: a6a9b8a124fce5adc383130f08213682d41360dcd1f274f71b33e154a055cc62
Pure Prop Address: TMJxKznQHdHfEU5fnQZwkd5AQ1SGfmDPbwC
Theory Prop Id: 4367f3d59340f9ed670f3765311f00f3089bb7fe2b92d37317898a53a96fbced
Theory Prop Address: TMJd1FMb8pL4Bst6TsZEDp74LjsFRSbxwiC

Sep2_set_of_pairs

Theorem 20.82 ∀X.∀Y : ι → ι.∀R : ι → ι → o.set_of_pairs (Sep2 X Y R). The proposition is identified by the following information:

Pure Prop Id: a60553fc65df7d42bd360fe1cd8b1f9fc8ccc08be19e4177f1ba046a7532ceca
Pure Prop Address: TMTsAuBK39fZ9F9TNnMkeSTj8FGHfLzf6dj
Theory Prop Id: 405a7213163c862c29aeb2e09d4e10989b3c7e06b28850ebc7378a653fbe4fdc
Theory Prop Address: TMH6hRBLodwvqfQVE5ky49zcXJLCo6tRMY5

Sep2_ext

Theorem 20.83

\begin{matrix}∀X.∀Y : ι → ι.∀RR′ : ι → ι → o.(∀x ∈ X.∀y ∈ Y x.R x y ⇔ R′ x y) → \\ Sep2 X Y R = Sep2 X Y R′. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 0d9958e6cc16ac9372dc8846011178fe201c3d9b19cdc001f26e2e162b34d938
Pure Prop Address: TMJACniaRX5HaiHuCE6bbMAHvHDcUhXZ2pH
Theory Prop Id: 94f11b19e5200e39967696f4804cf43ac713a68f6e15ca6369756da68d2d967b
Theory Prop Address: TMRjiAegumXRHuqWAazF89LwDqZTyyMSddG

lam_ext_sub

Theorem 20.84 ∀X.∀F G : ι → ι.(∀x ∈ X.F x = G x) → (λx ∈ X.F x) ⊆ (λx ∈ X.G x). The proposition is identified by the following information:

20.6. PAIRS AS TUPLES 185

Pure Prop Id: b2e26f98453afefcb6f3f49b69c601bd95524ea1ad1932d9fcbd17ae8e3732d6
Pure Prop Address: TMTQkRNn8PD1f7MWUC1oifnJZfbsstmDxEp
Theory Prop Id: 79c629a63ae9a61f24fd859e86bcc2c5d7619d2e635836bd41d82acd839a0640
Theory Prop Address: TMGxXnao732sc8NYXxP9KnXrb3jXwMstnZZ

lam_ext

Theorem 20.85 ∀X.∀F G : ι → ι.(∀x ∈ X.F x = G x) → (λx ∈ X.F x) = (λx ∈ X.G x). The proposition is identified by the following information:

Pure Prop Id: 2a894ae4a36b338440c9116b7d1b5a156eee0387a8f218922dbee9a1525dd98f
Pure Prop Address: TMczyezqH3mj8KqKko9PzTFrNnQbEq2hbYj
Theory Prop Id: 15b7126fd92862295ccc8704bc2c31eecaf79f06db2c59eff164b6dc9e9b9d46
Theory Prop Address: TMVLMDciL8LE8HAPbirRFgBN1S6reWhpgHE

lam_eta

Theorem 20.86 ∀X.∀F : ι → ι.(λx ∈ X.(λx ∈ X.F x) x) = (λx ∈ X.F x). The proposition is identified by the following information:

Pure Prop Id: 19cd9ab03cf4c749ed30f30ed2ed505f627543fbe9a6a2e5eb19aa3c7b0d1b1b
Pure Prop Address: TMTTDo5Up8FsTSMC9bQsVxqiR8BLjPdvHGD
Theory Prop Id: 9103bc442270a76ca14900757db09b5b74bccb4b76e1e17b7ef9f7dd674fc8f9
Theory Prop Address: TMMRnP58fZgQnvj38xKhFcWRj7vE5fPQZAr

tuple_2_eta

Theorem 20.87 ∀xy.(λi ∈ 2.(x, y) i) = (x, y). The proposition is identified by the following information:

Pure Prop Id: 87a4aae7ea69d72da1affd33a6e6fdff7bb4f0d789de45c146aa1dfd53b9aeda
Pure Prop Address: TMLTia4kgENRM1HdxJL6n2c4optHfFxVxdR
Theory Prop Id: 29558e38c4342ce7594d397f1a575f3b2c5f24e1032c87cf9c8a8a79b6832c83
Theory Prop Address: TMGGubBp6qMsgQrvqroY5peGfyYATfj7oYx

Definition 20.12 We define lam2 to be λXY F.λx ∈ X.λy ∈ Y x.F x y of type ι → (ι → ι) → (ι → ι → ι) → ι identified by the following information:

Pure Object Id: f73bdcf4eeb9bdb2b6f83670b8a2e4309a4526044d548f7bfef687cb949027eb
Pure Object Address: TMYPta1FoiMuP1GN6PBin1Xx2bXh9F4jZu5
Theory Object Id: 1af26ca6648a0a5f2f504ed4d407dacb2d1295021bfab70e16a756e32019be58
Theory Object Address: TMFfLmxP1LgSqtznbFPEPY5Ne2h6kro5fyD

beta2

Theorem 20.88

∀X.∀Y : ι → ι.∀F : ι → ι → ι.∀x ∈ X.∀y ∈ Y x.lam2 X Y F x y = F x y.

The proposition is identified by the following information:

Pure Prop Id: 2de6dedd220a397af6370f72b7a1335bcfa1475e0d9eb3d9cc7ff55075b114b9
Pure Prop Address: TMagrn8mKPRRmZwMivWkUiffvhd5tm79Win
Theory Prop Id: 4bb51dcd166eb4dab05760eec9ee7a9c07acd7688f394f84fc37ce9a362ea7f7
Theory Prop Address: TMGMZXJkR32EUzV9izhKBE2MEoS3MKPLbXm

lam2_ext

Theorem 20.89

\begin{matrix}∀X.∀Y : ι → ι.∀F G : ι → ι → ι.(∀x ∈ X.∀y ∈ Y x.F x y = G x y) → \\ lam2 X Y F = lam2 X Y G. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 2fcf8329eb8970256dd334aa201b65fa25e730316c3d42fb8e24bbfb8baa1da6
Pure Prop Address: TMWA1B5WDTBTsPxXTq3a4ASUuADPsARiKwo
Theory Prop Id: 36ee1989db22297f701d46985e3f5b8484beb0cc98783f705be675646c00e858
Theory Prop Address: TMQRLHS3cYoRzXTpwWXkAqwX3mPwkxYT46Y

20.7 Encodings of Functions and Predicates as Sets

Definition 20.13 We define encode_u to be lam of type ι → (ι → ι) → ι identified by the following information:

Pure Object Id: 8acbb2b5de4ab265344d713b111d82b0048c8a4bf732a67ad35f1054a4eb4642
Pure Object Address: TMMDLyvQS3eoiP32KBXjSQyuvc4keN5yMmX
Theory Object Id: d64bcbbefe13f9fc0035d2317f017d9ee3aeb3de90b836ed7cbe11d46f478799
Theory Object Address: TMS2X4GvB3DZiVdQjy5ZeDZQVVCDuJBZvgW

Definition 20.14 We define decode_u to be ap of type ι → ι → ι identified by the following information:

Pure Object Id: c7aaa670ef9b6f678b8cf10b13b2571e4dc1e6fde061d1f641a5fa6c30c09d46
Pure Object Address: TMRkKHvzv8Tta9AhRXsVE4k7dQ68XWymCoD
Theory Object Id: a4d82c5a3643a7e174676c313ef995e0307b3b56f57e8ae761b7c6033bbc40db
Theory Object Address: TMULGKR83zvLzsaT5wtaLqEme8A8TV9uVcG

Definition 20.15 We define encode_b to be λXF.lam2 X (λ_.X) F of type ι → (ι → ι → ι) → ι identified by the following information:

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 187

Pure Object Id: 4c89a6c736b15453d749c576f63559855d72931c3c7513c44e12ce14882d2fa7
Pure Object Address: TMEnk3YrwKupP9z6sTwXokwBz2J6VVGp5Zh
Theory Object Id: 2857218138584f5287cfe4ade2be20096822070b538af7d77aac47d32304f2b6
Theory Object Address: TMEomjpi2NLe2mLYiaW5hqvmjvtSXDYVCri

Definition 20.16 We define decode_b to be λF xy.F x y of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 1024fb6c1c39aaae4a36f455b998b6ed0405d12e967bf5eae211141970ffa4fa
Pure Object Address: TMLJXUAwxDCzt7HeMEhmMtFjPFH82BuahcJ
Theory Object Id: 3b93ea7cf603f82962cfaa76e280e5218df781b7ff8b501b4dea38f2c76f0ce4
Theory Object Address: TMdai21pMbN5LhRjuzG8fbyq6jLXZp8xvQD

Definition 20.17 We define encode_p to be λXP.Sep X P of type ι → (ι → o) → ι identified by the following information:

Pure Object Id: f336a4ec8d55185095e45a638507748bac5384e04e0c48d008e4f6a9653e9c44
Pure Object Address: TMUtfXxDDLj9Tk47vMVXxn7X4YsuGxPH9iG
Theory Object Id: 7008ec0ee19d11cc67c5f11635c82852ccb27360b8bd8431676b0c7e03219428
Theory Object Address: TMU5xEoVJ4JK7LsxiB8DMm5ouqBu3Rgy3CL

Definition 20.18 We define decode_p to be λP x.x ∈ P of type ι → ι → o identified by the following information:

Pure Object Id: 02231a320843b92b212e53844c7e20e84a5114f2609e0ccf1fe173603ec3af98
Pure Object Address: TMNmYnbPvxg781LaE377972CwGGRP6A4t2c
Theory Object Id: e9507346effb55f2b66cdfb801523074014fb64eeccb8e226d8fdc9359337830
Theory Object Address: TMEnXjKVptqvfEYytFygN49BmBFhyNV8YJ8

Definition 20.19 We define encode_r to be λXR.Sep2 X (λ_.X) R of type ι → (ι → ι → o) → ι identified by the following information:

Pure Object Id: 17bc9f7081d26ba5939127ec0eef23162cf5bbe34ceeb18f41b091639dd2d108
Pure Object Address: TMVAvkv75vEbx8iL4LYfx77aKBqGpyF2x3q
Theory Object Id: e6e31b91e856a7c568a0d85625bf50833d9a6ba3696590b494c9be80206757f7
Theory Object Address: TMaAgBFCWYRYoi83L9vFaz2ee8AkoYAQ7dn

Definition 20.20 We define decode_r to be λRxy.(x, y) ∈ R of type ι → ι → ι → o identified by the following information:

Pure Object Id: f2f91589fb6488dd2bad3cebb9f65a57b7d7f3818091dcc789edd28f5d0ef2af
Pure Object Address: TMUxUtfWjzRHvsq1Hh2nzZmnFZn4W4k5yZ1
Theory Object Id: b93a65581af63e84497eb9a1ce7d86376ff948ca78630d50cb4fb879fe8b35a6
Theory Object Address: TMGH5mJWrbQCiFMwfcAubZ4xHfJdMSJxuWk

Definition 20.21 We define encode_c to be

λXC.Sep (Power X) (λU.(C (λx.x ∈ U)))

of type ι → ((ι → o) → o) → ι identified by the following information:

188 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Object Id: 02824c8d211e3e78d6ae93d4f25c198a734a5de114367ff490b2821787a620fc
Pure Object Address: TMPoUwriPcHMpXKUSPr2n5eeCVHPwVTvJjh
Theory Object Id: abcb4f0b23ce219b19f08deedb28f9946fc52a200cd5ab9c101ffc24d294a926
Theory Object Address: TMdFVb6jHzinhqHU8R1E9GETd9GtqL67GFs

Definition 20.22 We define decode_c to be

λCU.∃V.(∀x.U x ⇔ x ∈ V) ∧ V ∈ C

of type ι → (ι → o) → o identified by the following information:

Pure Object Id: 47a1eff65bbadf7400d8f2532141a437990ed7a8581fea1db023c7edd06be32c
Pure Object Address: TMGRAT91AwN8Jcefxc45YTgTK8w2jHVUmwH
Theory Object Id: 3edd4a21f511f37d2fa9b701d6bceefc3ec4979b0aa937da2224b9712a600388
Theory Object Address: TMKMbEUcTZ35ASSvnJwrdeRpZrpSFw3akQn

decode_encode_u

Theorem 20.90

∀X.∀F : ι → ι.∀x ∈ X.decode_u (encode_u X F) x = F x. The proposition is identified by the following information:

Pure Prop Id: 95991de564f22218049f5c01a2054c9f531e2a0e7f89348f4cd3cce56269ee6a
Pure Prop Address: TMW5SkUgFpNvxZtwNs5tigJKy4uVF4aLEX4
Theory Prop Id: b5182dff18aa9200707f42f128e18ba010c3c5b0a3a598682cfd9fe3e36def5d
Theory Prop Address: TMUK7trdHToYkqBA9FA7nyBfnGCJVnzMrdt

encode_u_ext

Theorem 20.91 ∀X.∀F F′ : ι → ι.(∀x ∈ X.F x = F′ x) → encode_u X F = encode_u X F′. The proposition is identified by the following information:

Pure Prop Id: 2a894ae4a36b338440c9116b7d1b5a156eee0387a8f218922dbee9a1525dd98f
Pure Prop Address: TMczyezqH3mj8KqKko9PzTFrNnQbEq2hbYj
Theory Prop Id: 15b7126fd92862295ccc8704bc2c31eecaf79f06db2c59eff164b6dc9e9b9d46
Theory Prop Address: TMVLMDciL8LE8HAPbirRFgBN1S6reWhpgHE

decode_encode_b

Theorem 20.92

∀X.∀F : ι → ι → ι.∀xy ∈ X.decode_b (encode_b X F) x y = F x y. The proposition is identified by the following information:

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 189

Pure Prop Id: 3ee73039a869eb8092f936deea2d3411e72305d82a14a6e988c4a1bf8d16ca83
Pure Prop Address: TMMn46yDXgo1kjEyPcuewexTo1Z9Ay42Wje
Theory Prop Id: 8d10ce96d1c3a115ca4a3c97096a0ef9b1241c7944342a034d52676ec1576ca9
Theory Prop Address: TMNxzXmNthWVuFuJrEbR2rZdFYrRPUUpZEg

encode_b_ext

Theorem 20.93 ∀X.∀F F′ : ι → ι → ι.(∀xy ∈ X.F x y = F′ x y) → encode_b X F = encode_b X F′. The proposition is identified by the following information:

Pure Prop Id: 332cea87dd4a05957a0e32b3f6ad9a7a65f7c156b384ab1468dabd57ed690e07
Pure Prop Address: TMF4RHp7VrPNSphZfAnfvL4aSLfmrbynJJb
Theory Prop Id: d010291fc325252d96b8617d5107ca33d0bd9906a978a831ac853d04850ba94b
Theory Prop Address: TMMdZbVStJuaWiLad5uZeQNMN7wid9XW5qy

decode_encode_p

Theorem 20.94

∀X.∀P : ι → o.∀x ∈ X.(decode_p (encode_p X P) x) = (P x). The proposition is identified by the following information:

Pure Prop Id: 76e0c4faba69952ec67742eeb94d927de40a5f192c9d6aecccd1035a84de0723
Pure Prop Address: TML6EPefmfw97vntVGgeVFz8HyFXfeVpHzC
Theory Prop Id: d58325ff2359f69c9e2bd75a02be4ef6854835056c014507b6c92d4e2cf7ceb3
Theory Prop Address: TMcADFusuPHtjARgdAGwzLTRfoHmoLdpnbx

encode_p_ext

Theorem 20.95 ∀X.∀P P′ : ι → o.(∀x ∈ X.P x ⇔ P′ x) → encode_p X P = encode_p X P′. The proposition is identified by the following information:

Pure Prop Id: 3b7abeffd741d27cffec5971d6a6a069502f5125137259639ec8c7fe9e08137e
Pure Prop Address: TMJ9Y8Nubbgp1Z251WmGmhd5ZqmvRxaibVa
Theory Prop Id: c103040e5a72e28e42a8b370fe0915a43c227b83d6405d91b72df648a194fbe7
Theory Prop Address: TMaHzJHmzSrfHeEaV6hRXwCRYPY5P2JHZy1

decode_encode_r

190 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS Theorem 20.96 ∀X.∀R : ι → ι → o.∀xy ∈ X.(decode_r (encode_r X R) x y) = (R x y). The proposition is identified by the following information:

Pure Prop Id: 1014a9bcf05c5f44d84ef0e62da543b87727aca195ad8496cd18d175889c170e
Pure Prop Address: TMGVNLSr6YarTJyvLJtRUNqkQhdbWL1MPdw
Theory Prop Id: 08a8a48da6a92bd5fc4224fbd113b9d97621ed1e4dd0440c1782a73cdced5b82
Theory Prop Address: TMKQ1xev2H6uLTz75vjFPQTwWGXdKuTWAw4

encode_r_ext

Theorem 20.97 ∀X.∀RR′ : ι → ι → o.(∀xy ∈ X.R x y ⇔ R′ x y) → encode_r X R = encode_r X R′. The proposition is identified by the following information:

Pure Prop Id: 7e28f4c2e174003767040fb758e5a7abd8bf6f6859c428b0cba2d3531a22173d
Pure Prop Address: TManVioQo3HkeWcxBTPLk7ZWsNqo8qG2Y11
Theory Prop Id: fa3b122d8b29e30bf435481e40c1cf6b59f126107a670fcc9198128c41d97b0c
Theory Prop Address: TMEyPUz1f5trR77TrCy5T6ALTtFw5QB8hNw

decode_encode_c

Theorem 20.98

∀X.∀C : (ι → o) → o.∀U : ι → o.(∀x.U x → x ∈ X) →

(decode_c (encode_c X C) U) = (C U).

The proposition is identified by the following information:

Pure Prop Id: 7a75630d7138237a35850129ec0899748c1ef7c52bef0bde27d23694b7dea800
Pure Prop Address: TMYedqhNDgqDPXcTzM355Rp53xYHrH9uyDf
Theory Prop Id: 880bfbe387f844851c05145a1db503fb36936d2294f75af51c5ce2785d96bf21
Theory Prop Address: TMG3Jf332EdHAKZvGPSKqfyZgKCYjmL7sQ3

encode_c_ext

Theorem 20.99 ∀X.∀CC′ : (ι → o) → o.(∀U : ι → o.(∀x.U x → x ∈ X) → (C U ⇔ C′ U)) →

encode_c X C = encode_c X C′.

The proposition is identified by the following information:

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 191

Pure Prop Id: baacc5826835bdf9ff6b14cd7397faa14ac4deec11a936d577ef7808babe0837
Pure Prop Address: TMKwAfPDG9pG12Fvp5oGJz7NXs7naikE7x8
Theory Prop Id: 0c5483759fc520ad24416714aabea399eb7cb84016cdcbc605ffec9d6504a0b0
Theory Prop Address: TMTNaV2fXyXXwapDqmZgoh9Bwtevm9wRREw

setprod_mon

Theorem 20.100 ∀XY : ι.X ⊆ Y → ∀ZW : ι.Z ⊆ W → X × Z ⊆ Y × W. The proposition is identified by the following information:

Pure Prop Id: 63d72a5736b6dad618fb3fe72b9bce7bd5535acd2498fd7efcd60df85fe65ef0
Pure Prop Address: TMa6kynUKR5kseqtucWFdECHsZBBaPS7mrj
Theory Prop Id: a74bcaef7267fce98f2a364fb6c5734d406f80cb027560a59947536a21433041
Theory Prop Address: TMRkBMu4VTR26BPV4YUTQHzEAUt27bx9EXj

setprod_mon0

Theorem 20.101 ∀XY : ι.X ⊆ Y → ∀Z : ι.X × Z ⊆ Y × Z. The proposition is identified by the following information:

Pure Prop Id: 7bd32d8dd36d89ace87f57f3e3d5d041a6b193856d3cabf95317325c73b3df3f
Pure Prop Address: TMP3EP5PaqbZ1cxhh8WmVQQLcsFCXTW7rCy
Theory Prop Id: 455a4284ff6e2959b9e8a657510f722883ee919b1b17703dac1942a7edbdda0c
Theory Prop Address: TMVWdnQ7qpSHvYcKK6yidsz39oPTDKBkq8r

setprod_mon1

Theorem 20.102 ∀X : ι.∀ZW : ι.Z ⊆ W → X × Z ⊆ X × W. The proposition is identified by the following information:

Pure Prop Id: 6e2e07464345fa4f1a371f688df924851a665ecd94aaed51223be1ae886bc23e
Pure Prop Address: TMSnywK9u9fk9Ed6jbYvBs1ZUC6LKFAgpKS
Theory Prop Id: a90d34c038dd8dbab84e007a80008f440fca47232d9907f639d52500513f5a4f
Theory Prop Address: TMPygssHhnaE9kaN6NAnhRiDZooV9D1npXA

pair_eta_Subq

Theorem 20.103 ∀w.pair (w 0) (w 1) ⊆ w. The proposition is identified by the following information:

Pure Prop Id: 4f3ebfd1a2003335ce7adcc7c9b187a22011f65ae156fa1445a702cf001c3927
Pure Prop Address: TMVusjsTB3ex7nLq2jbiMtM6jhKF3AJBbiD
Theory Prop Id: 74e00933da8b7173ceedb24a78d36aedd8271dc642ef0da29ab7608538b9bcb4
Theory Prop Address: TMFvqxkSHYjcmeNXXjANHaP3wFYajKhHUMA

Sigma_eta

Theorem 20.104 ∀X : ι.∀Y : ι → ι.∀z ∈ (Σx ∈ X.Y x).pair (z 0) (z 1) = z. The proposition is identified by the following information:

192 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 49ff5151b49302ac99558664cc438f0a4d40498c14aadb3e399282edcabda3fb
Pure Prop Address: TMSCgjHPA66KEFpj15NkePtPi8MUx8TeT4Z
Theory Prop Id: 8049133218c094e36a326d0302747b6faed7820279c4ca43b4aaaba59da8d86f
Theory Prop Address: TMZAcyQRmA4RUYzXLLDvy9X9mq3N9qQySRD

ReplEq_setprod_ext

Theorem 20.105

\begin{matrix}∀XY.∀F G : ι → ι → ι.(∀x ∈ X.∀y ∈ Y.F x y = G x y) → \\ \{F (w 0) (w 1)|w ∈ X × Y \} = \{G (w 0) (w 1)|w ∈ X × Y \}. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: c486ab09b581a313204dc303b15e3611e31121022211233d7418fa02f6cc7299
Pure Prop Address: TMWwkDTMCqBE34kbrXdyyAhqVm2sbfTQjce
Theory Prop Id: f689a07de22b0aec4db3cba58ed366ef7f1c7bf74bbf6b2e727fd32a368b68eb
Theory Prop Address: TMTDNNDk6TRtimfgiJDfHre2Zghrr4cfc5k

tuple_p_3_tuple

Theorem 20.106 ∀xyz : ι.tuple_p 3 (x, y, z). The proposition is identified by the following information:

Pure Prop Id: cf993987b43146748735706a7857f006cbd148697f781826b45bb9c65766a0d4
Pure Prop Address: TMF2RMvq3PMYzBY3dD3WXb7CLziCt6USMZD
Theory Prop Id: 77393568486dac4f6831171dad84d01e9b6df6e1d91b218fcdd823e87841c937
Theory Prop Address: TMR9fqGfQKrPPzAcNDvk14YN5xCHfLnGWnB

tuple_p_4_tuple

Theorem 20.107 ∀xyzw : ι.tuple_p 4 (x, y, z, w). The proposition is identified by the following information:

Pure Prop Id: 5cb9dd366dfb4bd3932546b2b6ce146f154c0ab4a373c38ab6422bfba5679085
Pure Prop Address: TMdpGdQnb2pjcxbUxX2XUCUMgbpPFfAHvzv
Theory Prop Id: 58c3b26de9e903a7f82ccf8e0237aeba24a5fc22ffdc521a5446cbca1243b942
Theory Prop Address: TMMXKVHZmRa3TWvKaoU4TWL4zLf8ocjRiox

Pi_Power_1

Theorem 20.108

∀X : ι.∀Y : ι → ι.(∀x ∈ X.Y x ∈ Power 1) → (Πx ∈ X.Y x) ∈ Power 1. The proposition is identified by the following information:

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 193

Pure Prop Id: 749337506bbff9c2bed1261a0faa5ea2bf92d0442dacccd5c665ddabd12a5e41
Pure Prop Address: TMSQqTJzh57zKzeJmXEtHFgtRKNkkoRZ5Yg
Theory Prop Id: 53185f562ff7551b58214aa425c4679c294ce17583608a17fded6fd0251b725e
Theory Prop Address: TMQ6JF9rZDAhvvypZvKH3NTaGvdGWRexq52

Pi_0_dom_mon

Theorem 20.109

\begin{matrix}∀XY : ι.∀A : ι → ι.X ⊆ Y → \\ (∀y ∈ Y.y ∉ X → 0 ∈ A y) → (Πx ∈ X.A x) ⊆ Πy ∈ Y.A y. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: afd68d2c59d656bb239ccbebabc9bd0a0bd90439f78158e146e8fc748463f7d5
Pure Prop Address: TMbgKzF8UHU8QuRUAoJPZtfPZpKdMV6fH4X
Theory Prop Id: 8b265dd368d5ae9935219b30f9ab92687d8f287fc776d2c4ff902bdf59efc063
Theory Prop Address: TMbVdN1MHHnjdbTQaqcsa3c43EUJLPRyXxf

Pi_cod_mon

Theorem 20.110 ∀X : ι.∀AB : ι → ι.(∀x ∈ X.A x ⊆ B x) → (Πx ∈ X.A x) ⊆ Πx ∈ X.B x. The proposition is identified by the following information:

Pure Prop Id: 3f6023959e78603c118adb6a4ff1b02d56955380be26e0747fae716b2cd95d97
Pure Prop Address: TMaeWxiHTomfFysYznzJfEDq1N349zj13nT
Theory Prop Id: c65ee3b2fffcc8ba8c7c4d2fef3d6aee20f82c17cf3c09dc7a7e72f8a251d98a
Theory Prop Address: TMZud88tiwxGZkbhtpgTUFSV8FPmskRGQHU

Pi_0_mon

Theorem 20.111

\begin{matrix}∀XY : ι.∀AB : ι → ι.(∀x ∈ X.A x ⊆ B x) → X ⊆ Y → \\ (∀y ∈ Y.y ∉ X → 0 ∈ B y) → (Πx ∈ X.A x) ⊆ Πy ∈ Y.B y. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 74a4777899bd48bce73a6950d5c843aa0ef4895865b762f7d6e9504173de2d2a
Pure Prop Address: TMQsGPc8N3Ubv5n8RJVHF7c8JKNSvH4P3Mv
Theory Prop Id: d876ad3e474910e13beae3e36658e1cba27e007339b5c367e122cea96ba5a530
Theory Prop Address: TMJpkgSLqGH7HNbMBRcohiUdmHzCQFD3orZ

setexp_2_eq

Theorem 20.112 ∀X : ι.X × X = X^{2}. The proposition is identified by the following information:

Pure Prop Id: ffa8ca03579eb26597d40ec371bf1e68b77c9e00e70e22aa75cb005b599928fa
Pure Prop Address: TMQo8m7mTFS5ULGe5RFn9ewAV6USEFmXGEe
Theory Prop Id: 97b6d296a52705b137946c7684a42f90e6dcdc16ccd27fcc6f843903334d6124
Theory Prop Address: TMU8ZQ356N7f1ur9PqrcyhFvUrEU8Uw3pCd

setexp_0_dom_mon

Theorem 20.113 ∀A : ι.0 ∈ A → ∀XY : ι.X ⊆ Y → A^{X} ⊆ A^{Y}. The proposition is identified by the following information:

Pure Prop Id: c82a77bea63c3045dd2d0390019427d36b3850e817a7c7d81c9d44dd83da9951
Pure Prop Address: TMQY2MPGRLH7DstKxCBvTt7ycuHKzG3Egta
Theory Prop Id: 320220221d2ff027b0b203b874e4695ae660ced55ac0790f282597f5e9613e9f
Theory Prop Address: TMdJC2UYVneWjpMTvmnbkFG2QrY6sZa7dzC

setexp_0_mon

Theorem 20.114 ∀XY AB : ι.0 ∈ B → A ⊆ B → X ⊆ Y → A^{X} ⊆ B^{Y}. The proposition is identified by the following information:

Pure Prop Id: 9626c863679499ee9efb3a97f6d2c94d448186ea2212527485c1494342277958
Pure Prop Address: TMSVKPxofDcRCRvbvF6CiPxy7w1U9GV3abU
Theory Prop Id: 90c509fbef3ca7822c8b20ec7e5a62fd62a043b9500aaa739b5118a3c994d408
Theory Prop Address: TMG4aTTfVaxVUnsqSgBoVVNATLoR7h38zjG

nat_in_setexp_mon

Theorem 20.115 ∀A : ι.0 ∈ A → ∀n.nat_p n → ∀m ∈ n.A^{m} ⊆ A^{n}. The proposition is identified by the following information:

Pure Prop Id: d4aff1269d336f4267535d4b630549f976211672cd2893c8f0d5f93ccc2a2c01
Pure Prop Address: TMHmDB4dHXevKqyYjk66EaYCALxGuURyksG
Theory Prop Id: 325485fa49bc41efd250c31eb18bded07de4228f53ab976ea68b483631dc46be
Theory Prop Address: TMX7YsW1eVF8qPSQpQudbRa9xp2Q6wGVEfD

tupleI0

Theorem 20.116 ∀XY x.x ∈ X → (0, x) ∈ (X, Y). The proposition is identified by the following information:

Pure Prop Id: a5d9e60bb578510eba8f8727ca81d4ad6c6ea416e3b1dec75e56736f2c7fbf28
Pure Prop Address: TMbWA1F32KHzES5haJKo6Ltd6Myvmdja6kj
Theory Prop Id: 7bad4e2f646b78cc8e4ab2dde15f4dad2279a633db69af2d130325d2df660497
Theory Prop Address: TMYRiZH6QeEcQSbTsa8ff35vULKtFPjDWGZ

tupleI1

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 195

Theorem 20.117 ∀XY y.y ∈ Y → (1, y) ∈ (X, Y). The proposition is identified by the following information:

Pure Prop Id: 5c347a6088d371c6a7be279b519d891f8b247a1dc86c7f1a5e4e91182d2cd741
Pure Prop Address: TMR457F4pvZZp7RcTd3TbZY6m59BdcrF2rh
Theory Prop Id: 5e3cb317730d8131182f7c240578e9b710c06861aa264924c853babd66ad53f1
Theory Prop Address: TMQKbRaumSCTKfCNkFDrzfYb7gjLPbdz9bg

tupleE

Theorem 20.118

∀XY z.z ∈ (X, Y) → (∃x ∈ X.z = (0, x)) ∨ (∃y ∈ Y.z = (1, y)).

The proposition is identified by the following information:

Pure Prop Id: 8b6d364ac254012dfb266e48ccd8017c4df7092dc82d2f6356aa98f2c113f819
Pure Prop Address: TMaevMuU9VPUjmzk93asQnRSf6s5giQjEGz
Theory Prop Id: 226332642373ce44237e3431b4e96d902d54149ae33bde4bedb64b30fb8d0fa2
Theory Prop Address: TMdn86EVVTVBkyXsXj9Vmq1oLxSWWr4xxfb

tuple_2_Sigma

Theorem 20.119 ∀X : ι.∀Y : ι → ι.∀x ∈ X.∀y ∈ Y x.(x, y) ∈ Σx ∈ X.Y x. The proposition is identified by the following information:

Pure Prop Id: d4a9d90bcc14a1607b933d2b6324f74a12e560f1ee23d66bbef65f0ab2300245
Pure Prop Address: TMTxDywb49ok9weqR5PA1YkH5HYVVQgfmLa
Theory Prop Id: de381fee5dd742f3e056c75fdc7abdebc389a9d213f325ce23fff6b0562da398
Theory Prop Address: TMYB1uFbN7KQg4EMgzCmiEUJxJcauCrH6gi

tuple_2_setprod

Theorem 20.120 ∀X : ι.∀Y : ι.∀x ∈ X.∀y ∈ Y.(x, y) ∈ X × Y. The proposition is identified by the following information:

Pure Prop Id: e907b80c4db444253db900e46741dbeb4c373f9357cc51990287b6b4bc4e0111
Pure Prop Address: TMU5XWQo6K7rWoGntpngjoB7dZXxPoSYoxC
Theory Prop Id: 1f9d7fcc7017d00b96708fc1b8cd4c299aede7855c9c051ad9ed1f17edf0696a
Theory Prop Address: TMQby1erMukDCSudbBu6DXLdSmpJnUwy4bN

tuple_Sigma_eta

Theorem 20.121 ∀X : ι.∀Y : ι → ι.∀z ∈ (Σx ∈ X.Y x).(z 0, z 1) = z. The proposition is identified by the following information:

Pure Prop Id: 7cc266621e49b9c431179e0045a15c6c2bd49d0bb6d49aefbb923becb4f3b9ee
Pure Prop Address: TMSgpyjZ757izmXroHBFU7jiFn39AxWAcPa
Theory Prop Id: d965e01dd42fa888cdd06e063200196d2578c975d5c706f981c47060524c38ec
Theory Prop Address: TMXQjnReFQQDxxkiWBRLot9EZBBJmxctC5k

196 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

apI2

Theorem 20.122 ∀fxy.(x, y) ∈ f → y ∈ f x. The proposition is identified by the following information:

Pure Prop Id: 334f53713d549322405aec659458426a25e4aa7e1dcba4fbfad411bfe0628835
Pure Prop Address: TMcAcmb8L7yHFzTFqsQ3a4kWhFMbRKovMNC
Theory Prop Id: 6f716abb65c7606ae6b5749d37dda052e685a4d4f7519d400bedcc1fcee23320
Theory Prop Address: TMTjJZzRce6va39d2z3pncuj6jrfGtJ3eoy

apE2

Theorem 20.123 ∀fxy.y ∈ f x → (x, y) ∈ f. The proposition is identified by the following information:

Pure Prop Id: 2a2c24116214dee00936cfdd3231d8a70a4ff6d3efda38a95a724e0e4cd155a1
Pure Prop Address: TMWSE1naRk6JPjqHbwkoVtBu4k9tumZgTyX
Theory Prop Id: 57702acf505aa564187ee95d1b240f6ea6befe178029989924da2df6cea2a42f
Theory Prop Address: TMGiJF872peSuqfKRH9jn9RcqbjuUAS8qbD

ap_const_0

Theorem 20.124 ∀x.0 x = 0. The proposition is identified by the following information:

Pure Prop Id: a4ea444af8365c1c911ed81357b063cf1823fcc07b1e9fa2848a6e26d14a5f23
Pure Prop Address: TMZQjyRKodf7sfVuPrUmzZLt88BXcuEKsxk
Theory Prop Id: f66b937fdac9351c50455abd7b2f8c4008f271f243b5a551b8020099b6ea59d9
Theory Prop Address: TMYDg1Trrx3692oVX9Md1Ss8h7pKgzipsXy

tuple_2_in_A_2

Theorem 20.125 ∀xyA.x ∈ A → y ∈ A → (x, y) ∈ A^{2}. The proposition is identified by the following information:

Pure Prop Id: 1fa999eab25ebfb7f69caf608dacd0faa4724522bf176de884702e54c99b0dfc
Pure Prop Address: TMTdPZ59TT37q3Gv7R8XAXdt6EShALVC2bx
Theory Prop Id: 234a4a6a1e3d22203eab3e75c58b409a92981de21f60b36b7bacdbe94c2f695e
Theory Prop Address: TMEgxar8usGXDWMn2dQrfqcaTBMsQmobYQL

tuple_2_bij_2

Theorem 20.126 ∀xy.x ∈ 2 → y ∈ 2 → x ≠ y → bij 2 2 (λi.(x, y) i). The proposition is identified by the following information:

Pure Prop Id: 1458e4c76e00e5e5ab03f1e51a59e8b2db6f5aedb0ec7b7437f01c606e57f61c
Pure Prop Address: TML944PLcXEhMqhaj1bSJ66jCqyeP9nidJV
Theory Prop Id: 5fae0dbc2404ab825ce386605b9b51bc9aeac0b473aaf79cb2ca0d2802ca5c3d
Theory Prop Address: TMTfcLL44Zqzsq91secQcUt9S5DKUtpL89s

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 197

tuple_3_eta

Theorem 20.127 ∀xyz.(λi ∈ 3.(x, y, z) i) = (x, y, z). The proposition is identified by the following information:

Pure Prop Id: 557b0ee79b2a52e8ce18d61ad94a7f15864471688ec9c794f1d5f24c9783005c
Pure Prop Address: TMXagwe8TteWaGYj5KgUBttDC1wW28KphHN
Theory Prop Id: b1962f9a855cfebb9323ea0e424ab44c88fc28f2f25e542d11602c8562b09544
Theory Prop Address: TMNJCyieXsgj2hu8gCm37y28P14rCCpCUyg

tuple_4_eta

Theorem 20.128 ∀xyzw.(λi ∈ 4.(x, y, z, w) i) = (x, y, z, w). The proposition is identified by the following information:

Pure Prop Id: 774296b519bf68aacdfe0445662dcdccfc6c74dcc7d3898d0507c5079d63d8f8
Pure Prop Address: TMQxXPgmA8bowKXArUpmiz87siaeSw2Q5MW
Theory Prop Id: e04fd23dfc3d7c5f1a127aba4c6547e3b393dfb6545e480f9953e26ca3c3fc0d
Theory Prop Address: TMYx47haX7KQ8PtiUHHnWXRWY2AfpDb5mNZ

20.7.1 Tuples with More Than 2 Components

Let x0, x1, x2 : ι be given.

tuple_3_0_eq

Theorem 20.129 (x0, x1, x2) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: 2299dd127ed9d00d183b81051f52579a3638dd91f46ef778aa50c442abc6b092
Pure Prop Address: TMFhwi4ZC3VWuBy42eEzHkLCbNwo3Ev8HEf
Theory Prop Id: 0a4f57728c3c5c761b448d44960ca6a828c0a440fd63d1aa6d9c2e57c0243787
Theory Prop Address: TMW5V1hAcqqC4skHEBzYR3ZiJGL8pvjwYWK

tuple_3_1_eq

Theorem 20.130 (x0, x1, x2) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 414e8bd1880695006637be5415198e0c43b53e9609ce62bb325610bdc38977f9
Pure Prop Address: TMX5TRv8W5gSMK3DnrehU1GjFmEMuBgMie9
Theory Prop Id: 14ea6626db4405895dcefd0542273be1017929dc5a35662b5fc7930a86d2bb05
Theory Prop Address: TMNBfanXBAVa8NNCyyTX1KCDoX1oZugTZeK

tuple_3_2_eq

Theorem 20.131 (x0, x1, x2) 2 = x2. The proposition is identified by the following information:

198 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

Pure Prop Id: 6b6b08fb7649ca4a49e5f256d73d38901cade1dbab78712081d4c92f5f63a44f
Pure Prop Address: TMUzP7TDpQtCwif8aAwohSj3JNWVX6TEdjc
Theory Prop Id: f69c06f91e4e8a435b4ac8fb4653343c5430c0f76186ed2c6b5a8285be49dd75
Theory Prop Address: TMQ7XSB36vKMD44joDM3WQNXQ5hqU7Zeu4r

Let x3 : ι be given.

tuple_4_0_eq

Theorem 20.132 (x0, x1, x2, x3) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: a6e3de0ca7e59eaaf3122d630187610de4188d6b0275524d8b75d2bc27abda32
Pure Prop Address: TMF5etreB3YRde69S2FeY8M1oVAWjjxQ6v5
Theory Prop Id: fe5627737874494b543777e809941dce8d7e9a8013c7963f2809078137e82277
Theory Prop Address: TMYGbLVThnvCWxfpjrsDHdPixQYq8AwiQAz

tuple_4_1_eq

Theorem 20.133 (x0, x1, x2, x3) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 733dff0a1d011b3d6a42f3265c3727b7801c71ce1aae4ed97fa0c7086d2a3fc6
Pure Prop Address: TMSYE1DSLMRdPHth8BZAHKadTikLP9Hp1dN
Theory Prop Id: bf73bc8d27290ee55ba023d0c035f7daaf4cf11c12a380d059c86f22816eb234
Theory Prop Address: TMX1ctd3QUcXQ7ntu5MvE9xqwY9fi3maxEg

tuple_4_2_eq

Theorem 20.134 (x0, x1, x2, x3) 2 = x2. The proposition is identified by the following information:

Pure Prop Id: ed4aa73355586a09081629dab3b75a5dec8a417c17ed0d84b8443443fb2785a4
Pure Prop Address: TMExys3qMfkBQ43ojYUvQ8d1LY1vZw1enfP
Theory Prop Id: 7d4422df851f3d7c9b2b10bc33820c35b9f3567e7a28a65da668a14c240d5d55
Theory Prop Address: TMNiHJsEoF9pvnK73fnpVv3rnvdPVJKtB7P

tuple_4_3_eq

Theorem 20.135 (x0, x1, x2, x3) 3 = x3. The proposition is identified by the following information:

Pure Prop Id: 6a238391dd9d73ec2a175b97bcd4c846967958d9fb539afaec96724c51bf7e9f
Pure Prop Address: TMTCp2qqDTin3jc37q6q6PLfMoJcCzaq48h
Theory Prop Id: 1a82d64dd6e2e5694e2d044a818489902c8025fada8fb5f3561aa926882f5af2
Theory Prop Address: TMHfo157hjhQzHLRL8e3AAL7NmnK4YXNyFv

Let x4 : ι be given.

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 199

tuple_5_0_eq

Theorem 20.136 (x0, x1, x2, x3, x4) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: f808b62bbfe0261c431302a1f9d628c853f68d912aa2f152472c2dfa90557d18
Pure Prop Address: TMGre8WKqnRD2NJPkoaA7jbLWUipgYm5jog
Theory Prop Id: 2eb5ddc5a16b3a7d9a1280b24d3e509bd011ce52296873af2d6879fb547774ea
Theory Prop Address: TMe1KWBCo857cwBq4Uy3dWDEAzMuSo7LxQm

tuple_5_1_eq

Theorem 20.137 (x0, x1, x2, x3, x4) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 5714821964a45eae8276b9e96c0d7f3dc70c4a421256104663fe4622cb6dda9d
Pure Prop Address: TMQcHzma6cxkMaJgGHj1SX9kiVwhL66i5aW
Theory Prop Id: 49b93f2b2cb80b89ed8e5dba6af36b0110694d89f4106c7b23eb3abdee76c4f2
Theory Prop Address: TMachUcuahLvMtFu7FZn7cYP4oDkK8ciBkE

tuple_5_2_eq

Theorem 20.138 (x0, x1, x2, x3, x4) 2 = x2. The proposition is identified by the following information:

Pure Prop Id: cfa53ec600e79400f6f667fa9b058c813cc71e4bebcbef4760d5e26acfb75b75
Pure Prop Address: TMFYhqnKU46a4x3LZyhSt4RajqBX6cEKbFx
Theory Prop Id: 3bb23a1a7686e9d933ead58faf0abbc08a769032e8988fd111a8d08413d02d1f
Theory Prop Address: TMWnPwZQBGvW4pee3GLW8LkfkhmmfRtVKmF

tuple_5_3_eq

Theorem 20.139 (x0, x1, x2, x3, x4) 3 = x3. The proposition is identified by the following information:

Pure Prop Id: 1b1587958694c5a5e88c0848a23a4e473c235eefdc4ef28de59de0e840fdd536
Pure Prop Address: TMaXA4j4fzMqVPAEs32Lkx3LxUCXkwLbpMV
Theory Prop Id: 527729cf29653b87109e805ea351973493674baa2ed19682f7fbb67c753e4707
Theory Prop Address: TMHJVjGa5dY6E8B16YUobHUQV7gxBZGTtkQ

tuple_5_4_eq

Theorem 20.140 (x0, x1, x2, x3, x4) 4 = x4. The proposition is identified by the following information:

Pure Prop Id: 4b873a60b686c40372a054db2e8bc63d7af0fe67ef2f26f88f9a9279bb14648f
Pure Prop Address: TMQuKuBaYQaVUjVyxJUQ247CUXS7YEvahrM
Theory Prop Id: 39fb6373be9c1ed3946575a5f4f3666171d7e8bfe812b3f8d57f6a9a6c5dafec
Theory Prop Address: TMG8m35MwfUjvACH7KLAnZSxxnizHeeC6py

Let x5 : ι be given.

tuple_6_0_eq

Theorem 20.141 (x0, x1, x2, x3, x4, x5) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: 09a916f7cab6d7d54eeebdf48131510af7e2286ceae22ad4b167cef732ec5024
Pure Prop Address: TMHkcrA1nz7EC1iKBwwwMHDsTRdph4WGCMe
Theory Prop Id: 431823a5e23f06055a6b8d3de1cd5baaf82ffaca75dc198d13b368e15c356496
Theory Prop Address: TMZfq1XThKjcGE26mvQn13MnXXuNnXLjT91

tuple_6_1_eq

Theorem 20.142 (x0, x1, x2, x3, x4, x5) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 29da6a2f4d3d0147e6f18d6c08aac5b7c54b66d6227fec45e9b37962c74efd62
Pure Prop Address: TMLK9MjtbDbJHdSXq3UFqNrBCpsMatyNCUJ
Theory Prop Id: 649eac8b3e9c6e791fc56cac42ca9787ec30ec00c1f8ecc5b68831e978a459c8
Theory Prop Address: TMcCvMHVhFKscbxwpj6BsdrkD4ne1ELkvgn

tuple_6_2_eq

Theorem 20.143 (x0, x1, x2, x3, x4, x5) 2 = x2. The proposition is identified by the following information:

Pure Prop Id: 8dde2d93ac41664c4b1687e280531801c7c9b7502d9ef12722abd166d16050e0
Pure Prop Address: TMaHcLc1izgmwaWPEBG57z2qAT17g26H4L7
Theory Prop Id: f750917ac06e451a6eecda8fca53336730cd931fdf493afb63413e3fe3aea0f9
Theory Prop Address: TMUHUCWa6x5Cs3iENyWQFdVm4DWTTXfjXKk

tuple_6_3_eq

Theorem 20.144 (x0, x1, x2, x3, x4, x5) 3 = x3. The proposition is identified by the following information:

Pure Prop Id: a5391e373cc4ee39be40eb3722e01b208d2935f48e4a422581e36b530e106751
Pure Prop Address: TMWnuyCP9Xhcj8WXFvvmExAgQv5q8FLMf87
Theory Prop Id: 9c6fc47f25c0dfa31aaf9660010b55a370cf936d297c0550428e6361c3846acb
Theory Prop Address: TMR4FTBgD9zS7EN9XzWafwrqhmFsUjwH57g

tuple_6_4_eq

Theorem 20.145 (x0, x1, x2, x3, x4, x5) 4 = x4. The proposition is identified by the following information:

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 201

Pure Prop Id: 454d77053132c71f1d965c99fa38da3d894908d33f7b7e16eaf6821ccc7266f8
Pure Prop Address: TMS87PUFZMv9WtLqYWCHZds6nUvxfnbUYRh
Theory Prop Id: 59ae63f950d447046bb6bae0eeeed28c4a7cf3348434b3fe2b85ed366aaccc5a
Theory Prop Address: TMYgmDm3q8F4Mm3PxvLRJqjhm53p6v4SckS

tuple_6_5_eq

Theorem 20.146 (x0, x1, x2, x3, x4, x5) 5 = x5. The proposition is identified by the following information:

Pure Prop Id: e53f7e86153a20747dc21bbf79ab7de024febeef898b88b7ca1b8a25f1dbe489
Pure Prop Address: TMdTTWmw4pKJNhk1pcAsNTBNcpR4rYNZdWx
Theory Prop Id: 3948d3857fe77f9ea2e9fa5db183819280b873fc0ff58efec01627914174b604
Theory Prop Address: TMcBoE6Vfa7iKDQFescgG8sF1w6NkoWVsRP

Let x6 : ι be given.

tuple_7_0_eq

Theorem 20.147 (x0, x1, x2, x3, x4, x5, x6) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: 2e6014a9fce0b27e920adaf5d4ce5cd70132df4f7186fe3ee541fbb88d55e5ff
Pure Prop Address: TMSw7zxqpUofUXu1iw13sy9eGNZFkJ83G8m
Theory Prop Id: 1dd20a0f28040bbf9f665874e170b2a7507504613edf15bfdaeccac483fe8d57
Theory Prop Address: TMFmBTCrMA6APNwYVkmVE5BR4uSANNxuCnr

tuple_7_1_eq

Theorem 20.148 (x0, x1, x2, x3, x4, x5, x6) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 50757afcd46384f164088eecd0890ea20e41d6faabc75520acbbcbc8caded1f9
Pure Prop Address: TMGz5g964HWbsbR8EdvqGJX38rYAsa8B6A4
Theory Prop Id: df1371c7546e44515aa1062d1787c0276481ea613ed0b0c96d52df80ee45960b
Theory Prop Address: TMJrw6pMykGM46pQg8E3npd2RCSbU1sxJdZ

tuple_7_2_eq

Theorem 20.149 (x0, x1, x2, x3, x4, x5, x6) 2 = x2. The proposition is identified by the following information:

Pure Prop Id: 42d292c18db7c0a80c38bd4ad63c9931011a99c7f28f1d3f98fc7be5ba02f6fa
Pure Prop Address: TMaDjeg9UWvnEZBxqcyDSjZqA9PdDXB6FdL
Theory Prop Id: f1f4ec953eb39d38b739d1cb8eb86ec8cb64944b9e496c426a73c3e9e8247f1e
Theory Prop Address: TMQmpwNMGPPbdwGTDnpPL4Bofnunn3sUWup

tuple_7_3_eq

Theorem 20.150 (x0, x1, x2, x3, x4, x5, x6) 3 = x3. The proposition is identified by the following information:

Pure Prop Id: 9bf8cc6fcb3bbff2fb13bd77a32b81fc0c78e82ac38fba8ecf898116272a76c6
Pure Prop Address: TMUya4w7djLwbukHFDyKwD1eFvHneTmmPTn
Theory Prop Id: 0ee6c00d403225b610cf3cbdcb67fe69f1b99a36ed115272a676b3ff5895ea20
Theory Prop Address: TMJdFNXdzogDAxgPCDUbrEfUH5qS2fodYvz

tuple_7_4_eq

Theorem 20.151 (x0, x1, x2, x3, x4, x5, x6) 4 = x4. The proposition is identified by the following information:

Pure Prop Id: 8462f55255364b442133232a99948ac976e6a6fb1fcad9753fd37e423292b072
Pure Prop Address: TMbPipbzrrjCW5jfNhEUnUim1TDD5mFRKh2
Theory Prop Id: a1e2a4c78071f6df819ba47d9b0dfd266b20490f5cfd23e24c39c68d7733ab6c
Theory Prop Address: TMSvveckTP4ZbhKhqsjNNf6HyqQpZtmtqeN

tuple_7_5_eq

Theorem 20.152 (x0, x1, x2, x3, x4, x5, x6) 5 = x5. The proposition is identified by the following information:

Pure Prop Id: d72f29ba20ccd83f14385aee069bd57c87c90fff8d2a86cf9c56d81df57ccf7a
Pure Prop Address: TMYnZE8WXPd7Lz6nNYaNwd53r9FLYVqbHot
Theory Prop Id: b3bff055493211cb1eaee1162d67c798ed21ac3305b1e4f92aea7ef0a84f21a3
Theory Prop Address: TMGq56172uFcBvMUViudP5XBkrjMQZDLoWW

tuple_7_6_eq

Theorem 20.153 (x0, x1, x2, x3, x4, x5, x6) 6 = x6. The proposition is identified by the following information:

Pure Prop Id: 6a6e0703f1e5cdb6bb4cdfe2feaf16914b1e19e04acbf2ec6f83392a9ba32159
Pure Prop Address: TMRnWwZbYhS2xiXxEBDwKjrHtL82AeLyj3v
Theory Prop Id: 3fd34b4049502dd532ee544c178b866077b61510f73540a979b4cb630631a075
Theory Prop Address: TMUYHfZMdobjQfNUAjkyHkmaeYViwqtCndh

Let x7 : ι be given.

tuple_8_0_eq

Theorem 20.154 (x0, x1, x2, x3, x4, x5, x6, x7) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: 4d531e28345c633c2002654dcc743a093af9ee8b7f68ff77c4f7267261a19d71
Pure Prop Address: TMckKDatqdQwJzLKWCavwMR6tjSPWLVCMWD
Theory Prop Id: 1b9fca46dbafdb59c9e774e473a1cbe61e9e6f78e7eaf15a1d24c3f6c61b8b52
Theory Prop Address: TMSNRtL24tamygfhPna4TbgXg6e1xm7U4SF

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 203

tuple_8_1_eq

Theorem 20.155 (x0, x1, x2, x3, x4, x5, x6, x7) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: a89a9e6dadd13a5330864d0051afdedff37a8d6c55b3991826097d7fd9e2a431
Pure Prop Address: TMcfnBEJnMxoVVovoBVi4dwC2PvUfkqu3eZ
Theory Prop Id: fafd80409a2da6a08db5bda00f3225829870d378adfb385c57fde0e89b0b986e
Theory Prop Address: TMPoe6wCmvUNubyq1MbpHDtwHtVKon77Awk

tuple_8_2_eq

Theorem 20.156 (x0, x1, x2, x3, x4, x5, x6, x7) 2 = x2. The proposition is identified by the following information:

Pure Prop Id: 49e82c53acd392cd8e7fdd61b280cc3953c508ee664a1617ecdc5a378d0dbc31
Pure Prop Address: TMduvfU8rksxh1TRJ5nqEAuDZZfnJ7orBPX
Theory Prop Id: c4f8325d16f1f37c6c1b8f3611beacf3e5027e0b4a964af75dd596b2fd3f3964
Theory Prop Address: TMYtGYjHksBMoXSBuCV2L7EZPAK41CBNa5B

tuple_8_3_eq

Theorem 20.157 (x0, x1, x2, x3, x4, x5, x6, x7) 3 = x3. The proposition is identified by the following information:

Pure Prop Id: 165a4eea8749fd9e02498a1ebda5ee12fe020e0fd7c18820b62a240cee4313ed
Pure Prop Address: TMVio5C24nLLNCMHHwF2Hj74cLzmmxeD1Vf
Theory Prop Id: 813d2d46172e67e30343892ae6f61699148d68c69c7af2ea5359c272e75dc1f6
Theory Prop Address: TMYr323e2X9sRx49CLZrkYn4vV85NQRyoRT

tuple_8_4_eq

Theorem 20.158 (x0, x1, x2, x3, x4, x5, x6, x7) 4 = x4. The proposition is identified by the following information:

Pure Prop Id: 4e1b495a5072ad1d3b879ed62eff2c100fb8f975c0ad2a6f54d301c81de26c44
Pure Prop Address: TMY5bJDeHJshTEqZj17jxPqa8JntKZiRsCe
Theory Prop Id: 4437b79893da40005550d241d6c12c7063f25afb64d237b52b0242a035218898
Theory Prop Address: TMMVNvpvjzLdjgRF292vzfCgLN1tRcuXph8

tuple_8_5_eq

Theorem 20.159 (x0, x1, x2, x3, x4, x5, x6, x7) 5 = x5. The proposition is identified by the following information:

Pure Prop Id: 563677657a81d01f8e8dd06ec4032e620ce92956af773ea886d10e8a18a0e37e
Pure Prop Address: TMP7vfkqYAkgGnygvPp1DhbmEUEpX2vcTC7
Theory Prop Id: 5078953fb1b78cf7ee83f76004c34c338e15e1b22a251c44b4c22a7e75b3e49d
Theory Prop Address: TMK3bws6NYHpsM12NM3kXJordSrkVZiwXdS

tuple_8_6_eq

Theorem 20.160 (x0, x1, x2, x3, x4, x5, x6, x7) 6 = x6. The proposition is identified by the following information:

Pure Prop Id: 8a30e81feba75452c73351f490aecb9e61529a2fb4beee2aea3f106abbec6783
Pure Prop Address: TMGVN5oXmWa1ZAx5JDJzTAxZNkAPUL18MWj
Theory Prop Id: 2e35ad0124fc4b59017241f14ed8cadd718350ce99b31f1f8010bd4433d2fc10
Theory Prop Address: TMVH2QvTzDv8bXohG5W1at4o2xEqMR3wTiN

tuple_8_7_eq

Theorem 20.161 (x0, x1, x2, x3, x4, x5, x6, x7) 7 = x7. The proposition is identified by the following information:

Pure Prop Id: 57dbf4f7ff71f1e9c347c26193ffe394feb41601e047c06c67f8288012465ab2
Pure Prop Address: TMULadmzcV4DX8JBCi7ogRsVnsU97NMsnBA
Theory Prop Id: 96777da9be70f7524c8eb35e7801fe9f0e20108d7bd0be0f5dc289bb821dc659
Theory Prop Address: TMcoMGrJAhGkKy8Louj6kpH3a7yNkTuMNYw

Let x8 : ι be given.

tuple_9_0_eq

Theorem 20.162 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 0 = x0. The proposition is identified by the following information:

Pure Prop Id: 26d3d48c9c93b7ec957763149ffa832050df3100f14d665f9a1c7a61f59e9925
Pure Prop Address: TMJMTn7StT21vzeC8qwYcWigwJ4kq5aY9oo
Theory Prop Id: 29c132099b2f6c53ca2f04ce3ca16cc8de2ea951e11fb4ac5dbd7fc184921f8c
Theory Prop Address: TMXgk5r2Wc1ku9pdXFHc89zP32xZ8hwx4kt

tuple_9_1_eq

Theorem 20.163 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 1 = x1. The proposition is identified by the following information:

Pure Prop Id: 3a3c1cb789c9f600657400101f3562fb437f82776a00c2a32c686549a63f8e42
Pure Prop Address: TMHkZbR5acwXro73RegaXWPvpdK2cEUGU44
Theory Prop Id: 6919b89ef581bce0dccbc7b8e0d02d831b4094ed9862095c50cb82f7fce1f138
Theory Prop Address: TMQE3e1rPeJMLKx8ndXckY8TL8GUjATgfDx

tuple_9_2_eq

Theorem 20.164 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 2 = x2. The proposition is identified by the following information:

20.7. ENCODINGS OF FUNCTIONS AND PREDICATES AS SETS 205

Pure Prop Id: 62a1ca05201c6214162f1b3abb7aac30f56939908278fa886379a9d4b8b82e80
Pure Prop Address: TMR3ZPXHQu2ywtSwNsvBLZjBq5B3bXZt1gr
Theory Prop Id: cbc94460959b856131086866189279bf769c522b562c6627b9f28426cbd337d6
Theory Prop Address: TMcHUnvscx8DjAsJLn8qhtXg6wp1Uwae864

tuple_9_3_eq

Theorem 20.165 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 3 = x3. The proposition is identified by the following information:

Pure Prop Id: ff32780d6432c453f8f483ee108e60697671b297f3692720ed857d111dfb71ff
Pure Prop Address: TMM8Wc2najKFrxPLc9rLMDHAPijUVkEZqgE
Theory Prop Id: 1c8999f1f6c222289e7f742f2e89913886228a37f06d003c45a34a2f39454710
Theory Prop Address: TMPrj3JT9vGnAt3EdTZh4hNCBJWF9jSUnCc

tuple_9_4_eq

Theorem 20.166 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 4 = x4. The proposition is identified by the following information:

Pure Prop Id: a6eff8bb384773eb50520928a17869c5a362f5a8aad62aefdbc76289e9a6d79f
Pure Prop Address: TMVD2U7o4KhyJ5vRSXBuCtRYNqiDUPGGdvV
Theory Prop Id: a8ebb556022b919699ce1585f44fe5382a8f90cbfb906757e003fee89ef48dc3
Theory Prop Address: TMYBL14qJDzq5tT7Kafcbuu2kZuud4Kxisj

tuple_9_5_eq

Theorem 20.167 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 5 = x5. The proposition is identified by the following information:

Pure Prop Id: c78a9b1e8f5a59a13fac52e14833a96eaca2cfa10c50958e33b15229b119fe88
Pure Prop Address: TMZwGSJLxkX6PKAVCtFGmZZJrSr2BqfJGTX
Theory Prop Id: d77f4e4c53456c2e683d10d9701b4877455bca6113ce67c8044a027158f9bda4
Theory Prop Address: TMYxcJb6i94M3w3oNmuZV5FueX4QnERLxfS

tuple_9_6_eq

Theorem 20.168 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 6 = x6. The proposition is identified by the following information:

Pure Prop Id: 6e88afe0e1aee7023c9f258a6001f4b275581eec0fa26c41b702c8ca1ff82aac
Pure Prop Address: TMWrYTvycmEwE7477Npqu6z4QaWkaJsxrK1
Theory Prop Id: 14cf05c1356df0137d6e6227fa3cf268b8268ba41d288eb04289088d348ca7fc
Theory Prop Address: TMbPUqUn32XqEuCscjPeu2f4QjNwqVJq1Vv

tuple_9_7_eq

Theorem 20.169 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 7 = x7. The proposition is identified by the following information:

Pure Prop Id: 810f4de4b017dc83221bcc8ff7116206998e3957bec3786295c9a0429ce531eb
Pure Prop Address: TMdA5ixqphhBmefBSSzHsamLt8GAzvT9yfo
Theory Prop Id: 6cad9319962321f56f773887d81f42aa4ad5b8eedcdcf7236252cca9d9f647ea
Theory Prop Address: TMbt5vtmatV53yku7m11vPWF23UU2d79D6y

tuple_9_8_eq

Theorem 20.170 (x0, x1, x2, x3, x4, x5, x6, x7, x8) 8 = x8. The proposition is identified by the following information:

Pure Prop Id: 8486de1f45eebeac976e64b40817e81d801358e1aa3eb4db3f60660662253a51
Pure Prop Address: TMbcVQrFnbaUtBNZg7rEVeHjsbRYhwsse9z
Theory Prop Id: 011a8763916b9f748447c0172bb5ffdb73d7eda2528b2a21e183952258975e18
Theory Prop Address: TMNGCEjuYZhPiHizVLu725KogvvA3BCGiiK

20.8 Notation for Sums and Products

Notation. We use Σx, …, y. as a binder notation corresponding to a term constructed using Sigma. We use × as a left associative infix operator corresponding to applying term setprod. We use Πx, …, y. as a binder notation corresponding to a term constructed using Pi. We use superscripts as notation for applying the term setexp.

tuple_3_in_A_3

Theorem 20.171 ∀xyzA.x ∈ A → y ∈ A → z ∈ A → (x, y, z) ∈ A^{3}. The proposition is identified by the following information:

Pure Prop Id: b6cd63778e5c307591e09f0dbe6709f3e2a0a25002362cc8bc25c497de01c8af
Pure Prop Address: TMJv4GJ6yA6E4zrTys8Fvo4pAxneFHykrEo
Theory Prop Id: 039f9b74048094fcf8f6a563d8ade3c5c8fae05574e684d2f7a1119e2bb00978
Theory Prop Address: TMcqM5egc3mW237ChXvW4EschnVxw2CaHsw

tuple_3_bij_3

Theorem 20.172

\begin{matrix}∀xyz.x ∈ 3 → y ∈ 3 → z ∈ 3 → x ≠ y → x ≠ z → y ≠ z → \\ bij 3 3 (λi.(x, y, z) i). \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 8fc73d4725ec781165c998d0b4398e6ba4fcf725b30d828af8a3419c83437f62
Pure Prop Address: TMHDdHVHUS4jN4k7phZYZXRAa2d6TcLAbNa
Theory Prop Id: 921a459779265cfc541794e752d792d1a82c2eaf3b4d1acdafdd92cbf66dab3e
Theory Prop Address: TMXTMvgV3GghUjq7QBj1XncxXwtTjahmJVW

20.8. NOTATION FOR SUMS AND PRODUCTS 207

tuple_4_in_A_4

Theorem 20.173

∀xyzwA.x ∈ A → y ∈ A → z ∈ A → w ∈ A → (x, y, z, w) ∈ A^{4}. The proposition is identified by the following information:

Pure Prop Id: 9dfa39b843ce4aacc1ee090d1483a1708c5dc6df066ae816e7a9ecee8013a642
Pure Prop Address: TMUtkQ3zWCfss9WPT6D8cb1egDHRfo1MVkn
Theory Prop Id: c23e38f69ea81031bbd8a5d922a52882ec9ad56393a4fac4d35a44bfa890dd02
Theory Prop Address: TMPzwDRXgLuZdfX4s8JwnPpH2jBU25hsWEm

tuple_4_bij_4

Theorem 20.174

∀xyzw.x ∈ 4 → y ∈ 4 → z ∈ 4 → w ∈ 4 →

x ≠ y → x ≠ z → x ≠ w → y ≠ z → y ≠ w → z ≠ w →

bij 4 4 (λi.(x, y, z, w) i).

The proposition is identified by the following information:

Pure Prop Id: 8d7a7d5ba57f64b10810f6a1009fb528d3588de74399da873640c14afb5cf725
Pure Prop Address: TMXDjtm9zxy9LdmBcvD1u1oaVds6NRkYyfJ
Theory Prop Id: d8db975b63b69912e055c4b2c24308cac5831ad7491c3f8bea7b38fe6767e7bc
Theory Prop Address: TMdGBeJemPTJ6p1TyGocFh5ejMH1SY1thm4

iff_refl

Theorem 20.175 ∀A : o.A ⇔ A. The proposition is identified by the following information:

Pure Prop Id: 01338dd4902fcd80dd89af75713f451fe1b1e4804229f05955b962fe6def6250
Pure Prop Address: TMSKz5gXPFakcgFRBJ5RJYfDQC2q8ocrFi1
Theory Prop Id: 7b71b52f0f8ece18ad34af695ce60729d9853085b816941378947b88e44cbf0a
Theory Prop Address: TML3Gcc3WbCE559uqmKxvpDahXs8iDJjURL

iff_sym

Theorem 20.176 ∀AB : o.(A ⇔ B) → (B ⇔ A). The proposition is identified by the following information:

Pure Prop Id: 00bac1cbdb77d159cd3e6b357772b77cfbf5fd08ad17791fd058a31c0475a3f6
Pure Prop Address: TMWtfJhtiuKuNHV2k6qcVXL3USFSvbCNETm
Theory Prop Id: 11f51b14cc9177b5bc57ce79c943b2dcdfe3d83f93b1a6336d2174ba4e9df0b7
Theory Prop Address: TMEgLJHQtw8TKqiFHfyA8RQk6wHyjMKHyrr

208 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS

iff_trans

Theorem 20.177 ∀ABC : o.(A ⇔ B) → (B ⇔ C) → (A ⇔ C). The proposition is identified by the following information:

Pure Prop Id: 363285ef27e2778afb838fe5fa3dd62807d7a90305f7def9e4fde9ad6aee1383
Pure Prop Address: TMZye8CSrfjLgw85BDLDes4EZvTCzY9U9wh
Theory Prop Id: c3b8f10058ec7ec17c37e487120493fd766e40d5ebea182f66ee1a52c306ddac
Theory Prop Address: TMGt6JXemXxv9o18nreYuNrNEaPaXkqRn3V

not_or_and_demorgan

Theorem 20.178 ∀AB : o.¬(A ∨ B) → ¬A ∧ ¬B. The proposition is identified by the following information:

Pure Prop Id: 8733b0a9ab0e00cdbd3a98177e398aa4cdb186d6fc7dc3377ad202c7a9841879
Pure Prop Address: TMRcU714fZk34v4xKvog8EUFH1iBy9go9dP
Theory Prop Id: 27db911b3442575d6174ef608b04eba2264a9ff43956cb9f48038590759a688b
Theory Prop Address: TMVWiPV5LBpzWhx13DG3yh1p7aPM8pSt1F5

and_not_or_demorgan

Theorem 20.179 ∀AB : o.¬A ∧ ¬B → ¬(A ∨ B). The proposition is identified by the following information:

Pure Prop Id: d4efd56c095b9a01570df661ec52e9fc1d3acf99dbb90a82e8047f781b21c525
Pure Prop Address: TMKRSDh9GPJPyX9ZtDqiV2KvjiUnNipaWCc
Theory Prop Id: 1b5ff58044cbc333c55074f5e5983bf4a0ac161bc254b95d032435918a88edba
Theory Prop Address: TMZ7VzgbGyQBuJB8KTDGonc6PtcMRxd2VSQ

not_ex_all_demorgan_i

Theorem 20.180 ∀P : ι → o.(¬∃x.P x) → ∀x.¬P x. The proposition is identified by the following information:

Pure Prop Id: 73523ea5da5dea92107627d5a291d82f8297690e14be35d05c010105c296529b
Pure Prop Address: TMMTCJ7GAVwzZ3Kw9d5Cbo8nEoqaWP6dBZs
Theory Prop Id: 075ddf1e62156846e54afd9c4c7a78cb61fa7d76482304fb50d121bd59d57fd5
Theory Prop Address: TMNGgYHHcc31AsZAqJEYPmSY4HMBaLLKAHS

not_all_ex_demorgan_i

Theorem 20.181 ∀P : ι → o.¬(∀x.P x) → ∃x.¬P x. The proposition is identified by the following information:

Pure Prop Id: 9bfa06e2e338dc6635e2549a3e17611c80217077a8b944a0478de74cd035b7e7
Pure Prop Address: TMdvAZELU6wRxcmyDehPqHUGeECff7oXd9k
Theory Prop Id: 4b147405ce7450475ec000fd6756e31ec6f80a467321e8614a031eb4efa82ae9
Theory Prop Address: TMReccPGDm1kZKJBaUcrWSWSEbaxPa45Qos

20.8. NOTATION FOR SUMS AND PRODUCTS 209

eq_or_nand

Theorem 20.182 or = (λxy : o.¬(¬x ∧ ¬y)). The proposition is identified by the following information:

Pure Prop Id: 24ac29991a11e5ff752e6de9a587fd6a214a662220d9d009904454a69ff0f25a
Pure Prop Address: TMaWRCFfz4Q1tpK7DeCGe2qubLQxDVTPHUN
Theory Prop Id: d212ef38daafdab7978d40905362a79d7c00ead847e06d6f47e2921708ce316e
Theory Prop Address: TMd6MEzYCjYvnk16kTfPmz5gduojxUSzocf

Definition 20.23 EpsR_i_i_1 is the opaque object of type (ι → ι → o) → ι identified by the following information:

Pure Object Id: 20c61c861cf1a0ec40aa6c975b43cd41a1479be2503a10765e97a8492374fbb0
Pure Object Address: TMdTiwgMp6N6HYyfybe9XUGgVm2baQ7rg6M
Theory Object Id: aced991b30f594da8af7187f84a33d664ee2260df28fd11d86b6d8b6281bc768
Theory Object Address: TMUkEyWxAV2ZVYdDpQxADf9KP7MV4bVGB3m

Definition 20.24 EpsR_i_i_2 is the opaque object of type (ι → ι → o) → ι identified by the following information:

Pure Object Id: eced574473e7bc0629a71e0b88779fd6c752d24e0ef24f1e40d37c12fedf400a
Pure Object Address: TMFeweZzkVerYMn5MZFuEXjq9udK79AoNif
Theory Object Id: 9087d981ba2801ae343ebf3a94c2e492b130dcee92d602d81eb4fa743b836009
Theory Object Address: TMG4jTkPFvT4pZQYm7XyoTAQ2rCoJWUGyE8

EpsR_i_i_12

Theorem 20.183

∀R : ι → ι → o.(∃xy.R x y) → R (EpsR_i_i_1 R) (EpsR_i_i_2 R).

The proposition is identified by the following information:

Pure Prop Id: 95a215a34ac047e6487e5f941e77b1193b239ef381850705e74d3f067c0bd706
Pure Prop Address: TMHei6pp5iZeH81iwXzRtjmAKAWBcbScwk3
Theory Prop Id: 444c78c63c5fa77e04db6105dd5977e47fec752e9fdc8aa48314397572063a6d
Theory Prop Address: TMM2ecoMZfXHpCELLyzJSaxKGa3nTPfMSpY

Definition 20.25 DescrR_i_io_1 is the opaque object of type

(ι → (ι → o) → o) → ι

identified by the following information:

Pure Object Id: 1d3fd4a14ef05bd43f5c147d7966cf05fd2fed808eea94f56380454b9a6044b2
Pure Object Address: TMYtxWX7Bmit31XQaJHqNRjz54fNehWemQx
Theory Object Id: 82aeccfa6a8d109c25ec362683dff6b3c19f69eb033bc53476eebc1307c2f73a
Theory Object Address: TMG2AhxUeoPURmtyqa7z6V67bTzk9kjG2Gb

210 CHAPTER 20. PAIRS, SUMS, FUNCTIONS AND PRODUCTS Definition 20.26 DescrR_i_io_2 is the opaque object of type

(ι → (ι → o) → o) → ι → o

identified by the following information:

Pure Object Id: 768eb2ad186988375e6055394e36e90c81323954b8a44eb08816fb7a84db2272
Pure Object Address: TMQ5YqiemA8Nk8YhMLn6FVrHVdoZLZYhFMU
Theory Object Id: 3f82bc7f6e996e5dd88799a9921cab56f8d25ed0bcfad1f9f0e2b2a2968fcab4
Theory Object Address: TMSA1F1Xty3y6u3fgqcMyVysuyvX9HKs1vN

DescrR_i_io_12

Theorem 20.184

∀R : ι → (ι → o) → o.

(∃x.(∃y : ι → o.R x y) ∧ (∀yz : ι → o.R x y → R x z → y = z)) →

R (DescrR_i_io_1 R) (DescrR_i_io_2 R).

The proposition is identified by the following information:

Pure Prop Id: a4fae2546427685ef36418f88478e6af3ac2e79bb359fbb0c2ffab0ae113217f
Pure Prop Address: TMXTFXfZpFuoocq9aqaPeoZwN93KkV2Yp2R
Theory Prop Id: da4f023fa8a6db67051918ac52618aa28cd3cd54056987f781a8085ae23273a1
Theory Prop Address: TMZVTm3MugdHj2HeAWLwWA24D1RpC1Kq1TU

Surreal Numbers I

21.1 Surreal Numbers as Predicates on Ordinals

Definition 21.1 We define PNoEq_to be λαpq.∀β ∈ α.p β ⇔ q β of type ι → (ι → o) → (ι → o) → o identified by the following information:

Pure Object Id: d7d95919a06c44c69c724884cd5a474ea8f909ef85aae19ffe4a0ce816fa65fd
Pure Object Address: TMWJ8MJ9pcvgRi55Dv3T4ZDyt4PLqinJMsL
Theory Object Id: 144b6539b9fe7bd5fe6597bae5a8aafdcf9b0b0be449f56a38d2ec86b003c9fa
Theory Object Address: TMFreHQ2BuPpMHdDn38YhqZKp88uZF4gkiP

PNoEq_ref_

Theorem 21.1 ∀α.∀p : ι → o.PNoEq_α p p. The proposition is identified by the following information:

Pure Prop Id: c27ebb17a66d8faafb90c6e2c087edacd9723fae48be3b2096fcbfb163cae3ba
Pure Prop Address: TMR5Xh2p5AFCBWLLpLnHCLb42vf2t6zTXDM
Theory Prop Id: df4cf66649d1fd1d26f29aadca2801400f850f3b4eafdc3be31d69bde8bff283
Theory Prop Address: TMVv8oi3PpMpj4sDpqjfj6YKhg5rdMeRTKn

PNoEq_sym_

Theorem 21.2

∀α.∀pq :ι → o.PNoEq_α p q→PNoEq_α q p.

The proposition is identified by the following information:

212 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: b813f3d183ae4292b34de6ea5ae07acbc9fd890e0ca37ba1d70d462219b6b8bc
Pure Prop Address: TMdRTKSCoAPR1yxWCt8NqSNPsCeHURA3SC1
Theory Prop Id: bfda0b2058ffa91747bd1ac3030674a6cc496315b46c8308434fba9aea3f2657
Theory Prop Address: TMLYYwg1Kxo3WZ71E4yBpKXxAE1s3vDazeX

PNoEq_tra_

Theorem 21.3

∀α.∀pqr : ι → o.PNoEq_α p q → PNoEq_α q r → PNoEq_α p r. The proposition is identified by the following information:

Pure Prop Id: dded4c9b3c97511b828af400470ae4f1a1e6f19deb142324786a79e7a5e93f13
Pure Prop Address: TMHc7CPnyGLonvb5EfKN1tAwcVMJrF3RQEa
Theory Prop Id: fc92f3d0615e8bb0c739baceeb42a209768755f252f7c3eec934cebe529e6569
Theory Prop Address: TMdXdRjndh2ZZ4bnALH7m92PyqRv7hAdFp5

PNoEq_antimon_

Theorem 21.4

∀pq : ι → o.∀α.ordinal α → ∀β ∈ α.PNoEq_α p q → PNoEq_β p q. The proposition is identified by the following information:

Pure Prop Id: 8868e6589767bf4a8cd7788af6bd77faeab57b18ffdcecbe07aba3ed7624aee9
Pure Prop Address: TMT6Y9Nfrocaqa6thWmudf8Un3qLyLduStm
Theory Prop Id: 094ae21814e156023eeffb5f103f193c98ceca6be42e5d08f1b636e235fc167a
Theory Prop Address: TMcxE6t2A2eCrodDrECpNG7uGeMtfeS2zeN

Definition 21.2 We define PNoLt_to be

λαpq.∃β ∈ α.PNoEq_β p q ∧ ¬p β ∧ q β

of type ι → (ι → o) → (ι → o) → o identified by the following information:

Pure Object Id: 34de6890338f9d9b31248bc435dc2a49c6bfcd181e0b5e3a42fbb5ad769ab00d
Pure Object Address: TMZD4o1Ego5dxANfVMw5J8RL7t5naNG9U6L
Theory Object Id: c95964975192a20b94bb694b4b9a7c55de7fce1d4a56cd9d0ed8d67ae58fb60e
Theory Object Address: TMG3oBNXZHp4KrZAdKCT2kf3DvDUqGNax9b

PNoLt_E_

Theorem 21.5

∀α.∀pq : ι → o.PNoLt_α p q → ∀R : o.

(∀β.β ∈ α → PNoEq_β p q → ¬p β → q β → R) → R.

The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 213

Pure Prop Id: 9ba5433aba8ff79cf5b29265ee6f028e168d178699edf447c85bbc8deeac2a4e
Pure Prop Address: TMPQB13m96524sAUWzy2GhLxsUb6akgizyg
Theory Prop Id: 45aec844f872697c4972ac0ebc6a05b529ce6188709968bbc3161616d02ce8ae
Theory Prop Address: TMNravm4aL128MRxYXGMQuCHaauCV7HVCoo

PNoLt_irref_

Theorem 21.6 ∀α.∀p : ι → o.¬PNoLt_α p p. The proposition is identified by the following information:

Pure Prop Id: fcb502c9cc3805a7a66fa875754cf8bf23ca05874f5dbdc9fa408a45bf6b32cd
Pure Prop Address: TMQ9CemPxrwhC7R2QenNg6KpFZuCjjKb8x2
Theory Prop Id: 987dff18ff1b92d203fd612706b6d76ae7437c60c295931c75665e68331e9fff
Theory Prop Address: TMYjm5nDdDXhStiue7yDo3jqogooZAR36uB

PNoLt_mon_

Theorem 21.7

∀pq : ι → o.∀α.ordinal α → ∀β ∈ α.PNoLt_β p q → PNoLt_α p q. The proposition is identified by the following information:

Pure Prop Id: b1c3cf515eec97e9e1b0860995940355d51e18df35bc3f66115f683ebee76539
Pure Prop Address: TMJDSNwcnhwkzoXr3Z8kRHFuHXmWJPaX3Ce
Theory Prop Id: d8d0f010ecea2f707f8d719d67b8d74d77b42bb87c9317a35e1a8a3e812312c5
Theory Prop Address: TMdFZnNNsv6frZTGhJdUhF23hzzvniYV8hv

PNoLt_trichotomy_or_

Theorem 21.8 ∀pq : ι → o.∀α.ordinal α → PNoLt_α p q ∨ PNoEq_α p q ∨ PNoLt_α q p. The proposition is identified by the following information:

Pure Prop Id: 6644084c8b8082d13eb4f918c0458a6a57e0e4c038c596a1395d3eb99cd50e79
Pure Prop Address: TMcyZEERpexRUZhwpuM9GSuX3Sk5rPkRLNj
Theory Prop Id: 8d3824c21ec2e23224adc53da96078403c04f5984cb1b5363e6033afb3557ba8
Theory Prop Address: TMdzU9ADKUux6zhuXTRcw9NBL5GrWiZFnL5

PNoLt_tra_

Theorem 21.9 ∀α.ordinal α → ∀pqr : ι → o.PNoLt_α p q → PNoLt_α q r → PNoLt_α p r. The proposition is identified by the following information:

214 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 3d7d91c93d4e62413c6d9dd981bb8b73fe7cceb7d203f0798d848121d5071de2
Pure Prop Address: TMLwvVmBwmxcKpTr11BfuohRJkbqDNeYgkK
Theory Prop Id: 44767a86f732b5f6b0f1ea4f2fd902fb4c657cd5b93d818dccf9adaf334006e4
Theory Prop Address: TMXUBoU9EyL1j4brgyEDkPSkDKW8vBUKaAk

Definition 21.3 PNoLt is the opaque object of type

ι → (ι → o) → ι → (ι → o) → o

identified by the following information:

Pure Object Id: 8f57a05ce4764eff8bc94b278352b6755f1a46566cd7220a5488a4a595a47189
Pure Object Address: TMQBApBYGx5gde2jLgpjV5KGX8BPSrspNY9
Theory Object Id: f2094ca61fcaf27eecff3b6b119f415eaf83537511fa1030a24f9fa1658338a9
Theory Object Address: TMUcNNa6THZvpai86WFEVkAxVdKBFzgWMLt

PNoLtI1

Theorem 21.10

∀αβ.∀pq :ι → o.PNoLt_(α ∩ β) p q→PNoLt α p β q.

The proposition is identified by the following information:

Pure Prop Id: c5735c7eb7f3308228886e03ec77196c4e828c0817e59a83eb6ea91b420edb3e
Pure Prop Address: TMS9TizXPog5PcwVnGnsKDk5q9DPz7TbhoH
Theory Prop Id: ecbc7189a71dd70627c550c8d9fdc53c6bd779fdc38db26b5b96dd819de399c8
Theory Prop Address: TMdBwxsaeBCn1Cz8duce8N385NuVba9M1iz

PNoLtI2

Theorem 21.11

∀αβ.∀pq : ι → o.α ∈ β → PNoEq_α p q → q α → PNoLt α p β q. The proposition is identified by the following information:

Pure Prop Id: 419b53c5b7f090cd6916ecca9011b3d8d19ae95bc60020cad45d8c92803c2005
Pure Prop Address: TMLoQH9MBcB6zQZDo5PkbRneUZBXQQRzDr8
Theory Prop Id: 030093820795845274dac213f6f565752cfe3de0e2652c9c76ca161494618286
Theory Prop Address: TMGXNx56uxbqYFaCe1PxneJjhYYGcLCyE8T

PNoLtI3

Theorem 21.12

∀αβ.∀pq : ι → o.β ∈ α → PNoEq_β p q → ¬p β → PNoLt α p β q. The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 215

Pure Prop Id: c392cb53f2ad2f2d109882679fc4b8a23c27eeb0ff9a5fef31347a885ba891da
Pure Prop Address: TMNZw7Cj3j1w5e5hoebGCVirnCRB3RXaNvW
Theory Prop Id: 4104429992b8f35592b8ceba357f6dc0dc7f2ba8a5f6b06451be23cdd6fb161b
Theory Prop Address: TMFhY7wPRcF4NknzzkXMPbsPrBbMZ2PZQb8

PNoLtE

Theorem 21.13

∀αβ.∀pq : ι → o.PNoLt α p β q → ∀R : o.
(PNoLt_(α ∩ β) p q → R) →
(α ∈ β → PNoEq_α p q → q α → R) →
(β ∈ α → PNoEq_β p q → ¬p β → R) → R.

The proposition is identified by the following information:

Pure Prop Id: f7e6e3e5f56759996f8e71b71ecc1cfeba2385a738e3535fb9f45c5f05a7ba7a
Pure Prop Address: TMPLU4MZci1RZyu9YajBNzBvAn2625YmMhz
Theory Prop Id: af36ceaf250752d7f1c85964b80e0b0df1acb7a0538c6e38f8cd6b431c530c5e
Theory Prop Address: TMbpk33jSr7sb7xpz9amLj1PFYcwRPKWBzJ

PNoLtE2

Theorem 21.14

∀α.∀pq :ι → o.PNoLt α p α q→PNoLt_α p q.

The proposition is identified by the following information:

Pure Prop Id: ad392c5c3af8700314707c033e9816397448a5fcd60aab607c4fa7af5a143745
Pure Prop Address: TMX9jn8QF6K3SxcZrS4opjiQZ9yxZ4mP3ju
Theory Prop Id: 7ef10485fe210dd748b752318940f7a2ccbe8a44d685b1525c931e391a9d5662
Theory Prop Address: TMX4uzbNMJMGDv9jTKuePDdqiWegaum41fi

PNoLt_irref

Theorem 21.15 ∀α.∀p : ι → o.¬PNoLt α p α p. The proposition is identified by the following information:

Pure Prop Id: ff1a80a33f6f089b5a40873efe47b5c845fd624b4250c5c7bb6d8c4de6bc398d
Pure Prop Address: TMYgpDGCHGRHDBZdPr12n2a5kcKFbuHdceA
Theory Prop Id: d4c3797f7ec820194be1429644bfb16565fc442db2d2480bac78b9f481bb2a97
Theory Prop Address: TMdk8u8CYrFAFVf8jmy3pdMCDAMd2SPuATi

PNoLt_trichotomy_or

216 CHAPTER 21. SURREAL NUMBERS I Theorem 21.16

∀αβ.∀pq : ι → o.ordinal α → ordinal β →

PNoLt α p β q ∨ α = β ∧ PNoEq_α p q ∨ PNoLt β q α p. The proposition is identified by the following information:

Pure Prop Id: 4f39e49b96938924ecee0c62b18b8054abc06386845a55af32ecbc5c07a60f7d
Pure Prop Address: TMcxeDW7jxsMf6iBBZyA1rMm28N9LGxrxAe
Theory Prop Id: 90b38722517f98c1ebbb09338d5ac216a2660c649969e817ce4af46864ab1174
Theory Prop Address: TMKtLGK85iJ16Tv3NbRNth1Uc13sd3VWcEC

PNoLtEq_tra

Theorem 21.17

∀αβ.ordinal α → ordinal β →

∀pqr : ι → o.PNoLt α p β q → PNoEq_β q r → PNoLt α p β r. The proposition is identified by the following information:

Pure Prop Id: 2e082ab6742a9e96182b14bc401ef1691287ea931ad64b2d9f8143dd38e72381
Pure Prop Address: TMKVFaHQYU7RMkic7ehXR1Hz5AxwJLND1Xs
Theory Prop Id: 2f2a828f72a2bc454fe6d454ef8a36f6aad539b02f33d5546be1b804baa1f49b
Theory Prop Address: TMVam3H7JTsW4GptAT9oJpfx8ncfn8EV3XV

PNoEqLt_tra

Theorem 21.18

∀αβ.ordinal α → ordinal β →

∀pqr : ι → o.PNoEq_α p q → PNoLt α q β r → PNoLt α p β r. The proposition is identified by the following information:

Pure Prop Id: fe144b0ff77f85ded00481bd97ae472212a28995d89fa3667ab8889dff197bb5
Pure Prop Address: TMVXYVUmzv2xFHz12tNLMKuyKAeKQiURGoM
Theory Prop Id: f076541f250b71cb6126517531b079c63b10d79f85685776f781bf746959df68
Theory Prop Address: TMRsHUfrwX3yw5eXZcC3S8S9yJKYP2vG94Z

PNoLt_tra

Theorem 21.19

∀αβγ.ordinal α → ordinal β → ordinal γ →

∀pqr : ι → o.PNoLt α p β q → PNoLt β q γ r → PNoLt α p γ r. The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 217

Pure Prop Id: b72127d429a420ed218c832bea2f3aaf8a09149da0276d6333614f557fff90a7
Pure Prop Address: TMV9P8LWZv2DWhZZm536XdWjUpfNXSoY83p
Theory Prop Id: 8eff85e4785607b84b3c70850a18baea404f11fd03cc04cfd5afa1cb189a0c5f
Theory Prop Address: TMGtkttdVbaT9NVAgziSNR2tFzasudosPWy

Definition 21.4 We define PNoLe to be

λαpβq.PNoLt α p β q ∨ α = β ∧ PNoEq_α p q

of type ι → (ι → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: ac6311a95e065816a2b72527ee3a36906afc6b6afb8e67199361afdfc9fe02e2
Pure Object Address: TMa2eyC7pjLoLi2qJAssPFeBwpm49h4vYjE
Theory Object Id: 770cf7219dc1ea3a0f1a555c8be6a16fbc83df72a88b338d5a9b6c5282933d86
Theory Object Address: TMHvhi4cZx4BpvWATQS4A5YGwuJKHFHnqZ8

PNoLeI1

Theorem 21.20

∀αβ.∀pq :ι → o.PNoLt α p β q→PNoLe α p β q.

The proposition is identified by the following information:

Pure Prop Id: 1f0528aa090a84e44d3bcd21d1f5c364665fee6e39da73097b72aa626f61a067
Pure Prop Address: TMMU61EKjZQ56vwqnq9EJcT3RZD4kTLXmEj
Theory Prop Id: 674a2cb5c37edfb9cea0bddf54687b384d613f51ce354dd54fc543730f626cc9
Theory Prop Address: TMHGpziusAgRkcNhfxaVjHNxAiJnaP9Aq4m

PNoLeI2

Theorem 21.21

∀α.∀pq :ι → o.PNoEq_α p q→PNoLe α p α q.

The proposition is identified by the following information:

Pure Prop Id: b453329022d5a4973fd4979e8276c334bd9855b0fcef38a490d0d3832d17f295
Pure Prop Address: TMRnggtbHEorcGeqrrjsVRdcj87SoZCNfE9
Theory Prop Id: 0b2157420981c9fb11d49eda5ed224af6f8a3582031030525c97a2cb5d7f2065
Theory Prop Address: TMEuEJtVZKAPqoRvcrAuffF91gTB2F6p1ur

PNoLe_ref

Theorem 21.22 ∀α.∀p : ι → o.PNoLe α p α p. The proposition is identified by the following information:

218 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 1b91bdb4dd7268a37d5bc57b8815bd8867df7f57b837c68a5a3cecba131ffe39
Pure Prop Address: TMZFLxhVzuJSDerZWEiaV9vKjwvZbajaqne
Theory Prop Id: 613386d1779902b3b59aa4f008e13853ef42f333df28e697121c9469f82e8a8d
Theory Prop Address: TMNTwN9qruHGfPxKb1SjuvQHFwhXyGnA1sj

PNoLe_antisym

Theorem 21.23

∀αβ.ordinal α → ordinal β →

∀pq : ι → o.PNoLe α p β q → PNoLe β q α p → α = β ∧ PNoEq_α p q. The proposition is identified by the following information:

Pure Prop Id: 934d7a99d54f73bf9a9e6317bc9decff59c4047034ee26980b50587b2dd95801
Pure Prop Address: TMNTPG6YJEwCxRs1V24gTDCmTzHzRvRc6gk
Theory Prop Id: 4e35944f68e317a6041fb50a04c326a498b0c1bc1e0ed7e8064b1a16614489aa
Theory Prop Address: TMLEM4Wo7kALZKsWnk4Wd5KjwhRhR5uC1HT

PNoLtLe_tra

Theorem 21.24

∀αβγ.ordinal α → ordinal β → ordinal γ →

∀pqr : ι → o.PNoLt α p β q → PNoLe β q γ r → PNoLt α p γ r. The proposition is identified by the following information:

Pure Prop Id: 1565862dfbb792dc415a313525571261e4fc5597f958d0f6839b147404b16497
Pure Prop Address: TMS5q2iXwJmrbrvpyaPEgKeFc4yT8unvrtd
Theory Prop Id: 09246b78705144b6b5679fa64a57f406c40b20b13b935e7a3c426c016425ec52
Theory Prop Address: TMP8YWTicgVUQEKtiP6hLWexaY1AtmqKF7Y

PNoLeLt_tra

Theorem 21.25

∀αβγ.ordinal α → ordinal β → ordinal γ →

∀pqr : ι → o.PNoLe α p β q → PNoLt β q γ r → PNoLt α p γ r. The proposition is identified by the following information:

Pure Prop Id: a260ca2dcf87c48e05a6f7916c83ca227958075cb681e56d4f97f095c7890142
Pure Prop Address: TMEugnvG1Bt7rUt9psKYMG4B3eXfMrHtrVq
Theory Prop Id: ad6971ea2f07aa1305f0c6bc480a834ada5728150db9419ad7ccd521ebe1ce04
Theory Prop Address: TMT7tULGek9mdvm2GJBcURRtAYaQxeEmHmE

PNoEqLe_tra

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 219 Theorem 21.26

∀αβ.ordinal α → ordinal β →

∀pqr : ι → o.PNoEq_α p q → PNoLe α q β r → PNoLe α p β r. The proposition is identified by the following information:

Pure Prop Id: 7d3ff80df17ec96e504b4964272829064a97b36a1e75174ea67fe631b2aa92ab
Pure Prop Address: TMWEkTHjED61EVaMYyFDrbsBT3uuPxsd2WS
Theory Prop Id: 002ebf808b3010e1129d1a3d0f3318345316603ff5e9f8a4752963619b88ad30
Theory Prop Address: TMX3rf1dBjD7HFURU7toJLUuPtwumdeBSD4

PNoLeEq_tra

Theorem 21.27

∀αβ.ordinal α → ordinal β →

∀pqr : ι → o.PNoLe α p β q → PNoEq_β q r → PNoLe α p β r. The proposition is identified by the following information:

Pure Prop Id: 2def03362104b85520032b762fc8d4c1d8df8e5ec52650b7c3b2e85cae9c2fcc
Pure Prop Address: TMJmHGCtRPZGoKvdPGHCWSTaoKXedqvvXcT
Theory Prop Id: ed30aa19eb8fe59fe9c61636687168d8be8ac512c5d98ff2581121ba1ffd0765
Theory Prop Address: TMRHBJiUqPVNK9rBz9VWXHRyyHjTuJD3RGU

PNoLe_tra

Theorem 21.28

∀αβγ.ordinal α → ordinal β → ordinal γ →

∀pqr : ι → o.PNoLe α p β q → PNoLe β q γ r → PNoLe α p γ r. The proposition is identified by the following information:

Pure Prop Id: 70b92b6662709f7dc5822fd97e53303a3a2d055ae004b41c4e645dce93c651bc
Pure Prop Address: TMHzFcB1asSRVnHkPCarNFE3z9eXcp4jGKN
Theory Prop Id: 4b557dc79b8f761dcdf6d3964db848fe773041a6fb41c452623328cec43d42d1
Theory Prop Address: TMRqHazfJDQBeT8YDQifbd4FBKaDUPSHefM

Definition 21.5 We define PNo_downc to be

λLαp.∃β.ordinal β ∧ ∃q : ι → o.L β q ∧ PNoLe α p β q

of type (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

220 CHAPTER 21. SURREAL NUMBERS I

Pure Object Id: 93dab1759b565d776b57c189469214464808cc9addcaa28b2dbde0d0a447f94d
Pure Object Address: TMFFjJ65YM9kJCjR5s8uzZawGiUXuQM1tjB
Theory Object Id: f6c96279232d9e9cca23ffe49c0c29166fada636c1195de7ffbd7fabd07ac1c7
Theory Object Address: TMbMkrzAgHfL7RAFKvYtv8vLTv3CdoSjov9

Definition 21.6 We define PNo_upc to be

λRαp.∃β.ordinal β ∧ ∃q : ι → o.R β q ∧ PNoLe β q α p

of type (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 5437127f164b188246374ad81d6b61421533c41b1be66692fb93fdb0130e8664
Pure Object Address: TMaAkmpSZzSfoMddDbQnVgH1TnDuodvBF4s
Theory Object Id: b97935067aeb5829f0c939603d3342813777d67c23748ee242158f33ac42ed50
Theory Object Address: TMdz4JctWDDKNbEgwhwYLZAiTPcFhWgefwa

PNoLe_downc

Theorem 21.29 ∀L : ι → (ι → o) → o.∀αβ.∀pq : ι → o.ordinal α → ordinal β → PNo_downc L α p → PNoLe β q α p → PNo_downc L β q.

The proposition is identified by the following information:

Pure Prop Id: 6b1f0d64dd061a5eba7c0342d7f3676187349916ec76342278678c84b205b24f
Pure Prop Address: TMWAyyBEGMhjikXgumQbCvdi5WRdMG4XswZ
Theory Prop Id: cd130e6e61b7fe7f0bfe1dd262ffaa5f86165a48fa9f622a9e0d11503c799ff3
Theory Prop Address: TMM4WeADC271qKWhMRuk4GonoM9GLTxaXM9

PNo_downc_ref

Theorem 21.30

∀L : ι → (ι → o) → o.∀α.ordinal α → ∀p : ι → o.

L α p → PNo_downc L α p.

