Primitive. The name Eps_i is a term of type (set → prop) → set.
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
In Proofgold the corresponding term root is c3f0de... and proposition id is 756d6e...
Definition. We define True to be ∀p : prop, p → p of type prop.
In Proofgold the corresponding term root is f81b39... and object id is 94e6a7...
Definition. We define False to be ∀p : prop, p of type prop.
In Proofgold the corresponding term root is 1db705... and object id is 266ad5...
Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.
In Proofgold the corresponding term root is f30435... and object id is 0dc3c7...
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
Definition. We define and to be λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 2ba7d0... and object id is 37da83...
Notation. We use ∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
Definition. We define or to be λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 957746... and object id is 86ae64...
Notation. We use ∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
Definition. We define iff to be λA B : prop ⇒ and (A → B) (B → A) of type prop → prop → prop.
In Proofgold the corresponding term root is 98aaaf... and object id is 6588e5...
Notation. We use ↔ as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
Variable A : SType
Definition. We define eq to be λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type A → A → prop.
Definition. We define neq to be λx y : A ⇒ ¬ eq x y of type A → A → prop.
End of Section Eq
Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.
Notation. We use ≠ as an infix operator with priority 502 and no associativity corresponding to applying term neq.
Beginning of Section FE
Variable A B : SType
Axiom. (func_ext) We take the following as an axiom:
∀f g : A → B, (∀x : A, f x = g x) → f = g
End of Section FE
Beginning of Section Ex
Variable A : SType
Definition. We define ex to be λQ : A → prop ⇒ ∀P : prop, (∀x : A, Q x → P) → P of type (A → prop) → prop.
End of Section Ex
Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex.
Axiom. (prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
In Proofgold the corresponding term root is 00ac81... and proposition id is 1d1e56...
Primitive. The name In is a term of type set → set → prop.
Notation. We use ∈ as an infix operator with priority 500 and no associativity corresponding to applying term In. Furthermore, we may write ∀ x ∈ A, B to mean ∀ x : set, x ∈ A → B.
Definition. We define Subq to be λA B ⇒ ∀x ∈ A, x ∈ B of type set → set → prop.
In Proofgold the corresponding term root is 81c0ef... and object id is 317007...
Notation. We use ⊆ as an infix operator with priority 500 and no associativity corresponding to applying term Subq. Furthermore, we may write ∀ x ⊆ A, B to mean ∀ x : set, x ⊆ A → B.
Axiom. (set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
In Proofgold the corresponding term root is b3b698... and proposition id is 21238a...
Axiom. (In_ind) We take the following as an axiom:
∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X
In Proofgold the corresponding term root is 045211... and proposition id is 20faa5...
Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and.
Primitive. The name Empty is a term of type set.
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x ∈ Empty
In Proofgold the corresponding term root is 4da931... and proposition id is c7b0d7...
Primitive. The name ⋃ is a term of type set → set.
Axiom. (UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is bf4c26... and proposition id is 40660f...
Primitive. The name 𝒫 is a term of type set → set.
Axiom. (PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
In Proofgold the corresponding term root is 746e5a... and proposition id is d67d71...
Primitive. The name Repl is a term of type set → (set → set) → set.
Notation. {B| x ∈ A} is notation for Repl A (λ x . B).
Axiom. (ReplEq) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} ↔ ∃x ∈ A, y = F x
In Proofgold the corresponding term root is 4f2357... and proposition id is 471122...
Definition. We define TransSet to be λU : set ⇒ ∀x ∈ U, x ⊆ U of type set → prop.
In Proofgold the corresponding term root is 538bb7... and object id is e69f92...
Definition. We define Union_closed to be λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type set → prop.
In Proofgold the corresponding term root is 57561d... and object id is e45ed9...
Definition. We define Power_closed to be λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type set → prop.
In Proofgold the corresponding term root is 8b9cc0... and object id is d2954e...
Definition. We define Repl_closed to be λU : set ⇒ ∀X : set, X ∈ U → ∀F : set → set, (∀x : set, x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U of type set → prop.
In Proofgold the corresponding term root is 5574bb... and object id is 513534...
Definition. We define ZF_closed to be λU : set ⇒ Union_closed U ∧ Power_closed U ∧ Repl_closed U of type set → prop.
In Proofgold the corresponding term root is 1bd4aa... and object id is e1eccf...
Primitive. The name UnivOf is a term of type set → set.
Axiom. (UnivOf_In) We take the following as an axiom:
∀N : set, N ∈ UnivOf N
In Proofgold the corresponding term root is efcc16... and proposition id is e805d4...
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
In Proofgold the corresponding term root is 053575... and proposition id is cf4f5c...
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
In Proofgold the corresponding term root is 770e35... and proposition id is 0e7363...
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U
In Proofgold the corresponding term root is 042dda... and proposition id is 1f7933...
Axiom. (FalseE) We take the following as an axiom:
False → ∀p : prop, p
In Proofgold the corresponding term root is 0dcfe3... and proposition id is f9a230...
Axiom. (TrueI) We take the following as an axiom:
True
In Proofgold the corresponding term root is f81b39... and proposition id is eb5cbf...
Axiom. (andI) We take the following as an axiom:
∀A B : prop, A → B → A ∧ B
In Proofgold the corresponding term root is 3c4f59... and proposition id is 3bf198...
Axiom. (andEL) We take the following as an axiom:
∀A B : prop, A ∧ B → A
In Proofgold the corresponding term root is c2f006... and proposition id is 718be4...
Axiom. (andER) We take the following as an axiom:
∀A B : prop, A ∧ B → B
In Proofgold the corresponding term root is f66b82... and proposition id is 5caea9...
Axiom. (orIL) We take the following as an axiom:
∀A B : prop, A → A ∨ B
In Proofgold the corresponding term root is 55be8c... and proposition id is 382673...
Axiom. (orIR) We take the following as an axiom:
∀A B : prop, B → A ∨ B
In Proofgold the corresponding term root is 3c63fc... and proposition id is 8d8e40...
Beginning of Section PropN
Variable P1 P2 P3 : prop
Axiom. (and3I) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
In Proofgold the corresponding term root is 23ac9c... and proposition id is cb999b...
Axiom. (and3E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
In Proofgold the corresponding term root is 445f76... and proposition id is 50fabe...
Axiom. (or3I1) We take the following as an axiom:
P1 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is e44f34... and proposition id is ef9071...
Axiom. (or3I2) We take the following as an axiom:
P2 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 8df6e1... and proposition id is 827a91...
Axiom. (or3I3) We take the following as an axiom:
P3 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 017a00... and proposition id is 3b7f5a...
Axiom. (or3E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
In Proofgold the corresponding term root is 5d069d... and proposition id is 2120c9...
Variable P4 : prop
Axiom. (and4I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
In Proofgold the corresponding term root is ed32cb... and proposition id is 906aaf...
Variable P5 : prop
Axiom. (and5I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
In Proofgold the corresponding term root is 041ced... and proposition id is 11fc47...
End of Section PropN
Axiom. (not_or_and_demorgan) We take the following as an axiom:
∀A B : prop, ¬ (A ∨ B) → ¬ A ∧ ¬ B
In Proofgold the corresponding term root is 8733b0... and proposition id is 27db91...
Axiom. (not_ex_all_demorgan_i) We take the following as an axiom:
∀P : set → prop, (¬ ∃x, P x) → ∀x, ¬ P x
In Proofgold the corresponding term root is 73523e... and proposition id is 075ddf...
Axiom. (iffI) We take the following as an axiom:
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
In Proofgold the corresponding term root is 7516b9... and proposition id is 4d966a...
Axiom. (iffEL) We take the following as an axiom:
∀A B : prop, (A ↔ B) → A → B
In Proofgold the corresponding term root is 0e94c6... and proposition id is 9a07f6...
Axiom. (iffER) We take the following as an axiom:
∀A B : prop, (A ↔ B) → B → A
In Proofgold the corresponding term root is 54702a... and proposition id is f5dfc0...
Axiom. (iff_refl) We take the following as an axiom:
∀A : prop, A ↔ A
In Proofgold the corresponding term root is 01338d... and proposition id is 7b71b5...
Axiom. (iff_sym) We take the following as an axiom:
∀A B : prop, (A ↔ B) → (B ↔ A)
In Proofgold the corresponding term root is 00bac1... and proposition id is 11f51b...
Axiom. (iff_trans) We take the following as an axiom:
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
In Proofgold the corresponding term root is 363285... and proposition id is c3b8f1...
Axiom. (eq_i_tra) We take the following as an axiom:
∀x y z, x = y → y = z → x = z
In Proofgold the corresponding term root is 193bbf... and proposition id is 7ecbc4...
Axiom. (f_eq_i) We take the following as an axiom:
∀f : set → set, ∀x y, x = y → f x = f y
In Proofgold the corresponding term root is b0871b... and proposition id is 2a60fd...
Axiom. (neq_i_sym) We take the following as an axiom:
∀x y, x ≠ y → y ≠ x
In Proofgold the corresponding term root is 5c0ec9... and proposition id is 1a9898...
Definition. We define nIn to be λx X ⇒ ¬ In x X of type set → set → prop.
In Proofgold the corresponding term root is 36808c... and object id is 253451...
Notation. We use ∉ as an infix operator with priority 502 and no associativity corresponding to applying term nIn.
Axiom. (Eps_i_ex) We take the following as an axiom:
∀P : set → prop, (∃x, P x) → P (Eps_i P)
In Proofgold the corresponding term root is be36f0... and proposition id is 6a2832...
Axiom. (pred_ext) We take the following as an axiom:
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
In Proofgold the corresponding term root is e9e64b... and proposition id is d4b2d3...
Axiom. (prop_ext_2) We take the following as an axiom:
∀p q : prop, (p → q) → (q → p) → p = q
In Proofgold the corresponding term root is a9ede2... and proposition id is a56fc7...
Axiom. (Subq_ref) We take the following as an axiom:
∀X : set, X ⊆ X
In Proofgold the corresponding term root is 4c5395... and proposition id is 2efbac...
Axiom. (Subq_tra) We take the following as an axiom:
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
In Proofgold the corresponding term root is 79accb... and proposition id is fed662...
Axiom. (Subq_contra) We take the following as an axiom:
∀X Y z : set, X ⊆ Y → z ∉ Y → z ∉ X
In Proofgold the corresponding term root is d15e87... and proposition id is bd5a2f...
Axiom. (EmptyE) We take the following as an axiom:
∀x : set, x ∉ Empty
In Proofgold the corresponding term root is 93a77a... and proposition id is 8349ab...
Axiom. (Subq_Empty) We take the following as an axiom:
∀X : set, Empty ⊆ X
In Proofgold the corresponding term root is a517cd... and proposition id is 2faec3...
Axiom. (Empty_Subq_eq) We take the following as an axiom:
∀X : set, X ⊆ Empty → X = Empty
In Proofgold the corresponding term root is 2dd959... and proposition id is 988691...
Axiom. (Empty_eq) We take the following as an axiom:
∀X : set, (∀x, x ∉ X) → X = Empty
In Proofgold the corresponding term root is 3de0fd... and proposition id is 8b915b...
Axiom. (UnionI) We take the following as an axiom:
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
In Proofgold the corresponding term root is 51b549... and proposition id is bade05...
Axiom. (UnionE) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is db4bc1... and proposition id is ab781b...
Axiom. (UnionE_impred) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
In Proofgold the corresponding term root is 8efd57... and proposition id is d621b2...
Axiom. (PowerI) We take the following as an axiom:
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
In Proofgold the corresponding term root is 7143d7... and proposition id is 3c48d3...
Axiom. (PowerE) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
In Proofgold the corresponding term root is 149030... and proposition id is 647070...
Axiom. (Empty_In_Power) We take the following as an axiom:
∀X : set, Empty ∈ 𝒫 X
In Proofgold the corresponding term root is d2656b... and proposition id is 487855...
Axiom. (Self_In_Power) We take the following as an axiom:
∀X : set, X ∈ 𝒫 X
In Proofgold the corresponding term root is f81ef2... and proposition id is 8336ff...
Axiom. (xm) We take the following as an axiom:
∀P : prop, P ∨ ¬ P
In Proofgold the corresponding term root is 62e4c3... and proposition id is 7b6737...
Axiom. (dneg) We take the following as an axiom:
∀P : prop, ¬ ¬ P → P
In Proofgold the corresponding term root is 852f34... and proposition id is b7cc63...
Axiom. (not_all_ex_demorgan_i) We take the following as an axiom:
∀P : set → prop, ¬ (∀x, P x) → ∃x, ¬ P x
In Proofgold the corresponding term root is 9bfa06... and proposition id is 4b1474...
Axiom. (eq_or_nand) We take the following as an axiom:
or = (λx y : prop ⇒ ¬ (¬ x ∧ ¬ y))
In Proofgold the corresponding term root is 24ac29... and proposition id is d212ef...
Primitive. The name exactly1of2 is a term of type prop → prop → prop.
In Proofgold the corresponding term root is 163602... and object id is a0b944...
Axiom. (exactly1of2_I1) We take the following as an axiom:
∀A B : prop, A → ¬ B → exactly1of2 A B
In Proofgold the corresponding term root is 8e1e45... and proposition id is 4f94fa...
Axiom. (exactly1of2_I2) We take the following as an axiom:
∀A B : prop, ¬ A → B → exactly1of2 A B
In Proofgold the corresponding term root is c9fd90... and proposition id is 6c1ad9...
Axiom. (exactly1of2_E) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
In Proofgold the corresponding term root is 29b901... and proposition id is c67434...
Axiom. (exactly1of2_or) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → A ∨ B
In Proofgold the corresponding term root is 18af73... and proposition id is e092f6...
Axiom. (ReplI) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
In Proofgold the corresponding term root is b146c5... and proposition id is f84982...
Axiom. (ReplE) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∃x ∈ A, y = F x
In Proofgold the corresponding term root is 311cdf... and proposition id is d07e78...
Axiom. (ReplE_impred) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∀p : prop, (∀x : set, x ∈ A → y = F x → p) → p
In Proofgold the corresponding term root is 31e90a... and proposition id is 34fefd...
Axiom. (ReplE') We take the following as an axiom:
∀X, ∀f : set → set, ∀p : set → prop, (∀x ∈ X, p (f x)) → ∀y ∈ {f x|x ∈ X}, p y
In Proofgold the corresponding term root is e374fe... and proposition id is d4b7b0...
Axiom. (Repl_Empty) We take the following as an axiom:
∀F : set → set, {F x|x ∈ Empty} = Empty
In Proofgold the corresponding term root is 2af2ac... and proposition id is 0835bc...
Axiom. (ReplEq_ext_sub) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}
In Proofgold the corresponding term root is eb4c1d... and proposition id is 56aad0...
Axiom. (ReplEq_ext) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}
In Proofgold the corresponding term root is 81eb90... and proposition id is 781846...
Axiom. (Repl_inv_eq) We take the following as an axiom:
∀P : set → prop, ∀f g : set → set, (∀x, P x → g (f x) = x) → ∀X, (∀x ∈ X, P x) → {g y|y ∈ {f x|x ∈ X}} = X
In Proofgold the corresponding term root is 968c7a... and proposition id is 9ce73d...
Axiom. (Repl_invol_eq) We take the following as an axiom:
∀P : set → prop, ∀f : set → set, (∀x, P x → f (f x) = x) → ∀X, (∀x ∈ X, P x) → {f y|y ∈ {f x|x ∈ X}} = X
In Proofgold the corresponding term root is e15a6b... and proposition id is 5faff0...
Primitive. The name If_i is a term of type prop → set → set → set.
In Proofgold the corresponding term root is b8ff52... and object id is 76c4c0...
Notation. if cond then T else E is notation corresponding to If_i type cond T E where type is the inferred type of T.
Axiom. (If_i_correct) We take the following as an axiom:
∀p : prop, ∀x y : set, p ∧ (if p then x else y) = x ∨ ¬ p ∧ (if p then x else y) = y
In Proofgold the corresponding term root is 2a17ae... and proposition id is b5c26b...
Axiom. (If_i_0) We take the following as an axiom:
∀p : prop, ∀x y : set, ¬ p → (if p then x else y) = y
In Proofgold the corresponding term root is fde1a8... and proposition id is a9abd3...
Axiom. (If_i_1) We take the following as an axiom:
∀p : prop, ∀x y : set, p → (if p then x else y) = x
In Proofgold the corresponding term root is b1aa3f... and proposition id is a93c2c...
Axiom. (If_i_or) We take the following as an axiom:
∀p : prop, ∀x y : set, (if p then x else y) = x ∨ (if p then x else y) = y
In Proofgold the corresponding term root is 3e1439... and proposition id is 4de742...
Primitive. The name UPair is a term of type set → set → set.
In Proofgold the corresponding term root is 742438... and object id is dc3134...
Notation. {x,y} is notation for UPair x y.
Axiom. (UPairE) We take the following as an axiom:
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
In Proofgold the corresponding term root is 5009cc... and proposition id is e67fc5...
Axiom. (UPairI1) We take the following as an axiom:
∀y z : set, y ∈ {y,z}
In Proofgold the corresponding term root is b1eea2... and proposition id is bf3649...
Axiom. (UPairI2) We take the following as an axiom:
∀y z : set, z ∈ {y,z}
In Proofgold the corresponding term root is 27455b... and proposition id is a0a5b8...
Primitive. The name Sing is a term of type set → set.
In Proofgold the corresponding term root is bd01a8... and object id is b7dfc1...
Notation. {x} is notation for Sing x.
Axiom. (SingI) We take the following as an axiom:
∀x : set, x ∈ {x}
In Proofgold the corresponding term root is 6591bd... and proposition id is 2f93be...
Axiom. (SingE) We take the following as an axiom:
∀x y : set, y ∈ {x} → y = x
In Proofgold the corresponding term root is 9341fe... and proposition id is 3b0dd8...
Primitive. The name binunion is a term of type set → set → set.
In Proofgold the corresponding term root is 5e1ac4... and object id is e14989...
Notation. We use ∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.
Axiom. (binunionI1) We take the following as an axiom:
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
In Proofgold the corresponding term root is 753b6e... and proposition id is e43c8d...
Axiom. (binunionI2) We take the following as an axiom:
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
In Proofgold the corresponding term root is b8821c... and proposition id is 959535...
Axiom. (binunionE) We take the following as an axiom:
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
In Proofgold the corresponding term root is 5a3195... and proposition id is 1ceeb0...
Axiom. (binunionE') We take the following as an axiom:
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
In Proofgold the corresponding term root is e5b951... and proposition id is 570dd1...
Axiom. (binunion_asso) We take the following as an axiom:
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
In Proofgold the corresponding term root is e0f701... and proposition id is 61dfba...
Axiom. (binunion_com_Subq) We take the following as an axiom:
∀X Y : set, X ∪ Y ⊆ Y ∪ X
In Proofgold the corresponding term root is ee8f5a... and proposition id is d84076...
Axiom. (binunion_com) We take the following as an axiom:
∀X Y : set, X ∪ Y = Y ∪ X
In Proofgold the corresponding term root is e20f4f... and proposition id is bed33e...
Axiom. (binunion_idl) We take the following as an axiom:
∀X : set, Empty ∪ X = X
In Proofgold the corresponding term root is 8dc616... and proposition id is 52cd24...
Axiom. (binunion_idr) We take the following as an axiom:
∀X : set, X ∪ Empty = X
In Proofgold the corresponding term root is 67e82f... and proposition id is d5a005...
Axiom. (binunion_Subq_1) We take the following as an axiom:
∀X Y : set, X ⊆ X ∪ Y
In Proofgold the corresponding term root is b2c658... and proposition id is a9b3a8...
Axiom. (binunion_Subq_2) We take the following as an axiom:
∀X Y : set, Y ⊆ X ∪ Y
In Proofgold the corresponding term root is 3cf229... and proposition id is 1507ce...
Axiom. (binunion_Subq_min) We take the following as an axiom:
∀X Y Z : set, X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z
In Proofgold the corresponding term root is cc32a5... and proposition id is d64da4...
Axiom. (Subq_binunion_eq) We take the following as an axiom:
∀X Y, (X ⊆ Y) = (X ∪ Y = Y)
In Proofgold the corresponding term root is f8ee53... and proposition id is c2d393...
Definition. We define SetAdjoin to be λX y ⇒ X ∪ {y} of type set → set → set.
In Proofgold the corresponding term root is 1f3a09... and object id is 5bdda8...
Notation. We now use the set enumeration notation {...,...,...} in general. If 0 elements are given, then Empty is used to form the corresponding term. If 1 element is given, then Sing is used to form the corresponding term. If 2 elements are given, then UPair is used to form the corresponding term. If more than elements are given, then SetAdjoin is used to reduce to the case with one fewer elements.
Primitive. The name famunion is a term of type set → (set → set) → set.
In Proofgold the corresponding term root is b3e3bf... and object id is 241349...
Notation. We use ⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.
Axiom. (famunionI) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀x y : set, x ∈ X → y ∈ F x → y ∈ ⋃x ∈ XF x
In Proofgold the corresponding term root is 0c23d0... and proposition id is e4f490...
Axiom. (famunionE) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
In Proofgold the corresponding term root is 48683a... and proposition id is ede6e5...
Axiom. (famunionE_impred) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∀p : prop, (∀x, x ∈ X → y ∈ F x → p) → p
In Proofgold the corresponding term root is ab2266... and proposition id is 562023...
Axiom. (famunion_Empty) We take the following as an axiom:
∀F : set → set, (⋃x ∈ EmptyF x) = Empty
In Proofgold the corresponding term root is fc1bdd... and proposition id is 22bd10...
Beginning of Section SepSec
Variable X : set
Variable P : set → prop
Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z)
Let F : set → set ≝ λx ⇒ if P x then x else z
Primitive. The name Sep is a term of type set.
In Proofgold the corresponding term root is f336a4... and object id is 7008ec...
End of Section SepSec
Notation. {x ∈ A | B} is notation for Sep A (λ x . B).
Axiom. (SepI) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
In Proofgold the corresponding term root is ea56bb... and proposition id is 147a6a...
Axiom. (SepE) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
In Proofgold the corresponding term root is 102830... and proposition id is 00899a...
Axiom. (SepE1) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
In Proofgold the corresponding term root is d83445... and proposition id is 6399b0...
Axiom. (SepE2) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
In Proofgold the corresponding term root is aa61ca... and proposition id is 104227...
Axiom. (Sep_Subq) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
In Proofgold the corresponding term root is 97d080... and proposition id is c17d45...
Axiom. (Sep_In_Power) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
In Proofgold the corresponding term root is 612c94... and proposition id is fba53e...
Primitive. The name ReplSep is a term of type set → (set → prop) → (set → set) → set.
In Proofgold the corresponding term root is ec807b... and object id is 9bdb56...
Notation. {B| x ∈ A, C} is notation for ReplSep A (λ x . C) (λ x . B).
Axiom. (ReplSepI) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀x : set, x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}
In Proofgold the corresponding term root is 35b8ec... and proposition id is aa7b2b...
Axiom. (ReplSepE) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∃x : set, x ∈ X ∧ P x ∧ y = F x
In Proofgold the corresponding term root is 415ff1... and proposition id is b1bc76...
Axiom. (ReplSepE_impred) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∀p : prop, (∀x ∈ X, P x → y = F x → p) → p
In Proofgold the corresponding term root is 5d26e1... and proposition id is 063cea...
Primitive. The name binintersect is a term of type set → set → set.
In Proofgold the corresponding term root is b2abd2... and object id is 2c68d8...
Notation. We use ∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.
Axiom. (binintersectI) We take the following as an axiom:
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
In Proofgold the corresponding term root is fd656d... and proposition id is 9081a0...
Axiom. (binintersectE) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
In Proofgold the corresponding term root is 463b5a... and proposition id is ca0636...
Axiom. (binintersectE1) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X
In Proofgold the corresponding term root is b12123... and proposition id is 51005e...
Axiom. (binintersectE2) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ Y
In Proofgold the corresponding term root is 39656f... and proposition id is 599de4...
Axiom. (binintersect_Subq_1) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ X
In Proofgold the corresponding term root is f296ad... and proposition id is 303833...
Axiom. (binintersect_Subq_2) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ Y
In Proofgold the corresponding term root is fcd5f5... and proposition id is 8b9fd2...
Axiom. (binintersect_Subq_eq_1) We take the following as an axiom:
∀X Y, X ⊆ Y → X ∩ Y = X
In Proofgold the corresponding term root is 277eec... and proposition id is ec1cc3...
Axiom. (binintersect_Subq_max) We take the following as an axiom:
∀X Y Z : set, Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y
In Proofgold the corresponding term root is 95624d... and proposition id is 64636b...
Axiom. (binintersect_com_Subq) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ Y ∩ X
In Proofgold the corresponding term root is 410255... and proposition id is 5977b8...
Axiom. (binintersect_com) We take the following as an axiom:
∀X Y : set, X ∩ Y = Y ∩ X
In Proofgold the corresponding term root is 1a2c6f... and proposition id is ea8a1c...
Primitive. The name setminus is a term of type set → set → set.
In Proofgold the corresponding term root is c68e5a... and object id is 913da1...
Notation. We use ∖ as an infix operator with priority 350 and no associativity corresponding to applying term setminus.
Axiom. (setminusI) We take the following as an axiom:
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
In Proofgold the corresponding term root is 0feeb8... and proposition id is 80db6a...
Axiom. (setminusE) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
In Proofgold the corresponding term root is 6774f4... and proposition id is 71763a...
Axiom. (setminusE1) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
In Proofgold the corresponding term root is 064bd5... and proposition id is d74baf...
Axiom. (setminus_Subq) We take the following as an axiom:
∀X Y : set, X ∖ Y ⊆ X
In Proofgold the corresponding term root is 5a497a... and proposition id is 66f982...
Axiom. (setminus_Subq_contra) We take the following as an axiom:
∀X Y Z : set, Z ⊆ Y → X ∖ Y ⊆ X ∖ Z
In Proofgold the corresponding term root is b1f440... and proposition id is 9a06b2...
Axiom. (setminus_In_Power) We take the following as an axiom:
∀A U, A ∖ U ∈ 𝒫 A
In Proofgold the corresponding term root is 062275... and proposition id is f503e1...
Axiom. (In_irref) We take the following as an axiom:
∀x, x ∉ x
In Proofgold the corresponding term root is 12629a... and proposition id is 7bfd46...
Axiom. (In_no2cycle) We take the following as an axiom:
∀x y, x ∈ y → y ∈ x → False
In Proofgold the corresponding term root is 6d0cb3... and proposition id is 344f30...
Primitive. The name ordsucc is a term of type set → set.
In Proofgold the corresponding term root is 65d883... and object id is 1452f7...
Axiom. (ordsuccI1) We take the following as an axiom:
∀x : set, x ⊆ ordsucc x
In Proofgold the corresponding term root is 5a2f15... and proposition id is 367d80...
Axiom. (ordsuccI2) We take the following as an axiom:
∀x : set, x ∈ ordsucc x
In Proofgold the corresponding term root is 2f6efd... and proposition id is 2be6be...
Axiom. (ordsuccE) We take the following as an axiom:
∀x y : set, y ∈ ordsucc x → y ∈ x ∨ y = x
In Proofgold the corresponding term root is 01c4ee... and proposition id is 3849f5...
Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.
Axiom. (neq_0_ordsucc) We take the following as an axiom:
∀a : set, 0 ≠ ordsucc a
In Proofgold the corresponding term root is 1b20cb... and proposition id is 04c410...
Axiom. (neq_ordsucc_0) We take the following as an axiom:
∀a : set, ordsucc a ≠ 0
In Proofgold the corresponding term root is 9b391e... and proposition id is 346ee0...
Axiom. (ordsucc_inj) We take the following as an axiom:
∀a b : set, ordsucc a = ordsucc b → a = b
In Proofgold the corresponding term root is 40cad6... and proposition id is 4869a1...
Axiom. (ordsucc_inj_contra) We take the following as an axiom:
∀a b : set, a ≠ b → ordsucc a ≠ ordsucc b
In Proofgold the corresponding term root is 791304... and proposition id is 66f4df...
Axiom. (In_0_1) We take the following as an axiom:
0 ∈ 1
In Proofgold the corresponding term root is e984b0... and proposition id is 3be80a...
Axiom. (In_0_2) We take the following as an axiom:
0 ∈ 2
In Proofgold the corresponding term root is 56dc2c... and proposition id is 5a202e...
Axiom. (In_1_2) We take the following as an axiom:
1 ∈ 2
In Proofgold the corresponding term root is 8704b3... and proposition id is 431e9e...
Definition. We define nat_p to be λn : set ⇒ ∀p : set → prop, p 0 → (∀x : set, p x → p (ordsucc x)) → p n of type set → prop.
In Proofgold the corresponding term root is 458be3... and object id is cfd8eb...
Axiom. (nat_0) We take the following as an axiom:
nat_p 0
In Proofgold the corresponding term root is 1617bc... and proposition id is 457a72...
Axiom. (nat_ordsucc) We take the following as an axiom:
∀n : set, nat_p n → nat_p (ordsucc n)
In Proofgold the corresponding term root is 87f001... and proposition id is 57e940...
Axiom. (nat_1) We take the following as an axiom:
nat_p 1
In Proofgold the corresponding term root is 2f83fa... and proposition id is 672d0d...
Axiom. (nat_2) We take the following as an axiom:
nat_p 2
In Proofgold the corresponding term root is a8be15... and proposition id is 9112e9...
Axiom. (nat_0_in_ordsucc) We take the following as an axiom:
∀n, nat_p n → 0 ∈ ordsucc n
In Proofgold the corresponding term root is 9b50b9... and proposition id is 251f3d...
Axiom. (nat_ordsucc_in_ordsucc) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, ordsucc m ∈ ordsucc n
In Proofgold the corresponding term root is 819399... and proposition id is 4c2c95...
Axiom. (nat_ind) We take the following as an axiom:
∀p : set → prop, p 0 → (∀n, nat_p n → p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is 5d06f5... and proposition id is ecdce8...
Axiom. (nat_inv_impred) We take the following as an axiom:
∀p : set → prop, p 0 → (∀n, nat_p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is eddc5f... and proposition id is 175e22...
Axiom. (nat_inv) We take the following as an axiom:
∀n, nat_p n → n = 0 ∨ ∃x, nat_p x ∧ n = ordsucc x
In Proofgold the corresponding term root is 5d4296... and proposition id is 4f4167...
Axiom. (nat_complete_ind) We take the following as an axiom:
∀p : set → prop, (∀n, nat_p n → (∀m ∈ n, p m) → p n) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is d32e5b... and proposition id is 1e1608...
Axiom. (nat_p_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, nat_p m
In Proofgold the corresponding term root is e88449... and proposition id is 03b874...
Axiom. (nat_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, m ⊆ n
In Proofgold the corresponding term root is 6bc6e1... and proposition id is cad6fe...
Axiom. (nat_ordsucc_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ ordsucc n, m ⊆ n
In Proofgold the corresponding term root is e42e58... and proposition id is bc868d...
Axiom. (Union_ordsucc_eq) We take the following as an axiom:
∀n, nat_p n → ⋃ (ordsucc n) = n
In Proofgold the corresponding term root is 87308c... and proposition id is e8cdeb...
Axiom. (cases_1) We take the following as an axiom:
∀i ∈ 1, ∀p : set → prop, p 0 → p i
In Proofgold the corresponding term root is 706605... and proposition id is c6f0c0...
Axiom. (cases_2) We take the following as an axiom:
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
In Proofgold the corresponding term root is 4693b2... and proposition id is 74563a...
Axiom. (cases_3) We take the following as an axiom:
∀i ∈ 3, ∀p : set → prop, p 0 → p 1 → p 2 → p i
In Proofgold the corresponding term root is 2a0ca2... and proposition id is 1a8a20...
Axiom. (neq_0_1) We take the following as an axiom:
0 ≠ 1
In Proofgold the corresponding term root is 9630fd... and proposition id is 74867a...
Axiom. (neq_1_0) We take the following as an axiom:
1 ≠ 0
In Proofgold the corresponding term root is 3a0478... and proposition id is 952fed...
Axiom. (neq_0_2) We take the following as an axiom:
0 ≠ 2
In Proofgold the corresponding term root is 055580... and proposition id is 9c9a3f...
Axiom. (neq_2_0) We take the following as an axiom:
2 ≠ 0
In Proofgold the corresponding term root is 4d54d8... and proposition id is 00b357...
Axiom. (neq_1_2) We take the following as an axiom:
1 ≠ 2
In Proofgold the corresponding term root is 1d088a... and proposition id is e5542e...
Axiom. (ZF_closed_E) We take the following as an axiom:
∀U, ZF_closed U → ∀p : prop, (Union_closed U → Power_closed U → Repl_closed U → p) → p
In Proofgold the corresponding term root is d795d4... and proposition id is c78c79...
Axiom. (ZF_Union_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, ⋃ X ∈ U
In Proofgold the corresponding term root is a7cdf1... and proposition id is 250496...
Axiom. (ZF_Power_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, 𝒫 X ∈ U
In Proofgold the corresponding term root is 5d781f... and proposition id is 7411dd...
Axiom. (ZF_Repl_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, ∀F : set → set, (∀x ∈ X, F x ∈ U) → {F x|x ∈ X} ∈ U
In Proofgold the corresponding term root is 312268... and proposition id is 18f8ac...
Axiom. (ZF_UPair_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x y ∈ U, {x,y} ∈ U
In Proofgold the corresponding term root is 54256a... and proposition id is ef42d3...
Axiom. (ZF_Sing_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x ∈ U, {x} ∈ U
In Proofgold the corresponding term root is 5d8a83... and proposition id is d02d66...
Axiom. (ZF_binunion_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X Y ∈ U, (X ∪ Y) ∈ U
In Proofgold the corresponding term root is 8a6240... and proposition id is 1bacfd...
Axiom. (ZF_ordsucc_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x ∈ U, ordsucc x ∈ U
In Proofgold the corresponding term root is 68a6ba... and proposition id is 946e61...