The proposition is identified by the following information:

Pure Prop Id: 347aa9ec835b739b3314c47b28ca0cd2ebf6b08b10135cc7cb91d5cdf39bed56
Pure Prop Address: TMT5dDg6pVSTQmj3sYdZvHj975JFuQSEf3R
Theory Prop Id: 69dfcb8516ec52d7595fe7c4aaf6805468aa6d2634d29ea2b8226ad567420a27
Theory Prop Address: TMUEbPAgDX5aYyR8xXjQWGAsxbj8SwH4QN4

PNo_upc_ref

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 221 Theorem 21.31

∀R : ι → (ι → o) → o.∀α.ordinal α → ∀p : ι → o.

R α p → PNo_upc R α p.

The proposition is identified by the following information:

Pure Prop Id: 4039424bd88802176a6cbb56b76abcf8b8a1f616a3d0721d28e58f9f096aa816
Pure Prop Address: TMRis4sdu68KHLQb77r68nGDtFxyhKXKcBM
Theory Prop Id: c4bf71e4c9e3609364a3a0c2ce3ae0a4befd6f3181cea4eec8656173a1b32b72
Theory Prop Address: TMFX2E2HrfER46NsW7mPNvJqqJZFcAuJx1i

PNoLe_upc

Theorem 21.32

∀R :ι → (ι → o) → o.∀αβ.∀pq :ι → o.ordinal α→ordinal β→

PNo_upc R α p→PNoLe α p β q→PNo_upc R β q.

The proposition is identified by the following information:

Pure Prop Id: 111b74916dd3b9d72613111538f033f3bb8b5dfe2d40082b76d45a031efb7cc7
Pure Prop Address: TMcxmbHaZztM29ZiLWZpgrqw4nZNUzfJYeS
Theory Prop Id: eb9ed5adca244da4ff7ae93d024a4823361cb4695e55f8fa465e7e14c460eae2
Theory Prop Address: TMJYddW3A9sz7atvMHDox6oK3Fz5ZXhVgw5

Definition 21.7 We define PNoLt_pwise to be

λLR.

∀γ.ordinal γ → ∀p : ι → o.L γ p →
∀δ.ordinal δ → ∀q : ι → o.R δ q →
PNoLt γ p δ q

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → o identified by the following information:

Pure Object Id: ae448bdfa570b11d4a656e4d758bc6703d7f329c6769a116440eb5cde2a99f6f
Pure Object Address: TMc96g2JPpBeTmDVeFvvSZTGX5Tb5262YGw
Theory Object Id: 12380893a7dd9882f429d48a7961599b1aa9e328a364d33120d9b2b3f2b04f70
Theory Object Address: TMQdLYLNu2LJgg1svc2g1R5FdDAS2iXTH2E

PNoLt_pwise_downc_upc

Theorem 21.33 ∀LR :ι → (ι → o) → o.PNoLt_pwise L R→ PNoLt_pwise (PNo_downc L) (PNo_upc R).

The proposition is identified by the following information:

222 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 2b8e3bd1c2f83410acc3eb16745b3c189b91f3d88eff48c21bc1fe2da62df2a9
Pure Prop Address: TMaupVE3uvrBXnhCNWJG4uidezoNueB9Pzn
Theory Prop Id: 2f244c622c329868222671ac992535b2ec8a107cab98a55451d89fc062755198
Theory Prop Address: TMSZ221fidpj3oVVEjh7AxtKFSvU4SKPRwZ

Definition 21.8 We define PNo_rel_strict_upperbd to be

λLαp.∀β ∈ α.∀q : ι → o.PNo_downc L β q → PNoLt β q α p

of type (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: dc1665d67ed9f420c2da68afc3e3be02ee1563d2525cbd6ab7412038cd80aaec
Pure Object Address: TMPiEDqG64KxZbAF6RkpdrzFu2A2HmgQJ6p
Theory Object Id: 3e1663abe9ff987d905d14ab1ac2db49036bac313db730a8f2e8fbb4db756c76
Theory Object Address: TMGu9LKc1Ew275sc36SoMWGV6V7yydJCxoa

Definition 21.9 We define PNo_rel_strict_lowerbd to be

λRαp.∀β ∈ α.∀q : ι → o.PNo_upc R β q → PNoLt α p β q

of type (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: e5a4b6b2a117c6031cd7d2972902a228ec0cc0ab2de468df11b22948daf56892
Pure Object Address: TMMtUywXSfQieSYmZXKhRaMVMvNBJ4mv5n1
Theory Object Id: ff3150c8fe3f147f482d147d4948a3adfa0f23616d03c877213e009957f55a8e
Theory Object Address: TMUamWcr8Mykvg8ZyWAHjTGbXtZ7LdvyWix

Definition 21.10 We define PNo_rel_strict_imv to be

λLRαp.PNo_rel_strict_upperbd L α p ∧ PNo_rel_strict_lowerbd R α p

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 64ce9962d0a17b3b694666ef1fda22a8c5635c61382ad7ae5a322890710bc5f8
Pure Object Address: TMFES6j6egEfn4hHem6CQT3DsNYsaxQYweU
Theory Object Id: 5b35fbff3a2443e9ad06778c49bcebadc3d05d5d8c2da1fb0fad3c9047e95755
Theory Object Address: TMXkoajkux97senphT28NAy8HKer2u4kPUc

PNoEq_rel_strict_upperbd

Theorem 21.34

∀L :ι → (ι → o) → o.∀α.ordinal α→∀pq :ι → o.PNoEq_α p q→

PNo_rel_strict_upperbd L α p→

PNo_rel_strict_upperbd L α q.

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 223

Pure Prop Id: bb65cf435cd1184f19969d40225b30f85ec7629c2a8b3845828f9cae4b8218cd
Pure Prop Address: TMbeVMWawJhuN1ZSdk7xfgpCpASxk2FoVZv
Theory Prop Id: db4b8216b0bda52674a1b262e30b3b493a88c2284eb09e35892c6464171da698
Theory Prop Address: TMR2fFPGsUbreZFxGSKzs2Ut4aBjCbtG8FU

PNo_rel_strict_upperbd_antimon

Theorem 21.35

∀L :ι → (ι → o) → o.∀α.ordinal α→∀p :ι → o.∀β ∈α.

PNo_rel_strict_upperbd L α p→

PNo_rel_strict_upperbd L β p.

The proposition is identified by the following information:

Pure Prop Id: 9986afc591c8630816ae93586b1d747b9e6a7d43426e319ea6699264ca845b9c
Pure Prop Address: TMFfTTpCGWMgKptEhmq4H5u6A1pxWeCHZWr
Theory Prop Id: ed53da6a8a942b087f817d785b79f257de9cb0f526d77897b831913b7bfb6d4f
Theory Prop Address: TMdLwNGovgqDXAVmQmmx97gnpUHqbCcATFy

PNoEq_rel_strict_lowerbd

Theorem 21.36

∀R :ι → (ι → o) → o.∀α.ordinal α→∀pq :ι → o.PNoEq_α p q→

PNo_rel_strict_lowerbd R α p→

PNo_rel_strict_lowerbd R α q.

The proposition is identified by the following information:

Pure Prop Id: 76080ee1804cd6f07353c102ecc22dbe6f88819fba1748db34678b3a3205ce51
Pure Prop Address: TMRaaoK9aPyvyCQEyS6u7CuxJeMRarcqdua
Theory Prop Id: 614e589383386bf6bffd56988d8ed052f11fa7717c9bfc928aa218b49ba14436
Theory Prop Address: TMN9uHHDyxEb7wZLV54NYgKUBnfgd8Syeiz

PNo_rel_strict_lowerbd_antimon

Theorem 21.37

∀R :ι → (ι → o) → o.∀α.ordinal α→∀p :ι → o.∀β ∈α.

PNo_rel_strict_lowerbd R α p→

PNo_rel_strict_lowerbd R β p.

The proposition is identified by the following information:

Pure Prop Id: 27f253b3d87846cda9c59a20ef82f3175140815d3b9625c2b5dfe1a332086817
Pure Prop Address: TMQ6kfs1zMTGEcR3v2oD9G2cmxvPeeGcygk
Theory Prop Id: 344b63ede082390378753476ed8b39a6cdcbc2522581fb25d8b3e2dbcd366fd5
Theory Prop Address: TMVV2kTQbx2zWgUFoViphDG4PSygzxibjvw

224 CHAPTER 21. SURREAL NUMBERS I

PNoEq_rel_strict_imv

Theorem 21.38

∀LR :ι → (ι → o) → o.∀α.ordinal α→∀pq :ι → o.PNoEq_α p q→

PNo_rel_strict_imv L R α p→

PNo_rel_strict_imv L R α q.

The proposition is identified by the following information:

Pure Prop Id: e3f251299cd1135da075622be47bde557bbba6cbac1e28d78bb914fdda99a216
Pure Prop Address: TMX9qa29bL95NMwjxy6sv4Ubp6sbR8ZXSAy
Theory Prop Id: 7ae13df8bdc5ae35409ee041f3bcf55387bae17d0117e5e6491d3cb5cb89d4f9
Theory Prop Address: TMWRDDHJba821Tbd4gG36AUGcRoP4wgdEtZ

PNo_rel_strict_imv_antimon

Theorem 21.39

∀LR :ι → (ι → o) → o.∀α.ordinal α→∀p :ι → o.∀β ∈α.

PNo_rel_strict_imv L R α p→

PNo_rel_strict_imv L R β p.

The proposition is identified by the following information:

Pure Prop Id: 2434dc0ac445139024cddcec7dd72380f70b6c60cbf7e3eea162062ef8ef30b5
Pure Prop Address: TMTzSVmWpr6G9xpGB9xRz9GHutphxoFHpm5
Theory Prop Id: 87e0ce7b41f81ec7cd5cb539c8d29a4ee84162503f0111c69851aa5cb6250f67
Theory Prop Address: TMdHPiqFbrqqnJZvUEgGaaNprnFuirYL2XJ

Definition 21.11 We define PNo_rel_strict_uniq_imv to be

λLRαp.PNo_rel_strict_imv L R α p∧

∀q : ι → o.PNo_rel_strict_imv L R α q → PNoEq_α p q

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 23c9d918742fceee39b9ee6e272d5bdd5f6dd40c5c6f75acb478f608b6795625
Pure Object Address: TMcVvujMz8D899TFXW92xtjqAjrJ7xZ5wns
Theory Object Id: 56ed1b924504c7b2800fa12637e1e0a522196c5a1a82a00af10c643a8a5e9e6f
Theory Object Address: TMJg15q9kSMojiP1K5gjvfgbDB2SFT15THF

Definition 21.12 We define PNo_rel_strict_split_imv to be

λLRαp.PNo_rel_strict_imv L R (ordsucc α) (λδ.p δ ∧ δ ≠ α)∧

PNo_rel_strict_imv L R (ordsucc α) (λδ.p δ ∨ δ = α)

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 225

Pure Object Id: 217d179d323c5488315ded19cb0c9352416c9344bfdfbb19613a3ee29afb980d
Pure Object Address: TMcNiZGAEhYLnFaHRVtxA3RMJ6rQRD68crC
Theory Object Id: dd9ceee1f10e584685cd8a5d181d107a00ffeb879bddd77bb943edc9d869a262
Theory Object Address: TMaSwGSK92j2THPci27ww2urWPVb7Y8PaRX

PNo_extend0_eq

Theorem 21.40 ∀α.∀p : ι → o.PNoEq_α p (λδ.p δ ∧ δ ≠ α). The proposition is identified by the following information:

Pure Prop Id: 2d441ed2f6faaa3204d9c074c159ce10040fd6680f3c4f69726e8dbc60efc1b8
Pure Prop Address: TMWBZZ2LzEnHQnhQ2EUjYaKDpHL6urz6cT8
Theory Prop Id: 5ed08e400c3976db41fd77c3d882770ae3d2bc85093f84c0060820655b4328c3
Theory Prop Address: TMPWoqhrnorzigPXmr4GmgWiGibdw73sjz8

PNo_extend1_eq

Theorem 21.41 ∀α.∀p : ι → o.PNoEq_α p (λδ.p δ ∨ δ = α). The proposition is identified by the following information:

Pure Prop Id: 8eb99c36aa09b197a65897ed80cdf2570eba4e719d286f36fc4bc09afd4f3689
Pure Prop Address: TMQEFMAonmrWaLGhniLUme237vwrY6AwX3L
Theory Prop Id: 4cac1a7bb678742b042b74050de5270cd3a104cca4e451080d85e20b33bff7dc
Theory Prop Address: TMGkMWYd35nspxAzshHYLLW93mSrEQa177W

PNo_rel_imv_ex

Theorem 21.42

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

(∃p : ι → o.PNo_rel_strict_uniq_imv L R α p) ∨(∃τ ∈ α.∃p : ι → o.PNo_rel_strict_split_imv L R tau p).

The proposition is identified by the following information:

Pure Prop Id: 3e3224100da33bad94d660a0cfa8843fa7519fd5ff75e4c23da58e5b68707714
Pure Prop Address: TMb7LjGHosQp5Zu2ZyhehFbFtpyZL2yxrkB
Theory Prop Id: 749b63b90f39e599387d66934130ae6ac3e08e4f9989ca927eec36e855e09fc8
Theory Prop Address: TMMKoMsquCSh1xogZaGrGcBP6CoCXQLGM1t

Definition 21.13 We define PNo_lenbdd to be

λαL.∀β.∀p : ι → o.L β p → β ∈ α

of type ι → (ι → (ι → o) → o) → o identified by the following information:

Pure Object Id: 009fa745aa118f0d9b24ab4efbc4dc3a4560082e0d975fd74526e3c4d6d1511f
Pure Object Address: TMYM1fGNqp8awnq5MVCDqBVSdy2y4KSQX4c
Theory Object Id: a452a633e2734cc4531287f806932a19db788c9ebf515fd5c65361a126fcccda
Theory Object Address: TMc7MFWWHbWct8gGwr9Jjykxysxajx7WxYR

226 CHAPTER 21. SURREAL NUMBERS I

PNo_lenbdd_strict_imv_extend0

Theorem 21.43

∀LR :ι → (ι → o) → o.∀α.ordinal α→PNo_lenbdd α L→ PNo_lenbdd α R→∀p :ι → o.PNo_rel_strict_imv L R α p→

PNo_rel_strict_imv L R (ordsucc α) (λδ.p δ ∧ δ ≠ α).

The proposition is identified by the following information:

Pure Prop Id: 17a11416cb52c3c428daf77fd88095938feda8129ca2851349863fa5a7f70fab
Pure Prop Address: TMZ7dVvvzt9aBgJceuie2x9CTj7zzynBVKW
Theory Prop Id: 69327e8aa4dfac05b5218d04f861172313490347f560cc3142dc11e85c3d1de9
Theory Prop Address: TMFuqsPkUmPBmPBLFudNnuZaej9uSQJFdHA

PNo_lenbdd_strict_imv_extend1

Theorem 21.44

∀LR :ι → (ι → o) → o.∀α.ordinal α→PNo_lenbdd α L→ PNo_lenbdd α R→∀p :ι → o.PNo_rel_strict_imv L R α p→

PNo_rel_strict_imv L R (ordsucc α) (λδ.p δ ∨ δ = α).

The proposition is identified by the following information:

Pure Prop Id: ad69ec2daa15891f8d7b060e461a3f1b3e63589a17d9fe0b50275b428c54760d
Pure Prop Address: TMHJrXcf4TaXkTgTPiRVvo9nkWreomZdRJn
Theory Prop Id: 5ff9936fa7da19e574c1e407d5dc4601e10f51aca435de66bb11cc9e7ada1df8
Theory Prop Address: TMPvUJKpgzdwMRuCFNU2GRtAfonYYsF38vv

PNo_lenbdd_strict_imv_split

Theorem 21.45

∀LR :ι → (ι → o) → o.∀α.ordinal α→PNo_lenbdd α L→ PNo_lenbdd α R→∀p :ι → o.PNo_rel_strict_imv L R α p→

PNo_rel_strict_split_imv L R α p.

The proposition is identified by the following information:

Pure Prop Id: 76ae4f9f794260b3f0f83c999d5560105286744cef38b1a0234287b13ad8db9e
Pure Prop Address: TMaW9z99uWRswWnxdDU9cvmKc6HtRaCtVRU
Theory Prop Id: f98998c738859a6ff5fbcf1701155ca36cbcb0caef56b13a9cff1b1c937b6ece
Theory Prop Address: TMFZtcp8Jrq2c5Vbiocbri3d5UhPuowpAaw

PNo_rel_imv_bdd_ex

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 227

Theorem 21.46

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→

∃β ∈ ordsucc α.∃p : ι → o.PNo_rel_strict_split_imv L R β p.

The proposition is identified by the following information:

Pure Prop Id: 9a98e4a421cb8eedc40a867b16cac1d2529c4dfabcc3ff3b451f9e220420f209
Pure Prop Address: TMRSz1Na7BW752MS6vDye8Y6Vo8SfeVhV1j
Theory Prop Id: 25bcd2bc2f9d34e476f5e18c7b51c9075ed93e802655cc620f888c8e1fd16a22
Theory Prop Address: TMQfEm1NLf1eHjLsYL7HyTLrAxiyrsEBsSQ

Definition 21.14 We define PNo_strict_upperbd to be

λLαp.∀β.ordinal β → ∀q : ι → o.L β q → PNoLt β q α p

of type (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 6913bd802b6ead043221cf6994931f83418bb29a379dc6f65a313e7daa830dcc
Pure Object Address: TMKnMYekteuHoeWstJHEmJ4JAdKPexnFA42
Theory Object Id: 3723abda991ef19534694f8f4cd239c37f64fc7c8d2f1cbae883cc9f9d0d69b3
Theory Object Address: TMRkoJnHGKhQmpq21wXvTR2RBkLqLSagigX

Definition 21.15 We define PNo_strict_lowerbd to be

λRαp.∀β.ordinal β → ∀q : ι → o.R β q → PNoLt α p β q

of type (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 87fac4547027c0dfeffbe22b750fcc4ce9638e0ae321aeaa99144ce5a5a6de3e
Pure Object Address: TMZjYeWB7UQog19EjFL4PmaNhnokNCuMGJV
Theory Object Id: 3606d45e76891d3b7e28745aa1b61986d255603f7abeb8f63a847b1ccc62b95d
Theory Object Address: TMXvc4Vsa77sgNvdnXny3CP9WCzsnoarmV1

Definition 21.16 We define PNo_strict_imv to be

λLRαp.PNo_strict_upperbd L α p ∧ PNo_strict_lowerbd R α p

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: ee97f6fc35369c0aa0ddf247ea2ee871ae4efd862b662820df5edcc2283ba7e3
Pure Object Address: TMbEV7hkJGaPtHGWT7FsCyrBxQjHqPm9Ct9
Theory Object Id: bc6c55650a1ceb99aefb5c3b39fed27e70fc45b811cc09f91f1fd23a0cfbe326
Theory Object Address: TMHAeW8tAPaNuMKzRy2t7UZyujPe3oz8chT

228 CHAPTER 21. SURREAL NUMBERS I

PNoEq_strict_upperbd

Theorem 21.47

∀L :ι → (ι → o) → o.∀α.ordinal α→∀pq :ι → o.PNoEq_α p q→

PNo_strict_upperbd L α p→PNo_strict_upperbd L α q.

The proposition is identified by the following information:

Pure Prop Id: 9b7b8e2913af3ee84872e0fd4fca5bfeb65de5ae332f50e8005ce0b40c0390d1
Pure Prop Address: TMMwWqqBxTemQcZaoYuQ5csnpPswRJuhRvB
Theory Prop Id: f28b04874428203d54a6576e08aded8841fea8787b265c71cffb0a98a190b2b1
Theory Prop Address: TMFSxkkKeWx8CjszfDWMQmeEXEdqcpXHZqL

PNoEq_strict_lowerbd

Theorem 21.48

∀R :ι → (ι → o) → o.∀α.ordinal α→∀pq :ι → o.PNoEq_α p q→

PNo_strict_lowerbd R α p→PNo_strict_lowerbd R α q.

The proposition is identified by the following information:

Pure Prop Id: 4db3f9364356674211f51fc9a183e838dff60a313c518ca7d65b13fd0698d5a1
Pure Prop Address: TMG1CzmnW8AfKeL7zGEwMEtn4b1ApJy6q1g
Theory Prop Id: 67fa73a4227fa42a9ed6db8fd115f876749a15c619bbceedaf8114345d2a228b
Theory Prop Address: TMFvqJhbAHTTaVpfbgHnnHh9Roc9bwF9YcE

PNoEq_strict_imv

Theorem 21.49

∀LR :ι → (ι → o) → o.∀α.ordinal α→∀pq :ι → o.PNoEq_α p q→

PNo_strict_imv L R α p→PNo_strict_imv L R α q.

The proposition is identified by the following information:

Pure Prop Id: 1ec3525cfde857873a05b1492e442895aa98e91b1a7a58db58116a8baa7863f3
Pure Prop Address: TMWLyiXz4WsN9qiqmYUTwRb7RrR4sGpvULn
Theory Prop Id: 5f582f95ff13676a0730e043b393f09bf9b302f7c01959e268fdbaeaaba7bd23
Theory Prop Address: TMZZL19r37MLXbo9tRES9F3kF1Cdd7rP1Vx

PNo_strict_upperbd_imp_rel_strict_upperbd

Theorem 21.50 ∀L :ι → (ι → o) → o.∀α.ordinal α→∀β ∈ordsucc α.∀p :ι → o. PNo_strict_upperbd L α p→PNo_rel_strict_upperbd L β p.

The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 229

Pure Prop Id: 2898f0821b6ee4c20a8875fb97191a1b12a50a1e963c11f9aa1a31c2bd46f8e2
Pure Prop Address: TMXJEXJKF7JrSAhmPixqwmaahA97oWAoeLL
Theory Prop Id: 3cdd3b4f20ab67c1be5329318323deafeea009f34f28baa01090782c3bdfdd0e
Theory Prop Address: TMbeJVU3GyejPAEuuCqdu9srTvCD2PVUAy8

PNo_strict_lowerbd_imp_rel_strict_lowerbd

Theorem 21.51 ∀R :ι → (ι → o) → o.∀α.ordinal α→∀β ∈ordsucc α.∀p :ι → o. PNo_strict_lowerbd R α p→PNo_rel_strict_lowerbd R β p.

The proposition is identified by the following information:

Pure Prop Id: 442af59a4f76dc1ad25be2cc8abd5af2847bff7cd5f1b08ec480d479fe6f9148
Pure Prop Address: TMMPUoKVYbXX3DBLQHFe5h9b2gv8eiAFaAb
Theory Prop Id: 04acc2c7664cb2baa2edfe5f073538b0cb855556d54d07337c5d7f3ccfcf5ab0
Theory Prop Address: TMPPvnYPwEv5KBauhbm93s9dRpjMraJ2RMs

PNo_strict_imv_imp_rel_strict_imv

Theorem 21.52

∀LR :ι → (ι → o) → o.∀α.ordinal α→∀β ∈ordsucc α.∀p :ι → o.

PNo_strict_imv L R α p→PNo_rel_strict_imv L R β p. The proposition is identified by the following information:

Pure Prop Id: f4fc71c90b443abe5f08ec2e1cbad2b8e178220ae7a28146d2debaa27c8dca9e
Pure Prop Address: TMUZW37y1v2yvXxB4qDRF8ycQ12jacn99PZ
Theory Prop Id: 46a6aacddbb0224f249308bdbd8f4155dd2574ab91755a3faeebd246dd1ac8da
Theory Prop Address: TMaWoAQeNZDif3qPS3T1yTknjm8pRRW8a4h

PNo_rel_split_imv_imp_strict_imv

Theorem 21.53

∀LR :ι → (ι → o) → o.∀α.ordinal α→∀p :ι → o.

PNo_rel_strict_split_imv L R α p→

PNo_strict_imv L R α p.

The proposition is identified by the following information:

Pure Prop Id: 72be36124095b1c454cfc103ace8e578f6da1f3537bed3a63dff48a5ee9e67e3
Pure Prop Address: TMGERrdQ6q2bYZah7DP6dYkxMqTrBkNs3qx
Theory Prop Id: adfdb01bca46911cd3ac3a75bf022e1e4e0316883433f9218fddedd9a12aceeb
Theory Prop Address: TMTEYb8yRqBikXo6Wt5MPUkSefJ4iSEG83i

ordinal_PNo_strict_imv

230 CHAPTER 21. SURREAL NUMBERS I

Theorem 21.54 ∀LR :ι → (ι → o) → o.∀α.ordinal α→∀p :ι → o.(∀β ∈ α.p β)→

(∀β.ordinal β → ∀q : ι → o.L β q → β ∈ α)→(∀β ∈ α.L β p)→

(∀β.ordinal β → ∀q : ι → o.¬R β q)→PNo_strict_imv L R α p. The proposition is identified by the following information:

Pure Prop Id: 6ac14c06740c6ba05ec4cd77fc370b4915f44952ab2e3e843377f3ae0f4ef372
Pure Prop Address: TMPF3ri47F9YTagy21jCuzhVZiWgwJhBagV
Theory Prop Id: d6a73528e480611cb4cc96b7f8a095233cd0b5ce7a4c96d973c70a42ac001c03
Theory Prop Address: TMTeL2CwHrU7yAsCK9SoepcoGiqPerjGfAP

PNo_lenbdd_strict_imv_ex

Theorem 21.55

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→

∃β ∈ ordsucc α.∃p : ι → o.PNo_strict_imv L R β p.

The proposition is identified by the following information:

Pure Prop Id: 9485d658371746bd448a504cec578379db903809a34d0f3d7395e9ad92f52bc5
Pure Prop Address: TMboc8u3wjuBmQSkALzv47nZjKdu9Ln6G1i
Theory Prop Id: a864a57867c4b8fa4046e9c687fd4f648d42ccbee4cef69cf3b1c28bb983c0eb
Theory Prop Address: TMXJMwoh8T8TYKwFJYwWGnhaeosMEU67QtG

Definition 21.17 We define PNo_least_rep to be λLRβp.ordinal β ∧ PNo_strict_imv L R β p∧

∀γ ∈ β.∀q : ι → o.¬PNo_strict_imv L R γ q

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 0711e2a692a3c2bdc384654882d1195242a019a3fc5d1a73dba563c4291fc08b
Pure Object Address: TMMrUbZHNUQojuJFRTNTGugMnkz3yaWyJjG
Theory Object Id: e4c707e63a9d11d1b61d0db26f0cf3155ea6db0eca2d2f3dbf812d36a70e6c70
Theory Object Address: TMdTiKpkh6t2DH3NuLyGfeGX79J5YUPNSBs

PNo_lenbdd_least_rep_ex

Theorem 21.56

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→
∃β.∃p : ι → o.PNo_least_rep L R β p.

The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 231

Pure Prop Id: 4541c94e58f35924a0da5fd80f4d2327c22d662287cf6c55c9962fcacd32b6c0
Pure Prop Address: TMV8v9NRJQArdep3t94cASip2uDW46XdwHJ
Theory Prop Id: fc95a5ba7826e45c22086496d12170955e43c43c454f7c838a38598d83edd933
Theory Prop Address: TMQu9hsS7YcjLuJMR9UBzRz4gDzHhbZw56N

Definition 21.18 We define PNo_least_rep2 to be

λLRβp.PNo_least_rep L R β p ∧ ∀x.x ∉ β → ¬p x

of type (ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: 1f6dc5b1a612afebee904f9acd1772cd31ab0f61111d3f74cfbfb91c82cfc729
Pure Object Address: TMTPNGLhRLywGV7b2VErKKRqsBtTRqSdE63
Theory Object Id: 09d320f50b226d6de787cb7f04c0ac3eedf43d3981566d46692b73e884662b9b
Theory Object Address: TMRAk8jd2vN252xuwfLzStCvRTrSjBsrXz3

PNo_strict_imv_pred_eq

Theorem 21.57

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→∀pq :ι → o.

PNo_least_rep L R α p→PNo_strict_imv L R α q→∀β ∈α.

p β ⇔ q β.

The proposition is identified by the following information:

Pure Prop Id: 5e7822daeca655e2242d474ec86dd9579effe366031b4f9b73ec5ca4a659a545
Pure Prop Address: TMZq9y1Uq2aENtjsxC2H2kmR5Sts8dSVeAn
Theory Prop Id: 72e14d37cd81d2a456f9878f785f26768d709f7848a4cf627865ba0c3662b1b3
Theory Prop Address: TMP8YWfHwKekxFLMQmDee1eb36HPVWfDCkn

PNo_lenbdd_least_rep2_exuniq2

Theorem 21.58

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→
∃β.(∃p : ι → o.PNo_least_rep2 L R β p)∧
(∀pq : ι → o.PNo_least_rep2 L R β p →
PNo_least_rep2 L R β q → p = q).

The proposition is identified by the following information:

Pure Prop Id: 5060a91293273a3d3416ab99070c6d1e15f0cbef0c24aa15399ee37c868f691c
Pure Prop Address: TMT7sUcy4ndNzr9t94F3xrCG638UAh62av3
Theory Prop Id: 8b62c56d475417d4d361eaf3c3c3680d80138732014122d77d426f4860f69081
Theory Prop Address: TMNmdxAHFQ1AVyvBStP6JJjmC4WocYjvTV2

232 CHAPTER 21. SURREAL NUMBERS I Definition 21.19 PNo_bd is the opaque object of type

(ι → (ι → o) → o) → (ι → (ι → o) → o) → ι

identified by the following information:

Pure Object Id: ed76e76de9b58e621daa601cca73b4159a437ba0e73114924cb92ec8044f2aa2
Pure Object Address: TMNBrUd5KSYmRWHN49oFdMFsUPTiSKvsa9Q
Theory Object Id: aa75efbc66bb94911a3c3b156ea65bbe1775870ce7a06e24c056f8d5cce3660d
Theory Object Address: TMV6kaag6YpB5pyMDkL4wftUcq2Diskvi7a

Definition 21.20 PNo_pred is the opaque object of type

(ι → (ι → o) → o) → (ι → (ι → o) → o) → ι → o

identified by the following information:

Pure Object Id: b2d51dcfccb9527e9551b0d0c47d891c9031a1d4ee87bba5a9ae5215025d107a
Pure Object Address: TMU5awsv39apDkXc2r3HzJRP37vuRruKsnf
Theory Object Id: 2efd18fb25499ea37fc2af1b7a0ffdd36e6abc401c42458854e4f4288f7ca530
Theory Object Address: TMStxoqH3zLresqVCQdXwQ55q36aVygMDeL

PNo_bd_pred_lem

Theorem 21.59

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→

PNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R).

The proposition is identified by the following information:

Pure Prop Id: 749b8f4e860ae9a289350b300230645c0611ffff60798a2d0cdfd7fe0476bd2a
Pure Prop Address: TMYJBHCdV3Ct6gBy3MMb8STjafZb3SviQ9v
Theory Prop Id: 5de4a33591f164add23849f5545d9322fa8885bd09e7ba1894ce774b2c9cab1a
Theory Prop Address: TMP6BAwWzoNKb47BSVYq4GJmL3WoFMQGbqf

PNo_bd_pred

Theorem 21.60

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→

PNo_least_rep L R (PNo_bd L R) (PNo_pred L R).

The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 233

Pure Prop Id: 42224b3374ae5d0ed32e9420c84f18f8ea84d9eb8daae7337d3905a179b65d65
Pure Prop Address: TMR9wZusZqBwrBJNUbqyESMWattt5sL41Jf
Theory Prop Id: 3309f6a3ddcb2319d01432fd7a6a7d3c05f3b9836222f520fb3ec000b2b7299f
Theory Prop Address: TMH4SyZ8pRU9WGJb2eb6yLYAQnwpF555Lo5

PNo_bd_ord

Theorem 21.61

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→ PNo_lenbdd α L→PNo_lenbdd α R→ordinal (PNo_bd L R).

The proposition is identified by the following information:

Pure Prop Id: b40a91e75a7dcc94d55b6763376f5f366ab4a140e5e8b0380656e21f2b419fb8
Pure Prop Address: TMZxHtMby9p5j448ESrb1iGeSp3QjaX8APB
Theory Prop Id: bc6e649a31db5eaff908ac0cc857d64018672dfb788fc1bfef50fbd2a45f597c
Theory Prop Address: TMYrVH5VdsuG5TzNCVy7afWTZ3eHsgNAjH3

PNo_bd_pred_strict_imv

Theorem 21.62

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→

PNo_lenbdd α L→PNo_lenbdd α R→

PNo_strict_imv L R (PNo_bd L R) (PNo_pred L R).

The proposition is identified by the following information:

Pure Prop Id: b6398cfc0d268e1265cb766e4a3c763db5b5aba648cb2602837dfed49a6a1373
Pure Prop Address: TMVx2aGZb6qmcJNfMJMoPgY2mzk5RmNZstN
Theory Prop Id: ee0997fab4c7c13a66f2eda4601d185456ed98ab1a2c0269f8b7a47879f0a300
Theory Prop Address: TMMZTkgpVapPLqBQTVn8NSVkQRWNow8gRiS

PNo_bd_least_imv

Theorem 21.63

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→ PNo_lenbdd α L→PNo_lenbdd α R→∀γ ∈PNo_bd L R.∀q :ι → o.

¬PNo_strict_imv L R γ q.

The proposition is identified by the following information:

Pure Prop Id: 8756915afa703c80eca22a6825041ae9983fe1efe0425560dd4d0e65970d8a91
Pure Prop Address: TMEsGe3wG8yXWS94JJYUXQFxvwvV78FGr2C
Theory Prop Id: 266277c31581e02700387ec68a784741f97e44197639a61abe33b4bce3c8eccb
Theory Prop Address: TMR3RGTyTYH8yzKLkmxhSTiEdgD4ZTKaapR

PNo_bd_In

Theorem 21.64

∀LR :ι → (ι → o) → o.PNoLt_pwise L R→∀α.ordinal α→ PNo_lenbdd α L→PNo_lenbdd α R→PNo_bd L R ∈ ordsucc α.

The proposition is identified by the following information:

Pure Prop Id: ad857b445d1196a6cfc19a14ac9d45dff7f3b25ed55d9764dcbb9d4d5efdea8d
Pure Prop Address: TMasNPNdz1vv1jt29FCFyQpkupqkrcwKcUy
Theory Prop Id: 4719e12b28711d9d989a9ed9afa168f501d090c55ff5f971c2a9c4b951a3a4c8
Theory Prop Address: TMZ1d5WRxZPZR9eT7BLzoFprMn6mNkfxFcZ

Definition 21.21 We define PNoCutL to be λαpβq.β ∈ α ∧ PNoLt β q α p of type ι → (ι → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: d272c756dc639d0c36828bd42543cc2a0d93082c0cbbe41ca2064f1cff38c130
Pure Object Address: TMW89BeZ4LwBAWUSvcHTTffq4NjA9eURfe5
Theory Object Id: 5c2353f1a9132e4b0bef01f9d3b48e3dd377089926d3ec56a9c1b5f62d18564e
Theory Object Address: TMXddSCNaHGduK9qNYrg5YeV7BsECCYxzJL

Definition 21.22 We define PNoCutR to be λαpβq.β ∈ α ∧ PNoLt α p β q of type ι → (ι → o) → ι → (ι → o) → o identified by the following information:

Pure Object Id: ff28d9a140c8b282282ead76a2a40763c4c4800ebff15a4502f32af6282a3d2e
Pure Object Address: TMQ43uDGqp316dHj1xW6txbDkdt4BerVFRg
Theory Object Id: 6be9d2eca9c688e6504e0251a1344ee158dafeb178c09a30d02481f120ec9fdd
Theory Object Address: TMJvHmHmkfm3PZnD7HiJxv464ZKQNcn3xK4

PNoCutL_lenbdd

Theorem 21.65 ∀α.∀p : ι → o.PNo_lenbdd α (PNoCutL α p). The proposition is identified by the following information:

Pure Prop Id: 2ce3cac589e0288fcb056a351d7b7f52ba86fb73ca736e136679ff9ef0ea1ac3
Pure Prop Address: TMVhtYXkotNZng4p4VbWeUpqAzgiFcEJAs2
Theory Prop Id: 8d7ea3b990fcb5a72b452b48eff84626931337892eb12d3a2f14176e6d33964d
Theory Prop Address: TMbipKXeU4Z2DY2jqE2CDGdVBX2xtErK7wW

PNoCutR_lenbdd

Theorem 21.66 ∀α.∀p : ι → o.PNo_lenbdd α (PNoCutR α p). The proposition is identified by the following information:

21.1. SURREAL NUMBERS AS PREDICATES ON ORDINALS 235

Pure Prop Id: abe446408190f6d442a1f2eab7727c53458665db08e199262714d89da47c7000
Pure Prop Address: TMNwdvruagyifCoqN3GpPqcvsoJxB98XesL
Theory Prop Id: bba4ed960136adb5c77796704f207374e63fcf341cd9a438a50561365d6f14a0
Theory Prop Address: TMcNY1d7D2m6KMq1LqqgqprtHuNXxok7xt6

PNoCut_pwise

Theorem 21.67

∀α.ordinal α→∀p :ι → o.

PNoLt_pwise (PNoCutL α p) (PNoCutR α p).

The proposition is identified by the following information:

Pure Prop Id: 8bb0692a31e5ad0bfd5013045a2fbf8c7889e948699261b72291eae629aa15fe
Pure Prop Address: TMJo1MdhfFXdWBivM8BPiej6KZWhKLTQmBM
Theory Prop Id: e9740986f55e09faff6e0e851bdf5251ed23276fc5747ca8a67afcdf70970cbb
Theory Prop Address: TMZSEknJujgUGg2MoFZkCMMNLvQFRFD7SYC

PNoCut_strict_imv

Theorem 21.68

∀α.ordinal α→∀p :ι → o.

PNo_strict_imv (PNoCutL α p) (PNoCutR α p) α p.

The proposition is identified by the following information:

Pure Prop Id: dd8b71e50df3e9a494df5a5572ce41f85b622aa24c1f3e1ff457aa4ba08718be
Pure Prop Address: TMKFURgHEo6iHpwwz6JnZNJdRMvj8XUTA7D
Theory Prop Id: 6e3ce431e17b20e770f06809ce5182eb248cf10acb5c38a200119fe0520c9057
Theory Prop Address: TMPNQHfdoNywBhUdkPA9hJADKARLduKe5aY

PNoCut_bd_eq

Theorem 21.69

∀α.ordinal α→∀p :ι → o.

PNo_bd (PNoCutL α p) (PNoCutR α p) = α.

The proposition is identified by the following information:

Pure Prop Id: 9dda7c4906426d138c81234f3d50f5bc4402ad268025cc5da24c9b6fb4397ad3
Pure Prop Address: TMVsjT6ktkVWYJDvH6eTEYnV2zKwFHA9xag
Theory Prop Id: 8b0d28708c0a512260761ccd7dc5373cc4fbff3ff56d623c12cc98c1db4e3747
Theory Prop Address: TMR6CRmoDWEZoHBXWxwGBZgx6D5qz3GMeMa

PNoCut_pred_eq

Theorem 21.70

∀α.ordinal α→∀p :ι → o.

PNoEq_α p (PNo_pred (PNoCutL α p) (PNoCutR α p)). The proposition is identified by the following information:

Pure Prop Id: 3c5499b31499c700a1fb0b68f063a4b755a2658fb92632c82d2f048d7170c0f5
Pure Prop Address: TMawNU4WyhEJede1jj4Sjss5i363KEbGi1F
Theory Prop Id: 95a2d638ee2907592549cba3eecd680371133a20f157bf39a1ec7d00b4682410
Theory Prop Address: TMRmNdzKhB7GZ9dgSsWj9tM5jWyTG3QtGrZ

21.2 Surreal Numbers as Sets

21.3 TaggedSets

Let tag : ι → ι be λα.SetAdjoin α {1}. Notation. We use −′ as a postfix operator corresponding to applying term tag.

not_TransSet_Sing1

Theorem 21.71 ¬TransSet {1}. The proposition is identified by the following information:

Pure Prop Id: 8f2d673650ebb57c98d6d8d2a5fc915f7f2d811cbaf7858a359e1b97bb6bde17
Pure Prop Address: TMJBscGcLCjPWBHB8QVUutxyACNMVYf5z2Q
Theory Prop Id: be9a70c02282395d672ab7c5e5f2551733683ad6270f097ab4c85edf46ad1a63
Theory Prop Address: TMWGSzNuxQmGiDUtn5MjVnUHh75ihtU4J6W

not_ordinal_Sing1

Theorem 21.72 ¬ordinal {1}. The proposition is identified by the following information:

Pure Prop Id: 99a3f6ae0ef1daee4a28fe3fcfd481e5a6573dfa220b080f34987386122d9c11
Pure Prop Address: TMUKBsrmU2hKQg3EmpkpHjq5Vz3jGSJqAGL
Theory Prop Id: db797718fa26a8f2ad7c62c261b77ebfd7b9de2b25f962911447dc5838d54b9c
Theory Prop Address: TMKxsifEH9Ph4ovj8YaJwts3LDF7brCPFZZ

tagged_not_ordinal

Theorem 21.73 ∀y.¬ordinal (y′). The proposition is identified by the following information:

Pure Prop Id: f9488c7d361c3059b61e3d76aaa5bb7df6cb13c210dcceb3c43c7b2388c0afea
Pure Prop Address: TMSUbK9326n5Ft376LsWWx4xfGaDcP52PD5
Theory Prop Id: aaf6e467102a87c8e9f73204645cc7de8b8c4dcd478bf528601bbcb36cf3a97d
Theory Prop Address: TMVBkLkYkcbcmG1H2dngcHEkcuQeJt8yAkf

tagged_notin_ordinal

Theorem 21.74 ∀αy.ordinal α → (y′) ∉ α. The proposition is identified by the following information:

Pure Prop Id: b603ba84d8e3122663fa360bfa699dad0ab4e1f633f670e4cda297f9ba5b0208
Pure Prop Address: TMFQrda9FfP9kMHatsRA4mRhspKL6dCBanm
Theory Prop Id: c0e2c350bcd1226986c6e7fdbfa0f2c19884df170a172b85b757f9de370e72a5
Theory Prop Address: TMVojt27k6SvjqAbUtQu8cJoDPMYpTdsKum

21.3. TAGGEDSETS 237

tagged_eqE_Subq

Theorem 21.75 ∀αβ.ordinal α → α′ = β′ → α ⊆ β. The proposition is identified by the following information:

Pure Prop Id: cf032f4043866a354e9e5c9a481f6041aefa5320fd3e30ce2a8a672c5cbdc19e
Pure Prop Address: TMTWKSZoPeTMh9eB7LoRwuk94Bhfw2wU5zb
Theory Prop Id: ef37f58abd6ded2b344094fa0f103848b972602d46465aa33548d6c5eeffd6ed
Theory Prop Address: TMdBj6QvMekfbBeqoyfr1PZdrzpAAeNC5Ek

tagged_eqE_eq

Theorem 21.76 ∀αβ.ordinal α → ordinal β → α′ = β′ → α = β. The proposition is identified by the following information:

Pure Prop Id: 1f7dc940e194a7544299b8233619e1f491b2dde8acde374d90cb863cd307b0b7
Pure Prop Address: TMKi9eMo4knWpH5ySxts7dqcS5g7CzxpzRQ
Theory Prop Id: ec1608fa42f468b541a15ac8a164117d2d5256fb753cb09aa41258ec590a5e00
Theory Prop Address: TMYDsoSZjgQSPetekXkj4s6GU5ub3XVJaWh

tagged_ReplE

Theorem 21.77

∀αβ.ordinal α→ordinal β→β′ ∈ {γ′|γ ∈ α}→β ∈ α.

The proposition is identified by the following information:

Pure Prop Id: a1d43c66e0e7a187f286a4910c3a8578502c854c09b9ed79fd73f53b4d089435
Pure Prop Address: TMNf9aFM1UsRXJkDzFmAMwGHYvakMgcytbx
Theory Prop Id: 70c25038dc2d795e7cd8a963a601a4e91c9ec37c890e90874b9e2ffe3758f817
Theory Prop Address: TMTYBnzfsAwJny6EQ3AaAXwwpjV8sXaQQm4

ordinal_notin_tagged_Repl

Theorem 21.78 ∀αY.ordinal α → α ∉ {y′|y ∈ Y }. The proposition is identified by the following information:

Pure Prop Id: 7f35b530bae383bd34f04f1aeb8e7f87cad1d17284bd4ad34bd07b9a9c0888ad
Pure Prop Address: TMEiGdaBPeGmtidArg3tTV7mQKNB6Kjsv29
Theory Prop Id: e7ff3b50c4b94c722012bd9c9578bf3faa3faed0b7e0a6728349051f41a649fb
Theory Prop Address: TMUFgGXyhtzJfcJ5jwFoArpbHCNJMjtGFEC

Definition 21.23 We define SNoElts_to be λα.α ∪ {β′|β ∈ α} of type ι → ι identified by the following information:

Pure Object Id: c0ec73850ee5ffe522788630e90a685ec9dc80b04347c892d62880c5e108ba10
Pure Object Address: TMchcPtmntTsG9a2jnVbo3rvhd4YRFGdXd1
Theory Object Id: 6f0c364e1773eaa440ffcfc12f522a8c0532bdf6abfa9b8337d5e824777d1834
Theory Object Address: TMX8tC9smhsjcxum9MHG47kxXC2911bY8fh

238 CHAPTER 21. SURREAL NUMBERS I

SNoElts_mon

Theorem 21.79 ∀αβ.α ⊆ β → SNoElts_α ⊆ SNoElts_β. The proposition is identified by the following information:

Pure Prop Id: 49a08e4bc4f84602c91d1206ee5c208423e9081b680beb0711763d769896fcfb
Pure Prop Address: TMcnJXLfgf6uG9ovz9CcN8AxtyhCcG7uj6R
Theory Prop Id: 85961174f4e845853804252a1ed8497fdf7987feab6458860152819829435ac1
Theory Prop Address: TMYbW9UULsAfXGSQRuG31aqjM693gccqxqg

Definition 21.24 We define SNo_to be

λαx.x ⊆ SNoElts_α ∧ ∀β ∈ α.exactly1of2 (β′ ∈ x) (β ∈ x) of type ι → ι → o identified by the following information:

Pure Object Id: 4ab7e4afd8b51df80f04ef3dd42ae08c530697f7926635e26c92eb55ae427224
Pure Object Address: TMWTZaSAkv4fbU4uw785MTxgHkscdn5nC4o
Theory Object Id: c16a736947afa02fd21f8d698f4b28435c6d1a029515699f6dd4c54682d31685
Theory Object Address: TMQpDMmMJpF7k2EcsWrxASKz43UULkewFnB

Definition 21.25 We define PSNo to be λαp.{β ∈ α|p β} ∪ {β′|β ∈ α, ¬p β} of type ι → (ι → o) → ι identified by the following information:

Pure Object Id: 3657ae18f6f8c496bec0bc584e1a5f49a6ce4e6db74d7a924f85da1c10e2722d
Pure Object Address: TMYCsiv2Uki9NzUmKkGH2kHd2zXGFWsYi5p
Theory Object Id: 0a835174a0cf9cb97b20e29b98340a75a9c5f19db25004ae0688b25995ff83b3
Theory Object Address: TMTsRL2Kcwjhfick7fiiXD612CpaX6QFJea

PNoEq_PSNo

Theorem 21.80

∀α.ordinal α→∀p :ι → o.PNoEq_α (λβ.β ∈ PSNo α p) p. The proposition is identified by the following information:

Pure Prop Id: c12e023a1bba6d9a9ec78087054b4057985c72eed826a9b312ba4f9091a65a61
Pure Prop Address: TMZtDRwJxRs7oFkrbsWnLiWufDfjc6F5M8R
Theory Prop Id: 688518e3201b270ffc6ae7f6eaea9a64dca9d99cd02f4c980cb094834f164104
Theory Prop Address: TMWKeiYteNuQ4ETYrWy5zjJ7ErjwUTNfcGU

SNo_PSNo

Theorem 21.81

∀α.ordinal α→∀p :ι → o.SNo_α (PSNo α p).

The proposition is identified by the following information:

21.3. TAGGEDSETS 239

Pure Prop Id: 96e4e2ec36857754b409849e1c10332c78d89197c28a888ac4de3c1e39dee14e
Pure Prop Address: TMHEHzxt38dr1Sf22Q4XwK3oTJfyqBauuXx
Theory Prop Id: f03db005e0c97b51d3b6d91f5cc5af548be05d0a792345091ee88d22daa20473
Theory Prop Address: TMUdwihfVPMkko4XmqQdHcyTjgzoGYZRGjh

SNo_PSNo_eta_

Theorem 21.82

∀αx.ordinal α→SNo_α x→x = PSNo α (λβ.β ∈ x).

The proposition is identified by the following information:

Pure Prop Id: 1642af79554e4cf7b9318de374b2c7edd186172454054d947f607438269c9496
Pure Prop Address: TMcuWpQkUmH1LMTde4kSNzc7YBDbA8sokbU
Theory Prop Id: 630119c550dd8b5c2e0bd53e2ad889d4d08f72bc50e97a63bd942e63a6413115
Theory Prop Address: TMdU6L38rj7HbtERSXfWaxWZ7ZTCBb4pt8p

Definition 21.26 SNo is the opaque object of type ι → o identified by the following information:

Pure Object Id: 11faa7a742daf8e4f9aaf08e90b175467e22d0e6ad3ed089af1be90cfc17314b
Pure Object Address: TMN8jh3ustcw3ZSbqUeiv8YSjeH79sfuWBi
Theory Object Id: 116b7641893e1ad3998a7adf6c1fab6d1343fb60d499aef43ca12a4f847ef6b9
Theory Object Address: TMcKkMrp1PUTPmiJaXVAxaYrixRdqgEjZje

SNo_SNo

Theorem 21.83 ∀α.ordinal α → ∀z.SNo_α z → SNo z. The proposition is identified by the following information:

Pure Prop Id: 929b7ac3599669289a870d939e634944e175e96faf30cb744f95d10af1ed5d72
Pure Prop Address: TMMDuEgEyuT5E1BibeHE8pfR45YK1o1Bthr
Theory Prop Id: a59341cc6c21bd7a5fd6bbf9d96c7141e1e9a5f9d341f307e50b304d6ed01584
Theory Prop Address: TMUJvhsT86hcmvoRoqef5N9GPieDsaTYJah

Definition 21.27 SNoLev is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 293b77d05dab711767d698fb4484aab2a884304256765be0733e6bd5348119e8
Pure Object Address: TMLgHPbDhEczHhzhujz1Y6647YGuRwkP7uG
Theory Object Id: 83dce3cb89857056a3ed5cb650ae31e37721c49405fcf77a3b272f46d3ab6b80
Theory Object Address: TMUZTu2xUcdtVkLHYv2sJFX8Yj2S6Jd5jnk

SNoLev_uniq_Subq

Theorem 21.84

∀xαβ.ordinal α→ordinal β→SNo_α x→SNo_β x→α ⊆ β. The proposition is identified by the following information:

240 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 2ca114eb623f9aa0591aadc7c354900445384cf0262c752fa0430e62b230d79a
Pure Prop Address: TMJjyz93md1dpAbsBqGeb8qeQUo8pB87vCg
Theory Prop Id: 65843d7b872576b70cff22d1e596ee268a921435cd21f9de56a360ecd581bb9a
Theory Prop Address: TMUMCHaN9FLoTQWs3Q4fEncBkHicwqi4FfL

SNoLev_uniq

Theorem 21.85

∀xαβ.ordinal α→ordinal β→SNo_α x→SNo_β x→α = β. The proposition is identified by the following information:

Pure Prop Id: 68f71dfd2001b27983e3c169b043d92f238faadc82a072f2d495cdc9761d2189
Pure Prop Address: TMaDQTErBA2zHWoN1hNMatRbHaH7eAJwMWF
Theory Prop Id: 5d96d48026e74cd817ee90319b23cc2b42e20eba8d81f83f0edfcc2159f6ea5e
Theory Prop Address: TMdvMVpBuYy5UApCwAKZdNQE9dn3JHxmPy6

SNoLev_prop

Theorem 21.86

∀x.SNo x→ordinal (SNoLev x) ∧ SNo_(SNoLev x) x.

The proposition is identified by the following information:

Pure Prop Id: a8afccaac35cd67cae38ea5c7f8db5efcc00f46db4396c04d7693f267b847aa8
Pure Prop Address: TMEuZs9WhvFjxtstDDHiS5zYgaamp73ip3m
Theory Prop Id: 21761c1d62638b04252039c756c82b2f54681ae14cef2e53b054c7eed1be1251
Theory Prop Address: TMTeCDaoMd1YLP8PGYRivc2WeuktskbT8SH

SNoLev_ordinal

Theorem 21.87 ∀x.SNo x → ordinal (SNoLev x). The proposition is identified by the following information:

Pure Prop Id: 7cacc5f1e879f627f484fd25c7b6c53a51432ebb794812d1bf85218be858870a
Pure Prop Address: TMYpWCPBF4fDYL83uuTLHroYN9eaJFa1gjQ
Theory Prop Id: f0e9f7da1d1eea44bc8152e3a3c9944f3397f3099eb97e82a027f58d2c354304
Theory Prop Address: TMNg3i1WfBeNE9GZ3VagfsKKYV6q3v2MqxJ

SNoLev_

Theorem 21.88 ∀x.SNo x → SNo_(SNoLev x) x. The proposition is identified by the following information:

Pure Prop Id: 4a10339549425e6a1bcdd71c2ab61157802ad96e57ac87b6d61abeeb14663bc2
Pure Prop Address: TMF7HEEExZuLEM6qBULBQ44p5UsmzSwGq3s
Theory Prop Id: 49cd5a80c0a8afed4364ca62e2e89621ac660e834b9bc67ddf46be5aeb6d8009
Theory Prop Address: TMHMibDhQbDodJRNsKTwFaztQh19uChMD8h

21.3. TAGGEDSETS 241

SNo_PSNo_eta

Theorem 21.89 ∀x.SNo x → x = PSNo (SNoLev x) (λβ.β ∈ x). The proposition is identified by the following information:

Pure Prop Id: 6ffefe7aa4ffb2d1b5711998134756f9e5e8326d515f3a9f165fa2d847ae71c8
Pure Prop Address: TMTaGgEBt5cPzRbi7CBcmpXvUfY5LgAtNjV
Theory Prop Id: 8c78704c4a4036ae8fdbd50b6aa2091682a9b6cc08d150bb5d09f8689293ae7d
Theory Prop Address: TMPZ6V2HX31zAKKRowAqxj1W7MhW2vtEkh3

SNoLev_PSNo

Theorem 21.90

∀α.ordinal α→∀p :ι → o.SNoLev (PSNo α p) = α.

The proposition is identified by the following information:

Pure Prop Id: 07f1eb48a9b85d0d91b1644c1d205b7b5a1b4598605dbc92bcc702a5f6548159
Pure Prop Address: TMS4B9MsCny6DcgEj6Q5jyZtqGjyadqVrXb
Theory Prop Id: 9c4890c1c16e11041619677ff7a9cab66327b15dc07aa9724197a3baf15864e4
Theory Prop Address: TMH875f6xnwhLVBj8FhdaBY27tPcA7tc93w

SNo_Subq

Theorem 21.91

∀xy.SNo x→SNo y→SNoLev x ⊆ SNoLev y→(∀α ∈ SNoLev x.α ∈ x ⇔ α ∈ y)
→x ⊆ y.

The proposition is identified by the following information:

Pure Prop Id: a244d79cdb92b7d27cf9364b054d3ab11f8fe1a80867c8aee4bfdbab426dc71b
Pure Prop Address: TMZCRMtpxoXjbsEUAwQVnA3o3i8r9w9chR7
Theory Prop Id: a428aff2b59424a8834bd98dcf9cccb10f9ccd50a36aac8cdb05bdd1f8b0dc7b
Theory Prop Address: TMVu9eEtBLYvjJeiHT4owg9NLx7ADETn241

Definition 21.28 We define SNoEq_to be

λαxy.PNoEq_α (λβ.β ∈ x) (λβ.β ∈ y)

of type ι → ι → ι → o identified by the following information:

Pure Object Id: 5f11e279df04942220c983366e2a199b437dc705dac74495e76bc3123778dd86
Pure Object Address: TMHgy5CsvqeCa98Aq7nixfsuoKHSwuUheGS
Theory Object Id: 12b5f61704c15020e2c024d9a3c4b44f386d2447a7bfd3a7ba2d9be7e534d79f
Theory Object Address: TMPsS6FU8RDY8bj6bhgs9hNWme6t8mxkcHv

242 CHAPTER 21. SURREAL NUMBERS I

SNoEq_I

Theorem 21.92 ∀αxy.(∀β ∈ α.β ∈ x ⇔ β ∈ y) → SNoEq_α x y. The proposition is identified by the following information:

Pure Prop Id: a718b1b74fe6e25f89cd15b443e1050c4627de6e63f7fc13aed6604a5dff174d
Pure Prop Address: TMaoxpA3u5BRdKqSyxRArRdQKxBCCb4Ynfd
Theory Prop Id: fdcb780aed4882ebdf83c46eeec00af5f3c06f91828ac6f34b8cd53807a3f3f0
Theory Prop Address: TMQV9XrhBpFPAX5ZadNCNzSub81tr3rR5pd

SNoEq_E

Theorem 21.93 ∀αxy.SNoEq_α x y → ∀β ∈ α.β ∈ x ⇔ β ∈ y. The proposition is identified by the following information:

Pure Prop Id: 322284f9b12d82137cb4547584023ca4089ff501f8b46d31017d6644359f67bc
Pure Prop Address: TMUx62argfoVk7QVYzx9FBiQWTGGzPgSkMr
Theory Prop Id: 76c08c80b9ae86c43f88454225e5c2d37189dab1f7a4aaaab54885b6d1d584d5
Theory Prop Address: TMcZ243SUoVmTGmz8GBWnNRSGSoQLcSekNM

SNoEq_E1

Theorem 21.94 ∀αxy.SNoEq_α x y → ∀β ∈ α.β ∈ x → β ∈ y. The proposition is identified by the following information:

Pure Prop Id: bc569f4fe8f8babfb01e30d1ce5330c3cd167c3a34a4d7f6865bbcb63ff0c246
Pure Prop Address: TMcpXWEGwJyBfkywuB1xBLvouHwBSaFs2Lu
Theory Prop Id: c702b67d3382793ff6831ae5d9948c42fcdf9959f67e4f34b70f105a4fa627c0
Theory Prop Address: TMKUKLNUuNym8woqSCRiDcMxAfsCb1eUBk1

SNoEq_E2

Theorem 21.95 ∀αxy.SNoEq_α x y → ∀β ∈ α.β ∈ y → β ∈ x. The proposition is identified by the following information:

Pure Prop Id: 04c55ef7be9110a6c332fe11ca098564ded45c1ae7f131199dc8a18eff3f728a
Pure Prop Address: TMa2w3rdW3KJYQkLRhjwBtZCzeLY2CsSzUt
Theory Prop Id: 06351353e81c1d6cb1d4bd1b327570332781fae5e00a0975bce060a995406d00
Theory Prop Address: TMTis8K7byF4GTjz4MpYG9BkyoJCzkUAJyW

SNoEq_antimon_

Theorem 21.96

∀α.ordinal α→∀β ∈α.∀xy.SNoEq_α x y→SNoEq_β x y. The proposition is identified by the following information:

21.3. TAGGEDSETS 243

Pure Prop Id: 8d4936ed8e712c1dc231fb4b91a23637000f913b108f46cd2edc43f5e8726ba6
Pure Prop Address: TMFyyANM4SzwuZGwLMdhVdAWuiVUHstHQug
Theory Prop Id: 95a70a9933c8b7e36b75adaa97b483d7f470c88d64871a9b077468393a7cee44
Theory Prop Address: TMG63bj4zRBfeAkNC86WKZ7gtUpSwCSprJR

SNo_eq

Theorem 21.97

∀xy.SNo x→SNo y→SNoLev x = SNoLev y→
SNoEq_(SNoLev x) x y→x = y.

The proposition is identified by the following information:

Pure Prop Id: b08bdc8e0533c6d335bfaefe372cf3eae5a6596181f4279fbee1bbbb96bd58d2
Pure Prop Address: TMMawfQkhohvkcRg13ZBDTRwn1DgFdvbZN8
Theory Prop Id: bf4ac3765cc4e77e69dfed6b06b79b2a135de381ed468aff0717b55ecd997219
Theory Prop Address: TMXZ1YaK7uq7CL6xjMR9UpcJae85o8Qg6an

Let ctag : ι → ι be λα.SetAdjoin α {2}. Notation. We use −′′ as a postfix operator corresponding to applying term ctag.

ctagged_not_ordinal

Theorem 21.98 ∀y.¬ordinal (y′′). The proposition is identified by the following information:

Pure Prop Id: 06ba78058810b2a66de61b42a989e96b9faf1830fe88331433cb796ec5e9efba
Pure Prop Address: TMSJ67Y6zbYZXw4KipNfSDFXS7nCowqPsfv
Theory Prop Id: 762f4db9b4131082faa31f45fdf77ff2ecd9cada9a2b8be09a7c87338296ffaa
Theory Prop Address: TMUUZwa9YUDXipGjVpau4QY3LpPAUNonrus

ctagged_notin_ordinal

Theorem 21.99 ∀αy.ordinal α → (y′′) ∉ α. The proposition is identified by the following information:

Pure Prop Id: a6d7ced029db2bf647e1bcf629a266dc8aa5350b597ac826ee1499c9488ef153
Pure Prop Address: TMJtxbUm4vTnZsFYwmfoJXkdyU9Pa5iDTMG
Theory Prop Id: 33eb91696d423e27f32912360405bcff4635722555c6e964c127195a36bc3b46
Theory Prop Address: TMU4pDKct3EFb68TQMdmzrbHxh4jXHXLAx4

Sing2_notin_SingSing1

Theorem 21.100 {2} ∉ {{1}}. The proposition is identified by the following information:

244 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: f44be8cc1b2eabcd18be5c1af7cc7c118087af9d39e568bff28319aa056768c9
Pure Prop Address: TMYYY1TCNHPhoU7ioo1yEjqPen8SD381J2H
Theory Prop Id: 4504e136e5be3235264222f1b4fc92b8c82895870a0d5af8fb0ca383d0aa0b93
Theory Prop Address: TMXydjLtD7zAbYMVdsbyZnBWtZSoHs13VUk

ctagged_notin_SNo

Theorem 21.101 ∀xy.SNo x → (y′′) ∉ x. The proposition is identified by the following information:

Pure Prop Id: e87bab002c13fd4c8f45d1fab441c351590059c0546ef7cb97b2f419eded12bb
Pure Prop Address: TMLi9ZahvNViXr94ACKuVosVxgffZSXBaQh
Theory Prop Id: fb05c7f6bcb781bf45159e3631adfc011ed3dcf028713a0a5e65ffa8442e7de6
Theory Prop Address: TMdvxEZ5qYtgLAENGC1mYi7PPenMSpHbbbG

ctagged_eqE_eq

Theorem 21.102 ∀xy.SNo x → SNo y → ∀u ∈ x.∀v ∈ y.u′′ = v′′ → u = v. The proposition is identified by the following information:

Pure Prop Id: 9635c2246bd5ce564e4af804699383bd45289a4ce4fef5acfb843ea16be935ac
Pure Prop Address: TMWuHzjRejZ3vCngDLVxyx22KfXifdUFgrX
Theory Prop Id: 7a7dec8b0cb27354ad3358e329e40af18501227e17ce9a3c573fb2ced3933325
Theory Prop Address: TMFudCkTrZ8qDZwsaKBW9Qsuj6AjLeAqYtM

Definition 21.29 We define SNo_pair to be λxy.x ∪ {u′′|u ∈ y} of type ι → ι → ι identified by the following information:

Pure Object Id: 0c801be26da5c0527e04f5b1962551a552dab2d2643327b86105925953cf3b06
Pure Object Address: TMQqNXnBZnkKSe8hed4os6KPYmLwpS71Jw7
Theory Object Id: 88bcd797eb0ac50799968a11f7c23c8ed718c981b7a3bb422ca738771da70d49
Theory Object Address: TMS1SBjNkN4Fv979vSCovhJ3Hm5LmNC8vSP

SNo_pair_prop_1

Theorem 21.103

∀x1y1x2y2.SNo x1→SNo x2→SNo_pair x1 y1 = SNo_pair x2 y2→
x1 = x2.

The proposition is identified by the following information:

Pure Prop Id: 55e11f2068e4bac205e7f81aa76c24ddba355ab4507585cc4d8d2d2f4087857f
Pure Prop Address: TMY6PgwrmwyiN6arDm1Yhg5JrTC3jXy8Z7g
Theory Prop Id: f423640149097053118cefcbb2f2260329493536ad3ae58e47efbdc5cf96a62f
Theory Prop Address: TMabWw9KKwUpjo1s2ey6gDkv7hUyHidq12D

SNo_pair_prop_2

21.4. SURREAL NUMBERS AS SETS II 245 Theorem 21.104

∀x1y1x2y2.SNo x1→SNo y1→SNo x2→SNo y2→
SNo_pair x1 y1 = SNo_pair x2 y2→y1 = y2.

The proposition is identified by the following information:

Pure Prop Id: 0bfff99d8cefbebcd6db7207b1609c8e7c6338fc2381f6c937e65ef66435fa02
Pure Prop Address: TMZQx8qdiMuWPMgQr92RnVZJPMwYU6eRoDU
Theory Prop Id: cc5febed625436e972c38a7e71012d4d0459f0651c13dba74988508db075847c
Theory Prop Address: TMboEtYmU1rhLndeftmE1xvAXa12uenk52T

SNo_pair_prop

Theorem 21.105

∀x1y1x2y2.SNo x1→SNo y1→SNo x2→SNo y2→

SNo_pair x1 y1 = SNo_pair x2 y2→x1 = x2 ∧ y1 = y2.

The proposition is identified by the following information:

Pure Prop Id: 0c1fab5088198fbd78c0308f1d0a7d3e7a9067d7ed242c17489a18a01a610489
Pure Prop Address: TMdh56g5Z6tVNavL7VJHdhuafzTH1Pmfs82
Theory Prop Id: 9f8e820a171395b7e1ea5696b5ba61536ab0a1efd23d479c1ad3f329425325a3
Theory Prop Address: TMYoD7ZXdXJGFLDcspavQztKyvhuDPUyktz

SNo_pair_0

Theorem 21.106 ∀x.SNo_pair x 0 = x. The proposition is identified by the following information:

Pure Prop Id: a24e9ca908c70bad12f23288cfeadb94cf8d4a42e36b819d9a269d2083a09ee7
Pure Prop Address: TMLieswQhd4ZG8isZSgFpxUfMX6vYzwXue7
Theory Prop Id: fe382ec7ec6028476f54901b6bab3e279d994b93eb8d072db22da7f5d6c039d6
Theory Prop Address: TMZuZFbfyTNy3vBnkV22aEUTbRJMeouwCtc

21.4 Surreal Numbers as Sets II

Definition 21.30 We define SNoLt to be

λxy.PNoLt (SNoLev x) (λβ.β ∈ x) (SNoLev y) (λβ.β ∈ y) of type ι → ι → o identified by the following information:

Pure Object Id: 46e7ed0ee512360f08f5e5f9fc40a934ff27cfd0c79d3c2384e6fb16d461bd95
Pure Object Address: TMQeBApbcRbbcKWZqVjr7EnqQ7iX7wkV5cx
Theory Object Id: 8c8aa9c7cd39e813a521dcca067bdc27aa59721372e00112d3a56e0af7f1c820
Theory Object Address: TMKc4rFp1uKxLq8fj5pXFZziozMvupbPz34

Notation. We use < as an infix operator corresponding to applying term SNoLt.

Definition 21.31 We define SNoLe to be

λxy.PNoLe (SNoLev x) (λβ.β ∈ x) (SNoLev y) (λβ.β ∈ y)

of type ι → ι → o identified by the following information:

Pure Object Id: ddf7d378c4df6fcdf73e416f8d4c08965e38e50abe1463a0312048d3dd7ac7e9
Pure Object Address: TMKo92SvXPoMheM1GrcBcnHBZDewaXgqJwX
Theory Object Id: 54f405ff1c52a47311b5896cb2435cbca5240391e95a4473bb33d4f389ef4b27
Theory Object Address: TMcgcuHLpN7pQTJ2oMAMaweT9JaE3NrXnSu

Notation. We use ≤ as an infix operator corresponding to applying term SNoLe.

SNoLtLe

Theorem 21.107 ∀xy.x < y → x ≤ y. The proposition is identified by the following information:

Pure Prop Id: c0fef1fff2b54737a2955837a95cd2fa4a51eaeab8bdcd197730a7dc32fdaa00
Pure Prop Address: TMcpCGuF3ij7f7Q9otaxmn58tuxgzTAEELE
Theory Prop Id: f09833f8c9ddb44132bc696213b112e392eb8293700f8a6493ef998a11ef5bf7
Theory Prop Address: TMWx22uWdLVXSdBruTx9wkh1hYKRqrGaxYL

SNoLeE

Theorem 21.108 ∀xy.SNo x → SNo y → x ≤ y → x < y ∨ x = y. The proposition is identified by the following information:

Pure Prop Id: 1c9c62734eefe76ad0f740a1f3c250414dfb8d30fa252d1ca61a69f245fe533c
Pure Prop Address: TMKZjxSSN6F5jwo8R4FwwTKANQJT8wVY6Zq
Theory Prop Id: 23cc476eb8b886081aab3ea46dc0b4ab0cbfb0a43df47487eade402bb40d21d1
Theory Prop Address: TMR8soSdPW2WyGNYANuCH2tdDGwy3XK3ZZX

SNoEq_ref_

Theorem 21.109 ∀αx.SNoEq_α x x. The proposition is identified by the following information:

Pure Prop Id: 259cc3291296e3db90ffb96d963d0421fb6016e7a41218d81b92cd846193aa78
Pure Prop Address: TMGJ1mcTbnMvajkoJpACBxXXWZPaQ116hPq
Theory Prop Id: 92ef044c4eddbbb417dc88765739163aa0b710f5b5646ffae3c372cffa7a575a
Theory Prop Address: TMWvCZMB6S5XhCvHow2Zc5MwWv32acftDJc

SNoEq_sym_

21.4. SURREAL NUMBERS AS SETS II 247 Theorem 21.110 ∀αxy.SNoEq_α x y → SNoEq_α y x. The proposition is identified by the following information:

Pure Prop Id: f8c61d4617dba1594bda7ef1b25d222315df97512407b3dc4d77c1e117f5ffae
Pure Prop Address: TMYtRm1GvHqLhau3uVcMnTfMYn5u2YmwqHi
Theory Prop Id: f756d24db9f6575a09cab35daef6bf28fa9b1b0101f4d013fedfef1e8c0f7741
Theory Prop Address: TMZPd22JbRT6sSfR5cHZF68img19UYG8kuM

SNoEq_tra_

Theorem 21.111

∀αxyz.SNoEq_α x y→SNoEq_α y z→SNoEq_α x z.

The proposition is identified by the following information:

Pure Prop Id: 0fff96e9fa087a01b4e627c5a43d8f712211ef77acbd62778d0d5fbdc9a2c11e
Pure Prop Address: TMGUGWpuTgNVZaVHrjqqx3B3dvveuU3xb1w
Theory Prop Id: 246d040775eff6f1c7748648c0c49f13cbd822ffecf62018af28e16107c8db95
Theory Prop Address: TMKhJm34Nbo1ihoXZ31EHQfG87mFoxCD2xF

SNoLtE

Theorem 21.112

∀xy.SNo x→SNo y→x < y→∀p :o.
(∀z.SNo z → SNoLev z ∈ SNoLev x ∩ SNoLev y →

SNoEq_(SNoLev z) z x → SNoEq_(SNoLev z) z y → x < z →

z < y → SNoLev z ∉ x → SNoLev z ∈ y → p)

→

(SNoLev x ∈ SNoLev y → SNoEq_(SNoLev x) x y → SNoLev x ∈ y → p)

→

(SNoLev y ∈ SNoLev x → SNoEq_(SNoLev y) x y → SNoLev y ∉ x → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 958711d28833ece2b40c3f3b22c5429d43fda518512b7e6cd75b687901aa585c
Pure Prop Address: TMQaPacdRra3VaMe2NYNa6hUY8TFNJ1no3E
Theory Prop Id: d12fc61e3017695d6fd60f1d88de8697d1f3b34fae6748b05bc36f7361185ea5
Theory Prop Address: TMScxkqv8rKobWueAxfgzCcwhkfhaGV9qHT

SNoLtI2

248 CHAPTER 21. SURREAL NUMBERS I Theorem 21.113

∀xy.SNoLev x ∈ SNoLev y→SNoEq_(SNoLev x) x y→SNoLev x ∈ y
→x < y.

The proposition is identified by the following information:

Pure Prop Id: 282b0114a7969ac0a9f2f7d7736f7948b656946dfe0bd61474b62b3399f08148
Pure Prop Address: TMSSV4UzJfz3EizcWCjQwJdMaAvo7dQx3od
Theory Prop Id: 82b82caa129216de39cbf73134cd56db7ca51386bfea22d4c8ddd6c051496ed1
Theory Prop Address: TMTsvbaNTUL6yQqdnrt1ngGJ6yz18T2WJx7

SNoLtI3

Theorem 21.114

∀xy.SNoLev y ∈ SNoLev x→SNoEq_(SNoLev y) x y→SNoLev y ∉ x
→x < y.

The proposition is identified by the following information:

Pure Prop Id: 006afc7cf0f802618dbec43d665276d0ee81d35772edbf373b3bd651342a9d37
Pure Prop Address: TMJ4VDCxrxGVAF5fK26vap4HDavQoHRMJ4m
Theory Prop Id: 938f67cc578e6abe82c5585824484d30503f3bcc5e3689cb6de279568d894afc
Theory Prop Address: TMX95fsRv9YXCLYg3BLQgUEsosPv95mRaVf

SNoLt_irref

Theorem 21.115 ∀x.¬SNoLt x x. The proposition is identified by the following information:

Pure Prop Id: d5ecf97e33d758999475a13ee895246c6ae7bc234feff76c2354f56f7b624826
Pure Prop Address: TMbVeK3yqN3QTUGSqR22RZ7ufquPqAg75TV
Theory Prop Id: 06fac5ec34e79c463fc32f8c6ed17874daafb68995b87ca166ab687f4266c6e8
Theory Prop Address: TMFNgRdNx48U2iQGe5ump9qwyA5gisWDyYk

SNoLt_trichotomy_or

Theorem 21.116 ∀xy.SNo x → SNo y → x < y ∨ x = y ∨ y < x. The proposition is identified by the following information:

Pure Prop Id: 35a4c8a73fc7c1c7cbab4f63f7e92de7f75eec2e52fca95b9fd47f0ab2de7ae7
Pure Prop Address: TMdfGBU7nCaoeRDa8DYFzE35sW9rF2BaYJT
Theory Prop Id: 8ad056deb63dd86dc6b1c668c0e389f14827e04273dcb3e1a8d1c1e74bad1877
Theory Prop Address: TMdykwY9bimsgzdU8rQy52x82fcLDNP8gqm

SNoLt_trichotomy_or_impred

21.4. SURREAL NUMBERS AS SETS II 249 Theorem 21.117

∀xy.SNo x→SNo y→∀p :o.(x < y → p)→(x = y → p)→(y < x → p)→p. The proposition is identified by the following information:

Pure Prop Id: ba04ef155f5e47607fdb0216fb6a06340f5cd9616eb6c817260c9f387b62270a
Pure Prop Address: TMV8mPnyLgKgikuN7NsNsQJao2G8A2ya29W
Theory Prop Id: 7588e3b81e8c26b5bdfa83b3a50190f9f22e3e7c278731520d46b07ea230bf45
Theory Prop Address: TMYEfgRm1btK3489ffptw9arr8NJ1tSuHYr

SNoLt_tra

Theorem 21.118 ∀xyz.SNo x → SNo y → SNo z → x < y → y < z → x < z. The proposition is identified by the following information:

Pure Prop Id: ac0e241054ec1b90968f95ad1b556eec114498a58651d565ae4cac1b019867c7
Pure Prop Address: TMddkASjPFXDxWgZwPsWeWLvopbdDModThm
Theory Prop Id: ccfdf7a1275e7ae74a5f6ecc7d8b2def5778c8e9002abdb744c67fd331be61e1
Theory Prop Address: TML6qNhmMdQ9xqiz61XAszuyuyC6VGsunnR

SNoLe_ref

Theorem 21.119 ∀x.SNoLe x x. The proposition is identified by the following information:

Pure Prop Id: c940c82ce0e40faca616524909eeb5303ed083f5290e54d8054cfbae1cf0ec2f
Pure Prop Address: TMWbEJ22hQKnoK2eVaoyNrJEUXEkgbcRBJf
Theory Prop Id: 83bd2d49035b19f0998a95f22664723ce6d3c960022e6125f8249aa45f2b1292
Theory Prop Address: TMKJtJGg7gcLDxHPUbGWPcUfzbs1mmF3Wki

SNoLe_antisym

Theorem 21.120 ∀xy.SNo x → SNo y → x ≤ y → y ≤ x → x = y. The proposition is identified by the following information:

Pure Prop Id: f9f7571d97142cd27d147c350951113898e84e19eedaf980f36ed322766e0514
Pure Prop Address: TMdEqGkU3vjpdRYetepi2YSrbnZ5VjCxYuZ
Theory Prop Id: 3abee4eacee94652f2f428c3dd42de2ad807637c433d8c2c868a6509d7a6427a
Theory Prop Address: TMMewh6x1J5khDBzMVayDe8pgeEGnJt2fPL

SNoLtLe_tra

Theorem 21.121 ∀xyz.SNo x → SNo y → SNo z → x < y → y ≤ z → x < z. The proposition is identified by the following information:

Pure Prop Id: 3bc66187a3c8c6c503fb4e0c28121c6f7b0e9fdb129f4989a4e877bda7ea2331
Pure Prop Address: TMFeQ529o5NY2zuHeApHyghETBCcdzFrerN
Theory Prop Id: ba8545b12dea14cf235ae9007eb85c28adfad81e6abd4a3c0afdc1263e529e5d
Theory Prop Address: TMWXyUVVCVjMxZQeS5PSqxmnt47eXXrnZjj

250 CHAPTER 21. SURREAL NUMBERS I

SNoLeLt_tra

Theorem 21.122 ∀xyz.SNo x → SNo y → SNo z → x ≤ y → y < z → x < z. The proposition is identified by the following information:

Pure Prop Id: b2178f178198787e74b3d9f71600d68ffeef7e217a11d962036dfd3e08b75293
Pure Prop Address: TMLec6uaS4DS4VwQAcpcPM3YPb74q9SMcDB
Theory Prop Id: 4a77a2a749de106eaac82a7c1e466ed95de5fc489f1820e8b49646e6b03c8d35
Theory Prop Address: TMK9rwT18dhoTbiBXjrvgGExGkmYmRnFyZK

SNoLe_tra

Theorem 21.123 ∀xyz.SNo x → SNo y → SNo z → x ≤ y → y ≤ z → x ≤ z. The proposition is identified by the following information:

Pure Prop Id: a0c5c60c2ceffb86861009ffeca522474ef473926e20770226f32c90efdc18d0
Pure Prop Address: TMKoicgHRM8yShdwoBNSAh9MEvz7q8PtkM9
Theory Prop Id: 0319892cfab3f54bb914dbb9f14645402420d106668b7793a5f85e69ecca72fc
Theory Prop Address: TMXd83ANWfT4zya3nqhmvzCfE2ez1NuqTf3

SNoLtLe_or

Theorem 21.124 ∀xy.SNo x → SNo y → x < y ∨ y ≤ x. The proposition is identified by the following information:

Pure Prop Id: f595ac154b5f6c53eca4ea196af34004c4273a448817a2233791468da205e94a
Pure Prop Address: TMQDjtGFE58mqHck4JTyF12SPbWSzrFAjzw
Theory Prop Id: 8965d2d632b616ea19510993aaf5d59d9c47af2a7d76be6d00d0b436c1309dc4
Theory Prop Address: TMXGgukiEQFsFuC6uSDXeRYCUnqF5inCRTa

SNoLt_PSNo_PNoLt

Theorem 21.125

∀αβ.∀pq :ι → o.ordinal α→ordinal β→PSNo α p < PSNo β q→

PNoLt α p β q.

The proposition is identified by the following information:

Pure Prop Id: 6e122de5852708a5ca608843957b6c9ff2dae5ef5222ddf994aad35720219c8f
Pure Prop Address: TMG8BGEKrgDCqPJMuPr9e9EppTdzunca7fq
Theory Prop Id: 535161c1205ef5061ddcd8bc9186161d3eacd8b5c97edac1dfa870877bb532ba
Theory Prop Address: TMNDvRwj9viGBk1WZotrsPW535EA9L5CVb6

PNoLt_SNoLt_PSNo

21.4. SURREAL NUMBERS AS SETS II 251

Theorem 21.126

∀αβ.∀pq :ι → o.ordinal α→ordinal β→PNoLt α p β q→ PSNo α p < PSNo β q.

The proposition is identified by the following information:

Pure Prop Id: 11b6ed978d93a259a246a64548a3100d22b113d8b2de70d4392bb571d4078f73
Pure Prop Address: TMXkNqDx5R69XDH3RVS3UbfdKQwDetvxzFa
Theory Prop Id: 6196d926339459b6e115949379593dc6e6a3f5922f1b60dd11abdd0193525e46
Theory Prop Address: TMSzfRKTbTeWkajRMYPfF6mjUqxx7M2QY5i

Definition 21.32 We define SNoCut to be

λLR.PSNo

(PNo_bd (λαp.ordinal α ∧ PSNo α p ∈ L) (λαp.ordinal α ∧ PSNo α p ∈ R)) (PNo_pred (λαp.ordinal α ∧ PSNo α p ∈ L) (λαp.ordinal α ∧ PSNo α p ∈ R))

of type ι → ι → ι identified by the following information:

Pure Object Id: ec849efeaf83b2fcdbc828ebb9af38820604ea420adf2ef07bb54a73d0fcb75b
Pure Object Address: TMRuDcB8ezdcHgdFjNy68Wt7HG9e2X1bxLT
Theory Object Id: 295bc4535cc47135a06fa73d8ca9d4df8cc9e2d68a2a449df25a754b7e9dc1a9
Theory Object Address: TMP7d7RzZhj9qAvayw3UXZSiTmHLbd188MH

Definition 21.33 We define SNoCutP to be

λLR.(∀x ∈ L.SNo x) ∧ (∀y ∈ R.SNo y) ∧ (∀x ∈ L.∀y ∈ R.x < y)

of type ι → ι → o identified by the following information:

Pure Object Id: c083d829a4633f1bc9aeab80859255a8d126d7c6824bf5fd520d6daabfbbe976
Pure Object Address: TMdYBKYErUYWZiKgnTvgUffZxEVBUVDaLvh
Theory Object Id: 775127720c53377db4bdf359fa2ddfb72fd332c9614eaf97b7db372fc3b6837a
Theory Object Address: TMF5e4Sx4fER7UgD1xGTzVfv1apiMjDYdoK

SNoCutP_SNoCut

Theorem 21.127

∀LR.SNoCutP L R→

SNo (SNoCut L R)∧ SNoLev (SNoCut L R) ∈

ordsucc ((⋃_{x∈L} ordsucc (SNoLev x)) ∪ (⋃_{y∈R} ordsucc (SNoLev y)))∧ (∀x ∈ L.x < SNoCut L R)∧

(∀y ∈ R.SNoCut L R < y)∧
(∀z.SNo z → (∀x ∈ L.x < z) → (∀y ∈ R.z < y) →

SNoLev (SNoCut L R) ⊆ SNoLev z∧ SNoEq_(SNoLev (SNoCut L R)) (SNoCut L R) z).

The proposition is identified by the following information:

Pure Prop Id: 9398b7aeb19cac8a975ba6fbd95f5b59201e4420910809555431c85cad9a916b
Pure Prop Address: TMU2SS6hDznnxkCmvyEBsebcgVDL1S1H2x2
Theory Prop Id: 66456cd00f96a66ea2d8766a84096f51528bb65bd260c0bd0ee0a6d609286810
Theory Prop Address: TMMddRAHYAMxfbYKpGLowF2aZGsCS8EmE5Q

SNoCutP_SNoCut_impred

Theorem 21.128

∀LR.SNoCutP L R→∀p :o. (SNo (SNoCut L R) → SNoLev (SNoCut L R) ∈

ordsucc ((⋃_{x∈L} ordsucc (SNoLev x)) ∪ (⋃_{y∈R} ordsucc (SNoLev y))) → (∀x ∈ L.x < SNoCut L R) → (∀y ∈ R.SNoCut L R < y) →

(∀z.SNo z → (∀x ∈ L.x < z) → (∀y ∈ R.z < y) →

SNoLev (SNoCut L R) ⊆ SNoLev z∧ SNoEq_(SNoLev (SNoCut L R)) (SNoCut L R) z) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 02c4ae1ea5a415b6ae779362a18ac7cc3070a6c3c3929db683bbe6328a699c64
Pure Prop Address: TMZsxeiR2ByApNxNBoTaPP9YjGAmivj7Pn1
Theory Prop Id: 5bcc8fbe0029984d17a2e449797a375c7e0a555e9ac3699de6cb6126199cfb8e
Theory Prop Address: TMLavVhWLmAEpRZmDjMboZyS2VSTuS99y2H

SNoCutP_L_0

Theorem 21.129 ∀L.(∀x ∈ L.SNo x) → SNoCutP L 0. The proposition is identified by the following information:

Pure Prop Id: 47329ccaff17b8de852f941e3476b6e2a6155b82363ef200acdb240a53b6fda3
Pure Prop Address: TMTjdd2ksvbLqB5FsqRdHpicLo8hBbx6oUB
Theory Prop Id: ba61e1cdd15d766dc08200012803abfdba4ca254c084a1b0a956a1cce0e38bec
Theory Prop Address: TMc8bJkCqfKyT4pUBfyifSdGoAoYMJU3VNT

SNoCutP_0_R

Theorem 21.130 ∀R.(∀x ∈ R.SNo x) → SNoCutP 0 R. The proposition is identified by the following information:

Pure Prop Id: 54eb826f8bd3be24b1ac7e770fe0c82339a00e86e3aa6ad66d6c724dfb7ef61d
Pure Prop Address: TMYBbiGfDG8mCyX4fru38L5jAWs9jdk1E22
Theory Prop Id: a06bcdb281c727dba6fa27a17f7f8cf4ed415ef56fdd112760f4ab9f181bdd15
Theory Prop Address: TMVtzjAnWmMCfmCH1umdVeisYihbHtUcePx

21.4. SURREAL NUMBERS AS SETS II 253

SNoCutP_0_0

Theorem 21.131 SNoCutP 0 0. The proposition is identified by the following information:

Pure Prop Id: 3b144eeddb1a1e1da010e706f7ae7f7d3118085be6517aa02ae9e9baaabdbf9e
Pure Prop Address: TMFvHEjaKbqPcs4FFJwfso2a1BUbAJ1s8cW
Theory Prop Id: a5144d076fe9899c5c9e19256598fb91665b7f4ccdb1496e795e8e54b2d4dbd1
Theory Prop Address: TMMq2uhqdU4HWokrwXJLACzASFLaAXQaemQ

SNoCut_0_0

Theorem 21.132 SNoCut 0 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 6a1f81c2c7f937ee3a0d8974a08070c8b8027c1f662f906b43877504c8d43386
Pure Prop Address: TMXsrRAvtrifruw52jjYBXPcZSqXpXBYS6i
Theory Prop Id: abbffb386e047e8707315aa435470d2a1b93eb4b65743a2e3d1e70c9bfac9822
Theory Prop Address: TMNDz4jiqD3F4kCxcNG6hy3DkR89EsDykLk

ordinal_SNoLt_In

Theorem 21.133 ∀αβ.ordinal α → ordinal β → α < β → α ∈ β. The proposition is identified by the following information:

Pure Prop Id: d6f066102af8887431439634f1b57a9b0617aba84ba72851e224f021d81ee1fa
Pure Prop Address: TMboKANLD7F64pb6YHEZed6jmzXMjejG8Hn
Theory Prop Id: de775bfed53c50ad2f8415b8603d1aa62bc9e72e6af350f83a6b199c85e37f40
Theory Prop Address: TMY34unv2doasrWYaYXyGVaCNDYHke47nRH

ordinal_SNoLe_Subq

Theorem 21.134 ∀αβ.ordinal α → ordinal β → α ≤ β → α ⊆ β. The proposition is identified by the following information:

Pure Prop Id: c7c906306e5db042974931e626682e6d001f1926ae5d425fb53e074ecbc4a4b6
Pure Prop Address: TMUky22qrn52gwZasuEjEB8QeZ16vSaUQda
Theory Prop Id: 7447913a7365f17ac2bd36550d83a144dba2081fdb9554f75855b60e9ab366d6
Theory Prop Address: TMVmvfhvMNFxdUKaz49Zx12oefEK5cmZnmc

Definition 21.34 We define SNoS_to be

λα.{x ∈ Power (SNoElts_α)|∃β ∈ α.SNo_β x}

of type ι → ι identified by the following information:

Pure Object Id: d5069aa583583f8fbe5d4de1ba560cc89ea048cae55ea56270ec3dea15e52ca0
Pure Object Address: TMYeiHkopAzUFpJHiX6DyQTk3GQ6WaqpPTK
Theory Object Id: 4680ceebcb1ec94f9ea6976bf96fcfc7658bb190d9a29c5379cc475cc08324a0
Theory Object Address: TMS7Rrpy3RhKZhHAm1BhXAbgLBRoK6QSvXq

254 CHAPTER 21. SURREAL NUMBERS I

SNoS_E

Theorem 21.135

∀α.ordinal α→∀x ∈SNoS_α.∃β ∈ α.SNo_β x.

The proposition is identified by the following information:

Pure Prop Id: 970ce204c4c603af34a26281cf79119bce7edf82baa409d90d59caebd267c095
Pure Prop Address: TMRdGNRDtsqoEYn5nNq2tqy2tRb7GnvaRGr
Theory Prop Id: bb731c8dff4704c5eeb508b1721d9396d9976279c8d5f0cd55142b3de33acf35
Theory Prop Address: TMTy8u2pBq8VtTTWH867d9E9g248LG9VRPU

21.5 TaggedSets2

Let tag : ι → ι be λα.SetAdjoin α {1}. Notation. We use −′ as a postfix operator corresponding to applying term tag.

SNoS_I

Theorem 21.136

∀α.ordinal α→∀x.∀β ∈α.SNo_β x→x ∈ SNoS_α.

The proposition is identified by the following information:

Pure Prop Id: f5227dbc4fc6188f3716bce1eb047d891fe917014fdf489301acb45e2c964b31
Pure Prop Address: TMMNh5GJRjfU6R9jYR9oSB51a82UR8JuQEx
Theory Prop Id: 04f24bbca0962b615517b9ad07d87ff686ee791137b3963827817df34a43adf4
Theory Prop Address: TMKCtpebj2a31twyQ52NMFQ1XPqjxuuqVSL

SNoS_I2

Theorem 21.137

∀xy.SNo x→SNo y→SNoLev x ∈ SNoLev y→x ∈ SNoS_(SNoLev y). The proposition is identified by the following information:

Pure Prop Id: 7a14c7d2ef8827bceb18bb81afdc45412f5d9fea9a904ca562bfb0f81d89d276
Pure Prop Address: TMJ7HAvGksViV5vVRG5CZDkJu7DwFRhbuix
Theory Prop Id: b4042d0ea055be8fc3da99d768f853e3498c984596760bcc2ea38e088f32c744
Theory Prop Address: TMKRKaYnYhhkyGaZmUUzgdvAQzZwXTCobPz

SNoS_Subq

21.5. TAGGEDSETS2 255 Theorem 21.138

∀αβ.ordinal α→ordinal β→α ⊆ β→SNoS_α ⊆ SNoS_β. The proposition is identified by the following information:

Pure Prop Id: 6828e7536cfb2fa9917d83ea810ef035d006db4740719d8a39bb37385dbd855d
Pure Prop Address: TMUi6zuABJzGjt2kXYnMDgkM42sqaRJMMSb
Theory Prop Id: d928b8db0b481798073f21b6dda22ac8c152f53325fda84e77c9b8ec06d1d6e3
Theory Prop Address: TMcJgHjSXytYqLvRxz9hc3gMG8529moaiYm

SNoLev_uniq2

Theorem 21.139 ∀α.ordinal α → ∀x.SNo_α x → SNoLev x = α. The proposition is identified by the following information:

Pure Prop Id: fbb46148e53cbe3acaacc8b0b0d7126ebd90194aeb82198e6da28510ce8036d1
Pure Prop Address: TMPvNq7g9BwshVJwKXtCkaoFEnCVHUogHas
Theory Prop Id: f48d47fd7fa9af8472cc826916e9dcead2eaad0845f212108c42b4b8599893ef
Theory Prop Address: TMUESvsETvw3i1PT8zcwLiWdQvbBBUe1Xw5

SNoS_E2

Theorem 21.140

∀α.ordinal α→∀x ∈SNoS_α.∀p :o.

(SNoLev x ∈ α → ordinal (SNoLev x) → SNo x → SNo_(SNoLev x) x → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: deb03168c1f55ab1d7d74277ebc4f8f33c87b28b16265d557f144f3a25032de8
Pure Prop Address: TMRabQVdMnYRsChGTYCN42TFfnEWrGyAXz2
Theory Prop Id: 62b490ef867368eaf329040a83310276c2ad2d046f5274d8c507fe860ea3918f
Theory Prop Address: TMb5fbntT2N1CnACiMUeKM3JYGV5og2W5ZC

SNoS_In_neq

Theorem 21.141 ∀w.SNo w → ∀x ∈ SNoS_(SNoLev w).x ≠ w. The proposition is identified by the following information:

Pure Prop Id: ab85fd9a0fdd5f7f96ca2f75d5fd279bf5e0232396c952796f3fc30b522d427d
Pure Prop Address: TMFyX2RhEpYBCTYG6ZNQWUrYk2pQxTkZujK
Theory Prop Id: d5dbd7ce116b1b3b16cbebfad9387d99e477865cfc62e88ffd6e229c2248d86f
Theory Prop Address: TMZeUEJeQ6MejSErRQj2VVEk9N7K2zaE9uV

SNoS_SNoLev

Theorem 21.142

∀z.SNo z→z ∈ SNoS_(ordsucc (SNoLev z)).

The proposition is identified by the following information:

Pure Prop Id: a9ba1fc0964d1c60416e8e5bf12e8514139434ea6fe325257b505b527b64eee1
Pure Prop Address: TMW8wR8JRK9MYLigu7413p9VCMwfzGT34AQ
Theory Prop Id: 8efb88111c118260d0d7ef1b5cdc5079c9b4a3be002c1cb750eb87a510583725
Theory Prop Address: TMbmaGPAkU5fqbmKhk6PvZkArv7GqJuJRmY

Definition 21.35 We define SNoL to be λz.{x ∈ SNoS_(SNoLev z)|x < z} of type ι → ι identified by the following information:

Pure Object Id: 8cd812b65c6c466abea8433d21a39d35b8d8427ee973f9bb93567a1402705384
Pure Object Address: TMUvMXBQ19xuZkaN1Qr1oL9x2RgVQBhPiEP
Theory Object Id: 48bcf841807be9a2a8a7fdaa4062b7f700ed225f5219ffd13f26782d239d11ae
Theory Object Address: TMGV9huV8KZCMkFQWMGc7fqW2LnTPhHVw9y

Definition 21.36 We define SNoR to be λz.{y ∈ SNoS_(SNoLev z)|z < y} of type ι → ι identified by the following information:

Pure Object Id: 73b2b912e42098857a1a0d2fc878581a3c1dcc1ca3769935edb92613cf441876
Pure Object Address: TMSvKHoWGSuy2M3PmTt5s6iktHABF7qYPcr
Theory Object Id: 0cfe870d225f8cba537d2fe6552d285b3601e6ba81a7b2c47d955f4d490e3aba
Theory Object Address: TMH8PQYnqVae2kcUxoh7NLmWhd5qkNUZU8E

SNoCutP_SNoL_SNoR

Theorem 21.143 ∀z.SNo z → SNoCutP (SNoL z) (SNoR z). The proposition is identified by the following information:

Pure Prop Id: 52486657c66c30f3a313a97245051168bef3ab087bda6a6d0afdc7a3a1f8843e
Pure Prop Address: TMQrzjSnfMHhEaKZxuYJx3zZLL9WfuFojym
Theory Prop Id: 7aba571de6f479a36c61b1efbc3a3061f907bdbc5585aca815e7b788f51bd0a2
Theory Prop Address: TMVyVwNCUniGutAsBRT2zh5MKBDjX715Vgi

SNoL_E

Theorem 21.144

∀x.SNo x→∀w ∈SNoL x.∀p :o.
(SNo w → SNoLev w ∈ SNoLev x → w < x → p)
→p.

21.5. TAGGEDSETS2 257

Pure Prop Id: cead4f50e23f6816dc9703016f0ed79b8bdbc32a7c09a0f9af78552aeebba9c0
Pure Prop Address: TMZmXPiB4BB3YvdBwZTD97bGggD7rSLF6eD
Theory Prop Id: 7ccd8b9dc0186ef905a8d93999fdc69eb492737ff71320a53317c135b5d63fb4
Theory Prop Address: TMYKcvufFZ2FLebZaXDWknwUKjkcdzGsQTZ

SNoR_E

Theorem 21.145

∀x.SNo x→∀z ∈SNoR x.∀p :o.

(SNo z → SNoLev z ∈ SNoLev x → x < z → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: efca05b2e45ac1d5303c49c52a3636406f36f5a77d8812e0200eaefc9712df2c
Pure Prop Address: TMbsBxRcWxxu6mkgykGDi4yRnEUFXwJJe4Z
Theory Prop Id: 7dca045bf869829b43204d36be8d7632a5061acf12e145a99d1cd8d9be1edfda
Theory Prop Address: TMUsPcK3wqDZwDie4ZMQqUBLgLnuT57PPnY

SNoL_SNoS

Theorem 21.146

∀x.SNo x→∀w ∈SNoL x.w ∈ SNoS_(SNoLev x).