Axiom. (nat_p_UnivOf_Empty) We take the following as an axiom:
∀n : set, nat_p n → n ∈ UnivOf Empty
In Proofgold the corresponding term root is c885b4... and proposition id is 645aee...
Primitive. The name ω is a term of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
Axiom. (omega_nat_p) We take the following as an axiom:
∀n ∈ ω, nat_p n
In Proofgold the corresponding term root is f44a30... and proposition id is c2c61b...
Axiom. (nat_p_omega) We take the following as an axiom:
∀n : set, nat_p n → n ∈ ω
In Proofgold the corresponding term root is 6a1b44... and proposition id is d5080f...
Axiom. (omega_ordsucc) We take the following as an axiom:
∀n ∈ ω, ordsucc n ∈ ω
In Proofgold the corresponding term root is 246d63... and proposition id is 17fb64...
Definition. We define ordinal to be λalpha : set ⇒ TransSet alpha ∧ ∀beta ∈ alpha, TransSet beta of type set → prop.
In Proofgold the corresponding term root is ee28d9... and object id is 53ee8f...
Axiom. (ordinal_TransSet) We take the following as an axiom:
∀alpha : set, ordinal alpha → TransSet alpha
In Proofgold the corresponding term root is 0dbaa3... and proposition id is c9e708...
Axiom. (ordinal_Empty) We take the following as an axiom:
ordinal Empty
In Proofgold the corresponding term root is ce6bea... and proposition id is 7d95d1...
Axiom. (ordinal_Hered) We take the following as an axiom:
∀alpha : set, ordinal alpha → ∀beta ∈ alpha, ordinal beta
In Proofgold the corresponding term root is a7e78b... and proposition id is 43b0f3...
Axiom. (TransSet_ordsucc) We take the following as an axiom:
∀X : set, TransSet X → TransSet (ordsucc X)
In Proofgold the corresponding term root is fc2157... and proposition id is ebf9e3...
Axiom. (ordinal_ordsucc) We take the following as an axiom:
∀alpha : set, ordinal alpha → ordinal (ordsucc alpha)
In Proofgold the corresponding term root is 0ba97f... and proposition id is ab0f7f...
Axiom. (nat_p_ordinal) We take the following as an axiom:
∀n : set, nat_p n → ordinal n
In Proofgold the corresponding term root is 274fc4... and proposition id is 47cd49...
Axiom. (ordinal_1) We take the following as an axiom:
ordinal 1
In Proofgold the corresponding term root is c94d41... and proposition id is e325a5...
Axiom. (ordinal_2) We take the following as an axiom:
ordinal 2
In Proofgold the corresponding term root is a57075... and proposition id is e64a0d...
Axiom. (omega_TransSet) We take the following as an axiom:
TransSet ω
In Proofgold the corresponding term root is 8e89e6... and proposition id is c79e75...
Axiom. (omega_ordinal) We take the following as an axiom:
ordinal ω
In Proofgold the corresponding term root is 3d621a... and proposition id is 177484...
Axiom. (ordsucc_omega_ordinal) We take the following as an axiom:
ordinal (ordsucc ω)
In Proofgold the corresponding term root is 66659c... and proposition id is ecfad7...
Axiom. (TransSet_ordsucc_In_Subq) We take the following as an axiom:
∀X : set, TransSet X → ∀x ∈ X, ordsucc x ⊆ X
In Proofgold the corresponding term root is d5d882... and proposition id is 3c2e45...
Axiom. (ordinal_ordsucc_In_Subq) We take the following as an axiom:
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ⊆ alpha
In Proofgold the corresponding term root is 57345e... and proposition id is b00304...
Axiom. (ordinal_trichotomy_or) We take the following as an axiom:
∀alpha beta : set, ordinal alpha → ordinal beta → alpha ∈ beta ∨ alpha = beta ∨ beta ∈ alpha
In Proofgold the corresponding term root is 078a34... and proposition id is 21f2d6...
Axiom. (ordinal_trichotomy_or_impred) We take the following as an axiom:
∀alpha beta : set, ordinal alpha → ordinal beta → ∀p : prop, (alpha ∈ beta → p) → (alpha = beta → p) → (beta ∈ alpha → p) → p
In Proofgold the corresponding term root is 7c0f0b... and proposition id is bb6d2d...
Axiom. (ordinal_In_Or_Subq) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → alpha ∈ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 70fe0d... and proposition id is 7b99b0...
Axiom. (ordinal_linear) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 3f30a5... and proposition id is 3b1fce...
Axiom. (ordinal_ordsucc_In_eq) We take the following as an axiom:
∀alpha beta, ordinal alpha → beta ∈ alpha → ordsucc beta ∈ alpha ∨ alpha = ordsucc beta
In Proofgold the corresponding term root is 500378... and proposition id is 0c19e7...
Axiom. (ordinal_lim_or_succ) We take the following as an axiom:
∀alpha, ordinal alpha → (∀beta ∈ alpha, ordsucc beta ∈ alpha) ∨ (∃beta ∈ alpha, alpha = ordsucc beta)
In Proofgold the corresponding term root is 4450c0... and proposition id is a12697...
Axiom. (ordinal_ordsucc_In) We take the following as an axiom:
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ∈ ordsucc alpha
In Proofgold the corresponding term root is f4b1ed... and proposition id is 8ab1dc...
Axiom. (ordinal_famunion) We take the following as an axiom:
∀X, ∀F : set → set, (∀x ∈ X, ordinal (F x)) → ordinal (⋃x ∈ XF x)
In Proofgold the corresponding term root is fc0359... and proposition id is b3f1e1...
Axiom. (ordinal_binintersect) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∩ beta)
In Proofgold the corresponding term root is 43d507... and proposition id is 025328...
Axiom. (ordinal_binunion) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∪ beta)
In Proofgold the corresponding term root is 861ae3... and proposition id is b24989...
Axiom. (ordinal_ind) We take the following as an axiom:
∀p : set → prop, (∀alpha, ordinal alpha → (∀beta ∈ alpha, p beta) → p alpha) → ∀alpha, ordinal alpha → p alpha
In Proofgold the corresponding term root is bb5575... and proposition id is 973cd8...
Axiom. (least_ordinal_ex) We take the following as an axiom:
∀p : set → prop, (∃alpha, ordinal alpha ∧ p alpha) → ∃alpha, ordinal alpha ∧ p alpha ∧ ∀beta ∈ alpha, ¬ p beta
In Proofgold the corresponding term root is df6c77... and proposition id is ab6fe1...
Definition. We define inj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 264b04... and object id is 442958...
Definition. We define bij to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 76bef6... and object id is f7d93b...
Axiom. (bijI) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → bij X Y f
In Proofgold the corresponding term root is a9064f... and proposition id is 3229f9...
Axiom. (bijE) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → ∀p : prop, ((∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → p) → p
In Proofgold the corresponding term root is 2cf278... and proposition id is cf0759...
Primitive. The name inv is a term of type set → (set → set) → set → set.
In Proofgold the corresponding term root is 896fa9... and object id is f3a015...
Axiom. (surj_rinv) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀w ∈ Y, ∃u ∈ X, f u = w) → ∀y ∈ Y, inv X f y ∈ X ∧ f (inv X f y) = y
In Proofgold the corresponding term root is ebf371... and proposition id is 6489d2...
Axiom. (inj_linv) We take the following as an axiom:
∀X, ∀f : set → set, (∀u v ∈ X, f u = f v → u = v) → ∀x ∈ X, inv X f (f x) = x
In Proofgold the corresponding term root is 18ecce... and proposition id is d16557...
Axiom. (bij_inv) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
In Proofgold the corresponding term root is ff1332... and proposition id is 69c537...
Axiom. (bij_id) We take the following as an axiom:
∀X, bij X X (λx ⇒ x)
In Proofgold the corresponding term root is a8cefb... and proposition id is 707f37...
Axiom. (bij_comp) We take the following as an axiom:
∀X Y Z : set, ∀f g : set → set, bij X Y f → bij Y Z g → bij X Z (λx ⇒ g (f x))
In Proofgold the corresponding term root is abf851... and proposition id is 4c9045...
Definition. We define equip to be λX Y : set ⇒ ∃f : set → set, bij X Y f of type set → set → prop.
In Proofgold the corresponding term root is eb4419... and object id is 7f2bf6...
Axiom. (equip_ref) We take the following as an axiom:
∀X, equip X X
In Proofgold the corresponding term root is cc7e80... and proposition id is c41fb1...
Axiom. (equip_sym) We take the following as an axiom:
∀X Y, equip X Y → equip Y X
In Proofgold the corresponding term root is f28954... and proposition id is 272f04...
Axiom. (equip_tra) We take the following as an axiom:
∀X Y Z, equip X Y → equip Y Z → equip X Z
In Proofgold the corresponding term root is 322bfb... and proposition id is d616c1...
Axiom. (equip_0_Empty) We take the following as an axiom:
∀X, equip X 0 → X = 0
In Proofgold the corresponding term root is 83ab2f... and proposition id is 4c101c...
Beginning of Section SchroederBernstein
Axiom. (KnasterTarski_set) We take the following as an axiom:
∀A, ∀F : set → set, (∀U ∈ 𝒫 A, F U ∈ 𝒫 A) → (∀U V ∈ 𝒫 A, U ⊆ V → F U ⊆ F V) → ∃Y ∈ 𝒫 A, F Y = Y
In Proofgold the corresponding term root is 2ffb9f... and proposition id is 31ef9d...
Axiom. (image_In_Power) We take the following as an axiom:
∀A B, ∀f : set → set, (∀x ∈ A, f x ∈ B) → ∀U ∈ 𝒫 A, {f x|x ∈ U} ∈ 𝒫 B
In Proofgold the corresponding term root is 6799e8... and proposition id is 43ed06...
Axiom. (image_monotone) We take the following as an axiom:
∀f : set → set, ∀U V, U ⊆ V → {f x|x ∈ U} ⊆ {f x|x ∈ V}
In Proofgold the corresponding term root is ac6fe8... and proposition id is f923c6...
Axiom. (setminus_antimonotone) We take the following as an axiom:
∀A U V, U ⊆ V → A ∖ V ⊆ A ∖ U
In Proofgold the corresponding term root is 67bd0e... and proposition id is f9b735...
Axiom. (SchroederBernstein) We take the following as an axiom:
∀A B, ∀f g : set → set, inj A B f → inj B A g → equip A B
In Proofgold the corresponding term root is e247a8... and proposition id is 03a2cd...
End of Section SchroederBernstein
Beginning of Section PigeonHole
Axiom. (PigeonHole_nat) We take the following as an axiom:
∀n, nat_p n → ∀f : set → set, (∀i ∈ ordsucc n, f i ∈ n) → ¬ (∀i j ∈ ordsucc n, f i = f j → i = j)
In Proofgold the corresponding term root is d756b7... and proposition id is 604a84...
Axiom. (PigeonHole_nat_bij) We take the following as an axiom:
∀n, nat_p n → ∀f : set → set, (∀i ∈ n, f i ∈ n) → (∀i j ∈ n, f i = f j → i = j) → bij n n f
In Proofgold the corresponding term root is cd3b8d... and proposition id is 44a08f...
End of Section PigeonHole
Definition. We define finite to be λX ⇒ ∃n ∈ ω, equip X n of type set → prop.
In Proofgold the corresponding term root is 04632d... and object id is 4c54a5...
Axiom. (finite_ind) We take the following as an axiom:
∀p : set → prop, p Empty → (∀X y, finite X → y ∉ X → p X → p (X ∪ {y})) → ∀X, finite X → p X
In Proofgold the corresponding term root is c53cb1... and proposition id is 19baca...
Axiom. (finite_Empty) We take the following as an axiom:
finite 0
In Proofgold the corresponding term root is c073c6... and proposition id is ec2c27...
Axiom. (adjoin_finite) We take the following as an axiom:
∀X y, finite X → finite (X ∪ {y})
In Proofgold the corresponding term root is def317... and proposition id is e6aa6d...
Axiom. (binunion_finite) We take the following as an axiom:
∀X, finite X → ∀Y, finite Y → finite (X ∪ Y)
In Proofgold the corresponding term root is fc26bb... and proposition id is 3365a6...
Axiom. (famunion_nat_finite) We take the following as an axiom:
∀X : set → set, ∀n, nat_p n → (∀i ∈ n, finite (X i)) → finite (⋃i ∈ nX i)
In Proofgold the corresponding term root is 7f709a... and proposition id is f84ed0...
Axiom. (Subq_finite) We take the following as an axiom:
∀X, finite X → ∀Y, Y ⊆ X → finite Y
In Proofgold the corresponding term root is 6b062c... and proposition id is e4e3ee...
Axiom. (TransSet_In_ordsucc_Subq) We take the following as an axiom:
∀x y, TransSet y → x ∈ ordsucc y → x ⊆ y
In Proofgold the corresponding term root is f80c6e... and proposition id is 341993...
Axiom. (exandE_i) We take the following as an axiom:
∀P Q : set → prop, (∃x, P x ∧ Q x) → ∀r : prop, (∀x, P x → Q x → r) → r
In Proofgold the corresponding term root is 083f82... and proposition id is b1b26e...
Axiom. (exandE_ii) We take the following as an axiom:
∀P Q : (set → set) → prop, (∃x : set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is c3e995... and proposition id is ed9d65...
Axiom. (exandE_iii) We take the following as an axiom:
∀P Q : (set → set → set) → prop, (∃x : set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is 028a8f... and proposition id is 80b3b0...
Axiom. (exandE_iiii) We take the following as an axiom:
∀P Q : (set → set → set → set) → prop, (∃x : set → set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is 7807d4... and proposition id is d1536a...
Beginning of Section Descr_ii
Variable P : (set → set) → prop
Primitive. The name Descr_ii is a term of type set → set.
In Proofgold the corresponding term root is 3bae39... and object id is 3a4ad2...
Hypothesis Pex : ∃f : set → set, P f
Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
Axiom. (Descr_ii_prop) We take the following as an axiom:
P Descr_ii
In Proofgold the corresponding term root is 45bfc0... and proposition id is e705d1...
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (set → set → set) → prop
Primitive. The name Descr_iii is a term of type set → set → set.
In Proofgold the corresponding term root is ca5fc1... and object id is 877e40...
Hypothesis Pex : ∃f : set → set → set, P f
Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
Axiom. (Descr_iii_prop) We take the following as an axiom:
P Descr_iii
In Proofgold the corresponding term root is 261c3d... and proposition id is 160421...
End of Section Descr_iii
Beginning of Section Descr_Vo1
Variable P : Vo 1 → prop
Primitive. The name Descr_Vo1 is a term of type Vo 1.
In Proofgold the corresponding term root is 615c0a... and object id is a4ecdd...
Hypothesis Pex : ∃f : Vo 1, P f
Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
Axiom. (Descr_Vo1_prop) We take the following as an axiom:
P Descr_Vo1
In Proofgold the corresponding term root is f550ef... and proposition id is 2213e2...
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : set → set
Primitive. The name If_ii is a term of type set → set.
In Proofgold the corresponding term root is 17057f... and object id is e29015...
Axiom. (If_ii_1) We take the following as an axiom:
p → If_ii = f
In Proofgold the corresponding term root is 31521e... and proposition id is 342a1f...
Axiom. (If_ii_0) We take the following as an axiom:
¬ p → If_ii = g
In Proofgold the corresponding term root is 6acab0... and proposition id is 36418b...
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : set → set → set
Primitive. The name If_iii is a term of type set → set → set.
In Proofgold the corresponding term root is 331476... and object id is dd7e2b...
Axiom. (If_iii_1) We take the following as an axiom:
p → If_iii = f
In Proofgold the corresponding term root is 2d62ee... and proposition id is 48bc10...
Axiom. (If_iii_0) We take the following as an axiom:
¬ p → If_iii = g
In Proofgold the corresponding term root is 9b77d9... and proposition id is d11201...
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set → (set → set) → set
Primitive. The name In_rec_i is a term of type set → set.
In Proofgold the corresponding term root is fac413... and object id is 93aac4...
Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (In_rec_i_eq) We take the following as an axiom:
∀X : set, In_rec_i X = F X In_rec_i
In Proofgold the corresponding term root is 110653... and proposition id is e8653b...
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set → (set → (set → set)) → (set → set)
Primitive. The name In_rec_ii is a term of type set → (set → set).
In Proofgold the corresponding term root is f3c9ab... and object id is 3841bd...
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (In_rec_ii_eq) We take the following as an axiom:
∀X : set, In_rec_ii X = F X In_rec_ii
In Proofgold the corresponding term root is f7aff6... and proposition id is 7bb1e9...
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set → (set → (set → set → set)) → (set → set → set)
Primitive. The name In_rec_iii is a term of type set → (set → set → set).
In Proofgold the corresponding term root is 9b3a85... and object id is 651077...
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (In_rec_iii_eq) We take the following as an axiom:
∀X : set, In_rec_iii X = F X In_rec_iii
In Proofgold the corresponding term root is 5f0574... and proposition id is a0709d...
End of Section EpsilonRec_iii
Beginning of Section NatRec
Variable z : set
Variable f : set → set → set
Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
Definition. We define nat_primrec to be In_rec_i F of type set → set.
In Proofgold the corresponding term root is 3be1c5... and object id is 1a64d5...
Axiom. (nat_primrec_r) We take the following as an axiom:
∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
In Proofgold the corresponding term root is c955cd... and proposition id is 9ba887...
Axiom. (nat_primrec_0) We take the following as an axiom:
nat_primrec 0 = z
In Proofgold the corresponding term root is 416eaf... and proposition id is be11a2...
Axiom. (nat_primrec_S) We take the following as an axiom:
∀n : set, nat_p n → nat_primrec (ordsucc n) = f n (nat_primrec n)
In Proofgold the corresponding term root is 797185... and proposition id is fe4c88...
End of Section NatRec
Beginning of Section NatArith
Definition. We define add_nat to be λn m : set ⇒ nat_primrec n (λ_ r ⇒ ordsucc r) m of type set → set → set.
In Proofgold the corresponding term root is afa8ae... and object id is 29768b...
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Axiom. (add_nat_0R) We take the following as an axiom:
∀n : set, n + 0 = n
In Proofgold the corresponding term root is 402bcb... and proposition id is 23277f...
Axiom. (add_nat_SR) We take the following as an axiom:
∀n m : set, nat_p m → n + ordsucc m = ordsucc (n + m)
In Proofgold the corresponding term root is a53523... and proposition id is e61e18...
Axiom. (add_nat_p) We take the following as an axiom:
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n + m)
In Proofgold the corresponding term root is a68eaa... and proposition id is 4c1d5d...
Axiom. (add_nat_1_1_2) We take the following as an axiom:
1 + 1 = 2
In Proofgold the corresponding term root is 174b87... and proposition id is 77f23f...
Definition. We define mul_nat to be λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type set → set → set.
In Proofgold the corresponding term root is 35a1ef... and object id is 80ccc6...
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Axiom. (mul_nat_0R) We take the following as an axiom:
∀n : set, n * 0 = 0
In Proofgold the corresponding term root is b2f56c... and proposition id is 1e5b2c...
Axiom. (mul_nat_SR) We take the following as an axiom:
∀n m : set, nat_p m → n * ordsucc m = n + n * m
In Proofgold the corresponding term root is 37df54... and proposition id is 6dd6fc...
Axiom. (mul_nat_p) We take the following as an axiom:
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n * m)
In Proofgold the corresponding term root is d63438... and proposition id is e910cb...
End of Section NatArith
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} ∪ {f x|x ∈ X}) of type set → set.
In Proofgold the corresponding term root is 8f0026... and object id is 7736ca...
Axiom. (Inj1_eq) We take the following as an axiom:
∀X : set, Inj1 X = {0} ∪ {Inj1 x|x ∈ X}
In Proofgold the corresponding term root is 73d189... and proposition id is b80457...
Axiom. (Inj1I1) We take the following as an axiom:
∀X : set, 0 ∈ Inj1 X
In Proofgold the corresponding term root is 3c2cc4... and proposition id is 4ccac3...
Axiom. (Inj1I2) We take the following as an axiom:
∀X x : set, x ∈ X → Inj1 x ∈ Inj1 X
In Proofgold the corresponding term root is b6f23b... and proposition id is 086475...
Axiom. (Inj1E) We take the following as an axiom:
∀X y : set, y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X, y = Inj1 x
In Proofgold the corresponding term root is d4177b... and proposition id is 259217...
Axiom. (Inj1NE1) We take the following as an axiom:
∀x : set, Inj1 x ≠ 0
In Proofgold the corresponding term root is 5e3197... and proposition id is 09bd10...
Axiom. (Inj1NE2) We take the following as an axiom:
∀x : set, Inj1 x ∉ {0}
In Proofgold the corresponding term root is 9e94b1... and proposition id is 3c4dd0...
Definition. We define Inj0 to be λX ⇒ {Inj1 x|x ∈ X} of type set → set.
In Proofgold the corresponding term root is 814321... and object id is 7d54be...
Axiom. (Inj0I) We take the following as an axiom:
∀X x : set, x ∈ X → Inj1 x ∈ Inj0 X
In Proofgold the corresponding term root is 7fc2e9... and proposition id is 565378...
Axiom. (Inj0E) We take the following as an axiom:
∀X y : set, y ∈ Inj0 X → ∃x : set, x ∈ X ∧ y = Inj1 x
In Proofgold the corresponding term root is 416684... and proposition id is 855213...
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|x ∈ X ∖ {0}}) of type set → set.
In Proofgold the corresponding term root is f202c1... and object id is 2e1675...
Axiom. (Unj_eq) We take the following as an axiom:
∀X : set, Unj X = {Unj x|x ∈ X ∖ {0}}
In Proofgold the corresponding term root is 3b1aa6... and proposition id is b4c852...
Axiom. (Unj_Inj1_eq) We take the following as an axiom:
∀X : set, Unj (Inj1 X) = X
In Proofgold the corresponding term root is e2cdfc... and proposition id is ae00e1...
Axiom. (Inj1_inj) We take the following as an axiom:
∀X Y : set, Inj1 X = Inj1 Y → X = Y
In Proofgold the corresponding term root is ce8b66... and proposition id is 3e918a...
Axiom. (Unj_Inj0_eq) We take the following as an axiom:
∀X : set, Unj (Inj0 X) = X
In Proofgold the corresponding term root is f3e39c... and proposition id is 3808ad...
Axiom. (Inj0_inj) We take the following as an axiom:
∀X Y : set, Inj0 X = Inj0 Y → X = Y
In Proofgold the corresponding term root is 662f53... and proposition id is 29f945...
Axiom. (Inj0_0) We take the following as an axiom:
Inj0 0 = 0
In Proofgold the corresponding term root is 171f47... and proposition id is 57f5af...
Axiom. (Inj0_Inj1_neq) We take the following as an axiom:
∀X Y : set, Inj0 X ≠ Inj1 Y
In Proofgold the corresponding term root is 21bb03... and proposition id is 2b14d8...
Definition. We define setsum to be λX Y ⇒ {Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y} of type set → set → set.
In Proofgold the corresponding term root is afe37e... and object id is 5b7e09...
Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.
Axiom. (Inj0_setsum) We take the following as an axiom:
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
In Proofgold the corresponding term root is 1c6408... and proposition id is 246ac5...
Axiom. (Inj1_setsum) We take the following as an axiom:
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
In Proofgold the corresponding term root is d47a0f... and proposition id is 2b4f94...
Axiom. (setsum_Inj_inv) We take the following as an axiom:
∀X Y z : set, z ∈ X + Y → (∃x ∈ X, z = Inj0 x) ∨ (∃y ∈ Y, z = Inj1 y)
In Proofgold the corresponding term root is 645a6c... and proposition id is a36400...
Axiom. (Inj0_setsum_0L) We take the following as an axiom:
∀X : set, 0 + X = Inj0 X
In Proofgold the corresponding term root is 0be1e7... and proposition id is 36fabd...
Axiom. (Subq_1_Sing0) We take the following as an axiom:
1 ⊆ {0}
In Proofgold the corresponding term root is aeae9a... and proposition id is abab2d...
Axiom. (Inj1_setsum_1L) We take the following as an axiom:
∀X : set, 1 + X = Inj1 X
In Proofgold the corresponding term root is f8d721... and proposition id is 45b1c8...
Axiom. (nat_setsum1_ordsucc) We take the following as an axiom:
∀n : set, nat_p n → 1 + n = ordsucc n
In Proofgold the corresponding term root is 3be3cc... and proposition id is 8fbdda...
Axiom. (setsum_0_0) We take the following as an axiom:
0 + 0 = 0
In Proofgold the corresponding term root is 5dcaeb... and proposition id is 4f3651...
Axiom. (setsum_1_0_1) We take the following as an axiom:
1 + 0 = 1
In Proofgold the corresponding term root is c2107f... and proposition id is e8609a...
Axiom. (setsum_1_1_2) We take the following as an axiom:
1 + 1 = 2
In Proofgold the corresponding term root is 9c89c0... and proposition id is ef3f41...
Beginning of Section pair_setsum
Let pair ≝ setsum
Definition. We define proj0 to be λZ ⇒ {Unj z|z ∈ Z, ∃x : set, Inj0 x = z} of type set → set.
In Proofgold the corresponding term root is 21a488... and object id is 9c5e4e...
Definition. We define proj1 to be λZ ⇒ {Unj z|z ∈ Z, ∃y : set, Inj1 y = z} of type set → set.
In Proofgold the corresponding term root is 7aba21... and object id is 7fd81e...
Axiom. (Inj0_pair_0_eq) We take the following as an axiom:
Inj0 = pair 0
In Proofgold the corresponding term root is 0fe601... and proposition id is 490c2a...
Axiom. (Inj1_pair_1_eq) We take the following as an axiom:
Inj1 = pair 1
In Proofgold the corresponding term root is 0bbde8... and proposition id is 8272d7...
Axiom. (pairI0) We take the following as an axiom:
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
In Proofgold the corresponding term root is 21a414... and proposition id is 34813b...
Axiom. (pairI1) We take the following as an axiom:
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
In Proofgold the corresponding term root is bdaab5... and proposition id is b5248b...
Axiom. (pairE) We take the following as an axiom:
∀X Y z, z ∈ pair X Y → (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)
In Proofgold the corresponding term root is aac610... and proposition id is 0afdeb...
Axiom. (pairE0) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
In Proofgold the corresponding term root is 042f30... and proposition id is f50b25...
Axiom. (pairE1) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
In Proofgold the corresponding term root is 505cfc... and proposition id is 30a181...
Axiom. (proj0I) We take the following as an axiom:
∀w u : set, pair 0 u ∈ w → u ∈ proj0 w
In Proofgold the corresponding term root is be55b9... and proposition id is 840c8d...
Axiom. (proj0E) We take the following as an axiom:
∀w u : set, u ∈ proj0 w → pair 0 u ∈ w
In Proofgold the corresponding term root is b9afa9... and proposition id is 5b7f58...
Axiom. (proj1I) We take the following as an axiom:
∀w u : set, pair 1 u ∈ w → u ∈ proj1 w
In Proofgold the corresponding term root is fbddb7... and proposition id is 595b3c...
Axiom. (proj1E) We take the following as an axiom:
∀w u : set, u ∈ proj1 w → pair 1 u ∈ w
In Proofgold the corresponding term root is 06e32e... and proposition id is c3c3e6...
Axiom. (proj0_pair_eq) We take the following as an axiom:
∀X Y : set, proj0 (pair X Y) = X
In Proofgold the corresponding term root is d546f3... and proposition id is 0c12fb...
Axiom. (proj1_pair_eq) We take the following as an axiom:
∀X Y : set, proj1 (pair X Y) = Y
In Proofgold the corresponding term root is 9f98a8... and proposition id is e0cb0f...
Definition. We define Sigma to be λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 8acbb2... and object id is d64bcb...
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Axiom. (pair_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, pair x y ∈ ∑x ∈ X, Y x
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Axiom. (Sigma_eta_proj0_proj1) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z ∧ proj0 z ∈ X ∧ proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is afe962... and proposition id is 77ba8e...
Axiom. (proj_Sigma_eta) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z
In Proofgold the corresponding term root is 8b31cd... and proposition id is f30f76...
Axiom. (proj0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
In Proofgold the corresponding term root is ae12a7... and proposition id is 506869...
Axiom. (proj1_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is 553db6... and proposition id is d79917...
Axiom. (pair_Sigma_E1) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → y ∈ Y x
In Proofgold the corresponding term root is c294a7... and proposition id is 81d8b0...
Axiom. (Sigma_E) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → ∃x ∈ X, ∃y ∈ Y x, z = pair x y
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Definition. We define setprod to be λX Y : set ⇒ ∑x ∈ X, Y of type set → set → set.
In Proofgold the corresponding term root is fc0b60... and object id is 107b6b...
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Let lam : set → (set → set) → set ≝ Sigma
Definition. We define ap to be λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type set → set → set.
In Proofgold the corresponding term root is c7aaa6... and object id is a4d82c...
Notation. When x is a set, a term x y is notation for ap x y.
Notation. λ x ∈ A ⇒ B is notation for the set Sigma A (λ x : set ⇒ B).
Notation. We now use n-tuple notation (a0,...,an-1) for n ≥ 2 for λ i ∈ n . if i = 0 then a0 else ... if i = n-2 then an-2 else an-1.
Axiom. (lamI) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, pair x y ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Axiom. (lamE) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = pair x y
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Axiom. (apI) We take the following as an axiom:
∀f x y, pair x y ∈ f → y ∈ f x
In Proofgold the corresponding term root is 64667c... and proposition id is 770816...
Axiom. (apE) We take the following as an axiom:
∀f x y, y ∈ f x → pair x y ∈ f
In Proofgold the corresponding term root is 85693a... and proposition id is c08565...
Axiom. (beta) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
In Proofgold the corresponding term root is 95991d... and proposition id is b5182d...
Axiom. (proj0_ap_0) We take the following as an axiom:
∀u, proj0 u = u 0
In Proofgold the corresponding term root is 588f37... and proposition id is d4f351...
Axiom. (proj1_ap_1) We take the following as an axiom:
∀u, proj1 u = u 1
In Proofgold the corresponding term root is d116b3... and proposition id is 2328b0...
Axiom. (pair_ap_0) We take the following as an axiom:
∀x y : set, (pair x y) 0 = x
In Proofgold the corresponding term root is ac34b9... and proposition id is bb0477...
Axiom. (pair_ap_1) We take the following as an axiom:
∀x y : set, (pair x y) 1 = y
In Proofgold the corresponding term root is 2274fc... and proposition id is cce3d2...
Axiom. (ap0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
In Proofgold the corresponding term root is 861a79... and proposition id is 1affcc...
Axiom. (ap1_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 1) ∈ (Y (z 0))
In Proofgold the corresponding term root is ad3bfe... and proposition id is 520af5...
Definition. We define pair_p to be λu : set ⇒ pair (u 0) (u 1) = u of type set → prop.
In Proofgold the corresponding term root is ade2bb... and object id is 5c8a35...
Axiom. (pair_p_I) We take the following as an axiom:
∀x y, pair_p (pair x y)
In Proofgold the corresponding term root is 06d9a2... and proposition id is 8f2fde...
Axiom. (Subq_2_UPair01) We take the following as an axiom:
2 ⊆ {0,1}
In Proofgold the corresponding term root is d90884... and proposition id is 94203e...
Axiom. (tuple_pair) We take the following as an axiom:
∀x y : set, pair x y = (x,y)
In Proofgold the corresponding term root is c122f6... and proposition id is 18198e...
Definition. We define Pi to be λX Y ⇒ {f ∈ 𝒫 (∑x ∈ X, ⋃ (Y x))|∀x ∈ X, f x ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 40f1c3... and object id is a7b1b0...
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Axiom. (PiI) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, (∀u ∈ f, pair_p u ∧ u 0 ∈ X) → (∀x ∈ X, f x ∈ Y x) → f ∈ ∏x ∈ X, Y x
In Proofgold the corresponding term root is 2b7ef4... and proposition id is 1f8f6f...
Axiom. (lam_Pi) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀F : set → set, (∀x ∈ X, F x ∈ Y x) → (λx ∈ X ⇒ F x) ∈ (∏x ∈ X, Y x)
In Proofgold the corresponding term root is c0b88f... and proposition id is 3733f8...
Axiom. (ap_Pi) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, ∀x : set, f ∈ (∏x ∈ X, Y x) → x ∈ X → f x ∈ Y x
In Proofgold the corresponding term root is 7c3617... and proposition id is cb962e...
Definition. We define setexp to be λX Y : set ⇒ ∏y ∈ Y, X of type set → set → set.
In Proofgold the corresponding term root is 1de7fc... and object id is 021ae0...
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Axiom. (pair_tuple_fun) We take the following as an axiom:
pair = (λx y ⇒ (x,y))
In Proofgold the corresponding term root is b02dfa... and proposition id is ed37d0...
Axiom. (lamI2) We take the following as an axiom:
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Beginning of Section Tuples
Variable x0 x1 : set
Axiom. (tuple_2_0_eq) We take the following as an axiom:
(x0,x1) 0 = x0
In Proofgold the corresponding term root is 42d452... and proposition id is ad9037...
Axiom. (tuple_2_1_eq) We take the following as an axiom:
(x0,x1) 1 = x1
In Proofgold the corresponding term root is 596569... and proposition id is 2c0ad1...
End of Section Tuples
Axiom. (ReplEq_setprod_ext) We take the following as an axiom:
∀X Y, ∀F G : set → set → set, (∀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}
In Proofgold the corresponding term root is c486ab... and proposition id is f689a0...
Axiom. (tuple_2_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, (x,y) ∈ ∑x ∈ X, Y x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Axiom. (tuple_2_setprod) We take the following as an axiom:
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
In Proofgold the corresponding term root is e907b8... and proposition id is 1f9d7f...
End of Section pair_setsum
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Primitive. The name DescrR_i_io_1 is a term of type (set → (set → prop) → prop) → set.
In Proofgold the corresponding term root is 1d3fd4... and object id is 82aecc...
Primitive. The name DescrR_i_io_2 is a term of type (set → (set → prop) → prop) → set → prop.
In Proofgold the corresponding term root is 768eb2... and object id is 3f82bc...
Axiom. (DescrR_i_io_12) We take the following as an axiom:
∀R : set → (set → prop) → prop, (∃x, (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) → R (DescrR_i_io_1 R) (DescrR_i_io_2 R)
In Proofgold the corresponding term root is a4fae2... and proposition id is da4f02...
Definition. We define PNoEq_ to be λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type set → (set → prop) → (set → prop) → prop.
In Proofgold the corresponding term root is d7d959... and object id is 144b65...
Axiom. (PNoEq_ref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
In Proofgold the corresponding term root is c27ebb... and proposition id is df4cf6...
Axiom. (PNoEq_sym_) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
In Proofgold the corresponding term root is b813f3... and proposition id is bfda0b...
Axiom. (PNoEq_tra_) We take the following as an axiom:
∀alpha, ∀p q r : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q r → PNoEq_ alpha p r
In Proofgold the corresponding term root is dded4c... and proposition id is fc92f3...
Axiom. (PNoEq_antimon_) We take the following as an axiom:
∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoEq_ alpha p q → PNoEq_ beta p q
In Proofgold the corresponding term root is 8868e6... and proposition id is 094ae2...
Definition. We define PNoLt_ to be λalpha p q ⇒ ∃beta ∈ alpha, PNoEq_ beta p q ∧ ¬ p beta ∧ q beta of type set → (set → prop) → (set → prop) → prop.
In Proofgold the corresponding term root is 34de68... and object id is c95964...
Axiom. (PNoLt_E_) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoLt_ alpha p q → ∀R : prop, (∀beta, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → q beta → R) → R
In Proofgold the corresponding term root is 9ba543... and proposition id is 45aec8...
Axiom. (PNoLt_irref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
In Proofgold the corresponding term root is fcb502... and proposition id is 987dff...
Axiom. (PNoLt_mon_) We take the following as an axiom:
∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoLt_ beta p q → PNoLt_ alpha p q
In Proofgold the corresponding term root is b1c3cf... and proposition id is d8d0f0...
Axiom. (PNoLt_trichotomy_or_) We take the following as an axiom:
∀p q : set → prop, ∀alpha, ordinal alpha → PNoLt_ alpha p q ∨ PNoEq_ alpha p q ∨ PNoLt_ alpha q p
In Proofgold the corresponding term root is 664408... and proposition id is 8d3824...
Axiom. (PNoLt_tra_) We take the following as an axiom:
∀alpha, ordinal alpha → ∀p q r : set → prop, PNoLt_ alpha p q → PNoLt_ alpha q r → PNoLt_ alpha p r
In Proofgold the corresponding term root is 3d7d91... and proposition id is 44767a...
Primitive. The name PNoLt is a term of type set → (set → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 8f57a0... and object id is f2094c...
Axiom. (PNoLtI1) We take the following as an axiom:
∀alpha beta, ∀p q : set → prop, PNoLt_ (alpha ∩ beta) p q → PNoLt alpha p beta q
In Proofgold the corresponding term root is c5735c... and proposition id is ecbc71...
Axiom. (PNoLtI2) We take the following as an axiom:
∀alpha beta, ∀p q : set → prop, alpha ∈ beta → PNoEq_ alpha p q → q alpha → PNoLt alpha p beta q
In Proofgold the corresponding term root is 419b53... and proposition id is 030093...
Axiom. (PNoLtI3) We take the following as an axiom:
∀alpha beta, ∀p q : set → prop, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → PNoLt alpha p beta q
In Proofgold the corresponding term root is c392cb... and proposition id is 410442...
Axiom. (PNoLtE) We take the following as an axiom:
∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → ∀R : prop, (PNoLt_ (alpha ∩ beta) p q → R) → (alpha ∈ beta → PNoEq_ alpha p q → q alpha → R) → (beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → R) → R
In Proofgold the corresponding term root is f7e6e3... and proposition id is af36ce...
Axiom. (PNoLt_irref) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
In Proofgold the corresponding term root is ff1a80... and proposition id is d4c379...
Axiom. (PNoLt_trichotomy_or) We take the following as an axiom:
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q ∨ PNoLt beta q alpha p
In Proofgold the corresponding term root is 4f39e4... and proposition id is 90b387...
Axiom. (PNoLtEq_tra) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoLt alpha p beta q → PNoEq_ beta q r → PNoLt alpha p beta r
In Proofgold the corresponding term root is 2e082a... and proposition id is 2f2a82...
Axiom. (PNoEqLt_tra) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLt alpha q beta r → PNoLt alpha p beta r
In Proofgold the corresponding term root is fe144b... and proposition id is f07654...
Axiom. (PNoLt_tra) We take the following as an axiom:
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is b72127... and proposition id is 8eff85...
Definition. We define PNoLe to be λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type set → (set → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is ac6311... and object id is 770cf7...
Axiom. (PNoLeI1) We take the following as an axiom:
∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → PNoLe alpha p beta q
In Proofgold the corresponding term root is 1f0528... and proposition id is 674a2c...
Axiom. (PNoLeI2) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
In Proofgold the corresponding term root is b45332... and proposition id is 0b2157...
Axiom. (PNoLe_ref) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
In Proofgold the corresponding term root is 1b91bd... and proposition id is 613386...
Axiom. (PNoLe_antisym) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ∀p q : set → prop, PNoLe alpha p beta q → PNoLe beta q alpha p → alpha = beta ∧ PNoEq_ alpha p q
In Proofgold the corresponding term root is 934d7a... and proposition id is 4e3594...
Axiom. (PNoLtLe_tra) We take the following as an axiom:
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLe beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is 156586... and proposition id is 09246b...
Axiom. (PNoLeLt_tra) We take the following as an axiom:
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is a260ca... and proposition id is ad6971...
Axiom. (PNoEqLe_tra) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLe alpha q beta r → PNoLe alpha p beta r
In Proofgold the corresponding term root is 7d3ff8... and proposition id is 002ebf...
Axiom. (PNoLe_tra) We take the following as an axiom:
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLe beta q gamma r → PNoLe alpha p gamma r
In Proofgold the corresponding term root is 70b92b... and proposition id is 4b557d...
Definition. We define PNo_downc to be λL alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, L beta q ∧ PNoLe alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 93dab1... and object id is f6c962...
Definition. We define PNo_upc to be λR alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, R beta q ∧ PNoLe beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 543712... and object id is b97935...
Axiom. (PNoLe_downc) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_downc L alpha p → PNoLe beta q alpha p → PNo_downc L beta q
In Proofgold the corresponding term root is 6b1f0d... and proposition id is cd130e...
Axiom. (PNo_downc_ref) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, L alpha p → PNo_downc L alpha p
In Proofgold the corresponding term root is 347aa9... and proposition id is 69dfcb...
Axiom. (PNo_upc_ref) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, R alpha p → PNo_upc R alpha p
In Proofgold the corresponding term root is 403942... and proposition id is c4bf71...
Axiom. (PNoLe_upc) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_upc R alpha p → PNoLe alpha p beta q → PNo_upc R beta q
In Proofgold the corresponding term root is 111b74... and proposition id is eb9ed5...
Definition. We define PNoLt_pwise to be λL R ⇒ ∀gamma, ordinal gamma → ∀p : set → prop, L gamma p → ∀delta, ordinal delta → ∀q : set → prop, R delta q → PNoLt gamma p delta q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → prop.
In Proofgold the corresponding term root is ae448b... and object id is 123808...
Axiom. (PNoLt_pwise_downc_upc) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → PNoLt_pwise (PNo_downc L) (PNo_upc R)
In Proofgold the corresponding term root is 2b8e3b... and proposition id is 2f244c...
Definition. We define PNo_rel_strict_upperbd to be λL alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_downc L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is dc1665... and object id is 3e1663...
Definition. We define PNo_rel_strict_lowerbd to be λR alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_upc R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is e5a4b6... and object id is ff3150...
Definition. We define PNo_rel_strict_imv to be λL R alpha p ⇒ PNo_rel_strict_upperbd L alpha p ∧ PNo_rel_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 64ce99... and object id is 5b35fb...
Axiom. (PNoEq_rel_strict_upperbd) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L alpha q
In Proofgold the corresponding term root is bb65cf... and proposition id is db4b82...
Axiom. (PNo_rel_strict_upperbd_antimon) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p
In Proofgold the corresponding term root is 9986af... and proposition id is ed53da...
Axiom. (PNoEq_rel_strict_lowerbd) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R alpha q
In Proofgold the corresponding term root is 76080e... and proposition id is 614e58...
Axiom. (PNo_rel_strict_lowerbd_antimon) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p
In Proofgold the corresponding term root is 27f253... and proposition id is 344b63...
Axiom. (PNoEq_rel_strict_imv) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R alpha q
In Proofgold the corresponding term root is e3f251... and proposition id is 7ae13d...
Axiom. (PNo_rel_strict_imv_antimon) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p
In Proofgold the corresponding term root is 2434dc... and proposition id is 87e0ce...
Definition. We define PNo_rel_strict_uniq_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R alpha p ∧ ∀q : set → prop, PNo_rel_strict_imv L R alpha q → PNoEq_ alpha p q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 23c9d9... and object id is 56ed1b...
Definition. We define PNo_rel_strict_split_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha) ∧ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 217d17... and object id is dd9cee...
Axiom. (PNo_extend0_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
In Proofgold the corresponding term root is 2d441e... and proposition id is 5ed08e...
Axiom. (PNo_extend1_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
In Proofgold the corresponding term root is 8eb99c... and proposition id is 4cac1a...
Axiom. (PNo_rel_imv_ex) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → (∃p : set → prop, PNo_rel_strict_uniq_imv L R alpha p) ∨ (∃tau ∈ alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R tau p)
In Proofgold the corresponding term root is 3e3224... and proposition id is 749b63...
Definition. We define PNo_lenbdd to be λalpha L ⇒ ∀beta, ∀p : set → prop, L beta p → beta ∈ alpha of type set → (set → (set → prop) → prop) → prop.
In Proofgold the corresponding term root is 009fa7... and object id is a452a6...
Axiom. (PNo_lenbdd_strict_imv_extend0) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha)
In Proofgold the corresponding term root is 17a114... and proposition id is 69327e...
Axiom. (PNo_lenbdd_strict_imv_extend1) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha)
In Proofgold the corresponding term root is ad69ec... and proposition id is 5ff993...
Axiom. (PNo_lenbdd_strict_imv_split) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_split_imv L R alpha p
In Proofgold the corresponding term root is 76ae4f... and proposition id is f98998...
Axiom. (PNo_rel_imv_bdd_ex) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R beta p
In Proofgold the corresponding term root is 9a98e4... and proposition id is 25bcd2...
Definition. We define PNo_strict_upperbd to be λL alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 6913bd... and object id is 3723ab...
Definition. We define PNo_strict_lowerbd to be λR alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 87fac4... and object id is 3606d4...
Definition. We define PNo_strict_imv to be λL R alpha p ⇒ PNo_strict_upperbd L alpha p ∧ PNo_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is ee97f6... and object id is bc6c55...
Axiom. (PNoEq_strict_upperbd) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_upperbd L alpha p → PNo_strict_upperbd L alpha q
In Proofgold the corresponding term root is 9b7b8e... and proposition id is f28b04...
Axiom. (PNoEq_strict_lowerbd) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_lowerbd R alpha p → PNo_strict_lowerbd R alpha q
In Proofgold the corresponding term root is 4db3f9... and proposition id is 67fa73...
Axiom. (PNoEq_strict_imv) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_imv L R alpha p → PNo_strict_imv L R alpha q
In Proofgold the corresponding term root is 1ec352... and proposition id is 5f582f...
Axiom. (PNo_strict_upperbd_imp_rel_strict_upperbd) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p
In Proofgold the corresponding term root is 2898f0... and proposition id is 3cdd3b...
Axiom. (PNo_strict_lowerbd_imp_rel_strict_lowerbd) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p
In Proofgold the corresponding term root is 442af5... and proposition id is 04acc2...
Axiom. (PNo_strict_imv_imp_rel_strict_imv) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p
In Proofgold the corresponding term root is f4fc71... and proposition id is 46a6aa...
Axiom. (PNo_rel_split_imv_imp_strict_imv) We take the following as an axiom:
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, PNo_rel_strict_split_imv L R alpha p → PNo_strict_imv L R alpha p
In Proofgold the corresponding term root is 72be36... and proposition id is adfdb0...
Axiom. (PNo_lenbdd_strict_imv_ex) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_strict_imv L R beta p
In Proofgold the corresponding term root is 9485d6... and proposition id is a864a5...
Definition. We define PNo_least_rep to be λL R beta p ⇒ ordinal beta ∧ PNo_strict_imv L R beta p ∧ ∀gamma ∈ beta, ∀q : set → prop, ¬ PNo_strict_imv L R gamma q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 0711e2... and object id is e4c707...
Definition. We define PNo_least_rep2 to be λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta → ¬ p x of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 1f6dc5... and object id is 09d320...
Axiom. (PNo_strict_imv_pred_eq) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → ∀p q : set → prop, PNo_least_rep L R alpha p → PNo_strict_imv L R alpha q → ∀beta ∈ alpha, p beta ↔ q beta
In Proofgold the corresponding term root is 5e7822... and proposition id is 72e14d...
Axiom. (PNo_lenbdd_least_rep2_exuniq2) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta, (∃p : set → prop, PNo_least_rep2 L R beta p) ∧ (∀p q : set → prop, PNo_least_rep2 L R beta p → PNo_least_rep2 L R beta q → p = q)
In Proofgold the corresponding term root is 5060a9... and proposition id is 8b62c5...
Primitive. The name PNo_bd is a term of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set.
In Proofgold the corresponding term root is ed76e7... and object id is aa75ef...
Primitive. The name PNo_pred is a term of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.
In Proofgold the corresponding term root is b2d51d... and object id is 2efd18...
Axiom. (PNo_bd_pred_lem) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R)
In Proofgold the corresponding term root is 749b8f... and proposition id is 5de4a3...
Axiom. (PNo_bd_pred) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep L R (PNo_bd L R) (PNo_pred L R)
In Proofgold the corresponding term root is 42224b... and proposition id is 3309f6...
Axiom. (PNo_bd_In) We take the following as an axiom:
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_bd L R ∈ ordsucc alpha
In Proofgold the corresponding term root is ad857b... and proposition id is 4719e1...
Beginning of Section TaggedSets
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Theorem. (not_TransSet_Sing1) The following is provable:
¬ TransSet {1}
In Proofgold the corresponding term root is 8f2d67... and proposition id is be9a70...
Proof:
Assume H1: TransSet {1}.
We prove the intermediate claim L1: 0 ∈ {1}.
An exact proof term for the current goal is H1 1 (SingI 1) 0 In_0_1.
Apply neq_0_1 to the current goal.
An exact proof term for the current goal is SingE 1 0 L1.
∎
Theorem. (not_ordinal_Sing1) The following is provable:
¬ ordinal {1}
In Proofgold the corresponding term root is 99a3f6... and proposition id is db7977...
Proof:
Assume H1.
Apply H1 to the current goal.
Assume H2 _.
An exact proof term for the current goal is not_TransSet_Sing1 H2.
∎
Theorem. (tagged_not_ordinal) The following is provable:
∀y, ¬ ordinal (y ')
In Proofgold the corresponding term root is f9488c... and proposition id is aaf6e4...
Proof:
Let y be given.
Assume H1: ordinal (y ').
We prove the intermediate claim L1: {1} ∈ y '.
We will prove {1} ∈ y ∪ {{1}}.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is SingI {1}.
Apply not_ordinal_Sing1 to the current goal.
An exact proof term for the current goal is ordinal_Hered (y ') H1 {1} L1.
∎
Theorem. (tagged_notin_ordinal) The following is provable:
∀alpha y, ordinal alpha → (y ') ∉ alpha
In Proofgold the corresponding term root is b603ba... and proposition id is c0e2c3...
Proof:
Let alpha and y be given.
Assume H1 H2.
Apply tagged_not_ordinal y to the current goal.
An exact proof term for the current goal is ordinal_Hered alpha H1 (y ') H2.
∎
Theorem. (tagged_eqE_Subq) The following is provable:
∀alpha beta, ordinal alpha → alpha ' = beta ' → alpha ⊆ beta
In Proofgold the corresponding term root is cf032f... and proposition id is ef37f5...
Proof:
Let alpha and beta be given.
Assume Ha He.
Let gamma be given.
Assume Hc: gamma ∈ alpha.
We prove the intermediate claim L1: gamma ∈ beta '.
rewrite the current goal using He (from right to left).
We will prove gamma ∈ alpha ∪ {{1}}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is Hc.
Apply binunionE beta {{1}} gamma L1 to the current goal.
Assume H1: gamma ∈ beta.
An exact proof term for the current goal is H1.
Assume H1: gamma ∈ {{1}}.
We will prove False.
Apply not_ordinal_Sing1 to the current goal.
rewrite the current goal using SingE {1} gamma H1 (from right to left).
An exact proof term for the current goal is ordinal_Hered alpha Ha gamma Hc.
∎
Theorem. (tagged_eqE_eq) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha ' = beta ' → alpha = beta
In Proofgold the corresponding term root is 1f7dc9... and proposition id is ec1608...
Proof:
Let alpha and beta be given.
Assume Ha Hb He.
An exact proof term for the current goal is set_ext alpha beta (tagged_eqE_Subq alpha beta Ha He) (tagged_eqE_Subq beta alpha Hb (λq ⇒ He (λu v ⇒ q v u))).
∎
Theorem. (tagged_ReplE) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → beta ' ∈ {gamma '|gamma ∈ alpha} → beta ∈ alpha
In Proofgold the corresponding term root is a1d43c... and proposition id is 70c250...
Proof:
Let alpha and beta be given.
Assume Ha Hb.
Assume H1: beta ' ∈ {gamma '|gamma ∈ alpha}.
Apply ReplE_impred alpha (λgamma ⇒ gamma ') (beta ') H1 to the current goal.
Let gamma be given.
Assume H2: gamma ∈ alpha.
Assume H3: beta ' = gamma '.
We prove the intermediate claim L2: beta = gamma.
An exact proof term for the current goal is tagged_eqE_eq beta gamma Hb (ordinal_Hered alpha Ha gamma H2) H3.
rewrite the current goal using L2 (from left to right).
An exact proof term for the current goal is H2.
∎
Theorem. (ordinal_notin_tagged_Repl) The following is provable:
∀alpha Y, ordinal alpha → alpha ∉ {y '|y ∈ Y}
In Proofgold the corresponding term root is 7f35b5... and proposition id is e7ff3b...
Proof:
Let alpha and Y be given.
Assume H1: ordinal alpha.
Assume H2: alpha ∈ {y '|y ∈ Y}.
Apply ReplE_impred Y (λy ⇒ y ') alpha H2 to the current goal.
Let y be given.
Assume H3: y ∈ Y.
Assume H4: alpha = y '.
Apply tagged_not_ordinal y to the current goal.
We will prove ordinal (y ').
rewrite the current goal using H4 (from right to left).
An exact proof term for the current goal is H1.
∎
Definition. We define SNoElts_ to be λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type set → set.
In Proofgold the corresponding term root is c0ec73... and object id is 6f0c36...
Theorem. (SNoElts_mon) The following is provable:
∀alpha beta, alpha ⊆ beta → SNoElts_ alpha ⊆ SNoElts_ beta
In Proofgold the corresponding term root is 49a08e... and proposition id is 859611...
Proof:
Let alpha and beta be given.
Assume H1: alpha ⊆ beta.
Let x be given.
Assume H2: x ∈ alpha ∪ {gamma '|gamma ∈ alpha}.
Apply binunionE alpha {gamma '|gamma ∈ alpha} x H2 to the current goal.
Assume H3: x ∈ alpha.
We will prove x ∈ beta ∪ {gamma '|gamma ∈ beta}.
Apply binunionI1 to the current goal.
Apply H1 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: x ∈ {gamma '|gamma ∈ alpha}.
We will prove x ∈ beta ∪ {gamma '|gamma ∈ beta}.
Apply binunionI2 to the current goal.
We will prove x ∈ {gamma '|gamma ∈ beta}.
Apply ReplE_impred alpha (λgamma ⇒ gamma ') x H3 to the current goal.
Let gamma be given.
Assume H4: gamma ∈ alpha.
Assume H5: x = gamma '.
rewrite the current goal using H5 (from left to right).
We will prove gamma ' ∈ {gamma '|gamma ∈ beta}.
An exact proof term for the current goal is ReplI beta (λgamma ⇒ gamma ') gamma (H1 gamma H4).
∎
Definition. We define SNo_ to be λalpha x ⇒ x ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ x) (beta ∈ x) of type set → set → prop.
In Proofgold the corresponding term root is 4ab7e4... and object id is c16a73...
Definition. We define PSNo to be λalpha p ⇒ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta} of type set → (set → prop) → set.
In Proofgold the corresponding term root is 3657ae... and object id is 0a8351...
Theorem. (PNoEq_PSNo) The following is provable:
∀alpha, ordinal alpha → ∀p : set → prop, PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo alpha p) p
In Proofgold the corresponding term root is c12e02... and proposition id is 688518...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Let p be given.
Let beta be given.
Assume Hb: beta ∈ alpha.
We will prove beta ∈ PSNo alpha p ↔ p beta.
Apply iffI to the current goal.
Assume H1: beta ∈ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta}.
Apply binunionE {beta ∈ alpha|p beta} {beta '|beta ∈ alpha, ¬ p beta} beta H1 to the current goal.
Assume H2: beta ∈ {beta ∈ alpha|p beta}.
An exact proof term for the current goal is SepE2 alpha p beta H2.
Assume H2: beta ∈ {beta '|beta ∈ alpha, ¬ p beta}.
We will prove False.
Apply ReplSepE_impred alpha (λbeta ⇒ ¬ p beta) (λx ⇒ x ') beta H2 to the current goal.
Let gamma be given.
Assume Hc: gamma ∈ alpha.
Assume H3: ¬ p gamma.
Assume H4: beta = gamma '.
Apply tagged_notin_ordinal alpha gamma Ha to the current goal.
We will prove gamma ' ∈ alpha.
rewrite the current goal using H4 (from right to left).
An exact proof term for the current goal is Hb.
Assume H1: p beta.
We will prove beta ∈ PSNo alpha p.
We will prove beta ∈ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta}.
Apply binunionI1 to the current goal.
Apply SepI to the current goal.
An exact proof term for the current goal is Hb.
An exact proof term for the current goal is H1.
∎
Theorem. (SNo_PSNo) The following is provable:
∀alpha, ordinal alpha → ∀p : set → prop, SNo_ alpha (PSNo alpha p)
In Proofgold the corresponding term root is 96e4e2... and proposition id is f03db0...
Proof:
Let alpha be given.
Assume Ha.
Let p be given.
We will prove PSNo alpha p ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ PSNo alpha p) (beta ∈ PSNo alpha p).
Apply andI to the current goal.
Let u be given.
Assume Hu: u ∈ PSNo alpha p.
We will prove u ∈ SNoElts_ alpha.
Apply binunionE {beta ∈ alpha|p beta} {beta '|beta ∈ alpha, ¬ p beta} u Hu to the current goal.
Assume H1: u ∈ {beta ∈ alpha|p beta}.
Apply SepE alpha p u H1 to the current goal.
Assume H2: u ∈ alpha.
Assume H3: p u.
We will prove u ∈ alpha ∪ {beta '|beta ∈ alpha}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
Assume H1: u ∈ {beta '|beta ∈ alpha, ¬ p beta}.
Apply ReplSepE_impred alpha (λbeta ⇒ ¬ p beta) (λbeta ⇒ beta ') u H1 to the current goal.
Let beta be given.
Assume H2: beta ∈ alpha.
Assume H3: ¬ p beta.
Assume H4: u = beta '.
We will prove u ∈ alpha ∪ {beta '|beta ∈ alpha}.
Apply binunionI2 to the current goal.
We will prove u ∈ {beta '|beta ∈ alpha}.
rewrite the current goal using H4 (from left to right).
An exact proof term for the current goal is ReplI alpha (λbeta ⇒ beta ') beta H2.
Let beta be given.
Assume H1: beta ∈ alpha.
We prove the intermediate claim Lbeta: ordinal beta.
An exact proof term for the current goal is ordinal_Hered alpha Ha beta H1.
We will prove exactly1of2 (beta ' ∈ PSNo alpha p) (beta ∈ PSNo alpha p).
Apply xm (p beta) to the current goal.
Assume H2: p beta.
Apply exactly1of2_I2 to the current goal.
We will prove beta ' ∉ PSNo alpha p.
Assume H3: beta ' ∈ PSNo alpha p.
Apply binunionE {beta ∈ alpha|p beta} {beta '|beta ∈ alpha, ¬ p beta} (beta ') H3 to the current goal.
Assume H4: beta ' ∈ {beta ∈ alpha|p beta}.
Apply SepE alpha p (beta ') H4 to the current goal.
Assume H5: beta ' ∈ alpha.
Assume H6: p (beta ').
We will prove False.
Apply tagged_not_ordinal beta to the current goal.
An exact proof term for the current goal is ordinal_Hered alpha Ha (beta ') H5.
Assume H4: beta ' ∈ {beta '|beta ∈ alpha, ¬ p beta}.
Apply ReplSepE_impred alpha (λbeta ⇒ ¬ p beta) (λbeta ⇒ beta ') (beta ') H4 to the current goal.
Let gamma be given.
Assume H5: gamma ∈ alpha.
Assume H6: ¬ p gamma.
Assume H7: beta ' = gamma '.
We prove the intermediate claim Lgamma: ordinal gamma.
An exact proof term for the current goal is ordinal_Hered alpha Ha gamma H5.
We prove the intermediate claim L1: beta = gamma.
An exact proof term for the current goal is tagged_eqE_eq beta gamma Lbeta Lgamma H7.
Apply H6 to the current goal.
We will prove p gamma.
rewrite the current goal using L1 (from right to left).
We will prove p beta.
An exact proof term for the current goal is H2.
We will prove beta ∈ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is SepI alpha p beta H1 H2.
Assume H2: ¬ p beta.
Apply exactly1of2_I1 to the current goal.
We will prove beta ' ∈ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta}.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is ReplSepI alpha (λbeta ⇒ ¬ p beta) (λbeta ⇒ beta ') beta H1 H2.
We will prove beta ∉ PSNo alpha p.
Assume H3: beta ∈ PSNo alpha p.
Apply binunionE {beta ∈ alpha|p beta} {beta '|beta ∈ alpha, ¬ p beta} beta H3 to the current goal.
Assume H4: beta ∈ {beta ∈ alpha|p beta}.
Apply SepE alpha p beta H4 to the current goal.
Assume H5: beta ∈ alpha.
Assume H6: p beta.
We will prove False.
An exact proof term for the current goal is H2 H6.
Assume H4: beta ∈ {beta '|beta ∈ alpha, ¬ p beta}.
Apply ReplSepE_impred alpha (λbeta ⇒ ¬ p beta) (λbeta ⇒ beta ') beta H4 to the current goal.
Let gamma be given.
Assume H5: gamma ∈ alpha.
Assume H6: ¬ p gamma.
Assume H7: beta = gamma '.
Apply tagged_not_ordinal gamma to the current goal.
We will prove ordinal (gamma ').
rewrite the current goal using H7 (from right to left).
An exact proof term for the current goal is Lbeta.
∎
Theorem. (SNo_PSNo_eta_) The following is provable:
∀alpha x, ordinal alpha → SNo_ alpha x → x = PSNo alpha (λbeta ⇒ beta ∈ x)
In Proofgold the corresponding term root is 1642af... and proposition id is 630119...
Proof:
Let alpha and x be given.
Assume Ha Hx.
Apply Hx to the current goal.
Assume Hx1: x ⊆ SNoElts_ alpha.
Assume Hx2: ∀beta ∈ alpha, exactly1of2 (beta ' ∈ x) (beta ∈ x).
Apply set_ext to the current goal.
We will prove x ⊆ PSNo alpha (λbeta ⇒ beta ∈ x).
Let u be given.
Assume Hu: u ∈ x.
Apply binunionE alpha {beta '|beta ∈ alpha} u (Hx1 u Hu) to the current goal.
Assume H1: u ∈ alpha.
We will prove u ∈ {beta ∈ alpha|beta ∈ x} ∪ {beta '|beta ∈ alpha, beta ∉ x}.
Apply binunionI1 to the current goal.
Apply SepI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is Hu.
Assume H1: u ∈ {beta '|beta ∈ alpha}.
Apply ReplE_impred alpha (λbeta ⇒ beta ') u H1 to the current goal.
Let beta be given.
Assume H2: beta ∈ alpha.
Assume H3: u = beta '.
We will prove u ∈ {beta ∈ alpha|beta ∈ x} ∪ {beta '|beta ∈ alpha, beta ∉ x}.
Apply binunionI2 to the current goal.
We will prove u ∈ {beta '|beta ∈ alpha, beta ∉ x}.
rewrite the current goal using H3 (from left to right).
We will prove beta ' ∈ {beta '|beta ∈ alpha, beta ∉ x}.
We prove the intermediate claim L2: beta ∉ x.
Assume H4: beta ∈ x.
Apply exactly1of2_E (beta ' ∈ x) (beta ∈ x) (Hx2 beta H2) to the current goal.
Assume H5: beta ' ∈ x.
Assume H6: beta ∉ x.
An exact proof term for the current goal is H6 H4.
Assume H4: beta ' ∉ x.
Assume H5: beta ∈ x.
Apply H4 to the current goal.
We will prove (beta ') ∈ x.
rewrite the current goal using H3 (from right to left).
We will prove u ∈ x.
An exact proof term for the current goal is Hu.
An exact proof term for the current goal is ReplSepI alpha (λbeta ⇒ beta ∉ x) (λbeta ⇒ beta ') beta H2 L2.
We will prove PSNo alpha (λbeta ⇒ beta ∈ x) ⊆ x.
Let u be given.
Assume Hu: u ∈ PSNo alpha (λbeta ⇒ beta ∈ x).
We will prove u ∈ x.
Apply binunionE {beta ∈ alpha|beta ∈ x} {beta '|beta ∈ alpha, beta ∉ x} u Hu to the current goal.
Assume H1: u ∈ {beta ∈ alpha|beta ∈ x}.
Apply SepE alpha (λbeta ⇒ beta ∈ x) u H1 to the current goal.
Assume H2: u ∈ alpha.
Assume H3: u ∈ x.
An exact proof term for the current goal is H3.
Assume H1: u ∈ {beta '|beta ∈ alpha, beta ∉ x}.
Apply ReplSepE_impred alpha (λbeta ⇒ beta ∉ x) (λbeta ⇒ beta ') u H1 to the current goal.
Let beta be given.
Assume H2: beta ∈ alpha.
Assume H3: beta ∉ x.
Assume H4: u = beta '.
Apply exactly1of2_E (beta ' ∈ x) (beta ∈ x) (Hx2 beta H2) to the current goal.
Assume H5: beta ' ∈ x.
Assume H6: beta ∉ x.
rewrite the current goal using H4 (from left to right).
An exact proof term for the current goal is H5.
Assume H4: beta ' ∉ x.
Assume H5: beta ∈ x.
We will prove False.
An exact proof term for the current goal is H3 H5.
∎
Definition. We define SNo to be λx ⇒ ∃alpha, ordinal alpha ∧ SNo_ alpha x of type set → prop.
In Proofgold the corresponding term root is 11faa7... and object id is 116b76...
Theorem. (SNo_SNo) The following is provable:
∀alpha, ordinal alpha → ∀z, SNo_ alpha z → SNo z
In Proofgold the corresponding term root is 929b7a... and proposition id is a59341...
Proof:
Let alpha be given.
Assume Ha.
Let z be given.
Assume Hz: SNo_ alpha z.
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha z.
We use alpha to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Ha.
An exact proof term for the current goal is Hz.
∎
Definition. We define SNoLev to be λx ⇒ Eps_i (λalpha ⇒ ordinal alpha ∧ SNo_ alpha x) of type set → set.
In Proofgold the corresponding term root is 293b77... and object id is 83dce3...
Theorem. (SNoLev_uniq_Subq) The following is provable:
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha ⊆ beta
In Proofgold the corresponding term root is 2ca114... and proposition id is 65843d...
Proof:
Let x, alpha and beta be given.
Assume Ha Hb Hax Hbx.
Let gamma be given.
Assume Hc: gamma ∈ alpha.
We will prove gamma ∈ beta.
Apply Hax to the current goal.
Assume Hax1 Hax2.
Apply Hbx to the current goal.
Assume Hbx1 Hbx2.
We prove the intermediate claim Lc: ordinal gamma.
An exact proof term for the current goal is ordinal_Hered alpha Ha gamma Hc.
Apply exactly1of2_or (gamma ' ∈ x) (gamma ∈ x) (Hax2 gamma Hc) to the current goal.
Assume H1: gamma ' ∈ x.
We prove the intermediate claim L1: gamma ' ∈ beta ∪ {delta '|delta ∈ beta}.
An exact proof term for the current goal is Hbx1 (gamma ') H1.
We prove the intermediate claim L2: gamma ' ∈ beta ∨ gamma ' ∈ {delta '|delta ∈ beta}.
An exact proof term for the current goal is binunionE beta {delta '|delta ∈ beta} (gamma ') L1.
Apply L2 to the current goal.
Assume H2: gamma ' ∈ beta.
We will prove False.
An exact proof term for the current goal is tagged_notin_ordinal beta gamma Hb H2.
Assume H2: gamma ' ∈ {delta '|delta ∈ beta}.
An exact proof term for the current goal is tagged_ReplE beta gamma Hb Lc H2.