The proposition is identified by the following information:

Pure Prop Id: 6e3a2292c3997c29e78fcb520858907ddc605dae52383648872fd6c7ce45f0c1
Pure Prop Address: TMLcRAnSRqhUGWvLLTk89wzM2sZ4wvddz98
Theory Prop Id: ded78aa4dc105de48885dd5e055230505548286bce7f04f3f5050316c2c04cd0
Theory Prop Address: TMPHFqjx1BZx94gsexg3cyERVChRcfu9aMh

SNoR_SNoS

Theorem 21.147

∀x.SNo x→∀z ∈SNoR x.z ∈ SNoS_(SNoLev x).

The proposition is identified by the following information:

Pure Prop Id: 98051c05991e35d56daa447cc7976259612d6048fa82f9a44da11c916b906848
Pure Prop Address: TMJ2p8noSCp43KaJq9tZFaAKHrVfWm26tto
Theory Prop Id: e13f497f5dab3650206c4362046a1ec477ed8e40d771a06657c3db32f3b247b0
Theory Prop Address: TMJz37dSJBFjqARSMYVzEuc7L1KYXuTUBFc

SNoL_SNoS_

258 CHAPTER 21. SURREAL NUMBERS I Theorem 21.148 ∀z.SNoL z ⊆ SNoS_(SNoLev z). The proposition is identified by the following information:

Pure Prop Id: 73f5b2cc52ef1daf3b3493d4b3a1b56e0f6a7fedf019873cb56ed5165fc129f6
Pure Prop Address: TMVShTkL3TuhDkq8iCTfj9xfEecNaq6hZTr
Theory Prop Id: 409713becd3f086d3979ea36c2db06311233c54602ab8b2acc36bc0e856661f0
Theory Prop Address: TMacyLTfdfpQiEh9iWRH45thaEffs457QES

SNoR_SNoS_

Theorem 21.149 ∀z.SNoR z ⊆ SNoS_(SNoLev z). The proposition is identified by the following information:

Pure Prop Id: 16bdfc8b4f768a6b37a41115958b385c5a1ec67f13570e242fd1933e158b499f
Pure Prop Address: TMSz3ALjKBot9ZKASXvvTfDJErg7bfASa2J
Theory Prop Id: 2fe1c08fc3923f123fffe6c7d8be4b5955d977da8e53d13d4d6c6f092bdcc16d
Theory Prop Address: TMbbztsQAvihxVwkJ9kvqAoNEe7VRXcgYMR

SNoL_I

Theorem 21.150

∀z.SNo z→∀x.SNo x→SNoLev x ∈ SNoLev z→x < z→x ∈ SNoL z. The proposition is identified by the following information:

Pure Prop Id: ddd91ba46f8cb49763a2e555070ff9cf0d25b89336d5efd26582ff2019e6e033
Pure Prop Address: TMFWNR9kjhSxmC1PpWAxELFPMCQrQDgS3ii
Theory Prop Id: 4f3d049c65e44bceaa7abebe992ac83d5e90855d0e811e33cecee9e4c04e2717
Theory Prop Address: TMV173uXJoeNM1as5esbiKuaj6zEgDhUJka

SNoR_I

Theorem 21.151

∀z.SNo z→∀y.SNo y→SNoLev y ∈ SNoLev z→z < y→y ∈ SNoR z. The proposition is identified by the following information:

Pure Prop Id: 97693c3a27a8f00a3262b80f853884f0e0d68e482cbff49cb642b3f14d256b71
Pure Prop Address: TMTPFvahbCyh9yFHZ4zMiRoweJ7ZXMnfLx8
Theory Prop Id: 383e88e28a8ea102937169b00da3f18efb2523746b17c0fc524867476a449c5a
Theory Prop Address: TMSrHC17ffyA2enfcx5o1B7iUJn1JV3jwSc

SNo_eta

Theorem 21.152 ∀z.SNo z → z = SNoCut (SNoL z) (SNoR z). The proposition is identified by the following information:

21.5. TAGGEDSETS2 259

Pure Prop Id: 3cc369ecf11e43c8a7a6cd9176f7edf2e2d6f77128e0bb8b46dd7e9adb3afd18
Pure Prop Address: TMMbZPXPQWq2cEQg1eagYj6NwweZYGCHjiJ
Theory Prop Id: dc9273d2d30d56ac9a4eff2b975bce32ce2e1e109d620f9318846e4176dda05b
Theory Prop Address: TMZQ7w25dF2ftfDkjDshscxccT898iSijZ7

SNoCutP_SNo_SNoCut

Theorem 21.153 ∀LR.SNoCutP L R → SNo (SNoCut L R). The proposition is identified by the following information:

Pure Prop Id: 9621aa2a6018b3eb1183066003e6daeb26c90761edfa02dd7e872bf831ccb4f8
Pure Prop Address: TMNKaoM4RaxsmapryqxboP9xADcYUVy5VVL
Theory Prop Id: ae46e89c23c7cd0b1fbccb0a1e97f383308cec395b5c73ec3d890f6d1641f001
Theory Prop Address: TMMkfWp7u1CPUEtc9sUMWAX6ZkHiFKL3Tsx

SNoCutP_SNoCut_L

Theorem 21.154 ∀LR.SNoCutP L R → ∀x ∈ L.x < SNoCut L R. The proposition is identified by the following information:

Pure Prop Id: 984a30c42556263d781f855feaa7c03ffd5961f0b9c95159ea8302b3ca16467c
Pure Prop Address: TMKfLyuDzXLNY7PtnVTeZe7eLvSaxAynvFi
Theory Prop Id: 4bca14807188033563b691a30256c970867594d77736618ddaeed7f6ead01b56
Theory Prop Address: TMcuZs8Sau5dRwSYifuF8HCBdKnRRmSKZYt

SNoCutP_SNoCut_R

Theorem 21.155 ∀LR.SNoCutP L R → ∀y ∈ R.SNoCut L R < y. The proposition is identified by the following information:

Pure Prop Id: 9161db8c6978c922fbaa8ef29650182e4dc42c4f7e4bbe22e34280baae014e7e
Pure Prop Address: TMdihdbyAcDNSyUkD3TdUvmSMiU3VY6TxqA
Theory Prop Id: 7c3d12a89f726793a0a60bd44097c71eaee25e353ad706ea323bc543dbf26c62
Theory Prop Address: TMPBd6BZmniMTRdKnM7fbpYBkR7PiBbKsPm

SNoCutP_SNoCut_fst

Theorem 21.156

∀LR.SNoCutP L R→∀z.SNo z→(∀x ∈ L.x < z)→(∀y ∈ R.z < y)→

SNoLev (SNoCut L R) ⊆ SNoLev z∧ SNoEq_(SNoLev (SNoCut L R)) (SNoCut L R) z.

The proposition is identified by the following information:

Pure Prop Id: b52eb75a2c789c7e4e20b3f5065952940fdfe39af85380d08e4788785181a8fa
Pure Prop Address: TMGpKvy9aLQxgW1tTVWtNezptYGvyfjG5Lj
Theory Prop Id: 8362f8acd554e760437ebf6fcb6ddf58a0b184474b06a457ee1b26d03f2fb41a
Theory Prop Address: TMKQruVQCojmwaMqYHkmFCxgxgAAJu5hamZ

260 CHAPTER 21. SURREAL NUMBERS I

SNoCut_Le

Theorem 21.157

∀L1R1L2R2.SNoCutP L1 R1→SNoCutP L2 R2→
(∀w ∈ L1.w < SNoCut L2 R2)→(∀z ∈ R2.SNoCut L1 R1 < z)→

SNoCut L1 R1 ≤ SNoCut L2 R2.

The proposition is identified by the following information:

Pure Prop Id: bb8cdcd2d5fd2f00037512dfe3c9dc90186e99a11943fa34839c21c6d21327f9
Pure Prop Address: TMNRFtNxDRZszFXoDioSFyUkctJH577Wgtm
Theory Prop Id: 4237313994e58f38bf2a6019231c10978a504fc59d522337bd78a558d606fc9d
Theory Prop Address: TMU9RWn6464F25Pra7jPLPqiPfwa8i21YQ3

SNoCut_ext

Theorem 21.158

∀L1R1L2R2.SNoCutP L1 R1→SNoCutP L2 R2→
(∀w ∈ L1.w < SNoCut L2 R2)→(∀z ∈ R1.SNoCut L2 R2 < z)→
(∀w ∈ L2.w < SNoCut L1 R1)→(∀z ∈ R2.SNoCut L1 R1 < z)→

SNoCut L1 R1 = SNoCut L2 R2.

The proposition is identified by the following information:

Pure Prop Id: c1078eaf6d9d767331c2adfc239cf6d058aa0386b0e05639ec15f00d6bc8651b
Pure Prop Address: TMQEDzfjypV741w6WAoTJECPB5vgdYS7NXj
Theory Prop Id: f66618e40670b959c354add07597e52634943b95783e7550a46245c83744c351
Theory Prop Address: TMQc8BkWk3YDRQ3ZDJXw13dYU48b624zt8h

SNoLt_SNoL_or_SNoR_impred

Theorem 21.159

∀xy.SNo x→SNo y→x < y→∀p :o.(∀z ∈ SNoL y.z ∈ SNoR x → p)→

(x ∈ SNoL y → p)→(y ∈ SNoR x → p)→p.

The proposition is identified by the following information:

Pure Prop Id: f3b15a8dbd76f9e501e20410a95606c6e4d8a53c81914a627374d46d3d907b2f
Pure Prop Address: TMRDwmSiYvVpLRjvbAK1MWfXAVsfvotprEu
Theory Prop Id: ffc0bf982d5e8f4c56f088c7cc742ee4ab02e0c490f0c7638cbc614dc1677aca
Theory Prop Address: TMcckGGW8ynGG29S7CCB8qwfJasDqZZoUqS

SNoL_or_SNoR_impred

21.5. TAGGEDSETS2 261 Theorem 21.160 ∀xy.SNo x→SNo y→∀p :o.(x = y → p)→(∀z ∈ SNoL y.z ∈ SNoR x → p)→

(x ∈ SNoL y → p)→(y ∈ SNoR x → p)→(∀z ∈ SNoR y.z ∈ SNoL x → p)→
(x ∈ SNoR y → p)→(y ∈ SNoL x → p)→p.

The proposition is identified by the following information:

Pure Prop Id: f8a5e25fc885d4cd85a70c63657ca7547fbf75b02f27b82464c042547cc677f6
Pure Prop Address: TMKsCmKUpLx2BhcK6jEDUspKVKvE5k5QiRa
Theory Prop Id: 68ba9809e987e498558a641b25900231b82622d983653583a201a59a4d1c0b55
Theory Prop Address: TMKWiFqjuwfNEiTTVN1gfvvjWVrJhQj1gqg

ordinal_SNo_

Theorem 21.161 ∀α.ordinal α → SNo_α α. The proposition is identified by the following information:

Pure Prop Id: 4731fd4a3d0693e855e835e22a21f5e5977b26765eba190ec58fe9daa4d2afe8
Pure Prop Address: TMWHKTo9dcgkBtGz7uaXaYTj4kgHyCDDyAP
Theory Prop Id: 7754e54755e12ec8adcd2c120c5d5474ec249b91da929ca1e7981dd9d443b092
Theory Prop Address: TMbYC7hUX1cj11jwA8NrmcFtWzGyRNNcP3L

ordinal_SNoL

Theorem 21.162 ∀α.ordinal α → SNoL α = SNoS_α. The proposition is identified by the following information:

Pure Prop Id: fef60879d9dcebd791ff77a26af98cae1b05821d9fe106e7d892b781e23e857d
Pure Prop Address: TMKCDKt4HyT8bYK3wPMhnwhhHEbQ5xacdeK
Theory Prop Id: 95823e3d326f1172e3884b6aa394d6e9d59d2ddf7578bca2e2f2aa935524fd70
Theory Prop Address: TMd2zwjPUXZL6Vy46VFW4gwpc2zig21PKRe

ordinal_SNoR

Theorem 21.163 ∀α.ordinal α → SNoR α = Empty. The proposition is identified by the following information:

Pure Prop Id: 07c69afcba017b5ee2b34caa8d7b084b72822e15942c9a589e012fc3354dadc3
Pure Prop Address: TMEszERJsS849AqzpPGdTFGPRhenRZSY14a
Theory Prop Id: f853d9f1e9d0ea7f8c0d0f6837e18502503c2b8c0a0f017d5f07a8be9f70fda3
Theory Prop Address: TMRXLgDuMtvBV5siWdFYrySgsWpvnxN9925

ordinal_SNoCutP

Theorem 21.164

∀α.ordinal α→SNoCutP (SNoS_α) Empty.

The proposition is identified by the following information:

262 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 1e4d0d44a7461a68881734e0e91b185e38404b2e73f6e59f99b11fe839407b3d
Pure Prop Address: TMW1UvmeHxUQLv7BE98ap334SUfCm8Fu73Y
Theory Prop Id: 0cfe8be75650c6e5a146099344f74fb66757e93b2cd2edac1ae4deefc984dc64
Theory Prop Address: TMRkZRJr4waasoDGCAVbUFmg6pCVbwcgmgu

ordinal_SNoCut_eta

Theorem 21.165

∀α.ordinal α→α = SNoCut (SNoS_α) Empty.

The proposition is identified by the following information:

Pure Prop Id: a115354a2357cc66a492e6c27636f873c89ca659eeae89203adba376b623519f
Pure Prop Address: TMSVcnu3qbhNj765CNa5JVutvQs8ZAUZbac
Theory Prop Id: 1f113a2b452bb89a7755a221d966a72de174752e7ff3e1d939ad96691e455817
Theory Prop Address: TMFKp4mHaoLW3mEp9qD9XgwpSzYRMvq34F2

SNo_0

Theorem 21.166 SNo 0. The proposition is identified by the following information:

Pure Prop Id: c8204b8b1882d371baace062a2960d93b16000698c7d6a09a142db659b3f1fca
Pure Prop Address: TMN5B5ciEPfW7oQwW9YQcm3vLvkEtUZSZnt
Theory Prop Id: 94a310917df9f628979dfd6d1313a09153628ffb779d20e4011addf6dc4f2d0d
Theory Prop Address: TMFnaFB9Pt5pY6PgwroCRcRZDgrkfSYBuSa

SNoLev_0

Theorem 21.167 SNoLev 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 45a0f7a396d9f9d63598766e2614400055684c37fc87764bd32b3d694a4f878e
Pure Prop Address: TMJQ1Ke47ELvhoXkPaZRPfgMk7d8RGCakfG
Theory Prop Id: 3527a4181b9135af63b4982294e24684d20c27740e7c8367f29e25ad12624fa4
Theory Prop Address: TMLKkLmPfzCPj82cSv1tmqe99GdjnMnFnvv

SNoL_0

Theorem 21.168 SNoL 0 = 0. The proposition is identified by the following information:

Pure Prop Id: c01c2ba737a470d613bec196a861574fc97eac3285aae68253f8ce0a1199ed06
Pure Prop Address: TMPzAPKfbgig2tHnBj3tDTW5gQHRQr85Y6m
Theory Prop Id: fdaa694c1dad3b51990c33828083df7857c7ce848cddbb249b15569df8cf4b8c
Theory Prop Address: TMVPonV6zmxAnnnAJAxaJ4wWH75boedRK8V

SNoR_0

21.5. TAGGEDSETS2 263 Theorem 21.169 SNoR 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 7329c7a0141d5faa6d6c735a4606f98929d053ab782ccfee0d527914d8d0bb82
Pure Prop Address: TMHH23zks5tKNiFan71Zdnpm67D36aSbUoZ
Theory Prop Id: 51075148dc5c48d42db4d40072bca52ac7fb82c5a788b39d6f6b7e7c69cd1ecc
Theory Prop Address: TMajtxBFtVd1rrqznKQdB7sqrYQ1JGnSYS3

SNoL_1

Theorem 21.170 SNoL 1 = 1. The proposition is identified by the following information:

Pure Prop Id: 84d688b29c78bb8b93ed259d6f9f639fde3ee97c6f1196e2a383bf07c901a937
Pure Prop Address: TMSoMwfGW9Mu39jcotcU6thi6J4LWYEx7bn
Theory Prop Id: 15aa793de666a844e874c1d586a19989e272dfba20c5493bdc382a6d38d3b874
Theory Prop Address: TMTBX6BysYj8xm3zeyBXfyKjENW9x7yzMib

SNoR_1

Theorem 21.171 SNoR 1 = 0. The proposition is identified by the following information:

Pure Prop Id: e6a83a40947575bc277f1b8c0862189045f237a6889acb55f060da536e5d0e57
Pure Prop Address: TMPLYyxJkuB1KiJuFytWk8RBxXzyup6DPfW
Theory Prop Id: 8da427f5c15d8ec84a8b68f3c128839fd941641f566c3d76a8fc06701af24cd2
Theory Prop Address: TMVQas5fNZ38HF4hftusV6FZz3JLDZMUN9n

SNo_max_SNoLev

Theorem 21.172

∀x.SNo x→(∀y ∈ SNoS_(SNoLev x).y < x)→SNoLev x = x. The proposition is identified by the following information:

Pure Prop Id: ff019e872fd598ef8a2a3be672f6bcc7cc7017530d8e62908bad15bf571b16b0
Pure Prop Address: TMTDt7E8q54KkcysEpcH7qcMKm5WwHpsKqT
Theory Prop Id: 3be24662ae052242ba894f8ccca93140cca84c843c9c37beb7a1560477952006
Theory Prop Address: TMaz8UPXxX1bFVUXeXSuQwyrGjmQjY46y6C

SNo_max_ordinal

Theorem 21.173

∀x.SNo x→(∀y ∈ SNoS_(SNoLev x).y < x)→ordinal x.

The proposition is identified by the following information:

264 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 0b230ced38578b81f36ac519fa85acee92a243a563fecccc69ef13c26cc93ac1
Pure Prop Address: TMJBDQRmWeDvJspje8G3NTmF5iRA5QFjvTX
Theory Prop Id: f28b9a03523768b82a0c58f1bbd347f928da765c1382c04c5d8307d059157531
Theory Prop Address: TMTYDjsbqqY8KJTxNg5X66HtVW1bn6sTmVP

Definition 21.37 We define SNo_extend0 to be

λx.PSNo (ordsucc (SNoLev x)) (λδ.δ ∈ x ∧ δ ≠ SNoLev x) of type ι → ι identified by the following information:

Pure Object Id: 997d9126455d5de46368f27620eca2e5ad3b0f0ecdffdc7703aa4aafb77eafb6
Pure Object Address: TMQChUQBGYBvAJNKavV49dHDgfu4qGjxHQV
Theory Object Id: ff4161d4bf42181c2a7fa1c1b810f9d4878846fc9af335cc282ffd21f440f9f4
Theory Object Address: TMbS5DXTFrYci5baCfTAfaQPuyL4WdRc1rC

Definition 21.38 We define SNo_extend1 to be

λx.PSNo (ordsucc (SNoLev x)) (λδ.δ ∈ x ∨ δ = SNoLev x) of type ι → ι identified by the following information:

Pure Object Id: 464e47790f44cd38285279f563a5a918d73224c78a9ef17ae1a46e62a95b2a41
Pure Object Address: TMKMddAcJS6FL2G9uiJereyb1tjokjtKa8Q
Theory Object Id: 8c551b76c450b237e1236b32a569a43ca6b32a062d1d2a013da3dc303569fa68
Theory Object Address: TMQVotiw6nRDP1Q877RTRTfvRtH76B8baK8

SNo_extend0_SNo_

Theorem 21.174

∀x.SNo x→SNo_(ordsucc (SNoLev x)) (SNo_extend0 x). The proposition is identified by the following information:

Pure Prop Id: d41d3ef86775d62e2fc9b3c9a272033d4ea2944b3c8af763707dfdef923c968f
Pure Prop Address: TMQd3GQMR6hK1kRRtEeaAkFX4ThA8PYbx4s
Theory Prop Id: c8f8ea39e4cdce2253d47eb0646052513914897eaed4338727a5cd163df2b423
Theory Prop Address: TMH2yeAxqLFK5UHLb9b8dQ6DAmWA6XAxYHL

SNo_extend1_SNo_

Theorem 21.175

∀x.SNo x→SNo_(ordsucc (SNoLev x)) (SNo_extend1 x). The proposition is identified by the following information:

21.5. TAGGEDSETS2 265

Pure Prop Id: b7b8669f74b82f6386406947ed40fe7ac42ab269bbd6ea1fa28e5c6678ff4ef4
Pure Prop Address: TMZZkw7fkUGWbN4TRGNYZKePU3uacXEAcNy
Theory Prop Id: 73bbf256bdb458b2821e16079f5ac4e447c94d365812f86f5c93f8bc91c0cf93
Theory Prop Address: TMXbECAsrsXbVHqfqLcWLMGLrA5ADZz1k4Y

SNo_extend0_SNo

Theorem 21.176 ∀x.SNo x → SNo (SNo_extend0 x). The proposition is identified by the following information:

Pure Prop Id: b353a2b441224520a178acfb9f0f817ca34603e17848371ce6ed746729e2d8a2
Pure Prop Address: TMWGxYp3JwTkkqSrsYrfogqPN1vb2Xorvh1
Theory Prop Id: 76562ca12828cc8acac6c03cb0db1ff686b05c1fa8f57098d309a8fb90aee94c
Theory Prop Address: TMQtywDaDPXseniJkxkpxbNAdKd38UE2dQ8

SNo_extend1_SNo

Theorem 21.177 ∀x.SNo x → SNo (SNo_extend1 x). The proposition is identified by the following information:

Pure Prop Id: fa323a2025df47e7721db069b72ed5eee27d1ee3571a8a8af9723ad120927c88
Pure Prop Address: TMSdMxvikAWSF7zWCcgb3FfvRNkDDipwxVq
Theory Prop Id: 5345a767fd64365f567ddd0c36e71e4585b5269ee70acf1e49fbec29ea6b20f8
Theory Prop Address: TMY25eBqvG7PbCF867BfH8wXMFyJ3vZVa3z

SNo_extend0_SNoLev

Theorem 21.178

∀x.SNo x→SNoLev (SNo_extend0 x) = ordsucc (SNoLev x). The proposition is identified by the following information:

Pure Prop Id: 26efcd107d12caee8e8ee10adebe189fb528ee43376423b43537d0d980b96057
Pure Prop Address: TMLE3zBjSiXJKVByLuNppcpDaWc5KxbJsij
Theory Prop Id: 5aa0afa406447583a2ffcc856c3c2270249851120d173ff811fd161bbb5476f1
Theory Prop Address: TMFDXAViZJBzqyvBjZTxryG6TwYcAsVpsez

SNo_extend1_SNoLev

Theorem 21.179

∀x.SNo x→SNoLev (SNo_extend1 x) = ordsucc (SNoLev x). The proposition is identified by the following information:

Pure Prop Id: 33cf6044be79e5a605f13a1596393129b0521a4bbea91a8e8df65c3029e399ba
Pure Prop Address: TMQdtthxiaHskccMdbmaNfQoxB2zGD8wrG3
Theory Prop Id: 5293470b79054a91a67c775285d231ecc230d1826978e4d088daf8390581d242
Theory Prop Address: TMKdFGjuMvwpN2qMjfXbBzHx5ZvwNJUoX9Y

266 CHAPTER 21. SURREAL NUMBERS I

SNo_extend0_nIn

Theorem 21.180 ∀x.SNo x → SNoLev x ∉ SNo_extend0 x. The proposition is identified by the following information:

Pure Prop Id: 217cffe317f555501a713a30362da2e786958f5eb0cb0d728c124c2cd996b872
Pure Prop Address: TMda5fYn6yDo4M7LQx8adXmDoCPdX1T8aKm
Theory Prop Id: 0dfd17ffb30a9a53795a0c34f15c21b135e90759a30b52930e461dc926cfeab8
Theory Prop Address: TMa2tRaEiFrQbXQteXN4nGzCFeoa2D3wqNJ

SNo_extend1_In

Theorem 21.181 ∀x.SNo x → SNoLev x ∈ SNo_extend1 x. The proposition is identified by the following information:

Pure Prop Id: ac9570cd8593fef418bb2538fbb3c4c48095739f9ed56bba38b9f759f88d20c5
Pure Prop Address: TMUeMNcoZvQFFpyFh5fDtrPSnekbcwi9gVi
Theory Prop Id: 8ca372c127a4d684d18db02fb51193ea2fd70537694faea2430652d07ea29128
Theory Prop Address: TMRnfDntKJYsu4fKMmT6ZDjzF5aDKBGdaar

SNo_extend0_SNoEq

Theorem 21.182

∀x.SNo x→SNoEq_(SNoLev x) (SNo_extend0 x) x.

The proposition is identified by the following information:

Pure Prop Id: bd4a270b5c6aeb7814b4a5a42ab775e0d391605bb207e9a7d4771b5f610877a7
Pure Prop Address: TMcVbrAftnCJg4KkPuCtBWErWfBTh4ncZk4
Theory Prop Id: 1299864bcabf2b68aae890ec79e8dac76d30b36a0b72f71d066855a7c6727939
Theory Prop Address: TMJYn96t53oDn1QogAPqAu66JqLUXXXMPDh

SNo_extend1_SNoEq

Theorem 21.183

∀x.SNo x→SNoEq_(SNoLev x) (SNo_extend1 x) x.

The proposition is identified by the following information:

Pure Prop Id: ba974f3d548b12b8a092b7b2c74ff719f439552818912e00e7423c8b93a1bd73
Pure Prop Address: TMF7TV22yna3gp5cUwPCUxbJ6xgGF2dzCxP
Theory Prop Id: d6f8bb86911f71c00536c62a9ff4f7dcb502510ff47cbe6fa6e321ad985965b8
Theory Prop Address: TMGZEXiMNjzk35x2Tqe624m5WkQJ1kfATtd

SNo_extend0_Lt

21.5. TAGGEDSETS2 267

Theorem 21.184 ∀x.SNo x → SNo_extend0 x < x. The proposition is identified by the following information:

Pure Prop Id: bf669a947f56543b6f408b3b0d257dd39de5956a56aa8d486869eea444c8ce02
Pure Prop Address: TMLC64iAmyHaHX6nQwFVRm3kwcL6vvU4P2u
Theory Prop Id: ea5fd1a237ed14511ef2281ac67e33e3f2c0c0bf15a8e4ea87ffd5dd2258d376
Theory Prop Address: TMdzN9ZkvjnY1AoibSuChkYwsx6anuPKYr4

SNo_extend1_Gt

Theorem 21.185 ∀x.SNo x → x < SNo_extend1 x. The proposition is identified by the following information:

Pure Prop Id: ca0a2d175740921eb82773f6f29e8a9d9ce71765975b1a2abc1810e62321e34d
Pure Prop Address: TMPGx4oU5xXH99g4hKGMXgJyQojSJAw44zk
Theory Prop Id: 1c8d55e5a1c7bc27e06b6ece03d9153ad522999d088945e7671e790cafc9153f
Theory Prop Address: TMSXKAxcPxJwDFbDGV7GdQ6dzrXb8t8Gtfp

Definition 21.39 We define eps_to be λn.{0} ∪ {(ordsucc m)′|m ∈ n} of type ι → ι identified by the following information:

Pure Object Id: 5e992a3fe0c82d03dd3d5c5fbd76ae4e09c16d4ce8a8f2bbff757c1617d1cb0b
Pure Object Address: TMPREfrMv4dVM6bYC6yGeUixtCX8s5WtcZH
Theory Object Id: ffee2b2b0138d17826ad706a96fc5a6a6ba8aae8fe09f0db5d0285ae918123b2
Theory Object Address: TMWgmEV2PgET7GQLdd5raFjcRBfsyaTXCuK

eps_ordinal_In_eq_0

Theorem 21.186 ∀nα.ordinal α → α ∈ eps_n → α = 0. The proposition is identified by the following information:

Pure Prop Id: 3046fd78a51853d1bb9786f97dc51dba530f419189bc9aeb3da38539fa9b461a
Pure Prop Address: TMWe1qmzHkSSwwJFgrwPJoCfonbTQ4efjAN
Theory Prop Id: 1190f379f72b87aef60bf145308c2529adc42bd0db7ec4e1673c896c2a1c59a7
Theory Prop Address: TMbpDBk3MTcVsKLqZPBPoYrpujcgPfiY5Wv

eps_0_1

Theorem 21.187 eps_0 = 1. The proposition is identified by the following information:

Pure Prop Id: 5bbce1fa3c1be09ebf6d116df2665c36b7b8f577503b6c121bb25e865d3b7238
Pure Prop Address: TMR59LWQ8moWJ5QPeadff1HFh2ASfoVG2gN
Theory Prop Id: d073f03f40b0eb3c8b0bbe33e8eba6928631935514fda8a0b3beeaf04316d10c
Theory Prop Address: TMQ9LV548ZAa1dSucp5vhC4hNL2DmRGxCUz

SNo__eps_

268 CHAPTER 21. SURREAL NUMBERS I Theorem 21.188

∀n ∈omega.SNo_(ordsucc n) (eps_n).

The proposition is identified by the following information:

Pure Prop Id: e9c4d9b771bd18106ae6b62b078902538b29d64f81f15926efc4103f8c00c9ba
Pure Prop Address: TMHkLsYZsJdew3vMRGoV4eMkozdKinHgebF
Theory Prop Id: 1c1428769c5baa199f9145d7fd4da4efa99062ddab3a11093d0b91eaddea75ef
Theory Prop Address: TMZLbVSbviYtp4AZzF9JbpEFvFr85s8f34z

SNo_eps_

Theorem 21.189 ∀n ∈ omega.SNo (eps_n). The proposition is identified by the following information:

Pure Prop Id: 9fb70039757911dbbd9302330aa8493957a0f839a3c637e3161bffc475cc017a
Pure Prop Address: TMJo1r2k8j9rELnHY45xjWz1j6aXZKmg9xf
Theory Prop Id: 006b0fb6e9891e00a73714a2337f71ea81cd68e1d83e1ffbe6ff6ac2e2ebf23c
Theory Prop Address: TMJdzF6hF6rddoE61FfB1UwP9FVejWKk4Nf

SNoLev_eps_

Theorem 21.190 ∀n ∈ omega.SNoLev (eps_n) = ordsucc n. The proposition is identified by the following information:

Pure Prop Id: 6891e35f6b0e6661135cf838e9a5fb81c80f47ec3798a2a896ac13c50244ab3b
Pure Prop Address: TMdgBWSxkt84FLUVcqjHhWJ7ceSWs9iyw1B
Theory Prop Id: 75efd151c44e95bbe9b4c51e56bc5088f7629f12f537ea75601bc7ea9eaf088d
Theory Prop Address: TMQsUJFDt5htkcABdDejoFJAbjZMEUCEAeH

SNo_eps_SNoS_omega

Theorem 21.191 ∀n ∈ omega.eps_n ∈ SNoS_omega. The proposition is identified by the following information:

Pure Prop Id: 736d2a5d04c2733a6003f64baced09360d5b4e3fca0641afbbf2823c83ba944a
Pure Prop Address: TMbuXJoMG16dBYE5bnEUW2STmFeuKyFfvUX
Theory Prop Id: 52da1852246ddad0d39ed6113b9f3532b2d250e38f24dfcfeb5bb5b3d673d7d3
Theory Prop Address: TMWwLLNiXwTEkwv5MiJv3H87ChU7x8J9ACC

SNo_eps_decr

Theorem 21.192 ∀n ∈ omega.∀m ∈ n.eps_n < eps_m. The proposition is identified by the following information:

Pure Prop Id: d6a46a372ecfabb245c4193957c4f0508d4998eb45deefc85c435af5ad5b3b94
Pure Prop Address: TMPjtSvp9n2PhtKoxct2FCPEyUEF8mWwigi
Theory Prop Id: 67c76a0faead3dccb86543d7cb1d37298cd531be3091b54b04b83935dc85398b
Theory Prop Address: TMFVwJv6i1e3XzsQSuzx3ek4mzSFTbbAQN6

21.6. ORDINALS ARE SURREAL NUMBERS 269

SNo_eps_pos

Theorem 21.193 ∀n ∈ omega.0 < eps_n. The proposition is identified by the following information:

Pure Prop Id: 8b75de3a77baa9b92adae44b583582eb3b82463db2c14f404402a2bde097e88e
Pure Prop Address: TMYSgyTwMAyFvd1rAZg1g3St39umQ4DAf3o
Theory Prop Id: 760b4aa07773231fa328d11d9a374bf04e0c43853a3127c676e77c310efc55af
Theory Prop Address: TMPXpny51o2j5XrPqyJcP3FHvVjLPiqhf29

SNo_pos_eps_Lt

Theorem 21.194

∀n.nat_p n→∀x ∈SNoS_n.0 < x→eps_n < x.

The proposition is identified by the following information:

Pure Prop Id: 575b837b657e6e7a9b06a3cf27feee72addd6cc960da74d3e2bbceddc997f016
Pure Prop Address: TMXiEk8xFJoDwT18FPGUHmWfruXdJYK5VR8
Theory Prop Id: d115f0869d3b5e92d6a510264e9d3861cd7160f6be7616adc5574ddeffb04570
Theory Prop Address: TMMiCk9Z6f7NptPRQzK4m6JDrmCyFfYUM7L

21.6 Ordinals are Surreal Numbers

ordinal_SNo

Theorem 21.195 ∀α.ordinal α → SNo α. The proposition is identified by the following information:

Pure Prop Id: ac39bc678da7d4e002759c32fe0757799548274fcde56c0639a1d25743d1fce3
Pure Prop Address: TMV1VGX8kBqSXJHN8BCPYuKwoW4HYVoSf7C
Theory Prop Id: 1eea33bee9fd0b26c50050d87b178445fa7eadba2d0a37fed8f158dba3c91ff0
Theory Prop Address: TMe1KCf7jS8uH1DVEWLEFPafjhTXSfbJGKi

ordinal_SNoLev

Theorem 21.196 ∀α.ordinal α → SNoLev α = α. The proposition is identified by the following information:

Pure Prop Id: 56f8b04f51175b2d3ad3ec438b868e145d9fb2c1d306f23dedc2aac5944efcc6
Pure Prop Address: TMZ8sa3eLx4fHfcaC1ehbCESTMyVoJwHZet
Theory Prop Id: fea5fc7aa601d4080a114440a967755767fa47253a68cef3729d30cee0317500
Theory Prop Address: TMd5j8WdJGFuG8mnG6FCTaEsK9H3VxS3rpz

ordinal_SNoLev_max

270 CHAPTER 21. SURREAL NUMBERS I Theorem 21.197 ∀α.ordinal α → ∀z.SNo z → SNoLev z ∈ α → z < α. The proposition is identified by the following information:

Pure Prop Id: 945794b25c51118cbb7056a200fc29b3beac4a0954ff35c7d46b8b10d9b490eb
Pure Prop Address: TMW5wuDBjEPgptrHgu3NBPzWzTbiLpoMHFG
Theory Prop Id: 51ee47d6504231b0399ea5f21fb49d3219caefd43f8b7ed5b1fa6612bbd4856e
Theory Prop Address: TMMaysrC6GCnCvik13kVigRPVk8Z3bWgCDj

ordinal_In_SNoLt

Theorem 21.198 ∀α.ordinal α → ∀β ∈ α.β < α. The proposition is identified by the following information:

Pure Prop Id: e37d00b84165adfad6c68f140a3928d30ae38bcb8bc0d296cd08e27ee667c03c
Pure Prop Address: TMKf8Fadm6Fho8o69tQZDnNutgimZmGKfvS
Theory Prop Id: ced78e7d1eec9d129092ce29e5de303b7466ca2b9aed7d07aa7b3a24c939f5c3
Theory Prop Address: TMX6tHgRu4ztsUaXjsy7UkVCDdBhk7fa6MQ

ordinal_SNoLev_max_2

Theorem 21.199

∀α.ordinal α→∀z.SNo z→SNoLev z ∈ ordsucc α→z ≤ α.

The proposition is identified by the following information:

Pure Prop Id: 72c5eb9b297f67eda46b4ae46e71bb272a5437c076df002405f197e27453a158
Pure Prop Address: TMdSZyUHH9SYckxVb8GyE6S3rr2iCAbjF1T
Theory Prop Id: c9a7312092068758770432c6a1ad0f802c09d7c8bd89c0d980c3a4006485a0bc
Theory Prop Address: TMLopX51i5ivHHzxeprUDyP4DpoUWZeRmxK

ordinal_Subq_SNoLe

Theorem 21.200 ∀αβ.ordinal α → ordinal β → α ⊆ β → α ≤ β. The proposition is identified by the following information:

Pure Prop Id: 7558ba76794eacd9296839fd46833c13e08c174ebbbd233a9e7e99572119dc6f
Pure Prop Address: TMYpoMN7TNv38QhXrPvAmv9XN6Jwf9myViH
Theory Prop Id: 15c58afcaec3772f6849ca921cc76f9d5a42f0a409f34f90856359657725bb35
Theory Prop Address: TMYJsKJKFxZReCyT6P9ehV99SK24YQvYPAK

SNo_etaE

Theorem 21.201

∀z.SNo z→∀p :o.

(∀LR.SNoCutP L R → (∀x ∈ L.SNoLev x ∈ SNoLev z) → (∀y ∈ R.SNoLev y ∈ SNoLev z) → z = SNoCut L R → p)

→p.

The proposition is identified by the following information:

21.7. SURREALRECI 271

Pure Prop Id: c2447381b74da897eaa58c3af9b1d3dde69eab20d337d33fba3e80ec78a2b254
Pure Prop Address: TMMaNjK5BsDv7w7K5kZPoSC8eHU2ZpBXnb3
Theory Prop Id: 109f1d5e9cb018c3887e73bfa9bd86c12f6e0da2cf252e4591c3433b7220949e
Theory Prop Address: TMZNw3WejhC9Mtc8ZSCkmGmtc6te5te6DBB

SNo_ind

Theorem 21.202

∀P :ι → o.
(∀LR.SNoCutP L R → (∀x ∈ L.P x) → (∀y ∈ R.P y) → P (SNoCut L R))→∀z.
SNo z→P z.

The proposition is identified by the following information:

Pure Prop Id: 5b4c39e8a208eba4acd367f82c3e27cebe26cfdc469fca07db4499164e4a2ce9
Pure Prop Address: TMbQGwDAo4c4Qz7pDSP1Qrz23DzxEXC3zuv
Theory Prop Id: 94cbe61705cabfd1ca5e0c74b060f6cbf1387230afda6d17336d57953552fc1a
Theory Prop Address: TMdM7tGeFyUKwEjhWyhtce31kgdVFHS9DFT

21.7 SurrealRecI

Let F : ι → (ι → ι) → ι be given. Definition 21.40 SNo_rec_i is the opaque object of type ι → ι identified by the following information:

Pure Object Id: be45dfaed6c479503a967f3834400c4fd18a8a567c8887787251ed89579f7be3
Pure Object Address: TMbX65FfYCsMQpKH5CQAuQZ7t7Xzon4LVUj
Theory Object Id: 230ba80e6902fe44eb4a2d0e953e841148a6b8917a639f0a1a7683bf1567f6ed
Theory Object Address: TMFnUMDQ6KoAKr1mcWtFHKKkSNoPUN4jaDz

Assume the following.

∀z.SNo z → ∀gh : ι → ι.(∀w ∈ SNoS_(SNoLev z).g w = h w) →

F z g = F z h (21.1)

SNo_rec_i_eq

Theorem 21.203

∀z.SNo z→SNo_rec_i z = F z SNo_rec_i.

The proposition is identified by the following information:

Pure Prop Id: d5ffa87a5e29f9bd7db6e5f48c1339a209c1528efb466143479de577f4c925cf
Pure Prop Address: TMFehK7LHhf71LYnf7ptxSG8U7UPEUDMtLB
Theory Prop Id: f547c8e0e249495edd7c8c51106c6527c237807df09c30ce4aae1861b7fd8a48
Theory Prop Address: TMWhuu65bzKbkqnknDmtEdbETx8pBAhrZni

272 CHAPTER 21. SURREAL NUMBERS I

21.8 SurrealRecII

Let F : ι → (ι → (ι → ι)) → (ι → ι) be given. Definition 21.41 SNo_rec_ii is the opaque object of type ι → (ι → ι) identified by the following information:

Pure Object Id: e148e62186e718374accb69cda703e3440725cca8832aed55c0b32731f7401ab
Pure Object Address: TMQRggiyUu4JmM1iVoMTTJfKCnuvGKk9Coq
Theory Object Id: c3f21860c6b33b173235815c3bb495555ca57af947caf72a4ecda2bb7534873b
Theory Object Address: TMSBXU8YNhPPW34DsxuTsKMzbSfmRAeMJrj

Assume the following.

∀z.SNo z → ∀gh : ι → (ι → ι).(∀w ∈ SNoS_(SNoLev z).g w = h w) →

F z g = F z h

(21.2)

SNo_rec_ii_eq

Theorem 21.204

∀z.SNo z→SNo_rec_ii z = F z SNo_rec_ii.

The proposition is identified by the following information:

Pure Prop Id: 39e866e1054ce982b862417ffa3743bc0e8c324d6dd60d1efde2ee400235eb8c
Pure Prop Address: TMNXn2h6mjf3y4hBzx8Mu2JUFAE9hznp2cd
Theory Prop Id: 1c5373d70e7017dcc19aae17ae9fa209e30fb1e64e38a8cf591d94eb9a8e2f3d
Theory Prop Address: TMHAFMpf4o9ixm66ZLoJTKWUsAPeGaCat6j

21.9 SurrealRec2

Let F : ι → ι → (ι → ι → ι) → ι be given. Definition 21.42 SNo_rec2 is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 7d10ab58310ebefb7f8bf63883310aa10fc2535f53bb48db513a868bc02ec028
Pure Object Address: TMJMVhuN9vEa6QPsu5PJo6CBxM3Wg6MgxXd
Theory Object Id: 9511104c67c657c2ef57d259b59e1c5cb547d2196f59208de8f81071fa6bc63e
Theory Object Address: TMaJ9EqUKsqh8zZDNMETchCY8mQJMz7evJt

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 273 Assume the following.

∀w.SNo w → ∀z.SNo z →

∀gh : ι → ι → ι.(∀x ∈ SNoS_(SNoLev w).∀y.SNo y → g x y = h x y) →

(∀y ∈ SNoS_(SNoLev z).g w y = h w y) →

F w z g = F w z h

(21.3)

SNo_rec2_eq

Theorem 21.205

∀w.SNo w→∀z.SNo z→SNo_rec2 w z = F w z SNo_rec2. The proposition is identified by the following information:

Pure Prop Id: e9ea41d7eb248cf4d9445fd415cfe25af80fb49c218714c2380e2789bcae97a8
Pure Prop Address: TMRDgSHaazow3ek3rzLN6xJFhW5ZdzoekKh
Theory Prop Id: ac9f2b2f651e85145eb6e3ce4dfeb2a9f0bd619baca63e2ceaa4a8bb59e326f5
Theory Prop Address: TMJGL7o2tFChKYPUZdXzaYEMbNA7fM43fNy

21.10 More Results about Surreal Numbers

SNo_ordinal_ind

Theorem 21.206

∀P :ι → o.(∀α.ordinal α → ∀x ∈ SNoS_α.P x)→(∀x.SNo x → P x). The proposition is identified by the following information:

Pure Prop Id: 5b75fbe4f3fb968cae8d18aaf8e357e80492153f9fe23005c02bed666cc4dfac
Pure Prop Address: TMZqP4gVUEbQmwPD8hDsBmcStaLjrwssf66
Theory Prop Id: dc2c6d6f6bb1c43d5bbae8d635e29154247bd45fc51b459b16c9487e8936aec8
Theory Prop Address: TMS5emF6ssGYWbwuHaDYXV6uwhf7edEugak

SNo_ordinal_ind2

Theorem 21.207

∀P :ι → ι → o.

(∀α.ordinal α → ∀β.ordinal β → ∀x ∈ SNoS_α.∀y ∈ SNoS_β.P x y)

→(∀xy.SNo x → SNo y → P x y).

The proposition is identified by the following information:

274 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: ba3315e237ce1d4df16c16a8bf756e2bde89de0b18851b815b96450f882c55c7
Pure Prop Address: TMJp9FY8VctR1fkJgGYxDdh1RLMeBqeS99x
Theory Prop Id: 4cd5acb49460e8a170f2661eb1292d8e3541c9317544fd5f2d82c4686e8763f0
Theory Prop Address: TMN4ezAqsQi2C1fr3tcPbKS8LNDQoV38DBY

SNo_ordinal_ind3

Theorem 21.208

∀P :ι → ι → ι → o.
(∀α.ordinal α → ∀β.ordinal β → ∀γ.ordinal γ →

∀x ∈ SNoS_α.∀y ∈ SNoS_β.∀z ∈ SNoS_γ.P x y z)

→(∀xyz.SNo x → SNo y → SNo z → P x y z).

The proposition is identified by the following information:

Pure Prop Id: fccdb6672f955c6006797333286f8360f778fdb5570223fd20d9a063212269c4
Pure Prop Address: TMHEmrGgS81dQJ2JGimKGXSByW4T5buGsKL
Theory Prop Id: dece8cbbffb58ddf8cbf8dcf83ca0a96a6ef9a5448651a114c5355e9cdb4f7d8
Theory Prop Address: TMSHTjXpVeYmg8pivxYP7m7ifgZn67S1TeJ

SNoLev_ind

Theorem 21.209

∀P :ι → o.(∀x.SNo x → (∀w ∈ SNoS_(SNoLev x).P w) → P x)→

(∀x.SNo x → P x).

The proposition is identified by the following information:

Pure Prop Id: 8b2b421134af1c1c1d16aec0622c8efee4fd7ec43176635e591cc48df421e73b
Pure Prop Address: TMdjvtesifW1vMYwLzwUFB8ubFQRxeDErX8
Theory Prop Id: 851a886ebf6c314dbc8a05caeee024ca4081f1783f55486fc9481e9aa423b8ed
Theory Prop Address: TMXG4oEhSvYEBk3fcCYLLydMW8jZPQjHCwg

SNoLev_ind2

Theorem 21.210

∀P :ι → ι → o.

(∀xy.SNo x → SNo y → (∀w ∈ SNoS_(SNoLev x).P w y) →

(∀z ∈ SNoS_(SNoLev y).P x z) →

(∀w ∈ SNoS_(SNoLev x).∀z ∈ SNoS_(SNoLev y).P w z) → P x y)

→∀xy.SNo x→SNo y→P x y.

The proposition is identified by the following information:

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 275

Pure Prop Id: 87eb9a4029d2214aeadffe89e3f00b768dc1f0ee672b65aa78beff4546554a8a
Pure Prop Address: TMPRSJVgigZNu3Ln5T8fN87EJE6rPBh6eyv
Theory Prop Id: 800d4cef7892d2b9082248440b178c0db19d92fa16e49449e7934ce2a5cffeed
Theory Prop Address: TMdh3vKvQM9UhZmdEnwz2CChQjV8xXPGZtD

SNoLev_ind3

Theorem 21.211

∀P :ι → ι → ι → o.
(∀xyz.SNo x → SNo y → SNo z →
(∀u ∈ SNoS_(SNoLev x).P u y z) →
(∀v ∈ SNoS_(SNoLev y).P x v z) →
(∀w ∈ SNoS_(SNoLev z).P x y w) →

(∀u ∈ SNoS_(SNoLev x).∀v ∈ SNoS_(SNoLev y).P u v z) → (∀u ∈ SNoS_(SNoLev x).∀w ∈ SNoS_(SNoLev z).P u y w) → (∀v ∈ SNoS_(SNoLev y).∀w ∈ SNoS_(SNoLev z).P x v w) →

(∀u ∈ SNoS_(SNoLev x).∀v ∈ SNoS_(SNoLev y).∀w ∈ SNoS_(SNoLev z).P u v w) →

P x y z)

→∀xyz.SNo x→SNo y→SNo z→P x y z.

The proposition is identified by the following information:

Pure Prop Id: f1d779fd469c0845a98e0f93f05143afba1386516af9d4603637aac985cd04d3
Pure Prop Address: TMSvcfsKu6jonEsCbEr8VWi4pk11xZ363HY
Theory Prop Id: ee16b90e02757ceeb6643978af1b3c0c4b10662feeea83cd645e3f428caa2f4e
Theory Prop Address: TMNAmAH2wAfd64peNfnqHxL8gBKfNxd1haR

SNo_1

Theorem 21.212 SNo 1. The proposition is identified by the following information:

Pure Prop Id: ac50d79d994bfab93af900919749bb0b00bab77dd02496312cb99636c84de723
Pure Prop Address: TMLD7rYjwi3XCzS9wBFoZquU2fwKZu8hJfp
Theory Prop Id: 9a733a1814c1f8aa807a0905095d4095b6b6c16bd5c1cbe8d42ff51d0476728e
Theory Prop Address: TMV4EbhEF9Fv43TrwcGbkmuL2KWwTbyEHzw

SNo_2

Theorem 21.213 SNo 2. The proposition is identified by the following information:

Pure Prop Id: 487591f799d85ccddbcd2b6f2e71c42ada69163cc874f37e171eabf04a2cf2c8
Pure Prop Address: TMLrvp9CHQ6VVsuUey7NKkjK9sekHgGeozC
Theory Prop Id: c524a421cfe10bd6c73ba8913472e276988f04c927f9d815d647884e770ef7e5
Theory Prop Address: TMR8t9mAk1WDjyns3E36LrbyZ4oDbwT7ZHW

276 CHAPTER 21. SURREAL NUMBERS I

SNo_omega

Theorem 21.214 SNo omega. The proposition is identified by the following information:

Pure Prop Id: f6584889c3b559c0c57d54079ab6b3db3900f8646787147aba128aae330d55b6
Pure Prop Address: TMNCVbGvT6eb68ZqscmNFNvebebAFmqdReB
Theory Prop Id: db7b5f8d12a3f6e404156d15406fde7ccaacb30fcba4c04ca7993e10e5bc5231
Theory Prop Address: TMGovFDv31hbwMQHyVzGRJD2AUNGQ55Jbau

SNoLt_0_1

Theorem 21.215 0 < 1. The proposition is identified by the following information:

Pure Prop Id: dd604718e43f44863c2707c951e0869b557b6fa3a70647ba47a17ec52b0029a2
Pure Prop Address: TMdgV1jBe6EnZgcQwB7jjy19eTiHyJdqRgk
Theory Prop Id: cb6392897bc04cb1365085f9f877dc153c53b66befbecc65b224bd4cc2f5ecd5
Theory Prop Address: TMJwENHJCRTJn9GM5HVohBSxeqaDYfTPidF

SNoLt_0_2

Theorem 21.216 0 < 2. The proposition is identified by the following information:

Pure Prop Id: 7be36fc38585a8b130ffed59f4b86612f5a054fb66623f2e4759235722d406ef
Pure Prop Address: TMaz8fiLqm9K7Mq8fWt23tudQikDLE3nVPq
Theory Prop Id: f469347bc0b459c89c083e9dfd7070a8dfb29072eefa324603e21760ae54f647
Theory Prop Address: TMHYveHQDjM9W8DaiRmfzZeA4rimU2E92Nx

SNoLt_1_2

Theorem 21.217 1 < 2. The proposition is identified by the following information:

Pure Prop Id: cc44e8f0924d8a19e44142d916783f27b6cc875dba07d11a95a303ffd59e0426
Pure Prop Address: TMUn8b7nVzm61CdAFH2Td6XCY5QtWtzLVFN
Theory Prop Id: 19d5e1b76af23ecf0247d684e971a9c96fb452c3323f62c2d67d14b22fa104d1
Theory Prop Address: TMQQHPaJv8GbFZhqEM1fStxij59TT2UfDi7

SNoLev_0_eq_0

Theorem 21.218 ∀x.SNo x → SNoLev x = 0 → x = 0. The proposition is identified by the following information:

Pure Prop Id: fe0bc4ac89543a28bb90f488d7033f084538a78425338f1c33f2d0f1095b8658
Pure Prop Address: TMLtxEcKxjJ93DcQ4fddYfsYH8i24duAMQB
Theory Prop Id: dfcf39481eba0eed8e498b098ea072eb9bf52e6e184da7a902a0d3cf7d28b799
Theory Prop Address: TMQQUQVuJAU1SouCAxFzxgEkDK6uVzMBbqL

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 277

restr_SNo_

Theorem 21.219

∀x.SNo x→∀α ∈SNoLev x.SNo_α (x ∩ SNoElts_α).

The proposition is identified by the following information:

Pure Prop Id: d784442182c6268660ac7e2ebd4ce1f49daf2eaccc2ca06df86cf30b983f10b3
Pure Prop Address: TMKL6VTKgr5Lj5NeDnwUjkGyX4ocPdGDcir
Theory Prop Id: e0eca8857b82373a353cb7cad078aeafe8a30a60da8a14c48961e659c679e16d
Theory Prop Address: TMXSGP5nqZiFAfXN6HwbUUCGasSdVszy7Vr

restr_SNo

Theorem 21.220

∀x.SNo x→∀α ∈SNoLev x.SNo (x ∩ SNoElts_α).

The proposition is identified by the following information:

Pure Prop Id: 930b5633114b598786190b7f3e7477c4a66a906c5e9f9eba7449686faeef6a32
Pure Prop Address: TMHnTUT8yBBnGwav9Bnqhxz5kqA5ggvvevz
Theory Prop Id: d141ededdd7718b828cf689076cf982e403f154afe9f7dc65662f9704bddf669
Theory Prop Address: TMSgjXpo7LeuFty8KqVjACKW1eSX1X9yniR

restr_SNoLev

Theorem 21.221

∀x.SNo x→∀α ∈SNoLev x.SNoLev (x ∩ SNoElts_α) = α. The proposition is identified by the following information:

Pure Prop Id: cae8b53a142dc9bd4d328e356d5ea30fde7a5099b334cda244e99187209b8b99
Pure Prop Address: TMS5WZkxehafSgkaqhUw2VJHacUTw9rwS1L
Theory Prop Id: 8b18538bf7d6bd1d9a848a9a9517a642deb5a218ec3d8a7caacd524d21dfd64a
Theory Prop Address: TMJN6jAeeeX5njqn5SeaWsGAgEsFLV3XR4u

restr_SNoEq

Theorem 21.222

∀x.SNo x→∀α ∈SNoLev x.SNoEq_α (x ∩ SNoElts_α) x. The proposition is identified by the following information:

278 CHAPTER 21. SURREAL NUMBERS I

Pure Prop Id: 829f8b1631e1372f4e65a822476f355e1c77a81a384b9cca1a4a8d552190fc4d
Pure Prop Address: TMRgDXhz8Afh4H4FduKLm7tdh5mLBCuj1VX
Theory Prop Id: 704f5aec2f423db02021a4327ff03449e11de8691265c76b45d8784294ccc34c
Theory Prop Address: TMZQtNrbPSFdXyaV4jvd6yZ7nzE191PFXRr

restr_SNo_SNoCut

Theorem 21.223

∀x.SNo x→∀α ∈SNoLev x.∀p :o.

(SNoCutP {w ∈ SNoL x|SNoLev w ∈ α} {z ∈ SNoR x|SNoLev z ∈ α} → x ∩ SNoElts_α = SNoCut {w ∈ SNoL x|SNoLev w ∈ α} {z ∈ SNoR x|SNoLev z ∈ α}

→ p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 89a30eaccf26f7f6a96eaa958a6ba761b57a0f92bbecd3515a4233b677745527
Pure Prop Address: TMEjpjmejvYuUCvTudHxbUChhzKXViX36A4
Theory Prop Id: 965ec873f3ef2f209dcc57482d884d08d8c7e0ccc428c255bb178794d09f6adc
Theory Prop Address: TMJDSz4hS22ToFR3pnjsU5VSzcSsLte1yVF

Definition 21.43 pack_e is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: dd8f2d332fef3b4d27898ab88fa5ddad0462180c8d64536ce201f5a9394f40dd
Pure Object Address: TMZWdNrTTWaDgSjUhPMeWi1cahDuEhSHGWm
Theory Object Id: 02a4a758b68524d04d4d1b1347b10db0c244e6a85b8f75e8e031fe71887a51ea
Theory Object Address: TMdVWrX6F4nw7pL1cx1YSLHmAWvAQPB5uyT

pack_e_0_eq

Theorem 21.224 ∀SX.∀c : ι.S = pack_e X c → X = S 0. The proposition is identified by the following information:

Pure Prop Id: 6ff73aba63bf83b93fdcd68f7f3b473ca425f7b7ecad18caa91ad3f3910771e8
Pure Prop Address: TMF3p8wywFFn4tbwQTETRnnWJvQHuGheapA
Theory Prop Id: 00a7dba7aeb27089ddbf38d6a79dbefbcda54788a79f6438fdd968c1c97ac5b5
Theory Prop Address: TMWi4vaXB8eNLtzU2oJt4rkxxuU1HmYd2Td

pack_e_0_eq2

Theorem 21.225 ∀X.∀c : ι.X = pack_e X c 0. The proposition is identified by the following information:

Pure Prop Id: e524616fe01439ccfbbdbe4f68ef616b835892e97fb13817028ebbf959c61f1f
Pure Prop Address: TMNJYAKDBjmHn7HXSNC26jwp1s41xqC3inP
Theory Prop Id: 5904c567fc4acc3953bc24974e3fd7987245b906799d3f6bc8ba4f7f25608950
Theory Prop Address: TMEr6GzQSymKEUjhZT6JqViiScEVyeyANgm

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 279

pack_e_1_eq

Theorem 21.226 ∀SX.∀c : ι.S = pack_e X c → c = S 1. The proposition is identified by the following information:

Pure Prop Id: 04f7f9a2df055819f0ee6e284dc3cf33cb7c8d5bca68b3910258bbbe11062c86
Pure Prop Address: TMcNmrUzTXZFjuycPrfPwZEqsoyBeYPtDUL
Theory Prop Id: a8d7da99aade4bcb64785eef8d10ca2392a20230bc7f92e5bde81d1b1470cddb
Theory Prop Address: TMdRWR7pSiRTAZ3CjpuNGMk5aFmhMjmguaF

pack_e_1_eq2

Theorem 21.227 ∀X.∀c : ι.c = pack_e X c 1. The proposition is identified by the following information:

Pure Prop Id: 9fdf3acdb0181694d27792d6b805cb1e854bc59fc0bff1362aa634222d642348
Pure Prop Address: TMMriir6BTpwadY3sQTwcJ6UtGGpeNzruFP
Theory Prop Id: 571456f6d7bd181f2b902d4b966ebc6e588e0d89fa9a4ded6ad4bc4671514eaf
Theory Prop Address: TMHqTypumR87KMZqxQykwqpxf2fLTDSKntV

pack_e_inj

Theorem 21.228

∀XX′.∀cc′.pack_e X c = pack_e X′ c′→X = X′ ∧ c = c′. The proposition is identified by the following information:

Pure Prop Id: df5adc30165a60003f242f92d1c8fe389ba6bf0ef0a6d92e54e192638323f57f
Pure Prop Address: TMbTR7tamHQov3r4vUKNrPZaM7iLEL5BaG9
Theory Prop Id: 0e751afc35195926201e850996d7021c60132bafee50cd3f7b50f532ad803b3f
Theory Prop Address: TMXLaAMy8oAeTZ5Pr75ZnuM248oWLnFMqtq

Definition 21.44 We define struct_e to be

λS.∀q : ι → o.(∀X : ι.∀c : ι.c ∈ X → q (pack_e X c)) → q S of type ι → o identified by the following information:

Pure Object Id: d335d40e85580857cc0be3657b910d5bce202e0fab6875adec93e2fc5e5e57e4
Pure Object Address: TMJWWWBc6dvdqjxZCTPUgLbN4aXWyUnq9bt
Theory Object Id: 59de6a6f7234e918742ecb64f84be2c8961660da62ee30ccf1365eab256ab3c0
Theory Object Address: TMHJ9Le7KbYPQ93t9yavSzmxfaU78ZA7CZo

pack_struct_e_I

Theorem 21.229 ∀X.∀c : ι.c ∈ X → struct_e (pack_e X c). The proposition is identified by the following information:

Pure Prop Id: 91b51f2c281e33c4a047079e824292cc1811682fb17e3669dabee45608c97de2
Pure Prop Address: TMakVN1dCuKrF2sSY6ifvjCSKdUn9w2h614
Theory Prop Id: cc54f6f84b4c1ee3a02364985ad73f8f87488877bb3adf95720686a7b855b888
Theory Prop Address: TMaSjR5JQ2VRSZZPHRxJNCLkwoNQtS9DD46

pack_struct_e_E1

Theorem 21.230 ∀X.∀c : ι.struct_e (pack_e X c) → c ∈ X. The proposition is identified by the following information:

Pure Prop Id: 59653abe494ff81cf90bdae82d53b1f9a94dfc97f7ea3cb81712c80a9135ea4c
Pure Prop Address: TMHRM45GC3aqyJTvtVy7hXGp2whYDye9QNF
Theory Prop Id: 550a75e54c585c80f080162a35f7e2478a8d8d392ae7b1a822044c132b01dd74
Theory Prop Address: TMXc4Tt9w6ou7568GQpZHWHDita8JYAQUaJ

struct_e_eta

Theorem 21.231 ∀S.struct_e S → S = pack_e (S 0) (S 1). The proposition is identified by the following information:

Pure Prop Id: fa919b3231deef7d4ef4a92653ceff69c800f9aa38bb03d41f4866f19342040d
Pure Prop Address: TMUMVefsUP57yd7HoBLRFkwtdmzbuWgpQ5A
Theory Prop Id: bdc4a9dce3edc9c64967e889aac3b081b436d30d2696fe495c04711a89f46239
Theory Prop Address: TMbpmGPq3LT3TFSbSD5SYAp9JKZX2TbM5cp

Definition 21.45 unpack_e_i is the opaque object of type ι → (ι → ι → ι) → ι identified by the following information:

Pure Object Id: 91fff70be2f95d6e5e3fe9071cbd57d006bdee7d805b0049eefb19947909cc4e
Pure Object Address: TMHRZBuSbhYF1JLr1Hs3TPKTCSk5Cr8Gnys
Theory Object Id: fe41183ac024053e002b76da086e29818c4cac3a9cbc0104c1287b011dd6db98
Theory Object Address: TMWRuKPw9UaH5vAfF8rtNPUaJrQyP5RPDCu

unpack_e_i_eq

Theorem 21.232

∀Φ :ι → ι → ι.∀X.∀c :ι.unpack_e_i (pack_e X c) Φ = Φ X c.

The proposition is identified by the following information:

Pure Prop Id: 0a69b9763b0d02450063b2cb0f7352cf1dd65f6d2949e2ca0b1eec691efca645
Pure Prop Address: TMXmwRkpi5UiyzYAcTTBq6WvNqKhaF2yCvj
Theory Prop Id: 98c7fcde5290487eba2c62e6af0c27ecf0bc59143da5bd14fca21628d4bed0ee
Theory Prop Address: TMacpHNzxrWLsqUVSD652GArvNuF3vDpqCP

Definition 21.46 unpack_e_o is the opaque object of type ι → (ι → ι → o) → o identified by the following information:

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 281

Pure Object Id: 8e748578991012f665a61609fe4f951aa5e3791f69c71f5a551e29e39855416c
Pure Object Address: TMPPKW3r2wphDdHkkrGqms7psPywvYzk8Zv
Theory Object Id: c263bd473f626626582658a793421c02e2c9f3611ce4c4c4ca819f56cc256993
Theory Object Address: TMUEDqJEoZmYehm2AX3Mv7koPW8daCwQirK

unpack_e_o_eq

Theorem 21.233

∀Φ :ι → ι → o.∀X.∀c :ι.unpack_e_o (pack_e X c) Φ = Φ X c. The proposition is identified by the following information:

Pure Prop Id: 77684bc447f76184367525d0a412859ca7b0df8f1ddaa773da8fd0cf59b16bd3
Pure Prop Address: TMJs57aPr7U7dVBfH8ahQ3aD8tBBNA1oDyb
Theory Prop Id: b57b69660e83e8ccaf201f80e31bf02a3b6acbbac6d5fee57edfebc26325a9eb
Theory Prop Address: TMcntFFtgaj5E7fiL6PwsfE3P6zHsyUTdwA

Definition 21.47 pack_u is the opaque object of type ι → (ι → ι) → ι identified by the following information:

Pure Object Id: 119d13725af3373dd508f147836c2eff5ff5acf92a1074ad6c114b181ea8a933
Pure Object Address: TMZ2wxaB4wuhubUPHPk1gaizEbWbJzhsYbT
Theory Object Id: 3c7a5062deb09ce886a722a6a9cf08f879843995de990dfc0630269e3b8b19fa
Theory Object Address: TMbVCB5AE2QsSb8QjhV8k61yAVcNXc6gybs

pack_u_0_eq

Theorem 21.234 ∀SX.∀F : ι → ι.S = pack_u X F → X = S 0. The proposition is identified by the following information:

Pure Prop Id: e90120fb66c6e33b66df727e70deffca1cecf1c815f69ee141bf89989b70edd3
Pure Prop Address: TMGVK5PJBrEie7g7JjvK6x8kgyoW4ZFtpwu
Theory Prop Id: 699268c2f39933c4fa055b801b3afb571c28045dae8bb5a6bbf0863f15e50660
Theory Prop Address: TMUSVbwK29VTMzLPbxR5gXuNs6fXBzN4CnG

pack_u_0_eq2

Theorem 21.235 ∀X.∀F : ι → ι.X = pack_u X F 0. The proposition is identified by the following information:

Pure Prop Id: 079fb39255923eff39682a3718d622c8a40d34674fdf95cc10f606920887dd81
Pure Prop Address: TMKXKSi4pZddPxmhpCtcE6jh66s6fafGkNB
Theory Prop Id: 7eff940d0a6d0f8a81104b51146a0c8a93448c75c898a3382cc9544744401e18
Theory Prop Address: TMLLQY93heT4wwbJsE7WPX6bYakekk5LE6a

pack_u_1_eq

282 CHAPTER 21. SURREAL NUMBERS I Theorem 21.236

∀SX.∀F :ι → ι.S = pack_u X F →∀x ∈X.F x = decode_u (S 1) x. The proposition is identified by the following information:

Pure Prop Id: 53d04c6be0df4b0a8a73f3e7962b6fd76f97d1b0e8aeb6753d8f2eceb28170d7
Pure Prop Address: TMRRG8USJ8fkv5kE7Bjm2NXEz8mHWTTU7BF
Theory Prop Id: 9e80d0e685e0714450401020d0dda11302329314e7b5acad3049662aa8eb2796
Theory Prop Address: TMGeytgzBv7rLRFtZPQUVZVT21LKHsmGiHi

pack_u_1_eq2

Theorem 21.237

∀X.∀F :ι → ι.∀x ∈X.F x = decode_u (pack_u X F 1) x. The proposition is identified by the following information:

Pure Prop Id: 7fca7f65973aa80d2bc3a1cb75163816f61e4d6d2dc4f5765d6e297e96abdae4
Pure Prop Address: TMF37qTtWa8UwmQcuk4KkUAWcW9y1oiAjSP
Theory Prop Id: c23d46052a337e70be0e9c4349faf01e97c334cd076ac5bb2026a88115fd2db1
Theory Prop Address: TMdriRWWtDBheXF7Muhs1C7R3CgVDTDwMUM

pack_u_inj

Theorem 21.238 ∀XX′.∀F F′ :ι → ι.pack_u X F = pack_u X′ F′→X = X′ ∧ ∀x ∈ X.F x = F′ x. The proposition is identified by the following information:

Pure Prop Id: 6b8b57fe8b77ff40918b3007f4711e989111aaafa13fccea532be661542391c6
Pure Prop Address: TMF2hVeN68NYWiNVPfgquFhA14Cd2WfFA7h
Theory Prop Id: 6db3de9d9bc5bed83a7ab61a9d1445e0d1cca2c76182e965cc3d32fa08f1982c
Theory Prop Address: TMP5sxvmCFkYyNJUa6v8DPY99xURLaEs34f

pack_u_ext

Theorem 21.239

∀X.∀F F′ :ι → ι.(∀x ∈ X.F x = F′ x)→pack_u X F = pack_u X F′. The proposition is identified by the following information:

Pure Prop Id: 1aa49317bb1aafb7249631676368bd94610841b4a1c159800bc86f2708d6bf6d
Pure Prop Address: TMTX8evwr1dHzX2eGgpZhxbk3s1kai52w5w
Theory Prop Id: 981b39139ef57621b1bf8329aab2dea9fd20d3e11c2d121f340018c1b8ec9ada
Theory Prop Address: TMVTvfHNDMHTzwo3D4fzu5ZXxnQc8m1VEZQ

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 283 Definition 21.48 We define struct_u to be λS.∀q : ι → o.(∀X.∀F : ι → ι.(∀x ∈ X.F x ∈ X) → q (pack_u X F)) → q S of type ι → o identified by the following information:

Pure Object Id: df14296f07f39c656a6467de607eb3ced299a130355447584c264061b5275b93
Pure Object Address: TML4zt5QzeK1oosb9zeqYtcvb5j6UrWUQ83
Theory Object Id: df0642555ff24c4e0093386810bb87b51992acb9f16b14c5ddacf37ee589c227
Theory Object Address: TMcCXGRFwbxwpFTxpWqSgdFmjnxeAe1DkZn

pack_struct_u_I

Theorem 21.240

∀X.∀F :ι → ι.(∀x ∈ X.F x ∈ X)→struct_u (pack_u X F). The proposition is identified by the following information:

Pure Prop Id: 9ac1bfd471c8e0c07b3a8b8d8489ca22d34c772c104098df03fdc0f478c661f1
Pure Prop Address: TMaiejfiKboUsvt1vPnNj7r4ZdJKsYHEBm9
Theory Prop Id: 9556db5aef8e77543f91bc07bf376517d9a6dee8137c27d45a1b83aa366667f6
Theory Prop Address: TMKaRisyF4e8WBZv1N8ynvGUUuSkJVtA9ad

pack_struct_u_E1

Theorem 21.241

∀X.∀F :ι → ι.struct_u (pack_u X F)→∀x ∈X.F x ∈ X.

The proposition is identified by the following information:

Pure Prop Id: 98bace69f4799a59fdccc70ae5066e40837f3561cd06571f111486811be16df0
Pure Prop Address: TMdoPRYZsVB9wjKk8Ri2fsV5T5ZW37aXDb2
Theory Prop Id: 7db00f022f515797b4fb0d821fe7489995399358fa823f779c3b601870d2a4d9
Theory Prop Address: TMPmBmMcF9nNqTcrtPRKZjJNYvhvAYDxtgc

struct_u_eta

Theorem 21.242

∀S.struct_u S→S = pack_u (S 0) (decode_u (S 1)).

The proposition is identified by the following information:

Pure Prop Id: 0e0fc8f8e80ea783e6affd9347bd4f6d0900540ce0323735068c9fd37da332b6
Pure Prop Address: TMbYJxMnDh5AP5uka6PSR5qdqZPbtHMnZT3
Theory Prop Id: 558efaeb63c323d6b2e00428a406b8ff6f02d42ae11b33350ebd4d63f3b15f18
Theory Prop Address: TMZbzGyYczQAU2uwNtWM8TsVekdsMpRhw5A

284 CHAPTER 21. SURREAL NUMBERS I Definition 21.49 unpack_u_i is the opaque object of type

ι → (ι → (ι → ι) → ι) → ι

identified by the following information:

Pure Object Id: 111dc52d4fe7026b5a4da1edbfb7867d219362a0e5da0682d7a644a362d95997
Pure Object Address: TMJ9S9HN4bipGWyfsvMApvMV12tYYa7YBTd
Theory Object Id: 9b53a2081c6da4b6da58cf2e2cd87b51c01cb3aad8d0198d435a1a60d5dae91a
Theory Object Address: TMb9dRUCqQE14noMaQR9cGoqGSrgHMch1D6

unpack_u_i_eq

Theorem 21.243

∀Φ :ι → (ι → ι) → ι.∀X.∀F :ι → ι.

(∀F′ : ι → ι.(∀x ∈ X.F x = F′ x) → Φ X F′ = Φ X F)→

unpack_u_i (pack_u X F) Φ = Φ X F .

The proposition is identified by the following information:

Pure Prop Id: 8381e9777cbe016dda8f9bcd84257b15371d3bdd4e26a59a561cc83844b28df2
Pure Prop Address: TMGPuFBSKYSnC5Db2gcefmM9REhBKV3ciSg
Theory Prop Id: 133f42e59e5b3675fa1a64ca68c346da0148a3bcb8db86ae506fe34387ed80cc
Theory Prop Address: TMMw9upsAnfEWn83orKSgX3D13dW8cXKsJ1

Definition 21.50 unpack_u_o is the opaque object of type

ι → (ι → (ι → ι) → o) → o

identified by the following information:

Pure Object Id: eb32c2161bb5020efad8203cd45aa4738a4908823619b55963cc2fd1f9830098
Pure Object Address: TMHXEyss3bgpR5WZwP9nbSJUGVKgpx8hR7D
Theory Object Id: 8b490cedc3faf1e839d758df391095fb43853467123acd68380ad2bbd213e5d7
Theory Object Address: TMbyJnusRWuvLh37RbmVLuA1HeDayMwKt84

unpack_u_o_eq

Theorem 21.244

∀Φ :ι → (ι → ι) → o.∀X.∀F :ι → ι.

(∀F′ : ι → ι.(∀x ∈ X.F x = F′ x) → Φ X F′ = Φ X F)→

unpack_u_o (pack_u X F) Φ = Φ X F .

The proposition is identified by the following information:

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 285

Pure Prop Id: a862c11904e8a98362cb2ca4a94b3e16aea77824a968c54d67fdef41dbf14208
Pure Prop Address: TMbGK4qZjEcqzu2hydjMJie8FfBg6EAAg2S
Theory Prop Id: 3daff9f51b582e2a164301f59f8428029a2d119064e12ffa327165a9612fbdfa
Theory Prop Address: TMJJCNJVx3YKK8m4XGTUTRRXe5o7Hf4aW9V

Definition 21.51 pack_b is the opaque object of type ι → (ι → ι → ι) → ι identified by the following information:

Pure Object Id: 855387af88aad68b5f942a3a97029fcd79fe0a4e781cdf546d058297f8a483ce
Pure Object Address: TMMVdjfJFDf8CbzqwNoQ3mUFc4V83Wt6E2j
Theory Object Id: 965d60e636461e17d0b088d4ef59f38d71f00297a7cae99d4a96cffeca6ece15
Theory Object Address: TMYH1EdF9fP7KtMSudvx2S1z6Dg93V2gFPZ

pack_b_0_eq

Theorem 21.245 ∀SX.∀F : ι → ι → ι.S = pack_b X F → X = S 0. The proposition is identified by the following information:

Pure Prop Id: 88fe38e1e3d1bdd87d28927800e97536c017c5b945ed6ee12835147827096678
Pure Prop Address: TMPXf5qRUESrQFHe3eEvNw8NfS5DMZgoyEd
Theory Prop Id: 71d9298b5c2a83f3a1c602ceb0f739f9dcf08025e28432169e400a5bcb826cf0
Theory Prop Address: TMPunAXGyfUGF19jJ33E8oNYMXCDFxt7kMg

pack_b_0_eq2

Theorem 21.246 ∀X.∀F : ι → ι → ι.X = pack_b X F 0. The proposition is identified by the following information:

Pure Prop Id: 149dd8442141d0c1c06226a4b7125eda7c7a5d109dbeccc3ea0c8d1839f3838d
Pure Prop Address: TMcbsnvdHZJbA47etXzNDVwQbBnkePS3ghy
Theory Prop Id: 9bf81b992e14b803ddd19aefb763a8770611c9c5fa17470effe652aab3c9183e
Theory Prop Address: TMUryUSP1YfaUUeoeisdwCd4DRwdF8qmqUb

pack_b_1_eq

Theorem 21.247

∀SX.∀F :ι → ι → ι.S = pack_b X F →∀xy ∈X.F x y = decode_b (S 1) x y.

The proposition is identified by the following information:

Pure Prop Id: d3a61aaa2c6fa85326d0d1fd27c8f573ecc708ec4d76ad1a1abe5e6bbbd6abd3
Pure Prop Address: TMdzKNDCM8ocaiRdtLkRwCDFiqddCD9st6w
Theory Prop Id: 21712e8eea1a4e0f13bd9f8e8190b72c53104fcbeadbebef78accf2b3b33b851
Theory Prop Address: TMS4NWDzzDuvWpiWKuvFVqLtaZ3ZcX8WtUc

pack_b_1_eq2

286 CHAPTER 21. SURREAL NUMBERS I Theorem 21.248

∀X.∀F :ι → ι → ι.∀xy ∈X.F x y = decode_b (pack_b X F 1) x y. The proposition is identified by the following information:

Pure Prop Id: 454eade5d6916e25226236256e11334f175b7f7cd644b120cdad3d7dd5c5e6e9
Pure Prop Address: TMWdWKh8rp339UeC9sDaWqD8eAVpLX7Z5iS
Theory Prop Id: d1a9a05acd0ffe92475f293616460f7d80da98dc26534cad8a38968507a1ae00
Theory Prop Address: TMLZ8co6rk5aziSxSmJeTMBZPFjrDuvVx1w

pack_b_inj

Theorem 21.249

∀XX′.∀F F′ :ι → ι → ι.pack_b X F = pack_b X′ F′→

X = X′ ∧ ∀xy ∈ X.F x y = F′ x y.

The proposition is identified by the following information:

Pure Prop Id: 4ac18a20f5c60c34580cd550af2aff402c9f508b648a114d3e5e2ea05ef2375a
Pure Prop Address: TMT3PAPfzsJZYdkFmXBPKRriTJ16RQbe8yt
Theory Prop Id: e7c2d83ccf845c0d90123e0d147039f6eba9174a7529219510a7421b37427b1f
Theory Prop Address: TMXz4wfFqKWJRm9mNE9tXUB4V3tWeTeWyYk

pack_b_ext

Theorem 21.250

∀X.∀F F′ :ι → ι → ι.(∀xy ∈ X.F x y = F′ x y)→

pack_b X F = pack_b X F′.

The proposition is identified by the following information:

Pure Prop Id: 6d6939a2b043243248ceb1ea23d9b40ec6bc8fb261736d1831d9e10f23783a31
Pure Prop Address: TMaHYCJSjfZiY7qjR4iN2RGbjhwh6EVLJYN
Theory Prop Id: 3ac95dba16cba3dbbc90bc1fca2e869e6936625f308470e76d356473cba7627a
Theory Prop Address: TMK2351yJNApHnAS7155Cuqed26ESErCWJa

Definition 21.52 We define struct_b to be λS.∀q : ι → o.(∀X : ι.∀F : ι → ι → ι.(∀xy ∈ X.F x y ∈ X) → q (pack_b X F)) → q S of type ι → o identified by the following information:

Pure Object Id: 54dbd3a1c5d7d09ab77e3b5f4c62ce5467ada0cf451d47ee29b7359cc44017cc
Pure Object Address: TMEse5C6BhcvrdqN1kLCmCVchTMXrGpcjTH
Theory Object Id: ec21dab5d2ad24a500e876b5370a93fef45587d8ab5c00a38135c1c00796c9eb
Theory Object Address: TMdmavgqTvcPWn1s1jXYNzYjU9nxbzEFY2H

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 287

pack_struct_b_I

Theorem 21.251

∀X.∀F :ι → ι → ι.(∀xy ∈ X.F x y ∈ X)→struct_b (pack_b X F). The proposition is identified by the following information:

Pure Prop Id: 83f82c2d40c9edc96f854042ffdcba40bb79fc137957f7a157a7a521b94a3b29
Pure Prop Address: TMcH7M63CKv6BPcLYnrSwWU8dgENp9N8WyC
Theory Prop Id: 0ca86fc209d99165c44f41c8c4242bd06e0bf4fe4ba50f5451a78c59d396264d
Theory Prop Address: TMLzvnmLiuZxd4WpcgtmT4HzkYeu9oJmPW3

pack_struct_b_E1

Theorem 21.252

∀X.∀F :ι → ι → ι.struct_b (pack_b X F)→∀xy ∈X.F x y ∈ X. The proposition is identified by the following information:

Pure Prop Id: e76b711f4da2dc7324a0ca8bbaa3e9a0a1776119d6fc68ddc487c19f1b3b6ef8
Pure Prop Address: TMUep5KWt973mJbiAFReu4N3fD7JzxVBrgG
Theory Prop Id: a2f525ee0bd09ac467ed896b71a426bb93c2628f11efb722b69cb07e276d724b
Theory Prop Address: TMHG4dTesLSYiYcwpuDT1tcpdD6a4fPzy5F

struct_b_eta

Theorem 21.253

∀S.struct_b S→S = pack_b (S 0) (decode_b (S 1)).

The proposition is identified by the following information:

Pure Prop Id: 536ed4571cfde24c5f0dc64b9e9ffd2b657f23afd3f82086d621030805fb5114
Pure Prop Address: TMPEVs55RYKYcnco5aHg1EiBp9jRoaqAA6G
Theory Prop Id: 575bcffb4afcd6c37db437a9a89df09d1b9daa8bc2638184256049e29841eeb2
Theory Prop Address: TMSYCVHcDX5UQYtD2RKHK5UWHY2zE6APwEV

Definition 21.53 unpack_b_i is the opaque object of type

ι → (ι → (ι → ι → ι) → ι) → ι

identified by the following information:

Pure Object Id: b3bb92bcc166500c6f6465bf41e67a407d3435b2ce2ac6763fb02fac8e30640e
Pure Object Address: TMNZmh6j71sWmZmpeCQLCVGUVXxgvXyatB5
Theory Object Id: b47c344128564d38b19aee94b230561e50981b9bc34aeabcd1027b52d36240d4
Theory Object Address: TMTJ1tRG35mQbCDWChzhKgGESVZZrLHceGP

288 CHAPTER 21. SURREAL NUMBERS I

unpack_b_i_eq

Theorem 21.254

∀Φ :ι → (ι → ι → ι) → ι.∀X.∀F :ι → ι → ι.

(∀F′ : ι → ι → ι.(∀xy ∈ X.F x y = F′ x y) → Φ X F′ = Φ X F)→

unpack_b_i (pack_b X F) Φ = Φ X F .

The proposition is identified by the following information:

Pure Prop Id: 0aa7882234fc6912b6b3e4f22d40592682f0ceb44c55fca9b6028e9957ed26ba
Pure Prop Address: TMPfkFXN7gTRe1Y9msSvdvswaFPKLKJsqsF
Theory Prop Id: 659ca7a7505887c37bfdaad5fbda8fffb6e2e6f602ffd247af88305eb1f4a862
Theory Prop Address: TMNHSFWaR67WngH7FgbwYKUvfmTNTF5riaM

Definition 21.54 unpack_b_o is the opaque object of type

ι → (ι → (ι → ι → ι) → o) → o

identified by the following information:

Pure Object Id: 0601c1c35ff2c84ae36acdecfae98d553546d98a227f007481baf6b962cc1dc8
Pure Object Address: TMFT42eJTKM7rrEafjCbY27VvguwAhgSpBS
Theory Object Id: bea45508f7d246468c085f3f7525bacaaeae7831061f927ae066064ce4ebadd6
Theory Object Address: TMcFFPrXTWYs7rZkur73nsizTsDbGiwvHnj

unpack_b_o_eq

Theorem 21.255

∀Φ :ι → (ι → ι → ι) → o.∀X.∀F :ι → ι → ι.

(∀F′ : ι → ι → ι.(∀xy ∈ X.F x y = F′ x y) → Φ X F′ = Φ X F)→

unpack_b_o (pack_b X F) Φ = Φ X F .

The proposition is identified by the following information:

Pure Prop Id: 36249217e9f4af2de97729e96889f2f46a4aee8abae10f4a7b6eeac7725e8565
Pure Prop Address: TMVcr4vjHzNUUr7gkxQ7RQc49JkgGGy3YGA
Theory Prop Id: d10e77888bbcf1e2f90eec0314e98757bb89a0d253b83086addf858363f79c3b
Theory Prop Address: TMXR7UKpoi2iubM8VrTd9gbFCVWHtV2vx6V

Definition 21.55 pack_p is the opaque object of type ι → (ι → o) → ι identified by the following information:

Pure Object Id: d5afae717eff5e7035dc846b590d96bd5a7727284f8ff94e161398c148ab897f
Pure Object Address: TMZKq2rBy4uupXawQMQLzwHsbyNXwKbrHtK
Theory Object Id: 327f9f7a1cd5c793e02cabe2c78bd943eebf7e17640b5569cb69b7c6e867d288
Theory Object Address: TMaXW2e5TX9atqycshV6aWebjBHBw3qBaDQ

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 289

pack_p_0_eq

Theorem 21.256 ∀SX.∀P : ι → o.S = pack_p X P → X = S 0. The proposition is identified by the following information:

Pure Prop Id: 3e2bc1d01adbc1e08fb9a40b80d1078f5686a19087b5dc6a4a28f13ce1ed245d
Pure Prop Address: TMcoBzsTRTrztFDNo8R9KfFyzrSSNLQi8J1
Theory Prop Id: ab13889079587ea5409a49e8143a5bc1d13215ca88c52957afee0795cb787e1c
Theory Prop Address: TMLxrx1gZmbDxCm2cf8kvHNCEjJYYTKG1cH

pack_p_0_eq2

Theorem 21.257 ∀X.∀P : ι → o.X = pack_p X P 0. The proposition is identified by the following information:

Pure Prop Id: 11696272b13bb134c1f814d0377d8f961c2444ec3276903b6197cb8b5c617a6e
Pure Prop Address: TMRPLYPYL8Gsv7Hv3ycXTxChYbDMhuZEaAd
Theory Prop Id: fac8f41e2043963665d5308100a90c3cd449fe7866184fd601f0987c8846c9b8
Theory Prop Address: TMWKukhfMDzMZttRystq1Qgu6vLp3c8H2Bp

pack_p_1_eq

Theorem 21.258

∀SX.∀P :ι → o.S = pack_p X P →∀x ∈X.P x = decode_p (S 1) x. The proposition is identified by the following information:

Pure Prop Id: e77d74a3d40ec36d1d39189b6f349ea6192ca304bb30166c7982e27ab5f4463d
Pure Prop Address: TMZx1CjFBL7v9V4kJAPoMUFc6MQWeEFyAxE
Theory Prop Id: 8d778314b90e2c5d3ca7e90a948c37b0116287bf590445df492647685304f11c
Theory Prop Address: TMSC2Fhr34MeQfa9SHEFBMjJpxN3mr5w7x6

pack_p_1_eq2

Theorem 21.259

∀X.∀P :ι → o.∀x ∈X.P x = decode_p (pack_p X P 1) x. The proposition is identified by the following information:

Pure Prop Id: edd77250b9082459c51c2404779121d28df39720c403c1ba615bdb98cea24155
Pure Prop Address: TMZkKdo5noTu4SrYs3Y9ypPqsY1t9ZBbkQj
Theory Prop Id: 675cc21279443bb463c127734eaa6a938e37ac42884e7189535cb2f3f9554e3f
Theory Prop Address: TMS3zfU1RcgvXFsbKoXKLfUi5bvRNheLzFV

pack_p_inj

290 CHAPTER 21. SURREAL NUMBERS I Theorem 21.260 ∀XX′.∀P P′ :ι → o.pack_p X P = pack_p X′ P′→X = X′ ∧ ∀x ∈ X.P x = P′ x. The proposition is identified by the following information:

Pure Prop Id: 447155bb83c20b64c9aa17c73431b30e27f1182f50ed0138f70ee0f5dd7546a7
Pure Prop Address: TMdpN21GTqfTxNJNdjzawg4vyzWqRoBABY4
Theory Prop Id: 123b7446fa7f51381b466df69493c943b07297a406a23d2837d853425f1b81c5
Theory Prop Address: TMZezSHn85gE3VL3kzSaUAnpeLhgzoznQxZ

pack_p_ext

Theorem 21.261

∀X.∀P P′ :ι → o.(∀x ∈ X.P x ⇔ P′ x)→pack_p X P = pack_p X P′. The proposition is identified by the following information:

Pure Prop Id: e20088be64022ef4177b25ae43ecbe098bcd4ec9889eafa67356e50f074d9b3c
Pure Prop Address: TMQHBKHTocgt95FDHHgmVG6terquvvMXAX7
Theory Prop Id: 6327591dd89a183819d2315a107277c5ff966837b4658a1900a8927c37586a95
Theory Prop Address: TMLRUuJpnLfZbAuashKkp6abZVYuyhErHms

Definition 21.56 We define struct_p to be

λS.∀q : ι → o.(∀X : ι.∀P : ι → o.q (pack_p X P)) → q S of type ι → o identified by the following information:

Pure Object Id: b927035076bdb3f2b856017733102a46ad6a0c52f1f4808b6c0cc6bde29328b6
Pure Object Address: TMRQShbXQEMogWKiAvjj4S23VY1PBpfLExm
Theory Object Id: 5b2817559944fc0fdd6a8750018508b650f31cb6af63d767ded0c8472309f282
Theory Object Address: TMHTvYZ6KkSUTR8CfNH2bmXGhb7SHNNrJWd

pack_struct_p_I

Theorem 21.262 ∀X.∀P : ι → o.struct_p (pack_p X P). The proposition is identified by the following information:

Pure Prop Id: cb62efe4da65c84588ce56c8e842510106a18f82c93832ddb1e2a2acbd57adcf
Pure Prop Address: TMUKcJ75mxMagfVFQFZmrB8tJdXNnpRBhMM
Theory Prop Id: d2435fc55b93c69cbafe0cf9620c74888bf5617bb7d8f276f90ee91956e13a67
Theory Prop Address: TMFUi9YSFkPJLd7uyyf4idZheyd2iSc8y4U

struct_p_eta

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 291 Theorem 21.263

∀S.struct_p S→S = pack_p (S 0) (decode_p (S 1)).

The proposition is identified by the following information:

Pure Prop Id: 28d313b94faf78be69428bf15c4bdc56ac3b3ede6ea32a5d46da14eb79f02556
Pure Prop Address: TMGg8n9XFwykSXD4FQ2Q9USayRvoceEmnsL
Theory Prop Id: b1c3ea7aa5b4e91e8d35f1c894d953beb735090d5a08ce1508df140c7731f22a
Theory Prop Address: TMGh2mx2CkQeB1Xsb3gcjYRXiW8X2Wmjik9

Definition 21.57 unpack_p_i is the opaque object of type

ι → (ι → (ι → o) → ι) → ι

identified by the following information:

Pure Object Id: dda44134499f98b412d13481678ca2262d520390676ab6b40e0cb0e0768412f6
Pure Object Address: TMd8rM7X2eKWiHzb3iCj8EMQSAmUxWqsTAS
Theory Object Id: 02aa6a6b1d35a6d31321d714052e48ac29520943baf0867d4b7c81047c04b1df
Theory Object Address: TMUMBD9eJTd7hghvMZ9kDCBBszGxHy9JpsM

unpack_p_i_eq

Theorem 21.264

∀Φ :ι → (ι → o) → ι.∀X.∀P :ι → o.

(∀P′ : ι → o.(∀x ∈ X.P x ⇔ P′ x) → Φ X P′ = Φ X P)→

unpack_p_i (pack_p X P) Φ = Φ X P .

The proposition is identified by the following information:

Pure Prop Id: 320b78af7b08593888681ec40a36b6b9beffa499779c5c98d98c054e07741e79
Pure Prop Address: TMLCGUfRCP9vQnp7GJB41cjD4Vrux4buaBB
Theory Prop Id: 2653932c0e7cc4c5374e965bdab745a870e265ec1529e1be48b856200430735c
Theory Prop Address: TMUXbW94tZQYgwiXWFCbc4aLdmWdtpXkDaR

Definition 21.58 unpack_p_o is the opaque object of type

ι → (ι → (ι → o) → o) → o

identified by the following information:

Pure Object Id: 30f11fc88aca72af37fd2a4235aa22f28a552bbc44f1fb01f4edf7f2b7e376ac
Pure Object Address: TMWT49Ai1jUqLkomFLu5DJZESHDY57rpkpp
Theory Object Id: d42ffe55ca758d1a2808e2bf09432336b6012dd98176642489f5b352163c839b
Theory Object Address: TML2ogMCitPSn4cnkwyxFywgE1nHMrQtk2b

unpack_p_o_eq

Theorem 21.265

∀Φ :ι → (ι → o) → o.∀X.∀P :ι → o.

(∀P′ : ι → o.(∀x ∈ X.P x ⇔ P′ x) → Φ X P′ = Φ X P)→

unpack_p_o (pack_p X P) Φ = Φ X P .

The proposition is identified by the following information:

Pure Prop Id: ff2e0a66eb757950b800c844729a119d800324b515bcf1715aa55d8b539d2bc7
Pure Prop Address: TMLWdSrJohgGH4MnsHuPzU99KXhMztfBiB8
Theory Prop Id: 91b507a8afecc71b5b3acab9f63a3baddd24e6d5cfff2801c72bf2ba365cebf8
Theory Prop Address: TMa2dnR7HWNn2drguqdGnbBgzCEo6PraPi3

Definition 21.59 pack_r is the opaque object of type ι → (ι → ι → o) → ι identified by the following information:

Pure Object Id: 217a7f92acc207b15961c90039aa929f0bd5d9212f66ce5595c3af1dd8b9737e
Pure Object Address: TMUctEexwaiGRFvuiMsx3Luij4T9hycQFR1
Theory Object Id: 653348825eafd6be52649491cbf9da0b90728ef636545f2fb600630bb8ede9d7
Theory Object Address: TMN3CVcVBoGZd48wWhbqzV4YvsURA2XpsYf

pack_r_0_eq

Theorem 21.266 ∀SX.∀R : ι → ι → o.S = pack_r X R → X = S 0. The proposition is identified by the following information:

Pure Prop Id: 1a61666ba8ed6054918adb678e03f4f0f4e52214f5fadb9c9f34e65f12236f35
Pure Prop Address: TMPmyRQDsArPocUGo2q8pp2H9XidTrdzpim
Theory Prop Id: aa16a193ecac4ece1fe16115e1798004924cd7546cb19979d6a048737c8c576c
Theory Prop Address: TMNVChZMMuxdfe2iosJrMwXRxkaeUcZ4kvc

pack_r_0_eq2

Theorem 21.267 ∀X.∀R : ι → ι → o.X = pack_r X R 0. The proposition is identified by the following information:

Pure Prop Id: e9f3791f72bbaf693f3addf117ed97ffe7d9c8870838c07f30d122aca64e046a
Pure Prop Address: TMYru4HRQErL3eWLnb1NqojeVchvXn9Xjn6
Theory Prop Id: 09249bf17addc69ba6832be8e5f9b66a70f21059e88e762e548b6a234e305fba
Theory Prop Address: TMWSrqYbCVb7RWvvsJJK5qwBoNom14geqRG

pack_r_1_eq

Theorem 21.268

∀SX.∀R :ι → ι → o.S = pack_r X R→∀xy ∈X.R x y = decode_r (S 1) x y.

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 293

Pure Prop Id: ae868a8d8478c46e95f6c4e452385c0ab754ed15d225e4264fb7670471ce9a56
Pure Prop Address: TMPd9JAAJq5xDzmCczLyPedvrgGVdat5jz5
Theory Prop Id: 0bcee669141558ba17e2cbffb4dad25f53e895091c050e4ba640aade835538ec
Theory Prop Address: TMP9bEFP2SXcQRXBRBqazJ4BAu3Ltzc4nFL

pack_r_1_eq2

Theorem 21.269

∀X.∀R :ι → ι → o.∀xy ∈X.R x y = decode_r (pack_r X R 1) x y. The proposition is identified by the following information:

Pure Prop Id: 673e8ce75e264e65efac5d4112b07bf2f956aba214e048e5976a0c7d4f26c86b
Pure Prop Address: TMJsAy3ztkMuG5mtkuA7BGdKzRUaKs2tppt
Theory Prop Id: bf564258305254769de120596e1680500b51275da445c8e6331abbea10f5cc33
Theory Prop Address: TMKdqYogPTdzaPhkJNLJz4RJYuFFHgZ97g5

pack_r_inj

Theorem 21.270

∀XX′.∀RR′ :ι → ι → o.pack_r X R = pack_r X′ R′→

X = X′ ∧ ∀xy ∈ X.R x y = R′ x y.

The proposition is identified by the following information:

Pure Prop Id: 7f2df97fb2f06e6818243ca4aa3a49505e174015422b4165e4c0e85797545d17
Pure Prop Address: TMYtfZhkJBGMfEfGyEJ4nLXSF6xrqbv7xVn
Theory Prop Id: 3cd95991edbbb560c316012eca8622649a1b54474a30700935008c0e6f36b321
Theory Prop Address: TMLB6JT8F1NtoYVen4ywitVmvi85JmSkvXH

pack_r_ext

Theorem 21.271

∀X.∀RR′ :ι → ι → o.(∀xy ∈ X.R x y ⇔ R′ x y)→

pack_r X R = pack_r X R′.

The proposition is identified by the following information:

Pure Prop Id: 2f6a43369281b532971499e36bcc96d546582f450723f54cdd7373e9206c145e
Pure Prop Address: TMFKB6PjAimGYtZ9yNw7JAPcySU1UJJ8zFa
Theory Prop Id: 94a5d9602cf57a3f5654fc82347cce52c34639293e929a5536fb236f16c4a584
Theory Prop Address: TMNkz5vYZPjFcLYn1uGSKSZPVSM8RjwnAry

Definition 21.60 We define struct_r to be

λS.∀q : ι → o.(∀X : ι.∀R : ι → ι → o.q (pack_r X R)) → q S of type ι → o identified by the following information:

294 CHAPTER 21. SURREAL NUMBERS I

Pure Object Id: 2c39441e10ec56a1684beb291702e235e2a9e44db516e916d62ce426108cfd26
Pure Object Address: TMcKwAb1Y239cNAKRVhgscrwaw8GNLJkYPp
Theory Object Id: 62b6df4a9733ff8dc3d14aedd832787d9847bd18b34fcf2c6bff876818a1847d
Theory Object Address: TMYJ8WJhMWwP81uepXsK3pFf4gfjynNf86z

pack_struct_r_I

Theorem 21.272 ∀X.∀R : ι → ι → o.struct_r (pack_r X R). The proposition is identified by the following information:

Pure Prop Id: b8a6bd24313c25cdf6f5d780f72c0aa8fc8d00573ca62ac10879aab64b058d71
Pure Prop Address: TMLYEZeYcGNJ17buuXRfr7jgkECAGmexZCE
Theory Prop Id: e305fee3cf88a27e70cab3757abe5760dbb2dfa53a2681cf0007eae77a1ac0a5
Theory Prop Address: TMN1pwx29rgBN5sqW8xRekY99KzuFAfTFfC

struct_r_eta

Theorem 21.273

∀S.struct_r S→S = pack_r (S 0) (decode_r (S 1)).

The proposition is identified by the following information:

Pure Prop Id: 84273bb2eff84eb8f3c9054433916df932828a737f7c3ca3ab694990398c765d
Pure Prop Address: TMV5JzEiDcbVXBiiQYZZiJfEwYjacKQ4MxV
Theory Prop Id: a9afdff67a397303a2b3281aa9aec54d7e19dc89f19756ffe0b41bed021a4e74
Theory Prop Address: TMMrznhQDCUW9B4yjAoMNMThp6vbuGof6jH

Definition 21.61 unpack_r_i is the opaque object of type

ι → (ι → (ι → ι → o) → ι) → ι

identified by the following information:

Pure Object Id: 3ace9e6e064ec2b6e839b94e81788f9c63b22e51ec9dec8ee535d50649401cf4
Pure Object Address: TMN9skP6MgqmkRJ4cwifVmq6ZbEz41jqRV2
Theory Object Id: 076245d35f8b67c6c189c5aef9cfc61ae9e0e18184f6385bcb6a3dbdd97661c4
Theory Object Address: TMJh6uB6VQPwpShTzHxgdReiKesNatUru1Z

unpack_r_i_eq

Theorem 21.274

∀Φ :ι → (ι → ι → o) → ι.∀X.∀R :ι → ι → o.

(∀R′ : ι → ι → o.(∀xy ∈ X.R x y ⇔ R′ x y) → Φ X R′ = Φ X R)→

unpack_r_i (pack_r X R) Φ = Φ X R.

The proposition is identified by the following information:

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 295

Pure Prop Id: 16a31beed02eceef7aa5ad8abd3e417faefd8f5add89a3ffc21c18be48b02225
Pure Prop Address: TMNMHd4hurTht88FezFD988n9Kwz38gbmMT
Theory Prop Id: 9d8a6e472c07d73be0ef8a14ffae4a0b986fb8834986991cc7f1aaabb3f3bce2
Theory Prop Address: TMNFGGSBwtHNzfP26abSeRhNQbNe6qxGKxv

Definition 21.62 unpack_r_o is the opaque object of type

ι → (ι → (ι → ι → o) → o) → o

identified by the following information:

Pure Object Id: e3e6af0984cda0a02912e4f3e09614b67fc0167c625954f0ead4110f52ede086
Pure Object Address: TMZmDPx4PJagAzuKD7GKyixmxfa7zoZhWLS
Theory Object Id: 6bcb7cdbf5bd449df3c8deb49a0d9cdf76962583716fcb71c59d999f4bcc3fca
Theory Object Address: TMR5CinBdVnunwxcK6781T5WjV1NMdFfYig

unpack_r_o_eq

Theorem 21.275

∀Φ :ι → (ι → ι → o) → o.∀X.∀R :ι → ι → o.

(∀R′ : ι → ι → o.(∀xy ∈ X.R x y ⇔ R′ x y) → Φ X R′ = Φ X R)→

unpack_r_o (pack_r X R) Φ = Φ X R.

The proposition is identified by the following information:

Pure Prop Id: 2033874915fc7ab80e92006c39e273588d52da2c880d44c7a1709b538dcbf75e
Pure Prop Address: TMbXxr57ewDhvFG8is2EuTAvHskooz91px9
Theory Prop Id: 52c8fc566349ffa2b8dd50a78e4a51cb1ce6221b8fc68c78bc7b7d1cfcec0507
Theory Prop Address: TMHK3aoepzguxxe9uvkDTJLq9eEjx81iFMe

Definition 21.63 pack_c is the opaque object of type ι → ((ι → o) → o) → ι identified by the following information:

Pure Object Id: cd75b74e4a07d881da0b0eda458a004806ed5c24b08fd3fef0f43e91f64c4633
Pure Object Address: TMd1myShkR5W1i73HwjDPr8KSegvcYRGNkj
Theory Object Id: 385b4fc06e188b2cbb0ea30c0d01cf4136c17844fda6436fa4bc50f1ca06705a
Theory Object Address: TMRLzP6f981VnW9r3th3udMRX2Ub6wT7if3

pack_c_0_eq

Theorem 21.276 ∀SX.∀C : (ι → o) → o.S = pack_c X C → X = S 0. The proposition is identified by the following information:

Pure Prop Id: 7a818cf8506cfd3b76666875ebb3fc61c82548b59fc9ee11d386b244c6479132
Pure Prop Address: TMazxFUxbSMsNHSVTNyhQVMvZQVYMtn4rey
Theory Prop Id: 33b154a9ab4ecbefbf89524fafe42b917315c09c6052ab1c51e598aafe09f4d8
Theory Prop Address: TMGfBTiq6Sx1weZYbUWpWq2HwMh165mnVTc

296 CHAPTER 21. SURREAL NUMBERS I

pack_c_0_eq2

Theorem 21.277 ∀X.∀C : (ι → o) → o.X = pack_c X C 0. The proposition is identified by the following information:

Pure Prop Id: e668a001a1c285f7e59f4c803795dd7d4779ae52441d54042fd5ffe717dddcce
Pure Prop Address: TMU1E9mEhTwbdJ719haDZxuihHr9UCvVdNH
Theory Prop Id: c88b55706909d090d96e8cc6cfa99496cd78e11afd38eec353935f17985d1464
Theory Prop Address: TMTQUoVGyRyem2GnrMo5TZzGhHbcftwsL1J

pack_c_1_eq

Theorem 21.278 ∀SX.∀C :(ι → o) → o.S = pack_c X C→∀U :ι → o.(∀x.U x → x ∈ X)→

C U = decode_c (S 1) U.

The proposition is identified by the following information:

Pure Prop Id: 5c09442310e12d4810ed9f3e048015114950d37b0e8b72105b9b1bb99cf8e28a
Pure Prop Address: TMQYfBviDQMhm9XsosJVREqZzaDe4FqcxjB
Theory Prop Id: 821fb2213762f86b1740f4279d2eab6d7fb93fd32c8ebe4a6391c58ea8623af8
Theory Prop Address: TMdmyWGVAiB85Sa4EovKeQWGmrrH97ZjkPg

pack_c_1_eq2

Theorem 21.279

∀X.∀C :(ι → o) → o.∀U :ι → o.(∀x.U x → x ∈ X)→

C U = decode_c (pack_c X C 1) U.

The proposition is identified by the following information:

Pure Prop Id: db75edb08a1a1b58545b4c2359316ff590916926eb8c93afcd90edcae6a56de8
Pure Prop Address: TMLMrphf4gsuDFSivQWRvVEok1B1ijTDDi3
Theory Prop Id: db8f0c9ec729f71005b0d39a0dd45dcc57e2e402d85bc7972239e4dcb3491445
Theory Prop Address: TMHHqUa2BTRQWnruKb9Gi2QmNFrYUVn9izm

pack_c_inj

Theorem 21.280 ∀XX′.∀CC′ :(ι → o) → o.pack_c X C = pack_c X′ C′→ X = X′ ∧ ∀U : ι → o.(∀x.U x → x ∈ X) → C U = C′ U.

The proposition is identified by the following information:

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 297

Pure Prop Id: b5342db4d933f411377c97f1764dfc463670f61e9e5c7d2e559cd02374533b74
Pure Prop Address: TMb2WjtZGnPxZepZzY1QdwFcCN59VzNjmQv
Theory Prop Id: 1616e034dd768bac9158e8d3698f121fc76f282512edd80730dffa3671b83dbb
Theory Prop Address: TMN41sVNtmGNFNseJjz7CtVxUTGca3zf8Fx

pack_c_ext

Theorem 21.281 ∀X.∀CC′ :(ι → o) → o.(∀U : ι → o.(∀x.U x → x ∈ X) → (C U ⇔ C′ U))→

pack_c X C = pack_c X C′.

The proposition is identified by the following information:

Pure Prop Id: 4c016262198de07b1a463845c2d49e035d3595ebda34bc6acc4687e728e1bf21
Pure Prop Address: TMKA12T7rWqWi37qnNX1Sdao9MdhMjth6Cy
Theory Prop Id: c56d9be727cc8fe44ba7dff7e4a52d3142c3847f950d436c7c3a985a836a38f3
Theory Prop Address: TMMNarnCqG3uNHZiwphmNNhpz7GKTtcnVzm

Definition 21.64 We define struct_c to be

λS.∀q : ι → o.(∀X : ι.∀C : (ι → o) → o.q (pack_c X C)) → q S of type ι → o identified by the following information:

Pure Object Id: 9e6b18b2a018f4fb700d2e18f2d46585471c8c96cd648d742b7dbda08c349b17
Pure Object Address: TMWcyWdA8JkYry5Pcnkkq5U1QPa36w6b1L5
Theory Object Id: 7d5f09aa0fe499776ea89c1aa772c888bce1fc0e89851f4347dec84efa0b4ea1
Theory Object Address: TMW7M6E5Xmwk23LWFWBgLSDEoAq8oj8xvjy

pack_struct_c_I

Theorem 21.282 ∀X.∀C : (ι → o) → o.struct_c (pack_c X C). The proposition is identified by the following information:

Pure Prop Id: 708ab1416bde59517467bf78b5c03e29243d5f7dcaab69b7f4984e6e5e44bd0c
Pure Prop Address: TMb4doWh2x5zNeTDHTUgTiSD9NtGRrxK76r
Theory Prop Id: 9e3725e9cd266ee0013e48acce207c124782c57dbfda0f074f6a8c2e7f15fc39
Theory Prop Address: TMacYJooBqqThT8EFbSspPBtC9gFg81RNZ5

struct_c_eta

Theorem 21.283

∀S.struct_c S→S = pack_c (S 0) (decode_c (S 1)).

The proposition is identified by the following information:

Pure Prop Id: e9a19baf027baa0b457c0b4ad53e06f86965db35b095e21bbfdcc856488ce205
Pure Prop Address: TMGzKMqTDCSUsm3E8XZBzrbDXqT5nLqjeH7
Theory Prop Id: 492cc612750919ec1d22e8bdc9fb6755693cce05b75a791a2964562629d0c741
Theory Prop Address: TMGqxPLPiHT83m18jBgJDFekewkT2SUQYiZ

Definition 21.65 unpack_c_i is the opaque object of type

ι → (ι → ((ι → o) → o) → ι) → ι

identified by the following information:

Pure Object Id: a01360cac9e6ba0460797b632a50d83b7fadb5eadb897964c7102639062b23ba
Pure Object Address: TMbQScLkWSLe1HYb5m6QUhTBUkH2XSZfNEv
Theory Object Id: fa69d4911c2be7aef9c6749803f6c7d2e452fdb2d239cc2c8efdd2104077e48a
Theory Object Address: TMZxmf46bwJxe9YDsDHKRKXQrDdkLoPAP4T

unpack_c_i_eq

Theorem 21.284

∀Φ :ι → ((ι → o) → o) → ι.∀X.∀C :(ι → o) → o.

(∀C′ : (ι → o) → o.(∀U : ι → o.(∀x.U x → x ∈ X) → (C U ⇔ C′ U)) →

Φ X C′ = Φ X C) →unpack_c_i (pack_c X C) Φ = Φ X C.

The proposition is identified by the following information:

Pure Prop Id: 1c35a1ec49283779d9dad9c87ccddab3c155a1e81dfd7c30d2e5bf413a4903ca
Pure Prop Address: TMJGk56zQ9ppwWePYzrdvyPXUCDe2sEqZ19
Theory Prop Id: 3a893b44e699dd6503cf16fc256ea51e13c996f8fdbfb14928fb14d405d95013
Theory Prop Address: TMY29KidZk9kJp7G1gonGvtrEyAKfE9y3mh

Definition 21.66 unpack_c_o is the opaque object of type

ι → (ι → ((ι → o) → o) → o) → o

identified by the following information:

Pure Object Id: 939baef108d0de16a824c79fc4e61d99b3a9047993a202a0f47c60d551b65834
Pure Object Address: TMZ52NsRLMZey7UxdWrY5r5LQnLT6warXH5
Theory Object Id: 8b56a8588064b5d7d1c47e8fcf2acc9e2ce6c5f269ba5d05ed77c59568ec81b1
Theory Object Address: TMZkMyksvwVPJJNYLovJrphjDV7ZP1519ja

unpack_c_o_eq

Theorem 21.285

∀Φ :ι → ((ι → o) → o) → o.∀X.∀C :(ι → o) → o.

(∀C′ : (ι → o) → o.(∀U : ι → o.(∀x.U x → x ∈ X) → (C U ⇔ C′ U)) → Φ X C′ = Φ X C) →unpack_c_o (pack_c X C) Φ = Φ X C.

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 299

Pure Prop Id: 5145ca206371461257e3dbbda8cd67e11446e95f699a7643af128584410c440e
Pure Prop Address: TMMcXtq6qovUwYmcRYmCXc121HsSojJs5fr
Theory Prop Id: ba3042e75e86ab022bb94d2eb8a4e5e23f21d516fccd571acb857200dcc5ac3d
Theory Prop Address: TMZ6bYNJXuLEh438fMAjBWK4p44Add2Frrj

Definition 21.67 canonical_elt is the opaque object of type (ι → ι → o) → ι → ι identified by the following information:

Pure Object Id: 24615c6078840ea77a483098ef7fecf7d2e10585bf1f31bd0c61fbaeced3ecb8
Pure Object Address: TMTxaocPHR7KaoAb7prHbkajsp6MUfshdsg
Theory Object Id: a18d95a2ab34761092101861924b9134534850495d74fd75a2443135e945ea6f
Theory Object Address: TMTEe99JMMf634KDuVmQ44i85VxLmTHvB9y

canonical_elt_rel

Theorem 21.286

∀R :ι → ι → o.∀x :ι.R x x→R x (canonical_elt R x).

The proposition is identified by the following information:

Pure Prop Id: 409f7f93609b8bbc78e9b27c91b8e8b15d16e673c0445c98fe5cbae6fe93e350
Pure Prop Address: TMQQtrdvb5MuDAgqNkxaL8Yxdiut4fDtB6a
Theory Prop Id: a82ff2dd9b739269814be3b2b90d4dcd5d7e5f49fd6fa941daaf8f2fac8b7e4b
Theory Prop Address: TMahFABoX7qUN4TKbs8SxWDdavcsNyazvny

canonical_elt_eq

Theorem 21.287 ∀R :ι → ι → o.per R→∀xy :ι.R x y→ canonical_elt R x = canonical_elt R y.

The proposition is identified by the following information:

Pure Prop Id: 77a0bb710d7f6fd22c2038b8e8a7aa53cb38b8625399eda5a768a92f86f94529
Pure Prop Address: TMNGDho8FciVfUN7CiWcH22sTikdZ3nEXxy
Theory Prop Id: a3a6535b47fa636b2a37d0fd9d2cbc572ab41c638fc41180620163a965a862a9
Theory Prop Address: TMPA9JRmixkhj8Ft97qJcZE9RMXboWNNtQA

canonical_elt_idem

Theorem 21.288

∀R :ι → ι → o.per R→∀x :ι.R x x→

canonical_elt R x = canonical_elt R (canonical_elt R x).

The proposition is identified by the following information:

Pure Prop Id: f360e8148073fe4e1c4b5a43fcb23f3a4230874ab3afbbbea65f8ac02b0385d5
Pure Prop Address: TMGnsgqbHzSxjuur9NxxHHErkCtdk6tipsC
Theory Prop Id: 0881fbf40b7b603f1f612763b2238f0020aa675fdfbc4d16389c60ebcc4735ed
Theory Prop Address: TMNx4jXjzYQJoqNMEri4pBsyV3LhSvqYYqt

Definition 21.68 quotient is the opaque object of type (ι → ι → o) → ι → o identified by the following information:

Pure Object Id: 185d8f16b44939deb8995cbb9be7d1b78b78d5fc4049043a3b6ccc9ec7b33b41
Pure Object Address: TMQgacYA6wSdteHEaGGZB5pfV8odVjnD3Ld
Theory Object Id: eceb7632185ce5062420ae1f8b27cdcf062b7c243c6d9b7856e5e2a4e937e2af
Theory Object Address: TMdVzy3y6H8DfNQEHsaYmphZTWHJpPCs1dv

quotient_prop1

Theorem 21.289 ∀R : ι → ι → o.∀x : ι.quotient R x → R x x. The proposition is identified by the following information:

Pure Prop Id: c9cf06df33dd5d9e1508e9c920dbee7c4fbe3ac5ca1796506c238279d7024d76
Pure Prop Address: TMEpRYaApsS9NBgYkPfxvKjtbKoMxckLJkg
Theory Prop Id: 8eea9432bbf60de4e1871846f2697db36eb0c3579081190809f73c887aa8932c
Theory Prop Address: TMHSpwoRWkRrT199Pr4Zw6NBfBmFt5FaZDd

quotient_prop2

Theorem 21.290

∀R :ι → ι → o.per R→∀xy :ι.quotient R x→quotient R y→R x y→x = y.

The proposition is identified by the following information:

Pure Prop Id: 54b14f4b32c6d1e65fbf61514c95ad09fa776454efdf44259705e3654a8de198
Pure Prop Address: TMWEzaUyoSBC2cv5ikpYBmh1Wxg9bwCWMaW
Theory Prop Id: c5f241148436a9448eb8c5fe8b100b16191b8a22b99dbbc971e37dd9e6442a9c
Theory Prop Address: TMSQAtmzR5PbKfWrMfYzL1xj4QFSrYQWPLq

Definition 21.69 canonical_elt_def is the opaque object of type

(ι → ι → o) → (ι → ι) → ι → ι

identified by the following information:

Pure Object Id: cd4f601256fbe0285d49ded42c4f554df32a64182e0242462437212fe90b44cd
Pure Object Address: TMXTWshnbJ4phtQfmjt8rT3VUcaRS5EkRgM
Theory Object Id: c6ce994fa2a7c7d9b9936df4d487c851d3b5a14f9d5b842e6d7950167473ab6e
Theory Object Address: TMZLpA4pBtPQmtYCGc2fcZVLLXtrZbwNRz2

canonical_elt_def_rel

21.10. MORE RESULTS ABOUT SURREAL NUMBERS 301 Theorem 21.291

∀R :ι → ι → o.∀d :ι → ι.∀x :ι.R x x→R x (canonical_elt_def R d x). The proposition is identified by the following information:

Pure Prop Id: 427d7106df945e01d1e010f6b5f54619d5e305852962b29fd074411773f7a5fb
Pure Prop Address: TMQseUYfkBpoXqbKv8vXzWb7zmRmZGRt6ho
Theory Prop Id: ba911f1fb82909fa0343a89423fd8007a1eea7266f3a64d4f344fc4454b45e97
Theory Prop Address: TMSAmmXFWnmPQSVvG6xXns5c29knAy7NoJL

canonical_elt_def_eq

Theorem 21.292 ∀R :ι → ι → o.per R→∀d :ι → ι.(∀xy : ι.R x y → d x = d y)→∀xy :ι.R x y→

canonical_elt_def R d x = canonical_elt_def R d y.

The proposition is identified by the following information:

Pure Prop Id: 08b2b96ed66c9cf40b455864a9805316a228566ea2f3dee98727b4ddf6fdee29
Pure Prop Address: TMdbiHGV2yacZL1C6aVnpN4nqxmFxeFRfVh
Theory Prop Id: 8b36f1b656a7909d9a962189ecd03e0e7de3681872a48796808275153f9b010f
Theory Prop Address: TMQkuYZZz1p5gpefQX7a59yAUSxmoaUZ2B1

canonical_elt_def_idem

Theorem 21.293 ∀R :ι → ι → o.per R→∀d :ι → ι.(∀xy : ι.R x y → d x = d y)→∀x :ι.R x x→ canonical_elt_def R d x = canonical_elt_def R d (canonical_elt_def R d x). The proposition is identified by the following information:

Pure Prop Id: 49efe82bbd3304683bf8464c2a34622dd4c9bffd857d2336f4501a2673d7d8f4
Pure Prop Address: TMFnpUHvtfsuux3kkiYp7792SjHSqdtGr2C
Theory Prop Id: c305bfd1e77d91b812fe3dc8843e58279f1bb98be2cbbed8b4fedf7a73f5d6ef
Theory Prop Address: TMKtsdWDnNkcBLMACntcRypwVFCS39wBpEg

Definition 21.70 quotient_def is the opaque object of type

(ι → ι → o) → (ι → ι) → ι → o

identified by the following information:

Pure Object Id: 612d4b4fd0d22dd5985c10cf0fed7eda4e18dce70710ebd2cd5e91acf3995937
Pure Object Address: TMHzGWujGt1Siv1ANFttshwyzCe6VYRGi4K
Theory Object Id: b18ffb4e6bef489f8b9eecd55fec81b0c0593f62e6a1386eb07f6c9250a2d493
Theory Object Address: TMQgkTfPUbwLT8jGwoRckaDnvCcP3Vd9PPz

302 CHAPTER 21. SURREAL NUMBERS I

quotient_def_prop0

Theorem 21.294

∀R :ι → ι → o.per R→∀d :ι → ι.∀x :ι.R x (d x)→x = d x→

quotient_def R d x.

The proposition is identified by the following information:

Pure Prop Id: b6730b9da18d165aefaece5098a92db470398156beb51a1a0c1278d4e36152dc
Pure Prop Address: TMVu99E7LXXZ91pqY2xzxjzevyHiyRuDk2L
Theory Prop Id: 1e2c729d64e136e7b405ff57ee6ac780a1b626f5f145e0dc7b9f1d1c894c154b
Theory Prop Address: TMUnYQqkYvFzvC8Rsud6AfF4JJ7DqXnWQxo

quotient_def_prop1

Theorem 21.295

∀R :ι → ι → o.∀d :ι → ι.∀x :ι.quotient_def R d x→R x x. The proposition is identified by the following information:

Pure Prop Id: 96ac55830548884573f3e141434b062bf9a4c5864092d3e70136808df96b01ad
Pure Prop Address: TMXLZPFcYNs9m56G1QCx9QktiHeUscd41Rw
Theory Prop Id: 0ba02cd229346ac664fd2fadc979d704d3e12575ce6ce56adbcbe00e2e55b9db
Theory Prop Address: TMT4xAb2VRQxAVPG9ZG39hM8kN2w5vHWSNB

quotient_def_prop2

Theorem 21.296

∀R :ι → ι → o.per R→∀d :ι → ι.(∀xy : ι.R x y → d x = d y)→∀xy :ι.

quotient_def R d x→quotient_def R d y→R x y→x = y. The proposition is identified by the following information:

Pure Prop Id: fd0707aa86d2b360392ba2fdfe2a63261a390d4e8aa5535320c6e57d0b01996e
Pure Prop Address: TMU2h1hwqUi1KnjLzjZCzhLM9nK9DZmiDRP
Theory Prop Id: 0dc38ca2997b0761433b4fc94c031c565be7652963fd19f4257dd60e41928762
Theory Prop Address: TMNeKehn9tffwmMaonuDau88ErbSbo4Wzam

Chapter 22 Explicit Number Structures

22.1 explicit_Nats

Let N : ι be given. Let base : ι be given. Let S : ι → ι be given. Definition 22.1 explicit_Nats is the opaque object of type o identified by the following information:

Pure Object Id: 3fb62f75e778736947d086a36d35ebe45a5d60bf0a340a0761ba08a396d4123a
Pure Object Address: TMVtLiLBPcSiAyimBNLrSsPPvNGxNnQTLgx
Theory Object Id: 499c1253e694386778e3a86aec4d8d86e3e968012315d84ff3d8771c7ca33ca6
Theory Object Address: TMYM3xBCn91GR8gYvvjmLwhSW8eWQhQUR5V

explicit_Nats_I

Theorem 22.1

(base ∈ N)→(∀m ∈ N.S m ∈ N)→(∀m ∈ N.S m ≠ base)→(∀mn ∈ N.S m = S n → m = n)→

(∀p : ι → o.p base → (∀m.p m → p (S m)) → (∀m ∈ N.p m))→explicit_Nats.

The proposition is identified by the following information:

Pure Prop Id: 1e6abb88cfbf0a3803f631d2180983c78469316b36f44986957690ca50046636
Pure Prop Address: TMS7xTLnefHd6AjUKGkQBpdVrdEKnFWi6GV
Theory Prop Id: 1c1fb7a66f9f17cd55170faa1827e232bbd7515b8fbb9a57a4ccf3c34eaa501a
Theory Prop Address: TMZjQDjppyHzU9Sm626Nv7EXMsNVVAqkaJT

explicit_Nats_E

304 CHAPTER 22. EXPLICIT NUMBER STRUCTURES

Theorem 22.2

∀q :o.

(explicit_Nats → (base ∈ N) → (∀m ∈ N.S m ∈ N) →

(∀m ∈ N.S m ≠ base) →

(∀mn ∈ N.S m = S n → m = n) →

(∀p : ι → o.p base → (∀m.p m → p (S m)) → (∀m ∈ N.p m)) → q)

→explicit_Nats→q.

The proposition is identified by the following information:

Pure Prop Id: 5de31c779c105b713f97c938846c5369a751bf11d4d1c91131667de452b85d52
Pure Prop Address: TMN27YWUyKn3QBjt1t4WK1VcstkTSsWazb6
Theory Prop Id: 80e228811834277628e910ca46e86ac23fcdf98d3b95ab2ee85c1f1697c9842b
Theory Prop Address: TMXQDL2aLuWPAFGh9xNGSkbkMuMSyVTuvG2

explicit_Nats_ind

Theorem 22.3 explicit_Nats→∀p :ι → o.p base→(∀m ∈ N.p m → p (S m))→∀m ∈N.p m. The proposition is identified by the following information:

Pure Prop Id: 72f6cf6a8db779d851cead9db4030c8cf14cdb66b3dba2256273ea5c82f6cf75
Pure Prop Address: TMRCzQGX3LAr7ohapsy9xQiuobUsWPbHmfG
Theory Prop Id: 3e626e6f6e97b94d761465a195a81a506097757289b1852c22484904835f624b
Theory Prop Address: TMLtcGPvoYCa4mMihX67t4Lm78GzLsmB2B6

Definition 22.2 explicit_Nats_primrec is the opaque object of type

ι → (ι → ι → ι) → ι → ι

identified by the following information:

Pure Object Id: a61e60c0704e01255992ecc776a771ad4ef672b2ed0f4edea9713442d02c0ba9
Pure Object Address: TMY3C6hyjHYFvPNDaQK728xpbMFkgTGPbUQ
Theory Object Id: 02adae4d88dbb61570b0e9d5d5ea2d21abfb3e9a8408276329ddc4e3b1fc2a9a
Theory Object Address: TMUhjTmqPCD758LckQJPJ6DHuAFTQeHLoqS

explicit_Nats_primrec_base

Theorem 22.4 ∀a.∀f :ι → ι → ι.explicit_Nats→ explicit_Nats_primrec a f base = a.

The proposition is identified by the following information:

22.2. OMEGA GIVES EXPLICIT NATURALS 305

Pure Prop Id: ec9dac528f5b9d80aeb2b078f81657da0725083292683597b6ed0f6bcda0033c
Pure Prop Address: TMJxuYA6EQJ2GxewDFKrBTK4gZX2qiFpNuM
Theory Prop Id: 18887ff3f12d4f358090a38d2eecb57a5531dabfe6535ad15a910933b602f16c
Theory Prop Address: TMGkBbmvyBif6wxjhQzqdSRLtTP5TBd5e9z

explicit_Nats_primrec_S

Theorem 22.5

∀a.∀f :ι → ι → ι.explicit_Nats→∀n ∈N.

explicit_Nats_primrec a f (S n) = f n (explicit_Nats_primrec a f n). The proposition is identified by the following information:

Pure Prop Id: c79ce9883e3c0c86ed88d57e78ed6269f94c26749772f51c6ad325234a554663
Pure Prop Address: TMSv5jrFNLuwkARow9Y11SWFGFXBqcbUbLK
Theory Prop Id: bf9939e2289e86288ccefbe010777a3faf16479bf896f3b41a2c5a5a73e561cb
Theory Prop Address: TMUpUUqwaxvwAg3jgXfPz8RdqaXuW1kMAxx

explicit_Nats_primrec_P

Theorem 22.6

explicit_Nats→∀P :ι → o.∀a.P a→∀f :ι → ι → ι. (∀n ∈ N.∀b.P b → P (f n b))→∀n ∈N. P (explicit_Nats_primrec a f n).

The proposition is identified by the following information:

Pure Prop Id: 7ffca5b3e4e2357fc59f35b26bcc159af8962fe6020bca14a2c9246bac7165e9
Pure Prop Address: TMPVTJWhZ2JZ6V3t3BHXTUQ7k318dSnYpfW
Theory Prop Id: a23b352d53e7ed810c1716d20ebfd8cf9fd43007e863a3265aa1009d6883d822
Theory Prop Address: TMSyXZfNTKDx39oHHCRdmhL4WfJE3gghudc

22.2 Omega gives Explicit Naturals

explicit_Nats_omega

Theorem 22.7 explicit_Nats omega 0 ordsucc. The proposition is identified by the following information:

Pure Prop Id: a3ba10e61c3b75d16024bffd1f732e7b90427037533d34ee2e091771480c74a6
Pure Prop Address: TMdGdgcJArCKDPYXVndpbm1SU3AEVwyfoDy
Theory Prop Id: 4d074b5527b3d4f670e267d1341927f6e1665b1b695a8fa1d7b296e3b59b616f
Theory Prop Address: TMN5Yqfwk5yD9vYjacHHUZqkgRCThYxgE6u

306 CHAPTER 22. EXPLICIT NUMBER STRUCTURES

22.3 explicit_Nats_zero

Let N : ι be given. Let zero : ι be given. Let S : ι → ι be given.

Definition 22.3 explicit_Nats_zero_plus is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 9683ebbbd2610b6b9f8f9bb32a63d9d3cf8c376a919e6989444d6d995da2aceb
Pure Object Address: TMEmJ2AUSSkSo3PgswMDx4FDuVNXWTFp2Xv
Theory Object Id: a492e421e4fb131a6ece5ab66d32b1664ac5750c65f21c1ee016f0b281db4c68
Theory Object Address: TMZHNQrttsjehBJHhEo49zc8mrT3L85obE5

Notation. We use + as a right associative infix operator corresponding to applying term explicit_Nats_zero_plus.

Definition 22.4 explicit_Nats_zero_mult is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 7cf43a3b8ce0af790f9fc86020fd06ab45e597b29a7ff1dbbe8921910d68638b
Pure Object Address: TMWE5mZjYWyApPQfEURTERyzUQDmzqxvrRZ
Theory Object Id: 11a63fc4e4e266de185d7864c9ae0a5059a920dc1af5eb33480e4b6257a39712
Theory Object Address: TMa5dmS6AU8GbHfa7BizTDCqP4sRj1UmdGg

Notation. We use ∗ as a right associative infix operator corresponding to applying term explicit_Nats_zero_mult. Assume the following.

explicit_Nats N zero S  (22.1)

explicit_Nats_zero_plus_N

Theorem 22.8 ∀nm ∈ N.n + m ∈ N. The proposition is identified by the following information:

Pure Prop Id: 56523af04adfa34d579665edd443f3c8c991a55c2e1d59fe86eb7cb534626054
Pure Prop Address: TMKvkMhHVgvBVAMZ58PR6NKqCWx4kMPZmfY
Theory Prop Id: d3decda97e93dc588cf7d1be70d082fee93bb6edd68e6c451ca3e4e4685258a4
Theory Prop Address: TMXZCAYSxCqnbikDjS9v4JyUbJQi41N3Beg

explicit_Nats_zero_plus_0L

Theorem 22.9 ∀m ∈ N.zero + m = m. The proposition is identified by the following information:

Pure Prop Id: eda4ea107573c4741f17b9e4cfc4ad2c0fca35856cb39a9f9d0c7082eeef8edc
Pure Prop Address: TMZFAJoCr87Twhz3ue2eaZWtjHZrDbriNS2
Theory Prop Id: 6cbaa15873e859402e00e7efd7313e6202c9580bb8dc07d3eaff7cb6dd528689
Theory Prop Address: TMXGbC3fvTNoibPqk8dSmxUYfFe6A2jTjVA

22.4. EXPLICIT_NATS_ONE 307

explicit_Nats_zero_plus_SL

Theorem 22.10 ∀nm ∈ N.S n + m = S (n + m). The proposition is identified by the following information:

Pure Prop Id: 4b64cc27697767078865ed93f819202d8095e44f34703cacb427d0ff277b7772
Pure Prop Address: TMUB4sk69KXoqr9KeeXuQvkBtdzEy5uGSGY
Theory Prop Id: 67b5266ae2b89bd4f290f8483d37fed2bec3392b27b55c35022e5621958309a4
Theory Prop Address: TMUxY6JJgbyA2wvfR5kWXPg76FJgumm8rc1

explicit_Nats_zero_mult_N

Theorem 22.11 ∀nm ∈ N.n ∗ m ∈ N. The proposition is identified by the following information:

Pure Prop Id: 242c80020d9e0983d413a8726b0f8dd31a5d6ed1c4a614a9e372f46e71b811d7
Pure Prop Address: TMPyVVnedkqiLXtyLzYCYMHVttNM5sfuXaF
Theory Prop Id: 4d99dbaf92926086a5db90b0b45da73422e998d71f9d9b53551ce90cc1bc2a97
Theory Prop Address: TMUjvpTR8zMFez8UREoFWmF1mfB2Ck1G9qd

explicit_Nats_zero_mult_0L

Theorem 22.12 ∀m ∈ N.zero ∗ m = zero. The proposition is identified by the following information:

Pure Prop Id: b80ff108406a373e38422d03ee6a993a3300f355e3972522d8834e8f32e727eb
Pure Prop Address: TMdPBQEb6r1jazTfaSsAkVi4HU5BuA9SgCr
Theory Prop Id: 8b905296c07181890992e4efb2371e7d4c2eacd4d79aa5f2ae5bee8d137d8e99
Theory Prop Address: TMFy6HpCwNacy45vXp7VgoS5A62a6koR3ib

explicit_Nats_zero_mult_SL

Theorem 22.13 ∀nm ∈ N.S n ∗ m = m + n ∗ m. The proposition is identified by the following information:

Pure Prop Id: ceea17fc60f9c29a157639cceb082002f336d7ada992c4bf24f10ec2107aaeef
Pure Prop Address: TMN5mn9NySXen2SeWDjUyYFPcUTDr2zeAhC
Theory Prop Id: 58cb3351db2fc2413401ac6beecc0c5aa8cb6e94862e9b0888a08fab43ce10ae
Theory Prop Address: TMYNaqKknxFESuZan8Ztx9TK21bLv9z72UQ

22.4 explicit_Nats_one

Let N : ι be given. Let one : ι be given. Let S : ι → ι be given.

Definition 22.5 explicit_Nats_one_plus is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: c14dd5291f7204df5484a3c38ca94107f29d636a3cdfbd67faf1196b3bf192d6
Pure Object Address: TMVj1gkGc7wpNZ6S9o6JmxmKVGCvLP3HmTd
Theory Object Id: a0dc24bec298fc49cdd175bfad0801c6dda7e88d2602021d5fc71cc01e269ee8
Theory Object Address: TMW7qLS9YPVnzwPSYkBdmmRwk5ZbPbLdnpL

Notation. We use + as a right associative infix operator corresponding to applying term explicit_Nats_one_plus.

Definition 22.6 explicit_Nats_one_mult is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: ec4f9ffffa60d2015503172b35532a59cebea3390c398d0157fd3159e693eb97
Pure Object Address: TMJBfR7vUcShqA32xB1n6LKgJDcAtK2PWgY
Theory Object Id: 7cbccf9874bf38cebb42cd5bbc730880796dbf9ee8b00e588914593a33f3616e
Theory Object Address: TMdYg5kd19jjyivNeoE1Fq5Z77fpmPN4Ghy

Notation. We use ∗ as a right associative infix operator corresponding to applying term explicit_Nats_one_mult.

Definition 22.7 explicit_Nats_one_exp is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: cbcee236e6cb4bea1cf64f58905668aa36807a85032ea58e6bb539f5721ff4c4
Pure Object Address: TMGRgxwxS6aYefXQew7xTarwkrNkekEv1vL
Theory Object Id: f451e86804defa9f4c49ef8b31319235986f06a57cc47e224e0e616e81349e41
Theory Object Address: TMYMj3b1DyJ9Lbh2dkML476LtBBK6RhxAGa

Notation. We use ↑ as a right associative infix operator corresponding to applying term explicit_Nats_one_exp. Assume the following.

explicit_Nats N one S  (22.2)

explicit_Nats_one_plus_N

Theorem 22.14 ∀nm ∈ N.n + m ∈ N. The proposition is identified by the following information:

Pure Prop Id: b870373860de2f7ad912a4078539010f085139c691292d7e7d3c34db1f0cdb6a
Pure Prop Address: TMMGxvc17zs14pSXUk5XHB547EgcLE5Qj1X
Theory Prop Id: 2031f14ef57b3e0a0f0bb3c5d4e8d7cc6c01908d84f8ae8a58422536a7ebd671
Theory Prop Address: TMKsochxf4GKDVGFjEfDV7zBLDZ3NZmTqN8

explicit_Nats_one_plus_1L

Theorem 22.15 ∀m ∈ N.one + m = S m. The proposition is identified by the following information:

Pure Prop Id: e0ca47b5062d5ffc6896d07746c59aaca4bc4acc06788ddcfa7d606c68f00a95
Pure Prop Address: TMSjCGdV6bWn9b6ayvqYSEN5ezLiifHBvqP
Theory Prop Id: b861add16dde10cbd273ebb51a08c3bde324096c7b892a0f54908292dbd67d14
Theory Prop Address: TMRmkTUjpfWnjUnbtZRbTeUiDDjnxz6PKXn

22.4. EXPLICIT_NATS_ONE 309

explicit_Nats_one_plus_SL

Theorem 22.16 ∀nm ∈ N.S n + m = S (n + m). The proposition is identified by the following information:

Pure Prop Id: d527faf460c66eb2a605937d7bc284a505c5e71e11b5819b3ebe45aa5ae3df30
Pure Prop Address: TMZrvHTgr8uHgFqVfwzAjaHWjcyWUe9ku88
Theory Prop Id: 272ea646fea8c08826956116e3b06f06b4f1a29035d7099f3e17dbff9743db74
Theory Prop Address: TMdJyqAKqfUoMVXYdkRynaBe5Y4UCycBCSi

explicit_Nats_one_mult_N

Theorem 22.17 ∀nm ∈ N.n ∗ m ∈ N. The proposition is identified by the following information:

Pure Prop Id: 71dc3009d99275dcb7507e7e7f7b53e10abcd721e2a9f4479dfb1d9626492798
Pure Prop Address: TMUF1CggpPWJLEjwmb369yXyFWWhKTPPS6u
Theory Prop Id: 6cea9937d528c3ed3844899b07adc36352b53c024fca3a3fb796832b3cb817eb
Theory Prop Address: TMLGNEexWqJMgReiwzYkhaf87Wi9rCre9U7

explicit_Nats_one_mult_1L

Theorem 22.18 ∀m ∈ N.one ∗ m = m. The proposition is identified by the following information:

Pure Prop Id: 248ac634eb92e6d0725101234c0f2912ffe95ed5d81f7568c5ebace9788fe8c6
Pure Prop Address: TMTS3c1HETRVj5fANAi3ixWKrd9G35hxXTk
Theory Prop Id: cf9d37fbd63485076deb649fa124b9d2400952c5423fe11ea9f085d236188b80
Theory Prop Address: TMTMa1DWHDPBUbN5Uh2GsHTp6jg2Ve8n4iF

explicit_Nats_one_mult_SL

Theorem 22.19 ∀nm ∈ N.S n ∗ m = m + n ∗ m. The proposition is identified by the following information:

Pure Prop Id: f260dfa960c7ea2ad1c80083a06eaed1460025b578976944e9d439072b54a833
Pure Prop Address: TMYCaevEfmMyUdnjxigfSRQFKL62bwDtseb
Theory Prop Id: 84048a4e0ac742533042895099975cda2d4ce7777ccfe9a38dfebad9d348452b
Theory Prop Address: TMT3bNHPRqWLxd5k165RfwDrLuFDDnTU4GR

explicit_Nats_one_exp_N

Theorem 22.20 ∀nm ∈ N.n^{m} ∈ N. The proposition is identified by the following information:

Pure Prop Id: e06153d7a791dd306b96ac5fa124b22a5e0e6f3c79dc892e8973612dbdbcf965
Pure Prop Address: TMbPBpFHKw4tdRPzKEnzY36NJm6Wo5NA1vq
Theory Prop Id: dc83c194406168e2322b693e0db9e9c2935f9b5890252230aac13dbba5dd04f4
Theory Prop Address: TMXt3nAqYNYZHtj3LbChY5GKH6imEYDSJJA

explicit_Nats_one_exp_1L

Theorem 22.21 ∀n ∈ N.n^{one} = n. The proposition is identified by the following information:

Pure Prop Id: 3135e1cbd0675df0f6c911e6f05dd6ea73bc4c458266e5d736412045200aae78
Pure Prop Address: TMd59g2sbanMoRmaQmddMbB1kEpMZkGsjHd
Theory Prop Id: 35371d46a876899103c3b0122a82b56515f75a0378f71012b5d765398f35d30b
Theory Prop Address: TMKNGaufMJdNVayeapLn2aUCzuEgW63wrfh

explicit_Nats_one_exp_SL

Theorem 22.22 ∀nm ∈ N.n^{(S m)} = n ∗ n^{m}. The proposition is identified by the following information:

Pure Prop Id: 7a9c28b365c7612ae00bcf5eae2460615c9a85f39e5734a2fe11b4129c6272d2
Pure Prop Address: TMbzoYWnxTt9ZksjQ6HvVxXro6tokwCYzCB
Theory Prop Id: 61fbfd0911e934537afe54a0c8fd8369d881dbd3b94b5b74c40a418954a55d7d
Theory Prop Address: TMJ3ASygyK5PDcYrS3MhKvqvt7Buzmpj1BC

Definition 22.8 We define explicit_Nats_one_lt to be

λmn.m ∈ N ∧ n ∈ N ∧ ∃k ∈ N.m + k = n

of type ι → ι → o identified by the following information:

Pure Object Id: a887fc6dd84a5c55d822e4ceb932c68ead74c9292ff8f29b32a732a2ee261b73
Pure Object Address: TMaZtSJJReNwaSNEigaPZg3GCC7FWiry1eM
Theory Object Id: d1d62283501ce0525b4885ac683b5f8872a63a79b85eaa1912e4259776dcde37
Theory Object Address: TMQmi5rw5Vmk7auDTyrnNtW84cLiTH49cT4

Definition 22.9 We define explicit_Nats_one_le to be

λmn.m ∈ N ∧ n ∈ N ∧ (m = n ∨ ∃k ∈ N.m + k = n)

of type ι → ι → o identified by the following information:

Pure Object Id: 8d7cd4bd6b239e3dc559142aa4d11d731c7a888de9faa3b9eeec2e7d5651c3fb
Pure Object Address: TMM32WFKaju25pJnuVLyA3borpF7YFkso2V
Theory Object Id: 0165a966c5de8b542de020a29a4531b3e01f1d9ab3a4aa7fba893a0bf28f0580
Theory Object Address: TMMNCaFpJUV2w4YZrBepTz2BXoitsuYhDUE

Notation. We use < as an infix operator corresponding to applying term explicit_Nats_one_lt. Notation. We use ≤ as an infix operator corresponding to applying term explicit_Nats_one_le.

22.5 explicit_Nats_transfer

Let N : ι be given. Let base : ι be given. Let S : ι → ι be given. Let N′ : ι be given. Let base′ : ι be given. Let S′ : ι → ι be given. Let f : ι → ι be given.

explicit_Nats_transfer

Theorem 22.23

explicit_Nats N base S→bij N N′ f→f base = base′→
(∀n ∈ N.f (S n) = S′ (f n))→explicit_Nats N′ base′ S′.

The proposition is identified by the following information:

Pure Prop Id: 3a37116910f44ef4e3c8921b5dcaba3ab229f567fd494a0d9f6d3fd6ac9823f8
Pure Prop Address: TMGVh4mHULAwdH24Rh3RJUPVxrmkg1EWVue
Theory Prop Id: 6c4761bc5fcc2f65480e8a6ac8bd7faf5d89a85926881bb7e4724dd397f0746d
Theory Prop Address: TMbddV6bagEzzT8wfXKJuLdPeN4XyXPbp2U

312 CHAPTER 22. EXPLICIT NUMBER STRUCTURES

Groups

23.1 AssocComm

Let R : ι be given. Let plus : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus.

AssocComm_identities

Theorem 23.1 (∀xy ∈ R.x + y ∈ R)→(∀xyz ∈ R.x + (y + z) = (x + y) + z)→(∀xy ∈ R.x + y = y + x)→∀p :o.

((∀xyz ∈ R.x + y + z = y + x + z) →

(∀xyz ∈ R.x + y + z = z + x + y) →

(∀xyzw ∈ R.(x + y) + (z + w) = (x + z) + (y + w)) →

(∀xyzw ∈ R.x + y + z + w = w + x + y + z) →

(∀xyzw ∈ R.x + y + z + w = z + w + x + y) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: c45653e3c80cd999509820d81f22ae3c47b5cd44421738f32eda320ba570c87e
Pure Prop Address: TMGvdvL6fepmXewTz9sJwhigZicrLoSXLZZ
Theory Prop Id: 40e443867ab4b8a14aafd8ab67f7ac4ae69383ab44ea2a4da6420b281d0d0628
Theory Prop Address: TMQfH9YcUAFRqr7yeAgZh2h8mk1nc3GYCto

23.2 Group1

Let G : ι be given.

314 CHAPTER 23. GROUPS

23.2.1 Group1Explicit

Let op : ι → ι → ι be given. Notation. We use ∗ as a right associative infix operator corresponding to applying term op.

Definition 23.1 We define explicit_Group to be

(∀ab ∈ G.a ∗ b ∈ G) ∧ (∀abc ∈ G.a ∗ (b ∗ c) = (a ∗ b) ∗ c)∧ ∃e ∈ G.(∀a ∈ G.e ∗ a = a ∧ a ∗ e = a)∧

(∀a ∈ G.∃b ∈ G.a ∗ b = e ∧ b ∗ a = e)

of type o identified by the following information:

Pure Object Id: 0edee4fe5294a35d7bf6f4b196df80681be9c05fa6bb1037edb8959fae208cea
Pure Object Address: TMKHXb8sJYXkkV8waqUHeCymoBGxrmp2eJ7
Theory Object Id: a0d4c8bd9f788ef787589b54b2745f33864b9386aad8d7b99e0c747d5cd502b8
Theory Object Address: TMNeD6sRqkAPQrbpVxeWHbWkWuomaavLf35

explicit_Group_identity_unique

Theorem 23.2 ∀ee′ ∈ G.(∀a ∈ G.e ∗ a = a) → (∀a ∈ G.a ∗ e′ = a) → e = e′. The proposition is identified by the following information:

Pure Prop Id: fe12d74789aa4b30c5b553f9b583cc7dac65508fc7b435345c330c9c50b7bda4
Pure Prop Address: TMX43i28gCdiXowFyXeB4bgH5QZTmM6g1nh
Theory Prop Id: 4b4d44f80512969adc0813a098b3d4bc6f42912746d2cd4de30c9b4dc1fd8c1d
Theory Prop Address: TMcvB6UWPmzCrw9QT1edWJq9i5gU4ihPnYh

Assume the following.

explicit_Group  (23.1)

Definition 23.2 We define explicit_Group_identity to be

Eps_i (λe.e ∈ G ∧ ((∀a ∈ G.e ∗ a = a ∧ a ∗ e = a) ∧ ∀a ∈ G.∃b ∈ G.a ∗ b = e ∧ b ∗ a = e))

of type ι identified by the following information:

Pure Object Id: 8a818c489559985bbe78176e9e201d0d162b3dd17ee57c97b44a0af6b009d53b
Pure Object Address: TMMh2NL7Q85JJXCFBeNsaMqMcMPaFhqKbfP
Theory Object Id: 84a58742704bbb5cd7767ccb82e978851abf4fb9bc5ab7c27851fdc99d2cd664
Theory Object Address: TMdVb1nKZjH4P19ib11PFY6Cka8NUiF79Re

Let e be explicit_Group_identity.

23.2. GROUP1 315

Definition 23.3 We define explicit_Group_inverse to be

λa.Eps_i (λb.b ∈ G ∧ (a ∗ b = e ∧ b ∗ a = e))

of type ι → ι identified by the following information:

Pure Object Id: e8cf29495c4bd3a540c463797fd27db8e2dfcc7e9bd1302dad04b40c11ce4bab
Pure Object Address: TMHFMiwrvo9LoSX8Wi5EZvS4ccFWXeWxrgx
Theory Object Id: a89687cfcbd45caa52b5cdf460d31291544f843439c0fb0cac9bd218223d0a98
Theory Object Address: TMJDvZFoNydRtF58jVXVmTRo5GVfkFbJTHb

Notation. We use −^{−} as a postfix operator corresponding to applying term explicit_Group_inverse.

explicit_Group_identity_prop

Theorem 23.3

e ∈ G ∧ ((∀a ∈ G.e ∗ a = a ∧ a ∗ e = a) ∧ ∀a ∈ G.∃b ∈ G.a ∗ b = e ∧ b ∗ a = e).

The proposition is identified by the following information:

Pure Prop Id: 068ffab150dc76c7de09d187943d2888faceaba1fbdc6449b322c8d4ef63ac4c
Pure Prop Address: TMM9Eh1qhaiVeZj5Wgcxuvug9QQBNRfsEoz
Theory Prop Id: c36841621d7b57d11c84af6cd0ed9c8fba75205040141ec30f15c1792884c83e
Theory Prop Address: TMWKjEwmnVGHJzPZq7tPzu7AEs2mtZfK8zJ

explicit_Group_identity_in

Theorem 23.4 e ∈ G. The proposition is identified by the following information:

Pure Prop Id: 696ec40c6c7d83676b8638c42bd7832dd823bca9c973ae6f9c4f06cfa23bfc08
Pure Prop Address: TMSoutuNF8eeWkkhpwkABkzngwXxdhPEeii
Theory Prop Id: 8cf78c6448b0a5a1eafb0e5c957793ae6ea2c4988a86e53cde584dd0291d61f3
Theory Prop Address: TMHmwXi5SHMeK3Us86CEgisvDGA9xNi341B

explicit_Group_identity_lid

Theorem 23.5 ∀a ∈ G.e ∗ a = a. The proposition is identified by the following information:

Pure Prop Id: 0a8b96c11b2713c863850c5c7d4857de37aa4fca73f7c304457b56e2f95e16e3
Pure Prop Address: TMbD19oggdaE7cVLwcWS7ssWyd1V2HcUnDW
Theory Prop Id: c412492ac795885c6900db2fe22e1dd552814f2557d6d79041f6191b3689beed
Theory Prop Address: TMPTJy8uG71biAxCkf8AvzGecNsAuAUaKnK

explicit_Group_identity_rid

Theorem 23.6 ∀a ∈ G.a ∗ e = a. The proposition is identified by the following information:

Pure Prop Id: f6f5808dcc0c330452a517b4ebf8456cd5238d7b46e8111ab177813e331aa8b8
Pure Prop Address: TMVgRDZiCVC1sCRnAp2nQ2asZP7Bjg3oe5e
Theory Prop Id: ac30373bfbf244191de0c19f090210f9903310858129ab85808112737072b614
Theory Prop Address: TMLhbf6oyewQ17oxmbzXFC3ejnSF86C5A6t

explicit_Group_identity_invex

Theorem 23.7 ∀a ∈ G.∃b ∈ G.a ∗ b = e ∧ b ∗ a = e. The proposition is identified by the following information:

Pure Prop Id: a48e2c7de86dcd94f3d0b7870a3a456284feaaf5d5c20e80a45b93114c6d50d5
Pure Prop Address: TMW4eSZUkr3refSrUsK582eNHJhJwavEYV6
Theory Prop Id: aee17204445b1eac60e7b66a3cdd8608a0182bd2a434fc82de1a92678591d2a3
Theory Prop Address: TMTWCgk93fRtERh1PSJNhibGca6EnEgcnfe

explicit_Group_inverse_prop

Theorem 23.8 ∀a ∈ G.a^{−} ∈ G ∧ (a ∗ a^{−} = e ∧ a^{−} ∗ a = e). The proposition is identified by the following information:

Pure Prop Id: 0f17df9988cfc01139158b40656a874d8f2118364e3b47b12b829509448c3c11
Pure Prop Address: TMNhTk43Kf2YzVaFqj6KragMTmudKkw3kGf
Theory Prop Id: 6781d5952539e6702dfb38b38d848b596c60f3f0e775d5ca957f11e8938fcfe3
Theory Prop Address: TMMJ2fwNJJe1W18rM6iSWAfyEcKkyVumryi

explicit_Group_inverse_in

Theorem 23.9 ∀a ∈ G.a^{−} ∈ G. The proposition is identified by the following information:

Pure Prop Id: 0f9bac9fe458ed7636280b89adbff7d9c00215f5f4364d4666a09f2fed583992
Pure Prop Address: TMJm8VUj9hoafxu2Z8k99ys52dNjtstV9hY
Theory Prop Id: c688e8369c999b2423047a981d3b5c198fb153c1d0f7e6dbee92f27ff38da254
Theory Prop Address: TMbVkHwsuxJFdmgosfxN8dJkV4edBZBQ8Xf

explicit_Group_inverse_rinv

Theorem 23.10 ∀a ∈ G.a ∗ a^{−} = e. The proposition is identified by the following information:

Pure Prop Id: 5faedae00b2ff3112d18fc67a44cc32e8836e3aefdb6f82dd4e11df8648381dc
Pure Prop Address: TMNsYoMt2v2YNvxeJ1MzMMweHHpoJi62UCS
Theory Prop Id: 6522bcdcc934d3748a0dfcbd204413a2501816629e1eddf53c5b4d871f78ca57
Theory Prop Address: TMc4sqjSkyuS9R2hVFzNDY16nePbzE5g4h8

explicit_Group_inverse_linv

Theorem 23.11 ∀a ∈ G.a^{−} ∗ a = e. The proposition is identified by the following information:

Pure Prop Id: 434e096992c877eb38298cea7cc6fcf3ec01d9afbbcbd3b0783a632c67f5deff
Pure Prop Address: TMTD5RfGMs4wShS2nexxcTshjmbsydNaghn
Theory Prop Id: 6b2fca7f46e1532922eaca0735e71402b9a2629f54eb3f3571f4ca378487902d
Theory Prop Address: TMExrQ2X7HUbLBQZmQuMUtKf8RDh4mQWHWu

explicit_Group_lcancel

Theorem 23.12 ∀abc ∈ G.a ∗ b = a ∗ c → b = c. The proposition is identified by the following information:

Pure Prop Id: 33b1b149cdeee23d3bb33fc0a6188a68ffa7bd8e41332d8be527ae438267f3d4
Pure Prop Address: TMX6h7qhm3NenK6T4aPYQn1M9426HgYTMLw
Theory Prop Id: 826b2e6618fa8fe5b8cff0da58e5a8f4627dcb7b4323cd9e55e91ce57f05280e
Theory Prop Address: TMSLWhCZYGhVwR8odE7nnVNZCRLJR4iruFv

explicit_Group_rcancel

Theorem 23.13 ∀abc ∈ G.a ∗ c = b ∗ c → a = b. The proposition is identified by the following information:

Pure Prop Id: ef05cefa790fecef7769c37f99ae58c2563aaa45db53a8bee86258c6b886132a
Pure Prop Address: TMWrcUsJHruC5nYp5yAKHRzmwgkFPmSofTL
Theory Prop Id: 953b77b05a77af06ccb9cc940e52712be361bd38ef21fd159599be906700fe65
Theory Prop Address: TMGiX5S1yoJfx8UWN5xD5FJqKeuhVYzgttD

explicit_Group_rinv_rev

Theorem 23.14 ∀ab ∈ G.a ∗ b = e → b = a^{−}. The proposition is identified by the following information:

Pure Prop Id: 891f7754b53564fca6f0343cad2dc494efc6e05e34c918bed88fcbddc533265f
Pure Prop Address: TMGNPsYapqknYHxQsZ2F5XoqR6d9prQyeeR
Theory Prop Id: a04aee87622e0b7dc5811e087ae8a1b75632125b99c1aa5cb6c6c3c9beceb1fc
Theory Prop Address: TMSE4wKdLGWD9NZm2jRusJ5H4TwxUYDkdzK

explicit_Group_inv_com

Theorem 23.15 ∀ab ∈ G.a ∗ b = e → b ∗ a = e. The proposition is identified by the following information:

Pure Prop Id: 663c8f6ce8987057ef1a016fc2d2eb826895e0f18480b99833b0f0990206672c
Pure Prop Address: TMaSbuU5FwcNTsVwQ6PtZ6jLP8LYTgUxsUb
Theory Prop Id: f1098fa01f9673e741bc27d240745fb3bfa00795281c4048f9003c7cc6b2f23d
Theory Prop Address: TMYsR6YLMhEDxsZDgKWug5MYnBqnSoiNUZg

explicit_Group_inv_rev2

Theorem 23.16 ∀ab ∈ G.(a ∗ b) ∗ (a ∗ b) = e → (b ∗ a) ∗ (b ∗ a) = e. The proposition is identified by the following information:

Pure Prop Id: 68dbad203c9e4e0b7716e921ed000127a9cf37e288fd5c525a1cdd3a75b5471f
Pure Prop Address: TML3EFwc6aWd8wgzpvyuEJ5Ygy5et6QXvtZ
Theory Prop Id: c5aeaedf7b6e943800ba4479c3f4fbb8514a2e20caea2699de720d8d40aec27b
Theory Prop Address: TMRZWZ4JLKtefjP9iqT3BqVemyLiAo6X7VF

Definition 23.4 We define explicit_abelian to be ∀ab ∈ G.a ∗ b = b ∗ a of type o identified by the following information:

Pure Object Id: 05d54a9f11f12daa0639e1df1157577f57d8b0c6ede098282f5eca14026a360f
Pure Object Address: TMSbRNehAadQ85Kiirs6CeWkovxtJVvZdLQ
Theory Object Id: afb9d62b583f9c34b2932e79c800886506cf83a6d1957f0f38ea4b6120525bf5
Theory Object Address: TMWoaUJJFHkS4LvMBAUyKZxmsGTDk35A5mR

23.2.2 Group1Explicit2

Let op : ι → ι → ι be given. Notation. We use ∗ as a right associative infix operator corresponding to applying term op.

Group1Explicit2RepIndep

Let op′ : ι → ι → ι be given. Notation. We use × as a right associative infix operator corresponding to applying term op′. Assume the following.

∀ab ∈ G.a ∗ b = a × b  (23.2)

explicit_Group_repindep_imp

Theorem 23.17

explicit_Group op→explicit_Group op′.

The proposition is identified by the following information:

Pure Prop Id: f5012ed8643556d71d97efacc96365a3e64c4075b67a87d5e41d6946ea117e15
Pure Prop Address: TMK2g6qJebDnBquMej7EVWnZiqivTV1697r
Theory Prop Id: 9d8852ec071692482dd632725271f66464e62a8187bc25f252ba7aa90d3a2c38
Theory Prop Address: TMUCkaJVFQzcD3xM53V7UGDtKmLnoAfRCF8

Let e be explicit_Group_identity op. Let e′ be explicit_Group_identity op′.

explicit_Group_identity_repindep

Theorem 23.18 explicit_Group op → e = e′. The proposition is identified by the following information:

Pure Prop Id: 598ff1a23c1fe4126915975f010569b525215d6ff2b98a5ab546dd324cbccaf2
Pure Prop Address: TMcA9q6i8Vw98yFDFEvLQqJQQv6mxymW78w
Theory Prop Id: bf4147f42cb6909b3123e55bbdd9abcd5e4500bf4c228aba6d90e85f3863fc07
Theory Prop Address: TMGGsZcTZdyUMrW4cjQ2TioN1uRYTKdww5G

Let inv be explicit_Group_inverse op. Let inv′ be explicit_Group_inverse op′.

explicit_Group_inverse_repindep

Theorem 23.19 explicit_Group op → ∀a ∈ G.inv a = inv′ a. The proposition is identified by the following information:

Pure Prop Id: 1b3e083dd9cf397ee3bc561cfa6db02990a3cf16bf8021b5b480a16f23aeb4b6
Pure Prop Address: TMGDnT1Aqj6HwvAs6czLDGM8YE7vWNHc8xc
Theory Prop Id: bce8beb409ddbfeedc408e86bebac1a0f9ce7b545243ee799f091e40547b0865
Theory Prop Address: TMcrPmRNvDAbfSmF1GFzJbteMNNZLQJGYpX

explicit_abelian_repindep_imp

Theorem 23.20 explicit_abelian op→explicit_abelian op′. The proposition is identified by the following information:

Pure Prop Id: ef5af4d7ebcdc54f16dcbd6735446df711e99129ddf89eacd2c278c216ee4783
Pure Prop Address: TMPnVDeP4gfRfuDBEtT7nJNFcLgKCXaRHWg
Theory Prop Id: 6dc553047a706e22ba94296abf5b0db1652898d70d735cf338004f26e3e4c8b4
Theory Prop Address: TMQeZC612yR6Xczxn9AVL8mF3Ukm9V2F2bL

23.2.3 Group1Explicit3RepIndep

Let op : ι → ι → ι be given. Notation. We use ∗ as a right associative infix operator corresponding to applying term op. Let op′ : ι → ι → ι be given. Notation. We use × as a right associative infix operator corresponding to applying term op′. Assume the following.

∀ab ∈ G.a ∗ b = a × b  (23.3)

explicit_Group_repindep

Theorem 23.21 explicit_Group op ⇔ explicit_Group op′. The proposition is identified by the following information:

Pure Prop Id: 4c880a72b808ad924e81f56a1ed90dd384b42d8c05b5a34106946de015681f8b
Pure Prop Address: TMYoUUzVknsDHuBdb4CTMFGTQJ8RavVXrEb
Theory Prop Id: f0f9a2f129898373c5727edb2324561e41b4d2ee82534892365f8605e104fa10
Theory Prop Address: TMdA43jg23cA8VmWUWe6knaNXVhz835Psjq

320 CHAPTER 23. GROUPS

explicit_abelian_repindep

Theorem 23.22 explicit_abelian op ⇔ explicit_abelian op′. The proposition is identified by the following information:

Pure Prop Id: 4bf05bb4c1ce5486164c944aa718fe7c76541eaa979f90733925e0df5da80b31
Pure Prop Address: TMK9RV3nDAmdTWLAZyJDHS6BozLokJ5gF8H
Theory Prop Id: 9c96138929f1974912de1c755238da49ddca4b64627b220e7717c3142c9bffbb
Theory Prop Address: TMJUpZdPq3Ps6CrXDg4pSuMXobz6F6aTgdA

23.3 Groups Encoded as a Set

Definition 23.5 We define Group to be

λG.struct_b G ∧ unpack_b_o G explicit_Group

of type ι → o identified by the following information:

Pure Object Id: 3bcfdf72871668bce2faf4af19b82f05b1ff30e94a64bbc83215c8462bc294ca
Pure Object Address: TMPgSXfSPyvihsFWLDa9H7w9DDaHAAqfKWh
Theory Object Id: 48ad758ca1725528973592bef2eca92dd3b4cc57fcc1cec0478a8f8e97376e86
Theory Object Address: TMVKTH5owDCXYfd1wFd2cm4yxGHnuLN5CXz

Definition 23.6 We define abelian_Group to be

λG.Group G ∧ unpack_b_o G explicit_abelian

of type ι → o identified by the following information:

Pure Object Id: c9c27ca28ffd069e766ce309b49a17df75d70111ce5b2b8e9d836356fb30b20a
Pure Object Address: TMZYJxov52ZDnfLrrgP436fnEtrVohjgYF3
Theory Object Id: 82b65a8b4d905ce66d83d96cd51c72493e71be4e889671636b43a03ef4993b11
Theory Object Address: TMKxwHVb8mydLee3wEkwZAbNhhw23ui2AP7

Group_unpack_eq

Theorem 23.23

∀G.∀op :ι → ι → ι.

unpack_b_o (pack_b G op) explicit_Group = explicit_Group G op. The proposition is identified by the following information:

Pure Prop Id: aceab134c9adba80930143837f535ea11b55ad2ca8eaf42a51a35901b0783c35
Pure Prop Address: TMd7GQcSo21uLBQ7A49xYT78iKSY3PhWQFB
Theory Prop Id: f34cbdd88f4d553a25bb8ac2222863ca028a51086ae3e0f2d11a1fe78e716303
Theory Prop Address: TMZ6K2cAEkL2uXg3fBxTcxoJSRUaJHicJmH

23.3. GROUPS ENCODED AS A SET 321

GroupI

Theorem 23.24

∀G.∀op :ι → ι → ι.explicit_Group G op→Group (pack_b G op). The proposition is identified by the following information:

Pure Prop Id: f59584e74c871bfb0a1036e424d097143949b1923d9616b3320eb90f399a91ed
Pure Prop Address: TMQUiQQxmEfgMnDao2t2u2KdoezR3zraGZY
Theory Prop Id: 7a80f33492e5b3d03f1841fd7ccbd5fb77c2faa6b30b5c242db4c0059fc6cdb9
Theory Prop Address: TMTYbJ6XVymYaZkodwF8XHTGRBpsABB8dxk

GroupE

Theorem 23.25

∀G.∀op :ι → ι → ι.Group (pack_b G op)→explicit_Group G op. The proposition is identified by the following information:

Pure Prop Id: f3f463245e1c5c9686fda5a34894e9f282a8f86a019ada8b14ef0e11b0976d07
Pure Prop Address: TMHNfqMd8twRy3pLuxLepizuzSei8dgrR2i
Theory Prop Id: 2239355bbc80224799ae9be8d5818f67132ab3353e50b12be897026966f41a8f
Theory Prop Address: TMUKjYUwFEKr2Brho6Mkc5ukMBKuMH2PP4M

abelian_Group_unpack_eq

Theorem 23.26

∀G.∀op :ι → ι → ι.

unpack_b_o (pack_b G op) explicit_abelian = explicit_abelian G op. The proposition is identified by the following information:

Pure Prop Id: 2346855e53255b79e76009eda280acf374dad7dcc5e72ab74d2f42201d79140a
Pure Prop Address: TMMHnLwWhH3nJZkk281hptKVfm4dKceyAwW
Theory Prop Id: 6dfe13367c39b0c86ac8656ad108a540b7a3f5c3364369c3dea90e33b297a4a2
Theory Prop Address: TMWfTBZCZjhKbb8QKapfLpSa2pyTTfAE71m

abelian_Group_E

Theorem 23.27

∀G.∀op :ι → ι → ι.abelian_Group (pack_b G op)→ Group (pack_b G op) ∧ explicit_abelian G op.

The proposition is identified by the following information:

Pure Prop Id: 0ddfdd60c899db663169e18602174fc26a195f4e543a49ea6ff9d85c51d1db1d
Pure Prop Address: TMFf47YDuvj4E1VPha1jb6n3UhV7xC9VtyU
Theory Prop Id: 6ec272a62e16860ef6fdf6e1827bf60d5f1da411baf1baa0a7b2319ae3c3300a
Theory Prop Address: TMQz71B8ZwyWMJ8Y7PhfRsqJmZ5rqBNeqbW

23.4 Group2

Let G : ι be given. Let op : ι → ι → ι be given. Notation. We use ∗ as a right associative infix operator corresponding to applying term op. Notation. We use −^{−} as a postfix operator corresponding to applying term explicit_Group_inverse G op. Let H : ι be given.

Definition 23.7 We define explicit_subgroup to be Group (pack_b H op) ∧ H ⊆ G of type o identified by the following information:

Pure Object Id: 4f273e5c6c57dff24b2d1ca4088c333f4bbd2647ab77e6a03beedd08f8f17eb9
Pure Object Address: TMNFW3sUiRGF4QZDyZs4U1ZFwLXYNZndEUA
Theory Object Id: a5189d9d38cb9e37388edfc6167f21fbbac2dae96c0d30a42796f7f45363ba49
Theory Object Address: TMUKd1esxsYP6brvBoQj3SzvUxCbvF2x7bC

Definition 23.8 We define explicit_normal to be ∀x ∈ G.{x ∗ a ∗ x^{−}|a ∈ H} ⊆ H of type o identified by the following information:

Pure Object Id: 1cb8f85f222724bb5ee5926c5c1c0da8acd0066ba3fa91c035fe2fb8d511a035
Pure Object Address: TMUp97JhNka3EMtq317xqSLU7QwwU9Rv3di
Theory Object Id: d7a9568bb9deb69b03d47d288813c29f08fcc8800ee11aed9df4f6a44df35147
Theory Object Address: TMdaRbf7U1umHdo2j1hDVsveTbdoBQj6QLo

Assume the following.

Group (pack_b G op)  (23.4)

Let e be explicit_Group_identity G op.

explicit_subgroup_test

Theorem 23.28

H ⊆ G→e ∈ H→(∀a ∈ H.a^{−} ∈ H)→(∀ab ∈ H.a ∗ b ∈ H)→explicit_subgroup.

The proposition is identified by the following information:

Pure Prop Id: a0764d61788853004917f998bf172f98ea252833c7ffffbd4cc8f9fc6d366640
Pure Prop Address: TMF72nFcvhbTi7RSPrKiVdQH7Zno6gqKNzX
Theory Prop Id: febaefb669b389c2e7af77327b7c126d3c2296175d831b5152819092c60df749
Theory Prop Address: TMSe5XgyjkAZsLZRAWWesUNSkSR5BgeN71j

Assume the following.

explicit_subgroup  (23.5)

Let e′ be explicit_Group_identity H op.

explicit_subgroup_identity_eq

Theorem 23.29 e = e′. The proposition is identified by the following information:

Pure Prop Id: d608de6b443a17bb92b076f03900648de80c1688b0a9b18c0f6f2a87d48e9494
Pure Prop Address: TMMi9FM87WTYChZcDCYSiR1mUgcxriaPpTH
Theory Prop Id: 29bcb214853a5b2b4c2ba9cd9171fe4cd7583e3c7526e1f14f5c9d35785657f7
Theory Prop Address: TMMErcjnubQrVeuBCzarecXxmRfNpUwDVWL

explicit_subgroup_inv_eq

Theorem 23.30

∀a ∈H.explicit_Group_inverse G op a = explicit_Group_inverse H op a.

The proposition is identified by the following information:

Pure Prop Id: 23d0fd40782cc5d113a6f8abbffe684eabb63f361450d6781bee3a513762cc1c
Pure Prop Address: TMSzo58obcQmEn5FMDAcyyCsxm1mW4emXNT
Theory Prop Id: c0bd13173f3bd1ccdf37652d679abb58404d427bcf1cb4bb3209c92633bb1cb0
Theory Prop Address: TMMes5AgQed1v3SDwaTYoyxExk9LqUvpPHw

explicit_abelian_normal

Theorem 23.31 explicit_abelian G op→explicit_normal. The proposition is identified by the following information:

Pure Prop Id: 07d5b2fc245834d75ec2a313ddb687a0a4ac1c4858adfb47fb56a2b4cdadfcbb
Pure Prop Address: TMWykJghtnE3YAmhYszqL4L2WUWyWs8bESa
Theory Prop Id: 142e7aa60e5da0bf6b3fb1a0a03445c8098ce5c7613ce5f3f698f2493d8311bb
Theory Prop Address: TMPfm754BwmkjxHRPzWnL1tdpxUusE7tyXy

23.5 Group3

Let HG : ι be given. Let op, op′ : ι → ι → ι be given. Notation. We use ∗ as a right associative infix operator corresponding to applying term op. Notation. We use −^{−} as a postfix operator corresponding to applying term explicit_Group_inverse G op. Notation. We use × as a right associative infix operator corresponding to applying term op′. Notation. We use −^{−} as a postfix operator corresponding to applying term explicit_Group_inverse G op′. Assume the following.

explicit_Group G op  (23.6)

Assume the following.

H ⊆ G  (23.7)

Assume the following.

∀ab ∈ G.a ∗ b = a × b  (23.8)

explicit_normal_repindep_imp

Theorem 23.32 explicit_normal G op H→explicit_normal G op′ H. The proposition is identified by the following information:

Pure Prop Id: 3085191447907ed870314c89a8d28cb2745242ad5fc8d8fac5e605f123406e41
Pure Prop Address: TMF9MV6sqG4RPotXsm4dDP2WeBrREMZMpDz
Theory Prop Id: 9e1540196415c21f2eff94a4d69154a46a0a7f369764af1fe8c67431af6bab59
Theory Prop Address: TMFwHf4QsoVfQnoVRPxuR1DDrZAUtc9aPLc

23.6 Subgroup as a Relation on Encoded Groups

Definition 23.9 We define subgroup to be

λHG.struct_b G ∧ struct_b H∧

unpack_b_o G (λG′op.unpack_b_o H

(λH′_.H = pack_b H′ op ∧ Group (pack_b H′ op) ∧ H′ ⊆ G′))

of type ι → ι → o identified by the following information:

Pure Object Id: 994c6976ade70b6acb23366d7ac2e5d6cf47e35ce09a6d71154e38fd46406ad1
Pure Object Address: TMLRkyR7sy5G519JM6FKX7WCKVJc2Xnbd7n
Theory Object Id: cb09cb23c96a2afb0a031b5795b5906505c2d764fd807be41d9c9ed8115f7b0e
Theory Object Address: TMGMFfq46mwEhdYGyNc2EdYoj7nzTA8Jomq

Notation. We use ≤ as an infix operator corresponding to applying term subgroup.

Definition 23.10 We define subgroup_index to be

λHG.unpack_b_i G

(λG′op.{n ∈ omega|∃f ∈ G′^{ordsucc n}.∀ij ∈ ordsucc n.i ≠ j → ∀ab ∈ H 0.op (f i) a ≠ op (f j) b})

of type ι → ι → ι identified by the following information:

Pure Object Id: 1dd206bf7d1aea060662f06e19ec29564e628c990cb17f043a496e180cf094e8
Pure Object Address: TMPcqQwKgtAXH5j1TdcLe1pH8wNiDMvYyEf
Theory Object Id: 7ebea62699ca1f7386cd7aa4aca981fd2c9635dc208c7d879649411943005d9d
Theory Object Address: TMHQwMsB4gE9yRryzdXDQJVWgJ17RM6aMp4

Definition 23.11 We define normal_subgroup to be

λHG.H ≤ G∧ unpack_b_o G

(λG′op.unpack_b_o H (λH′_.explicit_normal G′ op H′))

of type ι → ι → o identified by the following information:

23.6. SUBGROUP AS A RELATION ON ENCODED GROUPS 325

Pure Object Id: 071c71e291b8c50d10f9d7f5544da801035d99ff3a68c9e5386b4087d22b5db2
Pure Object Address: TMd7xWz8FWuB8j5DzNNXPoPYjwwcw9iGFfh
Theory Object Id: e2b2818e909567261bb44f9241b012f127d9ff02c3dd0d03ad5c7d708a41ee32
Theory Object Address: TMUQxZTG9uZr52Krep3t5yfZUjXQwpCSMDD

pack_b_subgroup_E

Theorem 23.33 ∀HG :ι.∀opHop :ι → ι → ι.pack_b H opH ≤ pack_b G op→ pack_b H opH = pack_b H op ∧ explicit_subgroup G op H.

The proposition is identified by the following information:

Pure Prop Id: 7125ffccb47b1cd6d83c70896cd0d77ebc0a5c375a659546c75d43ff0d142474
Pure Prop Address: TMUbUKT8Uymhwq1sqJm6JGztSHZZZywozop
Theory Prop Id: 071cdd18db9355616c94b43f39efdd84fb6821f364314b2cb4f40e9bd1675033
Theory Prop Address: TMaQ7YHCEZFXsZgr3AL8pKTCzucZbFxr6jW

subgroup_E

Theorem 23.34

∀HG.H ≤ G→∀q :ι → ι → o.

(∀HG.∀op : ι → ι → ι.(∀ab ∈ G.op a b ∈ G) →

Group (pack_b H op) → H ⊆ G → q (pack_b H op) (pack_b G op))

→q H G.

The proposition is identified by the following information:

Pure Prop Id: 8d2e3a1f4dd63f83ee62c1339fd126e053185c4e48fc3cc95c24bf2ccac05098
Pure Prop Address: TMYiapjALEQg8YorewqjMBBDNZSfdKWuMUd
Theory Prop Id: 664898a15a2a44476b34fcd20187cb6b36a7be22f8e5a6208da45360a43d39bf
Theory Prop Address: TMR3yhY1JLi3dUUZzQR9GUXSZroHRWT43WE

abelian_group_normal_subgroup

Theorem 23.35

∀G.abelian_Group G→∀H.H ≤ G→normal_subgroup H G. The proposition is identified by the following information:

Pure Prop Id: 4382dad503528fbf8173cf9e47b154aca06b57bf02b22eb6830a50e23b94f865
Pure Prop Address: TMTKNLNxcPh9EVds4cHdQmAhf5w3KjMpXZk
Theory Prop Id: 31f5b8608189f9d346c09a8ca7beb6a0955a20f1ee768bbad1c3a8d8c9eff610
Theory Prop Address: TMRsrDZNbLqAcJgVD7iy9CvAaGQiKeavic1

subgroup_transitive

Theorem 23.36 ∀KHG.K ≤ H → H ≤ G → K ≤ G. The proposition is identified by the following information:

Pure Prop Id: c84fb0f00761163c826038c45b5a080d551a4af82e13ecdbec4190f7d264c8c6
Pure Prop Address: TMRVsYwzXoyQoBSWcrLFgzK5m4LjCaiomjX
Theory Prop Id: 026e9055ebb50e4182d43a6f9cdfa349ac1e29a7844be46cdc9b70e9aa507fce
Theory Prop Address: TMQpoQCnMNzGQ5c655fPoRUpabq3NF9LuZs

23.7 Group4

Let A : ι be given. Let G be {f ∈ A^{A}|bij A A (λx.f x)}. Let op be λfg : ι.λx ∈ A.g (f x). Notation. We use ∗ as a right associative infix operator corresponding to applying term op. Let id be λx ∈ A.x.

explicit_Group_symgroup

Theorem 23.37 explicit_Group G op. The proposition is identified by the following information:

Pure Prop Id: 75533e8a735e606e28b8daa28ccee75f6b02e88687f5737d356a9e287a3636e0
Pure Prop Address: TMKpf7yzXWNWoB7pTFZ9CSXQc1WgFWKDum8
Theory Prop Id: 2004dca00796c3260ddfc226ff73cd52a9ab5d9464afb109a5e9626139500ed8
Theory Prop Address: TMUjucvp4GP3FkV7CeMpFR5TdJdrvtmChGd

explicit_Group_symgroup_id_eq

Theorem 23.38 explicit_Group_identity G op = id. The proposition is identified by the following information:

Pure Prop Id: 21cf5cbe5a190e3996df417982f48894c7fcd25e942984729664b5b8bc6e0914
Pure Prop Address: TMZFox5vkuvToBeKHtjvAXUWRxd5zPHKZEF
Theory Prop Id: fce19ddc04b4e28eca8656cf372366102a641140d6d39cdbbd0e47fd939dd647
Theory Prop Address: TMXkzvHQA9cxQakivoT7RYrtHpaUzWEVmaX

explicit_Group_symgroup_inv_eq

Theorem 23.39

∀f ∈G.explicit_Group_inverse G op f = (λx ∈ A.inv A (λx.f x) x).

The proposition is identified by the following information:

Pure Prop Id: 569ca8a98ff593e668e81c3c75a51082eb69d41932f6f1b3eb2d602a460fcb77
Pure Prop Address: TMSEXyQTYWdXvSSZeGpr2AiQNQYfM8psrG6
Theory Prop Id: 9b2215cf55968e6869adb6aba2a45d58065e1806de069a6c1f8eba372d33e15e
Theory Prop Address: TMKp3LXskT1sHdd7hevEZu5XUkBG1q5YCR9

Let B : ι be given. Let H be {f ∈ A^{A}|bij A A (λx.f x) ∧ ∀x ∈ B.f x = x}.

explicit_subgroup_symgroup_fixing

Theorem 23.40 B ⊆ A → explicit_subgroup G op H. The proposition is identified by the following information:

Pure Prop Id: bbacee9bc55d4bb9d8a095e058cf6c04648223d4da35a401d88ff34482c4bfa6
Pure Prop Address: TMWhKgrPUBPXmguhqdDXCXSxNBeVZqTWqt7
Theory Prop Id: 9bbf1dff7d86d24b11b8fc750a9c1358c88d26f302226f1a0f7af5c4b779e747
Theory Prop Address: TMQPazgEs2ZAxG7K58ru8aJWKhEasAJanUc

23.8. GROUPS OF PERMUTATIONS 327

23.8 Groups of Permutations

Definition 23.12 We define symgroup to be

λA.pack_b {f ∈ A^{A}|bij A A (λx.f x)} (λfg.λx ∈ A.g (f x))

of type ι → ι identified by the following information:

Pure Object Id: b5e0e5eb78d27c7fd8102e3c6741350d4444905a165833aa81250cef01a1acea
Pure Object Address: TMZELiVF4MBqaUvQtfAy3PdnvoJi11ixgrf
Theory Object Id: 5a86a668a5aa4ec3f8070179712ef77612eb82d194a9d09eba866f04a67f1bb9
Theory Object Address: TMLr56c3xH1KtYDpD7smvX5TNYc9XxLWkp7

Definition 23.13 We define symgroup_fixing to be λAB.pack_b {f ∈ A^{A}|bij A A (λx.f x) ∧ ∀x ∈ B.f x = x} (λfg.λx ∈ A.g (f x)) of type ι → ι → ι identified by the following information:

Pure Object Id: 683becb457be56391cd0ea1cefb5016e2185efebd92b7afd41cd7b3a4769ac57
Pure Object Address: TMVDj8UpCzz8NDMvJKFnUV46Ac7avCYjHaA
Theory Object Id: 3ec65f6d528b1474770fc9b11779c3256c697e2ffcb88e012e04a72be7cb29e1
Theory Object Address: TMFNyn7aj1wSxx41MtoZKZ7b1VE1wtZVNQz

Group_symgroup

Theorem 23.41 ∀A.Group (symgroup A). The proposition is identified by the following information:

Pure Prop Id: 990875f1186a5c690222b6df93ca46ef542493573c3ec402b83ecaa74a396595
Pure Prop Address: TMUqQYsqDvdQsVAxhFJHuJf8qqVjQaX4xft
Theory Prop Id: aa2f4e58437f0a2d91605cef8477180bffe4c75c9f5c4c510c1123991d31284d
Theory Prop Address: TMQ1kebtip2R8A29YT8cryCxBqcSdnZRkFP

Group_symgroup_fixing

Theorem 23.42 ∀AB.B ⊆ A → Group (symgroup_fixing A B). The proposition is identified by the following information:

Pure Prop Id: 4c494d5b78b3e2f30e56cfecea27ca083d44f00a781f32acffbe38a3616a3bd0
Pure Prop Address: TMVs3oeD3WthGX2etWEzZB5jM5vfjmrzdv8
Theory Prop Id: 79320ab5a30ce81496e65d23083aa7bf3b2669673cf4d3f1da6f4cd56754af32
Theory Prop Address: TMJxYyCcFvZLw4yZZSr8PNBQfMkiujqpSn7

subgroup_symgroup_fixing

Theorem 23.43 ∀AB.B ⊆ A→symgroup_fixing A B ≤ symgroup A. The proposition is identified by the following information:

Pure Prop Id: ea4ea2cb60a4490fa7dd3a8d1121a24afe0c4f02b1e69057fa3ac1cbaa488ce1
Pure Prop Address: TMUgyamfautEr7neHo5s168PJsHpp3YAAxn
Theory Prop Id: e7086b5067a4adfaef6ac76b57d03ca28dd628c134faaba9dd82bdbe075035ba
Theory Prop Address: TMNMREeoQruvS2q4Z9DbBPztZroZJLfFJtw

328 CHAPTER 23. GROUPS

subgroup_symgroup_fixing2

Theorem 23.44

∀ABC.C ⊆ B→B ⊆ A→symgroup_fixing A B ≤ symgroup_fixing A C. The proposition is identified by the following information:

Pure Prop Id: 5b06faf92f0eabd16f89b9270b775c6a00de6b3dbc7910aba9398f6587c745ba
Pure Prop Address: TMFXcztsEWX26DZ48NuWwuvxCKRxMqxgYEw
Theory Prop Id: 93e5e57f39b365e446a7903858b9899067c88e034d90efdc581b4175c2a760c5
Theory Prop Address: TMVocWTn2DD7Q6zJVrVcTrKtjJ566LnziCr

nonnormal_subgroup

Theorem 23.45 ∃HG.Group G ∧ H ≤ G ∧ ¬normal_subgroup H G. The proposition is identified by the following information:

Pure Prop Id: 4af1eff8f561099d5165fbd9166ec526c420d387fa5843d448689ae34c64a1fd
Pure Prop Address: TMaJXbhnzpZH2dG2PZNn1vcD2Zrzy99NNUq
Theory Prop Id: f4475069cf93e4f1ed33d4f2b63efb7114719af9f10d271d932e038b09e2f797
Theory Prop Address: TMK3fNtE5vExNjZ8gT49kLhgbW7s8Mujkhm

Definition 23.14 We define normal_subgroup_equiv to be λGNab.unpack_b_o G (λGop.a ∈ G ∧ b ∈ G ∧ op a (explicit_Group_inverse G op b) ∈ N 0) of type ι → ι → ι → ι → o identified by the following information:

Pure Object Id: 8d6a0ebb123230b9386afac5f3888ea2c3009a7caabad13e57153f9a737c8d0b
Pure Object Address: TMRASNP9xsTmfEHZVz86EScPbtm9LE7RB3s
Theory Object Id: 6de522f05b2ef6abc7f410d16bdebf811c59bb4ef7f25e5a80615298e85ecef2
Theory Object Address: TMHp4buRe3CwdShZ32fRxcFLnxds4BLiMTy

Definition 23.15 We define quotient_Group to be

λGN.unpack_b_i G

(λG′op.pack_b {a ∈ G′|quotient (normal_subgroup_equiv G N) a}

(λab.canonical_elt (normal_subgroup_equiv G N) (op a b))) of type ι → ι → ι identified by the following information:

Pure Object Id: 971902aba30554ada3e97388da3b25c677413e2b4df617edba4112889c123612
Pure Object Address: TMRMVZTtqZEm7ZeJHfgYoXZDgFGDJRPPgV8
Theory Object Id: ef67eb20b92ec20f2b10d37a0da1f0c16b334205004d4b4bdc4b498bcd06a015
Theory Object Address: TMJprm3Y7UHzW78sFefvKVSjQLMnLJqyagm

Definition 23.16 We define trivial_Group_p to be λG.Group G ∧ ∀xy ∈ G 0.x = y of type ι → o identified by the following information:

Pure Object Id: 759fd51778137326dda8cf75ce644e3ec8608dfa7e59c54b364069dd6cf6dd0d
Pure Object Address: TMXgRAihjx6p6LVB2gjGrWcrkQxxPLzMt3u
Theory Object Id: 4a843e3be3a632d671082290bcb4335685122c8f5b5168ab2de4fffb849942c4
Theory Object Address: TMMyfsrQ9kVyrT23cGp37GGgZap9HT6THXK

Definition 23.17 We define solvable_Group_p to be

λG.∃n ∈ omega.∃Gseq.(∀i ∈ ordsucc n.Group (Gseq i))∧ (∀i ∈ n.normal_subgroup (Gseq (ordsucc i)) (Gseq i))∧

(∀i ∈ n.abelian_Group (quotient_Group (Gseq i) (Gseq (ordsucc i))))∧ G = Gseq 0 ∧ trivial_Group_p (Gseq n)

of type ι → o identified by the following information:

Pure Object Id: 49803500fa75d5030dddbb3d7a18d03eeebfdd720d57630629384cec6d8b3afc
Pure Object Address: TMX5XpRdUwhyCYB1UE83KHG1aehAuoESN6w
Theory Object Id: da0c1bb253cd8664d898fe71430bbdd8202b6943c0866c96597b014889d43008
Theory Object Address: TMNb2P5ANiNoTvE4HeSJnPNGJDNyZCjuTD5

Definition 23.18 We define Group_carrier to be λGs.Gs 0 of type ι → ι identified by the following information:

Pure Object Id: f0bb69e74123475b6ecce64430b539b64548b0b148267ea6c512e65545a391a1
Pure Object Address: TMY8sbcPiVs9VL311KPHpbm67U6sLM9JvF7
Theory Object Id: 9211e41eaf81bb8c152618cbe35230be60d736fcd5684c0b7bdc5ccc1ff70181
Theory Object Address: TMZQE6tPrFvJMaBcLrcMn85ZgscSeFbm4pT

Definition 23.19 We define Group_op to be λGs.decode_b (Gs 1) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 6968cd9ba47369f21e6e4d576648c6317ed74b303c8a2ac9e3d5e8bc91a68d9d
Pure Object Address: TMYTM1QQGqgVyRM9unR8CLL2scwRHYvV3c3
Theory Object Id: ed06aa34b4112f8507370b46c1c5c7de7f2f9af4f24b831d45893ad3066e994e
Theory Object Address: TMRiY7KU6BpU28KD19iwgBDLsQ5XyfpcbYL

23.9 Group2

Let Gs : ι be given. Let Gs′ : ι be given. Let G : ι be Group_carrier Gs. Notation. We use ∗ as a right associative infix operator corresponding to applying term Group_op Gs. Let G′ : ι be Group_carrier Gs′. Notation. We use × as a right associative infix operator corresponding to applying term Group_op Gs′.

Definition 23.20 We define Group_Hom to be

λg.Group Gs ∧ Group Gs′ ∧ g ∈ G′^{G} ∧ ∀ab ∈ G.g (a ∗ b) = g a × g b

of type ι → o identified by the following information:

Pure Object Id: 163d1cc9e169332d8257b89d93d961798c6847300b6756e818c4906f0e37c37f
Pure Object Address: TMG33bGq8oXGhbwW9bdnHoQwrAnZGRnqbDo
Theory Object Id: 04a4493700bf4496fc9785ab649cc965ed6f6509b5bedeb92b3cfd0295c6ab1d
Theory Object Address: TMbt3jCCTS2GpiirQWPe7YD1PGZoafrvFeD

Definition 23.21 We define Group_Iso to be λg.Group_Hom g ∧ bij G G′ (λx.g x) of type ι → o identified by the following information:

Pure Object Id: 3bc46030cc63aa36608ba06b3b00b696e1a1eb1de5346ff216ca33108220fbda
Pure Object Address: TMNoSwhDXv7bcUbtfNVM1cTexdvsoLbdLJQ
Theory Object Id: 3578401f9ab8d370c4c75e61d9c1049d50026478ddeaef7cb2f37a857cf8c8cd
Theory Object Address: TMLoxwKVttJ32tGGRhbrdMTf4SuCUFyYCkX

Definition 23.22 We define Group_Isomorphic to be ∃g.Group_Iso g of type o identified by the following information:

Pure Object Id: 727fd1e7d5e38e7145387c0a329b4de2b25f2de5d67989f11e8921bc96f4b6bd
Pure Object Address: TMPtRTKnaVTXR2dtDWrcvKMrZogxkw6v58N
Theory Object Id: 8b3f77534efc12910b019fd5efaa573bd03c8fbe3aaf9677848757f8741adaf1
Theory Object Address: TMKSQinvAqsVekHU9Jcu1GLLV1RU4ESajgh

Rings

24.1 explicit_Ring

Let R : ι be given. Let zero : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult.

Definition 24.1 explicit_Ring is the opaque object of type o identified by the following information:

Pure Object Id: aa9e02604aeaede16041c30137af87a14a6dd9733da1e227cc7d6b966907c706
Pure Object Address: TMKWfmiqi9982enMMW61SmX2e5KJYZZRvW1
Theory Object Id: 9c479e9dd34e64c77a730debb88d122fdd5adfceda11ca779159e45aed2bc6dc
Theory Object Address: TMJC4zWDzgdV3bp53NDJM4PjrbuESuq67yR

explicit_Ring_I

Theorem 24.1

(∀xy ∈ R.x + y ∈ R)→(∀xyz ∈ R.x + (y + z) = (x + y) + z)→(∀xy ∈ R.x + y = y + x)→
zero ∈ R→(∀x ∈ R.zero + x = x)→(∀x ∈ R.∃y ∈ R.x + y = zero)→(∀xy ∈ R.x ∗ y ∈ R)→
(∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z)→(∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z)→
(∀xyz ∈ R.(x + y) ∗ z = x ∗ z + y ∗ z)→explicit_Ring.

The proposition is identified by the following information:

Pure Prop Id: 22c5848914731e1acb3f2bdd3b5bea1e0b842aaaa63773bdad8403ec9062cf0b
Pure Prop Address: TMbhVQoNomR8RucZMffmjCGsSENbsaFffzF
Theory Prop Id: c833f7cffb3a99d6a671445f80dbc49660a3082bc66d24ff01289ad8615ffb3b
Theory Prop Address: TMWeeT9b2ScxzB9LDbXgaQkiZTzmNTSjyAf

332 CHAPTER 24. RINGS

explicit_Ring_E

Theorem 24.2

∀q :o.
(explicit_Ring → (∀xy ∈ R.x + y ∈ R) →

(∀xyz ∈ R.x + (y + z) = (x + y) + z) → (∀xy ∈ R.x + y = y + x) →

(zero ∈ R) → (∀x ∈ R.zero + x = x) →

(∀x ∈ R.∃y ∈ R.x + y = zero) →

(∀xy ∈ R.x ∗ y ∈ R) →

(∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z) →
(∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z) →
(∀xyz ∈ R.(x + y) ∗ z = x ∗ z + y ∗ z) → q)

→explicit_Ring→q.

The proposition is identified by the following information:

Pure Prop Id: 950aaf3139b76e70ba05068465d32a78e3f118d81f4d1e94b22cced92de5bfd0
Pure Prop Address: TMQfB6fMj33EVzRKYCyD87MzvedosSwVhJD
Theory Prop Id: 4b8365d9c8e0b8b788552aff606ae36d533041b7da02b2cec22a70afcfbd1597
Theory Prop Address: TMJpTMc6Ws4ojK4kibHze5L8RFuTnDjA4Dk

Definition 24.2 explicit_Ring_minus is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 2f43ee814823893eab4673fb76636de742c4d49e63bd645d79baa4d423489f20
Pure Object Address: TMaKRKGs3HSLGHhE6tiRaiCAnrUSperUAZx
Theory Object Id: 39f82b17ca020bfc8f422ff1f573e4b4d16cb568809ce43d0dd8180c479bac80
Theory Object Address: TMHJ1R7iBuvhhAf2k1Cz8ffCtedaaEHuKX8

Notation. We use −− as a prefix operator corresponding to applying term explicit_Ring_minus.

explicit_Ring_minus_prop

Theorem 24.3 explicit_Ring → ∀x ∈ R.−x ∈ R ∧ x + −x = zero. The proposition is identified by the following information:

Pure Prop Id: 1505204a877b431f2568dc7ac6d12545244a55ed333849833f1c46b8183d336e
Pure Prop Address: TMRsPoCLMDVwLK3rcKy5p43qryZKDLTiRyB
Theory Prop Id: 611ba38f9c0f91f22d522aa1088c3a0d2e970bdac08517c046f57e9723dea677
Theory Prop Address: TMEt9CzbdZzeQX9Zc7KR9eWrwfT3fpbr79B

explicit_Ring_minus_clos

24.1. EXPLICIT_RING 333

Theorem 24.4 explicit_Ring → ∀x ∈ R.−x ∈ R. The proposition is identified by the following information:

Pure Prop Id: f7fd70d2da1e992a18a6daffc8e53b6d72df3587a6d38db3c311d3d07eadeb34
Pure Prop Address: TMMXdMN1Ek9HzsMkM2k8YJ2yoC8vLSVSxaF
Theory Prop Id: 407c5021d140befb145ee35be8f062958adbf49ed32321837f9bdfd8cbf8bbf3
Theory Prop Address: TMFPcMzjC3fh84y93frtUFrQHpvWkhBH7dU

explicit_Ring_minus_R

Theorem 24.5 explicit_Ring → ∀x ∈ R.x + −x = zero. The proposition is identified by the following information:

Pure Prop Id: 50d8d21d95cbe4b19e35eceb1a27541f47149bceed25c6463ea708464904f821
Pure Prop Address: TMX1UUFqGBXBNndnGRpkvSGovMpgc9gU2DF
Theory Prop Id: f36bcb516e35dbe4a659c52f22434f2db5232803762843e14d255d39ff7cfe6f
Theory Prop Address: TMZLsVX3DiMMGuzxPY5SdVeZeJrSUrUPMu1

explicit_Ring_minus_L

Theorem 24.6 explicit_Ring → ∀x ∈ R.−x + x = zero. The proposition is identified by the following information:

Pure Prop Id: 0f88f84d640e937f6b72601b5dab987366579483fa4d481bcdda23f4a4ccf24c
Pure Prop Address: TMKnaY97FsvKqr4oNHqfkqDUzfWR5EDriF8
Theory Prop Id: 265b233b823a9b5403a7808c8d9fe40e67443a240aa436ac39880031c4b97c6b
Theory Prop Address: TMJDyjNAiaJZm8otfnLTZeaKs3qBbk4a5En

explicit_Ring_plus_cancelL

Theorem 24.7 explicit_Ring → ∀xyz ∈ R.x + y = x + z → y = z. The proposition is identified by the following information:

Pure Prop Id: 0d0971cc93610ddafc2ae881eca4b9cc1f517b4d2883cf319421aee337430896
Pure Prop Address: TMZdhjuSoUtSbh7ZariasMQr3GnYBuCj2DP
Theory Prop Id: 29aba412161a15a184e51ae63d56496c3de978192ab699d167c460212c724412
Theory Prop Address: TMF4XmuUfy2u5jbYgngzAWNZqQjoXEqyx7T

explicit_Ring_plus_cancelR

Theorem 24.8 explicit_Ring → ∀xyz ∈ R.x + z = y + z → x = y. The proposition is identified by the following information:

Pure Prop Id: faa841b125d630a9a1f491a07ed3a5d60de8dd20716db81e18b6f781bdc4684b
Pure Prop Address: TMcgiwGBhdYyV5TGZfhy58toP2rsnQBF51d
Theory Prop Id: db44c76a5e03e020f6aecd4b489349da20eea01ed3ccb5c1e7aba14d3b807824
Theory Prop Address: TMPsdV2xq6oy1c4yFNQy1vMJWTw1hohLykV

explicit_Ring_minus_invol

Theorem 24.9 explicit_Ring → ∀x ∈ R.−−x = x. The proposition is identified by the following information:

Pure Prop Id: d53c5c6286ff24824884d7e268b83ab80bc56b72e3ac4a5d7f66f254d781235c
Pure Prop Address: TMRSgzKJYUXvGzJtkNSKyvXy2ejXYwUhA9g
Theory Prop Id: ae91428e6764c3c11715f8090decf3cb92e118e24893f18575481d2e6e141b7f
Theory Prop Address: TMb8n5HKH6hot5PBfq3FJxG7Uy6tzs4YLHa

24.2 explicit_Ring_with_id

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult.