Assume H1: gamma ∈ x.
We prove the intermediate claim L1: gamma ∈ beta ∪ {delta '|delta ∈ beta}.
An exact proof term for the current goal is Hbx1 gamma H1.
We prove the intermediate claim L2: gamma ∈ beta ∨ gamma ∈ {delta '|delta ∈ beta}.
An exact proof term for the current goal is binunionE beta {delta '|delta ∈ beta} gamma L1.
Apply L2 to the current goal.
Assume H2: gamma ∈ beta.
An exact proof term for the current goal is H2.
Assume H2: gamma ∈ {delta '|delta ∈ beta}.
We will prove False.
An exact proof term for the current goal is ordinal_notin_tagged_Repl gamma beta Lc H2.
∎
Theorem. (SNoLev_uniq) The following is provable:
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha = beta
In Proofgold the corresponding term root is 68f71d... and proposition id is 5d96d4...
Proof:
Let x, alpha and beta be given.
Assume Ha Hb Hax Hbx.
Apply set_ext to the current goal.
An exact proof term for the current goal is SNoLev_uniq_Subq x alpha beta Ha Hb Hax Hbx.
An exact proof term for the current goal is SNoLev_uniq_Subq x beta alpha Hb Ha Hbx Hax.
∎
Theorem. (SNoLev_prop) The following is provable:
∀x, SNo x → ordinal (SNoLev x) ∧ SNo_ (SNoLev x) x
In Proofgold the corresponding term root is a8afcc... and proposition id is 21761c...
Proof:
Let x be given.
Assume Hx: SNo x.
An exact proof term for the current goal is Eps_i_ex (λalpha ⇒ ordinal alpha ∧ SNo_ alpha x) Hx.
∎
Theorem. (SNoLev_ordinal) The following is provable:
∀x, SNo x → ordinal (SNoLev x)
In Proofgold the corresponding term root is 7cacc5... and proposition id is f0e9f7...
Proof:
Let x be given.
Assume Hx.
Apply SNoLev_prop x Hx to the current goal.
Assume H1 _.
An exact proof term for the current goal is H1.
∎
Theorem. (SNoLev_) The following is provable:
∀x, SNo x → SNo_ (SNoLev x) x
In Proofgold the corresponding term root is 4a1033... and proposition id is 49cd5a...
Proof:
Let x be given.
Assume Hx.
Apply SNoLev_prop x Hx to the current goal.
Assume _ H1.
An exact proof term for the current goal is H1.
∎
Theorem. (SNo_PSNo_eta) The following is provable:
∀x, SNo x → x = PSNo (SNoLev x) (λbeta ⇒ beta ∈ x)
In Proofgold the corresponding term root is 6ffefe... and proposition id is 8c7870...
Proof:
Let x be given.
Assume Hx.
Apply SNoLev_prop x Hx to the current goal.
Assume Hx1: ordinal (SNoLev x).
Assume Hx2: SNo_ (SNoLev x) x.
Apply SNo_PSNo_eta_ to the current goal.
We will prove ordinal (SNoLev x).
An exact proof term for the current goal is Hx1.
We will prove SNo_ (SNoLev x) x.
An exact proof term for the current goal is Hx2.
∎
Theorem. (SNoLev_PSNo) The following is provable:
∀alpha, ordinal alpha → ∀p : set → prop, SNoLev (PSNo alpha p) = alpha
In Proofgold the corresponding term root is 07f1eb... and proposition id is 9c4890...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Let p be given.
We prove the intermediate claim L1: SNo_ alpha (PSNo alpha p).
An exact proof term for the current goal is SNo_PSNo alpha Ha p.
We prove the intermediate claim L2: SNo (PSNo alpha p).
We will prove ∃beta, ordinal beta ∧ SNo_ beta (PSNo alpha p).
We use alpha to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Ha.
An exact proof term for the current goal is L1.
Apply SNoLev_prop (PSNo alpha p) L2 to the current goal.
Assume H2: ordinal (SNoLev (PSNo alpha p)).
Assume H3: SNo_ (SNoLev (PSNo alpha p)) (PSNo alpha p).
An exact proof term for the current goal is SNoLev_uniq (PSNo alpha p) (SNoLev (PSNo alpha p)) alpha H2 Ha H3 L1.
∎
Theorem. (SNo_Subq) The following is provable:
∀x y, SNo x → SNo y → SNoLev x ⊆ SNoLev y → (∀alpha ∈ SNoLev x, alpha ∈ x ↔ alpha ∈ y) → x ⊆ y
In Proofgold the corresponding term root is a244d7... and proposition id is a428af...
Proof:
Let x and y be given.
Assume Hx Hy H1 H2.
Apply SNoLev_ x Hx to the current goal.
Assume Hx2a: x ⊆ SNoElts_ (SNoLev x).
Assume Hx2b: ∀beta ∈ SNoLev x, exactly1of2 (beta ' ∈ x) (beta ∈ x).
Apply SNoLev_ y Hy to the current goal.
Assume Hy2a: y ⊆ SNoElts_ (SNoLev y).
Assume Hy2b: ∀beta ∈ SNoLev y, exactly1of2 (beta ' ∈ y) (beta ∈ y).
Let u be given.
Assume Hu: u ∈ x.
We prove the intermediate claim L1: u ∈ SNoLev x ∪ {beta '|beta ∈ SNoLev x}.
An exact proof term for the current goal is Hx2a u Hu.
Apply binunionE (SNoLev x) {beta '|beta ∈ SNoLev x} u L1 to the current goal.
Assume H3: u ∈ SNoLev x.
Apply H2 u H3 to the current goal.
Assume H4 _.
An exact proof term for the current goal is H4 Hu.
Assume H3: u ∈ {beta '|beta ∈ SNoLev x}.
Apply ReplE_impred (SNoLev x) (λgamma ⇒ gamma ') u H3 to the current goal.
Let beta be given.
Assume H4: beta ∈ SNoLev x.
Assume H5: u = beta '.
We prove the intermediate claim L3: beta ∈ SNoLev y.
An exact proof term for the current goal is H1 beta H4.
Apply exactly1of2_E (beta ' ∈ y) (beta ∈ y) (Hy2b beta L3) to the current goal.
Assume H6: beta ' ∈ y.
Assume H7: beta ∉ y.
We will prove u ∈ y.
rewrite the current goal using H5 (from left to right).
An exact proof term for the current goal is H6.
Assume H6: beta ' ∉ y.
Assume H7: beta ∈ y.
We will prove False.
Apply exactly1of2_E (beta ' ∈ x) (beta ∈ x) (Hx2b beta H4) to the current goal.
Assume H8: beta ' ∈ x.
Assume H9: beta ∉ x.
Apply H9 to the current goal.
Apply H2 beta H4 to the current goal.
Assume _ H10.
An exact proof term for the current goal is H10 H7.
Assume H8: beta ' ∉ x.
Assume H9: beta ∈ x.
Apply H8 to the current goal.
We will prove beta ' ∈ x.
rewrite the current goal using H5 (from right to left).
An exact proof term for the current goal is Hu.
∎
Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type set → set → set → prop.
In Proofgold the corresponding term root is 5f11e2... and object id is 12b5f6...
Theorem. (SNoEq_I) The following is provable:
∀alpha x y, (∀beta ∈ alpha, beta ∈ x ↔ beta ∈ y) → SNoEq_ alpha x y
In Proofgold the corresponding term root is a718b1... and proposition id is fdcb78...
Proof:
Let alpha, x and y be given.
Assume Hxy.
An exact proof term for the current goal is Hxy.
∎
Theorem. (SNo_eq) The following is provable:
∀x y, SNo x → SNo y → SNoLev x = SNoLev y → SNoEq_ (SNoLev x) x y → x = y
In Proofgold the corresponding term root is b08bdc... and proposition id is bf4ac3...
Proof:
Let x and y be given.
Assume Hx Hy H1 H2.
Apply set_ext to the current goal.
We will prove x ⊆ y.
Apply SNo_Subq x y Hx Hy to the current goal.
We will prove SNoLev x ⊆ SNoLev y.
rewrite the current goal using H1 (from right to left).
Apply Subq_ref to the current goal.
An exact proof term for the current goal is H2.
We will prove y ⊆ x.
Apply SNo_Subq y x Hy Hx to the current goal.
We will prove SNoLev y ⊆ SNoLev x.
rewrite the current goal using H1 (from left to right).
Apply Subq_ref to the current goal.
Let alpha be given.
Assume H3: alpha ∈ SNoLev y.
We will prove alpha ∈ y ↔ alpha ∈ x.
Apply iff_sym to the current goal.
We will prove alpha ∈ x ↔ alpha ∈ y.
Apply H2 alpha to the current goal.
We will prove alpha ∈ SNoLev x.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is H3.
∎
End of Section TaggedSets
Definition. We define SNoLt to be λx y ⇒ PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.
In Proofgold the corresponding term root is 46e7ed... and object id is 8c8aa9...
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Definition. We define SNoLe to be λx y ⇒ PNoLe (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.
In Proofgold the corresponding term root is ddf7d3... and object id is 54f405...
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (SNoLtLe) The following is provable:
∀x y, x < y → x ≤ y
In Proofgold the corresponding term root is c0fef1... and proposition id is f09833...
Proof:
Let x and y be given.
Assume H1.
We will prove PNoLe (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y).
Apply PNoLeI1 to the current goal.
An exact proof term for the current goal is H1.
∎
Theorem. (SNoLeE) The following is provable:
∀x y, SNo x → SNo y → x ≤ y → x < y ∨ x = y
In Proofgold the corresponding term root is 1c9c62... and proposition id is 23cc47...
Proof:
Let x and y be given.
Assume Hx Hy.
Assume H1: PNoLe (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y).
Apply H1 to the current goal.
Assume H2: PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y).
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2.
Apply H2 to the current goal.
Assume H2: SNoLev x = SNoLev y.
Assume H3: PNoEq_ (SNoLev x) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
Apply orIR to the current goal.
An exact proof term for the current goal is SNo_eq x y Hx Hy H2 H3.
∎
Theorem. (SNoEq_sym_) The following is provable:
∀alpha x y, SNoEq_ alpha x y → SNoEq_ alpha y x
In Proofgold the corresponding term root is f8c61d... and proposition id is f756d2...
Proof:
Let alpha, x and y be given.
An exact proof term for the current goal is PNoEq_sym_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
∎
Theorem. (SNoEq_tra_) The following is provable:
∀alpha x y z, SNoEq_ alpha x y → SNoEq_ alpha y z → SNoEq_ alpha x z
In Proofgold the corresponding term root is 0fff96... and proposition id is 246d04...
Proof:
Let alpha, x, y and z be given.
An exact proof term for the current goal is PNoEq_tra_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) (λbeta ⇒ beta ∈ z).
∎
Theorem. (SNoLtE) The following is provable:
∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀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
In Proofgold the corresponding term root is 958711... and proposition id is d12fc6...
Proof:
Let x and y be given.
Assume Hx: SNo x.
Assume Hy: SNo y.
Assume Hxy: PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y).
Let p be given.
Assume Hp1 Hp2 Hp3.
We prove the intermediate claim LLx: ordinal (SNoLev x).
Apply SNoLev_ordinal x to the current goal.
An exact proof term for the current goal is Hx.
We prove the intermediate claim LLy: ordinal (SNoLev y).
Apply SNoLev_ordinal y to the current goal.
An exact proof term for the current goal is Hy.
Apply PNoLtE (SNoLev x) (SNoLev y) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) Hxy to the current goal.
Assume H1: PNoLt_ (SNoLev x ∩ SNoLev y) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
Apply PNoLt_E_ (SNoLev x ∩ SNoLev y) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) H1 to the current goal.
Let alpha be given.
Assume Ha: alpha ∈ SNoLev x ∩ SNoLev y.
Assume H2: PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
Assume H3: alpha ∉ x.
Assume H4: alpha ∈ y.
Apply binintersectE (SNoLev x) (SNoLev y) alpha Ha to the current goal.
Assume Ha1: alpha ∈ SNoLev x.
Assume Ha2: alpha ∈ SNoLev y.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is ordinal_Hered (SNoLev x) LLx alpha Ha1.
Set z to be the term PSNo alpha (λbeta ⇒ beta ∈ x).
We prove the intermediate claim L1: SNo_ alpha z.
Apply SNo_PSNo to the current goal.
An exact proof term for the current goal is La.
We prove the intermediate claim L2: SNo z.
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha z.
We use alpha to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is La.
An exact proof term for the current goal is L1.
Apply SNoLev_prop z L2 to the current goal.
Assume Hz1: ordinal (SNoLev z).
Assume Hz2: SNo_ (SNoLev z) z.
We prove the intermediate claim L3: SNoLev z = alpha.
An exact proof term for the current goal is SNoLev_uniq z (SNoLev z) alpha Hz1 La Hz2 L1.
We prove the intermediate claim L4: SNoEq_ alpha z x.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ z) (λbeta ⇒ beta ∈ x).
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo alpha (λbeta ⇒ beta ∈ x)) (λbeta ⇒ beta ∈ x).
Apply PNoEq_PSNo to the current goal.
An exact proof term for the current goal is La.
We prove the intermediate claim L5: SNoEq_ alpha z y.
Apply SNoEq_tra_ alpha z x y L4 to the current goal.
We will prove SNoEq_ alpha x y.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
An exact proof term for the current goal is H2.
Apply Hp1 z L2 to the current goal.
We will prove SNoLev z ∈ SNoLev x ∩ SNoLev y.
rewrite the current goal using L3 (from left to right).
An exact proof term for the current goal is Ha.
We will prove SNoEq_ (SNoLev z) z x.
rewrite the current goal using L3 (from left to right).
An exact proof term for the current goal is L4.
We will prove SNoEq_ (SNoLev z) z y.
rewrite the current goal using L3 (from left to right).
An exact proof term for the current goal is L5.
We will prove x < z.
We will prove PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev z) (λbeta ⇒ beta ∈ z).
rewrite the current goal using L3 (from left to right).
We will prove PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) alpha (λbeta ⇒ beta ∈ z).
Apply PNoLtI3 to the current goal.
We will prove alpha ∈ SNoLev x.
An exact proof term for the current goal is Ha1.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ z).
Apply PNoEq_sym_ to the current goal.
An exact proof term for the current goal is L4.
We will prove alpha ∉ x.
An exact proof term for the current goal is H3.
We will prove z < y.
We will prove PNoLt (SNoLev z) (λbeta ⇒ beta ∈ z) (SNoLev y) (λbeta ⇒ beta ∈ y).
rewrite the current goal using L3 (from left to right).
We will prove PNoLt alpha (λbeta ⇒ beta ∈ z) (SNoLev y) (λbeta ⇒ beta ∈ y).
Apply PNoLtI2 to the current goal.
We will prove alpha ∈ SNoLev y.
An exact proof term for the current goal is Ha2.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ z) (λbeta ⇒ beta ∈ y).
An exact proof term for the current goal is L5.
We will prove alpha ∈ y.
An exact proof term for the current goal is H4.
We will prove SNoLev z ∉ x.
rewrite the current goal using L3 (from left to right).
An exact proof term for the current goal is H3.
We will prove SNoLev z ∈ y.
rewrite the current goal using L3 (from left to right).
An exact proof term for the current goal is H4.
Assume H1: SNoLev x ∈ SNoLev y.
Assume H2: PNoEq_ (SNoLev x) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
Assume H3: SNoLev x ∈ y.
Apply Hp2 to the current goal.
We will prove SNoLev x ∈ SNoLev y.
An exact proof term for the current goal is H1.
We will prove SNoEq_ (SNoLev x) x y.
An exact proof term for the current goal is H2.
We will prove SNoLev x ∈ y.
An exact proof term for the current goal is H3.
Assume H1: SNoLev y ∈ SNoLev x.
Assume H2: PNoEq_ (SNoLev y) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
Assume H3: SNoLev y ∉ x.
Apply Hp3 to the current goal.
We will prove SNoLev y ∈ SNoLev x.
An exact proof term for the current goal is H1.
We will prove SNoEq_ (SNoLev y) x y.
An exact proof term for the current goal is H2.
We will prove SNoLev y ∉ x.
An exact proof term for the current goal is H3.
∎
Theorem. (SNoLtI2) The following is provable:
∀x y, SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → x < y
In Proofgold the corresponding term root is 282b01... and proposition id is 82b82c...
Proof:
Let x and y be given.
Assume H1 H2 H3.
We will prove PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y).
Apply PNoLtI2 to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
An exact proof term for the current goal is H3.
∎
Theorem. (SNoLtI3) The following is provable:
∀x y, SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → x < y
In Proofgold the corresponding term root is 006afc... and proposition id is 938f67...
Proof:
Let x and y be given.
Assume H1 H2 H3.
We will prove PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y).
Apply PNoLtI3 to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
An exact proof term for the current goal is H3.
∎
Theorem. (SNoLt_irref) The following is provable:
∀x, ¬ SNoLt x x
In Proofgold the corresponding term root is d5ecf9... and proposition id is 06fac5...
Proof:
Let x be given.
An exact proof term for the current goal is PNoLt_irref (SNoLev x) (λbeta ⇒ beta ∈ x).
∎
Theorem. (SNoLt_trichotomy_or) The following is provable:
∀x y, SNo x → SNo y → x < y ∨ x = y ∨ y < x
In Proofgold the corresponding term root is 35a4c8... and proposition id is 8ad056...
Proof:
Let x and y be given.
Assume Hx Hy.
Apply PNoLt_trichotomy_or (SNoLev x) (SNoLev y) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) (SNoLev_ordinal x Hx) (SNoLev_ordinal y Hy) to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply or3I1 to the current goal.
An exact proof term for the current goal is H1.
Assume H1.
Apply H1 to the current goal.
Assume H2: SNoLev x = SNoLev y.
Assume H3: PNoEq_ (SNoLev x) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
Apply or3I2 to the current goal.
An exact proof term for the current goal is SNo_eq x y Hx Hy H2 H3.
Assume H1.
Apply or3I3 to the current goal.
An exact proof term for the current goal is H1.
∎
Theorem. (SNoLt_trichotomy_or_impred) The following is provable:
∀x y, SNo x → SNo y → ∀p : prop, (x < y → p) → (x = y → p) → (y < x → p) → p
In Proofgold the corresponding term root is ba04ef... and proposition id is 7588e3...
Proof:
Let x and y be given.
Assume Hx Hy.
Let p be given.
Assume Hp1 Hp2 Hp3.
Apply SNoLt_trichotomy_or x y Hx Hy to the current goal.
Assume H.
Apply H to the current goal.
An exact proof term for the current goal is Hp1.
An exact proof term for the current goal is Hp2.
An exact proof term for the current goal is Hp3.
∎
Theorem. (SNoLt_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x < y → y < z → x < z
In Proofgold the corresponding term root is ac0e24... and proposition id is ccfdf7...
Proof:
Let x, y and z be given.
Assume Hx Hy Hz Hxy Hyz.
We will prove PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev z) (λbeta ⇒ beta ∈ z).
An exact proof term for the current goal is PNoLt_tra (SNoLev x) (SNoLev y) (SNoLev z) (SNoLev_ordinal x Hx) (SNoLev_ordinal y Hy) (SNoLev_ordinal z Hz) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) (λbeta ⇒ beta ∈ z) Hxy Hyz.
∎
Theorem. (SNoLe_ref) The following is provable:
∀x, SNoLe x x
In Proofgold the corresponding term root is c940c8... and proposition id is 83bd2d...
Proof:
Let x be given.
An exact proof term for the current goal is PNoLe_ref (SNoLev x) (λbeta ⇒ beta ∈ x).
∎
Theorem. (SNoLe_antisym) The following is provable:
∀x y, SNo x → SNo y → x ≤ y → y ≤ x → x = y
In Proofgold the corresponding term root is f9f757... and proposition id is 3abee4...
Proof:
Let x and y be given.
Assume Hx Hy Hxy Hyx.
Apply PNoLe_antisym (SNoLev x) (SNoLev y) (SNoLev_ordinal x Hx) (SNoLev_ordinal y Hy) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) Hxy Hyx to the current goal.
Assume H1: SNoLev x = SNoLev y.
Assume H2: PNoEq_ (SNoLev x) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y).
An exact proof term for the current goal is SNo_eq x y Hx Hy H1 H2.
∎
Theorem. (SNoLtLe_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x < y → y ≤ z → x < z
In Proofgold the corresponding term root is 3bc661... and proposition id is ba8545...
Proof:
Let x, y and z be given.
Assume Hx Hy Hz Hxy Hyz.
An exact proof term for the current goal is PNoLtLe_tra (SNoLev x) (SNoLev y) (SNoLev z) (SNoLev_ordinal x Hx) (SNoLev_ordinal y Hy) (SNoLev_ordinal z Hz) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) (λbeta ⇒ beta ∈ z) Hxy Hyz.
∎
Theorem. (SNoLeLt_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y < z → x < z
In Proofgold the corresponding term root is b2178f... and proposition id is 4a77a2...
Proof:
Let x, y and z be given.
Assume Hx Hy Hz Hxy Hyz.
An exact proof term for the current goal is PNoLeLt_tra (SNoLev x) (SNoLev y) (SNoLev z) (SNoLev_ordinal x Hx) (SNoLev_ordinal y Hy) (SNoLev_ordinal z Hz) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) (λbeta ⇒ beta ∈ z) Hxy Hyz.
∎
Theorem. (SNoLe_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y ≤ z → x ≤ z
In Proofgold the corresponding term root is a0c5c6... and proposition id is 031989...
Proof:
Let x, y and z be given.
Assume Hx Hy Hz Hxy Hyz.
An exact proof term for the current goal is PNoLe_tra (SNoLev x) (SNoLev y) (SNoLev z) (SNoLev_ordinal x Hx) (SNoLev_ordinal y Hy) (SNoLev_ordinal z Hz) (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) (λbeta ⇒ beta ∈ z) Hxy Hyz.
∎
Theorem. (SNoLtLe_or) The following is provable:
∀x y, SNo x → SNo y → x < y ∨ y ≤ x
In Proofgold the corresponding term root is f595ac... and proposition id is 8965d2...
Proof:
Let x and y be given.
Assume Hx Hy.
Apply SNoLt_trichotomy_or x y Hx Hy to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1: x < y.
Apply orIL to the current goal.
An exact proof term for the current goal is H1.
Assume H1: x = y.
Apply orIR to the current goal.
We will prove y ≤ x.
rewrite the current goal using H1 (from left to right).
We will prove y ≤ y.
Apply SNoLe_ref to the current goal.
Assume H1: y < x.
Apply orIR to the current goal.
We will prove y ≤ x.
Apply SNoLtLe to the current goal.
An exact proof term for the current goal is H1.
∎
Theorem. (SNoLt_PSNo_PNoLt) The following is provable:
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PSNo alpha p < PSNo beta q → PNoLt alpha p beta q
In Proofgold the corresponding term root is 6e122d... and proposition id is 535161...
Proof:
Let alpha, beta, p and q be given.
Assume Ha Hb.
We will prove PNoLt (SNoLev (PSNo alpha p)) (λgamma ⇒ gamma ∈ PSNo alpha p) (SNoLev (PSNo beta q)) (λgamma ⇒ gamma ∈ PSNo beta q) → PNoLt alpha p beta q.
rewrite the current goal using SNoLev_PSNo alpha Ha p (from left to right).
rewrite the current goal using SNoLev_PSNo beta Hb q (from left to right).
Assume H1: PNoLt alpha (λgamma ⇒ gamma ∈ PSNo alpha p) beta (λgamma ⇒ gamma ∈ PSNo beta q).
Apply PNoEqLt_tra alpha beta Ha Hb p (λgamma ⇒ gamma ∈ PSNo alpha p) q to the current goal.
We will prove PNoEq_ alpha p (λgamma ⇒ gamma ∈ PSNo alpha p).
Apply PNoEq_sym_ to the current goal.
Apply PNoEq_PSNo to the current goal.
An exact proof term for the current goal is Ha.
We will prove PNoLt alpha (λgamma ⇒ gamma ∈ PSNo alpha p) beta q.
Apply PNoLtEq_tra alpha beta Ha Hb (λgamma ⇒ gamma ∈ PSNo alpha p) (λgamma ⇒ gamma ∈ PSNo beta q) q to the current goal.
An exact proof term for the current goal is H1.
We will prove PNoEq_ beta (λgamma ⇒ gamma ∈ PSNo beta q) q.
Apply PNoEq_PSNo to the current goal.
An exact proof term for the current goal is Hb.
∎
Theorem. (PNoLt_SNoLt_PSNo) The following is provable:
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q → PSNo alpha p < PSNo beta q
In Proofgold the corresponding term root is 11b6ed... and proposition id is 6196d9...
Proof:
Let alpha, beta, p and q be given.
Assume Ha Hb.
Assume H1: PNoLt alpha p beta q.
We will prove PNoLt (SNoLev (PSNo alpha p)) (λgamma ⇒ gamma ∈ PSNo alpha p) (SNoLev (PSNo beta q)) (λgamma ⇒ gamma ∈ PSNo beta q).
rewrite the current goal using SNoLev_PSNo alpha Ha p (from left to right).
rewrite the current goal using SNoLev_PSNo beta Hb q (from left to right).
We will prove PNoLt alpha (λgamma ⇒ gamma ∈ PSNo alpha p) beta (λgamma ⇒ gamma ∈ PSNo beta q).
Apply PNoEqLt_tra alpha beta Ha Hb (λgamma ⇒ gamma ∈ PSNo alpha p) p (λgamma ⇒ gamma ∈ PSNo beta q) to the current goal.
We will prove PNoEq_ alpha (λgamma ⇒ gamma ∈ PSNo alpha p) p.
Apply PNoEq_PSNo to the current goal.
An exact proof term for the current goal is Ha.
We will prove PNoLt alpha p beta (λgamma ⇒ gamma ∈ PSNo beta q).
Apply PNoLtEq_tra alpha beta Ha Hb p q (λgamma ⇒ gamma ∈ PSNo beta q) to the current goal.
An exact proof term for the current goal is H1.
We will prove PNoEq_ beta q (λgamma ⇒ gamma ∈ PSNo beta q).
Apply PNoEq_sym_ to the current goal.
Apply PNoEq_PSNo to the current goal.
An exact proof term for the current goal is Hb.
∎
Definition. We define SNoCut to be λL R ⇒ PSNo (PNo_bd (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) (PNo_pred (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) of type set → set → set.
In Proofgold the corresponding term root is ec849e... and object id is 295bc4...
Definition. We define SNoCutP to be λL R ⇒ (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y) of type set → set → prop.
In Proofgold the corresponding term root is c083d8... and object id is 775127...
Theorem. (SNoCutP_SNoCut) The following is provable:
∀L R, SNoCutP L R → SNo (SNoCut L R) ∧ SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (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)
In Proofgold the corresponding term root is 9398b7... and proposition id is 66456c...
Proof:
Let L and R be given.
Assume HC: SNoCutP L R.
Apply HC to the current goal.
Assume HC.
Apply HC to the current goal.
Assume HL: ∀x ∈ L, SNo x.
Assume HR: ∀y ∈ R, SNo y.
Assume HLR: ∀x ∈ L, ∀y ∈ R, x < y.
Set L' to be the term λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L of type set → (set → prop) → prop.
Set R' to be the term λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R of type set → (set → prop) → prop.
Set tau to be the term PNo_bd L' R'.
Set w to be the term PNo_pred L' R'.
Set alpha to be the term ⋃x ∈ Lordsucc (SNoLev x).
Set beta to be the term ⋃y ∈ Rordsucc (SNoLev y).
We will prove SNo (SNoCut L R) ∧ SNoLev (SNoCut L R) ∈ ordsucc (alpha ∪ beta) ∧ (∀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 ∧ PNoEq_ (SNoLev (SNoCut L R)) (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z)).
We prove the intermediate claim LLR: PNoLt_pwise L' R'.
Let gamma be given.
Assume Hc: ordinal gamma.
Let p be given.
Assume H1: L' gamma p.
Apply H1 to the current goal.
Assume _ H1.
Let delta be given.
Assume Hd: ordinal delta.
Let q be given.
Assume H2: R' delta q.
Apply H2 to the current goal.
Assume _ H2.
We will prove PNoLt gamma p delta q.
Apply SNoLt_PSNo_PNoLt gamma delta p q Hc Hd to the current goal.
We will prove PSNo gamma p < PSNo delta q.
An exact proof term for the current goal is HLR (PSNo gamma p) H1 (PSNo delta q) H2.
We prove the intermediate claim La: ordinal alpha.
Apply ordinal_famunion L (λx ⇒ ordsucc (SNoLev x)) to the current goal.
Let x be given.
Assume Hx: x ∈ L.
We will prove ordinal (ordsucc (SNoLev x)).
Apply ordinal_ordsucc to the current goal.
Apply SNoLev_ordinal to the current goal.
We will prove SNo x.
An exact proof term for the current goal is HL x Hx.
We prove the intermediate claim Lb: ordinal beta.
Apply ordinal_famunion R (λy ⇒ ordsucc (SNoLev y)) to the current goal.
Let y be given.
Assume Hy: y ∈ R.
We will prove ordinal (ordsucc (SNoLev y)).
Apply ordinal_ordsucc to the current goal.
Apply SNoLev_ordinal to the current goal.
We will prove SNo y.
An exact proof term for the current goal is HR y Hy.
We prove the intermediate claim Lab: ordinal (alpha ∪ beta).
Apply ordinal_linear alpha beta La Lb to the current goal.
We will prove alpha ⊆ beta → ordinal (alpha ∪ beta).
rewrite the current goal using Subq_binunion_eq alpha beta (from left to right).
We will prove alpha ∪ beta = beta → ordinal (alpha ∪ beta).
Assume H1.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is Lb.
We will prove beta ⊆ alpha → ordinal (alpha ∪ beta).
rewrite the current goal using Subq_binunion_eq beta alpha (from left to right).
We will prove beta ∪ alpha = alpha → ordinal (alpha ∪ beta).
Assume H1.
rewrite the current goal using binunion_com (from left to right).
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is La.
We prove the intermediate claim LLab: PNo_lenbdd (alpha ∪ beta) L'.
Let gamma be given.
Let p be given.
Assume H1.
Apply H1 to the current goal.
Assume H1: ordinal gamma.
Assume H2: PSNo gamma p ∈ L.
We will prove gamma ∈ alpha ∪ beta.
Apply binunionI1 to the current goal.
We will prove gamma ∈ ⋃x ∈ Lordsucc (SNoLev x).
Apply famunionI L (λx ⇒ ordsucc (SNoLev x)) (PSNo gamma p) gamma to the current goal.
We will prove PSNo gamma p ∈ L.
An exact proof term for the current goal is H2.
We will prove gamma ∈ ordsucc (SNoLev (PSNo gamma p)).
rewrite the current goal using SNoLev_PSNo gamma H1 p (from left to right).
We will prove gamma ∈ ordsucc gamma.
Apply ordsuccI2 to the current goal.
We prove the intermediate claim LRab: PNo_lenbdd (alpha ∪ beta) R'.
Let gamma be given.
Let p be given.
Assume H1.
Apply H1 to the current goal.
Assume H1: ordinal gamma.
Assume H2: PSNo gamma p ∈ R.
We will prove gamma ∈ alpha ∪ beta.
Apply binunionI2 to the current goal.
We will prove gamma ∈ ⋃y ∈ Rordsucc (SNoLev y).
Apply famunionI R (λy ⇒ ordsucc (SNoLev y)) (PSNo gamma p) gamma to the current goal.
We will prove PSNo gamma p ∈ R.
An exact proof term for the current goal is H2.
We will prove gamma ∈ ordsucc (SNoLev (PSNo gamma p)).
rewrite the current goal using SNoLev_PSNo gamma H1 p (from left to right).
We will prove gamma ∈ ordsucc gamma.
Apply ordsuccI2 to the current goal.
Apply PNo_bd_pred L' R' LLR (alpha ∪ beta) Lab LLab LRab to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1: ordinal tau.
Assume H2: PNo_strict_imv L' R' tau w.
Assume H3: ∀gamma ∈ tau, ∀q : set → prop, ¬ PNo_strict_imv L' R' gamma q.
Apply H2 to the current goal.
Assume H4: PNo_strict_upperbd L' tau w.
Assume H5: PNo_strict_lowerbd R' tau w.
We prove the intermediate claim LNoC: SNo (SNoCut L R).
We will prove SNo (PSNo tau w).
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha (PSNo tau w).
We use tau to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
We will prove SNo_ tau (PSNo tau w).
Apply SNo_PSNo tau to the current goal.
We will prove ordinal tau.
An exact proof term for the current goal is H1.
We prove the intermediate claim LLleveqtau: SNoLev (SNoCut L R) = tau.
An exact proof term for the current goal is SNoLev_PSNo tau H1 (PNo_pred L' R').
We prove the intermediate claim LLbdtau: tau ∈ ordsucc (alpha ∪ beta).
An exact proof term for the current goal is PNo_bd_In L' R' LLR (alpha ∪ beta) Lab LLab LRab.
We prove the intermediate claim LLbd: SNoLev (SNoCut L R) ∈ ordsucc (alpha ∪ beta).
rewrite the current goal using LLleveqtau (from left to right).
An exact proof term for the current goal is LLbdtau.
We prove the intermediate claim LLecw: PNoEq_ tau (λgamma ⇒ gamma ∈ SNoCut L R) w.
We will prove PNoEq_ tau (λgamma ⇒ gamma ∈ PSNo tau w) w.
An exact proof term for the current goal is PNoEq_PSNo tau H1 w.
We prove the intermediate claim LLC: ordinal (SNoLev (SNoCut L R)).
Apply SNoLev_ordinal to the current goal.
An exact proof term for the current goal is LNoC.
We prove the intermediate claim LL: ∀x ∈ L, x < SNoCut L R.
Let x be given.
Assume Hx: x ∈ L.
We will prove x < SNoCut L R.
We will prove x < PSNo tau w.