Definition 24.3 explicit_Ring_with_id is the opaque object of type o identified by the following information:

Pure Object Id: 51b1b6cf343b691168d1f26c39212233b95f9ae7efa3be71d53540eaa3c849ab
Pure Object Address: TMdaa5S4BKNjpqS1g4ZsN2vwScQyufyRHC2
Theory Object Id: 3383bfe28df5b99c25f43d596c0ef0217f346f7b4b9482bb0cbae34091149924
Theory Object Address: TMPc9cZFPkXaxhQPst97YbsFqjZe58N9wQs

explicit_Ring_with_id_I

Theorem 24.10

(∀xy ∈ R.x + y ∈ R)→(∀xyz ∈ R.x + (y + z) = (x + y) + z)→(∀xy ∈ R.x + y = y + x)→
zero ∈ R→(∀x ∈ R.zero + x = x)→(∀x ∈ R.∃y ∈ R.x + y = zero)→(∀xy ∈ R.x ∗ y ∈ R)→
(∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z)→(one ∈ R)→(one ≠ zero)→(∀x ∈ R.one ∗ x = x)→
(∀x ∈ R.x ∗ one = x)→(∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z)→
(∀xyz ∈ R.(x + y) ∗ z = x ∗ z + y ∗ z)→explicit_Ring_with_id.

The proposition is identified by the following information:

Pure Prop Id: 2811338562140c0cd2270de2c5b8b90a2220baae01b626fc5a2384dc56f6ee59
Pure Prop Address: TMJooYZ4CrgUzVo3Bzu48i5pHmBC9ifXNhY
Theory Prop Id: 0109ea8d1cdf78b4c4198b9dc0579e38153af90331137e279dca9bb43be8a24b
Theory Prop Address: TMZPzA5o83etUX2nE84DdwyKQ7dHD7hyBLA

explicit_Ring_with_id_E

Theorem 24.11

∀q :o.

(explicit_Ring_with_id → (∀xy ∈ R.x + y ∈ R) → (∀xyz ∈ R.x + (y + z) = (x + y) + z) → (∀xy ∈ R.x + y = y + x) → (zero ∈ R) → (∀x ∈ R.zero + x = x) → (∀x ∈ R.∃y ∈ R.x + y = zero) → (∀xy ∈ R.x ∗ y ∈ R) → (∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R.one ∗ x = x) → (∀x ∈ R.x ∗ one = x) → (∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z) → (∀xyz ∈ R.(x + y) ∗ z = x ∗ z + y ∗ z) → q) →explicit_Ring_with_id→q.

The proposition is identified by the following information:

Pure Prop Id: 8ccd42001003ed1d73a68a98715061082c13cf9b4045c5ad0972d6d2a4abc865
Pure Prop Address: TMRhQLAyL1ePgWjSAjbaKVJM3aPA2Kad84X
Theory Prop Id: 94e4966c9127d4ec26511ff21d9016555b1c64c09b5a25b9a0ff7b14ca157b51
Theory Prop Address: TML8qDN4dhpVpcJvnzRzd6pGv3x7VGoQ4ZE

explicit_Ring_with_id_Ring

Theorem 24.12 explicit_Ring_with_id→explicit_Ring R zero plus mult. The proposition is identified by the following information:

Pure Prop Id: de7598000bb3e8d9d0c079d8c393c0647add3611485f554beb972cd63b0c3e18
Pure Prop Address: TMN4a8Jp21Suwa418T4HpRTPuYS9i9K2pG9
Theory Prop Id: 6333af3f823898e2e5ef5587ca605674e78e08fc142b015e8dde0012a1de2ec4
Theory Prop Address: TMc6f6BJAGYUkRp6vVLE5Qq4oekR9n7gJ85

Notation. We use −− as a prefix operator corresponding to applying term explicit_Ring_minus R zero plus mult.

explicit_Ring_with_id_minus_clos

Theorem 24.13 explicit_Ring_with_id → ∀x ∈ R.−x ∈ R. The proposition is identified by the following information:

336 CHAPTER 24. RINGS

Pure Prop Id: 5a4877220e6d8a5f449750b45b954c43d7bb295e3e890da80fd7c2235434f467
Pure Prop Address: TMPQuSyrrDPcLZfuACKu4c53aSTWev3qgN5
Theory Prop Id: 3cfc7850da9c3580b1c5ea83a321bd1140e03b38529cd3d45e38826d00c9ef51
Theory Prop Address: TMWoZMxoSrCdixGNZ9VBKwUMqr5CAaYD96g

explicit_Ring_with_id_minus_R

Theorem 24.14

explicit_Ring_with_id→∀x ∈R.x + −x = zero.

The proposition is identified by the following information:

Pure Prop Id: f46952af2450086f002e85d2d6e0556f7f24acc2118735482f95a69d45fa52ab
Pure Prop Address: TMHCSKcKw6w715v3zWTNQjH8owToUBSXdea
Theory Prop Id: 8841ef802d29f2a91dcb38ac2f2ee8e696c5b4536b7328b1511b09500d2d0012
Theory Prop Address: TMWtKvbayU8SfroSFkejQVC6W6nbnfrvb1B

explicit_Ring_with_id_minus_L

Theorem 24.15

explicit_Ring_with_id→∀x ∈R.−x + x = zero.

The proposition is identified by the following information:

Pure Prop Id: e8d98642ee9931149e776bcdddbfcdf98466a43b2db827516266be45bd30d348
Pure Prop Address: TMb7CRWhEaMLL4W5WF5t5amnUDdRobFSVb9
Theory Prop Id: 165cdf21cd0f400b4f6c03be7116abc9566fabac1d816fc347de2a525694e3b9
Theory Prop Address: TMX35b4kkmEtvQXdySEQugbu9V37WDaikHr

explicit_Ring_with_id_plus_cancelL

Theorem 24.16

explicit_Ring_with_id→∀xyz ∈R.x + y = x + z→y = z.

The proposition is identified by the following information:

Pure Prop Id: 71bef6786a50fbc6a6a273f4a9eff4374f26d88994c8365b72e0daa0e23143e3
Pure Prop Address: TMKJ2P9pdxXugWYcPhKAPqtjdx8qCYhWmMY
Theory Prop Id: 85138e19f6a6723c0ea841496568c64f40c18e1db10b2723c6d77f8596e268df
Theory Prop Address: TMVGif5zhTjdUyXp5anvQUtBGG5rfbX7ddZ

explicit_Ring_with_id_plus_cancelR

24.2. EXPLICIT_RING_WITH_ID 337 Theorem 24.17

explicit_Ring_with_id→∀xyz ∈R.x + z = y + z→x = y.

The proposition is identified by the following information:

Pure Prop Id: 187848b39ef969f3fde507982eafd051de7ab1e0f840087b77e121e96cd2c1fc
Pure Prop Address: TMYzuoTxJr6MhTadsdKwqXhkYK3ur8MvhKJ
Theory Prop Id: 302fc924e5436eb9d2e5f9b7edefcaf0847335356058020c631fbda9c01bc7b1
Theory Prop Address: TMGbuMdJUtXoPXDJ5sUq7cTdniqN9CM7pRd

explicit_Ring_with_id_minus_invol

Theorem 24.18 explicit_Ring_with_id → ∀x ∈ R.−−x = x. The proposition is identified by the following information:

Pure Prop Id: 7e2ce6a225ab10fc23adde10859d367f9dbabbf928990919a5bc29910205476e
Pure Prop Address: TMQK76BB3wvByKH1DMd1mwhQ7mBUo3KzPwY
Theory Prop Id: b433b46a9cbc815a0d2612acd26255f6df34bacc5c54530277b91926d351f631
Theory Prop Address: TMTzGdyieMSypyUHY9Q64TiWWdbBynCRuoF

explicit_Ring_with_id_minus_one_In

Theorem 24.19 explicit_Ring_with_id → −one ∈ R. The proposition is identified by the following information:

Pure Prop Id: 85b0236407eef117b01e63557a6f6a217a4948757c19b01f2fb5174081d84eed
Pure Prop Address: TMXLf2xnGebu3EyRH1esAtDb4r35PFjLqhZ
Theory Prop Id: aed03cc170861db941e2cd313b3ec44f90bf47a77311cec49b28c6e3e4d48540
Theory Prop Address: TMGRtpEhwtc8eg1XWzU4Sk8czRAk24M35AH

explicit_Ring_with_id_zero_multR

Theorem 24.20

explicit_Ring_with_id→∀x ∈R.x ∗ zero = zero.

The proposition is identified by the following information:

Pure Prop Id: 4a62e9fcb1d64a183b6295fd4ed4aba9eb09b3e0d5f275fcd8cc12c94aa442d0
Pure Prop Address: TMcnjcPqg8wqwQFAz8NTDCL51KPVyypCAig
Theory Prop Id: abe208295f59516f79837f5f471c2dda2bf8f33f4cf65bcfd158530eff7010c3
Theory Prop Address: TMbdUb6eVFEUNrJSreGzs2bAZr6eyyrhWfd

explicit_Ring_with_id_zero_multL

338 CHAPTER 24. RINGS Theorem 24.21

explicit_Ring_with_id→∀x ∈R.zero ∗ x = zero.

The proposition is identified by the following information:

Pure Prop Id: 46fb0e1fe2518ba1e903c00b5d02eb550a82f2839ac2bc021f0537f349aec4cb
Pure Prop Address: TMLkHw8gKwxRm3iSphg4bG4HnxhbaKjTqE4
Theory Prop Id: 8352e717b959e3341c61f0f05eabef91e77b66df445e9d75d2b708e96dd469f5
Theory Prop Address: TMW7Z4P7jE8uDrrpvsx73yL58TmGFqNpuwk

explicit_Ring_with_id_minus_mult

Theorem 24.22

explicit_Ring_with_id→∀x ∈R.−x = (−one) ∗ x.

The proposition is identified by the following information:

Pure Prop Id: 9d127a8ccb192ce461b0057b3348e4222c87b1c14666f7af75a91213930b4bdd
Pure Prop Address: TMdw33XLkkeMx1fCtBccHF6mmrg1MHhJCTk
Theory Prop Id: a85fe0130867e7b63136e642b5d12bba6d678e84770dedb29cddcd2d54c237c1
Theory Prop Address: TMXjFmdfwUTe58v3XycBBnWLN3ecVdoynv2

explicit_Ring_with_id_mult_minus

Theorem 24.23

explicit_Ring_with_id→∀x ∈R.−x = x ∗ (−one).

The proposition is identified by the following information:

Pure Prop Id: 8837bf363dd2a389c6264ecdd6145862e06c0d860307051430ec19cdd534d488
Pure Prop Address: TMakKZcgjKjCdtVgW3SZdCgaxwFVjQbR5Rr
Theory Prop Id: c61eb6196305d87d5e1ff179d767abbaeb21b0e821b3a9aa65c5e08b2da13c70
Theory Prop Address: TMWyT8XGK3tNij3sgsKDRvCMa7UVxEV34Sd

explicit_Ring_with_id_minus_one_square

Theorem 24.24

explicit_Ring_with_id→(−one) ∗ (−one) = one.

The proposition is identified by the following information:

Pure Prop Id: f6b3aca4e35bed72696bb58194e762fe71df22af5aa2ba4a20eb3cb185e29174
Pure Prop Address: TMbKTvnVWYvy7JDi45TnZqMzLQeGTT488j7
Theory Prop Id: f15095975e8203242ccee867846db7d1ff22f185cd0eb5d42934be01a000ec8a
Theory Prop Address: TMXsC4FvKMruMUkYRjfKG4VMBJC1cbu9rU9

explicit_Ring_with_id_minus_square

Theorem 24.25

explicit_Ring_with_id→∀x ∈R.(−x) ∗ (−x) = x ∗ x.

The proposition is identified by the following information:

Pure Prop Id: cf6988a1ad6d9a8623c0b2b14f24faadb38aef31eae52d0060f2892b40f1403e
Pure Prop Address: TMctyY5gHQHkN8MBCzjY1dQYHmXfR3DRMpt
Theory Prop Id: 218d33fbe8087bda46c6ffc2373aac19b542cc5eca4aa7125ccf7b7701998f41
Theory Prop Address: TMWeA74hJJNZgWcZWB1H6PzQVHCow6mJYFu

Definition 24.4 We define explicit_Ring_with_id_exp_nat to be

λxn.nat_primrec one (λ_r.x ∗ r) n

of type ι → ι → ι identified by the following information:

Pure Object Id: da14f3f79323b9ad1fb102c951a6ba616694bc14e5602b46a53c3b0e4928a55e
Pure Object Address: TMH1FdiM8AidR1Vbxis6wfjL483LJqfQGSF
Theory Object Id: 8058d9275d3dce79f0324a94ee62abfb6077060a9ed0a22dff3624c09ede793f
Theory Object Address: TMPvyvYkA2Szt3fWkneXYns8V2R3guxHXSK

Notation. We use ↑ as a right associative infix operator corresponding to applying term explicit_Ring_with_id_exp_nat.

Definition 24.5 We define explicit_Ring_with_id_eval_poly to be

λncsx.nat_primrec zero (λmr.cs m ∗ x^{m} + r) n

of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 7e80cba90c488f36c81034790c30418111d63919ffb3a9b47a18fd1817ba628e
Pure Object Address: TMJ6rFroSnnr3YyYokqB1fFwqTDMseMA9Pc
Theory Object Id: a93ca6874dd0fc9d10e725ebdc680764de2c751fe2955a812a3550b87334ae51
Theory Object Address: TMMeQp2kjHK2x54q5VmuFKELPoVkugf6jb6

24.3 explicit_Ring_with_id_RepIndep2

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let plus′, mult′ : ι → ι → ι be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Assume the following.

∀ab ∈ R.a + b = a ⊕ b  (24.1)

Assume the following.

∀ab ∈ R.a ∗ b = a × b  (24.2)

explicit_Ring_with_id_repindep

Theorem 24.26

explicit_Ring_with_id R zero one plus mult
⇔ explicit_Ring_with_id R zero one plus′ mult′.

The proposition is identified by the following information:

Pure Prop Id: a5dfcc96cb00f2205c3d5f2ac6a23140d6df64432d0b8b0a6e7cbe9701a3a2b6
Pure Prop Address: TMM9oEeg2JCz3Bqz65W4NLj1nv1sKS8dasp
Theory Prop Id: 67e2ce901a576edc313e31090eebfa992581693662f54b580c5d20f054153db7
Theory Prop Address: TMaNX5eBf7WxzhXPJWRFWthj8sKB2ruu2kf

24.4 explicit_CRing_with_id

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult.

Definition 24.6 explicit_CRing_with_id is the opaque object of type o identified by the following information:

Pure Object Id: 83e7eeb351d92427c0d3bb2bd24e83d0879191c3aa01e7be24fb68ffdbb9060c
Pure Object Address: TMKfVuNP6dqFRwExvJ9WyXcYZ3NfhsZyNwK
Theory Object Id: d75af9f3e353a0cff15b19ee5f3b63b1f8cb7ac51e67f0cc5138140b1f6e3fdd
Theory Object Address: TMMxvjm7y4j3vySsRsh1Va6jygpFMDeRtdb

explicit_CRing_with_id_I

Theorem 24.27

(∀xy ∈ R.x + y ∈ R)→(∀xyz ∈ R.x + (y + z) = (x + y) + z)→(∀xy ∈ R.x + y = y + x)→
zero ∈ R→(∀x ∈ R.zero + x = x)→(∀x ∈ R.∃y ∈ R.x + y = zero)→(∀xy ∈ R.x ∗ y ∈ R)→
(∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z)→(∀xy ∈ R.x ∗ y = y ∗ x)→(one ∈ R)→(one ≠ zero)→
(∀x ∈ R.one ∗ x = x)→(∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z)→

explicit_CRing_with_id.

The proposition is identified by the following information:

24.4. EXPLICIT_CRING_WITH_ID 341

Pure Prop Id: 889096929a3004fb09e1dfb86cc3b031dce78b5ce5b2ab064f27905045936a27
Pure Prop Address: TMRNEU81o8JKA2Eh2gUrS2b8pFv5FoeCm3s
Theory Prop Id: 679bfe2ab11592d4d80ec52820c46e5c80300e609e19aa4dc28ae56e77854229
Theory Prop Address: TMKhUWhmjdKqdspZLxVwpVyWxcqC5gtPAPX

explicit_CRing_with_id_E

Theorem 24.28

∀q :o.

(explicit_CRing_with_id → (∀xy ∈ R.x + y ∈ R) → (∀xyz ∈ R.x + (y + z) = (x + y) + z) → (∀xy ∈ R.x + y = y + x) → (zero ∈ R) → (∀x ∈ R.zero + x = x) → (∀x ∈ R.∃y ∈ R.x + y = zero) → (∀xy ∈ R.x ∗ y ∈ R) → (∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z) → (∀xy ∈ R.x ∗ y = y ∗ x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R.one ∗ x = x) → (∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z) → q) →explicit_CRing_with_id→q.

The proposition is identified by the following information:

Pure Prop Id: faabb6a45219704de2f0e3fef26389bffec0af08019c3749078473ced2cb3d11
Pure Prop Address: TMMxE19BawJennoso8H8tRbuhZucCddTgHK
Theory Prop Id: efd3a525fb8c1361391f2c1f17911c282c72ed050492dba2ca64c199693f7866
Theory Prop Address: TMJJgUD4bReLi7AP1TsY9TnoFikcsgyuy3G

explicit_CRing_with_id_Ring_with_id

Theorem 24.29

explicit_CRing_with_id→

explicit_Ring_with_id R zero one plus mult.

The proposition is identified by the following information:

Pure Prop Id: fbe43396cb1d43136aff0d303b13abd85b1ea630689e55ec6a2b988778f24e48
Pure Prop Address: TMQFMJBWxHz51Y8fYrL48pNMGk7r3p4736r
Theory Prop Id: 9be399985767fb38ca0d7d6e6d6b473480388de775deabbb47204f82989a81c7
Theory Prop Address: TMSCnJQ4PTpCxDB9v4DeaGQZuAAtWyEj5cy

explicit_CRing_with_id_Ring

342 CHAPTER 24. RINGS Theorem 24.30

explicit_CRing_with_id→explicit_Ring R zero plus mult. The proposition is identified by the following information:

Pure Prop Id: 0a1028f53dfffdb66a5bc8450b30b06f47584d5a8ebb7659caccc2431eab79cd
Pure Prop Address: TMWR4HYXqHQbNAtCPpCW4WyX1g6sEgeqM81
Theory Prop Id: d4833d0cb5524bc30358a3bb532e4e71813c9a5770dc5a75251e671d49156b0c
Theory Prop Address: TMW54V1xY2GACHQtTgGxeQnSjcgYE13QF5h

Notation. We use −− as a prefix operator corresponding to applying term explicit_Ring_minus R zero plus mult.

explicit_CRing_with_id_minus_clos

Theorem 24.31 explicit_CRing_with_id → ∀x ∈ R.−x ∈ R. The proposition is identified by the following information:

Pure Prop Id: 3ae85ff26a5f71ad4869ca5b097a67259c6de609119efeb4bdc60ed927184f6f
Pure Prop Address: TMdwWexqPB1YB6dqTefAgb1hYBMdY5om5oU
Theory Prop Id: 0765b182f4deb4a94a40f4675d75a50065d365c9670a0ab0f49e558d0f85993d
Theory Prop Address: TMJ4pj8CmckG9Wt5bG8Q52MwrXF25dyk3kb

explicit_CRing_with_id_minus_R

Theorem 24.32

explicit_CRing_with_id→∀x ∈R.x + −x = zero.

The proposition is identified by the following information:

Pure Prop Id: bb27a98e4f753c9ce2de93efd3a6c3a66fcfa71c58a26ed9feff5958784d9524
Pure Prop Address: TMLxMqEk88QyHzY7bcXNWKPPvM2GmbJkpT8
Theory Prop Id: 60d4131fe989a7c001f81787ae3ed44b78153d1f852ce54a3d4141b279c5c457
Theory Prop Address: TML4H1hcc9CreDDS8GT4dGMNCmQRxHBJUkG

explicit_CRing_with_id_minus_L

Theorem 24.33

explicit_CRing_with_id→∀x ∈R.−x + x = zero.

The proposition is identified by the following information:

Pure Prop Id: 50baa96d414935b78c8004fe8327833fdc95dbb21e677a91ab1cf2c6b0fe04ca
Pure Prop Address: TMaiKT6dJBUA6GYD8CEipgDz31FuNhAsVPZ
Theory Prop Id: f84540b0cdb980f22b39fce9eca7602a4d61e86a70a72452220292d78080cb9f
Theory Prop Address: TMWLryjTxXP9PfTDUBQTJVrMoHtCqMf2Toq

24.4. EXPLICIT_CRING_WITH_ID 343

explicit_CRing_with_id_plus_cancelL

Theorem 24.34

explicit_CRing_with_id→∀xyz ∈R.x + y = x + z→y = z. The proposition is identified by the following information:

Pure Prop Id: 6f6e462ce78421c4a368b9489c238c0a56b788c33381051bc30b2344cc6fc9dd
Pure Prop Address: TMYtj8Hh526XhqURvJhj6jZ4hduf9SfkVLZ
Theory Prop Id: 1efdbbbd0d4626f6ed562dd6cf586bde2ad4047d77bf071a55596278332fd212
Theory Prop Address: TMHKn5Uz9yj1C4oNmL951CdC7QTuio1h7e7

explicit_CRing_with_id_plus_cancelR

Theorem 24.35

explicit_CRing_with_id→∀xyz ∈R.x + z = y + z→x = y. The proposition is identified by the following information:

Pure Prop Id: f7bef7cef51fefac0a8a968cd48339390194bc43db609d321297622f290f854b
Pure Prop Address: TMXr6FaGNVmrtogzEG7TRA9wchey2iGQPw1
Theory Prop Id: 251f9b25df85d0dc24f0e3a92c389f8e0d2abf089dabb0942376c7f836fada37
Theory Prop Address: TMNaQYvw47cG7Q58RrnEKj8rrfammDMrR62

explicit_CRing_with_id_minus_invol

Theorem 24.36 explicit_CRing_with_id → ∀x ∈ R.−−x = x. The proposition is identified by the following information:

Pure Prop Id: ee28f648bb60af4ab802e7550db486a9b5f33435a2ddcfad83e57e7aeb38b90c
Pure Prop Address: TMPixMacvRwLbJEEYF3Tu3BXhGhgayHXYiq
Theory Prop Id: 01d19d582f4cd5e2997e85901f8234fd2844833c0ea422c25b30fb4cb7dcfba6
Theory Prop Address: TMaM5mPayy7LFWdA8ncy5w63W2sh9AEjLdV

explicit_CRing_with_id_minus_one_In

Theorem 24.37 explicit_CRing_with_id → −one ∈ R. The proposition is identified by the following information:

Pure Prop Id: 7608d9188365499939cb1fe551eecd223870593a73f2f473c50fdf0812b607fe
Pure Prop Address: TMbawiEWXmjkJn3cNvBUYBL28Cpc9LDQEGD
Theory Prop Id: 954fe6bb57ae5b479c5d7182e7aa2ca52a9d14d7226f7296a1bc495ed1c16908
Theory Prop Address: TMXjU4EzqoWzujVJA3GSZFqn43jFuMhi3Sf

explicit_CRing_with_id_zero_multR

344 CHAPTER 24. RINGS Theorem 24.38

explicit_CRing_with_id→∀x ∈R.x ∗ zero = zero.

The proposition is identified by the following information:

Pure Prop Id: 9f49c5a1afdc52b7b5433d4d3367501e3b8caf0412846c4cf5f182b8c277de85
Pure Prop Address: TMG7cDqVVyiPU92qFsQzXosDNLq1NTSreqP
Theory Prop Id: 0c4690feb0a694a7e1bb0c027da7c70d01be7f914b759b801e5e9fdb4cf100c9
Theory Prop Address: TMSRjqxP1jRu2t7GgGMpBgEXn8ivZQWM9um

explicit_CRing_with_id_zero_multL

Theorem 24.39

explicit_CRing_with_id→∀x ∈R.zero ∗ x = zero.

The proposition is identified by the following information:

Pure Prop Id: 0a092c2a6dd656ce3cf2cbab1ff88bd05ddbfb27fe53be361696f86c4312822c
Pure Prop Address: TMMgnbsVdPLJ3iwrHZPnZiErXzyVeTyZRU2
Theory Prop Id: 924526e9dfa6060f77cadce478465ee75ea541ecc409f563b2ffe02f103abfd9
Theory Prop Address: TMMFp4RVmjUKJA9f9cUetYE3usofxBj3uuC

explicit_CRing_with_id_minus_mult

Theorem 24.40

explicit_CRing_with_id→∀x ∈R.−x = (−one) ∗ x.

The proposition is identified by the following information:

Pure Prop Id: 2f95e375de8d8d7d33d150a3a832a3f28d3f4736facce19ec50f7bf1278b5de7
Pure Prop Address: TMc3H8URNXwHnJda77991VAXe6HmNqtnjWB
Theory Prop Id: fd44c89e251ce67e81ce86520a199d0039c96c2d1f0bcfba1e2da4ebecd8add4
Theory Prop Address: TMU9zhbrpi1cxTikjTTd5CSMd1G6Kst4j5w

explicit_CRing_with_id_mult_minus

Theorem 24.41

explicit_CRing_with_id→∀x ∈R.−x = x ∗ (−one).

The proposition is identified by the following information:

Pure Prop Id: 960b3084b8e49beeea6746780f075296d84ba89bfe0c23064365d8255960dfa6
Pure Prop Address: TMT8SWemm7sAFAGRv6i8SpVrYKufCRcTNkj
Theory Prop Id: 58dd34b96bd3da4dda59ac8f027f0144f9aa188ef626da02bf86692dc0f5a89c
Theory Prop Address: TMXMyqEZX2K4g8LgCYyD3LRKSt3urgcv9Gt

explicit_CRing_with_id_minus_one_square

Theorem 24.42

explicit_CRing_with_id→(−one) ∗ (−one) = one.

The proposition is identified by the following information:

Pure Prop Id: 9027680ad43bbdc05b9ac2585988bf9bca1f56435309fcf8330934d3a4e55242
Pure Prop Address: TMKHMtgM4RegJL7J6H3aDWWmZYyuG6dGt4F
Theory Prop Id: 27e1d09ca5cd60ad129f6b679f7062bed626f17e6e37987c960df24c3b5ef58b
Theory Prop Address: TMJQU82As7FBLwuBdVYR5vt1kuAL1jNF9cr

explicit_CRing_with_id_minus_square

Theorem 24.43

explicit_CRing_with_id→∀x ∈R.(−x) ∗ (−x) = x ∗ x.

The proposition is identified by the following information:

Pure Prop Id: cc64ff002d9d1fee374c40859d2a8b13e6a2e01c0daf12ebb732af37740d2369
Pure Prop Address: TMNzA9VqA3PfzJZ7FjCXuhvJBqkuXeLycnt
Theory Prop Id: cba06f4f773cf52216bcff465b0e35eeda22efd81812f03664505820cd4d217b
Theory Prop Address: TMaX7AFX3UeZnZX4SBaoUAegDF4y8TjJD2i

24.5 explicit_CRing_with_id_RepIndep2

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let plus′, mult′ : ι → ι → ι be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Assume the following.

∀ab ∈ R.a + b = a ⊕ b  (24.3)

Assume the following.

∀ab ∈ R.a ∗ b = a × b  (24.4)

explicit_CRing_with_id_repindep

346 CHAPTER 24. RINGS Theorem 24.44

explicit_CRing_with_id R zero one plus mult
⇔ explicit_CRing_with_id R zero one plus′ mult′.

The proposition is identified by the following information:

Pure Prop Id: ac2d59ca206daa22783237146a060cbb078337457a0a1298597e6fcae4da5f00
Pure Prop Address: TMJtWqjqttZHff5TE3g13Vjiu7BhtNnvzfB
Theory Prop Id: 6531240d433090d437bf90709c4ea3cb3ed4e135a6c52863192d1c5d24238853
Theory Prop Address: TMYe8NzBzkkswszXrNbVN74ksiBLPimnNkq

24.6 Packing two Binary Operations and a Con-

stant into a Set

Definition 24.7 pack_b_b_e is the opaque object of type

ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι

identified by the following information:

Pure Object Id: c9ca029e75ae9f47e0821539f84775cc07258db662e0b5ccf4a423c45a480766
Pure Object Address: TMcBhLLE6SjWU8YK3zBnZTiJ441rzSVQZ4f
Theory Object Id: 21d4d088757e7386d904218652e603fec7ce704c78740c732cc6f2bc8d0f15b9
Theory Object Address: TMT1KQaS6hSV4qnwVuKuKkuNrS6Em96tqXN

Definition 24.8 struct_b_b_e is the opaque object of type ι → o identified by the following information:

Pure Object Id: 755e4e2e4854b160b8c7e63bfc53bb4f4d968d82ce993ef8c5b5c176c4d14b33
Pure Object Address: TMVRzggDuRW6mHLPpsN6eZtf5G2DQ1nm6ug
Theory Object Id: 3be2a81c29bbdf8205c87f99ef9048632fba2bced44a1e8c51e6c55f41781fad
Theory Object Address: TMYHoLuYL7HNg4NL1sS21yHunHjH3Z1FY6X

Definition 24.9 unpack_b_b_e_i is the opaque object of type

ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι) → ι

identified by the following information:

Pure Object Id: 7004278ccd490dc5f231c346523a94ad983051f8775b8942916378630dd40008
Pure Object Address: TMQ74XmyoP47CnXDEBHKEFtMzk18jjqu6dt
Theory Object Id: 52a106befe8ad370345a8e6ab30f464040cc96b3bd3f78b38dc7599fc5e7de99
Theory Object Address: TMPRW7TWzKwNSZnPUoDa8Aj3P5cvnhJZi47

24.6. PACKING TWO BINARY OPERATIONS AND A CONSTANT INTO A SET347 Definition 24.10 unpack_b_b_e_o is the opaque object of type

ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → o) → o

identified by the following information:

Pure Object Id: b94d6880c1961cc8323e2d6df4491604a11b5f2bf723511a06e0ab22f677364d
Pure Object Address: TMVcrr5WR2rjHynX8x5DLY3reHfggeDReXu
Theory Object Id: 7b0871aa00521156ece9021408011d93cc22a3a3d8be2ffab32e46cde2c7017a
Theory Object Address: TMRmp4Vwef25JwbPLckTqR8B8PxPtoyHXZ1

Definition 24.11 We define Ring to be

λR.struct_b_b_e R∧

unpack_b_b_e_o R (λR plus mult zero.explicit_Ring R zero plus mult)

of type ι → o identified by the following information:

Pure Object Id: 61bbe03b7930b5642e013fa138d7c8e2353bab0324349fce28b7cbce8edce56a
Pure Object Address: TMVG5WKysVhbYYd6hg9dUY2aCKLj1h9qkSs
Theory Object Id: ffaf640f43cef8fec01ec428ab080a16b7b8c9fe29fc01c79b9a62fc32f9e1ab
Theory Object Address: TMVey7JB9XreF71vPMXWv4MfPYVqXDJNqi5

Definition 24.12 We define Ring_minus to be λRx.unpack_b_b_e_i R (λR plus mult zero.explicit_Ring_minus R zero plus mult x) of type ι → ι → ι identified by the following information:

Pure Object Id: e5405bc7309f2aa971bcab97aadae821ba1e89825a425c7bf10000998c5266c9
Pure Object Address: TMGB3zNFrM8bRch4GNc7TY8X7Y5CDFkWQMu
Theory Object Id: c725983606b8ac83202398408bea35fe4ce0b6c34d2d5411b8a4d54f4b73256f
Theory Object Address: TMGzuX373w32Vo8i8c1MYDeruu5iffk14jV

Definition 24.13 pack_b_b_e_e is the opaque object of type

ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι

identified by the following information:

Pure Object Id: 6859fb13a44929ca6d7c0e598ffc6a6f82969c8cfe5edda33f1c1d81187b9d31
Pure Object Address: TMHt74X3xjgCgvDbTBeJDSgoSU6NBFqLo3t
Theory Object Id: cf1c51f21701c7f90f031360baccab3d9f9cc6bcbb438ab90424d004540beecf
Theory Object Address: TMY3Jeq6qFWZoiMV17oUBbd2wzL4nynScQo

Definition 24.14 struct_b_b_e_e is the opaque object of type ι → o identified by the following information:

348 CHAPTER 24. RINGS

Pure Object Id: 7685bdf4f6436a90f8002790ede7ec64e03b146138f7d85bc11be7d287d3652b
Pure Object Address: TMMq6LTJYr6FfwYvoFthxRCCWMJeJ3PPcbc
Theory Object Id: 93fa62123840de71d70cd336cd76c191a66b2a255f74ff9d886c0ce7bd229764
Theory Object Address: TMKaRBHLAH7X2upT8ygEK87KwVQkCv88tVc

Definition 24.15 unpack_b_b_e_e_i is the opaque object of type

ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι) → ι

identified by the following information:

Pure Object Id: 0708055ba3473c2f52dbd9ebd0f0042913b2ba689b64244d92acea4341472a1d
Pure Object Address: TMPWkrRKKhARHw7yZVhmEDea2FrJHMb9vQ2
Theory Object Id: e7d717c6dc1d45afea7f0ffa393d0e0f9d8162bd1099d59986e7a90eae668a3d
Theory Object Address: TMPQgdNg6eUMGGtvqYwfi9ZggR6mb3tYxPU

Definition 24.16 unpack_b_b_e_e_o is the opaque object of type

ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → o) → o

identified by the following information:

Pure Object Id: 1bcc21b2f13824c926a364c5b001d252d630f39a0ebe1f8af790facbe0f63a11
Pure Object Address: TMLUiT5tyYDHruUZiDx7gvU1hj4TEKsQReX
Theory Object Id: 3325ddb79a400302ecc0a153bdd9c21b9425489ac6ae7b8d16a00c9f9386f079
Theory Object Address: TMUV6wYXdEfCF2Zrhg4Q7de7PEc2arc5QiC

Definition 24.17 We define Ring_with_id to be λR.struct_b_b_e_e R∧ unpack_b_b_e_e_o R

(λR plus mult zero one.explicit_Ring_with_id R zero one plus mult) of type ι → o identified by the following information:

Pure Object Id: e87205d21afb0774d1a02a5d882285c7554f7b824ee3d6ff0a0c0be48dac31d6
Pure Object Address: TMMXH8sbGq7TCys16vmazU94M6z99RUZJ4e
Theory Object Id: 5b1d1fc429c4ad1b59f175a48a3a442a9707b676450c6ff34ff21d9e21b47f80
Theory Object Address: TMHpeNjPXgpyDcsaGcgo2ZDP3VuziQZ8U4x

Ring_with_id_unpack_eq

Theorem 24.45

∀R.∀plus mult :ι → ι → ι.∀zero one.
unpack_b_b_e_e_o (pack_b_b_e_e R plus mult zero one)

(λR plus mult zero one.explicit_Ring_with_id R zero one plus mult)

= explicit_Ring_with_id R zero one plus mult.

The proposition is identified by the following information:

24.6. PACKING TWO BINARY OPERATIONS AND A CONSTANT INTO A SET349

Pure Prop Id: e125aaa25b8e15f0cb5142f069cd1158c31c0a22e2ba188ed03651f2b50f7f69
Pure Prop Address: TMYhZgSfEnPXAcYrzYgAs5j3TvrmvSQUYwo
Theory Prop Id: 6ff04cdbf65065a1048fb4ed5c01d06bc600971c026c0e88f07528dee5a7753b
Theory Prop Address: TMHgUEdaJ2656FZdQffXPyg9M1r2869W3gG

Definition 24.18 We define CRing_with_id to be

λR.struct_b_b_e_e R∧ unpack_b_b_e_e_o R

(λR plus mult zero one.explicit_CRing_with_id R zero one plus mult)

of type ι → o identified by the following information:

Pure Object Id: 38dd910342442210ba796f0e96ab9a7e6b36c17352f74ba8c9c2db4da2aebf0e
Pure Object Address: TMQgd2PoDs8mxEHY4Lw1N9XM2MuPgtY8d9w
Theory Object Id: a1021e16f50fc0f2bd0b04593e51a03b9457694823e7aa3bc57cf1efe615c1f7
Theory Object Address: TMbRJRM2RQRfp7E2CEx4AJhw2m5Gxt5HthV

CRing_with_id_unpack_eq

Theorem 24.46

∀R.∀plus mult :ι → ι → ι.∀zero one.
unpack_b_b_e_e_o (pack_b_b_e_e R plus mult zero one)

(λR plus mult zero one.explicit_CRing_with_id R zero one plus mult) = explicit_CRing_with_id R zero one plus mult.

The proposition is identified by the following information:

Pure Prop Id: 36ff594b81ff0f9289211e517dfadc9825f390bdb5273a1c1da1b64814838ac3
Pure Prop Address: TMGSuWhhM3Pypq6dXURWZnpYUJPYcx7vkJ2
Theory Prop Id: 7707613ab3233c231329185e9013759e858082c5e03f957f33f5a266d7c92331
Theory Prop Address: TMFpD2MTMvgWJJuwcM5sqPZKwuope6VFnQg

Definition 24.19 We define Ring_of_Ring_with_id to be

λR.unpack_b_b_e_e_i R (λR plus mult zero one.pack_b_b_e R plus mult zero)

of type ι → ι identified by the following information:

Pure Object Id: 9f7755a326e730a1c98395c05032d068f1d3a5762700ae73598c3c57a6bd51ea
Pure Object Address: TMJ3Jckg9eHwYuoWNm91qRwiRDGjP1Wv1u3
Theory Object Id: 119ffb040b7ae9e440cde0cba75f90e17ecab1fe0ecf8a8ace0582b41a80ab5e
Theory Object Address: TMSZtFuoH5AXFQXg4j9tyqGVoKFDW7ouQ8H

CRing_with_id_is_Ring_with_id

Theorem 24.47 ∀R.CRing_with_id R → Ring_with_id R. The proposition is identified by the following information:

Pure Prop Id: 815b955355c5b9a3a4bba1901f861f2e42a8aba0b2e264541eb5237d2b7defed
Pure Prop Address: TMUZNUomruVanp8ziSGJjGo6mnjbL8FcKLU
Theory Prop Id: e157b3b577d0c147bf7e94da6f8ae23dc7ce8e6040bdb6473b6afe038837f0b1
Theory Prop Address: TMPJAdDaLrQhhspajeStvBhg4f1PCFGYJTp

350 CHAPTER 24. RINGS

Explicit Reals

25.1 explicit_Reals

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult.

Definition 25.1 explicit_Field is the opaque object of type o identified by the following information:

Pure Object Id: 32dcc27d27b5003a86f8b13ba9ca16ffaec7168a11c5d9850940847c48148591
Pure Object Address: TMbJPQHnQf1mY3onQrsrhqCR24yapErnf21
Theory Object Id: 0020dc091a770bc26640d405438dea5752e187e4f46e3ea423aa06fbf99169cd
Theory Object Address: TMctGNriWgkHut2Cii7sum1tGGmeJ1TQf2z

explicit_Field_I

Theorem 25.1

(∀xy ∈ R.x + y ∈ R)→(∀xyz ∈ R.x + (y + z) = (x + y) + z)→(∀xy ∈ R.x + y = y + x)→
zero ∈ R→(∀x ∈ R.zero + x = x)→(∀x ∈ R.∃y ∈ R.x + y = zero)→(∀xy ∈ R.x ∗ y ∈ R)→
(∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z)→(∀xy ∈ R.x ∗ y = y ∗ x)→(one ∈ R)→(one ≠ zero)→
(∀x ∈ R.one ∗ x = x)→(∀x ∈ R.x ≠ zero → ∃y ∈ R.x ∗ y = one)→
(∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z)→explicit_Field.

The proposition is identified by the following information:

Pure Prop Id: 3ccc525dac877738bde04411aa5b2802762d08d54c0801938fbdb890674a0002
Pure Prop Address: TMdJ9fH9ZyRuWnaU1MAQEZ1tDGUfPtL2cec
Theory Prop Id: 51ce1195007755d317528633fca294788000229835b66559cc3f47f888639540
Theory Prop Address: TMKFU5TXhW1AbfR8o181j3pjJGyWs64AHai

explicit_Field_E

Theorem 25.2

∀q :o.

(explicit_Field → (∀xy ∈ R.x + y ∈ R) → (∀xyz ∈ R.x + (y + z) = (x + y) + z) → (∀xy ∈ R.x + y = y + x) → (zero ∈ R) → (∀x ∈ R.zero + x = x) → (∀x ∈ R.∃y ∈ R.x + y = zero) → (∀xy ∈ R.x ∗ y ∈ R) → (∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z) → (∀xy ∈ R.x ∗ y = y ∗ x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R.one ∗ x = x) → (∀x ∈ R.x ≠ zero → ∃y ∈ R.x ∗ y = one) → (∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z) → q →explicit_Field→q.

The proposition is identified by the following information:

Pure Prop Id: bb3db9eded830bf9d92132c0016be1d1a259929bceda5bdf57b4f64fcba43184
Pure Prop Address: TMXJPRqy8mBTRo6EPJpYzNS1rE86xJ2ABjo
Theory Prop Id: 6c4aa1700c811949523aab7956f68644700f9d543e3915471880f5956881b35e
Theory Prop Address: TMcxwBH2nbJB8F5LxJwz4A2as2v9Ejx9zoD

Definition 25.2 explicit_Field_minus is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 5be570b4bcbe7fefd36c2057491ddcc7b11903d8d98ca54d9b37db60d1bf0f7e
Pure Object Address: TMMPF8zir9fkfAtJFGxe63EcXTEPLHDtBA4
Theory Object Id: 7a10c2aa3843a7152cec6319a52f2f9ff01bb82b441aed667fd99a775dab9980
Theory Object Address: TMYcEMbnauFAYpJxp2DpEB5nscfUnNcsbZy

Notation. We use −− as a prefix operator corresponding to applying term explicit_Field_minus.

explicit_Field_minus_prop

Theorem 25.3 explicit_Field → ∀x ∈ R.−x ∈ R ∧ x + −x = zero. The proposition is identified by the following information:

25.1. EXPLICIT_REALS 353

Pure Prop Id: dfdfb501a72ee10c72ad8933d6f1bd3d1d25a2009a655a5683cf351539aa70bc
Pure Prop Address: TMRiffBv5SK7Dv3GZ8Qc3WgwrC1MxCSpHC7
Theory Prop Id: be5356f76e63b2d1a7cc20f125e927198f6005afc05464565ca4304115a8e435
Theory Prop Address: TMKQfrW9q8xeEbq3YsFtzRuMpXJcsRviQhq

explicit_Field_minus_clos

Theorem 25.4 explicit_Field → ∀x ∈ R.−x ∈ R. The proposition is identified by the following information:

Pure Prop Id: 19192856a86a06ac1d56d571d172b3dd60e77b95edea0e5a195a7849ea9fdd1d
Pure Prop Address: TMLDrc2xd6CY9VqXqvmt3SZypH79h73yzYt
Theory Prop Id: 853f414afb3b6db2964375b75fc2bdfa06eba2d8a9cf8c7993e79be8c16c241c
Theory Prop Address: TMMd78EndbrgewWKg7cug4aFmuNLcdvLETi

explicit_Field_minus_R

Theorem 25.5 explicit_Field → ∀x ∈ R.x + −x = zero. The proposition is identified by the following information:

Pure Prop Id: 16bd02b0d5c539b89ab6c3b290f99da455cb893382173ba554e3a9499850d228
Pure Prop Address: TMWcY3CmCi3C7TchQ2ZWaU2w6UYvY5EyRBX
Theory Prop Id: e00e5ccf74077747e0399d61c172ce48cfbd48eee9e6d1a7ccc3d55adb0814dd
Theory Prop Address: TMJphf6VbWpCYsxdqnUTWXSj422gfKJmxEo

explicit_Field_minus_L

Theorem 25.6 explicit_Field → ∀x ∈ R.−x + x = zero. The proposition is identified by the following information:

Pure Prop Id: 1e484c596156e799c8b1be02b2f184d891d04b570a626b0f353d58fb19b04a98
Pure Prop Address: TMZbY42rURTo9ZUysPTCKb3htjwFZXbEw1T
Theory Prop Id: 3c1cf77f15777c38163c09cb6f75c87835fac72965b443566fab8a7a40e9665f
Theory Prop Address: TMYDLj9BQtJLVtxTosQ1aMz4YRJvYdWceaV

explicit_Field_plus_cancelL

Theorem 25.7 explicit_Field → ∀xyz ∈ R.x + y = x + z → y = z. The proposition is identified by the following information:

Pure Prop Id: b9cc3f21432ad1dcc45524e4ac02e5182053a3f7477ca552e87c5132962558cb
Pure Prop Address: TMLW7dUp3ubrm2KQb1j1qqXX7SiHc2FMGYH
Theory Prop Id: da9a6bc0e051dd48f33393f832d94859ae59087f843e5647b7d48be40b16752c
Theory Prop Address: TMGDxEVrun2MVRqLiWkibA4xgq5mXZYRefM

explicit_Field_plus_cancelR

Theorem 25.8 explicit_Field → ∀xyz ∈ R.x + z = y + z → x = y. The proposition is identified by the following information:

354 CHAPTER 25. EXPLICIT REALS

Pure Prop Id: 8e7410c2408dd9fc6349ca4ae571c9cfc54c4cdbd647f030a422a0f138d33342
Pure Prop Address: TMLaB8tvdGpW8DwHyHVQH6BkFA5qABz62Xw
Theory Prop Id: 148a14870a33e0b1322a50fde11a6ff30e6ed7d2cf59e7b846b0a4fece952554
Theory Prop Address: TMdwAiT4yYz4w2xxTmzsG9vifx1CtnVpbbK

explicit_Field_minus_invol

Theorem 25.9 explicit_Field → ∀x ∈ R.−−x = x. The proposition is identified by the following information:

Pure Prop Id: d9245973a06673bbaca0fe2244ae2b76cb719974a7a1a5be04c9786c8ccd521a
Pure Prop Address: TMSFpQWK1zrTS89dhaG5MbKGZ3CzKuw2Hez
Theory Prop Id: e4a26d2ce9a3a793e18ddeb34b9e5f5689c952c20c234bbeaccb1e36b6ed3934
Theory Prop Address: TMSuivsorWRWJ8izeRUNUJJmCYigXx7CwUM

explicit_Field_minus_one_In

Theorem 25.10 explicit_Field → −one ∈ R. The proposition is identified by the following information:

Pure Prop Id: 3dea9f1e7c156a2c2a714f0dcbe2bb48e13a8f0786132f71784621f4566ba3de
Pure Prop Address: TMKembKWo9bsc7rvXjS3b9tPDernwnfN69C
Theory Prop Id: c3141394b9d6650476539d3de9c24c2094fafd277233c838e05eece229179706
Theory Prop Address: TMYYz1ipvpJRoYVLBTTHwW5inLAqHEZJNZj

explicit_Field_zero_multR

Theorem 25.11 explicit_Field → ∀x ∈ R.x ∗ zero = zero. The proposition is identified by the following information:

Pure Prop Id: f6ea109454bd63c383487f79cde58c39e60809206cdeecfb0cf5c33ba7a39a78
Pure Prop Address: TMc9K3rxpDhdcV4fiwudpCAvCbo4EgSX2FM
Theory Prop Id: cddbcdbd4bc296b7c12e6e905456a622a658af38644ec81e091b58a94847904f
Theory Prop Address: TMTpJuV9Dgs6cEeCYFcQ71NUT9zLc55yQ6G

explicit_Field_zero_multL

Theorem 25.12 explicit_Field → ∀x ∈ R.zero ∗ x = zero. The proposition is identified by the following information:

Pure Prop Id: a6219e7c7ce408f1a4ed2d0bbb6897dd8a32479b32efbe9caa5735d0b3fe8fa2
Pure Prop Address: TMbHRokQ9EFeZHPhTtnpAgJYqraYBTYVR9D
Theory Prop Id: bf3f6a5eb75519377a3a0ab3681dee400b8bb378f33c7dc4dcd864f874a61fca
Theory Prop Address: TMWNakNQV1Rea25NYbFjsGGZVZqZY7C1sHo

explicit_Field_minus_mult

Theorem 25.13 explicit_Field → ∀x ∈ R.−x = (−one) ∗ x. The proposition is identified by the following information:

25.1. EXPLICIT_REALS 355

Pure Prop Id: b725ff0fba328afa96eaf8d8cee4c89ac94d1e750fd3006aba3b6f6f6dc8d6fa
Pure Prop Address: TMTsU5wxazHBfmvDRU6dzpmJe6y8taJi1zr
Theory Prop Id: 6c32dff82b3955c923f8ad0af52f709a8d9750b5932c46a80584bf6fe86d4079
Theory Prop Address: TMG8caBgEaUDhsJtbdypcFcgYXKH7VumZf2

explicit_Field_minus_one_square

Theorem 25.14 explicit_Field → (−one) ∗ (−one) = one. The proposition is identified by the following information:

Pure Prop Id: 33687daadedd9048924a66186e9b1d11d2849f1c4c403989aa1dde1a2c8d1839
Pure Prop Address: TMRYgfX4tEirAsP8NiXBD231RunzNSeTGaQ
Theory Prop Id: 81b69b4f65a19abefefc96e03e44e28ccab7aa2d8fdd98ec035a25ceeb9f9c01
Theory Prop Address: TMWNc4GkfcpRjYQn5ss6k1srmNCX8pS3RTW

explicit_Field_minus_square

Theorem 25.15 explicit_Field → ∀x ∈ R.(−x) ∗ (−x) = x ∗ x. The proposition is identified by the following information:

Pure Prop Id: db0127263baf3b745b4b535f56b1f7c123744b30c2db578852e18d3b417f004c
Pure Prop Address: TMTV85Vo7eTyFBiu5YCvAEyGtkJGhZnmztE
Theory Prop Id: cc20f4507a4b8ab01b79ded3da8b0945e39532ee96a70c6ef10bab5a6e176273
Theory Prop Address: TMFLG2o5FZ3crQm2waGZhJKk41t1KZmfnsC

explicit_Field_minus_zero

Theorem 25.16 explicit_Field → −zero = zero. The proposition is identified by the following information:

Pure Prop Id: 677fffd7ef5af74d4eaf3075a3db01ed4e7685052f86b5da66de86c40ac0029c
Pure Prop Address: TMbHTa1NSQxgViQGA5LgQhajneJu1oTW2n7
Theory Prop Id: d58ae486546dede085e27a56fa6854b755e9bfc2f0b3c78aa184624c4d4535e2
Theory Prop Address: TMHxghZtCQFNpCShXEagGPGXBetd1ZxTcCB

explicit_Field_dist_R

Theorem 25.17 explicit_Field → ∀xyz ∈ R.(x + y) ∗ z = x ∗ z + y ∗ z. The proposition is identified by the following information:

Pure Prop Id: de79567ee223c72fabfe843b7d7526a3a12428317b799a662ff4ce3ad76f9472
Pure Prop Address: TMMz7KwL6xBwXeCscdhDKUbgZEeMqZXz2n1
Theory Prop Id: 2a896a6970c2a6f3d291e0b5a6330cabecc63ba17b6f9f6b4338f1a0e03a08a0
Theory Prop Address: TMXM5285AmLUf7pvii4Ym8YsRizGaSngE2Z

explicit_Field_minus_plus_dist

Theorem 25.18 explicit_Field → ∀xy ∈ R.−(x + y) = −x + −y. The proposition is identified by the following information:

356 CHAPTER 25. EXPLICIT REALS

Pure Prop Id: 7c28f73f41fb9362c802db194a80e809ab40b5fcbf86889187a241fb04c45f07
Pure Prop Address: TMMbYxMRKezKTJauH5tFeT6YcdVqNqaoNyb
Theory Prop Id: 3477df803cb462a1c67676532b809a8085f5965e8eded20de22c081f8b380c1f
Theory Prop Address: TMJEy6ntH8yniiQ1XpVapukWC9af7szg259

explicit_Field_minus_mult_L

Theorem 25.19 explicit_Field → ∀xy ∈ R.(−x) ∗ y = −(x ∗ y). The proposition is identified by the following information:

Pure Prop Id: d4042ab05af8694ed9f50ea48d73fd8241fe9e096a57f64ca3767ec2ee8622b8
Pure Prop Address: TMMDqKcMu6spUHkosKvvDfVjBChv657VpPo
Theory Prop Id: 2c1cad10f4a09ca8c4864cae7b040bf84788c98182553b04deca95d667a74739
Theory Prop Address: TMb8pKtSFLQTYNxrQHoTXXs92fMEGh22svD

explicit_Field_minus_mult_R

Theorem 25.20 explicit_Field → ∀xy ∈ R.x ∗ (−y) = −(x ∗ y). The proposition is identified by the following information:

Pure Prop Id: 1c896cfcf91413c6c45f35892ce57731b0adad535619672d8857f0281523657a
Pure Prop Address: TMHHkG7bnn6fLtxgsof8vd6CC3u39fp5sYn
Theory Prop Id: 46e01b365cf641485dae65460ad1681ecaab879cc9dd36e6f2dc58b3a94a1cd0
Theory Prop Address: TMPM6UjftWZ8u7qznYUZoTw3xxFGQZUT3ce

explicit_Field_square_zero_inv

Theorem 25.21 explicit_Field → ∀x ∈ R.x ∗ x = zero → x = zero. The proposition is identified by the following information:

Pure Prop Id: 18939b3ff00eabb11a4a28ae19cc0aac425c72704752d525860f80dfe5375212
Pure Prop Address: TMNAKKNruWjcXkrAW9Rb6BxwgJg9jKX4Lbd
Theory Prop Id: fe699ea22147eff4c7086ff7bf4954794c5e61d9d3e22c587a86681568f2423d
Theory Prop Address: TMP55pVZRC8XUcxBEuENBHAqrd7cau5uiY4

explicit_Field_mult_zero_inv

Theorem 25.22

explicit_Field→∀xy ∈R.x ∗ y = zero→x = zero ∨ y = zero. The proposition is identified by the following information:

Pure Prop Id: 48f3d685f5c19ffb943e4e92bf1143cd300504313d7025cbcd85d11fa57e29aa
Pure Prop Address: TMMoGSv87AzopLCYaS7tpk3DNioBcaw6e6Z
Theory Prop Id: d90b744848de4a3c083b3ce57e25e069c0d9b625f137f9e544dde929638e2794
Theory Prop Address: TManTc7e6UWwgQhhz3U1c3VVG946TvLTxR9

Let leq : ι → ι → o be given. Notation. We use ≤ as an infix operator corresponding to applying term leq.

25.1. EXPLICIT_REALS 357

Definition 25.3 explicit_OrderedField is the opaque object of type o identified by the following information:

Pure Object Id: a18f7bca989a67fb7dc6df8e6c094372c26fa2c334578447d3314616155afb72
Pure Object Address: TMSMYTTginXTnkcVZtpjAKPAaL7BnqqgRds
Theory Object Id: eea7fafdf682c37d08f8505d8389845b96987326a92b0823dac083b00eda754c
Theory Object Address: TMUVEidBNjgBhKRvbJD1JJ2kD28DQi7APKy

explicit_OrderedField_I

Theorem 25.23

explicit_Field→(∀xyz ∈ R.x ≤ y → y ≤ z → x ≤ z)→
(∀xy ∈ R.x ≤ y ∧ y ≤ x ⇔ x = y)→
(∀xy ∈ R.x ≤ y ∨ y ≤ x)→(∀xyz ∈ R.x ≤ y → x + z ≤ y + z)→

(∀xy ∈ R.zero ≤ x → zero ≤ y → zero ≤ x ∗ y)→explicit_OrderedField.

The proposition is identified by the following information:

Pure Prop Id: 80eb1d0d6d1ae59e7716b600362a8edadf3a3d46606eefe17eb4379546b5b255
Pure Prop Address: TMVbN3H3LfjoJc5JN3kFVvxfeyGkhWvjBwf
Theory Prop Id: 71586c9655f0d1781fbbb7e18105c89958843a5ccb8e7deeffe6b9e62b1b1e09
Theory Prop Address: TMTVD1npwX9pr1azvDC9toE4RLXrqq7MsyC

explicit_OrderedField_E

Theorem 25.24

∀q :o.

(explicit_OrderedField → explicit_Field → (∀xyz ∈ R.x ≤ y → y ≤ z → x ≤ z) → (∀xy ∈ R.x ≤ y ∧ y ≤ x ⇔ x = y) →

(∀xy ∈ R.x ≤ y ∨ y ≤ x) →

(∀xyz ∈ R.x ≤ y → x + z ≤ y + z) → (∀xy ∈ R.zero ≤ x → zero ≤ y → zero ≤ x ∗ y) → q)

→explicit_OrderedField→q.

The proposition is identified by the following information:

Pure Prop Id: 1c269ba4e553561a1cc6de2a41b6545ec8ac75e48e3c6cb23b2b6a324a64952b
Pure Prop Address: TMVNKTpEnn5zTRZtpuHVXwYgJDgVq64jhQb
Theory Prop Id: 5c835e8cb20f579d989cd542e60e75f77e9b09c2976f6070c05a17cfca4d407a
Theory Prop Address: TMaCBRHe1Qvk5uX74Rit6toSqNPaYfiTK4D

explicit_OrderedField_minus_leq

358 CHAPTER 25. EXPLICIT REALS Theorem 25.25 explicit_OrderedField → ∀xy ∈ R.x ≤ y → −y ≤ −x. The proposition is identified by the following information:

Pure Prop Id: 0d0f5cad7897b1fc9859fb0fb353f392e444d5b3b7e1db20a4f125e0e21ce6bf
Pure Prop Address: TMMwfqJswL3g7Cij23YHpxZpvqT7KfRqNMe
Theory Prop Id: ee304f2ce86cb10f1be68e56828aad0317279e773397f75e93647653c83be151
Theory Prop Address: TMMzgVBuTQhmpTV2B9ybeJX6tiEkbUXhbe3

explicit_OrderedField_square_nonneg

Theorem 25.26 explicit_OrderedField → ∀x ∈ R.zero ≤ x ∗ x. The proposition is identified by the following information:

Pure Prop Id: ebf414f2c6ff9848d35121369f85472ab25e8af5cc2728af838e40787e67ac6f
Pure Prop Address: TMV8VUHDk12iC7xpNnNDLjZD9BtuYf7ESyK
Theory Prop Id: 891e8662d8b95f38bd875c846330ed8d495c4375ade2fe5aac04a6d4c9daa3d3
Theory Prop Address: TMJ9LeAui7MyNuXXvPgJDye1dzUzpzKZanv

explicit_OrderedField_sum_squares_nonneg

Theorem 25.27 explicit_OrderedField→∀xy ∈R.zero ≤ x ∗ x + y ∗ y. The proposition is identified by the following information:

Pure Prop Id: 7596b76c40f5977c5d12346563941ea1bd17dedf0fdc6d4677b21fd667046cd0
Pure Prop Address: TMXYnd1YrHHrvjWcoZ6XYMkRe7mUtnUsHMB
Theory Prop Id: d9447e2431709c9a2271ba5d7c8d2c42675d962980e8d0c6a945b47a84e3f7f2
Theory Prop Address: TMR3ypJ2YAEpeYUYejDaWcqzkgc8VHN9BYU

explicit_OrderedField_sum_nonneg_zero_inv

Theorem 25.28

explicit_OrderedField→∀xy ∈R.zero ≤ x→zero ≤ y→x + y = zero→

x = zero ∧ y = zero.

The proposition is identified by the following information:

Pure Prop Id: 072f56b723bef4ade29f644b20f32711dd9f8872653a4603191255a02ed12aa5
Pure Prop Address: TMbqPNKUAZHZZvbj8kXQXLfL56CpAozuS97
Theory Prop Id: 5f6c78eb0411a6fb156cf3b2848ebb0c759d7a6a6b653d97dec27052916e94e5
Theory Prop Address: TMXUo7T7pnNdX24nuWLNaNSHmUgB2UFnGt4

explicit_OrderedField_sum_squares_zero_inv

Theorem 25.29 explicit_OrderedField→∀xy ∈R.x ∗ x + y ∗ y = zero→x = zero ∧ y = zero. The proposition is identified by the following information:

25.1. EXPLICIT_REALS 359

Pure Prop Id: 3475f408fb7b3dec70f825cb0332b28e033d25bc6e4cbd6a9b382bbd3d531484
Pure Prop Address: TMGde3gx4w4Uh9Awwi1qwfiWQ4CBJfJAtS1
Theory Prop Id: 5736f60e1ddc12d7525825b06f528d3d62fe1e6481630bd0d8e030f123c54b60
Theory Prop Address: TMG76tdh9rhRwpDqWUkbVYQDxYySTKTuYYE

explicit_OrderedField_leq_refl

Theorem 25.30 explicit_OrderedField → ∀x ∈ R.x ≤ x. The proposition is identified by the following information:

Pure Prop Id: 6d89bac73c97399406da5761a4e05ca142b2af3c9ea45f75c5674037832d0352
Pure Prop Address: TMJPTfciQTSdnS3SFhTmePpEG61KkYyXyh8
Theory Prop Id: 915e8fb0a5a50db0b4874494c5e1fbe27f4a9e913998db7bdbe6335c2782cef8
Theory Prop Address: TMUeqaV9ny8ooGx2mDKgZUyR36krjn7twu6

explicit_OrderedField_leq_antisym

Theorem 25.31 explicit_OrderedField→∀xy ∈R.x ≤ y→y ≤ x→x = y. The proposition is identified by the following information:

Pure Prop Id: 92c2c21b46e137352ea2b13877fba3bccf89bb379a6f9cb05d766c8493fbc378
Pure Prop Address: TMdF3uckvYNFsZLo2N2AUXRf961iqhpb4nY
Theory Prop Id: 2313aae88e7fee79d5231f7afbc5542b995b3af3ba7cc654b0066c73120152d4
Theory Prop Address: TMNmtcpgnXCEWQdjZhJoxe1nhrEE87jGdwk

explicit_OrderedField_leq_tra

Theorem 25.32 explicit_OrderedField→∀xyz ∈R.x ≤ y→y ≤ z→x ≤ z. The proposition is identified by the following information:

Pure Prop Id: 2d02678a2209fd726235ddb09c2aa9290b48375519487773c818f9d57860c62b
Pure Prop Address: TMcsuosBpLY4AFhGspmWx2rN1cH12b2XPqe
Theory Prop Id: 642b84d1691774bc188f593803d0d85f1012bda7c66ff4ef994e821e3570a030
Theory Prop Address: TMNgAZiMJeVFoYAuFpPJez3pVME9zrth8Xd

explicit_OrderedField_leq_zero_one

Theorem 25.33 explicit_OrderedField → zero ≤ one. The proposition is identified by the following information:

Pure Prop Id: 4f14f3aee0dd811396902d2e346ed9447d586fecfa610cb7f80d8c018b78445c
Pure Prop Address: TMcxPXzviwMyx5FiSvb9JnufFyGPAdVhiA2
Theory Prop Id: a0dbd6bf9eb8bf6f8529771cbf53aa3be49e04449a97763b5e01ea7d7ea2a870
Theory Prop Address: TMGr9TnXkfhuvkbv1dMj5JtTZ8RSfzoP95A

Definition 25.4 We define lt to be λxy.x ≤ y ∧ x ≠ y of type ι → ι → o identified by the following information:

360 CHAPTER 25. EXPLICIT REALS

Pure Object Id: d326e5351e117c7374d055c5289934dc3a6e79d66ed68f1c6e23297feea0148e
Pure Object Address: TMGFyhEzMLotVozJF1yiunrkGLuFdfjWkng
Theory Object Id: f91dfd2380597226f5c8900812832b2a5a67ae6568fe2b2a605848bd8d22540a
Theory Object Address: TMcecisvAF6JJ66jafWS8ZR87w7SMBwYATJ

Notation. We use < as an infix operator corresponding to applying term lt. Definition 25.5 natOfOrderedField_p is the opaque object of type ι → o identified by the following information:

Pure Object Id: f1c45e0e9f9df75c62335865582f186c3ec3df1a94bc94b16d41e73bf27899f9
Pure Object Address: TMUGM8F7ocg9jGYhPZVwzCfNfPzPmAanw4Z
Theory Object Id: f89652d7f0e1787e89129826561eeccb4f5558676c7910ea999306a32285e218
Theory Object Address: TMFAmRM27u8JLyZ7Jf1HAC8DeSELjytd8Ua

Let N be {n ∈ R|natOfOrderedField_p n}. Let Npos be {n ∈ N|n ≠ zero}.

explicit_Nats_natOfOrderedField

Theorem 25.34

explicit_OrderedField→explicit_Nats N zero (λm.m + one). The proposition is identified by the following information:

Pure Prop Id: 9cb324808fc5cf72d2d653a2418ab30cb9b85476e00be35fcd9b57f1bb93ad72
Pure Prop Address: TMWcMhihbJqy1w5uYhuM3Es9nEdDsCM8SJn
Theory Prop Id: 8d1a312e7d98b7df2205e4cfa93c99a5f6dfa4019921c333b9d177cc70aca004
Theory Prop Address: TMZXnwRs21hpHmTpXy9R1JnG8n2ykq4d1cS

explicit_PosNats_natOfOrderedField

Theorem 25.35

explicit_OrderedField→explicit_Nats Npos one (λm.m + one). The proposition is identified by the following information:

Pure Prop Id: 36987abea2f716243c8b427e45e8b2e1ae58f4dd9e1dd000b423f2e6f4bc48a0
Pure Prop Address: TMWy3odsbnhVAitSJ5hzGBv9T8fmV4ZP3S7
Theory Prop Id: 7b42ab4ebcb5779370bac576623544325cab108670d64b30ebe19743a3fa9054
Theory Prop Address: TMZEMUgieWVVNR5ysQcJpnbXL1KpnZvBLMh

Let Z be {n ∈ R|−n ∈ Npos ∨ n = zero ∨ n ∈ Npos}. Definition 25.6 We define explicit_OrderedField_rationalp to be

λx.∃n ∈ Z.∃m ∈ Npos.m ∗ x = n

of type ι → o identified by the following information:

Pure Object Id: bacdefb214268e55772b42739961f5ea4a517ab077ed4b168f45f4c3d4703ec4
Pure Object Address: TMS9Jk69fwzyerTVyhC5cLftFrz8QwxR86G
Theory Object Id: 6449a4f053483876d454bb93a1566cd7b4d24332ae6e750097943c2e903e0426
Theory Object Address: TMdsiVayo81K6oYHmqnGdSi5QMPLt3MnzCJ

Let Q be {x ∈ R|explicit_OrderedField_rationalp x}.

explicit_OrderedField_Npos_props

Theorem 25.36

explicit_OrderedField→∀p :o. (Npos ⊆ R → explicit_Nats Npos one (λm.m + one) → one ∈ Npos → (∀m ∈ Npos.m + one ≠ one) →

(∀m ∈ Npos.∀q : ι → o.q one → (∀n ∈ Npos.q (n + one)) → q m) →

(∀nm ∈ Npos.explicit_Nats_one_plus Npos one (λm.m + one) n m = n + m) → (∀nm ∈ Npos.explicit_Nats_one_mult Npos one (λm.m + one) n m = n ∗ m) →

(∀nm ∈ Npos.n + m ∈ Npos) →

(∀nm ∈ Npos.n ∗ m ∈ Npos) → p)
→p.

The proposition is identified by the following information:

Pure Prop Id: 70d4a40bcc65e646c44adc246e6e8230f11d26380216c98eef3ce0f533c5b967
Pure Prop Address: TMWVNiuNPNFnodbziNN8nLxyxLMnKWrd8wJ
Theory Prop Id: ccf61651d2adec6894aedbbc427dceb7844e2399f21269514640c7e6094ad427
Theory Prop Address: TMFByiQaDpiDUkPeJbt5Bp3Vmn1hJbuuYP1

explicit_OrderedField_Z_props

Theorem 25.37

explicit_OrderedField→∀p :o. ((∀n ∈ Npos.−n ∈ Z) →

zero ∈ Z → Npos ⊆ Z → Z ⊆ R →

(∀n ∈ Z.∀q : o.(−n ∈ Npos → q) → (n = zero → q) → (n ∈ Npos → q) → q) → one ∈ Z → −one ∈ Z →

(∀m ∈ Z.−m ∈ Z) →
(∀nm ∈ Z.n + m ∈ Z) →
(∀nm ∈ Z.n ∗ m ∈ Z) → p)
→p.

362 CHAPTER 25. EXPLICIT REALS

Pure Prop Id: c4e83c38fecfc73fc1d09e7d004296072c59b1e4f2df931d758d29e6794ea856
Pure Prop Address: TMXcg17hazxWWPbWnFd6LixgqsqD72qWt13
Theory Prop Id: 5d739c339801f054de73227f43deae05f5146a6f72e3301a34806a65741eee8f
Theory Prop Address: TMdyXYYsofQoayXp7LxqNscqFCZMBXVaVYV

explicit_OrderedField_Q_props

Theorem 25.38

explicit_OrderedField→∀p :o. (Q ⊆ R →

(∀x ∈ Q.∀q : o.(x ∈ R → ∀n ∈ Z.∀m ∈ Npos.m ∗ x = n → q) → q) →

(∀x ∈ R.∀n ∈ Z.∀m ∈ Npos.m ∗ x = n → x ∈ Q) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 469ebe93febae512d65ef077bb259d3d61a42e332d5dc0f218b4ef51c641e3e4
Pure Prop Address: TMTZyHSuTABLn2UE5U2gcU2Ezo5LNbr4p3D
Theory Prop Id: d268c341664267b91610963ba224b4a1a9c0f9f442a16df66f09d45352a4151a
Theory Prop Address: TMds236CmEKbqpnTaTJdR2MqiniGBAQsAPi

Definition 25.7 explicit_Reals is the opaque object of type o identified by the following information:

Pure Object Id: 2c81615a11c9e3e301f3301ec7862ba122acea20bfe1c120f7bdaf5a2e18faf4
Pure Object Address: TMNQkXLgjhMYMywv7BnJ8Buh3QQLNR7YL5B
Theory Object Id: f814a2e6f2e99b382ad540cb48300cde1d9995f4b6b2b5af7856369fd2644732
Theory Object Address: TMRfnwEvCdNWYa7FnDpZ9E82K56Dx1FvxWm

explicit_Reals_I

Theorem 25.39

explicit_OrderedField→
(∀xy ∈ R.zero < x → zero ≤ y → ∃n ∈ N.y ≤ n ∗ x)→

(∀ab ∈ R^{N}.(∀n ∈ N.a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n) →

∃x ∈ R.∀n ∈ N.a n ≤ x ∧ x ≤ b n) →explicit_Reals.

The proposition is identified by the following information:

Pure Prop Id: 409e2333061c2a93e2645163cc79fc5d5af155c8a5d27793194dc767da89abbb
Pure Prop Address: TMV9L8eBVS4dZTimiRu7xTnjDG7k6PkcS8x
Theory Prop Id: 5c0c700ed594f122d6868607cbbbb42d933731fe7421678006f15345427a750c
Theory Prop Address: TMFrB18DrFskuCZs1RC1v2N6mv4m2xbxyyh

explicit_Reals_E

25.1. EXPLICIT_REALS 363 Theorem 25.40

∀q :o.

(explicit_Reals → explicit_OrderedField → (∀xy ∈ R.zero < x → zero ≤ y → ∃n ∈ N.y ≤ n ∗ x) →

(∀ab ∈ R^{N}.(∀n ∈ N.a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n) →

∃x ∈ R.∀n ∈ N.a n ≤ x ∧ x ≤ b n)

→ q)

→explicit_Reals→q.

The proposition is identified by the following information:

Pure Prop Id: 21d90e2433a10558f7145bc3be6f01fa8745c28bffc25ad66bd1a18358b3b240
Pure Prop Address: TMGdZwHMdLeLHxRbHQCHJtCbKDNcLtG3W8L
Theory Prop Id: 4da66ca8d1ae9ed2c059a4f5bc10b24ce6225436b72100092ed91505c18552ff
Theory Prop Address: TMXm1gKZKK9VncpZdpgFvZ65WUBmGNKHtic

explicit_Reals_characteristic_0

Theorem 25.41

explicit_Reals→∀n ∈omega.nat_primrec one (λ_r.plus one r) n ≠ zero. The proposition is identified by the following information:

Pure Prop Id: 0212c2930b33db49e6a4620cb445fdddf720a725cafcc5cdfd73aba5653545a7
Pure Prop Address: TMQD99B4xAaxuUtBVEmerL9nB3AVPpeVrb5
Theory Prop Id: a77c5ab95a85405cab2dc8eb8cf6de3db3512e2461a8c18fdbf2c5586157d584
Theory Prop Address: TMcXpA2UzrnRUBX4icewcaFFnK1yHVkVbAY

364 CHAPTER 25. EXPLICIT REALS

Rings II

Definition 26.1 We define CRing_with_id_carrier to be λRs.Rs 0 of type ι → ι identified by the following information:

Pure Object Id: f0bb69e74123475b6ecce64430b539b64548b0b148267ea6c512e65545a391a1
Pure Object Address: TMY8sbcPiVs9VL311KPHpbm67U6sLM9JvF7
Theory Object Id: 9211e41eaf81bb8c152618cbe35230be60d736fcd5684c0b7bdc5ccc1ff70181
Theory Object Address: TMZQE6tPrFvJMaBcLrcMn85ZgscSeFbm4pT

Definition 26.2 We define CRing_with_id_plus to be λRs.decode_b (Rs 1) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 6968cd9ba47369f21e6e4d576648c6317ed74b303c8a2ac9e3d5e8bc91a68d9d
Pure Object Address: TMYTM1QQGqgVyRM9unR8CLL2scwRHYvV3c3
Theory Object Id: ed06aa34b4112f8507370b46c1c5c7de7f2f9af4f24b831d45893ad3066e994e
Theory Object Address: TMRiY7KU6BpU28KD19iwgBDLsQ5XyfpcbYL

Definition 26.3 We define CRing_with_id_mult to be λRs.decode_b (Rs 2) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 4b4b94d59128dfa2798a2e72c886a5edf4dbd4d9b9693c1c6dea5a401c53aec4
Pure Object Address: TMQSTL5b71ddbjnkjnGXfRZ61uXWmMGwJHD
Theory Object Id: 7eb39ed695c795e4451a9252d011d377d0f4bdf0bfe6193f7a31ceb58f687a40
Theory Object Address: TMaJmzoi8UnDNTAXrAvZxYbTeEdKV3viygg

Definition 26.4 We define CRing_with_id_zero to be λRs.Rs 3 of type ι → ι identified by the following information:

Pure Object Id: 61f7303354950fd70e647b19c486634dc22d7df65a28d1d6893007701d7b3df4
Pure Object Address: TMJzMmYjL1HdXDWMSRf7c775tmGDc6osaRL
Theory Object Id: b74fb66a398c6c647d135ea9283353963f66d301d2c1ee126b203679f2071094
Theory Object Address: TMHQo21kvxiD8yab3ek4ATBz2PBfwvqsmaJ

Definition 26.5 We define CRing_with_id_one to be λRs.Rs 4 of type ι → ι identified by the following information:

Pure Object Id: 57e734ae32adddc1f13e75e7bab22241e5d2c12955ae233ee90f53818028312a
Pure Object Address: TMV8UDBZRqv4jocV6zt17aP89PcYugzQkoU
Theory Object Id: b2fa78f7f58b95f4bc4f46ac268c3a959090897e56338c64bd84a73bc5fc2551
Theory Object Address: TMWFb9uZk2JEesS9quTe9a6Ww4ND9VcbBJA

26.1 CRing_with_id

Let Rs : ι be given. Assume the following.

CRing_with_id Rs  (26.1)

Let R : ι be CRing_with_id_carrier Rs. Let zero : ι be CRing_with_id_zero Rs. Let one : ι be CRing_with_id_one Rs. Notation. We use + as a right associative infix operator corresponding to applying term CRing_with_id_plus Rs. Notation. We use ∗ as a right associative infix operator corresponding to applying term CRing_with_id_mult Rs.

CRing_with_id_eta

Theorem 26.1

Rs = pack_b_b_e_e R (CRing_with_id_plus Rs) (CRing_with_id_mult Rs) zero one.

The proposition is identified by the following information:

Pure Prop Id: 19895bd57673dd78940963d6fcc41b374e3914364fcfea397ea66c6fbf664993
Pure Prop Address: TMGYeUaTmA7FDSErSMoGpiYWt7ZcZFFdt71
Theory Prop Id: d44bbbcb20f869f08519c1b4ae75e3aa5154362f4fff11b557d1e7203da89831
Theory Prop Address: TMJMAQYbxKT8R2VWNpVMaASpWXQHbY8dnP7

CRing_with_id_explicit_CRing_with_id

Theorem 26.2

explicit_CRing_with_id R zero one (CRing_with_id_plus Rs) (CRing_with_id_mult Rs).

The proposition is identified by the following information:

Pure Prop Id: a3ff783737e4d8958b2aa7415fe17473670a67c35da28139dff7bae3f4e94735
Pure Prop Address: TMVhXNKTXiit8PdcpXCyihn8b4FTsC3k2ej
Theory Prop Id: 56ddec707e12204d0adbd96c68e0500a77d81ad8852f7ce843d1a89f076bed97
Theory Prop Address: TMEtWQ6ajg8K49oJHzYUKgUUcZG8mV9k9CD

CRing_with_id_zero_In

Theorem 26.3 zero ∈ R. The proposition is identified by the following information:

Pure Prop Id: d1c133446d084c259b4e26e8455730b6db88c0480b50cfe8c1548ebaac397411
Pure Prop Address: TMUd78nxJNekmpXHWS1gxBmHFVqBbVXqURF
Theory Prop Id: 42b83047b5470f202412a31f5819c2bcfd6ce9af636f261d3171012400a8362c
Theory Prop Address: TMHL6R5QUphtzjmu3bzsfeJdq6c3Yemfgg6

26.1. CRING_WITH_ID 367

CRing_with_id_one_In

Theorem 26.4 one ∈ R. The proposition is identified by the following information:

Pure Prop Id: 2fa10721fd9208ab1e1eda4402ab18280e5076b055a04f23501ee00c86ae3bd6
Pure Prop Address: TMSn9V5PkmSZNhoiJE1BXRDWi4AyqGo5tEs
Theory Prop Id: a53700e2c16d4d39a135c22a89b655a707721c850f4b02f791f7b5962324450e
Theory Prop Address: TMHiL2AUU8VvaHXP7gKgkrGeZd2daCDT76Y

CRing_with_id_plus_clos

Theorem 26.5 ∀xy ∈ R.x + y ∈ R. The proposition is identified by the following information:

Pure Prop Id: 66b3347c3b4cc8181cb587a10e6fdddaf8f1db0dfaedebf691206c43b5454f81
Pure Prop Address: TMaPdKX5KfZy52M1fsgS698CBznuSXjAAQq
Theory Prop Id: d2fae7d02aba4468a0d3e7ae865a7175216a2064cc2a83eac8524bf761ae8665
Theory Prop Address: TMSuXgDfq9vWRJXbJvdfjSGgSqjU13XUvKZ

CRing_with_id_mult_clos

Theorem 26.6 ∀xy ∈ R.x ∗ y ∈ R. The proposition is identified by the following information:

Pure Prop Id: ef950216b478ee5ac786dcd5b26724cf2b7cca1507f263cf09aa4d17c6a4cfdc
Pure Prop Address: TMRK1cL4HUwW4gVfA7fWYbzoNAqvrXFqkd6
Theory Prop Id: 81665379575211b266779e952c3b4b18fd01a04f7da8eee59fb234fffb1db011
Theory Prop Address: TMbGpmv6AAcSZfaNEN3rm5PX6sRS59HttJi

CRing_with_id_plus_assoc

Theorem 26.7 ∀xyz ∈ R.x + (y + z) = (x + y) + z. The proposition is identified by the following information:

Pure Prop Id: 454cc66248a33dc6e34bc2d7c3406f6ea1034419dd0abe5e4915b38af1bdd1da
Pure Prop Address: TMQjX1emD1dnLgUKfE941oVUGCfDQyMUpXT
Theory Prop Id: f2fe8f25a1fc149692ba6c28a8f4d1838e62305907ba78cf398ef5c4730d85f2
Theory Prop Address: TMca3P6Uj8MFLwDfmKJw24JLqCBdBiTo2ge

CRing_with_id_plus_com

Theorem 26.8 ∀xy ∈ R.x + y = y + x. The proposition is identified by the following information:

Pure Prop Id: e1ec1adb066b85b12e0506e4fc204093d7e6ada60496967a6728d34eff569554
Pure Prop Address: TMPUNCr5oiXd8PKxvprAqJ3n8qdBby2zkwy
Theory Prop Id: 6b52cfd60b1677666cfc2a418ff72124083d95553587c923415f05e9d2660cfc
Theory Prop Address: TMYoTWDw5hq6S5t6qWKJAQSbseTkUH3rGXK

368 CHAPTER 26. RINGS II

CRing_with_id_zero_L

Theorem 26.9 ∀x ∈ R.zero + x = x. The proposition is identified by the following information:

Pure Prop Id: f35d5e2cb4c5f2f6924a8a216db8a83b2419462902ed4931aafb8448de773de7
Pure Prop Address: TMMv4dCHoLPd63XLsyBEpxxQatyYjuAN6Zd
Theory Prop Id: f987b8eb4b1d1c03ffa91d2acf41d423fbb463f0f6ff54054d915245384b9664
Theory Prop Address: TMZiPp7aZHNZCiWPSvz3S9vEUNYuY1ef5Ce

CRing_with_id_plus_inv

Theorem 26.10 ∀x ∈ R.∃y ∈ R.x + y = zero. The proposition is identified by the following information:

Pure Prop Id: 39fc1073cf2b1717e43d5e37e722ec6dc5e2762de4efea8b60fcd83168259e8b
Pure Prop Address: TMJppCMEt4Wjhgfzc6osCgs8pyrPW7A2t3q
Theory Prop Id: 8dbe3e89f0ed03c869c48edbd9729c9e9e07f83c48e1fc41cf4ce63726a55703
Theory Prop Address: TMGEihSCgFbz19W1WMiKWFqirbHZRxBnwEk

CRing_with_id_mult_assoc

Theorem 26.11 ∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z. The proposition is identified by the following information:

Pure Prop Id: fa25d807da6b86aa2872ac8f4b6ce9af0dde8c126b739a180e2292a5ad0b7dc0
Pure Prop Address: TMTGLRHnb5NytZEW9BtTjmjgzSoYUdX3Lq4
Theory Prop Id: 09949174d310e1abf6b516643ecbf69fb2cf7e3fb36f2b3cc266eaffaa5cca61
Theory Prop Address: TMUmkRGj1nHwhjroNxwFzc3LpQcb2T4SgJr

CRing_with_id_mult_com

Theorem 26.12 ∀xy ∈ R.x ∗ y = y ∗ x. The proposition is identified by the following information:

Pure Prop Id: 8fc88e1ed947c8d1d020e3f1b2c9ae9d85ce481324a851dcaf7ac3e82d522fb3
Pure Prop Address: TMUSB3uan5mCMRByLcZb9Vf6XGJLEqvGAx2
Theory Prop Id: 993072235b78d59978f6d2d7cf20206ad064f19386577afc365ea4ffde99e012
Theory Prop Address: TMbZ4NWw66ZTcKKd4JXSjKenmXtKidd92WT

CRing_with_id_one_neq_zero

Theorem 26.13 one ≠ zero. The proposition is identified by the following information:

Pure Prop Id: cedcbebb970be20c670e00da3c0b87ab77a367e70926a4ed9f074b2be9008982
Pure Prop Address: TMEvCCzeCPquYLSEU7q1oMEzpWzipFajv3Q
Theory Prop Id: ec7109346aee40725228c175e17b9274815d864e0391c1d1a02806855544bd46
Theory Prop Address: TMX7e8dpinoTbP3eGKDcZggchnnnzddqYNo

26.1. CRING_WITH_ID 369

CRing_with_id_one_L

Theorem 26.14 ∀x ∈ R.one ∗ x = x. The proposition is identified by the following information:

Pure Prop Id: a7a0f95242895f6eb543a18d764276740cde1e7c97815da94792785f6d715289
Pure Prop Address: TMMDR8jBbmxxtSxnqMkUtFxddjJTP4p9AoR
Theory Prop Id: 47ab0ad2e91b36e068e15b85e2ae4556a99d2d2131d1a80be5fbc3485fcd84a8
Theory Prop Address: TMZnbveBuCsuGB8UqSdxmfU8pxPepufVw7K

CRing_with_id_distr_L

Theorem 26.15 ∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z. The proposition is identified by the following information:

Pure Prop Id: 3c94070ae4c81515b83396b01a1abf18f8fb887ede6f676ce628a4dd5c373902
Pure Prop Address: TMRDgjKhiZf81J84szN54xfkhAS79YaDiGY
Theory Prop Id: aca955a68afb3f852001bc0ad5cef20f02c96080f226c19c097a0acaeca82731
Theory Prop Address: TMYojyDz1DaZZMWZU9guJSxD1TDFN4qyKXy

Definition 26.6 We define CRing_with_id_omega_exp to be

λx.nat_primrec one (λkr.x ∗ r)

of type ι → ι → ι identified by the following information:

Pure Object Id: b079d3c235188503c60077e08e7cb89a17aa1444006788a8377caf62b43ea1ea
Pure Object Address: TMQDSzK4zGei1WaFxwYxfp5T1fYD9frJ1iE
Theory Object Id: 5d35489a6173669c2175e151eba6c7c0d9bd86c956b8a3cbee0b8aa433d69a7a
Theory Object Address: TMQgksRuPWDCMf6N4hRujxQQ6p1EcFejvED

Notation. We use superscripts as notation corresponding to applying term CRing_with_id_omega_exp.

CRing_with_id_omega_exp_0

Theorem 26.16 ∀x.x^{0} = one. The proposition is identified by the following information:

Pure Prop Id: 8b9c8bf0cedf602ad64ec9fda9404cd08eaea17ebf9730500f9d73a231c12db0
Pure Prop Address: TMKBtqa6CgMy36XtRCUkiY5qZML3T3omNvz
Theory Prop Id: 64d60c82466552e43c260e08ac60fa038064ba97a7b299601bb5ac348d15e92d
Theory Prop Address: TMU64mwxKSqZr95cwqeqp8gLtWdfbUdGauU

CRing_with_id_omega_exp_S

Theorem 26.17 ∀x.∀n ∈ omega.x^{(ordsucc n)} = x ∗ x^{n}. The proposition is identified by the following information:

370 CHAPTER 26. RINGS II

Pure Prop Id: 14d493dd404e3f9509a3156f2272e3c51abbd469b9e5a348093995321cdc02c7
Pure Prop Address: TMaPKnDcmp98SwqmHGsP6Upmm4LVc7HBYeY
Theory Prop Id: b501f7792ce7723a2549037b7664ccee4a1f7319ecbe7ec95d81d30ee5d64db7
Theory Prop Address: TMWJS21wHv8n2fKeHv1VWF6sSsyo6xbsLgo

CRing_with_id_omega_exp_1

Theorem 26.18 ∀x ∈ R.x^{1} = x. The proposition is identified by the following information:

Pure Prop Id: 730a1c203f7c249e3caf69aa7f389915ff525f30f92b1343d270b5df775e11ff
Pure Prop Address: TMWRcHBBm5BHyxkidxbDUj99jsLbDdcXpMk
Theory Prop Id: d2a867dea7b48e2f9e3089f5db74c3d8470de6f670f9209a78e953d06761fa8e
Theory Prop Address: TMZwVw12XXo388yfUGycKftBC8osJcNPCHv

CRing_with_id_omega_exp_clos

Theorem 26.19 ∀x ∈ R.∀n ∈ omega.x^{n} ∈ R. The proposition is identified by the following information:

Pure Prop Id: 9a2b5725819918b6748494bfa537f50d0ca70b39d8c1e8ac3ac748fbeccec3f9
Pure Prop Address: TMXNCAZsnHcpmBbcTyxVYrdZgESRVaFBJ13
Theory Prop Id: a996c7c3b925951b4e2d7bbb01fcc9986e58880420715d9c42b7d108ffc16761
Theory Prop Address: TMc79AxZxe6P15NYTfHWGFMuKcAwNRAnJsW

Definition 26.7 We define CRing_with_id_eval_poly to be

λncsx.nat_primrec zero (λmr.cs m ∗ x^{m} + r) n

of type ι → ι → ι → ι identified by the following information:

Pure Object Id: b84ca72faca3a10b9b75c38d406768819f79488fbad8d9bee8b765c45e35823c
Pure Object Address: TMUwUn45GuDKgSXWDvDMPcovuhY4uP1dZdG
Theory Object Id: 808af0dde5838cc9e9caab3e4c3a545899005ebfaec703c548a438c00a522768
Theory Object Address: TMTHaSMuk9FXH4zmYumMT1hCMdrHTXWYL3V

CRing_with_id_eval_poly_clos

Theorem 26.20

∀n ∈omega.∀cs ∈R^{n}.∀x ∈R.CRing_with_id_eval_poly n cs x ∈ R. The proposition is identified by the following information:

Pure Prop Id: e579be29fc0b739e1e606a16df30fd76865f842f7859988a490019039a140bfb
Pure Prop Address: TMSCLopfevPZB8YYCtqzjREB3sftpBvTih7
Theory Prop Id: b40a29975dbc63c22e581f918f9362b2bd0b03c27af7e307e9ac129426d92073
Theory Prop Address: TMZXvJWRJta8nKp7RX1mQcytfayt5Sk89bB

Explicit Reals II and Fields

27.1 explicit_Reals

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Notation. We use −− as a prefix operator corresponding to applying term explicit_Field_minus R zero one plus mult. Let leq : ι → ι → o be given. Notation. We use ≤ as an infix operator corresponding to applying term leq.

Let N be {n ∈ R|natOfOrderedField_p R zero one plus mult leq n}.

Let Npos be {n ∈ N|n ≠ zero}.

Let Z be {n ∈ R|−n ∈ Npos ∨ n = zero ∨ n ∈ Npos}.

Let Q be {x ∈ R|explicit_OrderedField_rationalp R zero one plus mult leq x}.

explicit_OrderedField_explicit_Field_Q

Theorem 27.1

explicit_OrderedField R zero one plus mult leq→
explicit_Field Q zero one plus mult.

The proposition is identified by the following information:

Pure Prop Id: af6a3ba60f48d142602c8b680df2961966f1dca1044d4270f1f29c8a72c59c21
Pure Prop Address: TMZT73ZmoriPCPWsoj29MFJq3aK7fa41ff9
Theory Prop Id: 63d4c7bfde12c187c511272b653382138f0591cd1d246d9947331032fefa5c86
Theory Prop Address: TMGnn2Z2r22We9QuK5hXYB6c9XHixx14xDr

explicit_OrderedField_sub

Theorem 27.2

explicit_OrderedField R zero one plus mult leq→∀R′ ⊆R.
zero ∈ R′→one ∈ R′→(∀xy ∈ R′.x + y ∈ R′)→(∀x ∈ R′.−x ∈ R′)→(∀xy ∈ R′.x ∗ y ∈ R′)
→(∀x ∈ R′.x ≠ zero → ∃y ∈ R′.x ∗ y = one)→
explicit_OrderedField R′ zero one plus mult leq.

The proposition is identified by the following information:

Pure Prop Id: 1bc19ee4bcca58ce8b4534af0101d211b4fc533dc4db649a1bb010477c1d0998
Pure Prop Address: TMF49guKUWG2dHumcM9HEv55USiZpXFGiDg
Theory Prop Id: 9966383d1240a949659da39081fa3506cca4e377a161c6ab8ff8c05865a2eab3
Theory Prop Address: TMLR4J2wofUrAHouvRF1dws9BDPmZefs7gr

explicit_Reals_sub

Theorem 27.3

explicit_OrderedField R zero one plus mult leq→∀R′ ⊆R.
zero ∈ R′→one ∈ R′→(∀xy ∈ R′.x + y ∈ R′)→(∀x ∈ R′.−x ∈ R′)→(∀xy ∈ R′.x ∗ y ∈ R′)
→(∀x ∈ R′.x ≠ zero → ∃y ∈ R′.x ∗ y = one)→
explicit_OrderedField R′ zero one plus mult leq.

The proposition is identified by the following information:

Pure Prop Id: 1bc19ee4bcca58ce8b4534af0101d211b4fc533dc4db649a1bb010477c1d0998
Pure Prop Address: TMF49guKUWG2dHumcM9HEv55USiZpXFGiDg
Theory Prop Id: 9966383d1240a949659da39081fa3506cca4e377a161c6ab8ff8c05865a2eab3
Theory Prop Address: TMLR4J2wofUrAHouvRF1dws9BDPmZefs7gr

27.1.1 explicit_Reals_Q_min_props

Let R′ : ι be given.

Let N′ be {n ∈ R′|natOfOrderedField_p R′ zero one plus mult leq n}.

Let Npos′ be {n ∈ N′|n ≠ zero}.

Let Z′ be

{n ∈ R′| explicit_Field_minus R′ zero one plus mult n ∈ Npos′
∨n = zero ∨ n ∈ Npos′}.

Let Q′ be {x ∈ R′|explicit_OrderedField_rationalp R′ zero one plus mult leq x}.

explicit_Reals_Q_min_props

Theorem 27.4

explicit_OrderedField R zero one plus mult leq→R′ ⊆ R→
explicit_Field R′ zero one plus mult→∀p :o.
((∀x ∈ R′.explicit_Field_minus R′ zero one plus mult x = −x) →
(∀x ∈ R′.−x ∈ R′) →
N = N′ → Npos = Npos′ → Z = Z′ → Q = Q′ → p)
→p.

The proposition is identified by the following information:

Pure Prop Id: 31679e3e73614c3b4e276ed2c828816866f95e9a6a5e2f4721f319e80f473912
Pure Prop Address: TMLPtuuQw2TanZmHDBpjYzPS2V2V5uY5Rv5
Theory Prop Id: 2ead4f461950b2019d11d00ddc1a62977823328b7794ed17307fb66dde3d0ccf
Theory Prop Address: TMU33ZE5CuhCrg8XK3QNyMjHM54qa6M4HFy

27.1.2 Q is Minimal as a Subfield of R

explicit_Reals_Q_min

Theorem 27.5

explicit_OrderedField R zero one plus mult leq→∀R′ ⊆R.
explicit_Field R′ zero one plus mult→Q ⊆ R′.

The proposition is identified by the following information:

Pure Prop Id: aa9362c28b55f5ede37059e938f69a0bdc1f5a2c40ebc1e5233c99544f022cfc
Pure Prop Address: TMTBsyn4r1w2A2v1wey6f5Xapri8dMAVYuY
Theory Prop Id: 4e94b5ab940a9c05d8ab92b169c0a80ff166d8866174805a6eccc7b684604233
Theory Prop Address: TMURjNJo39iyQq3Vy1vrzEGuvvAuAkto3iD

27.2 explicit_Field_transfer

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let R′ : ι be given. Let zero′, one′ : ι be given. Let plus′, mult′ : ι → ι → ι be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Let f : ι → ι be given.

explicit_Field_transfer

Theorem 27.6

explicit_Field R zero one plus mult→bij R R′ f→
f zero = zero′→f one = one′→(∀xy ∈ R.f (x + y) = f x ⊕ f y)→
(∀xy ∈ R.f (x ∗ y) = f x × f y)→
explicit_Field R′ zero′ one′ plus′ mult′.

The proposition is identified by the following information:

Pure Prop Id: ef2cec44c7db17e7db32410fd90c52ba3fa20b7b18117cf540a99f8cbc44c289
Pure Prop Address: TMH4MjCpSyBbECQsWMxidZBXrRgD3jxBeiR
Theory Prop Id: ac0144b31bec8f30802ed9e79cf8dd729ee2b3186952c3cb53f3e3d279b5d6e2
Theory Prop Address: TMHe1ubYe2eN5E7tR9x9YatesmwdpCoqbZC

27.3 explicit_Field_RepIndep2

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let plus′, mult′ : ι → ι → ι be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Assume the following.

∀ab ∈ R.a + b = a ⊕ b  (27.1)

Assume the following.

∀ab ∈ R.a ∗ b = a × b  (27.2)

explicit_Field_repindep

Theorem 27.7

explicit_Field R zero one plus mult ⇔ explicit_Field R zero one plus′ mult′.

The proposition is identified by the following information:

Pure Prop Id: 54840abeaa24e09c7beaa1936a5defc6c4ea07ae3fb67135a78d8477a50986cc
Pure Prop Address: TMVPNiv37XT25SiNai41zYPSyiPfKQk6XPN
Theory Prop Id: e0eee3acfed1c638458013101d378b56d17393576fd269927418992a1b7bcbc9
Theory Prop Address: TMRdcfuetQSDQ7zq6iYdygkjYWKXS2v6SE2

27.4. FIELDS ENCODED AS SETS 375

27.4 Fields Encoded as Sets

Definition 27.1 We define Field to be

λF.struct_b_b_e_e F ∧ unpack_b_b_e_e_o F

(λQ plus mult zero one.explicit_Field Q zero one plus mult)

of type ι → o identified by the following information:

Pure Object Id: c7b84fda1101114e9994926a427c957f42f89ae1918086df069cb1a030519b46
Pure Object Address: TMaCGKsNBVewLEqMyTdCeiKd5uAgK8kFPjY
Theory Object Id: 622c30c9ddf18abf87387ddef4f26b6be7b92f631ff5e6b4f5eabaac5839554b
Theory Object Address: TMNqKgkVbs1Np95FXHhyjgcT3UAD21waXUL

Field_unpack_eq

Theorem 27.8

∀R.∀plus mult :ι → ι → ι.∀zero one.
unpack_b_b_e_e_o (pack_b_b_e_e R plus mult zero one)

(λR plus mult zero one.explicit_Field R zero one plus mult) = explicit_Field R zero one plus mult.

The proposition is identified by the following information:

Pure Prop Id: 0da03c63995752d95dbfb4f0ee50676ef20f4ec29763384de9d7a7d83ec6a852
Pure Prop Address: TMG8uPxmuweznLRMtKvhzoTZwhtZUZ4zZ9d
Theory Prop Id: 8e102fafcd372a62e72fe55218203e2c116e3761df378e4406f47cf88eb7c593
Theory Prop Address: TMbAKTnaoDqhKZZ3k6KRGTBNAbKxCGKqeLg

Field_is_CRing_with_id

Theorem 27.9 ∀F.Field F → CRing_with_id F. The proposition is identified by the following information:

Pure Prop Id: 26b551faed44acd5217dd94eab68e99a166869a3dfd0d404203d4386512d45c8
Pure Prop Address: TMH41wnx33D5QctmfmPNsPdyGr5hZzzsYVV
Theory Prop Id: e3b51a97808ddcd82d668dd97352b7379f2a20e8f4731608e340953d0b7c4cf0
Theory Prop Address: TMQA24cveY5wc5Ys2Z97RVQgobE2zAHEJMR

27.5 explicit_OrderedField_transfer

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let leq : ι → ι → o be given. Notation. We use ≤ as an infix operator corresponding to applying term leq. Let R′ : ι be given. Let zero′, one′ : ι be given. Let plus′, mult′ : ι → ι → ι be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Let leq′ : ι → ι → o be given. Let f : ι → ι be given.

explicit_OrderedField_transfer

Theorem 27.10

explicit_OrderedField R zero one plus mult leq→bij R R′ f
→f zero = zero′→f one = one′→(∀xy ∈ R.f (x + y) = f x ⊕ f y)→
(∀xy ∈ R.f (x ∗ y) = f x × f y)→(∀xy ∈ R.x ≤ y ⇔ leq′ (f x) (f y))→
explicit_OrderedField R′ zero′ one′ plus′ mult′ leq′.

The proposition is identified by the following information:

Pure Prop Id: d8a397eb2d41fc3df32161a85d5f88f8e1a1f828c3996623964e0cef70186f61
Pure Prop Address: TMPpLE4XGZTmn26HRXT5ka3vz3Konq3gJec
Theory Prop Id: 5e1306d3daaaa9e52a4c86d25354e58525141369b79522baa7699df2d0fe77ea
Theory Prop Address: TMau3sL5KTvBtt4AUk19sXEx1vw2u1rpj2V

27.6 Selectors for Fields

Definition 27.2 We define Field_carrier to be λF s.F s 0 of type ι → ι identified by the following information:

Pure Object Id: f0bb69e74123475b6ecce64430b539b64548b0b148267ea6c512e65545a391a1
Pure Object Address: TMY8sbcPiVs9VL311KPHpbm67U6sLM9JvF7
Theory Object Id: 9211e41eaf81bb8c152618cbe35230be60d736fcd5684c0b7bdc5ccc1ff70181
Theory Object Address: TMZQE6tPrFvJMaBcLrcMn85ZgscSeFbm4pT

Definition 27.3 We define Field_plus to be λF s.decode_b (F s 1) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 6968cd9ba47369f21e6e4d576648c6317ed74b303c8a2ac9e3d5e8bc91a68d9d
Pure Object Address: TMYTM1QQGqgVyRM9unR8CLL2scwRHYvV3c3
Theory Object Id: ed06aa34b4112f8507370b46c1c5c7de7f2f9af4f24b831d45893ad3066e994e
Theory Object Address: TMRiY7KU6BpU28KD19iwgBDLsQ5XyfpcbYL

27.7. FIELD 377

Definition 27.4 We define Field_mult to be λF s.decode_b (F s 2) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 4b4b94d59128dfa2798a2e72c886a5edf4dbd4d9b9693c1c6dea5a401c53aec4
Pure Object Address: TMQSTL5b71ddbjnkjnGXfRZ61uXWmMGwJHD
Theory Object Id: 7eb39ed695c795e4451a9252d011d377d0f4bdf0bfe6193f7a31ceb58f687a40
Theory Object Address: TMaJmzoi8UnDNTAXrAvZxYbTeEdKV3viygg

Definition 27.5 We define Field_zero to be λF s.F s 3 of type ι → ι identified by the following information:

Pure Object Id: 61f7303354950fd70e647b19c486634dc22d7df65a28d1d6893007701d7b3df4
Pure Object Address: TMJzMmYjL1HdXDWMSRf7c775tmGDc6osaRL
Theory Object Id: b74fb66a398c6c647d135ea9283353963f66d301d2c1ee126b203679f2071094
Theory Object Address: TMHQo21kvxiD8yab3ek4ATBz2PBfwvqsmaJ

Definition 27.6 We define Field_one to be λF s.F s 4 of type ι → ι identified by the following information:

Pure Object Id: 57e734ae32adddc1f13e75e7bab22241e5d2c12955ae233ee90f53818028312a
Pure Object Address: TMV8UDBZRqv4jocV6zt17aP89PcYugzQkoU
Theory Object Id: b2fa78f7f58b95f4bc4f46ac268c3a959090897e56338c64bd84a73bc5fc2551
Theory Object Address: TMWFb9uZk2JEesS9quTe9a6Ww4ND9VcbBJA

Definition 27.7 Field_minus is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 9072a08b056e30edab702a9b7c29162af6170c517da489a9b3eab1a982fdb325
Pure Object Address: TMSqYJKBR2z1AqDFGucgVbSMyDGaQ2vHXdt
Theory Object Id: 15e172a9ace52b84fbc8c26210b492b9383ddf56708a72b2b40566b8848bddcf
Theory Object Address: TMN39qFAba46WDLPG5mn6arc4RJWRpLYRxY

27.7 Field

Let F s : ι be given. Assume the following.

Field F s  (27.3)

Let F : ι be Field_carrier F s. Let zero : ι be Field_zero F s. Let one : ι be Field_one F s. Notation. We use + as a right associative infix operator corresponding to applying term Field_plus F s. Notation. We use ∗ as a right associative infix operator corresponding to applying term Field_mult F s. Notation. We use −− as a prefix operator corresponding to applying term Field_minus F s.

Field_eta

378 CHAPTER 27. EXPLICIT REALS II AND FIELDS Theorem 27.11

F s = pack_b_b_e_e F (Field_plus F s) (Field_mult F s) zero one. The proposition is identified by the following information:

Pure Prop Id: d8b340ddeffe429e1fbbc93f3ec159869e95202bb0548fa01ab79999624db595
Pure Prop Address: TMcWSufBdeKRFXPYdLCveXwHxkWQ1G8Ku3s
Theory Prop Id: 9308bd595b390a905ea8a71bb9251cf9b4ee7546ed5158a721de4d6eec1991fd
Theory Prop Address: TMHvjr39A4G916GqBKMAajZKdrtBFnoqgcY

Field_explicit_Field

Theorem 27.12

explicit_Field F zero one (Field_plus F s) (Field_mult F s). The proposition is identified by the following information:

Pure Prop Id: 63dbf45fdb4fede703bd1ce1ee23508eef7fbbd8640d566b8ed583677d390af1
Pure Prop Address: TMYH7ErtGiAdfW9CrDDJ623kNniytKGVcjx
Theory Prop Id: cb57dbd436ae765672438d8ff24477520b196a981db8feb13f2698511c06c6f2
Theory Prop Address: TMLZFScv2FGGQMRvkH7kAbY42toRcVj4bqQ

Field_zero_In

Theorem 27.13 zero ∈ F. The proposition is identified by the following information:

Pure Prop Id: 5d2a12ae03ea09b1b00364de5be96f9417eba1807387eef0593c4686271f8b43
Pure Prop Address: TMV5NCa6Wx7gnzRhWo5AeVT8cTwstJid2DP
Theory Prop Id: 124d417e36ffc4ecd962d65565094a8b6a908ebccf5fab90ad3c52962d4c8899
Theory Prop Address: TMJTmqQWrFwKLaFx3qyrzQjDtPmc6sNEAEL

Field_one_In

Theorem 27.14 one ∈ F. The proposition is identified by the following information:

Pure Prop Id: 44022c6821b37c9bd64d3412ecf0090eea66e80eebb2def47c0ea8452f269ff2
Pure Prop Address: TMNtPhTBrbXJwXqr9MRjQjBCeakFLxdm4rA
Theory Prop Id: 54c548fa567255c0074c7815ff32044eca656b312bcbb87e75ab72ac4e037c4c
Theory Prop Address: TMaxye1k4K8yC8XLPLqCw4nFNFVXQMfGNmy

Field_plus_clos

Theorem 27.15 ∀xy ∈ F.x + y ∈ F. The proposition is identified by the following information:

27.7. FIELD 379

Pure Prop Id: 6cb9ab847c2ae424b9e4a853861f83b1519ca940a43ca706c3f58543aa68e6f8
Pure Prop Address: TMSL8rTfsH4EGhTbnt4mmV2f3XC1kRDSuhc
Theory Prop Id: 1404ed2bc6fc9cb6d24916d53178afcbbae9052d74dbddd08abb219ef3cc6988
Theory Prop Address: TMU5Koik3TBZQ4TJ7EcUGYpqbJxJUvF15eS

Field_mult_clos

Theorem 27.16 ∀xy ∈ F.x ∗ y ∈ F. The proposition is identified by the following information:

Pure Prop Id: 90491b3584ca5d345c64ad706ac87ca12bdae4cf2f13a8b47ed70d427919cb6a
Pure Prop Address: TMXeFVMheHpzd5M1At9SBCuvJfvr5e1F4vX
Theory Prop Id: 865f20b7a4b6f98b4c54586f84b21600d7e07243207713b57fa40e2ec5eda5ff
Theory Prop Address: TMLt6bowMxotb2XT7CkV7ag3q8KNznCkvSg

Field_plus_assoc

Theorem 27.17 ∀xyz ∈ F.x + (y + z) = (x + y) + z. The proposition is identified by the following information:

Pure Prop Id: 49e71c4d43cf0fdaf6727db033c4e9e44f74cc50c7f11e8f287078c9de4186d6
Pure Prop Address: TMctJ6QSciEPYRwyuAGW91t49FWX5iwGskB
Theory Prop Id: 5dac6c802dc58a4009410054756eb7533311da03f206d571ca13b54b7de62f48
Theory Prop Address: TMF9ZdDcMZBL4YbPNXREiZ4z6Ej4bQZGiAt

Field_plus_com

Theorem 27.18 ∀xy ∈ F.x + y = y + x. The proposition is identified by the following information:

Pure Prop Id: 3b445936b40abf6f0f573b3695af272cd8f98957cd9977a5b0ca3f6dadd52baf
Pure Prop Address: TMcSdkyySZ2JMLtoyeiiuVnq14co8TxbPgp
Theory Prop Id: 27a761d026aec44ac48a86b672c56e46b77dfbf7622acf1151a03c5a486af018
Theory Prop Address: TMM9QF6dQ6HnJnFroaJHaH635ak1Jn8Jx1o

Field_zero_L

Theorem 27.19 ∀x ∈ F.zero + x = x. The proposition is identified by the following information:

Pure Prop Id: e0daa0e57cb9935241117668f353fd724bcb6173088571de6e8928a74ded74c9
Pure Prop Address: TMTonQi48JamuRgckkERNUD5uLVijoYmgL3
Theory Prop Id: 37d0579521bcd82e296de371d607f6dbe718d874c53486483fbe1d86059ae1e2
Theory Prop Address: TMUpYjiRJxYVxfan4X4iVYUVm3AfpaMR9Tf

Field_plus_inv

Theorem 27.20 ∀x ∈ F.∃y ∈ F.x + y = zero. The proposition is identified by the following information:

380 CHAPTER 27. EXPLICIT REALS II AND FIELDS

Pure Prop Id: 919ce52a6adadf0cc8094ecbe60076ee5858155672dd27ab2180e7e6c7bbe1f6
Pure Prop Address: TMN39v7z6H2CoUPdUhfYEYjvy3Zw4eENYPq
Theory Prop Id: 262504b07a3f4e1940e4234afa8b6cb0e82cce09ab141e4e0d8b9675a6e62edd
Theory Prop Address: TMJdMzeW7T3rZffNfiYy8hAcNdvCNvWPow9

Field_mult_assoc

Theorem 27.21 ∀xyz ∈ F.x ∗ (y ∗ z) = (x ∗ y) ∗ z. The proposition is identified by the following information:

Pure Prop Id: 08143128b200fbed930f0fff6e90f4cb05098f19ae900fd1f1cb0beb563e2c09
Pure Prop Address: TMc9dXbETfWeBa2b8YnDKbhLw7uaVyQboHe
Theory Prop Id: bad75ea2350024fa534d6b728d0eba5cf28e28fbe493d52077a0cc7bedc82c9d
Theory Prop Address: TMLXQH9iTPeEeNTAWWLcpo5cJTA4AgEWJuq

Field_mult_com

Theorem 27.22 ∀xy ∈ F.x ∗ y = y ∗ x. The proposition is identified by the following information:

Pure Prop Id: b067a98764ad614c2eab7437092942718dbd9dd9a388f583ce6d32e68bd7dd64
Pure Prop Address: TMPegBnUirvhxDN9mWgopPcPibU15GBD2aT
Theory Prop Id: 95c74fc682d34ffea07fca550e5dfd7573581c19dd74de76bca295caa9119383
Theory Prop Address: TMSy5iWokESs8xWZnGcamGMUMENTfP5k3Jh

Field_one_neq_zero

Theorem 27.23 one ≠ zero. The proposition is identified by the following information:

Pure Prop Id: 0cd6b9f7a2a8864851fc03b62eca322da80ca6a538426cf27bba6634e3656726
Pure Prop Address: TMbSs4K7qkmEkJCui8RQ5ucU6Bxp9AhXSiP
Theory Prop Id: 266cd947601e72d416352cfdc17094f13320ebf353611367683f3cf2c946d097
Theory Prop Address: TMUvPmKRiKPF1BAM9e9AAYevwEYsUoNceXf

Field_one_L

Theorem 27.24 ∀x ∈ F.one ∗ x = x. The proposition is identified by the following information:

Pure Prop Id: f43915f1f2955af2eb66b15dad875d94d73d7b7b9834e7c9adb0f55df5881fe9
Pure Prop Address: TMXMDwpgqy9TopEv1E4DfdDaXfqz9tA6hLH
Theory Prop Id: 013427804b41d47918f24b864b24aab70dfc31113ba98edcd33612349510178b
Theory Prop Address: TMZ8XZoWUvugHdtZXDcG1tmeGAV5J4gRkxN

Field_mult_inv_L

Theorem 27.25 ∀x ∈ F.x ≠ zero → ∃y ∈ F.x ∗ y = one. The proposition is identified by the following information:

Pure Prop Id: 159c70606656e548df27dc9f2989d005a799bc59dfc48530c9aaef4d88bfe55a
Pure Prop Address: TMPGyXrYchHMpQoxBotw4fTsy9qkcod1cXb
Theory Prop Id: 99224d179a2cb454d745622ccfd66d3d60660e4bde41c61da98aa4dd3e938ee4
Theory Prop Address: TMH8ZcNMHvhfoVar9hvT3VD2o3DW9jyeJdr

Field_distr_L

Theorem 27.26 ∀xyz ∈ F.x ∗ (y + z) = x ∗ y + x ∗ z. The proposition is identified by the following information:

Pure Prop Id: 250384aaa9ecf10fbcf4ebefa3dc489b41d3cb10f656500151d49a9053cfe4f9
Pure Prop Address: TMZiuCgz7mgpvcfSTTHjcdzaaeZFaZEUrv8
Theory Prop Id: 0ebf99ce21204a4f57fb5783b2803ac92906cc63624dd2cc67cc70a1353aeee0
Theory Prop Address: TMUjioKdcupXefDDTsQSWE2Vw19Nna8245U

Definition 27.8 Field_div is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 33f36e749d7a3683affaed574c634802fe501ef213c5ca5e7c8dc0153464ea3e
Pure Object Address: TMR6i3AfWJEnAVs3vStqGHLoyVAtt7HuhVZ
Theory Object Id: 3a617adfbc7761f4bff8c1a67cc5a2811928c372828b963f2809216d145f0056
Theory Object Address: TMKD8yCQcW3giMss49AdhftfjbmKvDhxqNd

Notation. We use ÷ as an infix operator corresponding to applying term Field_div.

Field_div_prop

Theorem 27.27 ∀x ∈ F.∀y ∈ F \ {zero}.x ÷ y ∈ F ∧ x = y ∗ (x ÷ y). The proposition is identified by the following information:

Pure Prop Id: 320029a8925ea4f20053afa631f90353310f8bd2a02fc1312664ae3d5e617bab
Pure Prop Address: TMLCRGM73NJSReYA8S8dFeUvsVHD5BezMmw
Theory Prop Id: 11f8a74d8832813bb0bf2e47adbaecaec90b4bb4ffc2917b900e65c5fdfcd21a
Theory Prop Address: TMFtSZaRxbLf7cg7s9eC7F24ERy3YpttcjS

Field_div_clos

Theorem 27.28 ∀x ∈ F.∀y ∈ F \ {zero}.x ÷ y ∈ F. The proposition is identified by the following information:

Pure Prop Id: ae15934f0050805acbad83a7ea9b08d39343e7275a2aa0544983f8c3029b68ab
Pure Prop Address: TMLj2rQAk7mQRGGwfxY9ab6r8GZpdh5nryZ
Theory Prop Id: f08d2cba3fd57f474e2e6a9ed26207be75354dfc72cde6a65408f169f301435a
Theory Prop Address: TMLDYbyYNgVKinoxufwLvutrLNRfnk66vDt

Field_mult_div

Theorem 27.29 ∀x ∈ F.∀y ∈ F \ {zero}.x = y ∗ (x ÷ y). The proposition is identified by the following information:

382 CHAPTER 27. EXPLICIT REALS II AND FIELDS

Pure Prop Id: 1bda12fe605e6023cf81fa8393926be18b348502472ef7b4c757036a807eff68
Pure Prop Address: TMb6WJncGKrL2byhXB2YuUpprymEHFctWHV
Theory Prop Id: c8ef92ac56a58978069a954b8be3acacbe73d63049ec632444d403ecbafb062b
Theory Prop Address: TMEsHD4dd1bPW6M8yUdXaW4UYoSXZQW37fk

Field_div_undef1

Theorem 27.30 ∀xy.x ∉ F → x ÷ y = 0. The proposition is identified by the following information:

Pure Prop Id: 20e2690180e6250c71da49027a7a363d806e4fdad713709dbf99337aaa83d216
Pure Prop Address: TMWhJ43K252biCbrmZvzzeVwqqpjcQVZBxU
Theory Prop Id: b5ccab25818eb5893b4528f4adc070775587f1a1b252b703a355452ef3ec25ef
Theory Prop Address: TMS9ggt9Tn8AYLjGKxejwDFj1op6j9f7TWs

Field_div_undef2

Theorem 27.31 ∀xy.y ∉ F → x ÷ y = 0. The proposition is identified by the following information:

Pure Prop Id: 08ea037ff82421470289f5932634b1ce136fcb6d1f8706fa6883561e9bff8d34
Pure Prop Address: TMUWRvAmiES3iybNvXn7sUoxZwKmagyQ8rC
Theory Prop Id: 6c03f1015d94454404a03499287892345ea00012a20efb26d3b80fcf490734cd
Theory Prop Address: TMLn5yRrKaaMpTjWyKDfNinNPu78LWX738s

Field_div_undef3

Theorem 27.32 ∀x.x ÷ zero = 0. The proposition is identified by the following information:

Pure Prop Id: 5c48b82d4fc362fe344e205c36ca6878bd5138f2eb8555efdd2826a31ede70e2
Pure Prop Address: TMcjdraN7C77qSyG5w9GJM3oWt4kL3HUmCG
Theory Prop Id: 24e90d5e4e5f1d4012652a7c973b6fdadb4b7a94e33c52e4c0945440a41f612e
Theory Prop Address: TMUNnuEej7hLE5MrRKfW9mQkKWFqHyvwbkp

Notation. We use superscripts as notation corresponding to applying term CRing_with_id_omega_exp F s.

Field_omega_exp_0

Theorem 27.33 ∀x.x^{0} = one. The proposition is identified by the following information:

Pure Prop Id: 4362c28e2db5e3383addcc627e5b9fc5c9d8d6e0b00cb0d2216109cbb0b3cfb0
Pure Prop Address: TMNwr244Poo2trbPpcVQmtxpagskYic6goM
Theory Prop Id: 4527b98489253d272cf17c4b7c4e4690a8ebdb756ee228c39e35e2396e350909
Theory Prop Address: TMKmcrh5e3aKCAfTomskajMoU8zcUuiFdRz

Field_omega_exp_S

Theorem 27.34 ∀x.∀n ∈ omega.x^{(ordsucc n)} = x ∗ x^{n}. The proposition is identified by the following information:

Pure Prop Id: d2e5103c06f38d501b5624ac09de58fe1199ed0428e0099e2cecd857b56c9463
Pure Prop Address: TMRxJHnciqiQFbYupzsqKnpymvXkgoFKbz2
Theory Prop Id: 414637f793aeaef595d4415aea905c8712dd128bdf3888d6a1b8b79c783f993f
Theory Prop Address: TMXUtYW7Vcrt9RLnqAwfxxVBZXyfpj7hTyU

Field_omega_exp_1

Theorem 27.35 ∀x ∈ F.x^{1} = x. The proposition is identified by the following information:

Pure Prop Id: 351bec92d644d3d3f57736c30184720a55a3206dd32c5c5de668a1a15d21cf70
Pure Prop Address: TMKtTCkPJ5sHx3gK7y8wuSN5Rfkt5hunA6A
Theory Prop Id: 97fa394baba04b04edef47f65fc4ec793503e5777b288b978d77b4c9bb64fe45
Theory Prop Address: TMTi7WjdmiDN2vrsRDu17s69nAUYnNDuowU

Field_omega_exp_clos

Theorem 27.36 ∀x ∈ F.∀n ∈ omega.x^{n} ∈ F. The proposition is identified by the following information:

Pure Prop Id: 026ac1d1f5ef29832893f6c7944103acad2da742c648c31b129e3d067c5334f3
Pure Prop Address: TMSeptsHCdwBLhRoQhW2RzWACc5328SueZM
Theory Prop Id: 8e8df3c69edbfdbdf46653edec4be7381ff02d8b6f0c54facfc9849a6b7ad1fb
Theory Prop Address: TMJ98yveoAeohCchdDEX5eSGLgbMb5NtmQ4

Field_eval_poly_clos

Theorem 27.37

∀n ∈omega.∀cs ∈F^{n}.∀x ∈F.CRing_with_id_eval_poly F s n cs x ∈ F .

The proposition is identified by the following information:

Pure Prop Id: 339c6079b932e105f62ee9a139090ec0c35d05a91b77324b7c9d00829f5e157a
Pure Prop Address: TMUHU4Kdwck1DovJNNC1BznGxQ9LfqrpcD2
Theory Prop Id: e3e5a122a52cb88471f32d0108c8ebfd779225b3f222aaa00e6b00105eb91521
Theory Prop Address: TMWrCNHLicENtgPnEmebmghuFpCDVv9acP4

Field_plus_cancelL

Theorem 27.38 ∀xyz ∈ F.x + y = x + z → y = z. The proposition is identified by the following information:

Pure Prop Id: 443b41e62519d1cf11ce79b28ce6902b1cb036b303feb0522d8233e77510b0da
Pure Prop Address: TMQYEe7aS7C6puTrXwtgHP1HzLv6iA9FCYV
Theory Prop Id: 65814d60a21e4514adfe8122a71e77983e1c014e2364364e2b428a8e889110d1
Theory Prop Address: TMPeQdMTjszQk8ZwjKbe2Yx9p1bhEuku2th

384 CHAPTER 27. EXPLICIT REALS II AND FIELDS

Field_plus_cancelR

Theorem 27.39 ∀xyz ∈ F.x + z = y + z → x = y. The proposition is identified by the following information:

Pure Prop Id: f16e614fd8bcca5468eff3f77a1ffdbdf61ce6894a9642616315fd8217fa6896
Pure Prop Address: TMHxcrqScEQVs2BM9LqdwdJ73QaGHTHjtAw
Theory Prop Id: 5f414f4ccd4d5d42eb071a720d4cfd7f8e2a864a6d3ffecc438de4a083e92c4d
Theory Prop Address: TMMUfLR9eAhuBx9dH38X5X7iGs36sxjomLw

Field_minus_eq

Theorem 27.40

∀x ∈F.

−x = explicit_Field_minus F zero one (Field_plus F s) (Field_mult F s) x. The proposition is identified by the following information:

Pure Prop Id: de7b3a103710cd73b730d9ab58f9ad4f2d88bd767fec9ca2edba308880936b16
Pure Prop Address: TMVexy4tBFivo588m7GQ84C5mQBJ4yZx98Z
Theory Prop Id: f6684a8b3c3ff9441214e19ef1db049db0227f25e3db1d329cfd06250e47f513
Theory Prop Address: TMGtBPcXh4HJHkfMAC1yckyY5RPTB4NACYW

Field_minus_undef

Theorem 27.41 ∀x.x ∉ F → −x = 0. The proposition is identified by the following information:

Pure Prop Id: 357d777053f15331bc00d78b8e12f89dc8130cc8c44f2c6131dcc55defa8ee0f
Pure Prop Address: TMTwnPHTSFHEiMGLciYU9XDmEYP1U9Z2eZN
Theory Prop Id: 60ed459a7799cd3e857363ae83693d0f4826d792577f17db36e25c07793bc296
Theory Prop Address: TMW6cjVsvwBeiH3JX9mV7y27yXLNaXoLkjr

Field_minus_clos

Theorem 27.42 ∀x ∈ F.−x ∈ F. The proposition is identified by the following information:

Pure Prop Id: f1becfbfee0ef8b26de040c7f18230ec7527ddcf884ada3c8941b9b5c6de9b87
Pure Prop Address: TMbwyWNKrfuYFG1TmqXNEzZJ4rHSH3Yvzwv
Theory Prop Id: d590d7bb3e9f5cb4915a63f72350aaa482789cd650f74ceaa143f1b01be467b9
Theory Prop Address: TMKy1fXBQm9Ci94udEjidssK8Da4zqwq6BX

Field_minus_R

Theorem 27.43 ∀x ∈ F.x + −x = zero. The proposition is identified by the following information:

27.7. FIELD 385

Pure Prop Id: 5cb54808b5cc0bfe1a9faf501b5e357f53bff36ec590e8f602fed76ba642d4b3
Pure Prop Address: TMR6vs3rpH6qDTpNibPSbTftrd3h5nzUygq
Theory Prop Id: 2391ef531b49ac2b610cb94e6726c81909131c9e4acf4d3aa6ee03f5bdeea688
Theory Prop Address: TMMcagzy7PNZ6o6f6y6gxJVCHKFabzvbY1n

Field_minus_L

Theorem 27.44 ∀x ∈ F.−x + x = zero. The proposition is identified by the following information:

Pure Prop Id: 7c66759f9e4abb07e2159e5e1557c1602f8cb0f7f38d7aeae89ee98bbc859ebe
Pure Prop Address: TMHN25GQoh4jYakdvcZYx96QdJXH514To2u
Theory Prop Id: b5cf76ad592ab4b5a7384338d4297cf97c6981dd0a45ac579115230625384cc5
Theory Prop Address: TMXuvPZ1qizoaTqpK2wbJgJGEjrtducR28f

Field_minus_invol

Theorem 27.45 ∀x ∈ F.−−x = x. The proposition is identified by the following information:

Pure Prop Id: e097d2ad2fa0d0d1bf0193b8e412f65b42e3e5c73733c262d1b7eeb1cbd57883
Pure Prop Address: TMMMeGBCnj2s3dzeT7Yfa9bpUfU46psmpSe
Theory Prop Id: 7f6c7d184b0f20e8f912398c740d072ae44f16dc17d5358d415b7f2a5ccdece3
Theory Prop Address: TMTzNG5pZe3QRegKNVBSsZp3PqJjsutAt1f

Field_minus_one_In

Theorem 27.46 −one ∈ F. The proposition is identified by the following information:

Pure Prop Id: fbd93d93f1671451fec2d592c753cd960b2e1288c4241e1fad9a57ee7a022c02
Pure Prop Address: TMHWKgvtYjVJyEYbaQY7sTTe4usgtgVj9LS
Theory Prop Id: 61a82a30b97b7f4fb97d4f6625d6e919a0010537baceadb9f9b571c21d50a9f8
Theory Prop Address: TMSwuqi7dSFVmvwt5crCA1gi7nDJn439QVp

Field_zero_multR

Theorem 27.47 ∀x ∈ F.x ∗ zero = zero. The proposition is identified by the following information:

Pure Prop Id: 45513e8074cff3c93f264cf92510bcaed6b453af70ec391191356622b3682292
Pure Prop Address: TMPYu6VMYJksB1pVtcZhxA1ZtBtBiyAQA3q
Theory Prop Id: fadc07fc14b79ffe08eb28c7b3dd0a9607e542c43123389eacca8922d9a3bbd6
Theory Prop Address: TMWgQLa4SJqx34swEHxJoxG1BrHPgR8ESyy

Field_zero_multL

Theorem 27.48 ∀x ∈ F.zero ∗ x = zero. The proposition is identified by the following information:

386 CHAPTER 27. EXPLICIT REALS II AND FIELDS

Pure Prop Id: 5ee5eb5fde0181d92b2c7f6d7a2a896cd74e3a750cdabbb27d94b1bb4aab908c
Pure Prop Address: TMLc5uhw7hxR8sbQGdBpQkCtjeGhbq6hy1L
Theory Prop Id: 3e7854299ba2aa2daea10889a8309962f1b1d6c32b7006fc1c40500925221173
Theory Prop Address: TMRcY5cpYprw5iQzhQcoGB6ndppCPbrhUtT

Field_minus_mult

Theorem 27.49 ∀x ∈ F.−x = (−one) ∗ x. The proposition is identified by the following information:

Pure Prop Id: 7cd676e4adffa30d0cf22cf26fcde8bfd203a61ee59f416b12fa62cfd93db1b7
Pure Prop Address: TMHF8RJu8ok4xK5Js75hyF7aTwTeYUktaWC
Theory Prop Id: 775b246ad68e751222cbcb9a5e49e184ed3019fcce6a74f91edd30125e3f0098
Theory Prop Address: TMQzhW2ohogP8qPJxruSSb4HgNFe1kHyJaE

Field_minus_one_square

Theorem 27.50 (−one) ∗ (−one) = one. The proposition is identified by the following information:

Pure Prop Id: f1d40df6baeeb2bbfbb7f411d9caeb48477cc6718458d7e70ca7bd1b7a78d1cc
Pure Prop Address: TMPY9Fe2gL8ytA1WVeKrmqRrbe6bXkndVWA
Theory Prop Id: 4f370c1adbdb484572d2f537b4a62640f4e6192dcbbfc4d7dee46fb09f3433e6
Theory Prop Address: TMYd1s86jR7deZ3QCmxu2FzfXcVu8tMBNyU

Field_minus_square

Theorem 27.51 ∀x ∈ F.(−x) ∗ (−x) = x ∗ x. The proposition is identified by the following information:

Pure Prop Id: a55d0e9b98e66f6d5322b6523e3e708815bb201d618aa58f328931f5f07d0f37
Pure Prop Address: TMN6DCGTRZLbdyYaDBJ9hD7em3K6sLzKydd
Theory Prop Id: 726593b03eefe34d9157a17b6e679b85bcf1865e151930e64fe462ee18a8a6a9
Theory Prop Address: TMNpHhjMNCtccTVy7oxGE8LEDQS3ikFDxRL

Field_minus_zero

Theorem 27.52 −zero = zero. The proposition is identified by the following information:

Pure Prop Id: f5214a9560f5568380d8be3826be07444fe57c851dd66aa09a5764efb8077718
Pure Prop Address: TMNcsvycuD4vkz4pfqQk1YGD8DMTSNJU4AH
Theory Prop Id: 25e6829a5bf06f2eaecafec005fea75bb87bdbfe0e161ec9b16912fa023dc76d
Theory Prop Address: TMXQJxi91EWkUHXgdLamfSNazD2x5oL6iHv

Field_dist_R

Theorem 27.53 ∀xyz ∈ F.(x + y) ∗ z = x ∗ z + y ∗ z. The proposition is identified by the following information:

27.7. FIELD 387

Pure Prop Id: deb7136831fde91f8aaab03eb0c31cdcadb1e244807de445094801e21f93cc47
Pure Prop Address: TMdNLPJWkse1Z9urNENt7q5Qeyf5LwWd49a
Theory Prop Id: 10fe6a61397ce15f1f074447b70bfba8150ab8b2d77f7129a3ca0d5dda084ce2
Theory Prop Address: TMNDDzYVTXFEChN6smH4YbceCPCtL4pBEz9

Field_minus_plus_dist

Theorem 27.54 ∀xy ∈ F.−(x + y) = −x + −y. The proposition is identified by the following information:

Pure Prop Id: 7d70513a02d66f0f56d24e5c8cab215e006349e8a3dbcd5fd887d9d1dcc56fe1
Pure Prop Address: TMKE1EAuG9tUBJEqHNho1ojpdVMUyyVrRkp
Theory Prop Id: cf09f8db6ef5dbb5d2ed9a9695a44c7323061de997ed2f25e7785eb1d93e4486
Theory Prop Address: TMRjUdfwmx5YiiXEHvkgNMeZ615jeu3ixtf

Field_minus_mult_L

Theorem 27.55 ∀xy ∈ F.(−x) ∗ y = −(x ∗ y). The proposition is identified by the following information:

Pure Prop Id: 58520e0833b7099a6940ba9408893dd9df17b42fbcadaeb89d0b3be3b1d4b459
Pure Prop Address: TMF5FPzDp822hueufQ3yiZiRPZucSMkbB1N
Theory Prop Id: 09040b49c949f9afb231e21ee54bb76add27e0f1944d83e5dfdea037fb2b5e78
Theory Prop Address: TMHvcKKRtR69YcV7eYTB3batvpaofCBGg4b

Field_minus_mult_R

Theorem 27.56 ∀xy ∈ F.x ∗ (−y) = −(x ∗ y). The proposition is identified by the following information:

Pure Prop Id: 0f87ffe9baf0ac37844ca25d819bbf2e6b56af4a7acba178604fef6b2c73975b
Pure Prop Address: TMNFhiGH735rxLAYHzqXjLVSVhiSUKYVJFj
Theory Prop Id: d3e4735bbc06ae8a25cb40640c44781c9e83add0c02c7c02058ad54bddb5536e
Theory Prop Address: TMbXkVYiquoTVr7qFbrQXZCThaRMHyD72Na

Field_square_zero_inv

Theorem 27.57 ∀x ∈ F.x ∗ x = zero → x = zero. The proposition is identified by the following information:

Pure Prop Id: 2b2bd6de108abc8d3794de6a31f5341040d00d973de79329c574098caed59e31
Pure Prop Address: TMZKCqVzNkAqM9J9D53Y9aBiDZvqkPQkdVB
Theory Prop Id: 08c2f8793e760cef2edbc292e690c746158bd7259df3295852d68c7214236e65
Theory Prop Address: TMUCadVDmPtHtGW9E9rogNfiQCpPtyuh4MU

Field_mult_zero_inv

Theorem 27.58 ∀xy ∈ F.x ∗ y = zero → x = zero ∨ y = zero. The proposition is identified by the following information:

Pure Prop Id: e191e1698c04a863c4f10266ada28dbd2db7490864c6c1c80f7dc9b6049a62ab
Pure Prop Address: TMY8AV6XaQZR4ezqWu2BT6WM441CDJBthsX
Theory Prop Id: fbeb3acafb828d177072b74d6982e4e3a3df773c746d4ea7168c3a03b93613bd
Theory Prop Address: TMYaR1LATsRePnNzvKfW1CMi8VJXyAhwf3e

27.8 Field2

Let F s : ι be given. Let F s′ : ι be given. Let F : ι be Field_carrier F s. Let zero : ι be Field_zero F s. Let one : ι be Field_one F s. Notation. We use + as a right associative infix operator corresponding to applying term Field_plus F s. Notation. We use ∗ as a right associative infix operator corresponding to applying term Field_mult F s. Let F′ : ι be Field_carrier F s′. Let zero′ : ι be Field_zero F s′. Let one′ : ι be Field_one F s′. Notation. We use ⊕ as a right associative infix operator corresponding to applying term Field_plus F s′. Notation. We use × as a right associative infix operator corresponding to applying term Field_mult F s′. Notation. We use −− as a prefix operator corresponding to applying term Field_minus F s. Notation. We use −− as a prefix operator corresponding to applying term Field_minus F s′.

Definition 27.9 subfield is the opaque object of type o identified by the following information:

Pure Object Id: 9216abdc1fcc466f5af2b17824d279c6b333449b4f283df271151525ba7c9aca
Pure Object Address: TMJhV7ybK4sNoy7dLu16XHkQMMh2u7ogDg9
Theory Object Id: 8b767b8278051b976eaa45775446a5c3d3fb3b6c01e0d6652741dd499114ae43
Theory Object Address: TMGfnnZhzcvAyc6uD3itbd4JQrSmEALeWf1

subfield_I

Theorem 27.59

Field F s→Field F s′→F ⊆ F′→zero = zero′→one = one′→
(∀ab ∈ F.a + b = a ⊕ b)→(∀ab ∈ F.a ∗ b = a × b)→subfield.

The proposition is identified by the following information:

Pure Prop Id: 7e727ad2e5128678d00eab2e3f41718c941d248ab8bc80d9290e8e449722a6cf
Pure Prop Address: TMM7f9u7WrEFj8YHR1zoL7g1oxSnnpx4nuA
Theory Prop Id: d7fa7daaa1fd31a6790debe2a61072593e062f53926bea2d384140301fd0f3c5
Theory Prop Address: TMTAZrzxHvVF3ivGCdEm8NHia2yU7XfKPxD

subfield_E

Theorem 27.60

subfield→∀p :o.
(Field F s → Field F s′ →
F ⊆ F′ → zero = zero′ → one = one′ →
(∀ab ∈ F.a + b = a ⊕ b) → (∀ab ∈ F.a ∗ b = a × b) → p)
→p.

The proposition is identified by the following information:

27.8. FIELD2 389

Pure Prop Id: d7e94964874d4552f7612152bc7b4fe8077a730735f362aadb7346233bbbc7eb
Pure Prop Address: TMTNHkoYxk53iGUuEwSw3EVWoAuX1WUvdw8
Theory Prop Id: e75e56a8a51307bd8de2cf55f809fee0d2458291f278ceed9218a3d441845a43
Theory Prop Address: TMNqv7RamAPGivDz6Nh19aHNfkFEMuvESP4

Definition 27.10 Field_Hom is the opaque object of type ι → o identified by the following information:

Pure Object Id: 597c2157fb6463f8c1c7affb6f14328b44b57967ce9dff5ef3c600772504b9de
Pure Object Address: TMGp1283AAzgHauWyMFM6fwpTKhT2UvW44v
Theory Object Id: 30bf3301fb81b8870c663685fc6bfa8593d92c4bb8d8baedadeac04223d7577b
Theory Object Address: TML6Zjm746Vd3N4sjVSwnNb4J5yLWD6e5Yj

Field_Hom_I

Theorem 27.61

∀g.Field F s→Field F s′→g ∈ F′^{F} →g zero = zero′→g one = one′→
(∀ab ∈ F.g (a + b) = g a ⊕ g b)→(∀ab ∈ F.g (a ∗ b) = g a × g b)→

Field_Hom g.

The proposition is identified by the following information:

Pure Prop Id: 31c68d4d1908803e35c198b4917b363aece27f736abc525d3be26b2bbfaf25ed
Pure Prop Address: TML1nC9DaTHq5nADe4PBSDz3pRDBKuM4coa
Theory Prop Id: 12959a75af826745c76c8cacac56b223b08b5dfa4b785d1f4dbe42ad5c2234c4
Theory Prop Address: TMFvSuVBfh1siZrHxBxDNdzASV8SPhqhWGz

Field_Hom_E

Theorem 27.62

∀g.Field_Hom g→∀p :o.
(Field F s → Field F s′ → g ∈ F′^{F} →
g zero = zero′ → g one = one′ →
(∀ab ∈ F.g (a + b) = g a ⊕ g b) →
(∀ab ∈ F.g (a ∗ b) = g a × g b) →
(∀a ∈ F.g (−a) = −g a) →
(∀a ∈ F.g a = zero′ → a = zero) →
(∀ab ∈ F.g a = g b → a = b) →

(∀a ∈ F.∀n ∈ omega.g (CRing_with_id_omega_exp F s a n)

= CRing_with_id_omega_exp F s′ (g a) n) → p)

→p.

The proposition is identified by the following information:

390 CHAPTER 27. EXPLICIT REALS II AND FIELDS

Pure Prop Id: 0ba6afb6588f84ed9233205763416bbcb48104160ae81e7c83aaa749df56f30a
Pure Prop Address: TMHyWBfPWgBD8fHLFmj2uHd8LiGwzKaPXbL
Theory Prop Id: 4bd91ea8df1144675fc91986c05fe9c6aa39bb2352629ee956d785fb210304d0
Theory Prop Address: TMd7scvhecxt55fo9HQefDSbDkqviHLYMQk

Field_Hom_inj

Theorem 27.63 ∀g.Field_Hom g → ∀ab ∈ F.g a = g b → a = b. The proposition is identified by the following information:

Pure Prop Id: 68a0d1b5c80c7bc045dfade09ce3a0bf9500817cee9a0014553c03d27df85b25
Pure Prop Address: TMb9HZYE9ntGsPKXkYbtsoVw7iJz9L9VS4R
Theory Prop Id: be594831fd63fc200596d913b2146b4087df4d853a5d72bbbec93372920c5c06
Theory Prop Address: TMPDzSNyQDmjStN75qX6xofhPvz45wh64pJ

27.9 Basic Results about Fields

subfield_refl

Theorem 27.64 ∀F s.Field F s → subfield F s F s. The proposition is identified by the following information:

Pure Prop Id: 0eb1f1396b31417610ee278fb6a8b8d8cdd140a8eedf731088c361bd94a2e7f0
Pure Prop Address: TMQXDmjHytbWQAqPJjgqXLMDq8SMfVBrMVy
Theory Prop Id: 463bacb9ffd8a7c1e580b20d409cb1f1185318cd2fcd9c728d149096a959c4f5
Theory Prop Address: TMbHGwwWjrPBorTYyXRmvsBmdT2tyD2ziXc

subfield_tra

Theorem 27.65

∀F sF s′F s′′.subfield F s F s′→subfield F s′ F s′′→subfield F s F s′′. The proposition is identified by the following information:

Pure Prop Id: 66345481ffe8c4fd4fbb8f011c765a0d87adab31bcd77023511f6539b528d18e
Pure Prop Address: TMM11DSqtT24FeA4LM8aLCBtypoFCW7Y25E
Theory Prop Id: 98a2debbf6e60c86f06666f4d0a11ec1170a4b003c3f6f6ac2a0dbc3da02a741
Theory Prop Address: TMGnVRkmmsKG3BTL5eNCc52GyUUvBzfgjLe

Definition 27.11 Field_extension_by_1 is the opaque object of type

ι → ι → ι → o

identified by the following information:

27.9. BASIC RESULTS ABOUT FIELDS 391

Pure Object Id: 15c0a3060fb3ec8e417c48ab46cf0b959c315076fe1dc011560870c5031255de
Pure Object Address: TMQQtdP8BXq6yrNPjAqW2TErHDrTMXXAnHf
Theory Object Id: 937edba4a112338c9bd442b454c3d9930703191cb45b86271c395bc54d538d82
Theory Object Address: TMKZ3FEYqeFEFtGCpBAd1kYajCJ4YnCDYCB

Field_extension_by_1_I

Theorem 27.66

∀F sF s′a.subfield F s F s′→
a ∈ Field_carrier F s′ \ Field_carrier F s→

(∀F s′′.subfield F s F s′′ → a ∈ Field_carrier F s′′ → subfield F s′ F s′′)

→Field_extension_by_1 F s F s′ a.

The proposition is identified by the following information:

Pure Prop Id: b32e90df057fb63c1ad17c2a471f552dc5c42a3b49ce631d182cc17768c8ce16
Pure Prop Address: TMTostbVVVe1Wc1NFks4m8cRXEwnEtFnycJ
Theory Prop Id: d5557b3eb7f49b4ddec41d07f1d23e0b1ce4f8e7657a1c751d122b54af9fa41b
Theory Prop Address: TMPm9E17AXXZrMjJG29R9UHEicjTbN2gajj

Field_extension_by_1_E

Theorem 27.67

∀F sF s′a.Field_extension_by_1 F s F s′ a→∀p :o. (subfield F s F s′ → a ∈ Field_carrier F s′ \ Field_carrier F s →

(∀F s′′.subfield F s F s′′ → a ∈ Field_carrier F s′′ → subfield F s′ F s′′) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 338a5505ed42a91cb5fa5ec6ab0cf4e76ba17cac12e25143d21eb1cd650c46da
Pure Prop Address: TMNZzAKhVJ6yFrUHhoMKiupzbWxqEfPoy4J
Theory Prop Id: beafbae6117b463b1efc7ff06cf528539b2a460a7d5e21e7589a7e1d18fa1642
Theory Prop Address: TMFdxk4XGCyjeLqB2oApjE5fghNi2pqqZLN

Definition 27.12 radical_field_extension is the opaque object of type ι → ι → o identified by the following information:

Pure Object Id: 4b1aa61ecf07fd27a8a97a4f5ac5a6df80f2d3ad5f55fc4cc58e6d3e76548591
Pure Object Address: TMchEBbJiTrZFrworejH2faUQ47yR2qjVUh
Theory Object Id: 816f6d770c8c6df637703bbfd5f2ee005e7474a4930bf0953ebe06dca050d7cf
Theory Object Address: TMZ6XfW5b4iotXcF9javzfEe6WjNPU9qL1w

radical_field_extension_I

Theorem 27.68

∀F sF s′.∀r ∈omega.∀F seq.F seq 0 = F s→F seq r = F s′→
(∀i ∈ ordsucc r.Field (F seq i))→
(∀i ∈ r.∃a ∈ Field_carrier (F seq (ordsucc i)).∃n ∈ omega.

CRing_with_id_omega_exp (F seq (ordsucc i)) a n ∈ Field_carrier (F seq i)∧

Field_extension_by_1 (F seq i) (F seq (ordsucc i)) a) →radical_field_extension F s F s′.

The proposition is identified by the following information:

Pure Prop Id: 4e422a257d6bf13415f526581580edf290299362d88de0af188ed0d16b294fc0
Pure Prop Address: TMaTmGPiv6voz3uCpKzxGcrY1beeHKnr1k9
Theory Prop Id: 6bd73bdf40cf6c2ecf949365a2d93d0d579cef5aa7319b45b404b1739d97e812
Theory Prop Address: TMb2ezs3LpjpMQYPEQvK9PUQE6aN6HjXb27

radical_field_extension_E

Theorem 27.69

∀F sF s′.radical_field_extension F s F s′→∀p :o.
(Field F s → Field F s′ → subfield F s F s′ →
∀r ∈ omega.∀F seq.F seq 0 = F s → F seq r = F s′ →
(∀i ∈ ordsucc r.Field (F seq i)) →
(∀i ∈ ordsucc r.∀j ∈ ordsucc i.subfield (F seq j) (F seq i)) →
(∀i ∈ r.∃a ∈ Field_carrier (F seq (ordsucc i)).∃n ∈ omega.

CRing_with_id_omega_exp (F seq (ordsucc i)) a n ∈ Field_carrier (F seq i)∧

Field_extension_by_1 (F seq i) (F seq (ordsucc i)) a) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 04aa436465678920b287c7c3b096ec8a8ad9f9429cfbf92c410bebfde430692d
Pure Prop Address: TMKr8e5rQhQ1BMfoiUXe73gTjPvh5qaip7V
Theory Prop Id: 89a63147cae765b2eabc9cc755f1a4a6c0b4b6c6791569bb6a8e0ed9bd00f0fd
Theory Prop Address: TMYAut45JZbtAUu8e69Mdhf8jq8dSnpBN3M

Definition 27.13 Field_automorphism_fixing is the opaque object of type

ι → ι → ι → o

identified by the following information:

27.9. BASIC RESULTS ABOUT FIELDS 393

Pure Object Id: 55cacef892af061835859ed177e5442518c93eb7ee28697cde3deaec5eafbf01
Pure Object Address: TMN78iPppPwcxR2NUi3dP5EVHayaxwFwfWr
Theory Object Id: 8e98c0485c8e7d57eb3687ec81bbf72649aea6f54b04d8a845c0664d3392f858
Theory Object Address: TMFfKJ2CTJMaiuRS28eKxgU3frsxPZBoTum

Field_automorphism_fixing_I

Theorem 27.70

∀KF f.subfield F K→Field_Hom K K f→(∀y ∈ K 0.∃x ∈ K 0.f x = y)→

(∀x ∈ F 0.f x = x)→Field_automorphism_fixing K F f.

The proposition is identified by the following information:

Pure Prop Id: 8f616dfe39b72caefcdc6aabcb73389242f81b1a7364f5444e6672294fc7a13d
Pure Prop Address: TMSoh1Y12vBR6CW3KxCXhcWQZd5A1FZq1cU
Theory Prop Id: b928875c73ac27a404acf23ab1966ddfcc532a64ee21fb30dd9d57314756729b
Theory Prop Address: TMaXJQsntpr39fhBWDxXZwk8s6e8PSpYjTR

Field_automorphism_fixing_E

Theorem 27.71

∀KF f.Field_automorphism_fixing K F f→∀p :o. (subfield F K → Field_Hom K K f →

(∀y ∈ K 0.∃x ∈ K 0.f x = y) → (∀x ∈ F 0.f x = x) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: abbbec85a4745b7e0f7fca7af4f6c54dc3a65291a67132d9d9848d4338a72eef
Pure Prop Address: TMa3L2FNtNGpWEaV5Ctdc4CtUWWGvbY4cRP
Theory Prop Id: b2509c4f43b6dbea462f25fd32e67cd06f5e46262cb519d918cbb958ec386a98
Theory Prop Address: TMHuLZV7jdk3tK6t9sNMSxXdPPK7H6k2VGW

Definition 27.14 We define lam_comp to be λAfg.λx ∈ A.f (g x) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 29d9e2fc6403a0149dee771fde6a2efc8a94f848a3566f3ccd60af2065396289
Pure Object Address: TMZBHh5NXVkgufPoqMzvVCpRFLW2WdfYiJJ
Theory Object Id: f47cda05aa5549792c42c687ced0171e637154b32dbda20a3dd2065d950abd6f
Theory Object Address: TMT1tRAwqVEGpkkvkUdLo93JTWkuSxofJxU

Definition 27.15 We define lam_id to be λA.λx ∈ A.x of type ι → ι identified by the following information:

Pure Object Id: 6271864c471837aeded4c4e7dc37b9735f9fc4828a34ac9c7979945da815a3c7
Pure Object Address: TMTjnAw5cKWKUVsHNJJeNN6CgmY8LpuZiYg
Theory Object Id: 86ccd74f57ceb4f87e01c0731683396877d457448dc5c342c35066e0a38f076b
Theory Object Address: TMPRHK8Wyojj7r7mPHtDjeU6KLQUQfd4mis

394 CHAPTER 27. EXPLICIT REALS II AND FIELDS

lam_comp_exp_In

Theorem 27.72 ∀ABC.∀f ∈ B^{A}.∀g ∈ C^{B}.lam_comp A g f ∈ C^{A}. The proposition is identified by the following information:

Pure Prop Id: a368c0a62f8223dcdf075df5f370b455c9b5fc97a3edf8807b478842be43c4f7
Pure Prop Address: TMGg18zcz7ghZbezgKuURXy33FhGmmAWbXm
Theory Prop Id: 4fede2b31459edc9b94436567d1a7a2010f2eee7b82f8cd7fbbdb2b6ad196ec4
Theory Prop Address: TMHzPAuFt6vTdQEr4t62NpVuDaya4x1nqZK

lam_id_exp_In

Theorem 27.73 ∀A.lam_id A ∈ A^{A}. The proposition is identified by the following information:

Pure Prop Id: a328f177c58333c407842c44b36358151a17648de6780a0fe94204d0656910eb
Pure Prop Address: TMG2mqqHqLB3YWR1XE3KpoehYhtcgwLGMsh
Theory Prop Id: 025740d37f99a1c2a8b9a6afd977237c7629216ef0f1579ec6fa5bc5a0b22968
Theory Prop Address: TMdqvc3zfrpqPDraF5Ttxg4ZWaLTYaoRK8c

lam_comp_assoc

Theorem 27.74

∀AB.∀f ∈B^{A}.∀gh.

lam_comp A h (lam_comp A g f) = lam_comp A (lam_comp B h g) f. The proposition is identified by the following information:

Pure Prop Id: 10d3c2043b57f47398d529842b2b4ef41092863b2ed292e467d739a5a5d48d05
Pure Prop Address: TMGNEh1ULegqm5w1qTFWfgYNKdQ2N5PWfL3
Theory Prop Id: 51ef710d91519b6ae4f1e6c1240232977314be4a5e218f254330dc6c7b0e7fb2
Theory Prop Address: TMX7HYhhWwwEmvKudmWCRyNh3vcLzY4xMPJ

lam_comp_id_L

Theorem 27.75 ∀AB.∀f ∈B^{A}.lam_comp A (lam_id B) f = f. The proposition is identified by the following information:

Pure Prop Id: e63b276d423c5ffa382e3d24fff1281fc5d23cdc2de80558f7664c37ab8eed29
Pure Prop Address: TMbj9gKjk5JQsbDmoPhorvNQU4Jq7DGrLwD
Theory Prop Id: 64750339f13b14b1eca32c23b90fa53b58a24aa24d334fb3e2b45602882f363c
Theory Prop Address: TMNXXqZFWBCWcQvn8czxiCArUPkXVyGAsqm

lam_comp_id_R

Theorem 27.76 ∀AB.∀f ∈B^{A}.lam_comp A f (lam_id A) = f. The proposition is identified by the following information:

27.9. BASIC RESULTS ABOUT FIELDS 395

Pure Prop Id: 0acec29b4958df486c134fef66fff2628d976311a127b9e66e47d31dce87a2b5
Pure Prop Address: TMMEAAXiHtjyoC9UixZUX89czksUW4L2AbP
Theory Prop Id: b8b0cd7ab3f2745fc54a89466dffe557184df20469638425a99e0d6af17cc7cf
Theory Prop Address: TMPrbzDNKGk5rH6cAEX3uuwsRXfYLypKbkq

Field_Hom_id

Theorem 27.77 ∀F.Field F →Field_Hom F F (lam_id (F 0)). The proposition is identified by the following information:

Pure Prop Id: e76108b7394a2d0ef4a23eeff5c2eff0a59bfd099b41a3efcf63c829da51cd0e
Pure Prop Address: TMMTreDckMCE8QUxnfjGRBucakRyKZDd5Uw
Theory Prop Id: 0a051badc75ad444016ac67529bbfa1ebe66ca5459cd8875e236240cb319fd29
Theory Prop Address: TMbgZLG5jAx7kLyryCTxrpfmKu9WndjBQKz

Field_Hom_comp

Theorem 27.78

∀F F′F′′gh.Field_Hom F F′ g→Field_Hom F′ F′′ h→
Field_Hom F F′′ (lam_comp (F 0) h g).

The proposition is identified by the following information:

Pure Prop Id: f96e7b8972ab6ddc476b85f43c26f21233bb2d05a4284718f987e06e5bb0a539
Pure Prop Address: TMXHqDN9vGwVX2JmvbvZuV4Jx7aQ6ptioS9
Theory Prop Id: c7176150678884e662532193f41ddefed41f8c69c784867e0e5e7599cfccc3d7
Theory Prop Address: TMFHd96nKERMA8xrf6a4PPnaFSZAyhMgQkd

Definition 27.16 We define Galois_Group to be λKF.pack_b {f ∈ K 0^{K 0}|Field_automorphism_fixing K F f} (lam_comp (K 0)) of type ι → ι → ι identified by the following information:

Pure Object Id: 831192b152172f585e3b6d9b53142cba7a2d8becdffe4e6337c037c027e01e21
Pure Object Address: TMYqxuc9rAFqZk2CRVHyc2y7pfNsCZU6equ
Theory Object Id: ca458a9929d10327f6d82d25599b1037bc98c2bdf117e527489c6a4d6ffdd167
Theory Object Address: TMTJqgfvRfobXKQitP5Ljdprk53Cpq27TKH

Galois_Group_0

Theorem 27.79

∀KF.

Galois_Group K F 0 = {f ∈ K 0^{K 0}|Field_automorphism_fixing K F f}. The proposition is identified by the following information:

Pure Prop Id: c937d2405bc02024b3013de5808dbe9e3504c142d376d7ff1ca27d63e12e562e
Pure Prop Address: TMP5RNZvDusyH2t1cf286riba2LdRzc1YJv
Theory Prop Id: efd193beaee2e86f35612249b45870a142e7e8c50719336225df8ef19e54b2d9
Theory Prop Address: TMHr9KDkArCoQ1vA2dUxzDwknQj6pDMv7LB

Galois_Group_Group

Theorem 27.80 ∀F K.subfield F K→Group (Galois_Group K F). The proposition is identified by the following information:

Pure Prop Id: 23e81a79edfd40eff87623569e69d5d6bd17fea035e30db29d3fa252dc9292df
Pure Prop Address: TMF8s32NGdaXaUxYsMdcKNoMmb1qtqso9ph
Theory Prop Id: ed4e90cd2b4fa72539381db20b559cb765274456c2523b29855fe21eefb0bf54
Theory Prop Address: TMZrV2L74MSsae3E5TMtqrcs8St33ZF5Hbo

27.10 explicit_Reals_transfer

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let leq : ι → ι → o be given. Notation. We use ≤ as an infix operator corresponding to applying term leq. Let R′ : ι be given. Let zero′, one′ : ι be given. Let plus′, mult′ : ι → ι → ι be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Let leq′ : ι → ι → o be given. Let f : ι → ι be given.

explicit_Reals_transfer

Theorem 27.81

explicit_Reals R zero one plus mult leq→bij R R′ f→
f zero = zero′→f one = one′→(∀xy ∈ R.f (x + y) = f x ⊕ f y)→
(∀xy ∈ R.f (x ∗ y) = f x × f y)→(∀xy ∈ R.x ≤ y ⇔ leq′ (f x) (f y))→
explicit_Reals R′ zero′ one′ plus′ mult′ leq′.

The proposition is identified by the following information:

Pure Prop Id: eab1221701cf6d740b9b72782e98f152a3f7f2cdf16638d049fc649d996fc29c
Pure Prop Address: TMYhyMVVJTvxXty5NxJVswTmqpdUcPwJ3PN
Theory Prop Id: 76534117383e5ea208dccdd5106698afc06e708ae05d009cb0e347dfd0cf2ae2
Theory Prop Address: TMY35GcU6u4DUqWLoorNqpUAYwMdFLABjvj

Surreal Numbers II

28.1 SurrealArithmetic

Definition 28.1 minus_SNo is the opaque object of type ι → ι identified by the following information:

Pure Object Id: 268a6c1da15b8fe97d37be85147bc7767b27098cdae193faac127195e8824808
Pure Object Address: TMcxvtkwUHDLCnnXkVE7rvLpYFkkCr9t57h
Theory Object Id: 8a6d8513b8fae7a11b1033553a028661380ee0d23337db3b5de7c37c00bf7d3a
Theory Object Address: TMNRx3zLagXCf6MQrfwBvyPXLa8o1ZnJoqf

Notation. We use −− as a prefix operator corresponding to applying term minus_SNo. Notation. We use ≤ as an infix operator corresponding to applying term SNoLe.

minus_SNo_eq

Theorem 28.1

∀x.SNo x→−x = SNoCut {−z|z ∈ SNoR x} {−w|w ∈ SNoL x}.

The proposition is identified by the following information:

Pure Prop Id: c9793e4a852e2a5d73a3008d3c0a88124864f04508e5f6cd84920975e8bddfc6
Pure Prop Address: TMFADt2W5uMVy5FYrgHvEE2vrdFR8Gnxb1b
Theory Prop Id: 570cba2e3229759ec994c333f1852d951e44a44817d76474c936f0b2783d17fb
Theory Prop Address: TMQTpg4CNj1EB19CWqcr8oBSbG7cvoVZJuJ

minus_SNo_prop1

398 CHAPTER 28. SURREAL NUMBERS II Theorem 28.2

∀x.SNo x→

SNo (−x) ∧ (∀u ∈ SNoL x.−x < −u) ∧ (∀u ∈ SNoR x.−u < −x)∧

SNoCutP {−z|z ∈ SNoR x} {−w|w ∈ SNoL x}.

The proposition is identified by the following information:

Pure Prop Id: 7c0e82df70ecda3be7a8153a63670c1b82bc354a3b3bb658b240db5bf6079ccc
Pure Prop Address: TMPFDUxTsU6Hgp2dqtAfHiUMzQdZC4s6Rpo
Theory Prop Id: f0b75f0c13ab8217a5ebfacdc1aebc7bacaf4f0bcf65609bd5bdfc26f85fe25e
Theory Prop Address: TMHp3P8GQ2PdxPN38fbjih9CXrrvAVxARmW

SNo_minus_SNo

Theorem 28.3 ∀x.SNo x → SNo (−x). The proposition is identified by the following information:

Pure Prop Id: 5cf7d9b9d43ef2ee9370e1aa6dded6d26ae3d971cbc3b561b6d07723f8215269
Pure Prop Address: TMdc5VANKtJFyfviMQyb75gDTg48GudNzRZ
Theory Prop Id: 744cc2ee9e53d674ba772a0aa685c56eb8e460e86e64854f7ce0be53b0da5fff
Theory Prop Address: TMQ1ogbeVfSeNvyG9vTPaMVZg2dqUSJyjSo

minus_SNo_Lt_contra

Theorem 28.4 ∀xy.SNo x → SNo y → x < y → −y < −x. The proposition is identified by the following information:

Pure Prop Id: 5ce52d7493ef6b5572b747bf671cea2e6e6f80ceb28c95e2123521dd504b0b0f
Pure Prop Address: TMYLjCwrhRZgQN5K5EwHjSFpA4jVNLAHCjn
Theory Prop Id: 3ebf6e39ebb99f5ddf5eee4fb81a248a684fd1274696b9741a23625f9df6dc52
Theory Prop Address: TMPCextugz7QEf4NpPTNxs5QRszhndWEXZL

minus_SNo_Le_contra

Theorem 28.5 ∀xy.SNo x → SNo y → x ≤ y → −y ≤ −x. The proposition is identified by the following information:

Pure Prop Id: c1b2259be1bdf45a1f93fe5cc80752b95adc141e37aea0fc5974e675529dbab0
Pure Prop Address: TMSdtEKoTTw4VsgYxJhGDbFrsEav7bgzwDz
Theory Prop Id: f8c8964a3e52fb3b06f65d0148f443ef72cb1a1710977f309628d62a6dc762ff
Theory Prop Address: TMR6iQfy2ketSGqusuoDGBAoZDHFdKZi5ei

minus_SNo_SNoCutP

Theorem 28.6

∀x.SNo x→SNoCutP {−z|z ∈ SNoR x} {−w|w ∈ SNoL x}. The proposition is identified by the following information:

28.1. SURREALARITHMETIC 399

Pure Prop Id: 99c6fcde5ba109860366fc8eb324200f6b134a75dc50371fa98ecbf13fe8dd9f
Pure Prop Address: TMPGzP3WvAeSXVX4PkvAXm9hswTwYn7Lx5a
Theory Prop Id: 5ad5e46892ec6dde1a8e632b515b72dd9624c207d92d7f5fc1b887ec45170a91
Theory Prop Address: TMYA5Ns3A1StK4eq1kfygXeX7mudw4n946H

minus_SNo_SNoCutP_gen

Theorem 28.7

∀LR.SNoCutP L R→SNoCutP {−z|z ∈ R} {−w|w ∈ L}.

The proposition is identified by the following information:

Pure Prop Id: e7650d8c5c9fb96894ef4334966c9c4043dc7b919ef49e218164db5101a12296
Pure Prop Address: TMcYQN83fYBS39M2EEEbDtftNsYeWHCkMAi
Theory Prop Id: 687fea2cc86a5e6dc4aaff906beb9303fa12e95620a78db83430a8be9dfd1bfc
Theory Prop Address: TMK1WeYsUqdVKtQ7kU3SZV4h8vJvGUyosuH

minus_SNo_Lev_lem1

Theorem 28.8

∀α.ordinal α→∀x ∈SNoS_α.SNoLev (−x) ⊆ SNoLev x.

The proposition is identified by the following information:

Pure Prop Id: 14986a810fa991813a940caff72106cdeb17e367c8dc9c6eeb88ab77ed7fb70c
Pure Prop Address: TMGTTrCpJEdP2JL7ZKjHkPvFjc3gfmdeYX5
Theory Prop Id: afc115b1b5f9ff0b6226e6d37a5708b45bf273bc776bf7816207943b0cf564fa
Theory Prop Address: TMcrjfrThFM8M6pTTZtnJdFZBn72x9XkSzf

minus_SNo_Lev_lem2

Theorem 28.9 ∀x.SNo x → SNoLev (−x) ⊆ SNoLev x. The proposition is identified by the following information:

Pure Prop Id: 87902034fac50c9d369be3b80426e4d2a737cf936787a4c0f6c8ad74e0843878
Pure Prop Address: TMbVFKSGdQJSr2kHyJUmNCjt7CTD1D6JKHf
Theory Prop Id: 19cf80ac01fc15c5bac2a1a546c9cfca35cf8eb2a455b3cf3fb66a2a7fe614f8
Theory Prop Address: TMQtKJ6HCGKKXSHeLSJwRyRPcKmC4ZiLZK5

minus_SNo_invol

Theorem 28.10 ∀x.SNo x → −−x = x. The proposition is identified by the following information:

Pure Prop Id: 44bfe89261da61b19b9a5cd41ede8430faa107c7822701da47ef38cb4549faa6
Pure Prop Address: TMc77Sp2dpzatNwekdynQs9vhSPfJgGXk1U
Theory Prop Id: cc64eeea172423c51379afadb3e01de9b7f2513ce938649ddcb32805f5bdafb0
Theory Prop Address: TMTt3PRTf9CD5ikrtEqSt2QevQVtKjQrFbz

400 CHAPTER 28. SURREAL NUMBERS II

minus_SNo_Lev

Theorem 28.11 ∀x.SNo x → SNoLev (−x) = SNoLev x. The proposition is identified by the following information:

Pure Prop Id: f65499af9040b6b3e15497cab281d7675bff566bb6d4fdd182cc0343568ae07e
Pure Prop Address: TMHMPna5msJBVV2aj6dRhmCueZSUBa2VhT5
Theory Prop Id: 476ae5e0d40774e983e1d64325be7c212ae84435e983e476cb99925933ecd248
Theory Prop Address: TMRdSSgdRywiPXPvi9bXyTwx2JCnJRrLkHS

minus_SNo_SNo_

Theorem 28.12

∀α.ordinal α→∀x.SNo_α x→SNo_α (−x).

The proposition is identified by the following information:

Pure Prop Id: 84605fa359f5f8a80873f561cdcf5e371180d9965ea1157048ba66f9af585e90
Pure Prop Address: TMZviY6yHgD2P1xafVTUWcBcjX291SfHuZq
Theory Prop Id: 14bd6d1aecca30a4c446e92cca90fa66fd7b0499acca03daf8cfa848740f428f
Theory Prop Address: TMTsLyGVKuoMiH43ymXqPkKMWYDN3HyfXHd

minus_SNo_SNoS_

Theorem 28.13

∀α.ordinal α→∀x.x ∈ SNoS_α→−x ∈ SNoS_α.

The proposition is identified by the following information:

Pure Prop Id: 12f1a5194619e18f763ea1432c60862d2c280bcc57c6b538e7af0b8563b3aef0
Pure Prop Address: TMNu6dbAKA4NtYuwXCDn3Z2xwPtwvtj71gH
Theory Prop Id: e10c92f39791767985ef5b9f5eed352a930cae37c683dfacbd6f47c7d2d23e93
Theory Prop Address: TMVtCJL3NR1TVjUhDpc8LMfGugbda1XLCG2

minus_SNoCut_eq_lem

Theorem 28.14

∀v.SNo v→∀LR.SNoCutP L R→v = SNoCut L R→
−v = SNoCut {−z|z ∈ R} {−w|w ∈ L}.

The proposition is identified by the following information:

Pure Prop Id: 29069a9700032f41c27cbce5d097d02ca500af7402feea86befd364a812bea18
Pure Prop Address: TMJJorcHcCvbte94Wk2mohN3ofdRRZi9USf
Theory Prop Id: 11eb03174b07f424338800b6644cd5921561dc680adb86b22f9254abda306195
Theory Prop Address: TMRBupH2YERv2pxoxNHvbLaoKkUEWp3LmVu

28.1. SURREALARITHMETIC 401

minus_SNoCut_eq

Theorem 28.15 ∀LR.SNoCutP L R→−SNoCut L R = SNoCut {−z|z ∈ R} {−w|w ∈ L}. The proposition is identified by the following information:

Pure Prop Id: 083ffe4c6522008e61c2b89e0e64c5db7b371419a23a3f14a519e0582688fb5b
Pure Prop Address: TMaD6A1fin9xGjs9gJk55ba3XE4iuTh2W7Y
Theory Prop Id: 54736f85eb44e8ebcd3e07eea3d33c79842c2ce605f7a6384f4ba14b9eec7d07
Theory Prop Address: TMGrpE8HLXgUszxGm966iTpad2EhS6XSckh

minus_SNo_Lt_contra1

Theorem 28.16 ∀xy.SNo x → SNo y → −x < y → −y < x. The proposition is identified by the following information:

Pure Prop Id: 226757cb052a4956017abe14687f3e1caa99a28fde325c1d779946bba7658526
Pure Prop Address: TMXXUYTV6XCxwBUgA7MgZTzj6H9ahkRPPTo
Theory Prop Id: 29872961c38defbc42168eadfbb395ff6b34bb794dfd9e81bcce6ad6374039e9
Theory Prop Address: TMEs4ZMsQaGiy3xaRcw9TpZk6KCqk5ye83u

minus_SNo_Lt_contra2

Theorem 28.17 ∀xy.SNo x → SNo y → x < −y → y < −x. The proposition is identified by the following information:

Pure Prop Id: c6f2dabf5db1946ff00666f185293c94dc027bfbc3c6b34ab842300e6fc61e10
Pure Prop Address: TMK1aCcPBp5isVNhukJBv5WwEVryhWR39im
Theory Prop Id: f419684217fb1e0747adc0f778ba066f701845079345dd8aaa40b5499c5cd992
Theory Prop Address: TMEu6PgqrG75bbqAH6z2eo8taFDwoWi72LL

minus_SNo_Lt_contra3

Theorem 28.18 ∀xy.SNo x → SNo y → −x < −y → y < x. The proposition is identified by the following information:

Pure Prop Id: 30b6d6ce0519f672dafb8da416d1f08cc773f7718ea0c0e59fe2b674a8a28eb5
Pure Prop Address: TMEiiHLSUJb67TTgfxay254xNKWCum3TL6U
Theory Prop Id: 20ebc3b1dcf4268fcbbda61e77e0b1aa0cd9de3c07e995cd027e1623022ab3a6
Theory Prop Address: TMdv9HV4i4ia6bWB1AEuc8hYPUVNAMAF4i3

minus_SNo_0

Theorem 28.19 −0 = 0. The proposition is identified by the following information:

402 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: 77bc6707cfcc559fd1f9509caeb31aaec5c6ea27e99983f5ed2a0b31aba16371
Pure Prop Address: TMExypvZyPTdXH5VP42X3hVoCRLY5xqMGSS
Theory Prop Id: 3fcc57eb71cf1d99ef0ab04ed19f5bc54b596714fb2b6d70ce2f61b05445433c
Theory Prop Address: TMS4JbQtTa5bFrxAcTD38DVaprpkd1UddT2

SNo_momega

Theorem 28.20 SNo (−omega). The proposition is identified by the following information:

Pure Prop Id: 9682bc4f449aab7850dab8b061240c04bbfa925a9af0c6d436b57571e88bf054
Pure Prop Address: TMbs8VnGZaJLV9PMfUU3ZA9UMWppyWZCbpk
Theory Prop Id: e72fb1696565b219d098fc491769ed67f15f1c716f9ba159513fa19c4a3cfb68
Theory Prop Address: TMQGDzKedUx3YyBnazoAQzc2BJYMagTe9aU

mordinal_SNo

Theorem 28.21 ∀α.ordinal α → SNo (−α). The proposition is identified by the following information:

Pure Prop Id: 67157695184d77aa292d2e91ae058aab5728c1aa262ad91430d98e01a61ecee5
Pure Prop Address: TMZCauMQgviQfBsquF7s2dXY5ovy2NZhJmB
Theory Prop Id: 533947d00d9f2f544511c3e8876c2d3bec39a85cf5de0507579619af91b1cfe3
Theory Prop Address: TMdpAbWGcEmUBU6mDwF1QUJ8J3YQKqAnC2W

mordinal_SNoLev

Theorem 28.22 ∀α.ordinal α → SNoLev (−α) = α. The proposition is identified by the following information:

Pure Prop Id: 872283abfa6f0dfbb1d72e382868b2cf0992ece74aabc0727d30681dd1c82466
Pure Prop Address: TMWf88dibCFMEwQnUShPjR7f5qzRTHihxjA
Theory Prop Id: 425f79bccb3002745cf4cfe3c4ee8e5f28fc81700abf759ffe8688c3ec559879
Theory Prop Address: TMJYCVXHmoi2fzEKN7sFk63txQj7WCuRVYU

mordinal_SNoLev_min

Theorem 28.23

∀α.ordinal α→∀z.SNo z→SNoLev z ∈ α→−α < z.

The proposition is identified by the following information:

Pure Prop Id: c28e6646c6374a60e623edaf3e98581ff82a2d308bc362c7ed683cb7ef763c9b
Pure Prop Address: TMGx6jRVew4w8A8QL13WubtT3gsqgBkjuKj
Theory Prop Id: a11cfa7f908447229087de2b3b5ed724471c04c5ace74fef64304120331db0b2
Theory Prop Address: TMcMHpMnYb8FFYn3kS1X4hRDwLCnUBPrPN2

mordinal_SNoLev_min_2

28.1. SURREALARITHMETIC 403

Theorem 28.24

∀α.ordinal α→∀z.SNo z→SNoLev z ∈ ordsucc α→−α ≤ z.

The proposition is identified by the following information:

Pure Prop Id: 2766c7ac03ddae43cd989f364bd3933e33016953b97822589263b8813a782baf
Pure Prop Address: TMdLNMofLk9MRXKxrVStAmXcnfLmpPf5ZLD
Theory Prop Id: a96d68eb40e4505dff800c19d3a67d927597a15491c19c2e7613c3653faa7592
Theory Prop Address: TMUbQC9sotj3Sdmk1rPhzFc2uGHxwQ7fTKG

minus_SNo_SNoS_omega

Theorem 28.25 ∀x ∈ SNoS_omega.−x ∈ SNoS_omega. The proposition is identified by the following information:

Pure Prop Id: daeba565dddc519103825c78e55714d95a0ae94548ea309e8d64ab2518ce1c95
Pure Prop Address: TMHUVaZspynr2H3jcHXqWoRHirGbNS7NfFb
Theory Prop Id: 0c19f5cfaa9b340bbf0166314f1046d3ad7c8558b64304949986f6ee62b7946e
Theory Prop Address: TMV7G2rh3rEAwiDk5QRMvAdQX4xEJpR6g26

Definition 28.2 add_SNo is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 127d043261bd13d57aaeb99e7d2c02cae2bd0698c0d689b03e69f1ac89b3c2c6
Pure Object Address: TMHBQoir8zBmsd18ceCptzsSPneX7e796Qh
Theory Object Id: f62f9ad0a4b78ce9ea1f70abc3f81bb4efd2f55019f5ea81fec6bff7d056a9c4
Theory Object Address: TMLfAbLP1JzEe3VPaGmScRvpGeCCLs2hBkM

Notation. We use + as a right associative infix operator corresponding to applying term add_SNo.

add_SNo_eq

Theorem 28.26

∀x.SNo x→∀y.SNo y→

x + y = SNoCut ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y})

({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y}).

The proposition is identified by the following information:

Pure Prop Id: 0c11282e349ac03ead817fec6965779c38db2bba16e6617397d7a00fec0025f2
Pure Prop Address: TMUsqiTidRPvWsTgAU6Pe8thY6esLYajg8X
Theory Prop Id: 437a9c143d85d0cbb514a028f14f083cc0f8d79aa47f3997255c796243fc0fe7
Theory Prop Address: TMGPPTS21hFunW6FDmNfEs1RLtzqupLkwJY

404 CHAPTER 28. SURREAL NUMBERS II

add_SNo_prop1

Theorem 28.27

∀xy.SNo x→SNo y→
SNo (x + y) ∧ (∀u ∈ SNoL x.u + y < x + y)∧
(∀u ∈ SNoR x.x + y < u + y)∧
(∀u ∈ SNoL y.x + u < x + y)∧
(∀u ∈ SNoR y.x + y < x + u)∧

SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y})

({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y}).

The proposition is identified by the following information:

Pure Prop Id: f3ece7fc5faeb9b5356d40372aef6e582964fa486d2e44e7f3eed3f05d52e3e0
Pure Prop Address: TMcpTrmdaf7f5xyc6e4BpSWAmxNsmf3E5yS
Theory Prop Id: 5a879f7365529b11a3ffce943b27ada13bc1e46e2af4e2258090e38d55606b7f
Theory Prop Address: TMRNpdhJ6PXJ334PSwBu4u3N4MRpHafuGvT

SNo_add_SNo

Theorem 28.28 ∀xy.SNo x → SNo y → SNo (x + y). The proposition is identified by the following information:

Pure Prop Id: c4a9a6e12a8ec5b5bebaa97ea0d920bd43eea74d9b3d0ffda15762cf19565969
Pure Prop Address: TMPtcdp9k9YMXhBQVaR4jVorJnSfZBVvKki
Theory Prop Id: a57095a9caa997f445ebde562cde8dd04d8e8a5fd2a57d1c3a9f8decf60ccdeb
Theory Prop Address: TMXsdqtKxXjtQCrsnZQNbBCQsbgVDjWXpMf

SNo_add_SNo_3

Theorem 28.29 ∀xyz.SNo x → SNo y → SNo z → SNo (x + y + z). The proposition is identified by the following information:

Pure Prop Id: f6ef07d3f7fb2565f3fe4732b883d4499d8442b69d6ae7d977010d18d4d54ecf
Pure Prop Address: TMZ4dtBp9Qizyu4nFxn7a2N9SQRVoE9Q4W8
Theory Prop Id: f2d78c7d8aa6405036ae26879362dfde9af9a15cdbbf11ba2bff0d754b7750ca
Theory Prop Address: TMVox3RUbkj9ZYffbdvRh7znGFXNfLVwPFV

SNo_add_SNo_4

Theorem 28.30

∀xyzw.SNo x→SNo y→SNo z→SNo w→SNo (x + y + z + w). The proposition is identified by the following information:

28.1. SURREALARITHMETIC 405

Pure Prop Id: d71f0eb42650c644bb0c4a2fbb29173cc3bf47fea9dd9401c0ba715333db1820
Pure Prop Address: TMF4oHEkLVa8KD2NcVmcFum9JhucpudfEb1
Theory Prop Id: 42037e0c85566de52aba1c0f97268637a493ef459b07be2723bfca5dc7b13a97
Theory Prop Address: TMXXq7JGxapVLMuBsmWzgDWpgqW72RtAMLV

add_SNo_Lt1

Theorem 28.31 ∀xyz.SNo x → SNo y → SNo z → x < z → x + y < z + y. The proposition is identified by the following information:

Pure Prop Id: 92232f18c39390c575ed22b7c92927a25b5ade867b5e2ed5f2cdf7e509909805
Pure Prop Address: TMLEq8tneuw32G568W1mBtwRTNNgQPCRYyz
Theory Prop Id: cb0738c7872b30b23cb65bbcd1ce88351f2466b21c1b5cc15f5b44ae581e28e0
Theory Prop Address: TMWTBYY2jenWVwUwvSromB8AgRWDVcJnbY6

add_SNo_Le1

Theorem 28.32 ∀xyz.SNo x → SNo y → SNo z → x ≤ z → x + y ≤ z + y. The proposition is identified by the following information:

Pure Prop Id: d8f37ed7006da1a1793317440a31b24b6f87f27cad9329c38e8a7ff9753d2dcc
Pure Prop Address: TMaVRrrSjDXkuboEtM4VK6QYgB6UVbfBcaG
Theory Prop Id: abcf7263a107c501b54d30826fb67a71bfd65aa0e1fa33d5cb94ba28bbd40502
Theory Prop Address: TMR5TeFwaZGZUnBsDVqT64ZqwgMewHGgCDF

add_SNo_Lt2

Theorem 28.33 ∀xyz.SNo x → SNo y → SNo z → y < z → x + y < x + z. The proposition is identified by the following information:

Pure Prop Id: 6ad46463d6f79423eff3ac9a3ac6201e44983b342ffb704d9cde5795efc8a7ec
Pure Prop Address: TMZabfrosx2WbihjxmG74tgmZaLS7LCKpAU
Theory Prop Id: 6f0941dc05860da5ece42171247edca54127ab818e70661d52995088db287957
Theory Prop Address: TMQWhSEEumkynNocoC5sSjUxscps15Jp64G

add_SNo_Le2

Theorem 28.34 ∀xyz.SNo x → SNo y → SNo z → y ≤ z → x + y ≤ x + z. The proposition is identified by the following information:

Pure Prop Id: 14975eae6645de30652349b64a88f0b2e35b84bc21be97aafc239516dd628b87
Pure Prop Address: TMdukf7SbuQ39yuQq8KCjup31RNEDuTLQnT
Theory Prop Id: 3f52ba378d30a817615bc0f6bf9697c00b83ce446cbe3de8fff3a6d92b6bd5de
Theory Prop Address: TMGdyU2zNNkb5T6LeAwxX74oWd2P7WYJJ8c

add_SNo_Lt3a

406 CHAPTER 28. SURREAL NUMBERS II Theorem 28.35 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x < z→y ≤ w→x + y < z + w. The proposition is identified by the following information:

Pure Prop Id: b2215f63878ba2a6fc09f12939fd6fc51dc580691897e8323370abb12c70f7bd
Pure Prop Address: TMYyEHQwicHmVL6MUyPu2QNJHpVu8pL5YVP
Theory Prop Id: 18ff91f30d21406ebb6b75675c0b1f7e8ad283cbf98775f53dd3e656a5fa522f
Theory Prop Address: TMSmpE2jeB2cyTEuG7Rq8sykhPLAACRxekP

add_SNo_Lt3b

Theorem 28.36 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x ≤ z→y < w→x + y < z + w. The proposition is identified by the following information:

Pure Prop Id: 5b4f92cfc6a3acaa35dd069d515cd4c025daa3b2bca2a7add6810dc336b0235f
Pure Prop Address: TMKXJb12jTxUrEE48Fsz3YxLGNM4wfBPXJW
Theory Prop Id: 8fe0d98eaddd0798a52530dbd10ab1d00f92e187620a37fb685f5830bd4c085c
Theory Prop Address: TMKS8vtMN5qWoxjxHv7NEEavnk6emGyiiMz

add_SNo_Lt3

Theorem 28.37 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x < z→y < w→x + y < z + w. The proposition is identified by the following information:

Pure Prop Id: 7141a86b460a8ecf3af113551c387396eea76973c00c0ae6d3a7a9f0806449bd
Pure Prop Address: TMNriuSM92c9RucKEto1JSRuMZvj8bjdnr5
Theory Prop Id: 55bdabe93657d73da3016bb38681d52b49a61dca9be7e4b0f99461db17aaf692
Theory Prop Address: TMKtJcmnivzZgTkMMwdV5vUEUxKGyMkrQFd

add_SNo_Le3

Theorem 28.38 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x ≤ z→y ≤ w→x + y ≤ z + w. The proposition is identified by the following information:

Pure Prop Id: 0be637ad324a3ea9caae189e47f4890cba93a45121b22ff1e10fad675f87f12c
Pure Prop Address: TMSVKSkQGVqbboR9prtEBre18xSpVMapMrK
Theory Prop Id: be66ca7a659e973fad15e883dbd8397986cfe73bd862e7b50ccf70a912a6e0b0
Theory Prop Address: TMdD3NhDM7mVSeEXJrP8Vh8HeE3of9UjkEu

28.1. SURREALARITHMETIC 407

add_SNo_SNoCutP

Theorem 28.39

∀xy.SNo x→SNo y→

SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y})

({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y}).

The proposition is identified by the following information:

Pure Prop Id: e2b7d74c79711fae6a14f4d67a701081855be3bb26fd923a62a391e04160bfae
Pure Prop Address: TMdKu7mpdNdzCSmXhFYuNuBjrmEsPPsqqGm
Theory Prop Id: f86b1a41eabc3959fd8be49bbc8865a0009802c350299879c6b3a1eb97530c7d
Theory Prop Address: TMR6c36epFgVd323wsqJDSNi9GvJ5dHX5bG

add_SNo_SNoCutP_gen

Theorem 28.40

∀LxRxLyRy.SNoCutP Lx Rx→SNoCutP Ly Ry→

SNoCutP

({w + SNoCut Ly Ry|w ∈ Lx} ∪ {SNoCut Lx Rx + w|w ∈ Ly}) ({z + SNoCut Ly Ry|z ∈ Rx} ∪ {SNoCut Lx Rx + z|z ∈ Ry}).

The proposition is identified by the following information:

Pure Prop Id: 075eaabfcc7432298a2db5f87aafbc0106915ca041c774d445bb94d0821d5824
Pure Prop Address: TMS5jpucdv4wNPKAJszA5iBP9k5qK9fGjf2
Theory Prop Id: 9afab2f26e1613aa776c9706130a926c7e2da1ffe974ae51ddb5a8b57c102cf0
Theory Prop Address: TMUA1C1f53xTBiuMvrMGVhLK8tVv3uALH6H

add_SNo_com

Theorem 28.41 ∀xy.SNo x → SNo y → x + y = y + x. The proposition is identified by the following information:

Pure Prop Id: 548814e9730b34ca53c6149a1239b003d299813e4e1610c765a8f32771db006e
Pure Prop Address: TMb6qo68fsewkpEkdhHBPyCDbKiaep9rQ9B
Theory Prop Id: f017e50fae3edfeeaedc58af7e2473b3c3008dd0d65d3dd902ce9902cfcedaed
Theory Prop Address: TMdQusEuxAaq4pjYGPyZyTsGeCDoe7ixEg5

add_SNo_0L

Theorem 28.42 ∀x.SNo x → 0 + x = x. The proposition is identified by the following information:

408 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: 107598652879bcb8baa57261d6be92d58926c75b0158561c7a0fbf824b5533ac
Pure Prop Address: TMPpszvpgSas6jmAy2oPhZkMYNxdnAa9bCr
Theory Prop Id: 446bada3d91ea404046f63fd5ef3b884131bc7c36182164979919c7306a98104
Theory Prop Address: TMLtAihTfZ4tMgum6Ca1thK1rJp885zUT7m

add_SNo_0R

Theorem 28.43 ∀x.SNo x → x + 0 = x. The proposition is identified by the following information:

Pure Prop Id: d8a186fe010013b2755c30ba0a202ddabfa4c545c10aac8b4e1364cac61cebd6
Pure Prop Address: TMTawHrBTSYsj41yX9x5qiLExuLJBg499YB
Theory Prop Id: 1a7fa1896336c0b5037e49fd8da8f614bd60fec53a52508d1c24eb4c32d2c081
Theory Prop Address: TMMmjuw778ta6y4jTCoCMS5fmu1gZ8B6AGF

add_SNo_minus_SNo_linv

Theorem 28.44 ∀x.SNo x → −x + x = 0. The proposition is identified by the following information:

Pure Prop Id: 42471917566847dbc969b69ad8eb5cfbb677e19d7005aaefa3ddbcd227fbc72c
Pure Prop Address: TMPWBgm88UpCF22WujXVVkVtCu8iciY3QTW
Theory Prop Id: 44e251b28aec6132ef7e2d0a0c34be7ac5c327d1039a329806f84dd65df681f5
Theory Prop Address: TML2RWMKucEcfgS9pHC3eNVJ6jgXjXCWKn2

SNo_add_SNo_3c

Theorem 28.45 ∀xyz.SNo x → SNo y → SNo z → SNo (x + y + −z). The proposition is identified by the following information:

Pure Prop Id: 104eb97ceeb39012baf2633f2295fcd8488c5da863f883a4f0bd3508d7022b82
Pure Prop Address: TMVNoU8QDePUm51AQC5k9QeJwu735iBxPvj
Theory Prop Id: 833be87302ff13765fe452b02d5b463172269368fd9039c3342e653a13738d59
Theory Prop Address: TMPvYhSo4MBSTDX3QY6YVnoEkB94k85UJj6

add_SNo_minus_SNo_rinv

Theorem 28.46 ∀x.SNo x → x + −x = 0. The proposition is identified by the following information:

Pure Prop Id: 9e8c44fbf607e617cb13942672b1aaa08da477c7c043bd3445e12df7880842e3
Pure Prop Address: TMYMYkykBhLq1apDzVyC2vJc6hiCZ65cEJD
Theory Prop Id: b2f9cf9ffdd19a14a7d49bd3eecd70f6ba950f3bae905926575b58dbe69c5611
Theory Prop Address: TMS4a619pMmyZvGY8DXex1wWCwRRFpTtwJ7

add_SNo_ordinal_SNoCutP

28.1. SURREALARITHMETIC 409 Theorem 28.47

∀α.ordinal α→∀β.ordinal β→

SNoCutP ({x + β|x ∈ SNoS_α} ∪ {α + x|x ∈ SNoS_β}) Empty. The proposition is identified by the following information:

Pure Prop Id: 2823e439cb46c064666c0dd60fea0fc706e8f3518725e3e71e9fe27292df89a0
Pure Prop Address: TMUPaueAJ3SYX5RiMrtRmYAsP3Gv5ibyiJr
Theory Prop Id: 507eab0935b7feec7b7116c01bd1ac27c74326982b1b48e41b6c44b87990749b
Theory Prop Address: TMT1Gz8K4EguVmo5NB2Woi7sw5sQBm85KE2

add_SNo_ordinal_eq

Theorem 28.48

∀α.ordinal α→∀β.ordinal β→

α + β = SNoCut ({x + β|x ∈ SNoS_α} ∪ {α + x|x ∈ SNoS_β}) Empty. The proposition is identified by the following information:

Pure Prop Id: cfe1f500f4a75d5fa0a4a8a1366f200cc39ec6e6179ec1c26eadd7b4611de086
Pure Prop Address: TMPQvhtityMjoibgERv3aQLgbEgRcqm7yXL
Theory Prop Id: ee8e7c5eccf8aee691d357e43b94c01938268230735907d55299cc3a35954d02
Theory Prop Address: TMQPK7b3tUhBwCLXYG48aj1mc232WYpgPJF

add_SNo_ordinal_ordinal

Theorem 28.49

∀α.ordinal α→∀β.ordinal β→ordinal (α + β).

The proposition is identified by the following information:

Pure Prop Id: e9309ac99999e13ed9ba4be238444a0ade5ba236e0ed518f2e0b024d43d638d7
Pure Prop Address: TMQJ6hARogVv6a8W1vmi4QLJdstmu7vvsko
Theory Prop Id: ee9c238cf553597102f9b9ee4da351edf8550e6a9298c9f211be331552a2cb2b
Theory Prop Address: TMRRB6iU5Y8ULDEdxSDS6ZdHUDYdozzKnx2

add_SNo_ordinal_SL

Theorem 28.50

∀α.ordinal α→∀β.ordinal β→

ordsucc α + β = ordsucc (α + β).

The proposition is identified by the following information:

410 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: d55bd8e80d14016f86e065ddbbdfb226571ce2f30e4aebe8b4e0aca9e5092472
Pure Prop Address: TMQnHkNidg5FWQLhSNenPvjDxCjnJdeF1bF
Theory Prop Id: 61ae57513d7783186f7327027f1807ee424c8d2af57399e9fb52c5685603f457
Theory Prop Address: TMcBJQnghsqzp4UQfnx4PBJi2B8VvW3agRn

add_SNo_ordinal_SR

Theorem 28.51

∀α.ordinal α→∀β.ordinal β→

α + ordsucc β = ordsucc (α + β).

The proposition is identified by the following information:

Pure Prop Id: e7378a22af92b7c147bbd7898baff5455a0ef96fa00ffdf898122a3bf428bdd3
Pure Prop Address: TMXcxTEQ97P4n3PTFXCCqgWyQErB5tL9Za7
Theory Prop Id: 0c1a6df8d4077eb378e14555b16968f4b38ec67e6f97bbeefe5a78c70e15a227
Theory Prop Address: TMLPo24YiNr9FHf939LUvMLAynxujJ8Qhjg

add_SNo_ordinal_InL

Theorem 28.52

∀α.ordinal α→∀β.ordinal β→∀γ ∈α.γ + β ∈ α + β.

The proposition is identified by the following information:

Pure Prop Id: 09a829e8ab6f31d46768fdcb929196886d6328bcf64c285f93a561a65c1398aa
Pure Prop Address: TMRVsTCaSviQYts1RQeAwzNWGiuAUb5kYUF
Theory Prop Id: 5acda8957c70e073f723185142d6e08818fa09b36f05b1bd211a60f1ba14fdf9
Theory Prop Address: TMVTmvXxRgnniTCwKdyop3YHpKGaSbz6KZo

add_SNo_ordinal_InR

Theorem 28.53

∀α.ordinal α→∀β.ordinal β→∀γ ∈β.α + γ ∈ α + β.

The proposition is identified by the following information:

Pure Prop Id: d291f44187e6051da6ee1d562ded685ef47c231d1bf1f5d37d7dda55a8e5ca3f
Pure Prop Address: TMNJwMYCWAjhoPBNGGAFssvdn2KaxT6X5zC
Theory Prop Id: 0f42b5bcdea755e828207f260bca45c0356328488660727bd8cab53262b609b3
Theory Prop Address: TMSm47BPo5yfVb687cN4t62bbsfGbbbXCiA

add_nat_add_SNo

Theorem 28.54 ∀nm ∈ omega.add_nat n m = n + m. The proposition is identified by the following information:

28.1. SURREALARITHMETIC 411

Pure Prop Id: 1701fc5ed7579abb5b201f5ec8a96acd2ac68046ad3176c036d134dea5669afe
Pure Prop Address: TMXMKYNwgEJbbeSZcXdj5a87FYKoc4K9apt
Theory Prop Id: 9df659a7b32d9f3afa10987ffbc9d9a84706cca922950d969a3d81f702e1b893
Theory Prop Address: TMGmoXE93EaXQanV3zMwj3Fp6HW7SWjG9xx

add_SNo_In_omega

Theorem 28.55 ∀nm ∈ omega.n + m ∈ omega. The proposition is identified by the following information:

Pure Prop Id: f45cbb9681681aaabac9270b640eb30171eb36f216ec7a99e5823ef45f94066a
Pure Prop Address: TMPib41CNAq7UTPnewz4ZBfR7FdVnYWEtRy
Theory Prop Id: eb6375250d942b2db2172dc601d939190506da07644a1cc5fdce38ab22ecf5fa
Theory Prop Address: TMP489nANG759rumydRqmyjQssxqD72fosB

add_SNo_SNoL_interpolate

Theorem 28.56

∀xy.SNo x→SNo y→∀u ∈SNoL (x + y).
(∃v ∈ SNoL x.u ≤ v + y) ∨ (∃v ∈ SNoL y.u ≤ x + v).