We prove the intermediate claim L1: SNo x.
An exact proof term for the current goal is HL x Hx.
We prove the intermediate claim LLx: ordinal (SNoLev x).
An exact proof term for the current goal is SNoLev_ordinal x L1.
We prove the intermediate claim L2: x = PSNo (SNoLev x) (λgamma ⇒ gamma ∈ x).
Apply SNo_PSNo_eta to the current goal.
An exact proof term for the current goal is L1.
We prove the intermediate claim L3: L' (SNoLev x) (λgamma ⇒ gamma ∈ x).
We will prove ordinal (SNoLev x) ∧ PSNo (SNoLev x) (λgamma ⇒ gamma ∈ x) ∈ L.
Apply andI to the current goal.
An exact proof term for the current goal is LLx.
rewrite the current goal using L2 (from right to left).
An exact proof term for the current goal is Hx.
We will prove x < PSNo tau w.
rewrite the current goal using L2 (from left to right).
Apply PNoLt_SNoLt_PSNo (SNoLev x) tau (λgamma ⇒ gamma ∈ x) w LLx H1 to the current goal.
We will prove PNoLt (SNoLev x) (λgamma ⇒ gamma ∈ x) tau w.
An exact proof term for the current goal is H4 (SNoLev x) LLx (λgamma ⇒ gamma ∈ x) L3.
We prove the intermediate claim LR: ∀y ∈ R, SNoCut L R < y.
Let y be given.
Assume Hy: y ∈ R.
We will prove SNoCut L R < y.
We will prove PSNo tau w < y.
We prove the intermediate claim L1: SNo y.
An exact proof term for the current goal is HR y Hy.
We prove the intermediate claim LLy: ordinal (SNoLev y).
An exact proof term for the current goal is SNoLev_ordinal y L1.
We prove the intermediate claim L2: y = PSNo (SNoLev y) (λgamma ⇒ gamma ∈ y).
Apply SNo_PSNo_eta to the current goal.
An exact proof term for the current goal is L1.
We prove the intermediate claim L3: R' (SNoLev y) (λgamma ⇒ gamma ∈ y).
We will prove ordinal (SNoLev y) ∧ PSNo (SNoLev y) (λgamma ⇒ gamma ∈ y) ∈ R.
Apply andI to the current goal.
An exact proof term for the current goal is LLy.
rewrite the current goal using L2 (from right to left).
An exact proof term for the current goal is Hy.
We will prove PSNo tau w < y.
rewrite the current goal using L2 (from left to right).
Apply PNoLt_SNoLt_PSNo tau (SNoLev y) w (λgamma ⇒ gamma ∈ y) H1 LLy to the current goal.
We will prove PNoLt tau w (SNoLev y) (λgamma ⇒ gamma ∈ y).
An exact proof term for the current goal is H5 (SNoLev y) LLy (λgamma ⇒ gamma ∈ y) L3.
Apply and5I to the current goal.
An exact proof term for the current goal is LNoC.
An exact proof term for the current goal is LLbd.
An exact proof term for the current goal is LL.
An exact proof term for the current goal is LR.
We will prove ∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ PNoEq_ (SNoLev (SNoCut L R)) (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z).
Let z be given.
Assume Hz: SNo z.
Assume H10: ∀x ∈ L, x < z.
Assume H11: ∀y ∈ R, z < y.
We prove the intermediate claim LLz: ordinal (SNoLev z).
Apply SNoLev_ordinal to the current goal.
An exact proof term for the current goal is Hz.
We prove the intermediate claim Lzimv: PNo_strict_imv L' R' (SNoLev z) (λgamma ⇒ gamma ∈ z).
We will prove PNo_strict_upperbd L' (SNoLev z) (λgamma ⇒ gamma ∈ z) ∧ PNo_strict_lowerbd R' (SNoLev z) (λgamma ⇒ gamma ∈ z).
Apply andI to the current goal.
Let gamma be given.
Assume Hc: ordinal gamma.
Let q be given.
Assume Hq: L' gamma q.
We will prove PNoLt gamma q (SNoLev z) (λgamma ⇒ gamma ∈ z).
Apply Hq to the current goal.
Assume Hq1: ordinal gamma.
Assume Hq2: PSNo gamma q ∈ L.
Apply SNoLt_PSNo_PNoLt gamma (SNoLev z) q (λgamma ⇒ gamma ∈ z) Hc LLz to the current goal.
We will prove PSNo gamma q < PSNo (SNoLev z) (λgamma ⇒ gamma ∈ z).
rewrite the current goal using SNo_PSNo_eta z Hz (from right to left).
We will prove PSNo gamma q < z.
An exact proof term for the current goal is H10 (PSNo gamma q) Hq2.
Let gamma be given.
Assume Hc: ordinal gamma.
Let q be given.
Assume Hq: R' gamma q.
We will prove PNoLt (SNoLev z) (λgamma ⇒ gamma ∈ z) gamma q.
Apply Hq to the current goal.
Assume Hq1: ordinal gamma.
Assume Hq2: PSNo gamma q ∈ R.
Apply SNoLt_PSNo_PNoLt (SNoLev z) gamma (λgamma ⇒ gamma ∈ z) q LLz Hc to the current goal.
We will prove PSNo (SNoLev z) (λgamma ⇒ gamma ∈ z) < PSNo gamma q.
rewrite the current goal using SNo_PSNo_eta z Hz (from right to left).
We will prove z < PSNo gamma q.
An exact proof term for the current goal is H11 (PSNo gamma q) Hq2.
We prove the intermediate claim LLznt: SNoLev z ∉ tau.
Assume H12: SNoLev z ∈ tau.
An exact proof term for the current goal is H3 (SNoLev z) H12 (λgamma ⇒ gamma ∈ z) Lzimv.
We prove the intermediate claim LLzlet: tau ⊆ SNoLev z.
Apply ordinal_In_Or_Subq (SNoLev z) tau LLz H1 to the current goal.
Assume H12: SNoLev z ∈ tau.
We will prove False.
An exact proof term for the current goal is LLznt H12.
Assume H12.
An exact proof term for the current goal is H12.
We will prove SNoLev (SNoCut L R) ⊆ SNoLev z ∧ PNoEq_ (SNoLev (SNoCut L R)) (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z).
rewrite the current goal using LLleveqtau (from left to right).
We will prove tau ⊆ SNoLev z ∧ PNoEq_ tau (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z).
Apply andI to the current goal.
We will prove tau ⊆ SNoLev z.
An exact proof term for the current goal is LLzlet.
We will prove PNoEq_ tau (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z).
Apply PNoLt_trichotomy_or_ w (λgamma ⇒ gamma ∈ z) tau H1 to the current goal.
Assume H12.
Apply H12 to the current goal.
Assume H12: PNoLt_ tau w (λgamma ⇒ gamma ∈ z).
We will prove False.
Apply H12 to the current goal.
Let delta be given.
Assume H13.
Apply H13 to the current goal.
Assume Hd: delta ∈ tau.
Assume H13.
Apply H13 to the current goal.
Assume H13.
Apply H13 to the current goal.
Assume H13: PNoEq_ delta w (λgamma ⇒ gamma ∈ z).
Assume H14: ¬ w delta.
Assume H15: delta ∈ z.
We prove the intermediate claim Ld: ordinal delta.
An exact proof term for the current goal is ordinal_Hered tau H1 delta Hd.
We prove the intermediate claim Lsd: ordinal (ordsucc delta).
An exact proof term for the current goal is ordinal_ordsucc delta Ld.
Set z0 to be the term λeps ⇒ eps ∈ z ∧ eps ≠ delta of type set → prop.
Set z1 to be the term λeps ⇒ eps ∈ z ∨ eps = delta of type set → prop.
We prove the intermediate claim Lnz0d: ¬ z0 delta.
Assume H10.
Apply H10 to the current goal.
Assume H11: delta ∈ z.
Assume H12: delta ≠ delta.
Apply H12 to the current goal.
Use reflexivity.
We prove the intermediate claim Lz1d: z1 delta.
We will prove delta ∈ z ∨ delta = delta.
Apply orIR to the current goal.
Use reflexivity.
Apply H3 delta Hd (λgamma ⇒ gamma ∈ z) to the current goal.
We will prove PNo_strict_imv L' R' delta (λgamma ⇒ gamma ∈ z).
Apply PNo_rel_split_imv_imp_strict_imv L' R' delta Ld (λgamma ⇒ gamma ∈ z) to the current goal.
We will prove PNo_rel_strict_split_imv L' R' delta (λgamma ⇒ gamma ∈ z).
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) z0 ∧ PNo_rel_strict_imv L' R' (ordsucc delta) z1.
Apply andI to the current goal.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) z0.
Apply PNoEq_rel_strict_imv L' R' (ordsucc delta) Lsd w z0 to the current goal.
We will prove PNoEq_ (ordsucc delta) w z0.
Let eps be given.
Assume He: eps ∈ ordsucc delta.
Apply ordsuccE delta eps He to the current goal.
Assume He1: eps ∈ delta.
Apply iff_trans (w eps) (eps ∈ z) (z0 eps) to the current goal.
An exact proof term for the current goal is H13 eps He1.
An exact proof term for the current goal is PNo_extend0_eq delta (λgamma ⇒ gamma ∈ z) eps He1.
Assume He1: eps = delta.
We will prove w eps ↔ z0 eps.
rewrite the current goal using He1 (from left to right).
We will prove w delta ↔ z0 delta.
Apply iffI to the current goal.
Assume H16: w delta.
We will prove False.
An exact proof term for the current goal is H14 H16.
Assume H16: z0 delta.
We will prove False.
An exact proof term for the current goal is Lnz0d H16.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) w.
Apply PNo_strict_imv_imp_rel_strict_imv L' R' tau H1 (ordsucc delta) to the current goal.
We will prove ordsucc delta ∈ ordsucc tau.
Apply ordinal_ordsucc_In tau H1 to the current goal.
An exact proof term for the current goal is Hd.
We will prove PNo_strict_imv L' R' tau w.
An exact proof term for the current goal is H2.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) z1.
Apply PNoEq_rel_strict_imv L' R' (ordsucc delta) Lsd (λgamma ⇒ gamma ∈ z) z1 to the current goal.
We will prove PNoEq_ (ordsucc delta) (λgamma ⇒ gamma ∈ z) z1.
Let eps be given.
Assume He: eps ∈ ordsucc delta.
Apply ordsuccE delta eps He to the current goal.
Assume He1: eps ∈ delta.
An exact proof term for the current goal is PNo_extend1_eq delta (λgamma ⇒ gamma ∈ z) eps He1.
Assume He1: eps = delta.
We will prove eps ∈ z ↔ z1 eps.
rewrite the current goal using He1 (from left to right).
We will prove delta ∈ z ↔ z1 delta.
Apply iffI to the current goal.
Assume _.
An exact proof term for the current goal is Lz1d.
Assume _.
An exact proof term for the current goal is H15.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) (λgamma ⇒ gamma ∈ z).
Apply PNo_strict_imv_imp_rel_strict_imv L' R' (SNoLev z) LLz (ordsucc delta) to the current goal.
We will prove ordsucc delta ∈ ordsucc (SNoLev z).
Apply ordinal_ordsucc_In (SNoLev z) LLz to the current goal.
We will prove delta ∈ SNoLev z.
Apply LLzlet to the current goal.
We will prove delta ∈ tau.
An exact proof term for the current goal is Hd.
We will prove PNo_strict_imv L' R' (SNoLev z) (λgamma ⇒ gamma ∈ z).
An exact proof term for the current goal is Lzimv.
Assume H12: PNoEq_ tau w (λgamma ⇒ gamma ∈ z).
Apply PNoEq_tra_ tau (λgamma ⇒ gamma ∈ SNoCut L R) w (λgamma ⇒ gamma ∈ z) to the current goal.
We will prove PNoEq_ tau (λgamma ⇒ gamma ∈ SNoCut L R) w.
An exact proof term for the current goal is LLecw.
We will prove PNoEq_ tau w (λgamma ⇒ gamma ∈ z).
An exact proof term for the current goal is H12.
Assume H12: PNoLt_ tau (λgamma ⇒ gamma ∈ z) w.
We will prove False.
Apply H12 to the current goal.
Let delta be given.
Assume H13.
Apply H13 to the current goal.
Assume Hd: delta ∈ tau.
Assume H13.
Apply H13 to the current goal.
Assume H13.
Apply H13 to the current goal.
Assume H13: PNoEq_ delta (λgamma ⇒ gamma ∈ z) w.
Assume H14: delta ∉ z.
Assume H15: w delta.
We prove the intermediate claim Ld: ordinal delta.
An exact proof term for the current goal is ordinal_Hered tau H1 delta Hd.
We prove the intermediate claim Lsd: ordinal (ordsucc delta).
An exact proof term for the current goal is ordinal_ordsucc delta Ld.
Set z0 to be the term λeps ⇒ eps ∈ z ∧ eps ≠ delta of type set → prop.
Set z1 to be the term λeps ⇒ eps ∈ z ∨ eps = delta of type set → prop.
We prove the intermediate claim Lnz0d: ¬ z0 delta.
Assume H10.
Apply H10 to the current goal.
Assume H11: delta ∈ z.
Assume H12: delta ≠ delta.
Apply H12 to the current goal.
Use reflexivity.
We prove the intermediate claim Lz1d: z1 delta.
We will prove delta ∈ z ∨ delta = delta.
Apply orIR to the current goal.
Use reflexivity.
Apply H3 delta Hd (λgamma ⇒ gamma ∈ z) to the current goal.
We will prove PNo_strict_imv L' R' delta (λgamma ⇒ gamma ∈ z).
Apply PNo_rel_split_imv_imp_strict_imv L' R' delta Ld (λgamma ⇒ gamma ∈ z) to the current goal.
We will prove PNo_rel_strict_split_imv L' R' delta (λgamma ⇒ gamma ∈ z).
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) z0 ∧ PNo_rel_strict_imv L' R' (ordsucc delta) z1.
Apply andI to the current goal.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) z0.
Apply PNoEq_rel_strict_imv L' R' (ordsucc delta) Lsd (λgamma ⇒ gamma ∈ z) z0 to the current goal.
We will prove PNoEq_ (ordsucc delta) (λgamma ⇒ gamma ∈ z) z0.
Let eps be given.
Assume He: eps ∈ ordsucc delta.
Apply ordsuccE delta eps He to the current goal.
Assume He1: eps ∈ delta.
An exact proof term for the current goal is PNo_extend0_eq delta (λgamma ⇒ gamma ∈ z) eps He1.
Assume He1: eps = delta.
We will prove eps ∈ z ↔ z0 eps.
rewrite the current goal using He1 (from left to right).
We will prove delta ∈ z ↔ z0 delta.
Apply iffI to the current goal.
Assume H16: delta ∈ z.
We will prove False.
An exact proof term for the current goal is H14 H16.
Assume H16: z0 delta.
We will prove False.
An exact proof term for the current goal is Lnz0d H16.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) (λgamma ⇒ gamma ∈ z).
Apply PNo_strict_imv_imp_rel_strict_imv L' R' (SNoLev z) LLz (ordsucc delta) to the current goal.
We will prove ordsucc delta ∈ ordsucc (SNoLev z).
Apply ordinal_ordsucc_In (SNoLev z) LLz to the current goal.
We will prove delta ∈ SNoLev z.
Apply LLzlet to the current goal.
We will prove delta ∈ tau.
An exact proof term for the current goal is Hd.
We will prove PNo_strict_imv L' R' (SNoLev z) (λgamma ⇒ gamma ∈ z).
An exact proof term for the current goal is Lzimv.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) z1.
Apply PNoEq_rel_strict_imv L' R' (ordsucc delta) Lsd w z1 to the current goal.
We will prove PNoEq_ (ordsucc delta) w z1.
Let eps be given.
Assume He: eps ∈ ordsucc delta.
Apply ordsuccE delta eps He to the current goal.
Assume He1: eps ∈ delta.
Apply iff_trans (w eps) (eps ∈ z) (z1 eps) to the current goal.
Apply iff_sym to the current goal.
An exact proof term for the current goal is H13 eps He1.
An exact proof term for the current goal is PNo_extend1_eq delta (λgamma ⇒ gamma ∈ z) eps He1.
Assume He1: eps = delta.
We will prove w eps ↔ z1 eps.
rewrite the current goal using He1 (from left to right).
We will prove w delta ↔ z1 delta.
Apply iffI to the current goal.
Assume _.
An exact proof term for the current goal is Lz1d.
Assume _.
An exact proof term for the current goal is H15.
We will prove PNo_rel_strict_imv L' R' (ordsucc delta) w.
Apply PNo_strict_imv_imp_rel_strict_imv L' R' tau H1 (ordsucc delta) to the current goal.
We will prove ordsucc delta ∈ ordsucc tau.
Apply ordinal_ordsucc_In tau H1 to the current goal.
An exact proof term for the current goal is Hd.
We will prove PNo_strict_imv L' R' tau w.
An exact proof term for the current goal is H2.
∎
Theorem. (SNoCutP_SNoCut_impred) The following is provable:
∀L R, SNoCutP L R → ∀p : prop, (SNo (SNoCut L R) → SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (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
In Proofgold the corresponding term root is 02c4ae... and proposition id is 5bcc8f...
Proof:
Let L and R be given.
Assume HLR.
Let p be given.
Assume Hp.
Apply SNoCutP_SNoCut L R HLR to the current goal.
Assume H.
Apply H to the current goal.
Assume H.
Apply H to the current goal.
Assume H.
Apply H to the current goal.
An exact proof term for the current goal is Hp.
∎
Theorem. (SNoCutP_L_0) The following is provable:
∀L, (∀x ∈ L, SNo x) → SNoCutP L 0
In Proofgold the corresponding term root is 47329c... and proposition id is ba61e1...
Proof:
Let L be given.
Assume H1.
We will prove (∀x ∈ L, SNo x) ∧ (∀y ∈ 0, SNo y) ∧ (∀x ∈ L, ∀y ∈ 0, x < y).
Apply and3I to the current goal.
An exact proof term for the current goal is H1.
Let y be given.
Assume Hy.
We will prove False.
An exact proof term for the current goal is EmptyE y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
We will prove False.
An exact proof term for the current goal is EmptyE y Hy.
∎
Theorem. (SNoCutP_0_R) The following is provable:
∀R, (∀x ∈ R, SNo x) → SNoCutP 0 R
In Proofgold the corresponding term root is 54eb82... and proposition id is a06bcd...
Proof:
Let R be given.
Assume H1.
We will prove (∀x ∈ 0, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ 0, ∀y ∈ R, x < y).
Apply and3I to the current goal.
Let x be given.
Assume Hx.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
∎
Theorem. (SNoCutP_0_0) The following is provable:
SNoCutP 0 0
In Proofgold the corresponding term root is 3b144e... and proposition id is a5144d...
Proof:
Apply SNoCutP_L_0 to the current goal.
Let x be given.
Assume Hx.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
∎
Definition. We define SNoS_ to be λalpha ⇒ {x ∈ 𝒫 (SNoElts_ alpha)|∃beta ∈ alpha, SNo_ beta x} of type set → set.
In Proofgold the corresponding term root is d5069a... and object id is 4680ce...
Theorem. (SNoS_E) The following is provable:
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∃beta ∈ alpha, SNo_ beta x
In Proofgold the corresponding term root is 970ce2... and proposition id is bb731c...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Let x be given.
Assume H1: x ∈ SNoS_ alpha.
We will prove ∃beta ∈ alpha, SNo_ beta x.
An exact proof term for the current goal is SepE2 (𝒫 (SNoElts_ alpha)) (λx ⇒ ∃beta ∈ alpha, SNo_ beta x) x H1.
∎
Beginning of Section TaggedSets2
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Theorem. (SNoS_I) The following is provable:
∀alpha, ordinal alpha → ∀x, ∀beta ∈ alpha, SNo_ beta x → x ∈ SNoS_ alpha
In Proofgold the corresponding term root is f5227d... and proposition id is 04f24b...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Let x be given.
Let beta be given.
Assume Hb: beta ∈ alpha.
Assume H1: SNo_ beta x.
Apply H1 to the current goal.
Assume H2: x ⊆ SNoElts_ beta.
Assume H3: ∀gamma ∈ beta, exactly1of2 (gamma ' ∈ x) (gamma ∈ x).
We will prove x ∈ SNoS_ alpha.
We will prove x ∈ {x ∈ 𝒫 (SNoElts_ alpha)|∃gamma ∈ alpha, SNo_ gamma x}.
Apply SepI to the current goal.
We will prove x ∈ 𝒫 (SNoElts_ alpha).
Apply PowerI to the current goal.
We will prove x ⊆ SNoElts_ alpha.
Apply Subq_tra x (SNoElts_ beta) (SNoElts_ alpha) H2 to the current goal.
We will prove SNoElts_ beta ⊆ SNoElts_ alpha.
Apply SNoElts_mon to the current goal.
Apply Ha to the current goal.
Assume Ha1 _.
An exact proof term for the current goal is Ha1 beta Hb.
We will prove ∃gamma ∈ alpha, SNo_ gamma x.
We use beta to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb.
An exact proof term for the current goal is H1.
∎
Theorem. (SNoS_I2) The following is provable:
∀x y, SNo x → SNo y → SNoLev x ∈ SNoLev y → x ∈ SNoS_ (SNoLev y)
In Proofgold the corresponding term root is 7a14c7... and proposition id is b4042d...
Proof:
Let x and y be given.
Assume Hx Hy Hxy.
An exact proof term for the current goal is SNoS_I (SNoLev y) (SNoLev_ordinal y Hy) x (SNoLev x) Hxy (SNoLev_ x Hx).
∎
Theorem. (SNoS_Subq) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → SNoS_ alpha ⊆ SNoS_ beta
In Proofgold the corresponding term root is 6828e7... and proposition id is d928b8...
Proof:
Let alpha and beta be given.
Assume Ha Hb Hab.
Let x be given.
Assume Hx: x ∈ SNoS_ alpha.
Apply SNoS_E alpha Ha x Hx to the current goal.
Let gamma be given.
Assume Hc.
Apply Hc to the current goal.
Assume Hc1: gamma ∈ alpha.
Assume Hc2: SNo_ gamma x.
An exact proof term for the current goal is SNoS_I beta Hb x gamma (Hab gamma Hc1) Hc2.
∎
Theorem. (SNoLev_uniq2) The following is provable:
∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNoLev x = alpha
In Proofgold the corresponding term root is fbb461... and proposition id is f48d47...
Proof:
Let alpha be given.
Assume Ha.
Let x be given.
Assume Hx.
Apply SNoLev_prop x (SNo_SNo alpha Ha x Hx) to the current goal.
Assume Hx1: ordinal (SNoLev x).
Assume Hx2: SNo_ (SNoLev x) x.
Apply SNoLev_uniq x to the current goal.
An exact proof term for the current goal is Hx1.
An exact proof term for the current goal is Ha.
An exact proof term for the current goal is Hx2.
An exact proof term for the current goal is Hx.
∎
Theorem. (SNoS_E2) The following is provable:
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∀p : prop, (SNoLev x ∈ alpha → ordinal (SNoLev x) → SNo x → SNo_ (SNoLev x) x → p) → p
In Proofgold the corresponding term root is deb031... and proposition id is 62b490...
Proof:
Let alpha be given.
Assume Ha.
Let x be given.
Assume Hx.
Let p be given.
Assume Hp.
Apply SNoS_E alpha Ha x Hx to the current goal.
Let beta be given.
Assume H1.
Apply H1 to the current goal.
Assume Hb: beta ∈ alpha.
Assume H1: SNo_ beta x.
We prove the intermediate claim Lb: ordinal beta.
An exact proof term for the current goal is ordinal_Hered alpha Ha beta Hb.
We prove the intermediate claim Lx: SNo x.
An exact proof term for the current goal is SNo_SNo beta Lb x H1.
Apply SNoLev_prop x Lx to the current goal.
Assume Hx1: ordinal (SNoLev x).
Assume Hx2: SNo_ (SNoLev x) x.
We prove the intermediate claim Lxb: SNoLev x = beta.
An exact proof term for the current goal is SNoLev_uniq2 beta Lb x H1.
We prove the intermediate claim Lxa: SNoLev x ∈ alpha.
rewrite the current goal using Lxb (from left to right).
An exact proof term for the current goal is Hb.
An exact proof term for the current goal is Hp Lxa Hx1 Lx Hx2.
∎
Theorem. (SNoS_In_neq) The following is provable:
∀w, SNo w → ∀x ∈ SNoS_ (SNoLev w), x ≠ w
In Proofgold the corresponding term root is ab85fd... and proposition id is d5dbd7...
Proof:
Let w be given.
Assume Hw: SNo w.
Let x be given.
Assume Hx: x ∈ SNoS_ (SNoLev w).
Assume Hxw: x = w.
Apply SNoLev_prop w Hw to the current goal.
Assume Hw1: ordinal (SNoLev w).
Assume Hw2: SNo_ (SNoLev w) w.
Apply SNoS_E2 (SNoLev w) Hw1 x Hx to the current goal.
Assume Hx1: SNoLev x ∈ SNoLev w.
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
We will prove False.
Apply In_irref (SNoLev x) to the current goal.
rewrite the current goal using Hxw (from left to right) at position 2.
An exact proof term for the current goal is Hx1.
∎
Theorem. (SNoS_SNoLev) The following is provable:
∀z, SNo z → z ∈ SNoS_ (ordsucc (SNoLev z))
In Proofgold the corresponding term root is a9ba1f... and proposition id is 8efb88...
Proof:
Let z be given.
Assume Hz: SNo z.
Apply SNoLev_prop z Hz to the current goal.
Assume Hz1: ordinal (SNoLev z).
Assume Hz2: SNo_ (SNoLev z) z.
An exact proof term for the current goal is SNoS_I (ordsucc (SNoLev z)) (ordinal_ordsucc (SNoLev z) Hz1) z (SNoLev z) (ordsuccI2 (SNoLev z)) Hz2.
∎
Definition. We define SNoL to be λz ⇒ {x ∈ SNoS_ (SNoLev z)|x < z} of type set → set.
In Proofgold the corresponding term root is 8cd812... and object id is 48bcf8...
Definition. We define SNoR to be λz ⇒ {y ∈ SNoS_ (SNoLev z)|z < y} of type set → set.
In Proofgold the corresponding term root is 73b2b9... and object id is 0cfe87...
Theorem. (SNoCutP_SNoL_SNoR) The following is provable:
∀z, SNo z → SNoCutP (SNoL z) (SNoR z)
In Proofgold the corresponding term root is 524866... and proposition id is 7aba57...
Proof:
Let z be given.
Assume Hz: SNo z.
Set L to be the term SNoL z.
Set R to be the term SNoR z.
We prove the intermediate claim LLz: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
We prove the intermediate claim L1: ∀x ∈ L, SNo x.
Let x be given.
Assume Hx: x ∈ L.
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha x.
We prove the intermediate claim L1a: x ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is SepE1 (SNoS_ (SNoLev z)) (λx ⇒ x < z) x Hx.
Apply SNoS_E (SNoLev z) LLz x L1a to the current goal.
Let beta be given.
Assume H1.
Apply H1 to the current goal.
Assume Hb: beta ∈ SNoLev z.
Assume H1: SNo_ beta x.
We use beta to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is ordinal_Hered (SNoLev z) LLz beta Hb.
An exact proof term for the current goal is H1.
We prove the intermediate claim L2: ∀y ∈ R, SNo y.
Let y be given.
Assume Hy: y ∈ R.
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha y.
We prove the intermediate claim L2a: y ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is SepE1 (SNoS_ (SNoLev z)) (λy ⇒ z < y) y Hy.
Apply SNoS_E (SNoLev z) LLz y L2a to the current goal.
Let beta be given.
Assume H1.
Apply H1 to the current goal.
Assume Hb: beta ∈ SNoLev z.
Assume H1: SNo_ beta y.
We use beta to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is ordinal_Hered (SNoLev z) LLz beta Hb.
An exact proof term for the current goal is H1.
We prove the intermediate claim L3: ∀x ∈ L, ∀y ∈ R, x < y.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
Apply SNoLt_tra x z y (L1 x Hx) Hz (L2 y Hy) to the current goal.
We will prove x < z.
An exact proof term for the current goal is SepE2 (SNoS_ (SNoLev z)) (λx ⇒ x < z) x Hx.
We will prove z < y.
An exact proof term for the current goal is SepE2 (SNoS_ (SNoLev z)) (λy ⇒ z < y) y Hy.
We will prove SNoCutP L R.
We will prove (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y).
Apply and3I to the current goal.
An exact proof term for the current goal is L1.
An exact proof term for the current goal is L2.
An exact proof term for the current goal is L3.
∎
Theorem. (SNoL_E) The following is provable:
∀x, SNo x → ∀w ∈ SNoL x, ∀p : prop, (SNo w → SNoLev w ∈ SNoLev x → w < x → p) → p
In Proofgold the corresponding term root is cead4f... and proposition id is 7ccd8b...
Proof:
Let x be given.
Assume Hx: SNo x.
Let w be given.
Assume Hw: w ∈ SNoL x.
Let p be given.
Assume Hp.
Apply SepE (SNoS_ (SNoLev x)) (λw ⇒ w < x) w Hw to the current goal.
Assume Hw1: w ∈ SNoS_ (SNoLev x).
Assume Hw2: w < x.
Apply SNoS_E2 (SNoLev x) (SNoLev_ordinal x Hx) w Hw1 to the current goal.
Assume Hw3: SNoLev w ∈ SNoLev x.
Assume Hw4: ordinal (SNoLev w).
Assume Hw5: SNo w.
Assume Hw6: SNo_ (SNoLev w) w.
An exact proof term for the current goal is Hp Hw5 Hw3 Hw2.
∎
Theorem. (SNoR_E) The following is provable:
∀x, SNo x → ∀z ∈ SNoR x, ∀p : prop, (SNo z → SNoLev z ∈ SNoLev x → x < z → p) → p
In Proofgold the corresponding term root is efca05... and proposition id is 7dca04...
Proof:
Let x be given.
Assume Hx: SNo x.
Let z be given.
Assume Hz: z ∈ SNoR x.
Let p be given.
Assume Hp.
Apply SepE (SNoS_ (SNoLev x)) (λz ⇒ x < z) z Hz to the current goal.
Assume Hz1: z ∈ SNoS_ (SNoLev x).
Assume Hz2: x < z.
Apply SNoS_E2 (SNoLev x) (SNoLev_ordinal x Hx) z Hz1 to the current goal.
Assume Hz3: SNoLev z ∈ SNoLev x.
Assume Hz4: ordinal (SNoLev z).
Assume Hz5: SNo z.
Assume Hz6: SNo_ (SNoLev z) z.
An exact proof term for the current goal is Hp Hz5 Hz3 Hz2.
∎
Theorem. (SNoL_SNoS_) The following is provable:
∀z, SNoL z ⊆ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 73f5b2... and proposition id is 409713...
Proof:
Let z be given.
An exact proof term for the current goal is Sep_Subq (SNoS_ (SNoLev z)) (λx ⇒ x < z).
∎
Theorem. (SNoR_SNoS_) The following is provable:
∀z, SNoR z ⊆ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 16bdfc... and proposition id is 2fe1c0...
Proof:
Let z be given.
An exact proof term for the current goal is Sep_Subq (SNoS_ (SNoLev z)) (λy ⇒ z < y).
∎
Theorem. (SNoL_SNoS) The following is provable:
∀x, SNo x → ∀w ∈ SNoL x, w ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 6e3a22... and proposition id is ded78a...
Proof:
Let x be given.
Assume Hx.
Let w be given.
Assume Hw: w ∈ SNoL x.
Apply SNoL_E x Hx w Hw to the current goal.
Assume Hw1: SNo w.
Assume Hw2: SNoLev w ∈ SNoLev x.
Assume Hw3: w < x.
We will prove w ∈ SNoS_ (SNoLev x).
An exact proof term for the current goal is SNoS_I2 w x Hw1 Hx Hw2.
∎
Theorem. (SNoR_SNoS) The following is provable:
∀x, SNo x → ∀z ∈ SNoR x, z ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 98051c... and proposition id is e13f49...
Proof:
Let x be given.
Assume Hx.
Let z be given.
Assume Hz: z ∈ SNoR x.
Apply SNoR_E x Hx z Hz to the current goal.
Assume Hz1: SNo z.
Assume Hz2: SNoLev z ∈ SNoLev x.
Assume Hz3: x < z.
We will prove z ∈ SNoS_ (SNoLev x).
An exact proof term for the current goal is SNoS_I2 z x Hz1 Hx Hz2.
∎
Theorem. (SNoL_I) The following is provable:
∀z, SNo z → ∀x, SNo x → SNoLev x ∈ SNoLev z → x < z → x ∈ SNoL z
In Proofgold the corresponding term root is ddd91b... and proposition id is 4f3d04...
Proof:
Let z be given.
Assume Hz.
Let x be given.
Assume Hx Hxz1 Hxz2.
We will prove x ∈ SNoL z.
We will prove x ∈ {x ∈ SNoS_ (SNoLev z)|x < z}.
Apply SepI to the current goal.
We will prove x ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is SNoS_I2 x z Hx Hz Hxz1.
We will prove x < z.
An exact proof term for the current goal is Hxz2.
∎
Theorem. (SNoR_I) The following is provable:
∀z, SNo z → ∀y, SNo y → SNoLev y ∈ SNoLev z → z < y → y ∈ SNoR z
In Proofgold the corresponding term root is 97693c... and proposition id is 383e88...
Proof:
Let z be given.
Assume Hz.
Let y be given.
Assume Hy Hyz Hzy.
We will prove y ∈ SNoR z.
We will prove y ∈ {y ∈ SNoS_ (SNoLev z)|z < y}.
Apply SepI to the current goal.
We will prove y ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is SNoS_I2 y z Hy Hz Hyz.
We will prove z < y.
An exact proof term for the current goal is Hzy.
∎
Theorem. (SNo_eta) The following is provable:
∀z, SNo z → z = SNoCut (SNoL z) (SNoR z)
In Proofgold the corresponding term root is 3cc369... and proposition id is dc9273...
Proof:
Let z be given.
Assume Hz: SNo z.
Set L to be the term SNoL z.
Set R to be the term SNoR z.
We prove the intermediate claim LLz: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
We prove the intermediate claim LC: SNoCutP L R.
An exact proof term for the current goal is SNoCutP_SNoL_SNoR z Hz.
Apply SNoCutP_SNoCut L R LC to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1: SNo (SNoCut L R).
Assume H2: SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))).
Assume H3: ∀x ∈ L, x < SNoCut L R.
Assume H4: ∀y ∈ R, SNoCut L R < y.
Assume H5: ∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ PNoEq_ (SNoLev (SNoCut L R)) (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z).
We prove the intermediate claim L4: ordinal (SNoLev (SNoCut L R)).
Apply SNoLev_ordinal to the current goal.
An exact proof term for the current goal is H1.
We prove the intermediate claim L5: ∀x ∈ L, x < z.
Let x be given.
Assume Hx: x ∈ L.
An exact proof term for the current goal is SepE2 (SNoS_ (SNoLev z)) (λx ⇒ x < z) x Hx.
We prove the intermediate claim L6: ∀y ∈ R, z < y.
Let y be given.
Assume Hy: y ∈ R.
An exact proof term for the current goal is SepE2 (SNoS_ (SNoLev z)) (λy ⇒ z < y) y Hy.
Apply H5 z Hz L5 L6 to the current goal.
Assume H6: SNoLev (SNoCut L R) ⊆ SNoLev z.
Assume H7: PNoEq_ (SNoLev (SNoCut L R)) (λgamma ⇒ gamma ∈ SNoCut L R) (λgamma ⇒ gamma ∈ z).
We prove the intermediate claim L7: SNoLev (SNoCut L R) = SNoLev z.
Apply ordinal_trichotomy_or (SNoLev (SNoCut L R)) (SNoLev z) L4 LLz to the current goal.
Assume H8.
Apply H8 to the current goal.
Assume H8: SNoLev (SNoCut L R) ∈ SNoLev z.
We will prove False.
Apply SNoLt_trichotomy_or z (SNoCut L R) Hz H1 to the current goal.
Assume H9.
Apply H9 to the current goal.
Assume H9: z < SNoCut L R.
Apply SNoLt_irref (SNoCut L R) to the current goal.
Apply H4 to the current goal.
We will prove SNoCut L R ∈ R.
We will prove SNoCut L R ∈ {y ∈ SNoS_ (SNoLev z)|z < y}.
Apply SepI to the current goal.
Apply SNoS_I (SNoLev z) LLz (SNoCut L R) (SNoLev (SNoCut L R)) H8 to the current goal.
We will prove SNo_ (SNoLev (SNoCut L R)) (SNoCut L R).
An exact proof term for the current goal is SNoLev_ (SNoCut L R) H1.
We will prove z < SNoCut L R.
An exact proof term for the current goal is H9.
Assume H9: z = SNoCut L R.
Apply In_irref (SNoLev z) to the current goal.
rewrite the current goal using H9 (from left to right) at position 1.
An exact proof term for the current goal is H8.
Assume H9: SNoCut L R < z.
Apply SNoLt_irref (SNoCut L R) to the current goal.
Apply H3 to the current goal.
We will prove SNoCut L R ∈ L.
We will prove SNoCut L R ∈ {x ∈ SNoS_ (SNoLev z)|x < z}.
Apply SepI to the current goal.
Apply SNoS_I (SNoLev z) LLz (SNoCut L R) (SNoLev (SNoCut L R)) H8 to the current goal.
We will prove SNo_ (SNoLev (SNoCut L R)) (SNoCut L R).
An exact proof term for the current goal is SNoLev_ (SNoCut L R) H1.
We will prove SNoCut L R < z.
An exact proof term for the current goal is H9.
Assume H8: SNoLev (SNoCut L R) = SNoLev z.
An exact proof term for the current goal is H8.
Assume H8: SNoLev z ∈ SNoLev (SNoCut L R).
We will prove False.
Apply In_irref (SNoLev z) to the current goal.
Apply H6 to the current goal.
An exact proof term for the current goal is H8.
We will prove z = SNoCut L R.
Use symmetry.
We will prove SNoCut L R = z.
Apply SNo_eq to the current goal.
We will prove SNo (SNoCut L R).
An exact proof term for the current goal is H1.
We will prove SNo z.
An exact proof term for the current goal is Hz.
We will prove SNoLev (SNoCut L R) = SNoLev z.
An exact proof term for the current goal is L7.
We will prove ∀alpha ∈ SNoLev (SNoCut L R), alpha ∈ SNoCut L R ↔ alpha ∈ z.
An exact proof term for the current goal is H7.
∎
Theorem. (SNoCutP_SNo_SNoCut) The following is provable:
∀L R, SNoCutP L R → SNo (SNoCut L R)
In Proofgold the corresponding term root is 9621aa... and proposition id is ae46e8...
Proof:
Let L and R be given.
Assume H1: SNoCutP L R.
Apply SNoCutP_SNoCut L R H1 to the current goal.
Assume H2 _.
Apply H2 to the current goal.
Assume H2 _.
Apply H2 to the current goal.
Assume H2 _.
Apply H2 to the current goal.
Assume H2 _.
An exact proof term for the current goal is H2.
∎
Theorem. (SNoCutP_SNoCut_L) The following is provable:
∀L R, SNoCutP L R → ∀x ∈ L, x < SNoCut L R
In Proofgold the corresponding term root is 984a30... and proposition id is 4bca14...
Proof:
Let L and R be given.
Assume H1: SNoCutP L R.
Apply SNoCutP_SNoCut L R H1 to the current goal.
Assume H2 _.
Apply H2 to the current goal.
Assume H2 _.
Apply H2 to the current goal.
Assume _ H2.
An exact proof term for the current goal is H2.
∎
Theorem. (SNoCutP_SNoCut_R) The following is provable:
∀L R, SNoCutP L R → ∀y ∈ R, SNoCut L R < y
In Proofgold the corresponding term root is 9161db... and proposition id is 7c3d12...
Proof:
Let L and R be given.
Assume H1: SNoCutP L R.
Apply SNoCutP_SNoCut L R H1 to the current goal.
Assume H2 _.
Apply H2 to the current goal.
Assume _ H2.
An exact proof term for the current goal is H2.
∎
Theorem. (SNoCutP_SNoCut_fst) The following is provable:
∀L R, 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
In Proofgold the corresponding term root is b52eb7... and proposition id is 8362f8...
Proof:
Let L and R be given.
Assume H1: SNoCutP L R.
Apply SNoCutP_SNoCut L R H1 to the current goal.
Assume _ H2.
An exact proof term for the current goal is H2.
∎
Theorem. (SNoCut_Le) The following is provable:
∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 ≤ SNoCut L2 R2
In Proofgold the corresponding term root is bb8cdc... and proposition id is 423731...
Proof:
Let L1, R1, L2 and R2 be given.
Assume HLR1 HLR2.
Assume H1: ∀w ∈ L1, w < SNoCut L2 R2.
Assume H2: ∀z ∈ R2, SNoCut L1 R1 < z.
Apply HLR1 to the current goal.
Assume HLR1a.
Apply HLR1a to the current goal.
Assume HLR1a: ∀x ∈ L1, SNo x.
Assume HLR1b: ∀y ∈ R1, SNo y.
Assume HLR1c: ∀x ∈ L1, ∀y ∈ R1, x < y.
Apply HLR2 to the current goal.
Assume HLR2a.
Apply HLR2a to the current goal.
Assume HLR2a: ∀x ∈ L2, SNo x.
Assume HLR2b: ∀y ∈ R2, SNo y.
Assume HLR2c: ∀x ∈ L2, ∀y ∈ R2, x < y.
Set alpha to be the term ⋃x ∈ L1ordsucc (SNoLev x).
Set beta to be the term ⋃y ∈ R1ordsucc (SNoLev y).
Set gamma to be the term ⋃x ∈ L2ordsucc (SNoLev x).
Set delta to be the term ⋃y ∈ R2ordsucc (SNoLev y).
Apply SNoCutP_SNoCut L1 R1 HLR1 to the current goal.
Assume H3.
Apply H3 to the current goal.
Assume H3.
Apply H3 to the current goal.
Assume H3.
Apply H3 to the current goal.
Assume H3: SNo (SNoCut L1 R1).
Assume H4: SNoLev (SNoCut L1 R1) ∈ ordsucc (alpha ∪ beta).
Assume H5: ∀x ∈ L1, x < SNoCut L1 R1.
Assume H6: ∀y ∈ R1, SNoCut L1 R1 < y.
Assume H7: ∀z, SNo z → (∀x ∈ L1, x < z) → (∀y ∈ R1, z < y) → SNoLev (SNoCut L1 R1) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L1 R1)) (SNoCut L1 R1) z.
Apply SNoCutP_SNoCut L2 R2 HLR2 to the current goal.
Assume H8.
Apply H8 to the current goal.
Assume H8.
Apply H8 to the current goal.
Assume H8.
Apply H8 to the current goal.
Assume H8: SNo (SNoCut L2 R2).
Assume H9: SNoLev (SNoCut L2 R2) ∈ ordsucc (gamma ∪ delta).
Assume H10: ∀x ∈ L2, x < SNoCut L2 R2.
Assume H11: ∀y ∈ R2, SNoCut L2 R2 < y.
Assume H12: ∀z, SNo z → (∀x ∈ L2, x < z) → (∀y ∈ R2, z < y) → SNoLev (SNoCut L2 R2) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L2 R2)) (SNoCut L2 R2) z.
Apply SNoLtLe_or (SNoCut L2 R2) (SNoCut L1 R1) H8 H3 to the current goal.
Assume H13: SNoCut L2 R2 < SNoCut L1 R1.
We will prove False.
Apply SNoLtE (SNoCut L2 R2) (SNoCut L1 R1) H8 H3 H13 to the current goal.
Let z be given.
Assume Hz1: SNo z.
Assume Hz2: SNoLev z ∈ SNoLev (SNoCut L2 R2) ∩ SNoLev (SNoCut L1 R1).
Assume Hz3: SNoEq_ (SNoLev z) z (SNoCut L2 R2).
Assume Hz4: SNoEq_ (SNoLev z) z (SNoCut L1 R1).
Assume Hz5: (SNoCut L2 R2) < z.
Assume Hz6: z < (SNoCut L1 R1).
Assume Hz7: SNoLev z ∉ (SNoCut L2 R2).
Assume Hz8: SNoLev z ∈ (SNoCut L1 R1).
We prove the intermediate claim LzL1: ∀x ∈ L1, x < z.
Let x be given.
Assume Hx: x ∈ L1.
Apply SNoLt_tra x (SNoCut L2 R2) z (HLR1a x Hx) H8 Hz1 to the current goal.
We will prove x < SNoCut L2 R2.
An exact proof term for the current goal is H1 x Hx.
We will prove SNoCut L2 R2 < z.
An exact proof term for the current goal is Hz5.
We prove the intermediate claim LzR1: ∀y ∈ R1, z < y.
Let y be given.
Assume Hy: y ∈ R1.
Apply SNoLt_tra z (SNoCut L1 R1) y Hz1 H3 (HLR1b y Hy) to the current goal.
We will prove z < SNoCut L1 R1.
An exact proof term for the current goal is Hz6.
We will prove SNoCut L1 R1 < y.
An exact proof term for the current goal is H6 y Hy.
Apply H7 z Hz1 LzL1 LzR1 to the current goal.
Assume H14: SNoLev (SNoCut L1 R1) ⊆ SNoLev z.
Assume _.
Apply In_irref (SNoLev z) to the current goal.
Apply H14 to the current goal.
We will prove SNoLev z ∈ SNoLev (SNoCut L1 R1).
An exact proof term for the current goal is binintersectE2 (SNoLev (SNoCut L2 R2)) (SNoLev (SNoCut L1 R1)) (SNoLev z) Hz2.
Assume H14: SNoLev (SNoCut L2 R2) ∈ SNoLev (SNoCut L1 R1).
Assume H15: SNoEq_ (SNoLev (SNoCut L2 R2)) (SNoCut L2 R2) (SNoCut L1 R1).
Assume H16: SNoLev (SNoCut L2 R2) ∈ (SNoCut L1 R1).
Set z to be the term SNoCut L2 R2.
We prove the intermediate claim LzR1: ∀y ∈ R1, z < y.
Let y be given.
Assume Hy: y ∈ R1.
Apply SNoLt_tra z (SNoCut L1 R1) y H8 H3 (HLR1b y Hy) to the current goal.
We will prove z < SNoCut L1 R1.
An exact proof term for the current goal is H13.
We will prove SNoCut L1 R1 < y.
An exact proof term for the current goal is H6 y Hy.
Apply H7 z H8 H1 LzR1 to the current goal.
Assume H17: SNoLev (SNoCut L1 R1) ⊆ SNoLev z.
Assume _.
Apply In_irref (SNoLev z) to the current goal.
Apply H17 to the current goal.
We will prove SNoLev z ∈ SNoLev (SNoCut L1 R1).
An exact proof term for the current goal is H14.
Assume H14: SNoLev (SNoCut L1 R1) ∈ SNoLev (SNoCut L2 R2).
Assume H15: SNoEq_ (SNoLev (SNoCut L1 R1)) (SNoCut L2 R2) (SNoCut L1 R1).
Assume H16: SNoLev (SNoCut L1 R1) ∉ (SNoCut L2 R2).
Set z to be the term SNoCut L1 R1.
We prove the intermediate claim LzL2: ∀x ∈ L2, x < z.
Let x be given.
Assume Hx: x ∈ L2.
Apply SNoLt_tra x (SNoCut L2 R2) z (HLR2a x Hx) H8 H3 to the current goal.
We will prove x < SNoCut L2 R2.
An exact proof term for the current goal is H10 x Hx.
We will prove SNoCut L2 R2 < z.
An exact proof term for the current goal is H13.
Apply H12 z H3 LzL2 H2 to the current goal.
Assume H17: SNoLev (SNoCut L2 R2) ⊆ SNoLev z.
Assume _.
Apply In_irref (SNoLev z) to the current goal.
Apply H17 to the current goal.
We will prove SNoLev z ∈ SNoLev (SNoCut L2 R2).
An exact proof term for the current goal is H14.
Assume H13: SNoCut L1 R1 ≤ SNoCut L2 R2.
An exact proof term for the current goal is H13.
∎
Theorem. (SNoCut_ext) The following is provable:
∀L1 R1 L2 R2, 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
In Proofgold the corresponding term root is c1078e... and proposition id is f66618...
Proof:
Let L1, R1, L2 and R2 be given.
Assume HLR1 HLR2.
Assume H1: ∀w ∈ L1, w < SNoCut L2 R2.
Assume H2: ∀z ∈ R1, SNoCut L2 R2 < z.
Assume H3: ∀w ∈ L2, w < SNoCut L1 R1.
Assume H4: ∀z ∈ R2, SNoCut L1 R1 < z.
We prove the intermediate claim LNLR1: SNo (SNoCut L1 R1).
An exact proof term for the current goal is SNoCutP_SNo_SNoCut L1 R1 HLR1.
We prove the intermediate claim LNLR2: SNo (SNoCut L2 R2).
An exact proof term for the current goal is SNoCutP_SNo_SNoCut L2 R2 HLR2.
Apply SNoLe_antisym (SNoCut L1 R1) (SNoCut L2 R2) LNLR1 LNLR2 to the current goal.
We will prove SNoCut L1 R1 ≤ SNoCut L2 R2.
An exact proof term for the current goal is SNoCut_Le L1 R1 L2 R2 HLR1 HLR2 H1 H4.
We will prove SNoCut L2 R2 ≤ SNoCut L1 R1.
An exact proof term for the current goal is SNoCut_Le L2 R2 L1 R1 HLR2 HLR1 H3 H2.
∎
Theorem. (SNoLt_SNoL_or_SNoR_impred) The following is provable:
∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → p
In Proofgold the corresponding term root is f3b15a... and proposition id is ffc0bf...
Proof:
Let x and y be given.
Assume Hx Hy Hxy.
Let p be given.
Assume Hp1 Hp2 Hp3.
Apply SNoLtE x y Hx Hy Hxy to the current goal.
Let z be given.
Assume Hz1 Hz2 _ _ Hz3 Hz4 _ _.
Apply binintersectE (SNoLev x) (SNoLev y) (SNoLev z) Hz2 to the current goal.
Assume Hz2a Hz2b.
Apply Hp1 z to the current goal.
We will prove z ∈ SNoL y.
An exact proof term for the current goal is SNoL_I y Hy z Hz1 Hz2b Hz4.
We will prove z ∈ SNoR x.
An exact proof term for the current goal is SNoR_I x Hx z Hz1 Hz2a Hz3.
Assume H1 _ _.
Apply Hp2 to the current goal.
An exact proof term for the current goal is SNoL_I y Hy x Hx H1 Hxy.
Assume H1 _ _.
Apply Hp3 to the current goal.
An exact proof term for the current goal is SNoR_I x Hx y Hy H1 Hxy.
∎
Theorem. (SNoL_or_SNoR_impred) The following is provable:
∀x y, SNo x → SNo y → ∀p : prop, (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
In Proofgold the corresponding term root is f8a5e2... and proposition id is 68ba98...
Proof:
Let x and y be given.
Assume Hx Hy.
Let p be given.
Assume Hp1 Hp2 Hp3 Hp4 Hp5 Hp6 Hp7.
Apply SNoLt_trichotomy_or_impred x y Hx Hy to the current goal.
Assume H1: x < y.
Apply SNoLt_SNoL_or_SNoR_impred x y Hx Hy H1 to the current goal.
An exact proof term for the current goal is Hp2.
An exact proof term for the current goal is Hp3.
An exact proof term for the current goal is Hp4.
Assume H1: x = y.
An exact proof term for the current goal is Hp1 H1.
Assume H1: y < x.
Apply SNoLt_SNoL_or_SNoR_impred y x Hy Hx H1 to the current goal.
Let z be given.
Assume H2 H3.
An exact proof term for the current goal is Hp5 z H3 H2.
An exact proof term for the current goal is Hp7.
An exact proof term for the current goal is Hp6.
∎
Theorem. (SNoL_SNoCutP_ex) The following is provable:
∀L R, SNoCutP L R → ∀w ∈ SNoL (SNoCut L R), ∃w' ∈ L, w ≤ w'
In Proofgold the corresponding term root is 3b4695... and proposition id is 6d8af6...
Proof:
Let L and R be given.
Assume HLR.
Set y to be the term SNoCut L R.
Let w be given.
Assume Hw: w ∈ SNoL y.
Apply dneg to the current goal.
Assume HC: ¬ ∃w' ∈ L, w ≤ w'.
Apply HLR to the current goal.
Assume H.
Apply H to the current goal.
Assume HL HR HLR'.
Apply SNoCutP_SNoCut_impred L R HLR to the current goal.
Assume H1: SNo y.
Assume H2: SNoLev y ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))).
Assume H3: ∀w ∈ L, w < y.
Assume H4: ∀z ∈ R, y < z.
Assume H5: ∀u, SNo u → (∀w ∈ L, w < u) → (∀z ∈ R, u < z) → SNoLev y ⊆ SNoLev u ∧ SNoEq_ (SNoLev y) y u.
Apply SNoL_E y H1 w Hw to the current goal.
Assume Hw1 Hw2 Hw3.
We prove the intermediate claim L1: SNoLev y ⊆ SNoLev w ∧ SNoEq_ (SNoLev y) y w.
Apply H5 w Hw1 to the current goal.
Let w' be given.
Assume Hw': w' ∈ L.
We will prove w' < w.
Apply SNoLtLe_or w' w (HL w' Hw') Hw1 to the current goal.
Assume H6.
An exact proof term for the current goal is H6.
Assume H6: w ≤ w'.
Apply HC to the current goal.
We use w' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hw'.
An exact proof term for the current goal is H6.
Let z be given.
Assume Hz: z ∈ R.
We will prove w < z.
Apply SNoLt_tra w y z Hw1 H1 (HR z Hz) Hw3 to the current goal.
We will prove y < z.
An exact proof term for the current goal is H4 z Hz.
Apply In_irref (SNoLev w) to the current goal.
We will prove SNoLev w ∈ SNoLev w.
Apply andEL (SNoLev y ⊆ SNoLev w) (SNoEq_ (SNoLev y) y w) L1 to the current goal.
We will prove SNoLev w ∈ SNoLev y.
An exact proof term for the current goal is Hw2.
∎
Theorem. (SNoR_SNoCutP_ex) The following is provable:
∀L R, SNoCutP L R → ∀z ∈ SNoR (SNoCut L R), ∃z' ∈ R, z' ≤ z
In Proofgold the corresponding term root is ca7be2... and proposition id is bb0b53...
Proof:
Let L and R be given.
Assume HLR.
Set y to be the term SNoCut L R.
Let z be given.
Assume Hz: z ∈ SNoR y.
Apply dneg to the current goal.
Assume HC: ¬ ∃z' ∈ R, z' ≤ z.
Apply HLR to the current goal.
Assume H.
Apply H to the current goal.
Assume HL HR HLR'.
Apply SNoCutP_SNoCut_impred L R HLR to the current goal.
Assume H1: SNo y.
Assume H2: SNoLev y ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))).
Assume H3: ∀w ∈ L, w < y.
Assume H4: ∀z ∈ R, y < z.
Assume H5: ∀u, SNo u → (∀w ∈ L, w < u) → (∀z ∈ R, u < z) → SNoLev y ⊆ SNoLev u ∧ SNoEq_ (SNoLev y) y u.
Apply SNoR_E y H1 z Hz to the current goal.
Assume Hz1 Hz2 Hz3.
We prove the intermediate claim L1: SNoLev y ⊆ SNoLev z ∧ SNoEq_ (SNoLev y) y z.
Apply H5 z Hz1 to the current goal.
Let w be given.
Assume Hw: w ∈ L.
We will prove w < z.
Apply SNoLt_tra w y z (HL w Hw) H1 Hz1 to the current goal.
We will prove w < y.
An exact proof term for the current goal is H3 w Hw.
We will prove y < z.
An exact proof term for the current goal is Hz3.
Let z' be given.
Assume Hz': z' ∈ R.
We will prove z < z'.
Apply SNoLtLe_or z z' Hz1 (HR z' Hz') to the current goal.
Assume H6.
An exact proof term for the current goal is H6.
Assume H6: z' ≤ z.
Apply HC to the current goal.
We use z' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hz'.
An exact proof term for the current goal is H6.
Apply In_irref (SNoLev z) to the current goal.
We will prove SNoLev z ∈ SNoLev z.
Apply andEL (SNoLev y ⊆ SNoLev z) (SNoEq_ (SNoLev y) y z) L1 to the current goal.
We will prove SNoLev z ∈ SNoLev y.
An exact proof term for the current goal is Hz2.
∎
Theorem. (ordinal_SNo_) The following is provable:
∀alpha, ordinal alpha → SNo_ alpha alpha
In Proofgold the corresponding term root is 4731fd... and proposition id is 7754e5...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
We will prove alpha ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ alpha) (beta ∈ alpha).
Apply andI to the current goal.
We will prove alpha ⊆ SNoElts_ alpha.
Let beta be given.
Assume Hb: beta ∈ alpha.
We will prove beta ∈ alpha ∪ {beta '|beta ∈ alpha}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is Hb.
We will prove ∀beta ∈ alpha, exactly1of2 (beta ' ∈ alpha) (beta ∈ alpha).
Let beta be given.
Assume Hb: beta ∈ alpha.
Apply exactly1of2_I2 to the current goal.
We will prove beta ' ∉ alpha.
Assume H1: beta ' ∈ alpha.
We will prove False.
Apply tagged_not_ordinal beta to the current goal.
We will prove ordinal (beta ').
An exact proof term for the current goal is ordinal_Hered alpha Ha (beta ') H1.
We will prove beta ∈ alpha.
An exact proof term for the current goal is Hb.
∎
Theorem. (ordinal_SNo) The following is provable:
∀alpha, ordinal alpha → SNo alpha
In Proofgold the corresponding term root is ac39bc... and proposition id is 1eea33...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
We will prove ∃beta, ordinal beta ∧ SNo_ beta alpha.
We use alpha to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Ha.
An exact proof term for the current goal is ordinal_SNo_ alpha Ha.
∎
Theorem. (ordinal_SNoLev) The following is provable:
∀alpha, ordinal alpha → SNoLev alpha = alpha
In Proofgold the corresponding term root is 56f8b0... and proposition id is fea5fc...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Apply SNoLev_prop alpha (ordinal_SNo alpha Ha) to the current goal.
Assume H1: ordinal (SNoLev alpha).
Assume H2: SNo_ (SNoLev alpha) alpha.
An exact proof term for the current goal is SNoLev_uniq alpha (SNoLev alpha) alpha H1 Ha H2 (ordinal_SNo_ alpha Ha).
∎
Theorem. (ordinal_SNoLev_max) The following is provable:
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → z < alpha
In Proofgold the corresponding term root is 945794... and proposition id is 51ee47...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Let z be given.
Assume Hz: SNo z.
Assume Hz2: SNoLev z ∈ alpha.
We prove the intermediate claim La1: SNo alpha.
An exact proof term for the current goal is ordinal_SNo alpha Ha.
We prove the intermediate claim La2: SNoLev alpha = alpha.
An exact proof term for the current goal is ordinal_SNoLev alpha Ha.
We will prove z < alpha.
Apply SNoLt_trichotomy_or z alpha Hz (ordinal_SNo alpha Ha) to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1: z < alpha.
An exact proof term for the current goal is H1.
Assume H1: z = alpha.
We will prove False.
Apply In_irref alpha to the current goal.
rewrite the current goal using La2 (from right to left) at position 1.
We will prove SNoLev alpha ∈ alpha.
rewrite the current goal using H1 (from right to left) at position 1.
We will prove SNoLev z ∈ alpha.
An exact proof term for the current goal is Hz2.
Assume H1: alpha < z.
We will prove False.
Apply SNoLtE alpha z La1 Hz H1 to the current goal.
Let w be given.
rewrite the current goal using La2 (from left to right).
Assume Hw: SNo w.
Assume H2: SNoLev w ∈ alpha ∩ SNoLev z.
Assume H3: SNoEq_ (SNoLev w) w alpha.
Assume H4: SNoEq_ (SNoLev w) w z.
Assume H5: alpha < w.
Assume H6: w < z.
Assume H7: SNoLev w ∉ alpha.
We will prove False.
Apply H7 to the current goal.
An exact proof term for the current goal is binintersectE1 alpha (SNoLev z) (SNoLev w) H2.
rewrite the current goal using La2 (from left to right).
Assume H2: alpha ∈ SNoLev z.
We will prove False.
An exact proof term for the current goal is In_no2cycle alpha (SNoLev z) H2 Hz2.
rewrite the current goal using La2 (from left to right).
Assume H2: SNoLev z ∈ alpha.
Assume H3: SNoEq_ (SNoLev z) alpha z.
Assume H4: SNoLev z ∉ alpha.
An exact proof term for the current goal is H4 H2.
∎
Theorem. (ordinal_SNoL) The following is provable:
∀alpha, ordinal alpha → SNoL alpha = SNoS_ alpha
In Proofgold the corresponding term root is fef608... and proposition id is 95823e...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
We prove the intermediate claim La1: SNo alpha.
An exact proof term for the current goal is ordinal_SNo alpha Ha.
We prove the intermediate claim La2: SNoLev alpha = alpha.
An exact proof term for the current goal is ordinal_SNoLev alpha Ha.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ SNoL alpha.
Apply SNoL_E alpha La1 x Hx to the current goal.
Assume Hx1: SNo x.
Assume Hx2: SNoLev x ∈ SNoLev alpha.
Assume Hx3: x < alpha.
We will prove x ∈ SNoS_ alpha.
rewrite the current goal using La2 (from right to left).
We will prove x ∈ SNoS_ (SNoLev alpha).
Apply SNoS_I2 x alpha Hx1 La1 to the current goal.
We will prove SNoLev x ∈ SNoLev alpha.
An exact proof term for the current goal is Hx2.
Let x be given.
Assume Hx: x ∈ SNoS_ alpha.
Apply SNoS_E2 alpha Ha x Hx to the current goal.
Assume Hx1: SNoLev x ∈ alpha.
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
We will prove x ∈ SNoL alpha.
Apply SNoL_I alpha La1 x Hx3 to the current goal.
We will prove SNoLev x ∈ SNoLev alpha.
rewrite the current goal using La2 (from left to right).
An exact proof term for the current goal is Hx1.
We will prove x < alpha.
An exact proof term for the current goal is ordinal_SNoLev_max alpha Ha x Hx3 Hx1.
∎
Theorem. (ordinal_SNoR) The following is provable:
∀alpha, ordinal alpha → SNoR alpha = Empty
In Proofgold the corresponding term root is 07c69a... and proposition id is f853d9...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
We prove the intermediate claim La1: SNo alpha.
An exact proof term for the current goal is ordinal_SNo alpha Ha.
We prove the intermediate claim La2: SNoLev alpha = alpha.
An exact proof term for the current goal is ordinal_SNoLev alpha Ha.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx: x ∈ SNoR alpha.
Apply SNoR_E alpha La1 x Hx to the current goal.
Assume Hx1: SNo x.
rewrite the current goal using La2 (from left to right).
Assume Hx2: SNoLev x ∈ alpha.
Assume Hx3: alpha < x.
We will prove False.
Apply SNoLt_irref x to the current goal.
Apply SNoLt_tra x alpha x Hx1 La1 Hx1 to the current goal.
We will prove x < alpha.
An exact proof term for the current goal is ordinal_SNoLev_max alpha Ha x Hx1 Hx2.
We will prove alpha < x.
An exact proof term for the current goal is Hx3.
∎
Theorem. (nat_p_SNo) The following is provable:
∀n, nat_p n → SNo n
In Proofgold the corresponding term root is 0644f3... and proposition id is 5fd930...
Proof:
Let n be given.
Assume Hn.
Apply ordinal_SNo to the current goal.
We will prove ordinal n.
Apply nat_p_ordinal to the current goal.
An exact proof term for the current goal is Hn.
∎
Theorem. (omega_SNo) The following is provable:
∀n ∈ ω, SNo n
In Proofgold the corresponding term root is d32269... and proposition id is d0f9c6...
Proof:
Let n be given.
Assume Hn.
Apply nat_p_SNo to the current goal.
Apply omega_nat_p to the current goal.
An exact proof term for the current goal is Hn.
∎
Theorem. (omega_SNoS_omega) The following is provable:
ω ⊆ SNoS_ ω
In Proofgold the corresponding term root is 4ba265... and proposition id is 2ccf7a...
Proof:
Let n be given.
Assume Hn: n ∈ ω.
Apply SNoS_I ω omega_ordinal n n to the current goal.
An exact proof term for the current goal is Hn.
We will prove SNo_ n n.
rewrite the current goal using ordinal_SNoLev n (nat_p_ordinal n (omega_nat_p n Hn)) (from right to left) at position 1.
We will prove SNo_ (SNoLev n) n.
Apply SNoLev_ to the current goal.
Apply omega_SNo to the current goal.
An exact proof term for the current goal is Hn.
∎
Theorem. (ordinal_In_SNoLt) The following is provable:
∀alpha, ordinal alpha → ∀beta ∈ alpha, beta < alpha
In Proofgold the corresponding term root is e37d00... and proposition id is ced78e...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Let beta be given.
Assume Hb: beta ∈ alpha.
We prove the intermediate claim Lb: ordinal beta.
An exact proof term for the current goal is ordinal_Hered alpha Ha beta Hb.
We prove the intermediate claim Lb1: SNo beta.
An exact proof term for the current goal is ordinal_SNo beta Lb.
We prove the intermediate claim Lb2: SNoLev beta = beta.
An exact proof term for the current goal is ordinal_SNoLev beta Lb.
Apply ordinal_SNoLev_max alpha Ha beta Lb1 to the current goal.
We will prove SNoLev beta ∈ alpha.
rewrite the current goal using Lb2 (from left to right).
An exact proof term for the current goal is Hb.
∎
Theorem. (ordinal_SNoLev_max_2) The following is provable:
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → z ≤ alpha
In Proofgold the corresponding term root is 72c5eb... and proposition id is c9a731...
Proof:
Let alpha be given.
Assume Ha: ordinal alpha.
Apply Ha to the current goal.
Assume Ha1 _.
Let z be given.
Assume Hz: SNo z.
Assume Hz2: SNoLev z ∈ ordsucc alpha.
We prove the intermediate claim La1: SNo alpha.
An exact proof term for the current goal is ordinal_SNo alpha Ha.
We prove the intermediate claim La2: SNoLev alpha = alpha.
An exact proof term for the current goal is ordinal_SNoLev alpha Ha.
Apply ordsuccE alpha (SNoLev z) Hz2 to the current goal.
Assume Hz3: SNoLev z ∈ alpha.
We will prove z ≤ alpha.
Apply SNoLtLe to the current goal.
We will prove z < alpha.
An exact proof term for the current goal is ordinal_SNoLev_max alpha Ha z Hz Hz3.
Assume Hz3: SNoLev z = alpha.
Apply dneg to the current goal.
Assume H1: ¬ (z ≤ alpha).
We prove the intermediate claim L1: ∀beta, ordinal beta → beta ∈ alpha → beta ∈ z.
Apply ordinal_ind to the current goal.
Let beta be given.
Assume Hb: ordinal beta.
Assume IH: ∀gamma ∈ beta, gamma ∈ alpha → gamma ∈ z.
Assume Hb2: beta ∈ alpha.
Apply dneg to the current goal.
Assume H2: beta ∉ z.
Apply H1 to the current goal.
Apply SNoLtLe to the current goal.