The proposition is identified by the following information:

Pure Prop Id: 97136316ef3ba2048fc844999ceaa34b07167a3712ce76f041d8382dbfc0b380
Pure Prop Address: TMFeNW7HkxH1nH7hkD7ZZwNKgZJpFXGy2jK
Theory Prop Id: 3cafc181e5e0dd3820d6f9d73c0ed6639ab60e79224ef8a2c5119732a877cd30
Theory Prop Address: TMMVRDZraKVB3LvPdW6kCBnJker3CjaSPfz

add_SNo_SNoR_interpolate

Theorem 28.57

∀xy.SNo x→SNo y→∀u ∈SNoR (x + y).
(∃v ∈ SNoR x.v + y ≤ u) ∨ (∃v ∈ SNoR y.x + v ≤ u).

The proposition is identified by the following information:

Pure Prop Id: 928f71993ee178fd38105f25f4ad163aeeb97d5ac1bbb4c218f4c3b81d4cd25f
Pure Prop Address: TMX122A6CkJ17L11g6XTAB5Z87Xwqzq3F4H
Theory Prop Id: 31fa9428aeddf723102cc0d5660dce0ce0e1e068f2bd5d6fd5d853418947a80e
Theory Prop Address: TMTKJ9smZabCsGAHoewvLNHykWiCKJuNchm

add_SNo_assoc

Theorem 28.58

∀xyz.SNo x→SNo y→SNo z→x + (y + z) = (x + y) + z.

The proposition is identified by the following information:

412 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: 39b331b6381b6e5ee82cce99630525f4cb0ca03008347a4acf301031da6002ce
Pure Prop Address: TMUConoLia3kWzpgXw8RzijjEGkNtvGJecF
Theory Prop Id: bc15ad70cce2a78c1f054e4e77fa522dd72c07d6620fb831284863016cf6e3da
Theory Prop Address: TMFHBgSLqG7xXui9hnNbAKuSE67kqnChBbz

add_SNo_cancel_L

Theorem 28.59 ∀xyz.SNo x → SNo y → SNo z → x + y = x + z → y = z. The proposition is identified by the following information:

Pure Prop Id: 0140969cf9285d05d4b30bf21c910adfed405122d410120abd72c11f5ab69789
Pure Prop Address: TMV4roc7BLNMpAapZh7KA7NtFj11mPqdozE
Theory Prop Id: 781256c452aceda44607ee4205e8ee86d1eea0b3f32743f53b342771142ce743
Theory Prop Address: TMHbLqftV2LcfDUv7HqxqMhm7CVsY7Ef2K8

add_SNo_cancel_R

Theorem 28.60 ∀xyz.SNo x → SNo y → SNo z → x + y = z + y → x = z. The proposition is identified by the following information:

Pure Prop Id: 3de43c040780ea5cd150c912ec06fac7c4f463b8b0a89ff1d2cba1c673d3b3d6
Pure Prop Address: TMLNfa9NDWM7B9HYGBe7xuczocSytHXKARC
Theory Prop Id: e62801806a6a72a797723a605192e13732faf66171abd8d9323db7972673a977
Theory Prop Address: TMVQQYhZPbjCpQtrWniV5GvvRBpj4GaPPzW

minus_add_SNo_distr

Theorem 28.61 ∀xy.SNo x → SNo y → −(x + y) = (−x) + (−y). The proposition is identified by the following information:

Pure Prop Id: 57ec9da45362cd7c9418befe3fb2ca772e55318bd76cfac89139baf603dd198c
Pure Prop Address: TMQTvHftZnQzgoUvhX3DhSiB9EYGcjbMEVt
Theory Prop Id: da8d5a3c4332f78b540e7fb374f3ba393f34ebef39aa08f03a43e8a0eb0ec48f
Theory Prop Address: TMMrTspPpgWYYtPV5GfMyFKZxbTpr6YpBF8

minus_add_SNo_distr_3

Theorem 28.62

∀xyz.SNo x→SNo y→SNo z→−(x + y + z) = −x + −y + −z. The proposition is identified by the following information:

Pure Prop Id: b074acb051db50b8aebd6c84b804f36ef95546c8c1d8b4a12fad41b6036b675d
Pure Prop Address: TMHG5Wzxu3ufeWeCBHAySJy5K1FsFRffozM
Theory Prop Id: 94dbf3383e5ad97c01e0494f7ad58f05eb86370106ca98f801505d5cc17d8e8c
Theory Prop Address: TMGDtLEgxm1wM1omKuqmT7qds1DiL446e61

add_SNo_Lev_bd

28.1. SURREALARITHMETIC 413 Theorem 28.63

∀xy.SNo x→SNo y→SNoLev (x + y) ⊆ SNoLev x + SNoLev y. The proposition is identified by the following information:

Pure Prop Id: 0f257c03f4bd792cfafdf5b6628524966280c12bc90c0b7152a74c1c453b0a97
Pure Prop Address: TMHLNim923vpzTQhz7fmKu2bm4ncVWUmWEE
Theory Prop Id: 0680effc26564a3378ad068703ae226003b02850f73e91331345c41424701bcf
Theory Prop Address: TMKzYXGUEBsqkz6Pd1y7TGUmq9kreetmiSK

add_SNo_SNoS_omega

Theorem 28.64 ∀xy ∈ SNoS_omega.x + y ∈ SNoS_omega. The proposition is identified by the following information:

Pure Prop Id: 9311032f024de4324b4cbbbc04cbbc6f507e4664b92e5daf8d2b539deaa0d877
Pure Prop Address: TMRdieTgkX8dUfZyNNYHypf4kTH3PoJW3Ms
Theory Prop Id: 6d16aa7a6b095f74f1b34091c0e9a002aa247761437c60a7c0261e4d1687a1d4
Theory Prop Address: TMVjF1tKn3421u44CyHtBcSNvHoJBqRKST1

add_SNo_minus_R2

Theorem 28.65 ∀xy.SNo x → SNo y → (x + y) + −y = x. The proposition is identified by the following information:

Pure Prop Id: 3e7b333f3cd5354e81de9e18e82adb1ea3101121fa6de8717d4d9dd7a8de2dfc
Pure Prop Address: TMbjFdWsQkwJkFdFKsygecXhLiJUroenET2
Theory Prop Id: 91f4efb884c7f253c44c5fd69b6ef119bd466cbf203a587487e6a292ebc37f9e
Theory Prop Address: TMdam15xQW7pR7unsMT5iH6SPdFATTi712m

add_SNo_Lt1_cancel

Theorem 28.66 ∀xyz.SNo x → SNo y → SNo z → x + y < z + y → x < z. The proposition is identified by the following information:

Pure Prop Id: e5c11e1cc1b83566aacf4247c67eda38cbd881bcf48b4a30fd81d1ae9be3ae25
Pure Prop Address: TMVGk6XaUoPX6ZGVPLEVZWzvzKkLektMYLP
Theory Prop Id: 2ec8aef4360c9ee2ffa1912943451de1852b1cf2af0400f6da375d69c467ae9e
Theory Prop Address: TMLJkdMCCCn6YVEdM7uNU6PFwazJo7w6REd

add_SNo_Lt2_cancel

Theorem 28.67 ∀xyz.SNo x → SNo y → SNo z → x + y < x + z → y < z. The proposition is identified by the following information:

Pure Prop Id: 9ecfc82c2e1b3fbbf14127fd056f50b5f4f4058c7b9c2da324e8e11162871620
Pure Prop Address: TMJyFpf59yM4gCj8nhUhoxe5MLMc9ZvK1hj
Theory Prop Id: d4afbdcdb563fce45f9fdb7e6d26326d06598fc21bee88b552fb7439b946957b
Theory Prop Address: TMF8QKcffVRxMAdSUMqMY8FSyHY2Pu3AYJc

414 CHAPTER 28. SURREAL NUMBERS II

add_SNo_assoc_4

Theorem 28.68 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x + y + z + w = (x + y + z) + w. The proposition is identified by the following information:

Pure Prop Id: b9919a859ce250ebed40a38e210d51ea56ad81bd3b548d2973c10fa4b77224fb
Pure Prop Address: TMNn2vs2scHFb5sPooErQtoePq5skwF4Rqs
Theory Prop Id: f2ec8cf0669d6b7f4ba80ac3f1c7fc3a38bc969d82b8ab446307e654aeb60211
Theory Prop Address: TMJdJZYxC1SYeUZNH9CnoUzpc5nbzLd4E1K

add_SNo_com_3_0_1

Theorem 28.69 ∀xyz.SNo x → SNo y → SNo z → x + y + z = y + x + z. The proposition is identified by the following information:

Pure Prop Id: 53adb5589660e895a568e553cecd4c833dfe11b8d4b0a59395733cf1473d5a4f
Pure Prop Address: TMan13aAzSS4boU3k6BCV2m9HpcAqzjMqYs
Theory Prop Id: 3c9cc8b1863d8757dc461ba3222ea23daf40651a26eb8bbaa4eb74d8ce5ecb34
Theory Prop Address: TMLUi74QYWaLowsgzo8D5KaFFmMGAF33Uov

add_SNo_com_4_inner_flat

Theorem 28.70

∀xyzw.SNo y→SNo z→SNo w→x + y + z + w = x + z + y + w. The proposition is identified by the following information:

Pure Prop Id: 353c422c4473fec6d253f64dc6011273518f6fd9b8666b1d04ac4b7a10b21676
Pure Prop Address: TMYagwq4AL9eS2satFuJrRGsXJQ7UaQ2Y2Z
Theory Prop Id: d32d7fe4fd859c93a5b059b561d61395eae0d2dc1771536899a51aa7290836c3
Theory Prop Address: TMJ7ALWqQknTtDK5LC67JaovH5kH3hMjM3S

add_SNo_com_3b_1_2

Theorem 28.71

∀xyz.SNo x→SNo y→SNo z→(x + y) + z = (x + z) + y.

The proposition is identified by the following information:

Pure Prop Id: 2ccf2bcc11ca672b64552dc368e1746da109c4c422678991b3ab7c3011849772
Pure Prop Address: TMPuoTqK9f79je6oZPXi6p9wEreuXeYKFpg
Theory Prop Id: f2dd80a1f9d4848b30eb661596f9b7355f21ffc0ff2b67d04df7791c06f27387
Theory Prop Address: TMXQdGRpcSTBjZ8pjJmoA2DCTzkK5tBAKzR

28.1. SURREALARITHMETIC 415

add_SNo_com_4_inner_mid

Theorem 28.72 ∀xyzw.SNo x→SNo y→SNo z→SNo w→(x + y) + (z + w) = (x + z) + (y + w). The proposition is identified by the following information:

Pure Prop Id: 635cb763d6883b1df327b97d296852bd72be35ebf27ba3197cc20d58b5db1923
Pure Prop Address: TMNBs9W7CDbVpEM2DJAD6LJaUJdzLs5mL2p
Theory Prop Id: e25ee6de8e445a677f9f91027aeb3dc6ab5acb07b724880668b83cb66a9d287c
Theory Prop Address: TMRqXvHGUrVc7Tj4T8TaayZuLEe7vs6QkAa

add_SNo_rotate_3_1

Theorem 28.73 ∀xyz.SNo x → SNo y → SNo z → x + y + z = z + x + y. The proposition is identified by the following information:

Pure Prop Id: 51654e504536bbcc3c5f757f3dcad5e306f61cb3f3113fe097e1d1fbba5aac97
Pure Prop Address: TMNHRTRCD9bsMPb3UbC8jvRrJ6N9ZH1Eh4N
Theory Prop Id: 340640be61f33faa3a7f153679d19bb2ce9291b5efec590860ba767ade73a9a3
Theory Prop Address: TMQNLhVAcRPvXzedEwagyyteqQKc581dhD5

add_SNo_rotate_4_1

Theorem 28.74 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x + y + z + w = w + x + y + z. The proposition is identified by the following information:

Pure Prop Id: 4bc7aa1aee0011ef665476863c178f590fbe4fbe6699dd7ccb934845846d8dd3
Pure Prop Address: TMaTstSL13vyKcGtLbTNn8eAVSGWCpDMRRP
Theory Prop Id: ce0a6b44cbcebe4b78d9bcbc4db2e540564f76969de5d18f22a263048dfe8518
Theory Prop Address: TMYrz7XuuFsQmh65FTDfV3W2dcW9j9WhNCe

add_SNo_rotate_5_1

Theorem 28.75

∀xyzwv.SNo x→SNo y→SNo z→SNo w→SNo v→
x + y + z + w + v = v + x + y + z + w.

The proposition is identified by the following information:

Pure Prop Id: 4b4b645a9de69d196bcfac0f564a76aba4c24d5518f13324b64621276d543128
Pure Prop Address: TMSSCZTFkPsvFM1mXZ7MP4aihGyFoBRfhdC
Theory Prop Id: 0714aabd91b09f4e30b2978cd4c05acb5975913fcf91782f98282a06ecc98e76
Theory Prop Address: TMKsqf29EQpew5TASzktuUb1DHqMTYo9542

416 CHAPTER 28. SURREAL NUMBERS II

add_SNo_rotate_5_2

Theorem 28.76

∀xyzwv.SNo x→SNo y→SNo z→SNo w→SNo v→
x + y + z + w + v = w + v + x + y + z.

The proposition is identified by the following information:

Pure Prop Id: 33b5cc61253fa4c067f352dd7d4f385b800a18a8d891459904b242726be1c98b
Pure Prop Address: TMNP66fS26FKAKuh9MEtRW9VDian3FYj9qr
Theory Prop Id: fec9d019d82128a582a52e0bc0e64cfc44a05d4f84f9952d8cb92946b031856d
Theory Prop Address: TMUguce5TL82mq83tgNMXbiPmFe3zcgvQFJ

add_SNo_minus_SNo_prop1

Theorem 28.77 ∀xy.SNo x → SNo y → −x + x + y = y. The proposition is identified by the following information:

Pure Prop Id: 2cd65e577fae467be069f5ab9a2bfa88c578085e70deca674f146625df01bb31
Pure Prop Address: TMNj1ZknUmnj8fyskPShRtyeBwzp2553NVN
Theory Prop Id: 598ee6b17ec7a3c47d15dffa959d384acc4d4d5bdbcd02790c13e2ff045e1440
Theory Prop Address: TMKE2xaN2B6wD8vGVfwUJaufhki7EhAkGfE

add_SNo_minus_SNo_prop2

Theorem 28.78 ∀xy.SNo x → SNo y → x + −x + y = y. The proposition is identified by the following information:

Pure Prop Id: 632a4d661bba9435289e33a84ddfd6e1ebe55256f17529d37fd7bf784a245260
Pure Prop Address: TMFUMMG5NWfvwVqV1VzWhtid5D1cW3ENDqd
Theory Prop Id: a8450da6ba6ffd91e010f90197162261ef21fb4e025e591e5137531e12d7ad70
Theory Prop Address: TMJQtEivAHnWG6vm64j8jL7rdfeCLJv59AH

add_SNo_minus_SNo_prop3

Theorem 28.79 ∀xyzw.SNo x→SNo y→SNo z→SNo w→(x + y + z) + (−z + w) = x + y + w. The proposition is identified by the following information:

Pure Prop Id: 038fdd7f7b41d5dbabb1c2b43f6ee18b5eb51c7b73b574795aa81a5f6e7185db
Pure Prop Address: TMQQTF6qMromiAnZNHGPbLUmDnoRJJsw184
Theory Prop Id: 0d4f2459906cf0deac11ccfa5d8ce0300b8a0e2e2e35908d116f376387aeaf89
Theory Prop Address: TMVtSJwaz4PKT2M9971YpjjmSqY2uXJQFDe

add_SNo_minus_SNo_prop4

28.1. SURREALARITHMETIC 417 Theorem 28.80 ∀xyzw.SNo x→SNo y→SNo z→SNo w→(x + y + z) + (w + −z) = x + y + w. The proposition is identified by the following information:

Pure Prop Id: 9303242681e65d4bb41186882b5827d87cfb757866a860dcd21e8d85c6540d00
Pure Prop Address: TMZxz5UVrwqNzRN8eXdiojfqVbJhkXRzfpt
Theory Prop Id: 4c3067acc33707997f49c35d6a0ac8a1b7845a5ed0454d39acbfe19ba57d0c1f
Theory Prop Address: TMWftxv2baJqYmzmKYPcnYgd25XrtWoXJST

add_SNo_minus_SNo_prop5

Theorem 28.81 ∀xyzw.SNo x→SNo y→SNo z→SNo w→(x + y + −z) + (z + w) = x + y + w. The proposition is identified by the following information:

Pure Prop Id: bdfd3ff33cf2f7c3beec961e55cb8c70e3e72bedc8e24a5079ae13e57cef924e
Pure Prop Address: TMbd9Mg6tPihAjiCRTdu2H9iHTJu8P6vdH9
Theory Prop Id: eeb587de80f26e1b6307d8991dce2c6ed3d960c0a1571b8722cff655214df698
Theory Prop Address: TMRmP8fuJtFdvTRXG6bJ2zsSkasVFLW8YiM

add_SNo_minus_Lt1

Theorem 28.82 ∀xyz.SNo x → SNo y → SNo z → x + −y < z → x < z + y. The proposition is identified by the following information:

Pure Prop Id: 643876090ca9e8c818bb1d1517026337499a8a61a6383863a9f6b90713b953f8
Pure Prop Address: TMHorPzXGJj6NLSDuVdotitbnKv8te63hAg
Theory Prop Id: 4033dc222689776035a38336a7f6ce157593e8786381729a207ecdc69f8508d1
Theory Prop Address: TMGgmNN1c8BXxQunMKeC7ctpH6iBXxvzUXy

add_SNo_minus_Lt1b

Theorem 28.83 ∀xyz.SNo x → SNo y → SNo z → x < z + y → x + −y < z. The proposition is identified by the following information:

Pure Prop Id: 5012b14b32060175e560f0d1e7e017da053c610000fa170aeaf365e6d6410d7c
Pure Prop Address: TMGs3ZQCrCC6dtPSMkxhL3RGUTqUer9SdGs
Theory Prop Id: 0b5cca84918d73e36a13fd959569d5c6a5d4a05c85317235433f59a19bc601a2
Theory Prop Address: TMPp2LmW4bmLHnLXjmQuZtQpoHAz4uTwF1i

add_SNo_minus_Lt2

Theorem 28.84 ∀xyz.SNo x → SNo y → SNo z → z < x + −y → z + y < x. The proposition is identified by the following information:

418 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: 76c543653b6da539ad0d8357582da1ab031b9383d2fae35a230e98de8946856e
Pure Prop Address: TMFL2J2Xi9knH1EYexCMF24iaPSDRahCQtP
Theory Prop Id: 3621456ff64800b7e93dff3968e21ba0755569c11a30efbf0360787c625c56dc
Theory Prop Address: TMSsbaxypo5H5nF8jvxTgrgpmxmVfGPHE3p

add_SNo_minus_Lt2b

Theorem 28.85 ∀xyz.SNo x → SNo y → SNo z → z + y < x → z < x + −y. The proposition is identified by the following information:

Pure Prop Id: 24d1a6c7194f01d2e61624e193908bc5ed6870b188d549fdbfafb81f3e69660b
Pure Prop Address: TMYLxBcGLaYBenPqsTbU48VLhTDu35ZvWin
Theory Prop Id: 68419210383aa804b0c97019a1190bf0a75fb45009b55d3f310cccb98a55f216
Theory Prop Address: TMLRXmb9LNiNLw5Y1urKn6AQHWWwgKREUkY

add_SNo_minus_Lt1b3

Theorem 28.86 ∀xyzw.SNo x→SNo y→SNo z→SNo w→x + y < w + z→x + y + −z < w. The proposition is identified by the following information:

Pure Prop Id: 218e4f25517f8ff1478046e3df782ae832d9f7e33e0b433bf05b9643a0208e85
Pure Prop Address: TMRSZGz7nE2AVHGmgMaepm9CAmfWzJ72Boq
Theory Prop Id: 8ea4a7cab08aeec1b407bd563474dd6766fbcf0794f34b125012f4f8d61f021f
Theory Prop Address: TMRrXMsdvggvcKCwoxKF8kLUbznUq5NaVmd

add_SNo_minus_Lt2b3

Theorem 28.87 ∀xyzw.SNo x→SNo y→SNo z→SNo w→w + z < x + y→w < x + y + −z. The proposition is identified by the following information:

Pure Prop Id: d9462b9396765bcb6e599c448b5b2a0e15bfeea62de5713637e71fe91bc14e06
Pure Prop Address: TMG6X9g8zK9X5PbsVyTtDpJgN73SyRkZBJ2
Theory Prop Id: 4bf57517c0115e8dae375c95d1c1bf403694f4ee328c9dfd9c34f22eca40d62b
Theory Prop Address: TMGSKHUM3KGisvSLGi1Zpy9MG4UtFxqFThU

add_SNo_Lt_subprop2

Theorem 28.88

∀xyzwuv.SNo x→SNo y→SNo z→SNo w→SNo u→SNo v→x + u < z + v→
y + v < w + u→x + y < z + w.

The proposition is identified by the following information:

28.1. SURREALARITHMETIC 419

Pure Prop Id: 848909edfa645bd860cf5cc6acec778e23e88c5a769fe3c959d4b3566fbb306b
Pure Prop Address: TMLUrjDQwdJ3cSJVaNX11wZPbmHmYJm2F4A
Theory Prop Id: 7f1b50ad173ad970e6eee06a0d1dedb182caf1cfff89406b30003a7afc38c6c3
Theory Prop Address: TMcJPm4yGSHS1kUSPRFgEHGenN7vG4V1Nzw

add_SNo_Lt_subprop3a

Theorem 28.89

∀xyzwua.SNo x→SNo y→SNo z→SNo w→SNo u→SNo a→x + z < w + a→
y + a < u→x + y + z < w + u.

The proposition is identified by the following information:

Pure Prop Id: 43322d509ad5f4223a3ea2d226e4b0ccb8c26b9a1175adae9d1d447e776bf7c0
Pure Prop Address: TMMYB2kCLPC8xoSmU63JvGfNSPFzUYuK6pv
Theory Prop Id: cde0c9b5a2910f46768762e1f3849c4e1b8cda57c9c1575aa1b9bf556baa627d
Theory Prop Address: TMPNtwtWveLBYLorWYj4fRoEpqz56R9oTm5

add_SNo_Lt_subprop3b

Theorem 28.90

∀xywuva.SNo x→SNo y→SNo w→SNo u→SNo v→SNo a→x + a < w + v→
y < a + u→x + y < w + u + v.

The proposition is identified by the following information:

Pure Prop Id: 9d5c37e2e6a26c6dc29e00cf8767d02fc77b30d09cebed0550f159cb9d1334c6
Pure Prop Address: TMakKb23RRtRLWKpGuoMNwnASjKertEftag
Theory Prop Id: b197b177985ff32095d16adfad88ff6b97c3c9885a5a1cadc19c5cb553b4345d
Theory Prop Address: TMduddfsYHhZALrRDzTwzypdKDVHK1k8VYh

add_SNo_Lt_subprop3c

Theorem 28.91

∀xyzwuabc.SNo x→SNo y→SNo z→SNo w→SNo u→SNo a→SNo b→

SNo c→x + a < b + c→y + c < u→b + z < w + a→x + y + z < w + u. The proposition is identified by the following information:

Pure Prop Id: 18461471e206fc8672b0e953edf289af2051148c543ab9c39752e75fd74b9902
Pure Prop Address: TMLDVH1R1XQXWmhFyN7kxcnxZ8vV1jj1gge
Theory Prop Id: f8cf2ba034b2ae66a187647fe084618a9c62057a7ab092702eb1355e51bfdb56
Theory Prop Address: TMNUyCkvtK5vp56fDjYpDWz4tisomP2mVS5

add_SNo_Lt_subprop3d

420 CHAPTER 28. SURREAL NUMBERS II

Theorem 28.92

∀xywuvabc.SNo x→SNo y→SNo w→SNo u→SNo v→SNo a→SNo b→

SNo c→x + a < b + v→y < c + u→b + c < w + a→x + y < w + u + v.

The proposition is identified by the following information:

Pure Prop Id: a447c6d932bea68dbde54b06fe58a08cf250a32d1e9e252f9aa646d7d450a31f
Pure Prop Address: TMGmodxxtB7938aypz74aGsuhYxLsXh57g5
Theory Prop Id: a1520a522ad3deb92a6746488e5cfc1dfb1d1fd6684fe79e87bba20cda4756b6
Theory Prop Address: TMLrgfSnAmTdRkGo9xWrngAvimcZVK9yU2d

ordinal_ordsucc_SNo_eq

Theorem 28.93 ∀α.ordinal α → ordsucc α = 1 + α. The proposition is identified by the following information:

Pure Prop Id: f0596d71a0270ee871aebb4abb6a468185b1e2d2deec29fda29af07ac94080e7
Pure Prop Address: TMKvpBUtRCutc7AQ2jjzLG87zkaCypGHwqw
Theory Prop Id: 36bf6dd6ade137d7a6431b1d2f74c22803884fe3bcc9f6b223f90c9587bc1fde
Theory Prop Address: TMM3U1W3pWWJb6gkvcZQNx2QDYDK4UxBhJx

add_SNo_omega_eps_Lt

Theorem 28.94

∀xy ∈SNoS_omega.x < y→∃n ∈ omega.x + eps_n < y.

The proposition is identified by the following information:

Pure Prop Id: 5a850b3a5f7e8e409aa7b3a8fde3eaf603b6c57ac0565694a8d02fbe81aa1496
Pure Prop Address: TMSCHeh28B5U2XMhbESr5SKSXVCskfzuwZx
Theory Prop Id: 0afdf612e70b6faf98df67163f00af4cacf8720792da2450f3f84ab2b65e27b4
Theory Prop Address: TMYHZueBTibUJWtZVQUjiVzfGCegt89U2VT

Definition 28.3 mul_SNo is the opaque object of type ι → ι → ι identified by the following information:

Pure Object Id: 48d05483e628cb37379dd5d279684d471d85c642fe63533c3ad520b84b18df9d
Pure Object Address: TMSw7TH4yUrNvqkxFpRdKgaGjhbMXKDtVKM
Theory Object Id: b14730ddd09967223586b7b73ba059d8fee46ef74b4d5c13b7276c084e8f3dea
Theory Object Address: TMdQsgnzh6YiiBs9ki4XhdobvDFEYV9QYKG

Notation. We use ∗ as a right associative infix operator corresponding to applying term mul_SNo.

mul_SNo_eq

Theorem 28.95

∀x.SNo x→∀y.SNo y→

x ∗ y = SNoCut ({(w 0) ∗ y + x ∗ (w 1) + −(w 0) ∗ (w 1)|w ∈ SNoL x × SNoL y}∪

{(z 0) ∗ y + x ∗ (z 1) + −(z 0) ∗ (z 1)|z ∈ SNoR x × SNoR y})
({(w 0) ∗ y + x ∗ (w 1) + −(w 0) ∗ (w 1)|w ∈ SNoL x × SNoR y}∪
{(z 0) ∗ y + x ∗ (z 1) + −(z 0) ∗ (z 1)|z ∈ SNoR x × SNoL y}).

The proposition is identified by the following information:

Pure Prop Id: 726bd964aa39e59e3a6693e9b7d4306b1accb9a6c62ae1891b050b131c4ea01f
Pure Prop Address: TMSnEDskXoZEWD5U2DVgxtTRUp75JLmq8fL
Theory Prop Id: 3525c5e458ce4f160f6f38c46b032eb43534d43481b8d0ff0b1096daa93e9080
Theory Prop Address: TMR2SBeeUCWzXPrVEFQprrZoRPjqi24Acwd

mul_SNo_eq_2

Theorem 28.96

∀xy.SNo x→SNo y→∀p :o.
(∀LR.(∀u.u ∈ L →

(∀q : o.(∀w0 ∈ SNoL x.∀w1 ∈ SNoL y.u = w0 ∗ y + x ∗ w1 + −w0 ∗ w1 → q) →

(∀z0 ∈ SNoR x.∀z1 ∈ SNoR y.u = z0 ∗ y + x ∗ z1 + −z0 ∗ z1 → q) → q))

→
(∀w0 ∈ SNoL x.∀w1 ∈ SNoL y.w0 ∗ y + x ∗ w1 + −w0 ∗ w1 ∈ L) →
(∀z0 ∈ SNoR x.∀z1 ∈ SNoR y.z0 ∗ y + x ∗ z1 + −z0 ∗ z1 ∈ L) →
(∀u.u ∈ R →

(∀q : o.(∀w0 ∈ SNoL x.∀z1 ∈ SNoR y.u = w0 ∗ y + x ∗ z1 + −w0 ∗ z1 → q) → (∀z0 ∈ SNoR x.∀w1 ∈ SNoL y.u = z0 ∗ y + x ∗ w1 + −z0 ∗ w1 → q) → q))

→

(∀w0 ∈ SNoL x.∀z1 ∈ SNoR y.w0 ∗ y + x ∗ z1 + −w0 ∗ z1 ∈ R) →
(∀z0 ∈ SNoR x.∀w1 ∈ SNoL y.z0 ∗ y + x ∗ w1 + −z0 ∗ w1 ∈ R) →
x ∗ y = SNoCut L R → p)
→p.

The proposition is identified by the following information:

Pure Prop Id: 97f72a94a57b9c976cabb7db6e1c9e782baccf6d99699b64f549f38a4a8cb50c
Pure Prop Address: TMLuSvvCfD9Ze7AKPowkWZPkeM2o9mvreGc
Theory Prop Id: d69cc6521bcd2b18a3af424dfe57b3ca7eccc3c292bd43c9f65e2a98dcebcffe
Theory Prop Address: TMRezz7usQBo1aanPbksRyAGNHbrpgWVes4

422 CHAPTER 28. SURREAL NUMBERS II

mul_SNo_prop_1

Theorem 28.97

∀x.SNo x→∀y.SNo y→∀p :o.
(SNo (x ∗ y) →

(∀u ∈ SNoL x.∀v ∈ SNoL y.u ∗ y + x ∗ v < x ∗ y + u ∗ v) → (∀u ∈ SNoR x.∀v ∈ SNoR y.u ∗ y + x ∗ v < x ∗ y + u ∗ v) → (∀u ∈ SNoL x.∀v ∈ SNoR y.x ∗ y + u ∗ v < u ∗ y + x ∗ v) → (∀u ∈ SNoR x.∀v ∈ SNoL y.x ∗ y + u ∗ v < u ∗ y + x ∗ v) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: 3018e48e1b0b8923d2bc9e30faf03586dced5941b18f9a31b0e91b0d91716d40
Pure Prop Address: TMdaGCethjpbbQ5f2zXiMzy1Jw6trWXkwEA
Theory Prop Id: 5e4d160028e2e0d69e57789a6c38f1230eca4d80e5f1e171fc8969764d28dd07
Theory Prop Address: TMZ8JpWhmNRNmu7xFhYLa8nAtcJESfVjnAP

SNo_mul_SNo

Theorem 28.98 ∀xy.SNo x → SNo y → SNo (x ∗ y). The proposition is identified by the following information:

Pure Prop Id: 7fa2e2dd6764040b3f888d899581ea742d829880456c04996546837d770370c5
Pure Prop Address: TMX6qVheSX36EwKWKF211NtEBN8GMqFFQjT
Theory Prop Id: 9fd082b661154955f669084ede91b91cef6249e30cbec7c7a954ac883b5ca01b
Theory Prop Address: TMTR3KoWZkq4VBDJf71KPEZUEwdxEBQ1SwZ

mul_SNo_eq_3

28.1. SURREALARITHMETIC 423

Theorem 28.99

∀xy.SNo x→SNo y→∀p :o.
(∀LR.SNoCutP L R →
(∀u.u ∈ L →

(∀q : o.(∀w0 ∈ SNoL x.∀w1 ∈ SNoL y.u = w0 ∗ y + x ∗ w1 + −w0 ∗ w1 → q) →

(∀z0 ∈ SNoR x.∀z1 ∈ SNoR y.u = z0 ∗ y + x ∗ z1 + −z0 ∗ z1 → q) → q))

→
(∀w0 ∈ SNoL x.∀w1 ∈ SNoL y.w0 ∗ y + x ∗ w1 + −w0 ∗ w1 ∈ L) →
(∀z0 ∈ SNoR x.∀z1 ∈ SNoR y.z0 ∗ y + x ∗ z1 + −z0 ∗ z1 ∈ L) →
(∀u.u ∈ R →

(∀q : o.(∀w0 ∈ SNoL x.∀z1 ∈ SNoR y.u = w0 ∗ y + x ∗ z1 + −w0 ∗ z1 → q) → (∀z0 ∈ SNoR x.∀w1 ∈ SNoL y.u = z0 ∗ y + x ∗ w1 + −z0 ∗ w1 → q) → q))

→

(∀w0 ∈ SNoL x.∀z1 ∈ SNoR y.w0 ∗ y + x ∗ z1 + −w0 ∗ z1 ∈ R) →
(∀z0 ∈ SNoR x.∀w1 ∈ SNoL y.z0 ∗ y + x ∗ w1 + −z0 ∗ w1 ∈ R) →
x ∗ y = SNoCut L R → p)
→p.

The proposition is identified by the following information:

Pure Prop Id: 51793959fac9070d8e2d9b035efecb4ad9c7ae3ca6da33d55ab3bf797bef3297
Pure Prop Address: TMaj5xJYDn5QmHnKGHqSMLP3XxL1wkJv3vo
Theory Prop Id: 720ba07dcca3e0725279be9e2c27ff2f73fa21fb411e18239df4e0915a5111b7
Theory Prop Address: TMWEwehT67Ew9qx9JCyHJdWR4Zn2VcBGx6y

mul_SNo_Lt

Theorem 28.100 ∀xyuv.SNo x→SNo y→SNo u→SNo v→u < x→v < y→u ∗ y + x ∗ v < x ∗ y + u ∗ v. The proposition is identified by the following information:

Pure Prop Id: 428b35f3bc30112249eab05959f8ad7ada3ebbb7987ced62d2fe3afde9af092b
Pure Prop Address: TMLpp8YEH3knCvvN7eHZnAuNDCMtsQzd14d
Theory Prop Id: c232011455c468168fd8e65dfaebb59e340331574a0089936e166d3a4fcf2f5d
Theory Prop Address: TMWuSHJpH8gNAX4DUjm9Wzk6tp7vPQphtfr

mul_SNo_Le

Theorem 28.101 ∀xyuv.SNo x→SNo y→SNo u→SNo v→u ≤ x→v ≤ y→u ∗ y + x ∗ v ≤ x ∗ y + u ∗ v. The proposition is identified by the following information:

424 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: d9a12c5ce8058d66cd1876f20d85be2a5ceec00d4bf569f3282bbee6cb91faab
Pure Prop Address: TMXSQ1sbRtw5pLGrB7wQb2i4sBb3bPWrE3x
Theory Prop Id: acb62030c6b82e1dd945365185266c95047e40137c030a2512820a86905108f6
Theory Prop Address: TMFCxz59dZUG7KrmMeKtLnrjcymT35p7rac

mul_SNo_SNoL_interpolate

Theorem 28.102

∀xy.SNo x→SNo y→∀u ∈SNoL (x ∗ y).

(∃v ∈ SNoL x.∃w ∈ SNoL y.u + v ∗ w ≤ v ∗ y + x ∗ w)∨ (∃v ∈ SNoR x.∃w ∈ SNoR y.u + v ∗ w ≤ v ∗ y + x ∗ w).

The proposition is identified by the following information:

Pure Prop Id: 7d1b52199f12ab9873941a7285388e03c30e7d2abddf759a627f6f95d4d06ea0
Pure Prop Address: TMd7FoqCiPFwP8mZudWTt9WbKfWC2KEFNpJ
Theory Prop Id: f460fc19e5496a545e8abbaf4188a0ddb17c95a931bb76236bbcc1ab7829c3da
Theory Prop Address: TMQhwb7bstgkPJXZU2FBTPqwJhZ6onk8qEw

mul_SNo_SNoL_interpolate_impred

Theorem 28.103

∀xy.SNo x→SNo y→∀u ∈SNoL (x ∗ y).∀p :o.

(∀v ∈ SNoL x.∀w ∈ SNoL y.u + v ∗ w ≤ v ∗ y + x ∗ w → p)→

(∀v ∈ SNoR x.∀w ∈ SNoR y.u + v ∗ w ≤ v ∗ y + x ∗ w → p)→p. The proposition is identified by the following information:

Pure Prop Id: 395b74682e5fe85f5a9dd02048bafdd0dde83a774140d2e273cde97e801116d8
Pure Prop Address: TMLLQmqPzFYwqdwY8PrqVGLuvrDLR95qgbq
Theory Prop Id: ad6f960961622055682865532b9e4b9576d116539cdde8aee2b4846f9ef3628d
Theory Prop Address: TMUjL5DBmrfFRCEH1KtPfFFFPNQ8LHK6fES

mul_SNo_SNoR_interpolate

Theorem 28.104

∀xy.SNo x→SNo y→∀u ∈SNoR (x ∗ y).

(∃v ∈ SNoL x.∃w ∈ SNoR y.v ∗ y + x ∗ w ≤ u + v ∗ w)∨ (∃v ∈ SNoR x.∃w ∈ SNoL y.v ∗ y + x ∗ w ≤ u + v ∗ w).

The proposition is identified by the following information:

Pure Prop Id: e1594953c98071d2fbe62599ce28adefdbe6f718c6c29a167036427051a6c7ae
Pure Prop Address: TMb5PPn6ZNjq3UQXjdnTK8NbeGxGWefm1uY
Theory Prop Id: b67a3574b63d9737488eb186c785d77ae947f04968d568abb2eaa58e430ac851
Theory Prop Address: TMUGZi51yni6VZeAy2PCP7s6JRSFQbEGocS

28.1. SURREALARITHMETIC 425

mul_SNo_SNoR_interpolate_impred

Theorem 28.105

∀xy.SNo x→SNo y→∀u ∈SNoR (x ∗ y).∀p :o.

(∀v ∈ SNoL x.∀w ∈ SNoR y.v ∗ y + x ∗ w ≤ u + v ∗ w → p)→

(∀v ∈ SNoR x.∀w ∈ SNoL y.v ∗ y + x ∗ w ≤ u + v ∗ w → p)→p. The proposition is identified by the following information:

Pure Prop Id: e5cae53a35ffe5b399ffd05d7bd80447303ca0d61aa9feca19313ed6bd26663b
Pure Prop Address: TMYf3j6h6sSAvpn3bX7i2vn59bPSs8V1Lyj
Theory Prop Id: d1fff1afdd85cf834265b58f478397a8d6a7bb46decf4e5fa0e45dfaca997f73
Theory Prop Address: TMGsDwijjDwd8s7TJ4BtBpPj7TywGA7uPvh

mul_SNo_zeroR

Theorem 28.106 ∀x.SNo x → x ∗ 0 = 0. The proposition is identified by the following information:

Pure Prop Id: d279f406aa4b69f7009533d48bdd2cde0236df1ead29105c36976e3dc31e0eea
Pure Prop Address: TMbh4sw15c3nkGcdW9gjqchsqeGnE6CLJka
Theory Prop Id: a4f4a68e9bdee9674a6a2323e1cb9216f16fac58db1b650aa99363d451432e40
Theory Prop Address: TMX3UwQTHomopN15s51CQVuoHBm4nfFok7n

mul_SNo_oneR

Theorem 28.107 ∀x.SNo x → x ∗ 1 = x. The proposition is identified by the following information:

Pure Prop Id: 1d2f1d16bda75bb4c0ed0799701f296b11ddee1eb905b339cd6220b0faf7d94c
Pure Prop Address: TMPJYXDtPMCDcjFFQegpZzooLCy6nB6AnVt
Theory Prop Id: 5477a4ee853ab494ceac03d98e85ac4588ce7bdd59886ad227b471fce11f4d89
Theory Prop Address: TMKdX9N9egsSZcy8hkKrdhm8AietaRG3z8M

mul_SNo_com

Theorem 28.108 ∀xy.SNo x → SNo y → x ∗ y = y ∗ x. The proposition is identified by the following information:

Pure Prop Id: 21f7d80766165f191325b0834d9de7e7be3a0dd1d07fa89c1f7a6020d3653a22
Pure Prop Address: TMWdJKtXqZ53gSQC2BVHdRbKeugBVoakiVX
Theory Prop Id: 83d5fab1574f51f9c204a181c7dfc8be42d76f66ec97dbb3470cdfdc46326b24
Theory Prop Address: TMdg6XgWdnxnzK7BDdwoHBgq7caePWS9hV1

mul_SNo_minus_distr

426 CHAPTER 28. SURREAL NUMBERS II Theorem 28.109 ∀xy.SNo x → SNo y → (−x) ∗ y = −x ∗ y. The proposition is identified by the following information:

Pure Prop Id: 9beaa01938101baaa11d2398155e5373cc0ae1d7998293c91f378b289f8e122f
Pure Prop Address: TMRSczXWoW4GcLzauYywq4qjgTNRs8DVfKY
Theory Prop Id: 3d0cd1fa56b07fbeb5efae9a27e18be23a0f0160374b403e50685ba3228a818a
Theory Prop Address: TMXe2FHAwQLWs8w3e4pZKSq3Ay1HjVYmAo9

mul_SNo_distrR

Theorem 28.110

∀xyz.SNo x→SNo y→SNo z→(x + y) ∗ z = x ∗ z + y ∗ z.

The proposition is identified by the following information:

Pure Prop Id: 6a23e513510b3a7b5a3392d09ddfa8bf723376a9ef4220bd708cc688d619d3f6
Pure Prop Address: TMEse8J6Aed49wCopbm1rWGiRVunCrST9NN
Theory Prop Id: 5f038253af67c067b7090c811a60073e88d35c5f778f03a6091776774636ffef
Theory Prop Address: TMYL91kZXWG9iK5FrgphNrfwHqoEcLWbXTo

mul_SNo_distrL

Theorem 28.111

∀xyz.SNo x→SNo y→SNo z→x ∗ (y + z) = x ∗ y + x ∗ z.

The proposition is identified by the following information:

Pure Prop Id: 4e7eba7fc4815da2d007f42429b70d6437766813fda9ea365ce905b2c09884fa
Pure Prop Address: TMSEGG8Zev8b3WSFiMRZbgEmXP278T6o11U
Theory Prop Id: 2a483403dad7cd65bfa69608ac938c5560f66186e39f7cfeafc037b7cf113feb
Theory Prop Address: TMXPqQpttxR52gPP1rKLuLFPGP1wExdFugR

mul_SNo_assoc

Theorem 28.112

∀xyz.SNo x→SNo y→SNo z→x ∗ (y ∗ z) = (x ∗ y) ∗ z.

The proposition is identified by the following information:

Pure Prop Id: 71a504fd63097789a8be85f133188fe0e6cdb7c87cd67c75d8f488d15233773b
Pure Prop Address: TMbdjdgGdSBzHFduhV2xBDmPQxmKwQt1tuM
Theory Prop Id: d322c17c41c9815d6f808e736a3c46d7cf138d7fb9de4bfebcd6ce05e597e568
Theory Prop Address: TMUDFiVX4hFhFSKJNGre9DQ2rsnsdEBsmWG

mul_nat_mul_SNo

Theorem 28.113 ∀nm ∈ omega.mul_nat n m = n ∗ m. The proposition is identified by the following information:

Pure Prop Id: da39d396fdbb6dc3abd4d8b61394e2ae8cd83bfda154cafeaa7b0d1ee0288b3b
Pure Prop Address: TMQtAZAzTub1xFgkzJYBwEnD6qz9UzUMmJp
Theory Prop Id: eb633595c5ae47b0fcc4717134102599b1f7892572e59291ab05562597e9fb82
Theory Prop Address: TMaXHxX9CkqxrxU1kbFsYKxJbdBZWsWsgbd

mul_SNo_In_omega

Theorem 28.114 ∀nm ∈ omega.n ∗ m ∈ omega. The proposition is identified by the following information:

Pure Prop Id: 7b6023e5c720f7f5082b318b3189f8c277b1498a0c34be6e13516ce9bce0a8f1
Pure Prop Address: TMbHqwvzDZAcjviA88kYptiX95LF6c3pw8m
Theory Prop Id: 8435a4b44154c9d6a5498ab9f17973fc5f75b3b8098f3884189ec529e738ffc5
Theory Prop Address: TMLcnoowy4SKdgLinmXu8vq7b1VLk5mBvRB

Definition 28.4 We define diadic_open to be

λX.X ⊆ SNoS_omega∧

∀x ∈ X.∃n ∈ omega.∀y ∈ SNoS_omega.x + −eps_n < y → y < x + eps_n → y ∈ X

of type ι → o identified by the following information:

Pure Object Id: 16a206dc52b4bb55b4a6c92782a08f06f95014669558c00b452a0ba01b887b1c
Pure Object Address: TMXnCV5QN4u4VL3jUYWi1Q5TKvaYVFiyMAy
Theory Object Id: f8473685b008a719f4b168e65e6b1cd5168f93a2f789ae2758a850f2b196b7db
Theory Object Address: TMFiXJkdoLrmEDAbDLC8bw1WgubyoT6m1rW

diadic_open_I

Theorem 28.115

∀X ⊆SNoS_omega.

(∀x ∈ X.∃n ∈ omega.∀y ∈ SNoS_omega.x + −eps_n < y → y < x + eps_n → y ∈ X) →diadic_open X.

The proposition is identified by the following information:

Pure Prop Id: 690598e6fd4ea25bb12e3b584dc4b13fe6c60343f666f02f1641e57297d2d891
Pure Prop Address: TMc3CK4vFU9yhYePJb36m6beBT5HxUzg5Dz
Theory Prop Id: c974cede2afec12b64c094aa129dc1b1d1a692b8afe4e7954cb66d9d2f5c2b77
Theory Prop Address: TMHeah5uo3Soo2iroPYRB2ntKum4R7fb8uT

Definition 28.5 We define SNoL_omega to be λx.{y ∈ SNoS_omega|y < x} of type ι → ι identified by the following information:

Pure Object Id: 5f401f6d84aeff20334d10330efdc9bb3770d636d6c18d803bce11a26288a60d
Pure Object Address: TMS5SZr3PAStrSpayzuJN2wkhBQBnxtsCKv
Theory Object Id: 2e3a675cbb0cd13246446f4a4b5819f9d0cd3a852470bdf73b86e4c9fa5f8aac
Theory Object Address: TMTDoEUQNzUxLj7ehiQCYBDF5vvzDtYJBZ7

Definition 28.6 We define SNoR_omega to be λx.{y ∈ SNoS_omega|x < y} of type ι → ι identified by the following information:

Pure Object Id: d24599d38d9bd01c24454f5419fe61db8cc7570c68a81830565fbfa41321648f
Pure Object Address: TMcZ1fLri43RwrwbvUWagasgXfWF6BJHVSg
Theory Object Id: 255a0c77460ecd33846983d06bff6d5687f5c12703ee773bd7ac29ee80876aee
Theory Object Address: TMZoQpNUd5MbGrEDKDP95Eu98wX9TZSnriY

diadic_open_SNoL_omega_I

Theorem 28.116

∀z.SNo z→(∀x ∈ SNoL_omega z.∃n ∈ omega.x + eps_n < z)→

diadic_open (SNoL_omega z).

The proposition is identified by the following information:

Pure Prop Id: f034141f72d6c4eb35d80a2119f7efda32b3a2da588de397b25001b9c4cc72d9
Pure Prop Address: TMZo5wo73TruUc3HVgkyKRi1Lm1wR18VgRA
Theory Prop Id: 7fb6824bab5f2011e10de03e8eea40ef57b1d040bc5a7600c6e49a17598be7f1
Theory Prop Address: TMKVw1RPWUuG98CDGajHDtbjjBEGV18MFAE

diadic_open_SNoR_omega_I

Theorem 28.117

∀z.SNo z→(∀x ∈ SNoR_omega z.∃n ∈ omega.z < x + −eps_n)→

diadic_open (SNoR_omega z).

The proposition is identified by the following information:

Pure Prop Id: 76200c785beb8c7d3f1ca8ae3c4cf9b46875e0cfc933a9cb2834cf5dfa366c76
Pure Prop Address: TMVnsa7wcgr6WYawj3FNHQPu1WxbQwjWAQf
Theory Prop Id: 58c46af97ae7d614a76a0d68417ebdde3cae5408341ae09dd687c865366186f0
Theory Prop Address: TMbr7DWfJLuoLhotsiMYBsLoSh3C8or2K85

Definition 28.7 We define real to be

{x ∈ SNoS_(ordsucc omega)| SNoL_omega x ≠ 0∧

SNoR_omega x ≠ 0∧

diadic_open (SNoL_omega x)∧ diadic_open (SNoR_omega x)}

of type ι identified by the following information:

Pure Object Id: 1bd1ac4ee8852db6f99d372f1c27ef21bca430b1c9c4c4d184e017f6a4d8d1b4
Pure Object Address: TMKMHojzxJK5coFqxQsQtHCgmSHKPCW9Vm9
Theory Object Id: 87ab7c83c1184f9f51c4168a5ec6a5ea9da4a8e2acb1fc82e0f19a7afc1617c4
Theory Object Address: TMYohVoF9p6nAvkXe1FijCMbejrCeRTmmEN

real_I

Theorem 28.118

∀x ∈SNoS_(ordsucc omega).SNoL_omega x ≠ 0→

SNoR_omega x ≠ 0→diadic_open (SNoL_omega x)→

diadic_open (SNoR_omega x)→x ∈ real.

The proposition is identified by the following information:

Pure Prop Id: 09fffc7b9e5d0b1d7f15cd56209324363defbb3ebe9c272e83a66e2590b7bd24
Pure Prop Address: TMUWdsBKSGjUkuMEbeQNbUqTtaf4Jfjg27G
Theory Prop Id: 594245216a8c68c4e6457b47578cb5437b65d219679a5836cb77b5ff224d6c9a
Theory Prop Address: TMGSVE4T9JHVEamf5z5kbyL8PNooxb94NMt

real_E

Theorem 28.119

∀x ∈real.∀p :o.

(x ∈ SNoS_(ordsucc omega) → SNoL_omega x ≠ 0 → SNoR_omega x ≠ 0 → diadic_open (SNoL_omega x) → diadic_open (SNoR_omega x) → p) →p.

The proposition is identified by the following information:

Pure Prop Id: 44069e352f5036a86127073abddc29a0ac5cf4eb6f4d81d6ec3a0209c2755103
Pure Prop Address: TMHRqA5xCUMEW6aiM8SkmMPejfbXcp8AuW2
Theory Prop Id: e2acb3ab0b286d7acb20f49882f08eb34a7bbd287abb4ee34541635153010085
Theory Prop Address: TMbuQMxV1gs1YZpGC9dAoKbCjDcsEGPDtg5

Subq_real_SNoS_ordsucc_omega

Theorem 28.120 real ⊆ SNoS_(ordsucc omega). The proposition is identified by the following information:

Pure Prop Id: 78ab2f997523a88bf0249dfd2d26b12d3641ee1e1b29b0fe0b6ecbe2bcb9ba5e
Pure Prop Address: TMTqU7cPcHtqdsT73PkgeBYFon4NVibV1rP
Theory Prop Id: 2858a333c0f86c675cb7a2fd1e8d4029fc7b67d5a2e36fcb017031c2a22d4db9
Theory Prop Address: TMKo5QEXCMn8QagibutbBTzBZSUDAxobgfW

430 CHAPTER 28. SURREAL NUMBERS II

Subq_SNoS_omega_real

Theorem 28.121 ∀x ∈ SNoS_omega.x ∈ real. The proposition is identified by the following information:

Pure Prop Id: db92898b58c469e9ad58b8b3729b507e8cadb7c632fb9757b051ec3b47524b67
Pure Prop Address: TMJL6KqW8mXtGGFbki5ao4x1Pu9AbFw59T2
Theory Prop Id: 7315809dd9bd2ef3a3617333ce251afbd32d098f391ab09694db0d0a3ed52701
Theory Prop Address: TMYg3xroKjVdTtceUrNgH6Jq8ZdKuyS3gD1

SNoCutP_SNoL_SNoR_omega

Theorem 28.122

∀x.SNo x→SNoCutP (SNoL_omega x) (SNoR_omega x). The proposition is identified by the following information:

Pure Prop Id: 15b221ba0c4ae0e05ed8cc21a7cdf61ac684525faddd1c477ed29fb7cbd9ee7b
Pure Prop Address: TMQ4q89ASduA9c9ftASiNBHz8pQAiMs6fvH
Theory Prop Id: 354d7d3952bf1c176fc5b59a37d8d3fde5ecdce30261e4d7e68896d0d7a6e87a
Theory Prop Address: TMLgzevvcXAVWAzGL5wtnbtt749rGGQR3aV

SNoS_ordsucc_omega_SNoL_SNoR_omega

Theorem 28.123

∀x ∈SNoS_(ordsucc omega). x = SNoCut (SNoL_omega x) (SNoR_omega x).

The proposition is identified by the following information:

Pure Prop Id: 0e7b9203cc682cd4470f749df0d7773bb56f7d49172a681a31039c99f4c6190e
Pure Prop Address: TMHfAHcvWTSrs5RmhM3t5QLf3y1Ps4dbWMu
Theory Prop Id: 1df44e924ecdb42a015d2758e58cb9ae0cb4cf2f67ebce9729fc086bad0fbdd4
Theory Prop Address: TMHNAhMLb3WzEVoWuDPHTbhjyEcTzN9mZJx

real_SNoL_SNoR_omega

Theorem 28.124

∀x ∈real.x = SNoCut (SNoL_omega x) (SNoR_omega x). The proposition is identified by the following information:

Pure Prop Id: e3f60c2768e445b24135a40107035ad21c03de45130bcda31e71da16caa4c99c
Pure Prop Address: TMKK4pjw5QfzGoVXPooBhrwXjMz5y7zwC83
Theory Prop Id: 11dba32c06250858d053b157288593cb4e9cc3b5e1cac811216e1643ff3caecb
Theory Prop Address: TMad5zGLRZhwq8fTQPe55TZwskUMePg3kMb

28.1. SURREALARITHMETIC 431

real_ex_diad_Lt

Theorem 28.125 ∀x ∈ real.∃w ∈ SNoS_omega.w < x. The proposition is identified by the following information:

Pure Prop Id: 8a1a82ac356e457f7d6fea412cc602e7bf973ae5a3d1fe392acbd6183e111060
Pure Prop Address: TMRmEwRJSZMnW9FxQdrwZVkpY4X3hZtgEsF
Theory Prop Id: 95fe1677ab448626991245b1f6c72063712786eb5fc15e18c8ad861be5a10d24
Theory Prop Address: TMWnkcpDQgaZXznHwxbP51Zpm4HsdCijkgK

real_ex_diad_Gt

Theorem 28.126 ∀x ∈ real.∃z ∈ SNoS_omega.x < z. The proposition is identified by the following information:

Pure Prop Id: edbfff85c909fe0b9cc4f471ef0cb3f4b241c89ff5fdc0aac267048126f79d55
Pure Prop Address: TMWbbSgeg81TLqCRrR2wnZxFJM2VyEnhoKs
Theory Prop Id: 8b82781e914ba17ea6121ff32dd4355d61bc2d5e58cf93f4dfc19f6be51aa543
Theory Prop Address: TMVen1Z8YEbACtaXxe8EHzxX42Fe1HVxEn9

mul_SNo_pos_pos

Theorem 28.127 ∀xy.SNo x → SNo y → 0 < x → 0 < y → 0 < x ∗ y. The proposition is identified by the following information:

Pure Prop Id: fda2502eea941a875df72ec6015f1570649b018b98f8184cd3c875919ea4099f
Pure Prop Address: TMMBQ62TWFg4vbNhYgkxMEhq8L4wrwpNeH4
Theory Prop Id: d984dcdb63a86f7c983729aa4499bd351e39b1649325ebc42012cca4a4f9dda6
Theory Prop Address: TMRy9goqWBW3Lrva8p4Hh4yhTMAF2u9LrbY

mul_SNo_pos_neg

Theorem 28.128 ∀xy.SNo x → SNo y → 0 < x → y < 0 → x ∗ y < 0. The proposition is identified by the following information:

Pure Prop Id: 4dec1e722a076d06ad4666c4d9e70af174b210cfaebca7581e05f1d86d354df3
Pure Prop Address: TMPctCrMmwtqdsDYtwTpmW8cn8ReSGEXW1d
Theory Prop Id: 697e77b4240d0fbd5154af82b65814d654a79f225b48542bc4a3d1b906b094d7
Theory Prop Address: TMbmWHtMpjybVdPJPF2YcxQPAiySRhKRtU7

mul_SNo_neg_pos

Theorem 28.129 ∀xy.SNo x → SNo y → x < 0 → 0 < y → x ∗ y < 0. The proposition is identified by the following information:

Pure Prop Id: 6f7ff58caeaa194bad28d270130f11e1db31bb6ab731d5ea0051c78e8fe144e4
Pure Prop Address: TMHNbtt4BUkPSLXdqKhLABgy4oEcpaoGeQF
Theory Prop Id: 6e7adce5ea23f07343a9f2808aa1618cc2fef1e19abfb8217fb9154a7c79b8f6
Theory Prop Address: TMX5KmwY5bfi7Ub8rgxQwDCpNPeAkJVtZsN

432 CHAPTER 28. SURREAL NUMBERS II

mul_SNo_neg_neg

Theorem 28.130 ∀xy.SNo x → SNo y → x < 0 → y < 0 → 0 < x ∗ y. The proposition is identified by the following information:

Pure Prop Id: 86d604810bc69f81920bb16b33d0acc3bedea0cfb10d12b4458a7d1fafe8a9d9
Pure Prop Address: TMaUyk4nPJDDrgihhHeRFKvQ2fABVtnDW2a
Theory Prop Id: 03517a8b5c20769332200cc5606cad376758ee52bb6086d3523cb789f19cee77
Theory Prop Address: TMZsT4fPodxbAqp3EhReFUneq4rHkVaHEZS

mul_SNo_nonzero

Theorem 28.131 ∀xy.SNo x → SNo y → x ≠ 0 → y ≠ 0 → x ∗ y ≠ 0. The proposition is identified by the following information:

Pure Prop Id: 62b0712aafda95c0ed43facad05fea37b6fb484147b8453d916f90fe017d591f
Pure Prop Address: TMGU71yz2XNvrMWpSrZ2VaZAXfAZoXYkpsg
Theory Prop Id: edd14a5db38e09a96214c426f3ca626167e570af71109dca57dc26ce0cdfae63
Theory Prop Address: TMH1iV5T59LgjrgGemBxMfYGnwBeTHPSpxr

minus_SNo_restr_SNo

Theorem 28.132

∀x.SNo x→∀α ∈SNoLev x.
(−x) ∩ SNoElts_α = −(x ∩ SNoElts_α).

The proposition is identified by the following information:

Pure Prop Id: 6e2d93e0dc17abea886b929fe5995e073d3fdcf446b9f988050524e8d71bd48f
Pure Prop Address: TMG933hdiaf584T7p9ZFRvPn9oiCEWieGQW
Theory Prop Id: 33a9b7427401c079a1d2742931e0769dcffafc114bd1bf1548f1b32dbd6c8dc6
Theory Prop Address: TMYmCxYkd92jjewaiC7sVwLTus2QgwYLten

minus_SNo_exactly1of2

Theorem 28.133

∀x.SNo x→∀α ∈SNoLev x.exactly1of2 (α ∈ x) (α ∈ −x).

The proposition is identified by the following information:

Pure Prop Id: 8867febe718634f83190f3f8c3dab70b7c761d564ddd1acb9cadda95bf892f1b
Pure Prop Address: TMbbMHfSdx97M7MMua7mnYgvKSs2grfdonk
Theory Prop Id: c2e4600c71315f1fd85398873234adada26e385afa312ce802aceb4cd7e1c2b3
Theory Prop Address: TMZRH9eUQQW9BQLZhXvbWDnscfgzh7sfosN

minus_SNo_In

Theorem 28.134 ∀x.SNo x → ∀α ∈ SNoLev x.α ∈ x → α ∉ −x. The proposition is identified by the following information:

Pure Prop Id: 634ffccfb2da872d1912604afc7221a4c2f97936c287ab1c4253071e1dfcfc61
Pure Prop Address: TMRJdHvUjKuqbmun53qwo4QgnKCJpLgQbyt
Theory Prop Id: 6ad85181a4e76fe3485b7a68f6c450251768534fcb5c6074cb42e2acf9661987
Theory Prop Address: TMHfvPTQWHBMA7BghiU2NwAbmLn8jDpkF9g

minus_SNo_nIn

Theorem 28.135 ∀x.SNo x → ∀α ∈ SNoLev x.α ∉ x → α ∈ −x. The proposition is identified by the following information:

Pure Prop Id: 42b82e1e669aebbf602bf5113f1d414db866cfe089bda3a8f91d980976cd7959
Pure Prop Address: TMcrVxUeXZKH9hKHxpogFheX7RW2QbWzvcH
Theory Prop Id: c24e982ace7dff228ddfbb28d50b4cddc7e5842b4c70d4190e10a022c64bebc0
Theory Prop Address: TMbXvwL69SpXNMdrP7cnTzE8ykxUg2LWeVZ

real_minus_SNo

Theorem 28.136 ∀x ∈ real.−x ∈ real. The proposition is identified by the following information:

Pure Prop Id: 5ba504335f94f103b4e4f4b4128f81159e7d57c6f6e51ddf1aac116c6bc3d844
Pure Prop Address: TMRu8g6RVBXBFKY32BG4WQRsE2JJUCAdD8A
Theory Prop Id: 54413c793a2a6cf0fcb3da3a35724c098c7e8b441474a0aac138fbd28ee3cb5c
Theory Prop Address: TMJ5SdaCycBTmAK8QT7hmjqUxcKbkFRkZSH

Definition 28.8 We define div_SNo to be

λxy.if y = 0 then 0 else Eps_i (λz.SNo z ∧ z ∗ y = x)

of type ι → ι → ι identified by the following information:

Pure Object Id: 16510b6e91dc8f8934f05b3810d2b54c286c5652cf26501797ea52c33990fa93
Pure Object Address: TMJtKhWrCvxZo4EySoUGxRLHGe8kxsrf5wV
Theory Object Id: 41b026a24f3f123a0f6f2e71ea4d39925a2b48bf6dedd1c902de3b86264323f1
Theory Object Address: TMXu1QtbM1e9R4bcCcqrQqVE2nbX4D1QdbG

Definition 28.9 We define exp_SNo_nat to be

λnm : ι.nat_primrec 1 (λ_r.n ∗ r) m

of type ι → ι → ι identified by the following information:

Pure Object Id: cc51438984361070fa0036749984849f690f86f00488651aabd635e92983c745
Pure Object Address: TMHssmmer5vtejxULVomtdegStRPMUKFbiM
Theory Object Id: ea81040eaa779da7e35317f42e478d807fb38e0dd71814599673b64763a57fba
Theory Object Address: TMPrqwjmsGkTo9k9ojpFo3twgRABMuqrPhD

Notation. We use superscripts as notation corresponding to applying term exp_SNo_nat.

434 CHAPTER 28. SURREAL NUMBERS II

28.2 Complex Surreals

Definition 28.10 We define CSNo to be

λz.∃x.SNo x ∧ ∃y.SNo y ∧ z = SNo_pair x y

of type ι → o identified by the following information:

Pure Object Id: c35281fa7c11775a593d536c7fec2695f764921632445fa772f3a2a45bdf4257
Pure Object Address: TMUD5RtiGmKRpxwvJihKA6DLDFMXAMhjUgo
Theory Object Id: 9f6440ba5a12befd32f7a9856b6769fa208407cf0c9b4747cb6036bb0d6250a3
Theory Object Address: TMLizygxxxRc6MNrrbecR4regbF9SnDnoL1

CSNo_I

Theorem 28.137

∀xy.SNo x→SNo y→CSNo (SNo_pair x y).

The proposition is identified by the following information:

Pure Prop Id: a092484c55b8291c75d0d36b92ada8b7272277bdbed2f8e85dd4647982d82a11
Pure Prop Address: TMLuqWx8Kn7Zr4K7kCY3vBtreAy6N2Gq5Ho
Theory Prop Id: b22a0800c64b20ca0a19ed9cf3cf1103cba24b4506c039ffdc78edc941d716bc
Theory Prop Address: TMSb6wFZKXobEGqJEzsAV83ZDCU7SEHmi81

CSNo_E

Theorem 28.138

∀z.CSNo z→∀p :ι → o.

(∀xy.SNo x → SNo y → z = SNo_pair x y → p (SNo_pair x y))→p z. The proposition is identified by the following information:

Pure Prop Id: d92da0243181c2c853df14146584be693f357edce1fe0d6dc91b3a43175cd9a5
Pure Prop Address: TMF2YxYA3n45UKAAPKZdwQUZ5pkopFy84Zo
Theory Prop Id: fdbc8d36959b5172de4f1d46ecd65e45290e18a238b0d3274dd0238e86f92e5a
Theory Prop Address: TMREupmRGKvMjoTNRuLkffjgfQYrtqyaxmd

SNo_CSNo

Theorem 28.139 ∀x.SNo x → CSNo x. The proposition is identified by the following information:

Pure Prop Id: 56121e05e33db2f68607348644f875d4c59e36314e4c7aade2ed7acf3bc097ea
Pure Prop Address: TMKSgGTcxX9dqXoEQ6Z8PkWivenDVtKxQTL
Theory Prop Id: d50db589d56adb9c8fd5479df494050c31aaa3429ec352b511a7b0a13aeea945
Theory Prop Address: TMR4DcyLohCpAbSXv9WH4Xcg5MLahA6HKCW

28.3. COMPLEX 435

28.3 Complex

Definition 28.11 We define Complex_i to be SNo_pair 0 1 of type ι identified by the following information:

Pure Object Id: d0c55cfc8a943f26e3abfa84ecab85911a27c5b5714cd32dcda81d104eb92c6e
Pure Object Address: TMRLLQsBvKWihFburT51dAV1GsLtPYwc9Zk
Theory Object Id: df0e4e103bfa2441c4757b4d5b0e56da78eeec142d6eb59cb57bf6f96d041b2e
Theory Object Address: TMWUiSAowU8u6wDqUE8abxMdTuajU7UtstH

Let i be Complex_i.

SNo_Complex_i

Theorem 28.140 CSNo i. The proposition is identified by the following information:

Pure Prop Id: 0b8add253d6a140458980890174b902430843878d8fbe0e4e45711dca51045e1
Pure Prop Address: TMTBCkaSu5M84GaWqNVMA2i4e55BjcK87XM
Theory Prop Id: 139a531c852859624509b40bb3d55cf6dc1c47b1c91e4e56557dd5eca9352643
Theory Prop Address: TMMieXaSc5pjuKyowxbkxfTqbz2F6SwKDZS

Definition 28.12 We define CSNo_Re to be

λz.Eps_i (λx.SNo x ∧ ∃y.SNo y ∧ z = SNo_pair x y)

of type ι → ι identified by the following information:

Pure Object Id: 9481cf9deb6efcbb12eccc74f82acf453997c8e75adb5cd83311956bcc85d828
Pure Object Address: TMbtVDb4DspXgdVe6YzPV3kDEEEwHY6ZCot
Theory Object Id: 78e2fa14ee2f6baf7a5e87660241882e064e475595b0b4d3abb32b22e16d4844
Theory Object Address: TMcoCaQ7XUy1bDRwyD8B1Hc9XvedDZEaj9Z

Definition 28.13 We define CSNo_Im to be

λz.Eps_i (λy.SNo y ∧ z = SNo_pair (CSNo_Re z) y)

of type ι → ι identified by the following information:

Pure Object Id: 5dad3f55c3f3177e2d18188b94536551b7bfd38a80850f4314ba8abb3fd78138
Pure Object Address: TMZBTXQFnkALw8eLrVDkBW9kDUBUNtJ9KV1
Theory Object Id: 995ec6771de10483744c42c2860b06ba0be54c9d0335cccb1a962e90378575b5
Theory Object Address: TMFEq3qfdWzjBymvGvoarLmQpoH3fgwe9yD

Let Re : ι → ι be CSNo_Re. Let Im : ι → ι be CSNo_Im. Let pa : ι → ι → ι be SNo_pair.

CSNo_Re1

436 CHAPTER 28. SURREAL NUMBERS II Theorem 28.141

∀z.CSNo z→SNo (Re z) ∧ ∃y.SNo y ∧ z = pa (Re z) y.

The proposition is identified by the following information:

Pure Prop Id: d1afc4d126dd465be260002c44748dfd6b52c42ef68b99e2aca9d39589415eff
Pure Prop Address: TMVD9LSTuGahX5mYEQHbkD3Aj4r6YjvDqav
Theory Prop Id: dda4fdb9868a65730eda625dcc1b25d74a368899640ab43b19a56dee292fd951
Theory Prop Address: TMdw63sz9FuEG3u34nTUC9xRWJSnjKoK5iM

CSNo_Re2

Theorem 28.142 ∀xy.SNo x → SNo y → Re (pa x y) = x. The proposition is identified by the following information:

Pure Prop Id: b2daba3d81b7c6ff9fbeddbdc673af753aea0654c6f0beb2e4150cec6aa3ff36
Pure Prop Address: TMbd79jCDur1j41vTRXKVgXL6tDG3dczCch
Theory Prop Id: c4a4431b7dc74e1c59f2d3f78882e9511898bb365f4441eb504d35a102c73f5d
Theory Prop Address: TMTPxuYePSpdrjC2hZR9YA4mNhhCM2obwvy

CSNo_Im1

Theorem 28.143

∀z.CSNo z→SNo (Im z) ∧ z = pa (Re z) (Im z).

The proposition is identified by the following information:

Pure Prop Id: 74e0f37f75ccb3555e54f4af34c332761bbf40f68215121dd3426a5fc1d57563
Pure Prop Address: TMbMFKS5usWyvTh28krP3yJ58GAo85ggE5u
Theory Prop Id: f920bd6c9ae2ef46e0da84205614a41dca1aa86484bc96ed97046e5cb2dcc706
Theory Prop Address: TMSHmgJJG9JSYax4SjR2fRxfC1xbTdvaFHp

CSNo_Im2

Theorem 28.144 ∀xy.SNo x → SNo y → Im (pa x y) = y. The proposition is identified by the following information:

Pure Prop Id: 2bc6171b4c1d4c23d2b8257ee568deebce1bdeeb5abadafad24f46402844bec8
Pure Prop Address: TMPfMNHVuYrdTndLNgbw7ssKmmTUSmVQhuq
Theory Prop Id: 8be49e5c0e40a6664121bff48d6f103b94ded1a85e8276f4270b2fecc4365f2d
Theory Prop Address: TMUySj1Kdt4rWdt8CvGuPiFauFwdwqGuBrt

CSNo_ReR

Theorem 28.145 ∀z.CSNo z → SNo (Re z). The proposition is identified by the following information:

Pure Prop Id: 2d8bafb4716aba45ba78940b437d476181b1832c88e0fdf73d1f44139d745aa6
Pure Prop Address: TMHWBdpBEx7e2XaqsRv4PexZFWK6mxntB5L
Theory Prop Id: 9eab1faa688e3e8a126798d1a4feec2d230170ca2c79e1f6a7cac4413ce087e6
Theory Prop Address: TMLjD3HKc5VtY2zV8JTsHHEn8qrJqupKsD9

CSNo_ImR

Theorem 28.146 ∀z.CSNo z → SNo (Im z). The proposition is identified by the following information:

Pure Prop Id: ab80bc51455c8401ea6cfc9fb8b50fce5202d296b70813e18ab3b4578724d01b
Pure Prop Address: TMHM4TWxDfhFey1F4K4ZfTkNgJ3aXPsHVnx
Theory Prop Id: f91d5d3724c3d6ffc90c6787c4369da0ef974adfbd2841122b8eea32364c3543
Theory Prop Address: TMaeNJLGGokLzoVMSdnkmyBJB7tx3aSEAk3

CSNo_ReIm

Theorem 28.147 ∀z.CSNo z → z = pa (Re z) (Im z). The proposition is identified by the following information:

Pure Prop Id: 445a11e5d038263fe95c6356046a0e1e52288c7a48a5d5c22900215dc15818bf
Pure Prop Address: TMbfkg4MdezPBN2Y4Qjkj5qcWqmWQ3YK3MM
Theory Prop Id: 520c8f693aef23e47d75aa0230b2dc952db5beee2eec2007112e6239da337856
Theory Prop Address: TMPQ5WLn58Hodm558JuPKepWo3qKYLgWAfo

CSNo_ReIm_split

Theorem 28.148

∀zw.CSNo z→CSNo w→Re z = Re w→Im z = Im w→z = w.

The proposition is identified by the following information:

Pure Prop Id: 1b6f8a62f72e786e4c9bf9f10ef1b663021c7dd462f9001118924d0c3eb05d56
Pure Prop Address: TMF22319Si8r5cP9dtLLXWhca498DPYaPfW
Theory Prop Id: e59d00e026008d1f4bb97a82ee1e249a3a6533f81c95228512a948d73a5948a1
Theory Prop Address: TMQQTkXaTqRkMfBzratppu8SFd9kJhSw1K9

Notation. We use −− as a prefix operator corresponding to applying term minus_SNo. Notation. We use + as a right associative infix operator corresponding to applying term add_SNo. Notation. We use ∗ as a right associative infix operator corresponding to applying term mul_SNo.

Definition 28.14 We define minus_CSNo to be λz.pa (−Re z) (−Im z) of type ι → ι identified by the following information:

Pure Object Id: 9c138ddc19cc32bbd65aed7e98ca568d6cf11af8ab01e026a5986579061198b7
Pure Object Address: TMLB9tLVVqyKeYKxjvFwaspknDz9iJsXi8C
Theory Object Id: e44642b3fe572281d00e78cdd9551e412cd2f9c70b5057cc266234cae61b142c
Theory Object Address: TMcRv5uzwYDFcF46XsKgFNN7RdY9tuZbeEK

Definition 28.15 We define add_CSNo to be λzw.pa (Re z + Re w) (Im z + Im w) of type ι → ι → ι identified by the following information:

Pure Object Id: 30acc532eaa669658d7b9166abf687ea3e2b7c588c03b36ba41be23d1c82e448
Pure Object Address: TMEg3b8dv5i7oFXmXe8scTrFUcM8PzJ4zpW
Theory Object Id: b095bc500f522f93df919a70a2f70782f5980aa96ce262ebe8358b56594a3c2f
Theory Object Address: TMaA7uRAA9rzybfgoRTMJhUqq2bqKoX4Wx4

Definition 28.16 We define mul_CSNo to be λzw.pa (Re z ∗ Re w + −(Im z ∗ Im w)) (Re z ∗ Im w + Im z ∗ Re w) of type ι → ι → ι identified by the following information:

Pure Object Id: e40da52d94418bf12fcea785e96229c7cfb23420a48e78383b914917ad3fa626
Pure Object Address: TMG6o42Tv7okyH5PRhCzCahxTbhTrJK9kr3
Theory Object Id: 9567b4289023a19d5a345a5792fabad48f116c4488984243ea731c878bfe55aa
Theory Object Address: TMKcJrsKgvdh8ShstdBRhZw6Tr7gLxdNL7q

Definition 28.17 We define div_CSNo to be

λxy.if y = 0 then 0 else Eps_i (λz.CSNo z ∧ mul_CSNo z y = x)

of type ι → ι → ι identified by the following information:

Pure Object Id: 98e51ea2719a0029e0eac81c75004e4edc85a0575ad3f06c9d547c11b354628c
Pure Object Address: TMcjyPavg5d3bDQmhU1GhdJgnZiMVguCn1Q
Theory Object Id: 144bb6125865eeb3fbeec27048fa8deb33d515116b6f667efc6200b817c7e681
Theory Object Address: TMTPDUmndBUSQQFuYNMpTVmCmdY9FA3BZtX

CSNo_minus_CSNo

Theorem 28.149 ∀z.CSNo z → CSNo (minus_CSNo z). The proposition is identified by the following information:

Pure Prop Id: 7ee6ad0a8aa89406d6b84b29278fb1765127152d723155b4a1c9d6f6c0fea591
Pure Prop Address: TMKJwYdVTJY7k1e8hMuzVYgoKvGEEEBByvU
Theory Prop Id: 309c1df653ed60931ce0009531575a033315729a35f70a72363bc4dc5a0b3d22
Theory Prop Address: TMFr3QPLTA3mRgec65aLawX7v5PhREet73h

SNo_Re

Theorem 28.150 ∀x.SNo x → Re x = x. The proposition is identified by the following information:

Pure Prop Id: 62c082375447ec523dbbb2ba00907a038d5e29d38b336bb7ebcfe5b9350e5b40
Pure Prop Address: TMUR6cn1xni6GYWDsvMbvTJXUokaMMtL4U4
Theory Prop Id: b4d75e236f52950d7d4585f33566679757107a6a43067a557b1b954b9c81d095
Theory Prop Address: TMcerjrejE4PH4juW3mMCPrs95Q3KqvPrwh

SNo_Im

Theorem 28.151 ∀x.SNo x → Im x = 0. The proposition is identified by the following information:

Pure Prop Id: 79fa36b0f877aa0ebc7f2b10cd692fd78dbc9d6316fc4e53d0c61bab998ecf30
Pure Prop Address: TMZ4xEKBB6n896d4JKvevEKTFR67E7xpLGS
Theory Prop Id: 045ef70364115765ca4038bb6cadac8b50b511da827132c78a50ec9ae776e935
Theory Prop Address: TMGyhJYv437WcR94jiMRHMwX2AwFbxWRWtp

Re_0

Theorem 28.152 Re 0 = 0. The proposition is identified by the following information:

Pure Prop Id: b67351949a46cdf795a5f976b27f975352929b68d5d5087280de8114752abcf2
Pure Prop Address: TMU3hZhd9p6dUuXdcCT5sFV8f9rU1SLuLvg
Theory Prop Id: 4d3237f583747a05847601b9321cb843fa14501467c9f468a5a30405685ab1cf
Theory Prop Address: TMPVj3k4kQSu82UN1JnuwzTMbxZYQkycJGG

Im_0

Theorem 28.153 Im 0 = 0. The proposition is identified by the following information:

Pure Prop Id: 9c8b687bb8d2707b766077cc814cd6418fa32bda09f73340d3b7538ba73166a2
Pure Prop Address: TMQ5sK61iFZnh5NvkYvBS6MNzDSzr4Q2mQx
Theory Prop Id: c115486aac15de92b350f5674a5898e90fd8a9590b7695815ace4de38d425d98
Theory Prop Address: TMR2c3XQQEW8f7iwU4myNJae1rZMRsRSBpo

Re_1

Theorem 28.154 Re 1 = 1. The proposition is identified by the following information:

Pure Prop Id: 88a2aa81be9b1d2f724a88816564d021c38a577fdbf421180a4c9a6b8d80afc7
Pure Prop Address: TMdGDE8JarQ1jbtW4A8oVRGior9g3VHcxwo
Theory Prop Id: a28b943c82a02609fdf4f3d340fd99c6f99f7fd4e1f24cf07c819165b82ba877
Theory Prop Address: TMR387PaXEC5B2kz1ZyEUXJc9eNwtkfqgH2

Im_1

Theorem 28.155 Im 1 = 0. The proposition is identified by the following information:

Pure Prop Id: 3b8fb5faff6bf916e912016e794c0366b38e4279d8fe1cdeeee5aa0afc03878d
Pure Prop Address: TMZBLCABN62himZDekvtrQPidyv7fcLegSk
Theory Prop Id: 1ca93cd098ccdc1ee3d42ee3ff16671c864fc5e68c4548d64594dc3f72490d3f
Theory Prop Address: TMXR9fNVoFqUzmgKeV3ME2ptYRiTSH3E7yW

Re_i

Theorem 28.156 Re i = 0. The proposition is identified by the following information:

Pure Prop Id: 8f12f0decf716ca445164b75da58be3f2a7555cb429371632eb027a71508c1d1
Pure Prop Address: TMK96CbwKZMULThVCgvnQBDjGHq6LXfbzfA
Theory Prop Id: bfd076483d6cacc95ee0f7ff33c3ceda810e4fef403a422c0ecef2104636ffdf
Theory Prop Address: TMWVmowUnC9iG9jmJkW8uaTKvWjDDPGhJaq

Im_i

Theorem 28.157 Im i = 1. The proposition is identified by the following information:

Pure Prop Id: 1795ca8f39eb6b9aa0312ca4743f676b7dad188738ed6523cb5b0e265558330b
Pure Prop Address: TMUhvnCxhioSCB7KmHbHrUrhDXFyjVzSfy7
Theory Prop Id: 6a700bf595245199596fb74a524dc20a87be2fa234d75a9425a486afde75e739
Theory Prop Address: TMV23ugJNe6xxnjymVV5jwq448H5U6cUMNz

add_SNo_add_CSNo

Theorem 28.158 ∀xy.SNo x → SNo y → x + y = add_CSNo x y. The proposition is identified by the following information:

Pure Prop Id: 7f45ee359a31d432da4768def1fc35bea04f7baa2682362d567be00cc83e019c
Pure Prop Address: TMPEXTCLw1a9KLNu4LjTEYeAiXpBwtm8LYN
Theory Prop Id: a9fb7d4440806134531895d72ae0d8c65453fa88d1b4a12beeccef25e3784bcc
Theory Prop Address: TMYERyzMgccEbP9urACTzpKQZ6NqHoJHCpt

CSNo_add_CSNo

Theorem 28.159

∀zw.CSNo z→CSNo w→CSNo (add_CSNo z w).

The proposition is identified by the following information:

Pure Prop Id: e57a7792a9adb12eaa7d6f4b276a0d7c46de087f9e650c63fa726890a32a5ea5
Pure Prop Address: TMTfa7wPMhSgYfANj3ojEMJAqmSXEbR5wUi
Theory Prop Id: 9f039b5cd4b098a77ad130de4441f49110a2a2055cf0ea279af6db8406a33f15
Theory Prop Address: TMRZouHFioSkxKnrxrt5ocqpagQGyQhPBxr

add_CSNo_0L

Theorem 28.160 ∀z.CSNo z → add_CSNo 0 z = z. The proposition is identified by the following information:

Pure Prop Id: e96f34c9cab6bff68069bd60d532d81645e33b9518db1f62bf0603eb2cb2b881
Pure Prop Address: TMVPvy3qhQSp4Z8EZHLGefCshoKUigGYBXD
Theory Prop Id: 9130336e1f65fce4abcbac21206949f4a80b64433ce8b55a026e39164bbe17ae
Theory Prop Address: TMJn8298DRnHjSHEgaMHnaaNn1aon8VyNao

28.3. COMPLEX 441

add_CSNo_0R

Theorem 28.161 ∀z.CSNo z → add_CSNo z 0 = z. The proposition is identified by the following information:

Pure Prop Id: 013f9d9fcb6813d280a54b0c238a0259bbf245f8c19e895056f52e4c3adeb31e
Pure Prop Address: TMMjUSCaFbW2xfSEQbTD6ikQsQfwYk4GFkA
Theory Prop Id: c676dd1073460165d7f272c397af95f64013350b90182b6212b134b88768602a
Theory Prop Address: TMSwnX5gYRWnQPynxcvBXkQVYgYNF2643Z5

add_CSNo_minus_CSNo_linv

Theorem 28.162

∀z.CSNo z→add_CSNo (minus_CSNo z) z = 0.

The proposition is identified by the following information:

Pure Prop Id: 87ef9663f32d233b6c1ad44f942d450a76582c2e2ac1a28a691747d6debc0bb4
Pure Prop Address: TMU75d85SiBY1nyEa7XDpioTuTb27YfaBGF
Theory Prop Id: 08935706a424eeef4a646b98c7dc667515cfe98f6a4e99488cf85e717d7bf96d
Theory Prop Address: TMWn3sP7p5YyoTSSYF8fydsrBPLkb9tjRKH

add_CSNo_minus_CSNo_rinv

Theorem 28.163

∀z.CSNo z→add_CSNo z (minus_CSNo z) = 0.

The proposition is identified by the following information:

Pure Prop Id: 350994f849f5d91eeb37d84462edfab4a847176479ae191377608bb65c929618
Pure Prop Address: TMRirTWCrP5cVCz5MNiHfJ1PfNDmQMnbMjY
Theory Prop Id: 268ed1aa6845aff2a06bc069ee87b3e46e06fa4ca30e8fe25c18f5b8541b8d69
Theory Prop Address: TMH6YAu7Qh6P2UjC47FxTcP6vhxmhZgf3V8

minus_SNo_minus_CSNo

Theorem 28.164 ∀x.SNo x → −x = minus_CSNo x. The proposition is identified by the following information:

Pure Prop Id: 95399dcec725f3eaf0d48ead791b18326af59f257c823337fdccdeb08631a78c
Pure Prop Address: TMc7tz6FA1Zoz96tsmZ9t1mi6UymeB4f5gM
Theory Prop Id: 23ad035b79e39c7c63924d15360f357008dcb6ef5d26f79c98ed7ee083b32498
Theory Prop Address: TMYnJa2FrK92ftPPxTYDAiWjQqyzmtH1euw

28.4 Complex II

Notation. We use −− as a prefix operator corresponding to applying term minus_CSNo. Notation. We use + as a right associative infix operator corresponding to applying term add_CSNo. Notation. We use ∗ as a right associative infix operator corresponding to applying term mul_CSNo. Notation. We use ÷ as an infix operator corresponding to applying term div_CSNo.

Definition 28.18 We define int to be omega ∪ {−n|n ∈ omega} of type ι identified by the following information:

Pure Object Id: 4daffb669546d65312481b5f945330815f8f5c460c7278516e497b08a82751e5
Pure Object Address: TMTGgpvNpXMeoR3NwZFd8y2xpBzTUSKgwbU
Theory Object Id: 50d206e99485ce72d7138ea33476258564bafcdd8979a0e0edf31a56964bb534
Theory Object Address: TMYfYSNfpYrjF2bVUtReGwX4aiKctLkRVZM

Definition 28.19 We define rational to be

ReplSep2 int (λ_.omega) (λnum den.den ≠ 0) (λnum den.num ÷ den)

of type ι identified by the following information:

Pure Object Id: 604584444cba0be7f74a050669bc3716573bcc6798d790d2a2e1d8203a2fd4d1
Pure Object Address: TMHgCpKhfWk6RuSojtz9kE5WKRRiXLgKQKG
Theory Object Id: cab0d274c7daec544f49542150159453f1d3ebe31677ddce588712380bdb3ffb
Theory Object Address: TMJ8qVVQVeuLH1QKNaSQ3YvEGoSB176XS43

Definition 28.20 We define Sum to be

λmnf.nat_primrec 0 (λkr.if k ∈ m then 0 else f k + r) (ordsucc n)

of type ι → ι → (ι → ι) → ι identified by the following information:

Pure Object Id: d8ee26bf5548eea4c1130fe0542525ede828afda0cef2c9286b7a91d8bb5a9bd
Pure Object Address: TMFy5sQDCD3u6eVG9houLmF2KaGQyzqJsCp
Theory Object Id: 81ddb08445e3612684413209359bef93d8b0cf5c8394caeca364d20434961abd
Theory Object Address: TMcbn2wMDtNVYfjf2vBVxsgPGPuUtbFsxBs

Definition 28.21 We define Prod to be

λmnf.nat_primrec 1 (λkr.if k ∈ m then 1 else f k ∗ r) (ordsucc n)

of type ι → ι → (ι → ι) → ι identified by the following information:

Pure Object Id: db3d34956afa98a36fc0bf6a163888112f6d0254d9942f5a36ed2a91a1223d71
Pure Object Address: TMdN8KB95hDDy7J7CZ25zug8XWiYBFRqyCp
Theory Object Id: da66d8ac9780f328f37f0b84d5376c156ffd8cf8b9eaf38731c6ac83d286cb0e
Theory Object Address: TMSX8CAGQArDkpR8V6Cb73nTcPCTkLUYXtA

28.5. INT 443

28.5 Int

Notation. We use + as a right associative infix operator corresponding to applying term add_SNo. Notation. We use ∗ as a right associative infix operator corresponding to applying term mul_SNo. Notation. We use −− as a prefix operator corresponding to applying term minus_SNo. Notation. We use ÷ as an infix operator corresponding to applying term div_SNo.

int_SNo_cases

Theorem 28.165

∀p :ι → o.(∀n ∈ omega.p n)→(∀n ∈ omega.p (−n))→∀x ∈int.p x.

The proposition is identified by the following information:

Pure Prop Id: 2504c05a08587fe0873ed45685efc417625f0a904281d653d757d643896f9a70
Pure Prop Address: TMUEfaEhnmjqzhHDct4aN4nYsgTMNHexc5Z
Theory Prop Id: a32835e1d8cbee7ebb59bff60b86d8d343a78a5a5b0a1a988fe8ae0752f9b0a9
Theory Prop Address: TMcPpJsLS8zQSHciyZymBNG61rHAnJoGbht

Subq_omega_int

Theorem 28.166 omega ⊆ int. The proposition is identified by the following information:

Pure Prop Id: c213ff287d87049b1e6a47a232f87c366800922741a9eeadb1d3ac2fbadaf052
Pure Prop Address: TMbZKRVRA2o1bkPt9NF2BWMQLKKJzDaq77o
Theory Prop Id: c3d99d24216bf364de569b6a9cd1de13b46ceb52421e78a517df07524d1308a5
Theory Prop Address: TMJyopLK6GuuCWqhP1XKASZh9iSwtdLiMs2

int_minus_SNo_omega

Theorem 28.167 ∀n ∈ omega.−n ∈ int. The proposition is identified by the following information:

Pure Prop Id: a66fb27a7b2af57722c6537d3983b55a12cc28475f1d8b8d9bdb1d857010e7af
Pure Prop Address: TMHaC3837eh4SUn1jL2MVZSY2HYJVykA2q8
Theory Prop Id: 1b4d50647e615e454ab958cf206bb3f4b349286064ed07e205cde10ee49537b0
Theory Prop Address: TMYLKPRR9oEXo9N8Z2YBDanBt7x7VLvfePj

int_minus_SNo

Theorem 28.168 ∀x ∈ int.−x ∈ int. The proposition is identified by the following information:

444 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: 66d7a7b7f8768657be1ea35e52473cc5e1846e635153a280e3783a8275062773
Pure Prop Address: TMSXiWmTFpmbC8vtTafxAwFeYuNrgZJj4vW
Theory Prop Id: daaadd58a6cfb975a0d9200d19c663526ca0a4951284ec16a5e11ae2e3fd7492
Theory Prop Address: TMV6sHDGtUc6XzQK9XLPBGFqCcMjQdPpFHz

int_add_SNo_lem

Theorem 28.169 ∀n ∈ omega.∀m.nat_p m → −n + m ∈ int. The proposition is identified by the following information:

Pure Prop Id: 6c976be5ae7c4eec61e1190f4b65a1cc39ebfb81542ae63578b69be42c01a06a
Pure Prop Address: TMLvPoJ9aN37XYyR84rMi7xqr8n8tsuhtfM
Theory Prop Id: 6fbb03f9ad1cac7ae41730f8a527dac94e9c83e918287c89f3787c4a3faac1a5
Theory Prop Address: TMPwH3A6N2tb87tNY5vbkqqydAuD5QR3EyF

int_add_SNo

Theorem 28.170 ∀xy ∈ int.x + y ∈ int. The proposition is identified by the following information:

Pure Prop Id: 02609f82bf442d61fb8c6410818e7e4cbf8c67c9ea08bffcbf8a77c06b0f784a
Pure Prop Address: TMFDVQawMT1qNmvZaWsWLAFRwNsZDvdEWd7
Theory Prop Id: a0aa5766d0cdd5590d613a2f72f025acfa458bc0cda8293f060de4d24de253c5
Theory Prop Address: TMag1esGeGDfnX5JdbJHct4N6L993dNSRki

int_mul_SNo

Theorem 28.171 ∀xy ∈ int.x ∗ y ∈ int. The proposition is identified by the following information:

Pure Prop Id: 3f000c087708670f2a9497d0587231f65beb85593b7ebdaf7909fe5e7d4b8c27
Pure Prop Address: TMbhP3nSug9dzpwnhRm7VfyD7KAzF5WfghL
Theory Prop Id: eefdfbfec3e66c305c9c9bc9c837268558851ac36fad12ee4b371f995afc9fd5
Theory Prop Address: TMbSE7nyU8X9azWYyGBkvTt8zRNBARSeYkH

Definition 28.22 We define divides_int to be

λmn.m ∈ int ∧ n ∈ int ∧ ∃k ∈ int.m ∗ k = n

of type ι → ι → o identified by the following information:

Pure Object Id: af5f6e65098725dbbf9014383760c67fe7961b5486f9b4fa4d3289ed424f0c1b
Pure Object Address: TMX9HJKhzi9AdyH6nXyJVdE4nKLJvyF9yNE
Theory Object Id: 5fa28bc7b02e3ca96f01706f3aa25b382b44df0d71c91f7a7c68aea0935e9273
Theory Object Address: TMT9KA9dVEU4Cq2ursSu7Ep2iNaWdDT8u12

Definition 28.23 We define equiv_int_mod to be

λmkn.m ∈ int ∧ k ∈ int ∧ n ∈ omega \ 1 ∧ divides_int (m + −k) n of type ι → ι → ι → o identified by the following information:

28.6. PACKING TWO OPERATIONS, A RELATION AND TWO CONSTANTS445

Pure Object Id: aea710807b7901288ae25daacdd1e52348dfbe79c9a58fabb5e91c68e15a1581
Pure Object Address: TMHGhKGHRdgbdvTgpBF85dWTWtVw1PHGZZM
Theory Object Id: a0bc48ffd10d4e9112ad9d98af3c6392a81e8ef3d019ce4b8ec86f77a5a49a7d
Theory Object Address: TMWqX7aoEj8tNfrfptrVaGaaFkknJoVdiPo

Definition 28.24 We define coprime_int to be λab.a ∈ int ∧ b ∈ int ∧ ∀x ∈ omega \ 1.divides_int x a → divides_int x b → x = 1 of type ι → ι → o identified by the following information:

Pure Object Id: 48ca77268e35362c65956d7fd4145d07598e7a91e7ef4a1e6af043bc1202b37f
Pure Object Address: TMRPbZHuWZc21V3zuKPZMGa165KDXe2rvuU
Theory Object Id: 76fc5b183d6bea5474424b2b6616dbb8c29591fb620aa364408cc43266fa15ec
Theory Object Address: TMXXHSe9mRiHSQ717podNGPQLawz24s1P1q

28.6 Packing Two Operations, a Relation and

Two Constants

Definition 28.25 pack_b_b_r_e_e is the opaque object of type

ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → o) → ι → ι → ι identified by the following information:

Pure Object Id: 8efb1973b4a9b292951aa9ca2922b7aa15d8db021bfada9c0f07fc9bb09b65fb
Pure Object Address: TMFRc8E8tg6JtPF2W7a5hcH7dKma4rhhjpe
Theory Object Id: a33bdffb44829c458189fcf9c1fc93f1abfb0007a9318b33fa18bab49932b69b
Theory Object Address: TMUoWdaNvqJBEvaP7AZFqh425UCXiK49xiT

pack_b_b_r_e_e_0_eq

Theorem 28.172

∀SX.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

S = pack_b_b_r_e_e X f g R c d→X = S 0.

The proposition is identified by the following information:

Pure Prop Id: f5a2203f23e7dd04fe8ca0e3fb972d0098d3812a6f2a49e5016460d7db993bf5
Pure Prop Address: TMRGip4djr41vccrTDQXNBbxCWrX8VbHcqJ
Theory Prop Id: 2c10ca5a1162991d07d3c553c21901d5abd7b71cb1ca8c024781096deed06657
Theory Prop Address: TMQjRgtSbPqLcPdbvF5gpcTyqr1VGnTtfgw

pack_b_b_r_e_e_0_eq2

446 CHAPTER 28. SURREAL NUMBERS II

Theorem 28.173

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

X = pack_b_b_r_e_e X f g R c d 0.

The proposition is identified by the following information:

Pure Prop Id: 7136b27641c7cc4e1a647a898e4ebf0836d4299f8da51ec9709fc246183a9280
Pure Prop Address: TMXU7ECTAkVR3MxsrCyo2KrDJ4eLHVp18Z2
Theory Prop Id: eb82f1296debb50aec472ad719325c4d415f74e68c24cbfeb6eb48e13b06fe27
Theory Prop Address: TMHkCAK4V2154ECdQyfXBNV3CMQ6datiFHq

pack_b_b_r_e_e_1_eq

Theorem 28.174

∀SX.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

S = pack_b_b_r_e_e X f g R c d→∀xy ∈X.

f x y = decode_b (S 1) x y.

The proposition is identified by the following information:

Pure Prop Id: ccc17a15fee3ff8e50e0a429d8fc3703e0d4322e680a51e8331926734edd0a78
Pure Prop Address: TMGv6tf3d83HWkjLGc4Bjfk2yL6bP1Y69Ea
Theory Prop Id: b982a46d670799d817f2c72da1c2d0fb616888b041e52fcb1bdd4cd4f36e4a15
Theory Prop Address: TMPFdacX88nXYB1oHJX48p69cohuPHPDyHj

pack_b_b_r_e_e_1_eq2

Theorem 28.175

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.∀xy ∈X.

f x y = decode_b (pack_b_b_r_e_e X f g R c d 1) x y. The proposition is identified by the following information:

Pure Prop Id: e738dbc94ff336d55dbd404f3523f553aed8269c125ce07c558108d936f8bf8e
Pure Prop Address: TMczpSJ93XaFjvH4vijE4cc2RjpSvLCBnZ1
Theory Prop Id: 3048f770c8c092a3cc8c55bb78a7d82f19565aac3ff6652ef9d55e6c0cfb0962
Theory Prop Address: TMGxoR9fiBCzsbqU2uTbeELHpxWJM9ikYSB

pack_b_b_r_e_e_2_eq

Theorem 28.176

∀SX.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

S = pack_b_b_r_e_e X f g R c d→∀xy ∈X.

g x y = decode_b (S 2) x y.

The proposition is identified by the following information:

28.6. PACKING TWO OPERATIONS, A RELATION AND TWO CONSTANTS447

Pure Prop Id: a822390210aa0eeae11feeec870412a9d3f167449db2716772c8d85d36fde3ad
Pure Prop Address: TMGuyMyeVJR3BDF1pJNw6KGpyy6D5ZNkVqb
Theory Prop Id: 2d8185611c6998b74706354379f7e37a3983d2acac54f2a80803b05199954f8f
Theory Prop Address: TMNWRvZ8TbLk6nt1L5aUcRmKEgigkF4bzbC

pack_b_b_r_e_e_2_eq2

Theorem 28.177

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.∀xy ∈X.

g x y = decode_b (pack_b_b_r_e_e X f g R c d 2) x y. The proposition is identified by the following information:

Pure Prop Id: 0904eb84a994cf26e71f4541979476936ad04090940f84ea7e3ae166191f386f
Pure Prop Address: TMZgHEgS5WRZZnmn2WtWxSCUaL6S8KAKuEU
Theory Prop Id: 514b7ec403837110f3c264073db0858ef3b66bcb89b034df3d99771f31f409f1
Theory Prop Address: TMPsaDHiTWNr1DquvQg49N7jxY2zPJAhBQD

pack_b_b_r_e_e_3_eq

Theorem 28.178

∀SX.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

S = pack_b_b_r_e_e X f g R c d→∀xy ∈X.

R x y = decode_r (S 3) x y.

The proposition is identified by the following information:

Pure Prop Id: 20b01269edbb5f190d76d15ed12389468b65b4f3a49a6747fc5a644de89692c4
Pure Prop Address: TMZCKwwQH2ns9uknt6pfpXLnMeYPTdrkKsA
Theory Prop Id: 543a595f6d03cb89cc51280f13fd9710e66da890237a5f792e79701620885eb7
Theory Prop Address: TMPLwektPtu4AjrXp4ghumX9jTbcGU7yeaD

pack_b_b_r_e_e_3_eq2

Theorem 28.179

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.∀xy ∈X.

R x y = decode_r (pack_b_b_r_e_e X f g R c d 3) x y. The proposition is identified by the following information:

Pure Prop Id: 84e5f9f9566add41b035a90f33b58ab881576661927e7112435c4cc63f5b9a82
Pure Prop Address: TMcKPp5Wtf1GqSMtPaoCwtuAz9G543UZXY7
Theory Prop Id: e6c65fe282c292809ca69c9ff83b219fff807abed4e4977ed7b6ca1bd889aeb5
Theory Prop Address: TMH676J4Qexg1jep9yQTFXu4vNi1TJ1qrZX

pack_b_b_r_e_e_4_eq

448 CHAPTER 28. SURREAL NUMBERS II Theorem 28.180

∀SX.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

S = pack_b_b_r_e_e X f g R c d→c = S 4.

The proposition is identified by the following information:

Pure Prop Id: 7c411fa1b904b5c366996dc15f10c49f1ca9b84a3441266671af3c8907575779
Pure Prop Address: TMSSAAEDYMUWREf5eqjEN4jmyCnfvyeobwE
Theory Prop Id: 4e1693db4de27914c071907d430af0401ef3bd6ac24367ac8791b6c5c512091e
Theory Prop Address: TMGzMNkeS7o5At8847MUjvdcHvigbTpaypB

pack_b_b_r_e_e_4_eq2

Theorem 28.181

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

c = pack_b_b_r_e_e X f g R c d 4.

The proposition is identified by the following information:

Pure Prop Id: 79c12fb164791187a04849be1011349dce3e15aad6a3329faac347fc335089f5
Pure Prop Address: TMFqB5mPoMCs8LrcpMtffiyDeommry4W3Wf
Theory Prop Id: 11df0febe6f1e3560b5041849da4dd2bdc05c8527c22c5617386567a92e57fa6
Theory Prop Address: TMct6V7NrbQ2y9kAcrjnEM2WVDGaQZrcgjK

pack_b_b_r_e_e_5_eq

Theorem 28.182

∀SX.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

S = pack_b_b_r_e_e X f g R c d→d = S 5.

The proposition is identified by the following information:

Pure Prop Id: 94b3c3a63e1902ef3037b87eaa7fcbbae94338cd1163cceeed376f1eb8ec7ef1
Pure Prop Address: TMHxq3wxovqDHYziUjPoyA45c2sAhm2UNoq
Theory Prop Id: cc915a2d36f6cc79c293654aab8fd3adb2808299e9b173a1c39bf3cdac0dfe5c
Theory Prop Address: TMSYohXBC39TSSsVBLMnYazqxd2k9SYkjSg

pack_b_b_r_e_e_5_eq2

Theorem 28.183

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

d = pack_b_b_r_e_e X f g R c d 5.

The proposition is identified by the following information:

28.6. PACKING TWO OPERATIONS, A RELATION AND TWO CONSTANTS449

Pure Prop Id: b0d3a42291179839984699106279142ceeb8c26475897616c224628ed6d51873
Pure Prop Address: TMcWSqvc4wwoPFxLaHnaqahY7ZNdALEXxz1
Theory Prop Id: 29ae629ba4f8c9f4ea56fe01e58c7e5adc9a3f4032832a6256e4172bc1857db1
Theory Prop Address: TMSx1WE5sJ7RC3YQth9f6oafrR3W5T8cUAW

pack_b_b_r_e_e_inj

Theorem 28.184 ∀XX′.∀ff′ :ι → ι → ι.∀gg′ :ι → ι → ι.∀RR′ :ι → ι → o.∀cc′ :ι.∀dd′ :ι. pack_b_b_r_e_e X f g R c d = pack_b_b_r_e_e X′ f′ g′ R′ c′ d′

→

X = X′∧
(∀xy ∈ X.f x y = f′ x y)∧
(∀xy ∈ X.g x y = g′ x y)∧
(∀xy ∈ X.R x y = R′ x y)∧
c = c′ ∧ d = d′.

The proposition is identified by the following information:

Pure Prop Id: 41bdc7cea27f40d41bd47e10070a9221fac56ee80059ed6ef1145c640f0a0696
Pure Prop Address: TMZzLqefgu7rLgWmGbXgEsJw9KyJ9HfnEfv
Theory Prop Id: 030a267e64cc2ab4e90b427a38635de36741b1d1f5f749729223836ad895b355
Theory Prop Address: TMFdLKT9s9pybGa29WS8rRsZtaZEB878Tqc

pack_b_b_r_e_e_ext

Theorem 28.185 ∀X.∀ff′ :ι → ι → ι.∀gg′ :ι → ι → ι.∀RR′ :ι → ι → o.∀c.∀d.

(∀xy ∈ X.f x y = f′ x y)→(∀xy ∈ X.g x y = g′ x y)→
(∀xy ∈ X.R x y ⇔ R′ x y)→

pack_b_b_r_e_e X f g R c d = pack_b_b_r_e_e X f′ g′ R′ c d. The proposition is identified by the following information:

Pure Prop Id: 65aff8c9424ee01cd07c15ae87a00a524f02d18e2ec143a5bba84026587bf7f3
Pure Prop Address: TMcK19CLntYiEUmEtDidiA4HpkWUA29DCY8
Theory Prop Id: 37b535ddadbc59098fcbd9eea3333e24e8d8ba92e8fa6adac43561bcb663a641
Theory Prop Address: TMSrpevWvuK3PWAB7qQDUur4T35U2y5PcFh

Definition 28.26 We define struct_b_b_r_e_e to be

λS.∀q : ι → o.(∀X : ι.∀f : ι → ι → ι.(∀xy ∈ X.f x y ∈ X) →

∀g : ι → ι → ι.(∀xy ∈ X.g x y ∈ X) →

∀R : ι → ι → o.∀c : ι.c ∈ X → ∀d : ι.d ∈ X →
q (pack_b_b_r_e_e X f g R c d))
→ q S

of type ι → o identified by the following information:

450 CHAPTER 28. SURREAL NUMBERS II

Pure Object Id: d542f60aebdbe4e018abe75d8586699fd6db6951a7fdc2bc884bfb42c0d77a22
Pure Object Address: TMXf978YVAAD5ZQex2ymHckFjvhRj8VqJG9
Theory Object Id: 4b668479c87844857651881d76dfae59eab34bb3676e61f76f6dd64b6060cf5d
Theory Object Address: TMKS3b6mms1vofrtcap5oMtL9Kdzd24UwRB

pack_struct_b_b_r_e_e_I

Theorem 28.186 ∀X.∀f :ι → ι → ι.(∀xy ∈ X.f x y ∈ X)→∀g :ι → ι → ι.(∀xy ∈ X.g x y ∈ X)→

∀R :ι → ι → o.∀c :ι.c ∈ X→∀d :ι.d ∈ X→ struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d).

The proposition is identified by the following information:

Pure Prop Id: 837b01cc10cf77ecfed9c05cb135bfee5d5785cb3eeb155cd67420a1e8117277
Pure Prop Address: TMdKBXs47VAuzq32DPCspAGpMJxWtP7HZgn
Theory Prop Id: 4e1d4bac0d2464119ef3ae868cd30477053be4e52d11f44a4826bb32b543f78a
Theory Prop Address: TMRfKTdq9BR53wNTCPty6ckkNQRPzNXtwXR

pack_struct_b_b_r_e_e_E1

Theorem 28.187 ∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d)→
∀xy ∈X.f x y ∈ X.

The proposition is identified by the following information:

Pure Prop Id: 33666944863c7073c5d24b456627c47939a6c292049de068c2dd0a2696024881
Pure Prop Address: TMbqCqWUV4Qr8yRDenPHW1SCXnMHwtzsEqR
Theory Prop Id: bce7cd0660e462218e87e9b6e9756ae8795d7e47a2034af9f01a233591cd4516
Theory Prop Address: TMTcgTJU3oXQrtzipjvwHThz1WBYVpT6Tr8

pack_struct_b_b_r_e_e_E2

Theorem 28.188 ∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d)→
∀xy ∈X.g x y ∈ X.

The proposition is identified by the following information:

Pure Prop Id: 8d7f819bc88f04c8fb38f9c45656b24be62467c1111be1ab6669a362d2c42bd9
Pure Prop Address: TMFq66jAxt6ToaMefzyqpZmo2jxWBS1L5KS
Theory Prop Id: a642313376049cb0e12a05cfd42eb6fe35a06fa7ac25ab566e3a668a2f094fa5
Theory Prop Address: TMKRXGcsc5vFBxDbQ738RFuErE5UAsvbXrX

28.6. PACKING TWO OPERATIONS, A RELATION AND TWO CONSTANTS451

pack_struct_b_b_r_e_e_E4

Theorem 28.189 ∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d)→
c ∈ X.

The proposition is identified by the following information:

Pure Prop Id: fe2c180cc0cf649c36df0e3a2f9dabf02603a9d163927935d608712fe908c0a0
Pure Prop Address: TMKkUcfVuzqdAAqixhrJdaDtwAiwP588iFb
Theory Prop Id: 4343532cd958d270bb174fb9ef0d486e82bcb7068472964cad87b3aa59268c36
Theory Prop Address: TMVkwckBqwmZ3ptsoBAtwgWpGJnBf1HEKJX

pack_struct_b_b_r_e_e_E5

Theorem 28.190 ∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.∀R :ι → ι → o.∀c :ι.∀d :ι.

struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d)→
d ∈ X.

The proposition is identified by the following information:

Pure Prop Id: 1eaba4d18e5ef9249157efebe811892c11f22eaccbdfaabe7207d0eed3610753
Pure Prop Address: TMN8VR6KEzNcbMfJUKCFWE9VrqsSAghpm2a
Theory Prop Id: d7fee779a369a0d4bd81cd98ec4b107047599cd8daebdc70621edcfba45ab303
Theory Prop Address: TMRKtcvjctfdAsskw9TbK95G7FLTa6F6w99

struct_b_b_r_e_e_eta

Theorem 28.191

∀S.struct_b_b_r_e_e S→
S =
pack_b_b_r_e_e (S 0) (decode_b (S 1)) (decode_b (S 2))
(decode_r (S 3)) (S 4) (S 5).

The proposition is identified by the following information:

Pure Prop Id: 08c6efad770174c53c57bf1abbcff7a0b238338e1a0a913fdb1e12ac817545eb
Pure Prop Address: TMGztHkuyt42zcYRkYwSMfmeqnNcm8EkxuB
Theory Prop Id: 3a4c426298d981750f4f7847d710686bde28aba7b880e1e16dc568078e493ab5
Theory Prop Address: TMdFRkJgJNNXSyM8pAQoyEqPBnQPHMehaWr

452 CHAPTER 28. SURREAL NUMBERS II Definition 28.27 unpack_b_b_r_e_e_i is the opaque object of type

ι → (ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → o) → ι → ι → ι) → ι identified by the following information:

Pure Object Id: 89fb5e380d96cc4a16ceba7825bfbc02dfd37f2e63dd703009885c4bf3794d07
Pure Object Address: TMV6sraHemVG7Q2TgN6oCsJmkmebyJuYVNd
Theory Object Id: 76f5f924237f946f8a80d7200b117ad49d05a912a831233242659e6f501bf978
Theory Object Address: TMRP69PDPah6ZkTeGTDbp6iVPip8vSRTsB8

unpack_b_b_r_e_e_i_eq

Theorem 28.192

∀Φ :ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → o) → ι → ι → ι.

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.

∀R :ι → ι → o.∀c :ι.∀d :ι.
(∀f′ : ι → ι → ι.(∀xy ∈ X.f x y = f′ x y) →
∀g′ : ι → ι → ι.(∀xy ∈ X.g x y = g′ x y) →
∀R′ : ι → ι → o.(∀xy ∈ X.R x y ⇔ R′ x y) →
Φ X f′ g′ R′ c d = Φ X f g R c d)
→

unpack_b_b_r_e_e_i (pack_b_b_r_e_e X f g R c d) Φ = Φ X f g R c d. The proposition is identified by the following information:

Pure Prop Id: b58f007e91d43a2c7a09dc0b595961936ec7da158803e7b98c8e478584d206ef
Pure Prop Address: TMPqkpAFSVWhtBYDqJsL7JfE4qYGQvWKVkp
Theory Prop Id: 9a7986d0258cb1c3270b60b6b454888095d2aa2a2a53d845b4756902a36876f7
Theory Prop Address: TMQfsPU4J48cc3XAFwauAcYbj5YSbwJVxJw

Definition 28.28 unpack_b_b_r_e_e_o is the opaque object of type

ι → (ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → o) → ι → ι → o) → o identified by the following information:

Pure Object Id: b3a2fc60275daf51e6cbe3161abaeed96cb2fc4e1d5fe26b5e3e58d0eb93c477
Pure Object Address: TMP75qXVTx8SpwHV1y6wefKnbFqVV7VPCV7
Theory Object Id: fc0d55fe8a09aad6902204ac4550a6671cc57b3b29e4c36503dea3e8cc818737
Theory Object Address: TMQVwZxiv8LXLzFAAtQUdESpWctYhNPKuLQ

unpack_b_b_r_e_e_o_eq

Theorem 28.193

∀Φ :ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → o) → ι → ι → o.

∀X.∀f :ι → ι → ι.∀g :ι → ι → ι.
∀R :ι → ι → o.∀c :ι.∀d :ι.
(∀f′ : ι → ι → ι.(∀xy ∈ X.f x y = f′ x y) →
∀g′ : ι → ι → ι.(∀xy ∈ X.g x y = g′ x y) →
∀R′ : ι → ι → o.(∀xy ∈ X.R x y ⇔ R′ x y) →
Φ X f′ g′ R′ c d = Φ X f g R c d)
→

unpack_b_b_r_e_e_o (pack_b_b_r_e_e X f g R c d) Φ = Φ X f g R c d.

The proposition is identified by the following information:

Pure Prop Id: b4ebc419abea98259a9b2fc6ee430a9eb6b66901e01778798caef327518ab73c
Pure Prop Address: TMWy9NbJuezT2C2YvSVs4gNoW6RxDCu36cJ
Theory Prop Id: 308ccf9882d7cb03559534a600b368145e2c7062734d3ea9574a2b00c5a6bbd1
Theory Prop Address: TMdarJDcWV77zy8PF3SrMpz5YVBUqiWCp7G

Definition 28.29 OrderedFieldStruct is the opaque object of type ι → o identified by the following information:

Pure Object Id: 98aa98f64160bebd5622706908b639f9fdfe4056f2678ed69e5b099b8dd7b888
Pure Object Address: TMEzWdp7RFmZMEiX1EsMMaPEcXjJJfCtETX
Theory Object Id: 1de6a120bae49aa4021a9a6626425398f0c8d34c17a449aca28902eba9cfe754
Theory Object Address: TMWMkhjM35KDgsppiv3Up8VXcf7pNB4BC6Q

28.7 explicit_OrderedField_RepIndep2

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Let leq : ι → ι → o be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let plus′, mult′ : ι → ι → ι be given. Let leq′ : ι → ι → o be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Assume the following.

∀ab ∈ R.a + b = a ⊕ b  (28.1)

Assume the following.

∀ab ∈ R.a ∗ b = a × b  (28.2)

Assume the following.

∀ab ∈ R.leq a b ⇔ leq′ a b  (28.3)

explicit_OrderedField_repindep

Theorem 28.194

explicit_OrderedField R zero one plus mult leq

⇔ explicit_OrderedField R zero one plus′ mult′ leq′.

The proposition is identified by the following information:

Pure Prop Id: aa6280ff9ecfb388daf95ebbc769d81f90cc6039e5299709d31d625891887057
Pure Prop Address: TMbsFvtTvqkK3SszQVqXzQ1UJEsUnVYiMZL
Theory Prop Id: be25ce3a19ac950fdaa6ea42fe6fcb818e428c8dc8aae4b4fbe47fa06d08e029
Theory Prop Address: TMdEcCwhu3D2kVgdgGVo1Cmg5W1sXuLX3HR

28.8 Unpacking Ordered Fields

OrderedFieldStruct_unpack_eq

Theorem 28.195

∀R.∀plus mult :ι → ι → ι.∀leq :ι → ι → o.∀zero one.
unpack_b_b_r_e_e_o (pack_b_b_r_e_e R plus mult leq zero one)

(λR plus mult leq zero one.explicit_OrderedField R zero one plus mult leq) = explicit_OrderedField R zero one plus mult leq.

The proposition is identified by the following information:

Pure Prop Id: 0f64cee1eaa56826e0a1df23c3c822f54eeeaf346766143d43f9cd8aaa5bcc05
Pure Prop Address: TMcaQWXF265MqyQdZxoLdbgCWynQwDq1QFz
Theory Prop Id: 6fe0fe98ce1675b6b7d6602c3080c16c71b29a61d4b004d43b5855aad4d9a43f
Theory Prop Address: TMXMSqH6fsZZwHid44hbcR3wDdTqU7Dm9HQ

Definition 28.30 We define RealsStruct to be

λR.struct_b_b_r_e_e R∧ unpack_b_b_r_e_e_o R

(λR plus mult leq zero one.explicit_Reals R zero one plus mult leq)

of type ι → o identified by the following information:

Pure Object Id: 9971dcf75f0995efe4245a98bcdf103e4713261d35432146405052f36f8234bf
Pure Object Address: TMJjjbmFf1HjinfhpdrL5vryqPNmEPLfGtv
Theory Object Id: 3d1a749dfea9bb7211b105adb25ae072c40841648e843f5cbc5483cd7b1257cd
Theory Object Address: TMQydd7NRsjg2p9tCMw9pgNmAnEKZZD616o

28.9 explicit_Reals_RepIndep2

Let R : ι be given. Let zero, one : ι be given. Let plus, mult : ι → ι → ι be given. Let leq : ι → ι → o be given. Notation. We use + as a right associative infix operator corresponding to applying term plus. Notation. We use ∗ as a right associative infix operator corresponding to applying term mult. Let plus′, mult′ : ι → ι → ι be given. Let leq′ : ι → ι → o be given. Notation. We use ⊕ as a right associative infix operator corresponding to applying term plus′. Notation. We use × as a right associative infix operator corresponding to applying term mult′. Assume the following.

∀ab ∈ R.a + b = a ⊕ b  (28.4)

Assume the following.

∀ab ∈ R.a ∗ b = a × b  (28.5)

Assume the following.

∀ab ∈ R.leq a b ⇔ leq′ a b  (28.6)

explicit_Reals_repindep

Theorem 28.196

explicit_Reals R zero one plus mult leq ⇔ explicit_Reals R zero one plus′ mult′ leq′.

The proposition is identified by the following information:

Pure Prop Id: e6777aed3a48a1eed98a6aa68ec40260a6fd7f77f99e3b2dfc8fd4213fbf51d5
Pure Prop Address: TMHRoUeAQwKBzV5DNUEnEtSGLyRQBCWi2oz
Theory Prop Id: e7a9656809ed03974b3a3e7265ecbe46e51f37029b92e73c457801fb7c611216
Theory Prop Address: TMav3v4dLGD2kzFGsBAHjnvy2SFfYuJuB5m

28.10 Unpacking Explicit Reals

RealsStruct_unpack_eq

Theorem 28.197

∀R.∀plus mult :ι → ι → ι.∀leq :ι → ι → o.∀zero one.
unpack_b_b_r_e_e_o (pack_b_b_r_e_e R plus mult leq zero one)

(λR plus mult leq zero one.explicit_Reals R zero one plus mult leq) = explicit_Reals R zero one plus mult leq.

Pure Prop Id: d753a2caecebad53fe06dc1e4aa5fc9c806b9df39093b6b31f3dfd7d2790d31f
Pure Prop Address: TMKP2fsixYdw8yYkEFeLRmmhRgMFNqBiCkE
Theory Prop Id: 1e4afd0f43b0a314315c5e7b2a4354fc8792a577d40f5ddbfe8054c2fd4bf78c
Theory Prop Address: TMHSDGonYJvmsYAvpMXetztjTkxJWrGWgqS

Definition 28.31 We define RealsStruct_carrier to be λRs.Rs 0 of type ι → ι identified by the following information:

Pure Object Id: f0bb69e74123475b6ecce64430b539b64548b0b148267ea6c512e65545a391a1
Pure Object Address: TMY8sbcPiVs9VL311KPHpbm67U6sLM9JvF7
Theory Object Id: 9211e41eaf81bb8c152618cbe35230be60d736fcd5684c0b7bdc5ccc1ff70181
Theory Object Address: TMZQE6tPrFvJMaBcLrcMn85ZgscSeFbm4pT

Definition 28.32 We define RealsStruct_plus to be λRs.decode_b (Rs 1) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 6968cd9ba47369f21e6e4d576648c6317ed74b303c8a2ac9e3d5e8bc91a68d9d
Pure Object Address: TMYTM1QQGqgVyRM9unR8CLL2scwRHYvV3c3
Theory Object Id: ed06aa34b4112f8507370b46c1c5c7de7f2f9af4f24b831d45893ad3066e994e
Theory Object Address: TMRiY7KU6BpU28KD19iwgBDLsQ5XyfpcbYL

Definition 28.33 We define RealsStruct_mult to be λRs.decode_b (Rs 2) of type ι → ι → ι → ι identified by the following information:

Pure Object Id: 4b4b94d59128dfa2798a2e72c886a5edf4dbd4d9b9693c1c6dea5a401c53aec4
Pure Object Address: TMQSTL5b71ddbjnkjnGXfRZ61uXWmMGwJHD
Theory Object Id: 7eb39ed695c795e4451a9252d011d377d0f4bdf0bfe6193f7a31ceb58f687a40
Theory Object Address: TMaJmzoi8UnDNTAXrAvZxYbTeEdKV3viygg

Definition 28.34 We define RealsStruct_leq to be λRs.decode_r (Rs 3) of type ι → ι → ι → o identified by the following information:

Pure Object Id: e59503fd8967d408619422bdebda4be08c9654014309b60880a3d7acf647e2b4
Pure Object Address: TMVpPCt6DnVyyzfVCfyhFErhhmLuHJWfLF5
Theory Object Id: e66070fe509c0522857fd1dec315953e7466e1fff9feba36b2b4e9f1f60f7ede
Theory Object Address: TMGzX1JqbMSdTKAjW1hbwGwKwhVDCcRKEhA

Definition 28.35 We define RealsStruct_zero to be λRs.Rs 4 of type ι → ι identified by the following information:

Pure Object Id: 57e734ae32adddc1f13e75e7bab22241e5d2c12955ae233ee90f53818028312a
Pure Object Address: TMV8UDBZRqv4jocV6zt17aP89PcYugzQkoU
Theory Object Id: b2fa78f7f58b95f4bc4f46ac268c3a959090897e56338c64bd84a73bc5fc2551
Theory Object Address: TMWFb9uZk2JEesS9quTe9a6Ww4ND9VcbBJA

Definition 28.36 We define RealsStruct_one to be λRs.Rs 5 of type ι → ι identified by the following information:

Pure Object Id: 7aa92281bc38360837d18b9ec38ff8359d55a8c6ef4349c5777aab707e79589b
Pure Object Address: TMG8fYn75tfBRqBJzcYzTPSyLmxrzMatNuC
Theory Object Id: e902384fa204aa548a7192aca2937826cba222865eef88ccb0f0dd747aff3878
Theory Object Address: TMF5LB2qHkXJqUPWHWovgb3u9mde2rLtjtB

28.10. UNPACKING EXPLICIT REALS 457 Definition 28.37 Field_of_RealsStruct is the opaque object of type ι → ι identified by the following information:

Pure Object Id: e1df2c6881a945043093f666186fa5159d4b31d68340b37bd2be81d10ba28060
Pure Object Address: TMYju5xqA4jiDon5Vrw64253Ybc4eYYipFW
Theory Object Id: ef27c4f9f215b0e5024d128458fefa23faae900a3a796c7655191a5180aa8fd2
Theory Object Address: TMS4LA1a2HRZdid71r9n3J7xvnqcWXoPRYU

Field_of_RealsStruct_0

Theorem 28.198

∀Rs.Field_of_RealsStruct Rs 0 = RealsStruct_carrier Rs. The proposition is identified by the following information:

Pure Prop Id: e3591bbc5c24313d727f99859f33bddaa24ed93f5e16fde0a5a27460f4deaede
Pure Prop Address: TMGpDdtdmCUs1jrGPV3mZk171Ru5G3eeAgA
Theory Prop Id: cb1fd7ac2423a50aaf869926ca1d1306d77aef6c4d3fa9f561447ef07b391cb8
Theory Prop Address: TMNmZN2BcVU3Bvv3THemmMQphYBi7VZEiK4

Field_of_RealsStruct_1

Theorem 28.199

∀Rs.∀xy ∈RealsStruct_carrier Rs.

Field_of_RealsStruct Rs 1 x y = RealsStruct_plus Rs x y. The proposition is identified by the following information:

Pure Prop Id: a29ae8db462729b7f524ef65c8169e19aa80b02b813f1538b50a4ea71cac00e4
Pure Prop Address: TMH9MkMZY6Nfy8CtfBXnKbGTTMMDYtF8dhg
Theory Prop Id: f4fd5c9274754011767f3f3a1963005a161b0e3450a8605917bb74939f42093a
Theory Prop Address: TMHPEQeFskKiibC12Eg5RfK8N4RmZgBPc6M

Field_of_RealsStruct_2

Theorem 28.200

∀Rs.∀xy ∈RealsStruct_carrier Rs.

Field_of_RealsStruct Rs 2 x y = RealsStruct_mult Rs x y. The proposition is identified by the following information:

Pure Prop Id: 144feb0ed55ae0791a3276243986374c09700d3d3d141752481d9130cc01a231
Pure Prop Address: TMQiqH6UGKCC2AY2aZmhwjBD1uigXrL9U5A
Theory Prop Id: b9ac92481e26e620bc1d7f8daba3ccb4dcffbac3703a966f7f6175c0996df7a2
Theory Prop Address: TMHw73bPqVnCDmcBgyog1NWbZ2kKZNjpM2F

Field_of_RealsStruct_3

Theorem 28.201

∀Rs.Field_of_RealsStruct Rs 3 = RealsStruct_zero Rs.

The proposition is identified by the following information:

Pure Prop Id: 3a0c651ac07600c96cb2b4b797025bc57c3f63713cb95551e7ff7570600dd9a6
Pure Prop Address: TMP2bK3Zy9P941RMigpsJeoKpTUvZdbu6jt
Theory Prop Id: 39ddb5b76ae60cb54984057402ef494c4fa1400d58d80b904f425a9a5e3609e3
Theory Prop Address: TMLDSxVyVzr1FPB6KncZtioj1PDrXC6M8SJ

Field_of_RealsStruct_4

Theorem 28.202

∀Rs.Field_of_RealsStruct Rs 4 = RealsStruct_one Rs.

The proposition is identified by the following information:

Pure Prop Id: f8ef8a9601b742e71cf054c41b20f6ec0dee405cf311308f2196ca2ae8742478
Pure Prop Address: TMHKLv75keSDVRMA8BVfAdqxSSSWoVHCjjK
Theory Prop Id: 79ba04006dd069d9bc477280db9ba7c3bf231bcc112d5aaa271be90cf51e8d30
Theory Prop Address: TMJTgymdjGtaV8G45xSwQDuSpRLtCjKY7LG

28.11 RealsStruct

Let Rs : ι be given. Assume the following.

RealsStruct Rs  (28.7)

Let R : ι be RealsStruct_carrier Rs. Let zero : ι be RealsStruct_zero Rs. Let one : ι be RealsStruct_one Rs. Notation. We use + as a right associative infix operator corresponding to applying term RealsStruct_plus Rs. Notation. We use ∗ as a right associative infix operator corresponding to applying term RealsStruct_mult Rs. Notation. We use ≤ as an infix operator corresponding to applying term RealsStruct_leq Rs.

RealsStruct_eta

Theorem 28.203

Rs =
pack_b_b_r_e_e R (RealsStruct_plus Rs) (RealsStruct_mult Rs)
(RealsStruct_leq Rs) zero one.

The proposition is identified by the following information:

28.11. REALSSTRUCT 459

Pure Prop Id: 17af86f27285e29398960ca5e9e3f6f45b69b7a9bdf827de09c0c669b7c75c6c
Pure Prop Address: TMWmf9GomThUEUvw4iwAFWDwRvBCKuKUJ81
Theory Prop Id: f1cea2ca03fd17235452510cf22a8e1709b9373fee61b9d92e62557ba14e4287
Theory Prop Address: TMbTji4uUoLpyNMKCgYne4Fdo2wyrpD2a2z

RealsStruct_explicit_Reals

Theorem 28.204 explicit_Reals R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs) (RealsStruct_leq Rs). The proposition is identified by the following information:

Pure Prop Id: d3877b9d93ebdb9c81bfb64ef324b904ac578f720458bd89b59d747bab8321ec
Pure Prop Address: TMbsdad9BHngHtYG5Eu56f41hLLyMhQ4wz9
Theory Prop Id: e96bfd03b3e326ebe235a70e085a6663531d7408960525dca468e6023497d823
Theory Prop Address: TMMM29TxtZ5QApu738fkTwRqJ5AxhqnNU1q

Field_of_RealsStruct_is_CRing_with_id

Theorem 28.205

CRing_with_id (Field_of_RealsStruct Rs).

The proposition is identified by the following information:

Pure Prop Id: 30ac8336eb76436c0b3b234f78de119af4f8aae671ed6946516746919bd40b57
Pure Prop Address: TMRwfFMfrGHNJdjEebj6AWUAsps8wxbBciR
Theory Prop Id: 8d5700aa263b7f06933873e9cd11f836e00e0138000b4d39f89edcf89a4b5b69
Theory Prop Address: TMYnS1DaswnJWAykA7gfcHRYUno5P7ZH1xt

Definition 28.38 We define RealsStruct_lt to be λxy.x ≤ y ∧ x ≠ y of type ι → ι → o identified by the following information:

Pure Object Id: 3a46d8baa691ebb59d2d6c008a346206a39f4baae68437520106f9673f41afc6
Pure Object Address: TMZU8qJtXFES85dykXqXncY6GKaYJb1UkwA
Theory Object Id: 1fb825962c4e3c207604d053fa6fdb0b78c4215dab034f2c783d256d7e08a3f3
Theory Object Address: TMcanZ7iZKUNsAhX6fjudSebaubcUhfXhth

Notation. We use < as an infix operator corresponding to applying term RealsStruct_lt.

explicit_Field_of_RealsStruct

Theorem 28.206

explicit_Field R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs). The proposition is identified by the following information:

460 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: d59c3f2e6f4d6724d3005e0bee526d84560fc597c159862117cd50437503db9a
Pure Prop Address: TMYsjqWb3PES4DyoFitce9WwQU2Rt5WHX1i
Theory Prop Id: 744fcb4f7b03a77f3f942ae46ec1dc0b6e66ac611a343175773867714b070aeb
Theory Prop Address: TMairfquc4KvRjg7Xc3pANWUih3uqyq8R5j

explicit_OrderedField_of_RealsStruct

Theorem 28.207

explicit_OrderedField R zero one (RealsStruct_plus Rs)

(RealsStruct_mult Rs) (RealsStruct_leq Rs).

The proposition is identified by the following information:

Pure Prop Id: 70df0d2f2fe64ff20992d794cafe8f45b02a15058699156c2ded116e9b0962e6
Pure Prop Address: TMUxQacVa4YowWpPGUq5zVNjPj5dQafMF9G
Theory Prop Id: afda6089023853c8239e43e59fcb8eb5e74e269802cfec8e72ac2071ca787656
Theory Prop Address: TMUSiQQFDe6Uiw3xmogsw37fkZT92vjPKrA

RealsStruct_OrderedField

Theorem 28.208 OrderedFieldStruct Rs. The proposition is identified by the following information:

Pure Prop Id: 3149fc73f3626eb90d8ffbc6259b266c543eaec7594c6c6c6503c7eab6bc76d6
Pure Prop Address: TMQXpz1uzda7WjR5D59goy6Npidv1Fh3ego
Theory Prop Id: 638a457d1b7e3a0558d1e4a8349534f593bad615e24d92349d002ae3afb8f22e
Theory Prop Address: TMGNWDRJRSAseCPGm2L1ufW5qeZCWTf1RTZ

Field_of_RealsStruct_1f

Theorem 28.209

(λxy : ι.Field_of_RealsStruct Rs 1 x y) = RealsStruct_plus Rs. The proposition is identified by the following information:

Pure Prop Id: 7affbde70101a9fb21231dde40bcdaf417f643cdb83ddc82db6176651037aac3
Pure Prop Address: TMe11GXyXEF6MGDABPzVWHtkhaqqa9Fkk9T
Theory Prop Id: fe164da026cbaa682ef9fe1136650ba3b8eb25224ebc8d70d1da787a38e26a69
Theory Prop Address: TMPP93FXoNZbKN3CDWyZUbmssxCbWqaUC2x

Field_of_RealsStruct_2f

Theorem 28.210

(λxy : ι.Field_of_RealsStruct Rs 2 x y) = RealsStruct_mult Rs. The proposition is identified by the following information:

28.11. REALSSTRUCT 461

Pure Prop Id: f983dc416a105a88e0acb3737c63f1e0d73c3c64c4520277a6e8d01d56bc243c
Pure Prop Address: TMGdwUUczg7R1h1T84CWJM1aykiAmFN7fNu
Theory Prop Id: 5715158dd199c38a59bbf5c3c0b1f41806ecc17815efa3fc7d133801a3cddea5
Theory Prop Address: TMMmVJkECh7QnA46fbLjGoemcoeBWqcFETd

explicit_Field_of_RealsStruct_2

Theorem 28.211

explicit_Field R (Field_of_RealsStruct Rs 3)

(Field_of_RealsStruct Rs 4) (decode_b (Field_of_RealsStruct Rs 1)) (decode_b (Field_of_RealsStruct Rs 2)).

The proposition is identified by the following information:

Pure Prop Id: cedc05dfb417fc06394a10542007a2f8ce55cc82ac0946b195b2c73fdfe9ade2
Pure Prop Address: TMNT1KjdwrJ9LD8Vnb1Xouv3qSy7pD9ivJ4
Theory Prop Id: edd0df751b28b776ab98f509eb6e9906b9ee240bf87903c0e4f8d10355c8de2c
Theory Prop Address: TMYv4TivoSVD8m2uZbe9Vwkjd9899mo2iPo

Field_Field_of_RealsStruct

Theorem 28.212 Field (Field_of_RealsStruct Rs). The proposition is identified by the following information:

Pure Prop Id: 0ec36b9dac36f4bf382d817f90965f789c23b007329c1f2a14b0014a91d92067
Pure Prop Address: TMZyZoKXReKnGyDuwVdpxzHCHi3LLUXkp3d
Theory Prop Id: 23c5218dab13d451320a4c008e8443b47261ff382e01d4a9eafa9b5032fd81d1
Theory Prop Address: TMR5Ubk1J3EVtZScLio9b8k4kf9rtQ6ncTX

RealsStruct_zero_In

Theorem 28.213 zero ∈ R. The proposition is identified by the following information:

Pure Prop Id: 005b3674102dca79680f3c66d82d0a566ec5ce1fbc87acc22a9a6ca9ebebf54b
Pure Prop Address: TMMBmyfDd54QzeVfnKcj1zfp6a52pbqPYQi
Theory Prop Id: 7813991fa3eec14bade682c678cb4927bd388c9ac91fba2e332415aa6f370b7f
Theory Prop Address: TMcHAxvwy4L9sxYGy2zcQMnxHoHXr9htE8A

RealsStruct_one_In

Theorem 28.214 one ∈ R. The proposition is identified by the following information:

Pure Prop Id: 1b33553dbfb8dd1c45ebfbfd7313edc634050f17c1b4547ca0b73c8154450951
Pure Prop Address: TMVFbWxvhhJfevVpKgMa2zFT5nstpC3SnzU
Theory Prop Id: 04a7a006f820df56470471035fcc3ca727dad5b96142181cf37a3fa7867815f3
Theory Prop Address: TMQ7UNpiYfxP56HgBYDN27D2wfmUQoY6GXz

462 CHAPTER 28. SURREAL NUMBERS II

RealsStruct_plus_clos

Theorem 28.215 ∀xy ∈ R.x + y ∈ R. The proposition is identified by the following information:

Pure Prop Id: d59fc8a19b30608e6133175e65740cbf406831dcc29d41398580efd3565b0304
Pure Prop Address: TMH8yov9d8x8DhvNqVPDNBwvVUPZbFnQY4N
Theory Prop Id: d047b1443a58c7267b47a6534919e45b97226c58a703aad09b0fcb5a5e1649e2
Theory Prop Address: TMMHW3MKm5vVqaUcstZAAHdswmLybKpv7jC

RealsStruct_mult_clos

Theorem 28.216 ∀xy ∈ R.x ∗ y ∈ R. The proposition is identified by the following information:

Pure Prop Id: b521d3f1983a1d40a297548001ef03c96963cfbb20d3944063f920b35cb4f799
Pure Prop Address: TMZECePNUD8jFwZt6i2o5f1wf4rK94xSn89
Theory Prop Id: cd4e02b42f650e37ee913b313e6820d2ad0116d56cfd6e5ef595d72f2bc261d6
Theory Prop Address: TMbyKuyhM9cra9Gm7KfgFoAyUN1YdzMLr52

RealsStruct_plus_assoc

Theorem 28.217 ∀xyz ∈ R.x + (y + z) = (x + y) + z. The proposition is identified by the following information:

Pure Prop Id: f3b3129567a8db11a6326406f56b94419e2d9818eac0f622df322d09400a6bcb
Pure Prop Address: TMJzLB6CoD1vut7yT35PBx1oo2PXUVc7Zig
Theory Prop Id: 8f95762ccd9f93efb2b813048852b496d88e5771e670e7c8330941bc2870621b
Theory Prop Address: TMTfndBh736RkfuCoAVb5EfY859TyC26KS3

RealsStruct_plus_com

Theorem 28.218 ∀xy ∈ R.x + y = y + x. The proposition is identified by the following information:

Pure Prop Id: 59c94c4de8b2476d2510dd4b09ddc432083d2bf7b7fb33b2024269e7cfc66637
Pure Prop Address: TMduLqT57Q1QdbQNsUt6QSDeqMnyE1sLKgk
Theory Prop Id: fe9682e564377a859f1f577491e17a474459086d1c02b7e016b4b3d5d708413c
Theory Prop Address: TMdBQiCvauMBPkmwE1yJizNxoC35UqZBmtT

RealsStruct_zero_L

Theorem 28.219 ∀x ∈ R.zero + x = x. The proposition is identified by the following information:

Pure Prop Id: 941604f00ed1e86b155b1b1cc752ab549a904f9ec545ac4e5e0501edea6bf9ae
Pure Prop Address: TMVWFuU6oaPoeDy7jmijYCvqgjMhDBfnAp6
Theory Prop Id: a028d8ecce4c4d08176730f488a7aa5d9e462795f24bc924a2776fee82844e1e
Theory Prop Address: TMTKfSnfXd5nWfTXC5qtkJxtoSvW9Z1vQDx

28.11. REALSSTRUCT 463

RealsStruct_mult_assoc

Theorem 28.220 ∀xyz ∈ R.x ∗ (y ∗ z) = (x ∗ y) ∗ z. The proposition is identified by the following information:

Pure Prop Id: a3751ec23c16fcb964095d434bd62fbff3f54ea6f367daa73dd6af90e7e15285
Pure Prop Address: TMVx6KqoHebZKqDiuAYv9dmMFLDaUM7Cqcs
Theory Prop Id: 9964f7eae81893737f8b8d14d0e37c2cbbffc7ba0883b2c17e9015b758de91f5
Theory Prop Address: TMHd9hTsZZPRjz6qz2H9V4kWSaeReQd6WyF

RealsStruct_mult_com

Theorem 28.221 ∀xy ∈ R.x ∗ y = y ∗ x. The proposition is identified by the following information:

Pure Prop Id: 917b61b0f80ba1c680830ac638f010b6d7ab9948b61b6bf2fa194f1559175899
Pure Prop Address: TMYHM6VL7di1iR2UetLF2MsoT7BqwWRXefM
Theory Prop Id: 56ee11d2a162693703d6398a6fa3b44e437c11cbc2034fd1d87a3616b79c425f
Theory Prop Address: TMM9uq46vBF85FpZjdJoCTa8ouco6ezSTf2

RealsStruct_one_neq_zero

Theorem 28.222 one ≠ zero. The proposition is identified by the following information:

Pure Prop Id: a175b20ec46a263f0237a1f588ae8b080a36967feae498f91cceed32f1f07bfa
Pure Prop Address: TMNKMBM19aBBHJYg384yRYAbPN53WVbc5ns
Theory Prop Id: 52ab44a448083c215a50ab8191899ef26e1fc7da0b367bd95ecbe48ce8e0c5ac
Theory Prop Address: TMPjniRmWmW6zxw8tiEspuizDHoc7XRo3VJ

RealsStruct_one_L

Theorem 28.223 ∀x ∈ R.one ∗ x = x. The proposition is identified by the following information:

Pure Prop Id: 25b4a10707d2aab2f4bcc2756fdd2e3abf00c1f705db1132de37a2fc6ea6f637
Pure Prop Address: TMMvdyQpdr8L3T42os9auY5Nm8FUhCGnw76
Theory Prop Id: a03ccb9f27af94b22542ec92ba2b8ba628d78ad3d57905ee68cf0d731ae618ef
Theory Prop Address: TMXFnDHcsSjoFcQzEX5A9GawZJ2M6iZkEV4

RealsStruct_distr_L

Theorem 28.224 ∀xyz ∈ R.x ∗ (y + z) = x ∗ y + x ∗ z. The proposition is identified by the following information:

Pure Prop Id: 7f277f99738c86e4cd0c17ce27f55256d4b83bac8c3ec324b94d179eb523992c
Pure Prop Address: TMMJoedUhPVrXxiACByyi6RA7wzPg2bP8AE
Theory Prop Id: 80b5a5b1ce589268084c8d0ff372432a6afa48599f754381e6ecf72024d11b73
Theory Prop Address: TMUbkahieeSRSSdoPifi8CERkncX96vdrCA

464 CHAPTER 28. SURREAL NUMBERS II

RealsStruct_leq_refl

Theorem 28.225 ∀x ∈ R.x ≤ x. The proposition is identified by the following information:

Pure Prop Id: 868cecbda36c1ae46534769ebbdb84ef47482569c02f1c634ef3b569a7a12c94
Pure Prop Address: TMUGH3okNJP2QDn6HWyznB2itjVp93X4N2M
Theory Prop Id: 16a3fb53c4f1ddec154a070ed94ab00c6d6f96a8618fb21ee9e7cf9b7c48e5b1
Theory Prop Address: TMNhQCrwQkgbWcYiN9bZNZmiH341kRj81fx

RealsStruct_leq_tra

Theorem 28.226 ∀xyz ∈ R.x ≤ y → y ≤ z → x ≤ z. The proposition is identified by the following information:

Pure Prop Id: 09ea87b2c8bb2a8ef1d5e4e3413f6ed17268c09df43a45348a5029aef49aec25
Pure Prop Address: TMWjFSH117sB2fX5RBJDyEzKLmyYTS2S17B
Theory Prop Id: 31d22cb5ee22963edf60c65a1536469fb0185ae68587cf700c23504bb704d4a9
Theory Prop Address: TMUUyEqPQ9BoJF9RsDg9cXEhU8SBZvmB7dv

RealsStruct_leq_antisym

Theorem 28.227 ∀xy ∈ R.x ≤ y → y ≤ x → x = y. The proposition is identified by the following information:

Pure Prop Id: 13d82edccef1b70d008353ac64c4821740a0880f70e0e680d356e6d9d079c525
Pure Prop Address: TMHzbKmyZCrxTuS3pfvy16rZJqPRa2zbWR4
Theory Prop Id: 0aa45faf2f614bd2d02eaabb71f298ba2905b9e906d084f527121abf9d477ea5
Theory Prop Address: TMNKwtGqpbPqDk86iVJJmWZZzeTFCETXo97

RealsStruct_leq_linear

Theorem 28.228 ∀xy ∈ R.x ≤ y ∨ y ≤ x. The proposition is identified by the following information:

Pure Prop Id: 8492e0db1af1c27447e1da622672c190501a41929ea1a5d824d99bf5c02e2cd4
Pure Prop Address: TMJpogMX6MJSnbVEpoyobdqAxQAvxSa6593
Theory Prop Id: 1f84f249a4223c91c6d7a2a24ab7e57bb4917d1c8b23d477391ddfe94e41e9a0
Theory Prop Address: TMN3dF6E7pq8oupzx7bGMNiMPz5ir8mq112

RealsStruct_leq_plus

Theorem 28.229 ∀xyz ∈ R.x ≤ y → x + z ≤ y + z. The proposition is identified by the following information:

Pure Prop Id: 8a4ab12e1315b912d8d64a150d7f14264405d48f8b09b19e281e9f7f5babc606
Pure Prop Address: TMcTUgJ6Y2J84VUxaEX5jmfrXQaXknJHnsS
Theory Prop Id: 37b323e19944cb2094fbf59f40ee6dc75941a5e8638cd782e29f1595ba2feb8d
Theory Prop Address: TMVdpCNvU6P7MxmUMqGyRQUPygZTUwsQj7w

28.11. REALSSTRUCT 465

RealsStruct_lt_leq

Theorem 28.230 ∀xy ∈ R.x < y → x ≤ y. The proposition is identified by the following information:

Pure Prop Id: 034e19d8815d7cac710f94e707f400dcc18e80e521efb31f4f07c41cc7eb551e
Pure Prop Address: TMRG27Re6ZfnLLrivhufYbciXkVrRqvBff7
Theory Prop Id: 8b5db0b565ec35e47e13bb761e5bf3d54bc087a8eea8a3da96873a105ea5b67d
Theory Prop Address: TMRAfQdUMEz2W8xXfP2Fp3jgBADZyt3d5q9

RealsStruct_lt_irref

Theorem 28.231 ∀x ∈ R.¬(x < x). The proposition is identified by the following information:

Pure Prop Id: 718b2a7630e43c2aa1a7ba78859688706a27263cc0e14ac10b19d2502d90d0be
Pure Prop Address: TMT3hWZ2i8mzcX5AV3nDJbM74z3kxuqnWTT
Theory Prop Id: 2c2cd678622bfb74b4ccf1f72f60b44aaeda5e1673f8577488821e96d3567025
Theory Prop Address: TMGLCxFW2fPYxLxs6VnBV5otswJNkwPZmso

RealsStruct_lt_leq_asym

Theorem 28.232 ∀xy ∈ R.x < y → ¬(y ≤ x). The proposition is identified by the following information:

Pure Prop Id: 036716b7031d9e9545f672350877cdd3cd606d761131cc3ce1039205806864f1
Pure Prop Address: TMJhZqMGmM2ycUqQT2qouBhNpEHfUJNVWVH
Theory Prop Id: 5cf7bdfe85d03b1346539e85cb1f326864fbad71343355b85a957631353d7744
Theory Prop Address: TMVm3xj3nwvohAUcnSzL6EYJStDpJiCLKGX

RealsStruct_leq_lt_asym

Theorem 28.233 ∀xy ∈ R.x ≤ y → ¬(y < x). The proposition is identified by the following information:

Pure Prop Id: 2cf72677c60b93322ea19a0826dab959dc3147d59f5d9d3fe5aa8b7174ee2ba1
Pure Prop Address: TMPymWMasfNM2Rji1AUDQcQM4LrH9uWx3Va
Theory Prop Id: eee248a189252e629bc193acfb88c7bba2acdb08489e25f8f10c512171761595
Theory Prop Address: TMcmYMME185KoSKgmuzQKpMQLXMeqigGBdb

RealsStruct_lt_asym

Theorem 28.234 ∀xy ∈ R.x < y → ¬(y < x). The proposition is identified by the following information:

Pure Prop Id: d64488d8f76ef862d95e7112eece4f8a3f3c67e4b05abe79c8e71640fdc45d88
Pure Prop Address: TMViyVGCbgDmrEXs1auawYkVTQNQjDYXtNt
Theory Prop Id: 95a9252b4e98f9f989fdb3147e32cecbad9ba3739992d1cda07bac3cf57c3017
Theory Prop Address: TMSE7BJFjaB53kpm9JzcemAjmx17n4miSVo

466 CHAPTER 28. SURREAL NUMBERS II

RealsStruct_lt_leq_tra

Theorem 28.235 ∀xyz ∈ R.x < y → y ≤ z → x < z. The proposition is identified by the following information:

Pure Prop Id: 27e379d62bccda6def02d503ec80dce5751e1b6c06ea0598c0e4f12313a0c0a4
Pure Prop Address: TMXQk3yFeunwa2tgkwDRkTAiC72gnc1kEbQ
Theory Prop Id: 0ad1d49b39850f537c0804836720afa6e45354a2506c9efdf082c5b1daef964a
Theory Prop Address: TMYGBds2bCxGu39zPdjB8uTHjqLijWHi2KU

RealsStruct_leq_lt_tra

Theorem 28.236 ∀xyz ∈ R.x ≤ y → y < z → x < z. The proposition is identified by the following information:

Pure Prop Id: a7fac80d5d695102c89d549a0ac57e21a1da6b01abfff6821d8671d05ac5bfe9
Pure Prop Address: TMWRFzWY9Ks96UCp5hHDSSwUHcp2kHabMiZ
Theory Prop Id: 17dbb16615d06d0e91aeb16bd8d187a094d35156dfc10c9a0cb26df2949b1a10
Theory Prop Address: TMTNZLqjCyTkXHcJDFnDct7KWxJxFjE1gTZ

RealsStruct_lt_tra

Theorem 28.237 ∀xyz ∈ R.x < y → y < z → x < z. The proposition is identified by the following information:

Pure Prop Id: db51e413c8dae5e58659c817cad674fc9603959ac8908d0d3173b2f974bf800b
Pure Prop Address: TMbyGzMwnbYf7Ef87fxfadSAny6Pk5qsDtt
Theory Prop Id: a20edd4147dc9434a2c2a9b874d21178efbdb0a783f8456ad92be39a4e9cc999
Theory Prop Address: TMcywXpnEowovbp6p7WFLjPSpRAsCXmnz7r

RealsStruct_lt_trich_impred

Theorem 28.238

∀xy ∈ R.∀p : o.(x < y → p) → (x = y → p) → (y < x → p) → p. The proposition is identified by the following information:

Pure Prop Id: 94de406929eca9f0fbfb4a2208de46b59aa03b0f73d1c028a0551f7cff85ac9c
Pure Prop Address: TMPYj9ARJvgSpTSMs4ztnxZZeSPKhuoUoz5
Theory Prop Id: f9cff737117c67a0fa5ded42ee2d5363ce2958c66b29c20fc47c132e3eb9efdd
Theory Prop Address: TMF5ZzdDFzFLMP7TYe4pY4t96duPiB7Gv7f

RealsStruct_lt_trich

Theorem 28.239 ∀xy ∈ R.x < y ∨ x = y ∨ y < x. The proposition is identified by the following information:

Pure Prop Id: 9b61418aa95cad0821b7b70b1ea6c48b07b043fc6317f9c66ece033f5aa59bf6
Pure Prop Address: TMboWZEUUV8p5yiVECUimfWvDD1YMWBhHDh
Theory Prop Id: 65d4e63207e42cdaab1884c4d8cdfd112007705d830e2a0ccb44f59f328c8223
Theory Prop Address: TMbrq68wUob61FbNk4jqcD83WaJgVRjAJoe

RealsStruct_leq_lt_linear

Theorem 28.240 ∀xy ∈ R.x ≤ y ∨ y < x. The proposition is identified by the following information:

Pure Prop Id: b5d0e7fc3cc6bce2c2a17bc42c6f157d981ae1ed7c08efd6ac7ca23f0356d3f0
Pure Prop Address: TMXnJJbrRwtujq4xzLDYXsAr5wYR1aqSr18
Theory Prop Id: 7c7a8089b86844735c6b2d3f4a519d71e25477cdc5a960264b601472a84efadc
Theory Prop Address: TMEmCCr1R4uUv68FnD6w9CyYspAKC65Udc3

Notation. We use −− as a prefix operator corresponding to applying term Field_minus (Field_of_RealsStruct Rs).

RealsStruct_minus_eq

Theorem 28.241

∀x ∈R.
−x =
explicit_Field_minus R (Field_of_RealsStruct Rs 3)

(Field_of_RealsStruct Rs 4) (decode_b (Field_of_RealsStruct Rs 1)) (decode_b (Field_of_RealsStruct Rs 2)) x.

The proposition is identified by the following information:

Pure Prop Id: 47bc6dfa278b5babc8da67e3a1e0b527ec1d23809b12328023a26ca5c7249583
Pure Prop Address: TMYZAfb32ApJmMaB3PnEm7Vj4oTBY46NBAk
Theory Prop Id: 66a3bbfbbb3361050ff6013b6efced8992a5edba8cbd9899558ae7cc61dbb8a6
Theory Prop Address: TMS87Bxs4mXuUt9jfG6JifxZNzvF7zzmfJt

RealsStruct_minus_clos

Theorem 28.242 ∀x ∈ R.−x ∈ R. The proposition is identified by the following information:

Pure Prop Id: fdd8e0f6e300511edced7df68d989a615e67aa1bc7de750f9a619a82debfc807
Pure Prop Address: TMWr5eXBuGtfE7oLQbomsUzc6fjwmfid258
Theory Prop Id: b7c738e349ab2d6f6e8ad276adafcf81dc9aca3628028a9f35cb26fc2d5a34ef
Theory Prop Address: TMVd1PVi5iMBiKL7rtZswTGDLj8iQbjZCSh

RealsStruct_minus_R

Theorem 28.243 ∀x ∈ R.x + −x = zero. The proposition is identified by the following information:

Pure Prop Id: 09095b96433691485a96edf8101177ad9b033cef032350fa0ce5069fae93de9f
Pure Prop Address: TMLSWdagYu788EzMjhuZP6Qm21mcP7gao1q
Theory Prop Id: d7fd806b6ae1b4b46446aada0c9c8d453b985568eb38cc1ba86372ba219d557d
Theory Prop Address: TMMMVeZuEoKYrJsCZYGX7isAfmKNaheKNC6

RealsStruct_minus_L

Theorem 28.244 ∀x ∈ R.−x + x = zero. The proposition is identified by the following information:

Pure Prop Id: 98ac5d2df769fab37af56d31b7a7b0a24ac969c2c72d9889077705d5c24ceabf
Pure Prop Address: TMKzv2RXJzXuZKiMMKDdny2ePWZafZ7U5ZX
Theory Prop Id: 23218f089518e8a515bb2026967940f961d8643db6282c71fb89bbac1b93ce39
Theory Prop Address: TMLhepvAwS8gbHygkWi1J1B7BRAUEkXdFje

RealsStruct_plus_cancelL

Theorem 28.245 ∀xyz ∈ R.x + y = x + z → y = z. The proposition is identified by the following information:

Pure Prop Id: 5ff5748a79dfce201548e0ea8e7fc3e8ed66fae54c0079d8409b468ec617420e
Pure Prop Address: TMXHN9o2PJFBMaNmp83BfCNYAonHHWiGFzX
Theory Prop Id: ff6953bf63e902e9883d98d6d01c20077d6e823216570f791adbe886fb641882
Theory Prop Address: TMcrA1J9U6fRWwNyvbHM8Pvu7xQqVhBf1Vp

RealsStruct_minus_eq2

Theorem 28.246

∀x ∈R.

−x = explicit_Field_minus R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs) x.

The proposition is identified by the following information:

Pure Prop Id: e8ae10f25c6af12d1958559a4f188545a0cb40752c041c3f8d11eecc9bcd7199
Pure Prop Address: TMTjWE5eDPyLEbX1dfg7EizwEtFGXzUQkDt
Theory Prop Id: a1211f03836b5eee939c778de3c1018416ca8ed1d3ee2b73d2266429b7c3e3e2
Theory Prop Address: TMaFALex8WPZq7Zo2cyTerKK2nzDvTo1quC

RealsStruct_plus_cancelR

Theorem 28.247 ∀xyz ∈ R.x + z = y + z → x = y. The proposition is identified by the following information:

28.11. REALSSTRUCT 469

Pure Prop Id: baf30bc9e3bd74b52dd88cb1eea019d239e8ac999133603bb6c0ffe240f0585d
Pure Prop Address: TMJ8DeDPrVjLKi4AWwmCV15PJt4zh9Qea36
Theory Prop Id: a5736933cee0a9aaae630453df6f585065561f78e9b7df048786845e7dddfdc1
Theory Prop Address: TMRycJqeTxMBEZCvL8J2Hpzh8k9oRbHARWu

RealsStruct_minus_invol

Theorem 28.248 ∀x ∈ R.−−x = x. The proposition is identified by the following information:

Pure Prop Id: 08c894cf89246f6d453037ab7581b8a4980b918261d561476b8d1c569c9972f4
Pure Prop Address: TMZQm7b2b6rZPGFcUA5n5jUA5xm2TjgdNtJ
Theory Prop Id: 58988336a2e2c10ce0f8c945854017783f1fdb55c1a40daf0cdb8f1a17da8153
Theory Prop Address: TMExht3DFRB3GYdQvpgYztYDMSMkBYJGNpx

RealsStruct_minus_one_In

Theorem 28.249 −one ∈ R. The proposition is identified by the following information:

Pure Prop Id: bd1d4f7909be8f917eab926743f5502922a8a810a93cbd00d6412952a908801c
Pure Prop Address: TMWc1PSLG2857QLtbTuiWKgfqKfe8q9nLcz
Theory Prop Id: 39bedc18953a944c01c33894b19ee20e55993b517d311d2daa46cd1165c05d3b
Theory Prop Address: TMR42g5o3sV9VSPe54YRCdkizri2QPBCjT6

RealsStruct_zero_multR

Theorem 28.250 ∀x ∈ R.x ∗ zero = zero. The proposition is identified by the following information:

Pure Prop Id: e8baa64a646894666f2b0b82b5918f7ed9318f8f8eee4f59dbbfba3c20d9ff21
Pure Prop Address: TMPdW3hMJTztdGUdkDRakYTSjbabprAc9Z6
Theory Prop Id: 88201a486135db74001498754cf27b8685565cca99912ee888c9482032e13640
Theory Prop Address: TMX7vtUzERwfALngARjEZ8VQG9g4HjUN3xS

RealsStruct_zero_multL

Theorem 28.251 ∀x ∈ R.zero ∗ x = zero. The proposition is identified by the following information:

Pure Prop Id: 751db0caf92b3fe28e704ad517da9de59bb6d3b21c4e04d1f9b4a77d601f5c33
Pure Prop Address: TMH5oywcudUDCPXjnkAtprfxnT8Mx3M4znP
Theory Prop Id: 15727a520c07b04ff12d5865f874e26ce2c8daf0a2613a88bd45313469d1bb82
Theory Prop Address: TMU6ughQfPFJrRB8wkmoxv7fyit1ENwF82x

RealsStruct_minus_mult

Theorem 28.252 ∀x ∈ R.−x = (−one) ∗ x. The proposition is identified by the following information:

470 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: d488fae83059b5b9d5f6500d2855e253c82b63de6ffcbcc7a9a08a9e90364a54
Pure Prop Address: TMYWs7B7tsm6BLxQZXktxsTeJ7uiY9hTMj7
Theory Prop Id: 221fb8b61d0263c90b7a1b788de3aeb55c2d1c2061e15c7cee6e1abf97657d4b
Theory Prop Address: TMXHd2hW4LazoJxHtfBqjWYQHGqzvfC1yW7

RealsStruct_minus_one_square

Theorem 28.253 (−one) ∗ (−one) = one. The proposition is identified by the following information:

Pure Prop Id: cab4aa58bf0267bee8cf3c2536213ef66d714059ef2218d7e6f1728f28f8ee99
Pure Prop Address: TMPnEwkVaRzmRUkEJ1woHFG1DZk5uMWQaEU
Theory Prop Id: 701edc74dbc3b46a566cc9743166e0ab300f6a34691eebe4339c03df61b41a75
Theory Prop Address: TMQvuBviqrrzJPYbWJNSTnSuiYbaqYvrUro

RealsStruct_minus_square

Theorem 28.254 ∀x ∈ R.(−x) ∗ (−x) = x ∗ x. The proposition is identified by the following information:

Pure Prop Id: 95c7432ee6391c9809080d41eed33f798d0d274f825cb91bde1c81c29c2d0fb2
Pure Prop Address: TMb7xqhMnUxosCkgLM9VrW3WNqF4se2wayn
Theory Prop Id: 93bb8340ea11964abd12d112440b8853a2dbc1264d1238cd0571b528ea47232d
Theory Prop Address: TMaYV6vuKLDWHanfeHhxV3h8HcDTY2T7wrM

RealsStruct_minus_zero

Theorem 28.255 −zero = zero. The proposition is identified by the following information:

Pure Prop Id: a65ac7f737bccdfc3753590b8bce334772b3b6919a1922509b3d0063820c713e
Pure Prop Address: TMTqKgfPgRZixBhk5JsKaM8ZeTqva2dbwrH
Theory Prop Id: 31d61336fe6999362705f5feadf4f373b68c3c731a34d56603a78e289700a6fd
Theory Prop Address: TMYiSLxX166trdrKLyA51oNquqKN3M3YbqS

RealsStruct_dist_R

Theorem 28.256 ∀xyz ∈ R.(x + y) ∗ z = x ∗ z + y ∗ z. The proposition is identified by the following information:

Pure Prop Id: 862fd843939bd3632046f172b4ceaf3d4e6b2698b28cb880f14e41743a84cf8f
Pure Prop Address: TMGHAMzokV28j6GZ5fDvRnCYLM1h8jYg2jh
Theory Prop Id: ce3537a1b799f75f41b619dd337ad2faa0ab92ffa22088d81b931f86215f45cb
Theory Prop Address: TMc4SuM9sH5ftzoRUZc5jyapjLYJmbwrTMH

RealsStruct_minus_plus_dist

Theorem 28.257 ∀xy ∈ R.−(x + y) = −x + −y. The proposition is identified by the following information:

28.11. REALSSTRUCT 471

Pure Prop Id: bbbec91faa7e9e78adb2960de270de19dc98814dd4909c4e5f8279063acf2549
Pure Prop Address: TMEpdMaKhWF6yhGZvGSnnsfdxW3JE2HhaX7
Theory Prop Id: e102afbd1d47d23a82a7d28fb39b18fed4f9d1368508a7d3169ed416e1e3ecac
Theory Prop Address: TMXSKFTjeXzq6D3jVPWKqcXCcbSQehf31Mv

RealsStruct_minus_mult_L

Theorem 28.258 ∀xy ∈ R.(−x) ∗ y = −(x ∗ y). The proposition is identified by the following information:

Pure Prop Id: 87a9ee20b9e8f4696f85fe723b6595c3796cd59ace6d3cc05a2f8d4d42e69893
Pure Prop Address: TMccDiAHv34RsKwiFv5Ze6HutuaTy3ivq71
Theory Prop Id: 145734116522a560429224bd70db12b8ab512947aa0e49a218ed22e31a0e7111
Theory Prop Address: TMTFyy4RHjnYmm6dmcA1XLwxqwZju5r6roQ

RealsStruct_minus_mult_R

Theorem 28.259 ∀xy ∈ R.x ∗ (−y) = −(x ∗ y). The proposition is identified by the following information:

Pure Prop Id: 1dfc6deb04af278e1661d3df5a3bdea0daab232c9302727cd688b9c76e430dbc
Pure Prop Address: TMXtdQduGF54FwdCwf1pD3PBBy4rK8RihuY
Theory Prop Id: 8cd9e0d875d43b1ddc9c929156b184121abe145cc15e0d102566046cd722f439
Theory Prop Address: TMSbRFVgjqs3EKV1FxZLbyQFWKwbK5ZxJ6C

RealsStruct_mult_zero_inv

Theorem 28.260 ∀xy ∈ R.x ∗ y = zero → x = zero ∨ y = zero. The proposition is identified by the following information:

Pure Prop Id: 52e521e4ea57136109ff9c68732a22d7e0fce2c81fe7e0333f27725db442259e
Pure Prop Address: TMSLZXtw9SNRfYRMKsJi6bSZm5LUozCpXP8
Theory Prop Id: ca0065eef41293f5e9cd0d9a5017dcd5aa4f3456b0df6f2e56a71ae1b77b12ff
Theory Prop Address: TMT3MLk7YqUD9iX8atswcLFDca5cKNVBF5B

RealsStruct_square_zero_inv

Theorem 28.261 ∀x ∈ R.x ∗ x = zero → x = zero. The proposition is identified by the following information:

Pure Prop Id: 8aa0cd6d0c435ca74fff47f90162b7c2c3d6b793b431dc4b57459872f365c838
Pure Prop Address: TMaEmRhRXsGfs29zHkD7Vcbp4X5kNcSAapY
Theory Prop Id: d4c42484e082d273bd7d0b5e37d7d640a925b2221ce532e2e8bf25c52fb9e418
Theory Prop Address: TMNUaVt1RUxY5s8TcGo5CqbLi9uuLiivewh

RealsStruct_minus_leq

Theorem 28.262 ∀xy ∈ R.x ≤ y → −y ≤ −x. The proposition is identified by the following information:

472 CHAPTER 28. SURREAL NUMBERS II

Pure Prop Id: f49a3223293a899b90e3f0e2b2ccb4c7c143e078381e7b41b18f20349f7da516
Pure Prop Address: TMGUQNAZwd9CygmSnCMdCF9RyBCaAERrmJ3
Theory Prop Id: 56adcd399875c337ec8eb70665c36638054ac2e7b9ce43b5d38d0fd6d09d41fa
Theory Prop Address: TMMV3gE1eKPX49sr7atFfzx8pAxwxmwav9U

RealsStruct_square_nonneg

Theorem 28.263 ∀x ∈ R.zero ≤ x ∗ x. The proposition is identified by the following information:

Pure Prop Id: 1f7e7600170f3d0edf48eb23103a7e83310e8bcbbb489afe89d2a550424bc433
Pure Prop Address: TMX92WctujuKb5AX5z7JnBAbwpRUF1F1Yrn
Theory Prop Id: 116beb08289e1dae25c8f46eb411554c9409ef2e3486a7b37ca72edeeaa82c18
Theory Prop Address: TMHDYtHgvcDVHrwczd4cftPoYYYa2bG9w9x

RealsStruct_sum_squares_nonneg

Theorem 28.264 ∀xy ∈ R.zero ≤ x ∗ x + y ∗ y. The proposition is identified by the following information:

Pure Prop Id: 55dc2f3f44253c88b10d4b601b85eff9350d5214097a99dfdfbb9ed33ba8e291
Pure Prop Address: TMFA9PQVdmjfUj74vevkFTa7eHQkM1Vw17x
Theory Prop Id: e739ff92218b151c98245ecc8ad2ef578e165ca921eee5b84841fa4b11c3c743
Theory Prop Address: TMSDysdaJ68HZgUqpWWzxnY91V1BU9Eprtp

RealsStruct_sum_nonneg_zero_inv

Theorem 28.265

∀xy ∈R.zero ≤ x→zero ≤ y→x + y = zero→x = zero ∧ y = zero. The proposition is identified by the following information:

Pure Prop Id: a1f474df8b36a92e0b58e7c0515850cfcf881b7ddb216e72d98b55d27f765f6b
Pure Prop Address: TMQqUDVayaoT53ydRzz8cF1HJtqHjdQhbha
Theory Prop Id: 789223bfb56d1d985a4558cb85e56b74044e5dfa7a47009d14691aa59d1c5e97
Theory Prop Address: TMSUNEckz8DH6CXDWbVK7G39vzxmKYfFe5u

RealsStruct_sum_squares_zero_inv

Theorem 28.266 ∀xy ∈ R.x ∗ x + y ∗ y = zero → x = zero ∧ y = zero. The proposition is identified by the following information:

Pure Prop Id: 64bfc4ce2dead3e98b809e881e851e8a7b8d4a5992eef4830fcbb2460cb33ae9
Pure Prop Address: TMb4XesQGWXCWm9yHHnypZ9bBes3f2tF3D7
Theory Prop Id: c2b9c84106f6aa374eab5069dd30d750baf9110b257653b6b4701e79424f840f
Theory Prop Address: TMcj8zjWByWFUKfZtiUpAAS1zrBiXhL76KL

RealsStruct_leq_zero_one

Theorem 28.267 zero ≤ one. The proposition is identified by the following information:

Pure Prop Id: 75a9e5ea0655f6a7b81f9562d536cc979c304fee58c31e40455766da8a0b8491
Pure Prop Address: TMdQ75HNiDrTsVaaDbMqwf25EtKrdnktWAd
Theory Prop Id: fc79c3810891aeb6dd0f786541364fa69b5d2c6cc14b14c578108424308f783f
Theory Prop Address: TMWFE2LGFJYTijtjpZrCpn7C7MW7CXxMUks

Definition 28.39 RealsStruct_N is the opaque object of type ι identified by the following information:

Pure Object Id: 5e5ba235d9b0b40c084850811a514efd2e7f76df45b4ca1be43ba33c41356d3b
Pure Object Address: TMbcMfzuQfUfq11QLM6oT11DkR9A7LKHyxL
Theory Object Id: 49868cd01ccaaa52966256a5255beef5cfc5d2b96374f80d5069a1170c2c4c78
Theory Object Address: TMKcBDt3UC9DqpmqhzUAWU2YJ3EuMHE5yEh

Let N be RealsStruct_N.

RealsStruct_Arch

Theorem 28.268 ∀xy ∈ R.zero < x → zero ≤ y → ∃n ∈ N.y ≤ n ∗ x. The proposition is identified by the following information:

Pure Prop Id: 3d2ba33455345a9c0ba6e7d86b15bcead0c25d5b93f9dfdb5a52b615ac9babfb
Pure Prop Address: TMWCft8x2BTk4nokUerxJK2X11JdeiVv565
Theory Prop Id: 1f0d5fe981bc213e8f88cb7c74a9bf90e20b53b02288f897c78dfc099ab3dd61
Theory Prop Address: TMZGPa1mwgbVzkFdodAv6RjSmb9g3Bhh3hq

RealsStruct_Compl

Theorem 28.269

\begin{matrix}∀ab ∈R^{N}.(∀n ∈ N.a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n)→ \\ ∃x ∈ R.∀n ∈ N.a n ≤ x ∧ x ≤ b n. \end{matrix}

The proposition is identified by the following information:

Pure Prop Id: 6a20f5ecaafcc7e432b7c8ac16d2d71517ad446546abe4f692d40edd5f175aab
Pure Prop Address: TMSCeZV9tj3Dc22WnMpqqUpBYCFvYAsqBkc
Theory Prop Id: bf5231ea148f4c4a8e2c8b3b3d3ac74a7a34a6f05496134d190b71e9d903d2eb
Theory Prop Address: TMZEYVRwMZfXMAwxS5gUKffKiFvE5u1BQFx

RealsStruct_natOfOrderedField

Theorem 28.270 explicit_Nats N zero (λm.m + one). The proposition is identified by the following information:

Pure Prop Id: 6b43192d1dea3e3fc85a6591e66c6a726082864fd516780b26a754b19d6118ef
Pure Prop Address: TMK4NwKLMJh9YDNuNjVZk7onpb3hJRVZcqP
Theory Prop Id: 49002473fe506d6b9c9c98fcd03504abba2716d63b6a552bc1c8b880dc2511ed
Theory Prop Address: TMUHoaa1stUMPCnj1neeSaXtn79MZ9CQ2cT

Definition 28.40 We define RealsStruct_Npos to be {n ∈ N|n ≠ zero} identified by the following information:

Pure Object Id: 18dcd68b13ac8ef27a4d8dfec8909abfa78ffbb33539bea51d3397809f365642
Pure Object Address: TMXGAJt8GRxM5LRMTFpsj1piXUY7f9nuK7P
Theory Object Id: f5d38003115cfc06f11ecf0699e5b6ecab522d53550fce92c2ecdc20371c12e5
Theory Object Address: TMMXxyZsUkoJFF2GqJHfHwyksqYKy5LeJWe

Let Npos be RealsStruct_Npos.

RealsStruct_PosNats_natOfOrderedField

Theorem 28.271 explicit_Nats Npos one (λm.m + one). The proposition is identified by the following information:

Pure Prop Id: 608937f38d90199679a353df12d26e9c24924a2849bcb600a52f62801cf8c8b1
Pure Prop Address: TMH3THbHi69Lk6JSBHKSnCQykxJcsLKkScy
Theory Prop Id: 7be7ef55dfa7ee734330d79a5f088736665cccd7e5615ce026a8457ca9690f34
Theory Prop Address: TMUwcaA4tPYCgfSCsdNGLV1J2Wj2UjJKYcT

Definition 28.41 RealsStruct_Z is the opaque object of type ι identified by the following information:

Pure Object Id: 736c836746de74397a8fa69b6bbd46fc21a6b3f1430f6e4ae87857bf6f077989
Pure Object Address: TMZnqja29W4HpFDK7otUSfGC3Eomh9qtUB8
Theory Object Id: 0610d20138c846183403e57b8540fd87c9759a321e2a636ed8b592691ebcc810
Theory Object Address: TMUL7MYE83ADM6VaAkyBjtEUt5rRp6pVBu3

Let Z be RealsStruct_Z.

Definition 28.42 RealsStruct_Q is the opaque object of type ι identified by the following information:

Pure Object Id: 255c25547da742d6085a5308f204f5b90d6ba5991863cf0b3e4036ef74ee35a2
Pure Object Address: TMQNHtJRR1iyQCK2yVPcXNmgjvYt4LJWjPH
Theory Object Id: 0a4db083012656ded8a542881fa97da3998c2c25649b64f9c88525f48f727d53
Theory Object Address: TMHUMp4nRyqMcV6TWWc9HRr7HJygDDY4KSh

Let Q be RealsStruct_Q.

RealsStruct_Npos_props

28.11. REALSSTRUCT 475 Theorem 28.272

∀p :o.
(Npos ⊆ R →
explicit_Nats Npos one (λm.m + one) →
one ∈ Npos →
(∀m ∈ Npos.m + one ≠ one) →

(∀m ∈ Npos.∀q : ι → o.q one → (∀n ∈ Npos.q (n + one)) → q m) → (∀nm ∈ Npos.explicit_Nats_one_plus Npos one (λm.m + one) n m = n + m) → (∀nm ∈ Npos.explicit_Nats_one_mult Npos one (λm.m + one) n m = n ∗ m) →

(∀nm ∈ Npos.n + m ∈ Npos) → (∀nm ∈ Npos.n ∗ m ∈ Npos) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: ac462c732947010e996e2195f9457d22c91ea98709503c70dfd886bb0ff05a86
Pure Prop Address: TMUvjp5vzLmJS9vokz3WxDwgYtiFC9YfGot
Theory Prop Id: a2de5a1838f8c2dedd49986ce8e9b16887f479df3ea69491847fdc1ff939d7e0
Theory Prop Address: TMLHJgX12h6ob8HcMEAHsRmu8Sv6z8hA4Vg

RealsStruct_Npos_R

Theorem 28.273 Npos ⊆ R. The proposition is identified by the following information:

Pure Prop Id: 008a7ec9f81b901d173d4bb67cd3b9f8fd59e4f1d6086a46e01df9b0379a24ac
Pure Prop Address: TMJ4VGyP4MLVJHuj3qvzEBau5k5S6d87mJp
Theory Prop Id: fdd42c49268c6b0f74f093dd11460a4c3f87341db2ac930dc62f4e370c959def
Theory Prop Address: TMFUveydbCfNR11n2X6XYqhXH9GGB3eCa7E

RealsStruct_one_Npos

Theorem 28.274 one ∈ Npos. The proposition is identified by the following information:

Pure Prop Id: 060e811ee11a2fdd622c33063c392284ab0e32d1798a3c22f69b0e2cc87e0bb3
Pure Prop Address: TMVQ8Vzx4rfk9PgrXA5DC8bupwA7Rc71ciY
Theory Prop Id: adbd044729f5641d3c3402b8dae736e674669ab85664a907a855a441004cb866
Theory Prop Address: TMV8pdo8evmeKAvbv1pzGuyB3FWK7iaDugd

RealsStruct_Z_props

476 CHAPTER 28. SURREAL NUMBERS II Theorem 28.275

∀p :o.
((∀n ∈ Npos.−n ∈ Z) →
zero ∈ Z → Npos ⊆ Z → Z ⊆ R →

(∀n ∈ Z.∀q : o.(−n ∈ Npos → q) → (n = zero → q) → (n ∈ Npos → q) → q) → one ∈ Z → −one ∈ Z →

(∀m ∈ Z.−m ∈ Z) →
(∀nm ∈ Z.n + m ∈ Z) →
(∀nm ∈ Z.n ∗ m ∈ Z) → p)
→p.

The proposition is identified by the following information:

Pure Prop Id: 017fe94adff66febe9607cfa8e37ab6ccbb7b674ee4fb54a05fdde026adcf09d
Pure Prop Address: TMSzNh6NafcQ7JF4kYewkBLEW5kuczQKhgd
Theory Prop Id: 5f0a7e775f83d554f276621ef295027944bb5559b7afbb223b8ddfc6b44e546b
Theory Prop Address: TMQPzwy4S512aDpt1mBbU3fhUfvi6C5QvWZ

RealsStruct_neg_Z

Theorem 28.276 ∀n ∈ Npos.−n ∈ Z. The proposition is identified by the following information:

Pure Prop Id: f9e62d63a307ece18bbd80aa48a23f6f4d17cc93e2d4ced458eb45a3505e2639
Pure Prop Address: TMZmeurK9YG3FXNye9gkxqjf5hSGk8queat
Theory Prop Id: 4d43ebd17d60a85f9c79d798f7b371ae521f4a7bff8c51313295f609080b380a
Theory Prop Address: TMbm6VFrtyUyFrUCb5N5iv3gJ2ebnXd96kk

RealsStruct_zero_Z

Theorem 28.277 zero ∈ Z. The proposition is identified by the following information:

Pure Prop Id: e63b5bd24a1e20be4e359c4763b33e6b62e4af0ee2eb346b98a874d8b75cdf72
Pure Prop Address: TMZZPadL877aUv7uy4BJ4ZVoUgXHzpSDS2p
Theory Prop Id: d617fb71b2447bb5050839c73d379af276584315cac62d0fc944a5d871c873b7
Theory Prop Address: TMQkD9oz6MNGEZp6SBtWuDTTfrftxMyra1p

RealsStruct_Npos_Z

Theorem 28.278 Npos ⊆ Z. The proposition is identified by the following information:

Pure Prop Id: 3b82b4daba391f17879dad1f437cec7ff368e07aaae1e064ae35d652ed72ec7b
Pure Prop Address: TMHtwXjtrWbWqU2d751pwq11Lj48hsv1V9i
Theory Prop Id: 654198ba4994329078fbb885ead1adc3e09120276bfd4095fccaf00cd2c50a5f
Theory Prop Address: TMQdysxG2P8Lkp3CTWLhBkp2LqNBQWWr6Ak

28.11. REALSSTRUCT 477

RealsStruct_Z_R

Theorem 28.279 Z ⊆ R. The proposition is identified by the following information:

Pure Prop Id: afb222581fed3232153f5f2d44da3bbf0ce181c57357472cf15a82a3d810681e
Pure Prop Address: TMPwERt3rdFJJxsDnxdhcQgF2MpMFsf3i1V
Theory Prop Id: 9b0ab1796c671927fbee291f23eb65871f651953d2f84aa13bc10861a90fcac1
Theory Prop Address: TMGkAuVyoZcR48zuQfsrquFkDgtyiNcdENZ

RealsStruct_Q_props

Theorem 28.280

∀p :o.
(Q ⊆ R →

(∀x ∈ Q.∀q : o.(x ∈ R → ∀n ∈ Z.∀m ∈ Npos.m ∗ x = n → q) → q) →

(∀x ∈ R.∀n ∈ Z.∀m ∈ Npos.m ∗ x = n → x ∈ Q) → p)

→p.

The proposition is identified by the following information:

Pure Prop Id: df79085a1a2f4f031815dcd379e51112884d05ae157fcb74a8357af5dbe034eb
Pure Prop Address: TMHAByNoEuG6wgDbba13PkCr4En34QBgFU2
Theory Prop Id: 61a4917a561c1dba953134725332117c4984e25d8b4082687743fbe846b3ddbb
Theory Prop Address: TMFXPuxaie6AGgwAYXkFdSq3ph8iYboGfCE

RealsStruct_Q_R

Theorem 28.281 Q ⊆ R. The proposition is identified by the following information:

Pure Prop Id: b58f7fc652d381159596cca7859b3e14f253ce70a6c3e82c028fd369e27be76d
Pure Prop Address: TMPV2a7aaKXBnCmnZWeiVvEJgyUM2fJ74rB
Theory Prop Id: 2ff2bd5001c8ea913870f94bda57e521ab15c3861a373ea631397059ee56519a
Theory Prop Address: TMExUYP3PJpDU1qNcRiaxE1KgojHSA3GBtE

RealsStruct_Z_Q

Theorem 28.282 Z ⊆ Q. The proposition is identified by the following information:

Pure Prop Id: c1725ba9fa1a646ac94926c0e35963caee38e306b5ea5c0b8035f91f3edf3415
Pure Prop Address: TMZAUPnvZW22Vu1BjruFvkzzd99KRdyCUdr
Theory Prop Id: a45e948eb32abcb85b27b170a391ad1b87a9608ce8a8b0a2ca2474cc443dc97e
Theory Prop Address: TMNdjiF5RG9zcoL2KHoDujJUXF1wJoxRKBu

Notation. We use ÷ as an infix operator corresponding to applying term Field_div (Field_of_RealsStruct Rs).

RealsStruct_div_clos

Theorem 28.283 ∀x ∈ R.∀y ∈ R \ {zero}.x ÷ y ∈ R. The proposition is identified by the following information:

Pure Prop Id: ec4148787b44fb5fb3cdf6e61529dd3330a1e03298e8bbb7f08346f738f7db84
Pure Prop Address: TMJcMfeW6RzhudXxfGUAHHUfTerg3RCmgMh
Theory Prop Id: d6d8aa472b7b2bd90954c9998de9b5a34d99b67efd39be56f8b6218f52cd2666
Theory Prop Address: TMJZqMkKWwmAcVqoAbsN2xJM74ofu6EF33P

RealsStruct_mult_div

Theorem 28.284 ∀x ∈ R.∀y ∈ R \ {zero}.x = y ∗ (x ÷ y). The proposition is identified by the following information:

Pure Prop Id: e2c1b9283ef45d7416d60f2ca96fe1d920af9c0d51788f43831368a679fa3ada
Pure Prop Address: TMX4EqbemoaxGX4DUARDb4VZRc5ZLJMnF3R
Theory Prop Id: b7ef46712de5c007a18453e0201b1b57274c06e0a53cd5a6ca5b05df985c4296
Theory Prop Address: TMds3wgEHVh1PGSrKhMkyaYaGRFs33AcAF5

RealsStruct_div_undef1

Theorem 28.285 ∀xy.x ∉ R → x ÷ y = 0. The proposition is identified by the following information:

Pure Prop Id: 6dfdbeeddd4c9724f299edd163a3a6873b04737b480b6dc8622825fe39c1c4dc
Pure Prop Address: TMc3no3gvLt4aCZ3tswEMYfjDATjdvWyatp
Theory Prop Id: 351a0c5a15c5c52ea191c34690d189cbd566f2471f004c24cbe5973f3f13e435
Theory Prop Address: TMKJ4U4k2tPg572kNK69NYJYjNPh6PL3NV7

RealsStruct_div_undef2

Theorem 28.286 ∀xy.y ∉ R → x ÷ y = 0. The proposition is identified by the following information:

Pure Prop Id: f6f4785d660ace5156add885c383a28b7bc3a99a73d8631604e4197e578adf19
Pure Prop Address: TMWAnh5soQNHhi3Ykqn6R2ibResPkSHppAi
Theory Prop Id: 38a3ac5076648eec3d5a9c2e50962482b5cbee9811940dc2427c63819c1ca1f1
Theory Prop Address: TMH1mYFWp41cUUu5C87Kq6Db727u6UDJH3K

RealsStruct_div_undef3

Theorem 28.287 ∀x.x ÷ zero = 0. The proposition is identified by the following information:

Pure Prop Id: 36b6fd6a7618c15898b200e117ddef4e3fe1f0afb35443199399f43ed8036266
Pure Prop Address: TMch6Tu6CKzcYBgw8HNboBazjtiajd7yott
Theory Prop Id: 269a3881564868f7961f0c10bad584f0aaaf8295d8a8d2960f2f7fd1e55d6fd8
Theory Prop Address: TMM4gQM96yMEzmMaUmB5ZXKT5bDGqjMZCBZ

Notation. We use superscripts as notation corresponding to applying term CRing_with_id_omega_exp (Field_of_RealsStruct Rs).

RealsStruct_omega_exp_0

Theorem 28.288 ∀x.x^{0} = one. The proposition is identified by the following information:

Pure Prop Id: 4f6df1679e5256f3dee7aa33e1d3edf948a982563cef834884ba4637ff642cfd
Pure Prop Address: TMPSGSuE8LXDV4V4pvTtsJEu5FNyH1pBH5q
Theory Prop Id: e9266c9ac7734a0908ed30b0412c09ac2fdda52a8a7deb8bfa1f329e7fb57079
Theory Prop Address: TMTnnud9giEgqcthVyEpMdJL9R2n8gzrP43

RealsStruct_omega_exp_S

Theorem 28.289 ∀x.∀n ∈ omega.x^{(ordsucc n)} = x ∗ x^{n}. The proposition is identified by the following information:

Pure Prop Id: 02959608ed47b98e7731a645df3547cd11df919eee7b8db3daea36611099ac2b
Pure Prop Address: TMVd93sgsKDaad1CxNYVmDVHrLcidvdTG4V
Theory Prop Id: f566c18daaed324edc80d61391474e1103d351c36f7ff173e024f2c1aafb71d1
Theory Prop Address: TMPtaovCUa2PSiq9AjqGhtJZYWsZ7ChGjWW

RealsStruct_omega_exp_1

Theorem 28.290 ∀x ∈ R.x^{1} = x. The proposition is identified by the following information:

Pure Prop Id: 68407b69e24d974f37306277e9073efc244d1ef9140fa0f4a8f4dd184e3a5a2a
Pure Prop Address: TMSx7vspXi4hq9Zfc3jM1CSRCdp3Zc9NGRU
Theory Prop Id: 0c96c808e943389302e04e16c0d612284ef50232f566ab8f9ee19bc68a0f29ee
Theory Prop Address: TMKrzNRVaVw4oMu8nZ96vwqWBiBXmMPztXd

RealsStruct_omega_exp_clos

Theorem 28.291 ∀x ∈ R.∀n ∈ omega.x^{n} ∈ R. The proposition is identified by the following information:

Pure Prop Id: eddf030a23ad3950942cdc2fbcd92ac5593fa9938ea1aae2da771538223f24df
Pure Prop Address: TMNd3EMECuhQHTRvrFDgzibwvZjig6hvURt
Theory Prop Id: ce17ebf726a60585318ead733cb7b5ab80320f5ea8b27994da5e5e3d5ce67456
Theory Prop Address: TMYuofpdnGFN8PpDrLruYivMbNZMsjVDEPo

Definition 28.43 RealsStruct_abs is the opaque object of type ι → ι identified by the following information:

480 CHAPTER 28. SURREAL NUMBERS II

Pure Object Id: fcf3391eeb9d6b26a8b41a73f701da5e27e74021d7a3ec1b11a360045a5f13ca
Pure Object Address: TMMiiibxSu8KWC8MJw2qM5UvvwMNYkqgGwE
Theory Object Id: 480d2ba6a3a558d8d5642b2e68a88cdc65a67cf6bfe18dd0574c2b3d39d69d05
Theory Object Address: TMS4FH6cQhUENopB1oi9sMyEapmb4SckEPg

RealsStruct_abs_clos

Theorem 28.292 ∀x ∈ R.RealsStruct_abs x ∈ R. The proposition is identified by the following information:

Pure Prop Id: 616c66d879b5e2f3a98bb920d8521cc473629f43225754cc970c502c02b4d70b
Pure Prop Address: TMUJhPP2L2evvXGruzGDgRSyLBqRLkJsvew
Theory Prop Id: 5a1facaf2b22cf8313c47ab5d00dd1d44a7f68dcc498ee31e588070e06e5ee9a
Theory Prop Address: TMPi25yLHBrYYUawSVuNTAdtuM1J4ztYFgp

RealsStruct_abs_nonneg_case

Theorem 28.293 ∀x ∈ R.zero ≤ x → RealsStruct_abs x = x. The proposition is identified by the following information:

Pure Prop Id: 883e07037d0bc45c4917edc7b4cb14370ef1994f68ceb7ca887b0471e7efe2ce
Pure Prop Address: TMMrHyd4NfDirtPZFyor6Lq39dp81qYpayF
Theory Prop Id: 6d71adbe5160105d6dc17d08068cceebbdc2513ee44ef22f79096d4f8800b735
Theory Prop Address: TMcT4sXVUoiheLTy9BFaCAh8cxDxrU7E51w

RealsStruct_abs_neg_case

Theorem 28.294 ∀x ∈ R.x < zero → RealsStruct_abs x = −x. The proposition is identified by the following information:

Pure Prop Id: e8df199bfa3379375d084412f37f48b94a51f0ab84bb93db7eee356000d6afef
Pure Prop Address: TMF7zwqh96P1YidqPVMDC2qQyYrNC6tSu2M
Theory Prop Id: dc229d0f2268fec24b49f688c19646ae7a2e4a681cd5a1de6e740b17385573ba
Theory Prop Address: TMFDA2267ptXsetvPjSP9B3kKZxjxiVe5eW

RealsStruct_abs_nonneg

Theorem 28.295 ∀x ∈ R.zero ≤ RealsStruct_abs x. The proposition is identified by the following information:

Pure Prop Id: 3bae2c2e1b39e5830ba31107cc5578a7b625b83fec87496a6ae294027caaff03
Pure Prop Address: TMFabfpzFQBWazTGbdRKSFsh8H6hm1abtqe
Theory Prop Id: 701a7053ca066eed86f0442240fc617b8c699f3d2e403833bad55bfc42fbe7db
Theory Prop Address: TMK2MLY5rofcEXZj5urmPDkpnsWbeYS4r1w

RealsStruct_abs_zero_inv

Theorem 28.296 ∀x ∈ R.RealsStruct_abs x = zero → x = zero. The proposition is identified by the following information:

28.11. REALSSTRUCT 481

Pure Prop Id: 1c1613207ea57ba5de38eb117ab32b134385a248879098c9d978b0898f59be2f
Pure Prop Address: TMaZstUdzVeUwd6dG1Lk3yajQ9jPpSDmhtx
Theory Prop Id: 480222f2b6e6e15591c55f3c22675fc071131e50cca95468ad1137f79bfc20cc
Theory Prop Address: TMSwQ4V24ucAD2y6Y2NzBFzWw24QhLA9R2Q

RealsStruct_dist_zero_eq

Theorem 28.297 ∀xy ∈ R.RealsStruct_abs (x + −y) = zero → x = y. The proposition is identified by the following information:

Pure Prop Id: 0342d3a1b4534f4505f3b6ec468772cbfcd3cabf927b470703dcc7905876e881
Pure Prop Address: TMSygnAwf3Z1Nmyt9mu1mUiAZNnmpc8dwZS
Theory Prop Id: d65f00cc24e6beb44968f1f8cd4eca4f2c99cc6a6e3fad31d5ee330fc5cba95b
Theory Prop Address: TMG37mDdnRwQ9CZaHY6Sj3WmgU6SMiTvErN

Definition 28.44 We define RealsStruct_divides to be λmn.∃k ∈ Npos.m ∗ k = n of type ι → ι → o identified by the following information:

Pure Object Id: 9b1a146e04de814d3707f41de0e21bf469d07ef4ce11ae2297f43acd3cb43219
Pure Object Address: TMSgvVAsMPWzeya7w14QXMLpm3oR2sg2Eaw
Theory Object Id: 26b0fdc348372e6f151cfb0d9a85fba282630eaa98d67d43c3e14d87ae4d7907
Theory Object Address: TMEjvmG5htm9fo5j3Pi2G5irTZNBxF7JyLe

Definition 28.45 We define RealsStruct_Primes to be {n ∈ N|one < n ∧ ∀m ∈ Npos.RealsStruct_divides m n → m = one ∨ m = n} identified by the following information:

Pure Object Id: 96c4c45b8c7f933c096a5643362b6a5eeb1d9e4d338d9854d4921130e6c8cdbe
Pure Object Address: TMZiAsgfRmCK6xaRVxCPcc2VzGp7DAXMkTD
Theory Object Id: 4f229104c32c2be9f21a21d40bf2c8848e077881ee9d18ea39563dd5055f3409
Theory Object Address: TMRZKCzNYB8mn6YTzXbD5KApbcoYvpYmMSv

Definition 28.46 We define RealsStruct_coprime to be λmn.∀k ∈ Npos.RealsStruct_divides k m → RealsStruct_divides k n → k = one of type ι → ι → o identified by the following information:

Pure Object Id: 4857fdedbdc33f33b2c088479c67bbbceae1afd39fca032cb085f9f741cedca6
Pure Object Address: TMbzphBsd7GLnrNR8Mi8GuGapc1Xgdxr2y6
Theory Object Id: 112ea9cf8a89d5ad797d290cd1c0b26c9e1d6d41ecead75d5c7c7c91bd7ae48a
Theory Object Address: TMX9ZHV8iS7puN2vhGXqUyRwbsJ5YunXJLK

Let Qs be pack_b_b_e_e Q (RealsStruct_plus Rs) (RealsStruct_mult Rs) zero one.

Field_RealsStruct_Q

Theorem 28.298 Field Qs. The proposition is identified by the following information:

Pure Prop Id: 8b600c18b435f30873008e1fd2e1a7754844b34c5ce89afb02dc5257a7d031f0
Pure Prop Address: TMLDM8xjNHCEb7tkdmX1V9EKpmqGB7zvxKT
Theory Prop Id: b55584754578f3ef26cef3b37955ee96ee21f6de68eca6b9d14e62a3e2800837
Theory Prop Address: TMH6n1ZFLmtKDSDJVw8XnR3Do4UFL4NSg1Q

Definition 28.47 We define RealsStruct_omega_embedding to be

nat_primrec zero (λ_r.r + one)

of type ι → ι identified by the following information:

Pure Object Id: 7e897f0ce092b0138ddab8aa6a2fc3352b0c70d9383c22f99340c202779c7a39
Pure Object Address: TMFS8k2m5BMHnDtneuBVexHr4VeDPJZVm2S
Theory Object Id: 8d1b1d11f8d7a646ada866cb891c4f97971a895e92e965d05e6be6a0bdc0a0ed
Theory Object Address: TMLEHo8yiHeRBmSAVenhHZym5uhudBeWgVA

Let emb : ι → ι be RealsStruct_omega_embedding.

RealsStruct_omega_embedding_N

Theorem 28.299 ∀n ∈ omega.emb n ∈ N. The proposition is identified by the following information:

Pure Prop Id: 25cb31475191d8d41f82ac1cc047f3940af8421d025ff7a7c78fb4dc2ffe1cbe
Pure Prop Address: TMPtZMNGEv6XXfgBKcBe8kTdZ3eDJkq7yCv
Theory Prop Id: 9c35b850fb3d8b0adb59d8c3695495c52fc50505407deb7cccb631635d035d6a
Theory Prop Address: TMXJrPBAAoNfUBWRhgmzF7gfY3kLs2rzR2q