We will prove z < alpha.
We prove the intermediate claim Lb1: SNo beta.
An exact proof term for the current goal is ordinal_SNo beta Hb.
We prove the intermediate claim Lb2: SNoLev beta = beta.
An exact proof term for the current goal is ordinal_SNoLev beta Hb.
Apply SNoLt_tra z beta alpha Hz Lb1 La1 to the current goal.
We will prove z < beta.
Apply SNoLtI3 to the current goal.
We will prove SNoLev beta ∈ SNoLev z.
rewrite the current goal using Lb2 (from left to right).
rewrite the current goal using Hz3 (from left to right).
We will prove beta ∈ alpha.
An exact proof term for the current goal is Hb2.
We will prove SNoEq_ (SNoLev beta) z beta.
rewrite the current goal using Lb2 (from left to right).
Let gamma be given.
Assume Hc: gamma ∈ beta.
We will prove gamma ∈ z ↔ gamma ∈ beta.
Apply iffI to the current goal.
Assume _.
An exact proof term for the current goal is Hc.
Assume _.
We will prove gamma ∈ z.
Apply IH gamma Hc to the current goal.
We will prove gamma ∈ alpha.
An exact proof term for the current goal is Ha1 beta Hb2 gamma Hc.
We will prove SNoLev beta ∉ z.
rewrite the current goal using Lb2 (from left to right).
We will prove beta ∉ z.
An exact proof term for the current goal is H2.
We will prove beta < alpha.
An exact proof term for the current goal is ordinal_In_SNoLt alpha Ha beta Hb2.
We prove the intermediate claim L2: alpha ⊆ z.
Let beta be given.
Assume Hb: beta ∈ alpha.
An exact proof term for the current goal is L1 beta (ordinal_Hered alpha Ha beta Hb) Hb.
We prove the intermediate claim L3: z = alpha.
Apply SNo_eq z alpha Hz La1 to the current goal.
We will prove SNoLev z = SNoLev alpha.
rewrite the current goal using La2 (from left to right).
An exact proof term for the current goal is Hz3.
We will prove SNoEq_ (SNoLev z) z alpha.
rewrite the current goal using Hz3 (from left to right).
Let beta be given.
Assume Hb: beta ∈ alpha.
We will prove beta ∈ z ↔ beta ∈ alpha.
Apply iffI to the current goal.
Assume _.
An exact proof term for the current goal is Hb.
Assume _.
An exact proof term for the current goal is L2 beta Hb.
Apply H1 to the current goal.
We will prove z ≤ alpha.
rewrite the current goal using L3 (from left to right).
We will prove alpha ≤ alpha.
Apply SNoLe_ref to the current goal.
∎
Theorem. (ordinal_Subq_SNoLe) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → alpha ≤ beta
In Proofgold the corresponding term root is 7558ba... and proposition id is 15c58a...
Proof:
Let alpha and beta be given.
Assume Ha Hb Hab.
We prove the intermediate claim L1: alpha ∈ ordsucc beta.
Apply ordinal_In_Or_Subq alpha beta Ha Hb to the current goal.
Assume H1: alpha ∈ beta.
Apply ordsuccI1 to the current goal.
An exact proof term for the current goal is H1.
Assume H1: beta ⊆ alpha.
We prove the intermediate claim L1a: alpha = beta.
Apply set_ext to the current goal.
An exact proof term for the current goal is Hab.
An exact proof term for the current goal is H1.
rewrite the current goal using L1a (from left to right).
Apply ordsuccI2 to the current goal.
We prove the intermediate claim La1: SNo alpha.
An exact proof term for the current goal is ordinal_SNo alpha Ha.
We prove the intermediate claim La2: SNoLev alpha = alpha.
An exact proof term for the current goal is ordinal_SNoLev alpha Ha.
Apply ordinal_SNoLev_max_2 beta Hb alpha La1 to the current goal.
We will prove SNoLev alpha ∈ ordsucc beta.
rewrite the current goal using La2 (from left to right).
An exact proof term for the current goal is L1.
∎
Theorem. (ordinal_SNoLt_In) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha < beta → alpha ∈ beta
In Proofgold the corresponding term root is d6f066... and proposition id is de775b...
Proof:
Let alpha and beta be given.
Assume Ha Hb.
Assume H1.
Apply ordinal_In_Or_Subq alpha beta Ha Hb to the current goal.
Assume H2.
An exact proof term for the current goal is H2.
Assume H2: beta ⊆ alpha.
We will prove False.
Apply SNoLt_irref alpha to the current goal.
We will prove alpha < alpha.
Apply SNoLtLe_tra alpha beta alpha (ordinal_SNo alpha Ha) (ordinal_SNo beta Hb) (ordinal_SNo alpha Ha) H1 to the current goal.
We will prove beta ≤ alpha.
An exact proof term for the current goal is ordinal_Subq_SNoLe beta alpha Hb Ha H2.
∎
Theorem. (omega_nonneg) The following is provable:
∀m ∈ ω, 0 ≤ m
In Proofgold the corresponding term root is ee67c9... and proposition id is eaf06b...
Proof:
Let m be given.
Assume Hm.
Apply ordinal_Subq_SNoLe 0 m ordinal_Empty (nat_p_ordinal m (omega_nat_p m Hm)) to the current goal.
We will prove 0 ⊆ m.
Apply Subq_Empty to the current goal.
∎
Theorem. (SNo_0) The following is provable:
SNo 0
In Proofgold the corresponding term root is c8204b... and proposition id is 94a310...
Proof:
An exact proof term for the current goal is ordinal_SNo 0 ordinal_Empty.
∎
Theorem. (SNo_1) The following is provable:
SNo 1
In Proofgold the corresponding term root is ac50d7... and proposition id is 9a733a...
Proof:
Apply ordinal_SNo to the current goal.
Apply nat_p_ordinal to the current goal.
An exact proof term for the current goal is nat_1.
∎
Theorem. (SNo_2) The following is provable:
SNo 2
In Proofgold the corresponding term root is 487591... and proposition id is c524a4...
Proof:
Apply ordinal_SNo to the current goal.
Apply nat_p_ordinal to the current goal.
An exact proof term for the current goal is nat_2.
∎
Theorem. (SNoLev_0) The following is provable:
SNoLev 0 = 0
In Proofgold the corresponding term root is 45a0f7... and proposition id is 3527a4...
Proof:
An exact proof term for the current goal is ordinal_SNoLev 0 ordinal_Empty.
∎
Theorem. (SNoCut_0_0) The following is provable:
SNoCut 0 0 = 0
In Proofgold the corresponding term root is 6a1f81... and proposition id is abbffb...
Proof:
Apply SNoCutP_SNoCut_impred 0 0 SNoCutP_0_0 to the current goal.
Assume H1: SNo (SNoCut 0 0).
rewrite the current goal using famunion_Empty (λx ⇒ ordsucc (SNoLev x)) (from left to right).
rewrite the current goal using binunion_idl 0 (from left to right).
Assume H2: SNoLev (SNoCut 0 0) ∈ 1.
Assume _ _ _.
We prove the intermediate claim L1: SNoLev (SNoCut 0 0) = 0.
Apply cases_1 (SNoLev (SNoCut 0 0)) H2 (λu ⇒ u = 0) to the current goal.
We will prove 0 = 0.
Use reflexivity.
Apply SNo_eq (SNoCut 0 0) 0 H1 SNo_0 to the current goal.
We will prove SNoLev (SNoCut 0 0) = SNoLev 0.
Use transitivity with and 0.
An exact proof term for the current goal is L1.
Use symmetry.
An exact proof term for the current goal is SNoLev_0.
We will prove SNoEq_ (SNoLev (SNoCut 0 0)) (SNoCut 0 0) 0.
rewrite the current goal using L1 (from left to right).
We will prove SNoEq_ 0 (SNoCut 0 0) 0.
Apply SNoEq_I to the current goal.
Let beta be given.
Assume Hb: beta ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE beta Hb.
∎
Theorem. (SNoL_0) The following is provable:
SNoL 0 = 0
In Proofgold the corresponding term root is c01c2b... and proposition id is fdaa69...
Proof:
Apply Empty_Subq_eq to the current goal.
We will prove SNoL 0 ⊆ Empty.
Let z be given.
Assume Hz: z ∈ SNoL 0.
We prove the intermediate claim Lz: z ∈ SNoS_ 0.
rewrite the current goal using SNoLev_0 (from right to left).
We will prove z ∈ SNoS_ (SNoLev 0).
An exact proof term for the current goal is SNoL_SNoS_ 0 z Hz.
Apply SNoS_E2 0 ordinal_Empty z Lz to the current goal.
Assume Hz1: SNoLev z ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE (SNoLev z) Hz1.
∎
Theorem. (SNoR_0) The following is provable:
SNoR 0 = 0
In Proofgold the corresponding term root is 7329c7... and proposition id is 510751...
Proof:
Apply Empty_Subq_eq to the current goal.
We will prove SNoR 0 ⊆ Empty.
Let z be given.
Assume Hz: z ∈ SNoR 0.
We prove the intermediate claim Lz: z ∈ SNoS_ 0.
rewrite the current goal using SNoLev_0 (from right to left).
We will prove z ∈ SNoS_ (SNoLev 0).
An exact proof term for the current goal is SNoR_SNoS_ 0 z Hz.
Apply SNoS_E2 0 ordinal_Empty z Lz to the current goal.
Assume Hz1: SNoLev z ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE (SNoLev z) Hz1.
∎
Theorem. (SNoL_1) The following is provable:
SNoL 1 = 1
In Proofgold the corresponding term root is 84d688... and proposition id is 15aa79...
Proof:
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ SNoL 1.
We will prove x ∈ 1.
Apply SNoL_E 1 SNo_1 x Hx to the current goal.
Assume Hxa: SNo x.
rewrite the current goal using ordinal_SNoLev 1 (nat_p_ordinal 1 nat_1) (from left to right).
Assume Hxb: SNoLev x ∈ 1.
Assume _.
We prove the intermediate claim L1: 0 = x.
Apply SNo_eq 0 x SNo_0 Hxa to the current goal.
We will prove SNoLev 0 = SNoLev x.
rewrite the current goal using SNoLev_0 (from left to right).
We will prove 0 = SNoLev x.
Apply cases_1 (SNoLev x) Hxb (λu ⇒ 0 = u) to the current goal.
We will prove 0 = 0.
Use reflexivity.
We will prove SNoEq_ (SNoLev 0) 0 x.
rewrite the current goal using SNoLev_0 (from left to right).
We will prove SNoEq_ 0 0 x.
Apply SNoEq_I to the current goal.
Let beta be given.
Assume Hb: beta ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE beta Hb.
rewrite the current goal using L1 (from right to left).
An exact proof term for the current goal is In_0_1.
Let x be given.
Assume Hx: x ∈ 1.
We will prove x ∈ SNoL 1.
Apply cases_1 x Hx (λx ⇒ x ∈ SNoL 1) to the current goal.
We will prove 0 ∈ SNoL 1.
Apply SNoL_I 1 SNo_1 0 SNo_0 to the current goal.
We will prove SNoLev 0 ∈ SNoLev 1.
rewrite the current goal using SNoLev_0 (from left to right).
rewrite the current goal using ordinal_SNoLev 1 (nat_p_ordinal 1 nat_1) (from left to right).
An exact proof term for the current goal is In_0_1.
We will prove 0 < 1.
An exact proof term for the current goal is ordinal_In_SNoLt 1 (nat_p_ordinal 1 nat_1) 0 In_0_1.
∎
Theorem. (SNoR_1) The following is provable:
SNoR 1 = 0
In Proofgold the corresponding term root is e6a83a... and proposition id is 8da427...
Proof:
An exact proof term for the current goal is ordinal_SNoR 1 (nat_p_ordinal 1 nat_1).
∎
Theorem. (SNo_max_SNoLev) The following is provable:
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → SNoLev x = x
In Proofgold the corresponding term root is ff019e... and proposition id is 3be246...
Proof:
Let x be given.
Assume Hx: SNo x.
Assume H2: ∀y ∈ SNoS_ (SNoLev x), y < x.
We prove the intermediate claim LLx1: ordinal (SNoLev x).
An exact proof term for the current goal is SNoLev_ordinal x Hx.
We prove the intermediate claim LLx2: SNo (SNoLev x).
An exact proof term for the current goal is ordinal_SNo (SNoLev x) LLx1.
We prove the intermediate claim L3: x ≤ SNoLev x.
Apply ordinal_SNoLev_max_2 (SNoLev x) LLx1 x Hx to the current goal.
We will prove SNoLev x ∈ ordsucc (SNoLev x).
Apply ordsuccI2 to the current goal.
Apply SNoLeE x (SNoLev x) Hx LLx2 L3 to the current goal.
Assume H3: x < SNoLev x.
We will prove False.
Apply SNoLtE x (SNoLev x) Hx LLx2 H3 to the current goal.
Let z be given.
Assume Hz: SNo z.
Assume Hz1: SNoLev z ∈ SNoLev x ∩ SNoLev (SNoLev x).
Assume Hz2: SNoEq_ (SNoLev z) z x.
Assume Hz3: SNoEq_ (SNoLev z) z (SNoLev x).
Assume Hz4: x < z.
Assume Hz5: z < SNoLev x.
Assume Hz6: SNoLev z ∉ x.
Assume Hz7: SNoLev z ∈ SNoLev x.
Apply SNoLt_irref z to the current goal.
Apply SNoLt_tra z x z Hz Hx Hz to the current goal.
We will prove z < x.
Apply H2 to the current goal.
We will prove z ∈ SNoS_ (SNoLev x).
Apply SNoS_I (SNoLev x) LLx1 z (SNoLev z) to the current goal.
We will prove SNoLev z ∈ SNoLev x.
An exact proof term for the current goal is Hz7.
We will prove SNo_ (SNoLev z) z.
Apply SNoLev_ to the current goal.
An exact proof term for the current goal is Hz.
We will prove x < z.
An exact proof term for the current goal is Hz4.
Assume H4: SNoLev x ∈ SNoLev (SNoLev x).
We will prove False.
Apply In_irref (SNoLev x) to the current goal.
rewrite the current goal using ordinal_SNoLev (SNoLev x) LLx1 (from right to left) at position 2.
An exact proof term for the current goal is H4.
Assume H4: SNoLev (SNoLev x) ∈ SNoLev x.
We will prove False.
Apply In_irref (SNoLev x) to the current goal.
rewrite the current goal using ordinal_SNoLev (SNoLev x) LLx1 (from right to left) at position 1.
An exact proof term for the current goal is H4.
Assume H3: x = SNoLev x.
Use symmetry.
An exact proof term for the current goal is H3.
∎
Theorem. (SNo_max_ordinal) The following is provable:
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → ordinal x
In Proofgold the corresponding term root is 0b230c... and proposition id is f28b9a...
Proof:
Let x be given.
Assume Hx: SNo x.
Assume H2: ∀y ∈ SNoS_ (SNoLev x), y < x.
We will prove ordinal x.
rewrite the current goal using SNo_max_SNoLev x Hx H2 (from right to left).
We will prove ordinal (SNoLev x).
An exact proof term for the current goal is SNoLev_ordinal x Hx.
∎
Theorem. (pos_low_eq_one) The following is provable:
∀x, SNo x → 0 < x → SNoLev x ⊆ 1 → x = 1
In Proofgold the corresponding term root is 7c84ef... and proposition id is db40ad...
Proof:
Let x be given.
Assume Hx Hxpos Hxlow.
Apply SNoLtE 0 x SNo_0 Hx Hxpos to the current goal.
Let y be given.
Assume Hy1: SNo y.
Assume Hy2: SNoLev y ∈ SNoLev 0 ∩ SNoLev x.
We will prove False.
Apply EmptyE (SNoLev y) to the current goal.
We will prove SNoLev y ∈ 0.
rewrite the current goal using ordinal_SNoLev 0 ordinal_Empty (from right to left).
We will prove SNoLev y ∈ SNoLev 0.
An exact proof term for the current goal is binintersectE1 (SNoLev 0) (SNoLev x) (SNoLev y) Hy2.
rewrite the current goal using ordinal_SNoLev 0 ordinal_Empty (from left to right).
Assume H1: 0 ∈ SNoLev x.
Assume _.
Assume H2: 0 ∈ x.
We prove the intermediate claim L1: SNoLev x = 1.
Apply set_ext to the current goal.
An exact proof term for the current goal is Hxlow.
Let n be given.
Assume Hn: n ∈ 1.
Apply cases_1 n Hn to the current goal.
We will prove 0 ∈ SNoLev x.
An exact proof term for the current goal is H1.
Apply SNo_eq x 1 Hx SNo_1 to the current goal.
We will prove SNoLev x = SNoLev 1.
rewrite the current goal using ordinal_SNoLev 1 (nat_p_ordinal 1 nat_1) (from left to right).
We will prove SNoLev x = 1.
An exact proof term for the current goal is L1.
We will prove SNoEq_ (SNoLev x) x 1.
rewrite the current goal using L1 (from left to right).
We will prove SNoEq_ 1 x 1.
Let n be given.
Assume Hn: n ∈ 1.
We will prove n ∈ x ↔ n ∈ 1.
Apply cases_1 n Hn (λn ⇒ n ∈ x ↔ n ∈ 1) to the current goal.
We will prove 0 ∈ x ↔ 0 ∈ 1.
Apply iffI to the current goal.
Assume _.
An exact proof term for the current goal is In_0_1.
Assume _.
An exact proof term for the current goal is H2.
Assume H1: SNoLev x ∈ SNoLev 0.
We will prove False.
Apply EmptyE (SNoLev x) to the current goal.
We will prove SNoLev x ∈ 0.
rewrite the current goal using ordinal_SNoLev 0 ordinal_Empty (from right to left).
We will prove SNoLev x ∈ SNoLev 0.
An exact proof term for the current goal is H1.
∎
Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∧ delta ≠ SNoLev x) of type set → set.
In Proofgold the corresponding term root is 997d91... and object id is ff4161...
Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∨ delta = SNoLev x) of type set → set.
In Proofgold the corresponding term root is 464e47... and object id is 8c551b...
Theorem. (SNo_extend0_SNo_) The following is provable:
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend0 x)
In Proofgold the corresponding term root is d41d3e... and proposition id is c8f8ea...
Proof:
Let x be given.
Assume Hx: SNo x.
Set alpha to be the term SNoLev x.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is SNoLev_ordinal x Hx.
An exact proof term for the current goal is SNo_PSNo (ordsucc alpha) (ordinal_ordsucc alpha La) (λdelta ⇒ delta ∈ x ∧ delta ≠ alpha).
∎
Theorem. (SNo_extend1_SNo_) The following is provable:
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend1 x)
In Proofgold the corresponding term root is b7b866... and proposition id is 73bbf2...
Proof:
Let x be given.
Assume Hx: SNo x.
Set alpha to be the term SNoLev x.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is SNoLev_ordinal x Hx.
An exact proof term for the current goal is SNo_PSNo (ordsucc alpha) (ordinal_ordsucc alpha La) (λdelta ⇒ delta ∈ x ∨ delta = alpha).
∎
Theorem. (SNo_extend0_SNo) The following is provable:
∀x, SNo x → SNo (SNo_extend0 x)
In Proofgold the corresponding term root is b353a2... and proposition id is 76562c...
Proof:
Let x be given.
Assume Hx.
An exact proof term for the current goal is SNo_SNo (ordsucc (SNoLev x)) (ordinal_ordsucc (SNoLev x) (SNoLev_ordinal x Hx)) (SNo_extend0 x) (SNo_extend0_SNo_ x Hx).
∎
Theorem. (SNo_extend1_SNo) The following is provable:
∀x, SNo x → SNo (SNo_extend1 x)
In Proofgold the corresponding term root is fa323a... and proposition id is 5345a7...
Proof:
Let x be given.
Assume Hx.
An exact proof term for the current goal is SNo_SNo (ordsucc (SNoLev x)) (ordinal_ordsucc (SNoLev x) (SNoLev_ordinal x Hx)) (SNo_extend1 x) (SNo_extend1_SNo_ x Hx).
∎
Theorem. (SNo_extend0_SNoLev) The following is provable:
∀x, SNo x → SNoLev (SNo_extend0 x) = ordsucc (SNoLev x)
In Proofgold the corresponding term root is 26efcd... and proposition id is 5aa0af...
Proof:
Let x be given.
Assume Hx.
An exact proof term for the current goal is SNoLev_uniq2 (ordsucc (SNoLev x)) (ordinal_ordsucc (SNoLev x) (SNoLev_ordinal x Hx)) (SNo_extend0 x) (SNo_extend0_SNo_ x Hx).
∎
Theorem. (SNo_extend1_SNoLev) The following is provable:
∀x, SNo x → SNoLev (SNo_extend1 x) = ordsucc (SNoLev x)
In Proofgold the corresponding term root is 33cf60... and proposition id is 529347...
Proof:
Let x be given.
Assume Hx.
An exact proof term for the current goal is SNoLev_uniq2 (ordsucc (SNoLev x)) (ordinal_ordsucc (SNoLev x) (SNoLev_ordinal x Hx)) (SNo_extend1 x) (SNo_extend1_SNo_ x Hx).
∎
Theorem. (SNo_extend0_nIn) The following is provable:
∀x, SNo x → SNoLev x ∉ SNo_extend0 x
In Proofgold the corresponding term root is 217cff... and proposition id is 0dfd17...
Proof:
Let x be given.
Assume Hx: SNo x.
Set alpha to be the term SNoLev x.
Assume H2: alpha ∈ PSNo (ordsucc alpha) (λdelta ⇒ delta ∈ x ∧ delta ≠ alpha).
Set p to be the term λdelta ⇒ delta ∈ x ∧ delta ≠ alpha of type set → prop.
Apply binunionE {beta ∈ ordsucc alpha|p beta} {beta '|beta ∈ ordsucc alpha, ¬ p beta} alpha H2 to the current goal.
Assume H3: alpha ∈ {beta ∈ ordsucc alpha|p beta}.
Apply SepE2 (ordsucc alpha) p alpha H3 to the current goal.
Assume _.
Assume H4: alpha ≠ alpha.
Apply H4 to the current goal.
Use reflexivity.
Assume H3: alpha ∈ {beta '|beta ∈ ordsucc alpha, ¬ p beta}.
Apply ReplSepE_impred (ordsucc alpha) (λbeta ⇒ ¬ p beta) (λx ⇒ x ') alpha H3 to the current goal.
Let beta be given.
Assume Hb: beta ∈ ordsucc alpha.
Assume H4: ¬ p beta.
Assume H5: alpha = beta '.
Apply tagged_not_ordinal beta to the current goal.
We will prove ordinal (beta ').
rewrite the current goal using H5 (from right to left).
We will prove ordinal alpha.
An exact proof term for the current goal is SNoLev_ordinal x Hx.
∎
Theorem. (SNo_extend1_In) The following is provable:
∀x, SNo x → SNoLev x ∈ SNo_extend1 x
In Proofgold the corresponding term root is ac9570... and proposition id is 8ca372...
Proof:
Let x be given.
Assume Hx: SNo x.
Set alpha to be the term SNoLev x.
Set p to be the term λdelta ⇒ delta ∈ x ∨ delta = alpha of type set → prop.
We will prove alpha ∈ {beta ∈ ordsucc alpha|p beta} ∪ {beta '|beta ∈ ordsucc alpha, ¬ p beta}.
Apply binunionI1 to the current goal.
Apply SepI to the current goal.
We will prove alpha ∈ ordsucc alpha.
Apply ordsuccI2 to the current goal.
We will prove alpha ∈ x ∨ alpha = alpha.
Apply orIR to the current goal.
Use reflexivity.
∎
Theorem. (SNo_extend0_SNoEq) The following is provable:
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend0 x) x
In Proofgold the corresponding term root is bd4a27... and proposition id is 129986...
Proof:
Let x be given.
Assume Hx: SNo x.
Set alpha to be the term SNoLev x.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is SNoLev_ordinal x Hx.
We will prove SNoEq_ alpha (SNo_extend0 x) x.
Set p to be the term λbeta ⇒ beta ∈ x of type set → prop.
Set q to be the term λbeta ⇒ beta ∈ x ∧ beta ≠ alpha of type set → prop.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) p.
Apply PNoEq_tra_ alpha (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q p to the current goal.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q.
Apply PNoEq_antimon_ (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q (ordsucc alpha) (ordinal_ordsucc alpha La) alpha (ordsuccI2 alpha) to the current goal.
We will prove PNoEq_ (ordsucc alpha) (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q.
An exact proof term for the current goal is PNoEq_PSNo (ordsucc alpha) (ordinal_ordsucc alpha La) q.
We will prove PNoEq_ alpha q p.
Apply PNoEq_sym_ to the current goal.
An exact proof term for the current goal is PNo_extend0_eq alpha p.
∎
Theorem. (SNo_extend1_SNoEq) The following is provable:
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend1 x) x
In Proofgold the corresponding term root is ba974f... and proposition id is d6f8bb...
Proof:
Let x be given.
Assume Hx: SNo x.
Set alpha to be the term SNoLev x.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is SNoLev_ordinal x Hx.
We will prove SNoEq_ alpha (SNo_extend1 x) x.
Set p to be the term λbeta ⇒ beta ∈ x of type set → prop.
Set q to be the term λbeta ⇒ beta ∈ x ∨ beta = alpha of type set → prop.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) p.
Apply PNoEq_tra_ alpha (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q p to the current goal.
We will prove PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q.
Apply PNoEq_antimon_ (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q (ordsucc alpha) (ordinal_ordsucc alpha La) alpha (ordsuccI2 alpha) to the current goal.
We will prove PNoEq_ (ordsucc alpha) (λbeta ⇒ beta ∈ PSNo (ordsucc alpha) q) q.
An exact proof term for the current goal is PNoEq_PSNo (ordsucc alpha) (ordinal_ordsucc alpha La) q.
We will prove PNoEq_ alpha q p.
Apply PNoEq_sym_ to the current goal.
An exact proof term for the current goal is PNo_extend1_eq alpha p.
∎
Theorem. (SNoLev_0_eq_0) The following is provable:
∀x, SNo x → SNoLev x = 0 → x = 0
In Proofgold the corresponding term root is fe0bc4... and proposition id is dfcf39...
Proof:
Let x be given.
Assume Hx Hx0.
Apply SNo_eq x 0 Hx SNo_0 to the current goal.
We will prove SNoLev x = SNoLev 0.
rewrite the current goal using SNoLev_0 (from left to right).
An exact proof term for the current goal is Hx0.
We will prove SNoEq_ (SNoLev x) x 0.
rewrite the current goal using Hx0 (from left to right).
Let alpha be given.
Assume Ha: alpha ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE alpha Ha.
∎
Theorem. (SNo_0_eq_0) The following is provable:
∀q, SNo_ 0 q → q = 0
In Proofgold the corresponding term root is 72efa8... and proposition id is 7bd751...
Proof:
Let q be given.
Assume H1: SNo_ 0 q.
Apply SNoLev_0_eq_0 to the current goal.
An exact proof term for the current goal is SNo_SNo 0 ordinal_Empty q H1.
An exact proof term for the current goal is SNoLev_uniq2 0 ordinal_Empty q H1.
∎
Definition. We define eps_ to be λn ⇒ {0} ∪ {(ordsucc m) '|m ∈ n} of type set → set.
In Proofgold the corresponding term root is 5e992a... and object id is ffee2b...
Theorem. (eps_ordinal_In_eq_0) The following is provable:
∀n alpha, ordinal alpha → alpha ∈ eps_ n → alpha = 0
In Proofgold the corresponding term root is 3046fd... and proposition id is 1190f3...
Proof:
Let n and alpha be given.
Assume Ha.
Assume H1: alpha ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionE {0} {(ordsucc m) '|m ∈ n} alpha H1 to the current goal.
Assume H2: alpha ∈ {0}.
An exact proof term for the current goal is SingE 0 alpha H2.
Assume H2: alpha ∈ {(ordsucc m) '|m ∈ n}.
We will prove False.
Apply ReplE_impred n (λm ⇒ (ordsucc m) ') alpha H2 to the current goal.
Let m be given.
Assume Hm: m ∈ n.
Assume H3: alpha = (ordsucc m) '.
Apply tagged_not_ordinal (ordsucc m) to the current goal.
We will prove ordinal ((ordsucc m) ').
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is Ha.
∎
Theorem. (eps_0_1) The following is provable:
eps_ 0 = 1
In Proofgold the corresponding term root is 5bbce1... and proposition id is d073f0...
Proof:
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ {0} ∪ {(ordsucc m) '|m ∈ 0}.
Apply binunionE {0} {(ordsucc m) '|m ∈ 0} x Hx to the current goal.
Assume Hx: x ∈ {0}.
rewrite the current goal using SingE 0 x Hx (from left to right).
We will prove 0 ∈ 1.
An exact proof term for the current goal is In_0_1.
Assume Hx: x ∈ {(ordsucc m) '|m ∈ 0}.
Apply ReplE_impred 0 (λm ⇒ (ordsucc m) ') x Hx to the current goal.
Let m be given.
Assume Hm: m ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE m Hm.
Let x be given.
Assume Hx: x ∈ 1.
Apply cases_1 x Hx (λx ⇒ x ∈ eps_ 0) to the current goal.
We will prove 0 ∈ {0} ∪ {(ordsucc m) '|m ∈ 0}.
Apply binunionI1 to the current goal.
Apply SingI to the current goal.
∎
Theorem. (SNo__eps_) The following is provable:
∀n ∈ ω, SNo_ (ordsucc n) (eps_ n)
In Proofgold the corresponding term root is e9c4d9... and proposition id is 1c1428...
Proof:
Let n be given.
Assume Hn.
We prove the intermediate claim Ln: nat_p n.
An exact proof term for the current goal is omega_nat_p n Hn.
We will prove eps_ n ⊆ SNoElts_ (ordsucc n) ∧ ∀m ∈ ordsucc n, exactly1of2 (m ' ∈ eps_ n) (m ∈ eps_ n).
Apply andI to the current goal.
Let x be given.
Assume Hx: x ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionE {0} {(ordsucc m) '|m ∈ n} x Hx to the current goal.
Assume Hx: x ∈ {0}.
rewrite the current goal using SingE 0 x Hx (from left to right).
We will prove 0 ∈ SNoElts_ (ordsucc n).
We will prove 0 ∈ ordsucc n ∪ {beta '|beta ∈ ordsucc n}.
Apply binunionI1 to the current goal.
We will prove 0 ∈ ordsucc n.
An exact proof term for the current goal is nat_0_in_ordsucc n Ln.
Assume Hx: x ∈ {(ordsucc m) '|m ∈ n}.
Apply ReplE_impred n (λm ⇒ (ordsucc m) ') x Hx to the current goal.
Let m be given.
Assume Hm: m ∈ n.
Assume Hxm: x = (ordsucc m) '.
We will prove x ∈ SNoElts_ (ordsucc n).
We will prove x ∈ ordsucc n ∪ {beta '|beta ∈ ordsucc n}.
Apply binunionI2 to the current goal.
We will prove x ∈ {beta '|beta ∈ ordsucc n}.
rewrite the current goal using Hxm (from left to right).
We will prove (ordsucc m) ' ∈ {beta '|beta ∈ ordsucc n}.
An exact proof term for the current goal is ReplI (ordsucc n) (λbeta ⇒ beta ') (ordsucc m) (nat_ordsucc_in_ordsucc n Ln m Hm).
Let m be given.
Assume Hm: m ∈ ordsucc n.
We prove the intermediate claim Lm: nat_p m.
An exact proof term for the current goal is nat_p_trans (ordsucc n) (nat_ordsucc n Ln) m Hm.
Apply nat_inv m Lm to the current goal.
Assume Hm: m = 0.
rewrite the current goal using Hm (from left to right).
Apply exactly1of2_I2 to the current goal.
We will prove 0 ' ∉ eps_ n.
Assume H1: 0 ' ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionE {0} {(ordsucc m) '|m ∈ n} (0 ') H1 to the current goal.
Assume H2: 0 ' ∈ {0}.
Apply EmptyE {1} to the current goal.
We will prove {1} ∈ 0.
rewrite the current goal using SingE 0 (0 ') H2 (from right to left) at position 2.
We will prove {1} ∈ 0 '.
We will prove {1} ∈ 0 ∪ {{1}}.
Apply binunionI2 to the current goal.
We will prove {1} ∈ {{1}}.
Apply SingI to the current goal.
Assume H2: 0 ' ∈ {(ordsucc m) '|m ∈ n}.
Apply ReplE_impred n (λm ⇒ (ordsucc m) ') (0 ') H2 to the current goal.
Let m be given.
Assume Hm: m ∈ n.
Assume H3: 0 ' = (ordsucc m) '.
Apply neq_0_ordsucc m to the current goal.
We will prove 0 = ordsucc m.
Apply tagged_eqE_eq to the current goal.
We will prove ordinal 0.
An exact proof term for the current goal is ordinal_Empty.
We will prove ordinal (ordsucc m).
An exact proof term for the current goal is (nat_p_ordinal (ordsucc m) (nat_ordsucc m (nat_p_trans n Ln m Hm))).
We will prove 0 ' = (ordsucc m) '.
An exact proof term for the current goal is H3.
We will prove 0 ∈ eps_ n.
We will prove 0 ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is SingI 0.
Assume H1.
Apply H1 to the current goal.
Let k be given.
Assume H1.
Apply H1 to the current goal.
Assume Hk: nat_p k.
Assume Hmk: m = ordsucc k.
We prove the intermediate claim Lm: nat_p m.
rewrite the current goal using Hmk (from left to right).
An exact proof term for the current goal is nat_ordsucc k Hk.
We prove the intermediate claim LSk: ordsucc k ∈ ordsucc n.
rewrite the current goal using Hmk (from right to left).
An exact proof term for the current goal is Hm.
We prove the intermediate claim Lk: k ∈ n.
Apply ordsuccE n (ordsucc k) LSk to the current goal.
Assume H2: ordsucc k ∈ n.
Apply nat_trans n Ln (ordsucc k) H2 to the current goal.
We will prove k ∈ ordsucc k.
Apply ordsuccI2 to the current goal.
Assume H2: ordsucc k = n.
rewrite the current goal using H2 (from right to left).
Apply ordsuccI2 to the current goal.
Apply exactly1of2_I1 to the current goal.
We will prove m ' ∈ eps_ n.
We will prove m ' ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionI2 to the current goal.
rewrite the current goal using Hmk (from left to right).
We will prove (ordsucc k) ' ∈ {(ordsucc m) '|m ∈ n}.
An exact proof term for the current goal is ReplI n (λk ⇒ (ordsucc k) ') k Lk.
We will prove m ∉ eps_ n.
Assume H1: m ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionE {0} {(ordsucc m) '|m ∈ n} m H1 to the current goal.
Assume H2: m ∈ {0}.
Apply EmptyE 0 to the current goal.
We will prove 0 ∈ 0.
rewrite the current goal using SingE 0 m H2 (from right to left) at position 2.
We will prove 0 ∈ m.
rewrite the current goal using Hmk (from left to right).
An exact proof term for the current goal is nat_0_in_ordsucc k Hk.
Assume H2: m ∈ {(ordsucc j) '|j ∈ n}.
Apply ReplE_impred n (λj ⇒ (ordsucc j) ') m H2 to the current goal.
Let j be given.
Assume Hj: j ∈ n.
Assume Hmj: m = (ordsucc j) '.
Apply tagged_not_ordinal (ordsucc j) to the current goal.
We will prove ordinal ((ordsucc j) ').
rewrite the current goal using Hmj (from right to left).
We will prove ordinal m.
An exact proof term for the current goal is nat_p_ordinal m Lm.
∎
Theorem. (SNo_eps_) The following is provable:
∀n ∈ ω, SNo (eps_ n)
In Proofgold the corresponding term root is 9fb700... and proposition id is 006b0f...
Proof:
Let n be given.
Assume Hn.
We prove the intermediate claim Ln: nat_p n.
An exact proof term for the current goal is omega_nat_p n Hn.
An exact proof term for the current goal is SNo_SNo (ordsucc n) (ordinal_ordsucc n (nat_p_ordinal n Ln)) (eps_ n) (SNo__eps_ n Hn).
∎
Theorem. (SNo_eps_1) The following is provable:
SNo (eps_ 1)
In Proofgold the corresponding term root is 17c276... and proposition id is 8b78d7...
Proof:
An exact proof term for the current goal is SNo_eps_ 1 (nat_p_omega 1 nat_1).
∎
Theorem. (SNoLev_eps_) The following is provable:
∀n ∈ ω, SNoLev (eps_ n) = ordsucc n
In Proofgold the corresponding term root is 6891e3... and proposition id is 75efd1...
Proof:
Let n be given.
Assume Hn.
We prove the intermediate claim Ln: nat_p n.
An exact proof term for the current goal is omega_nat_p n Hn.
Apply SNoLev_uniq2 (ordsucc n) (ordinal_ordsucc n (nat_p_ordinal n Ln)) (eps_ n) (SNo__eps_ n Hn) to the current goal.
∎
Theorem. (SNo_eps_SNoS_omega) The following is provable:
∀n ∈ ω, eps_ n ∈ SNoS_ ω
In Proofgold the corresponding term root is 736d2a... and proposition id is 52da18...
Proof:
Let n be given.
Assume Hn.
Apply SNoS_I ω omega_ordinal (eps_ n) (ordsucc n) to the current goal.
We will prove ordsucc n ∈ ω.
An exact proof term for the current goal is omega_ordsucc n Hn.
We will prove SNo_ (ordsucc n) (eps_ n).
An exact proof term for the current goal is SNo__eps_ n Hn.
∎
Theorem. (SNo_eps_decr) The following is provable:
∀n ∈ ω, ∀m ∈ n, eps_ n < eps_ m
In Proofgold the corresponding term root is d6a46a... and proposition id is 67c76a...
Proof:
Let n be given.
Assume Hn.
Let m be given.
Assume Hm.
We prove the intermediate claim Lnn: nat_p n.
An exact proof term for the current goal is omega_nat_p n Hn.
We prove the intermediate claim Lmn: nat_p m.
An exact proof term for the current goal is nat_p_trans n Lnn m Hm.
We prove the intermediate claim Lm: m ∈ ω.
An exact proof term for the current goal is nat_p_omega m Lmn.
Apply SNoLtI3 to the current goal.
We will prove SNoLev (eps_ m) ∈ SNoLev (eps_ n).
rewrite the current goal using SNoLev_eps_ m Lm (from left to right).
rewrite the current goal using SNoLev_eps_ n Hn (from left to right).
We will prove ordsucc m ∈ ordsucc n.
An exact proof term for the current goal is nat_ordsucc_in_ordsucc n Lnn m Hm.
We will prove SNoEq_ (SNoLev (eps_ m)) (eps_ n) (eps_ m).
rewrite the current goal using SNoLev_eps_ m Lm (from left to right).
We will prove SNoEq_ (ordsucc m) (eps_ n) (eps_ m).
Let k be given.
Assume Hk: k ∈ ordsucc m.
We will prove k ∈ eps_ n ↔ k ∈ eps_ m.
We prove the intermediate claim Lk: ordinal k.
An exact proof term for the current goal is nat_p_ordinal k (nat_p_trans (ordsucc m) (nat_ordsucc m Lmn) k Hk).
Apply iffI to the current goal.
Assume H1: k ∈ eps_ n.
rewrite the current goal using eps_ordinal_In_eq_0 n k Lk H1 (from left to right).
We will prove 0 ∈ eps_ m.
We will prove 0 ∈ {0} ∪ {(ordsucc j) '|j ∈ m}.
Apply binunionI1 to the current goal.
Apply SingI to the current goal.
Assume H1: k ∈ eps_ m.
rewrite the current goal using eps_ordinal_In_eq_0 m k Lk H1 (from left to right).
We will prove 0 ∈ eps_ n.
We will prove 0 ∈ {0} ∪ {(ordsucc j) '|j ∈ n}.
Apply binunionI1 to the current goal.
Apply SingI to the current goal.
We will prove SNoLev (eps_ m) ∉ eps_ n.
rewrite the current goal using SNoLev_eps_ m Lm (from left to right).
Assume H1: ordsucc m ∈ eps_ n.
Apply neq_ordsucc_0 m to the current goal.
We will prove ordsucc m = 0.
An exact proof term for the current goal is eps_ordinal_In_eq_0 n (ordsucc m) (ordinal_ordsucc m (nat_p_ordinal m Lmn)) H1.
∎
Theorem. (SNo_eps_pos) The following is provable:
∀n ∈ ω, 0 < eps_ n
In Proofgold the corresponding term root is 8b75de... and proposition id is 760b4a...
Proof:
Let n be given.
Assume Hn.
Apply SNoLtI2 0 (eps_ n) to the current goal.
We will prove SNoLev 0 ∈ SNoLev (eps_ n).
rewrite the current goal using SNoLev_0 (from left to right).
rewrite the current goal using SNoLev_eps_ n Hn (from left to right).
We will prove 0 ∈ ordsucc n.
An exact proof term for the current goal is nat_0_in_ordsucc n (omega_nat_p n Hn).
We will prove SNoEq_ (SNoLev 0) 0 (eps_ n).
rewrite the current goal using SNoLev_0 (from left to right).
Let alpha be given.
Assume Ha: alpha ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE alpha Ha.
We will prove SNoLev 0 ∈ eps_ n.
rewrite the current goal using SNoLev_0 (from left to right).
We will prove 0 ∈ eps_ n.
We will prove 0 ∈ {0} ∪ {(ordsucc m) '|m ∈ n}.
Apply binunionI1 to the current goal.
Apply SingI to the current goal.
∎
Theorem. (SNo_pos_eps_Lt) The following is provable:
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → eps_ n < x
In Proofgold the corresponding term root is 72974f... and proposition id is 771b0e...
Proof:
Let n be given.
Assume Hn: nat_p n.
Let x be given.
Assume Hx: x ∈ SNoS_ (ordsucc n).
Assume Hxpos: 0 < x.
We prove the intermediate claim Ln: n ∈ ω.
An exact proof term for the current goal is nat_p_omega n Hn.
Apply SNoS_E2 (ordsucc n) (nat_p_ordinal (ordsucc n) (nat_ordsucc n Hn)) x Hx to the current goal.
Assume Hx1: SNoLev x ∈ ordsucc n.
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
We will prove eps_ n < x.
Apply SNoLt_trichotomy_or_impred (eps_ n) x (SNo_eps_ n Ln) Hx3 to the current goal.
Assume H2: eps_ n < x.
An exact proof term for the current goal is H2.
Assume H2: eps_ n = x.
We will prove False.
Apply In_irref (ordsucc n) to the current goal.
rewrite the current goal using SNoLev_eps_ n Ln (from right to left) at position 1.
We will prove SNoLev (eps_ n) ∈ ordsucc n.
rewrite the current goal using H2 (from left to right).
An exact proof term for the current goal is Hx1.
Assume H2: x < eps_ n.
We will prove False.
Apply SNoLtE x (eps_ n) Hx3 (SNo_eps_ n Ln) H2 to the current goal.
Let z be given.
Assume Hz1: SNo z.
Assume Hz2: SNoLev z ∈ SNoLev x ∩ SNoLev (eps_ n).
Assume Hz3: SNoEq_ (SNoLev z) z x.
Assume Hz4: SNoEq_ (SNoLev z) z (eps_ n).
Assume Hz5: x < z.
Assume Hz6: z < eps_ n.
Assume Hz7: SNoLev z ∉ x.
Assume Hz8: SNoLev z ∈ eps_ n.
We prove the intermediate claim Lz0: z = 0.
Apply SNoLev_0_eq_0 z Hz1 to the current goal.
An exact proof term for the current goal is eps_ordinal_In_eq_0 n (SNoLev z) (SNoLev_ordinal z Hz1) Hz8.
Apply SNoLt_irref x to the current goal.
We will prove x < x.
Apply SNoLt_tra x z x Hx3 Hz1 Hx3 Hz5 to the current goal.
We will prove z < x.
rewrite the current goal using Lz0 (from left to right).
An exact proof term for the current goal is Hxpos.
Assume H1: SNoLev x ∈ SNoLev (eps_ n).
Assume H2: SNoEq_ (SNoLev x) x (eps_ n).
Assume H3: SNoLev x ∈ (eps_ n).
We prove the intermediate claim Lx0: x = 0.
Apply SNoLev_0_eq_0 x Hx3 to the current goal.
An exact proof term for the current goal is eps_ordinal_In_eq_0 n (SNoLev x) Hx2 H3.
Apply SNoLt_irref x to the current goal.
rewrite the current goal using Lx0 (from left to right) at position 1.
An exact proof term for the current goal is Hxpos.
rewrite the current goal using SNoLev_eps_ n Ln (from left to right).
Assume H1: ordsucc n ∈ SNoLev x.
We will prove False.
An exact proof term for the current goal is In_no2cycle (SNoLev x) (ordsucc n) Hx1 H1.
∎
Theorem. (SNo_pos_eps_Le) The following is provable:
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc (ordsucc n)), 0 < x → eps_ n ≤ x
In Proofgold the corresponding term root is f1e297... and proposition id is 6658de...
Proof:
Let n be given.
Assume Hn.
Let x be given.
Assume Hx: x ∈ SNoS_ (ordsucc (ordsucc n)).
Assume Hxpos: 0 < x.
We prove the intermediate claim Ln: n ∈ ω.
An exact proof term for the current goal is nat_p_omega n Hn.
Apply SNoS_E2 (ordsucc (ordsucc n)) (nat_p_ordinal (ordsucc (ordsucc n)) (nat_ordsucc (ordsucc n) (nat_ordsucc n Hn))) x Hx to the current goal.
Assume Hx1: SNoLev x ∈ ordsucc (ordsucc n).
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
We will prove eps_ n ≤ x.
Apply SNoLtLe_or x (eps_ n) Hx3 (SNo_eps_ n Ln) to the current goal.
Assume H2: x < eps_ n.
We will prove False.
Apply SNoLtE x (eps_ n) Hx3 (SNo_eps_ n Ln) H2 to the current goal.
Let z be given.
Assume Hz1: SNo z.
Assume Hz2: SNoLev z ∈ SNoLev x ∩ SNoLev (eps_ n).
Assume Hz3: SNoEq_ (SNoLev z) z x.
Assume Hz4: SNoEq_ (SNoLev z) z (eps_ n).
Assume Hz5: x < z.
Assume Hz6: z < eps_ n.
Assume Hz7: SNoLev z ∉ x.
Assume Hz8: SNoLev z ∈ eps_ n.
We prove the intermediate claim Lz0: z = 0.
Apply SNoLev_0_eq_0 z Hz1 to the current goal.
An exact proof term for the current goal is eps_ordinal_In_eq_0 n (SNoLev z) (SNoLev_ordinal z Hz1) Hz8.
Apply SNoLt_irref x to the current goal.
We will prove x < x.
Apply SNoLt_tra x z x Hx3 Hz1 Hx3 Hz5 to the current goal.
We will prove z < x.
rewrite the current goal using Lz0 (from left to right).
An exact proof term for the current goal is Hxpos.
Assume H1: SNoLev x ∈ SNoLev (eps_ n).
Assume H2: SNoEq_ (SNoLev x) x (eps_ n).
Assume H3: SNoLev x ∈ (eps_ n).
We prove the intermediate claim Lx0: x = 0.
Apply SNoLev_0_eq_0 x Hx3 to the current goal.
An exact proof term for the current goal is eps_ordinal_In_eq_0 n (SNoLev x) Hx2 H3.
Apply SNoLt_irref x to the current goal.
rewrite the current goal using Lx0 (from left to right) at position 1.
An exact proof term for the current goal is Hxpos.
rewrite the current goal using SNoLev_eps_ n Ln (from left to right).
Assume H3: ordsucc n ∈ SNoLev x.
We will prove False.
Apply ordsuccE (ordsucc n) (SNoLev x) Hx1 to the current goal.
Assume H4: SNoLev x ∈ ordsucc n.
An exact proof term for the current goal is In_no2cycle (SNoLev x) (ordsucc n) H4 H3.
Assume H4: SNoLev x = ordsucc n.
Apply In_irref (SNoLev x) to the current goal.
rewrite the current goal using H4 (from left to right) at position 1.
An exact proof term for the current goal is H3.
Assume H2: eps_ n ≤ x.
An exact proof term for the current goal is H2.
∎
Theorem. (eps_SNo_eq) The following is provable:
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → SNoEq_ (SNoLev x) (eps_ n) x → ∃m ∈ n, x = eps_ m
In Proofgold the corresponding term root is 8c2587... and proposition id is 8515ac...
Proof:
Let n be given.
Assume Hn.
Let x be given.
Assume Hx1: x ∈ SNoS_ (ordsucc n).
Assume Hx2: 0 < x.
Assume Hx3: SNoEq_ (SNoLev x) (eps_ n) x.
Apply SNoS_E2 (ordsucc n) (nat_p_ordinal (ordsucc n) (nat_ordsucc n Hn)) x Hx1 to the current goal.
Assume Hx1a: SNoLev x ∈ ordsucc n.
Assume Hx1b: ordinal (SNoLev x).
Assume Hx1c: SNo x.
Assume Hx1d: SNo_ (SNoLev x) x.
We prove the intermediate claim L1: nat_p (SNoLev x).
Apply nat_p_trans (ordsucc n) (nat_ordsucc n Hn) to the current goal.
We will prove SNoLev x ∈ ordsucc n.
An exact proof term for the current goal is Hx1a.
Apply nat_inv (SNoLev x) L1 to the current goal.
Assume H1: SNoLev x = 0.
We will prove False.
We prove the intermediate claim L2: x = 0.
An exact proof term for the current goal is SNoLev_0_eq_0 x Hx1c H1.
Apply SNoLt_irref x to the current goal.
rewrite the current goal using L2 (from left to right) at position 1.
An exact proof term for the current goal is Hx2.
Assume H1.
Apply H1 to the current goal.
Let m be given.
Assume H1.
Apply H1 to the current goal.
Assume Hm1: nat_p m.
Assume Hm2: SNoLev x = ordsucc m.
We use m to witness the existential quantifier.
Apply andI to the current goal.
We will prove m ∈ n.
Apply nat_ordsucc_trans n Hn (SNoLev x) Hx1a to the current goal.
We will prove m ∈ SNoLev x.
rewrite the current goal using Hm2 (from left to right).
Apply ordsuccI2 to the current goal.
We will prove x = eps_ m.
Apply SNo_eq x (eps_ m) Hx1c (SNo_eps_ m (nat_p_omega m Hm1)) to the current goal.
We will prove SNoLev x = SNoLev (eps_ m).
rewrite the current goal using SNoLev_eps_ m (nat_p_omega m Hm1) (from left to right).
An exact proof term for the current goal is Hm2.
We will prove SNoEq_ (SNoLev x) x (eps_ m).
Apply SNoEq_tra_ (SNoLev x) x (eps_ n) (eps_ m) to the current goal.
Apply SNoEq_sym_ to the current goal.
An exact proof term for the current goal is Hx3.
We will prove SNoEq_ (SNoLev x) (eps_ n) (eps_ m).
rewrite the current goal using Hm2 (from left to right).
We will prove SNoEq_ (ordsucc m) (eps_ n) (eps_ m).
Apply SNoEq_I to the current goal.
Let k be given.
Assume Hk: k ∈ ordsucc m.
We prove the intermediate claim L3: ordinal k.
An exact proof term for the current goal is nat_p_ordinal k (nat_p_trans (ordsucc m) (nat_ordsucc m Hm1) k Hk).
Apply iffI to the current goal.
Assume H2: k ∈ eps_ n.
rewrite the current goal using eps_ordinal_In_eq_0 n k L3 H2 (from left to right).
We will prove 0 ∈ eps_ m.
We will prove 0 ∈ {0} ∪ {SetAdjoin (ordsucc k) {1}|k ∈ m}.
Apply binunionI1 to the current goal.
Apply SingI to the current goal.
Assume H2: k ∈ eps_ m.
rewrite the current goal using eps_ordinal_In_eq_0 m k L3 H2 (from left to right).
We will prove 0 ∈ eps_ n.
We will prove 0 ∈ {0} ∪ {SetAdjoin (ordsucc k) {1}|k ∈ n}.
Apply binunionI1 to the current goal.
Apply SingI to the current goal.
∎
Theorem. (eps_SNoCutP) The following is provable:
∀n ∈ ω, SNoCutP {0} {eps_ m|m ∈ n}
In Proofgold the corresponding term root is 545873... and proposition id is 30edf9...
Proof:
Let n be given.
Assume Hn.
Set L to be the term {0}.
Set R to be the term {eps_ m|m ∈ n}.
We prove the intermediate claim Ln: nat_p n.
An exact proof term for the current goal is omega_nat_p n Hn.
We will prove (∀w ∈ L, SNo w) ∧ (∀z ∈ R, SNo z) ∧ (∀w ∈ L, ∀z ∈ R, w < z).
Apply and3I to the current goal.
Let w be given.
Assume Hw.
rewrite the current goal using SingE 0 w Hw (from left to right).
An exact proof term for the current goal is SNo_0.
Let z be given.
Assume Hz.
Apply ReplE_impred n eps_ z Hz to the current goal.
Let m be given.
Assume Hm1: m ∈ n.
Assume Hm2: z = eps_ m.
rewrite the current goal using Hm2 (from left to right).
Apply SNo_eps_ to the current goal.
We will prove m ∈ ω.
Apply nat_p_omega to the current goal.
An exact proof term for the current goal is nat_p_trans n Ln m Hm1.
Let w be given.
Assume Hw.
Let z be given.
Assume Hz.
Apply ReplE_impred n eps_ z Hz to the current goal.
Let m be given.
Assume Hm1: m ∈ n.
Assume Hm2: z = eps_ m.
rewrite the current goal using SingE 0 w Hw (from left to right).
rewrite the current goal using Hm2 (from left to right).
We will prove 0 < eps_ m.
Apply SNo_eps_pos to the current goal.
We will prove m ∈ ω.
Apply nat_p_omega to the current goal.
An exact proof term for the current goal is nat_p_trans n Ln m Hm1.
∎
Theorem. (eps_SNoCut) The following is provable:
∀n ∈ ω, eps_ n = SNoCut {0} {eps_ m|m ∈ n}
In Proofgold the corresponding term root is 06d6c5... and proposition id is eba0c6...
Proof:
Let n be given.
Assume Hn.
Set L to be the term {0}.
Set R to be the term {eps_ m|m ∈ n}.
We prove the intermediate claim Ln: nat_p n.
An exact proof term for the current goal is omega_nat_p n Hn.
We prove the intermediate claim L1: SNoCutP L R.
An exact proof term for the current goal is eps_SNoCutP n Hn.
We prove the intermediate claim LRS: ∀z ∈ R, SNo z.
Apply L1 to the current goal.
Assume H _.
Apply H to the current goal.
Assume _ H.
An exact proof term for the current goal is H.
We prove the intermediate claim L2: (⋃x ∈ Lordsucc (SNoLev x)) = 1.
Apply set_ext to the current goal.
Let k be given.
Assume Hk.
Apply famunionE_impred L (λx ⇒ ordsucc (SNoLev x)) k Hk to the current goal.
Let w be given.
Assume Hw: w ∈ L.
rewrite the current goal using SingE 0 w Hw (from left to right).
rewrite the current goal using SNoLev_0 (from left to right).
Assume H2: k ∈ 1.
We will prove k ∈ 1.
An exact proof term for the current goal is H2.
Let i be given.
Assume Hi.
Apply cases_1 i Hi (λi ⇒ i ∈ ⋃x ∈ Lordsucc (SNoLev x)) to the current goal.
We will prove 0 ∈ ⋃x ∈ Lordsucc (SNoLev x).
Apply famunionI L (λx ⇒ ordsucc (SNoLev x)) 0 0 (SingI 0) to the current goal.
We will prove 0 ∈ ordsucc (SNoLev 0).
rewrite the current goal using SNoLev_0 (from left to right).
An exact proof term for the current goal is In_0_1.
We prove the intermediate claim L3: n ≠ 0 → (⋃y ∈ Rordsucc (SNoLev y)) = ordsucc n.
Assume Hn0: n ≠ 0.
Apply set_ext to the current goal.
Let k be given.
Assume Hk.
Apply famunionE_impred R (λy ⇒ ordsucc (SNoLev y)) k Hk to the current goal.
Let z be given.
Assume Hz: z ∈ R.
Apply ReplE_impred n eps_ z Hz to the current goal.
Let m be given.
Assume Hm1: m ∈ n.
Assume Hm2: z = eps_ m.
rewrite the current goal using Hm2 (from left to right).
rewrite the current goal using SNoLev_eps_ m (nat_p_omega m (nat_p_trans n Ln m Hm1)) (from left to right).
Assume H2: k ∈ ordsucc (ordsucc m).
We will prove k ∈ ordsucc n.
We prove the intermediate claim L3a: ordsucc m ∈ ordsucc n.
Apply ordinal_ordsucc_In to the current goal.
We will prove ordinal n.
Apply nat_p_ordinal n to the current goal.
An exact proof term for the current goal is Ln.
An exact proof term for the current goal is Hm1.
We prove the intermediate claim L3b: ordsucc m ⊆ ordsucc n.
An exact proof term for the current goal is nat_trans (ordsucc n) (nat_ordsucc n Ln) (ordsucc m) L3a.
Apply ordsuccE (ordsucc m) k H2 to the current goal.
Assume H3: k ∈ ordsucc m.
Apply L3b to the current goal.
An exact proof term for the current goal is H3.
Assume H3: k = ordsucc m.
rewrite the current goal using H3 (from left to right).
An exact proof term for the current goal is L3a.
Let i be given.
Assume Hi: i ∈ ordsucc n.
We will prove i ∈ ⋃y ∈ Rordsucc (SNoLev y).
Apply nat_inv n Ln to the current goal.
Assume H2: n = 0.
We will prove False.
An exact proof term for the current goal is Hn0 H2.
Assume H2.
Apply H2 to the current goal.
Let n' be given.
Assume H2.
Apply H2 to the current goal.
Assume Hn'1: nat_p n'.
Assume Hn'2: n = ordsucc n'.
Apply famunionI R (λy ⇒ ordsucc (SNoLev y)) (eps_ n') i to the current goal.
We will prove eps_ n' ∈ R.
Apply ReplI to the current goal.
We will prove n' ∈ n.
rewrite the current goal using Hn'2 (from left to right).
Apply ordsuccI2 to the current goal.
We will prove i ∈ ordsucc (SNoLev (eps_ n')).
rewrite the current goal using SNoLev_eps_ n' (nat_p_omega n' Hn'1) (from left to right).
We will prove i ∈ ordsucc (ordsucc n').
rewrite the current goal using Hn'2 (from right to left).
An exact proof term for the current goal is Hi.
We prove the intermediate claim L4: (⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y)) = ordsucc n.
Apply xm (n = 0) to the current goal.
Assume H2: n = 0.
We prove the intermediate claim L4a: R = 0.
Apply Empty_eq to the current goal.
Let z be given.
Assume Hz.
Apply ReplE_impred n eps_ z Hz to the current goal.
Let m be given.
rewrite the current goal using H2 (from left to right).
Assume Hm1: m ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE m Hm1.
rewrite the current goal using L4a (from left to right).
rewrite the current goal using famunion_Empty (λy ⇒ ordsucc (SNoLev y)) (from left to right).
We will prove (⋃x ∈ Lordsucc (SNoLev x)) ∪ 0 = ordsucc n.
rewrite the current goal using binunion_idr (from left to right).
We will prove (⋃x ∈ Lordsucc (SNoLev x)) = ordsucc n.
rewrite the current goal using H2 (from left to right).
An exact proof term for the current goal is L2.
Assume H2: n ≠ 0.
rewrite the current goal using L3 H2 (from right to left).
rewrite the current goal using Subq_binunion_eq (from right to left).
rewrite the current goal using L2 (from left to right).
rewrite the current goal using L3 H2 (from left to right).
We will prove 1 ⊆ ordsucc n.
Let i be given.
Assume Hi.
Apply cases_1 i Hi (λi ⇒ i ∈ ordsucc n) to the current goal.
We will prove 0 ∈ ordsucc n.
Apply nat_0_in_ordsucc to the current goal.
An exact proof term for the current goal is Ln.
Apply SNoCutP_SNoCut_impred L R L1 to the current goal.
Assume H1: SNo (SNoCut L R).
rewrite the current goal using L4 (from left to right).
Assume H2: SNoLev (SNoCut L R) ∈ ordsucc (ordsucc n).
Assume H3: ∀w ∈ L, w < SNoCut L R.
Assume H4: ∀z ∈ R, SNoCut L R < z.
Assume H5: ∀u, SNo u → (∀w ∈ L, w < u) → (∀z ∈ R, u < z) → SNoLev (SNoCut L R) ⊆ SNoLev u ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) u.
We prove the intermediate claim L5: SNo (eps_ n).
An exact proof term for the current goal is SNo_eps_ n Hn.
We prove the intermediate claim L6: ∀w ∈ L, w < eps_ n.
Let w be given.
Assume Hw.
rewrite the current goal using SingE 0 w Hw (from left to right).
We will prove 0 < eps_ n.
An exact proof term for the current goal is SNo_eps_pos n Hn.
We prove the intermediate claim L7: ∀z ∈ R, eps_ n < z.
Let z be given.
Assume Hz.
Apply ReplE_impred n eps_ z Hz to the current goal.
Let m be given.
Assume Hm1: m ∈ n.
Assume Hm2: z = eps_ m.
rewrite the current goal using Hm2 (from left to right).
We will prove eps_ n < eps_ m.
An exact proof term for the current goal is SNo_eps_decr n Hn m Hm1.
Apply H5 (eps_ n) L5 L6 L7 to the current goal.
Assume H6: SNoLev (SNoCut L R) ⊆ SNoLev (eps_ n).
Assume H7: SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) (eps_ n).
Use symmetry.
Apply SNo_eq (SNoCut L R) (eps_ n) H1 L5 to the current goal.
We will prove SNoLev (SNoCut L R) = SNoLev (eps_ n).
rewrite the current goal using SNoLev_eps_ n Hn (from left to right).
We will prove SNoLev (SNoCut L R) = ordsucc n.
Apply ordsuccE (ordsucc n) (SNoLev (SNoCut L R)) H2 to the current goal.
Assume H8: SNoLev (SNoCut L R) ∈ ordsucc n.
We will prove False.
We prove the intermediate claim L8: eps_ n < SNoCut L R.
Apply SNo_pos_eps_Lt n (omega_nat_p n Hn) (SNoCut L R) to the current goal.
We will prove SNoCut L R ∈ SNoS_ (ordsucc n).
Apply SNoS_I (ordsucc n) (nat_p_ordinal (ordsucc n) (nat_ordsucc n (omega_nat_p n Hn))) (SNoCut L R) (SNoLev (SNoCut L R)) H8 to the current goal.
We will prove SNo_ (SNoLev (SNoCut L R)) (SNoCut L R).
An exact proof term for the current goal is SNoLev_ (SNoCut L R) H1.
We will prove 0 < SNoCut L R.
Apply H3 to the current goal.
Apply SingI to the current goal.
Apply SNoLtE (eps_ n) (SNoCut L R) (SNo_eps_ n Hn) H1 L8 to the current goal.
Let z be given.
Assume Hz1: SNo z.
Assume Hz2: SNoLev z ∈ SNoLev (eps_ n) ∩ SNoLev (SNoCut L R).
Assume Hz3: SNoEq_ (SNoLev z) z (eps_ n).
Assume Hz4: SNoEq_ (SNoLev z) z (SNoCut L R).
Assume Hz5: eps_ n < z.
Assume Hz6: z < SNoCut L R.
Assume Hz7: SNoLev z ∉ eps_ n.
Assume Hz8: SNoLev z ∈ SNoCut L R.
We prove the intermediate claim L9: ∀w ∈ L, w < z.
Let w be given.
Assume Hw.
rewrite the current goal using SingE 0 w Hw (from left to right).
We will prove 0 < z.
Apply SNoLt_tra 0 (eps_ n) z SNo_0 (SNo_eps_ n Hn) Hz1 to the current goal.
An exact proof term for the current goal is SNo_eps_pos n Hn.
An exact proof term for the current goal is Hz5.
We prove the intermediate claim L10: ∀v ∈ R, z < v.
Let v be given.
Assume Hv.
Apply SNoLt_tra z (SNoCut L R) v Hz1 H1 (LRS v Hv) Hz6 to the current goal.
We will prove SNoCut L R < v.
Apply H4 to the current goal.
An exact proof term for the current goal is Hv.
Apply H5 z Hz1 L9 L10 to the current goal.
Assume H9: SNoLev (SNoCut L R) ⊆ SNoLev z.
We will prove False.
Apply In_irref (SNoLev z) to the current goal.
Apply H9 to the current goal.
An exact proof term for the current goal is binintersectE2 (SNoLev (eps_ n)) (SNoLev (SNoCut L R)) (SNoLev z) Hz2.
rewrite the current goal using SNoLev_eps_ n Hn (from left to right).
Assume H9: ordsucc n ∈ SNoLev (SNoCut L R).
We will prove False.
An exact proof term for the current goal is In_no2cycle (SNoLev (SNoCut L R)) (ordsucc n) H8 H9.
rewrite the current goal using SNoLev_eps_ n Hn (from left to right).
Assume H9: SNoLev (SNoCut L R) ∈ ordsucc n.
Assume H10: SNoEq_ (SNoLev (SNoCut L R)) (eps_ n) (SNoCut L R).
Assume H11: SNoLev (SNoCut L R) ∉ eps_ n.
We prove the intermediate claim L11: ∃m ∈ n, SNoCut L R = eps_ m.
Apply eps_SNo_eq n (omega_nat_p n Hn) (SNoCut L R) to the current goal.
We will prove SNoCut L R ∈ SNoS_ (ordsucc n).
Apply SNoS_I (ordsucc n) (nat_p_ordinal (ordsucc n) (nat_ordsucc n (omega_nat_p n Hn))) (SNoCut L R) (SNoLev (SNoCut L R)) to the current goal.
We will prove SNoLev (SNoCut L R) ∈ ordsucc n.
An exact proof term for the current goal is H9.
We will prove SNo_ (SNoLev (SNoCut L R)) (SNoCut L R).
Apply SNoLev_ to the current goal.
An exact proof term for the current goal is H1.
We will prove 0 < SNoCut L R.
Apply H3 to the current goal.
Apply SingI to the current goal.
We will prove SNoEq_ (SNoLev (SNoCut L R)) (eps_ n) (SNoCut L R).
An exact proof term for the current goal is H10.
Apply L11 to the current goal.
Let m be given.
Assume H12.
Apply H12 to the current goal.
Assume Hm1: m ∈ n.
Assume Hm2: SNoCut L R = eps_ m.
Apply SNoLt_irref (eps_ m) to the current goal.
We will prove eps_ m < eps_ m.
rewrite the current goal using Hm2 (from right to left) at position 1.
We will prove SNoCut L R < eps_ m.
Apply H4 to the current goal.
We will prove eps_ m ∈ {eps_ m|m ∈ n}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hm1.
Assume H8: SNoLev (SNoCut L R) = ordsucc n.
An exact proof term for the current goal is H8.
An exact proof term for the current goal is H7.
∎
End of Section TaggedSets2
Theorem. (SNo_etaE) The following is provable:
∀z, SNo z → ∀p : prop, (∀L R, SNoCutP L R → (∀x ∈ L, SNoLev x ∈ SNoLev z) → (∀y ∈ R, SNoLev y ∈ SNoLev z) → z = SNoCut L R → p) → p
In Proofgold the corresponding term root is c24473... and proposition id is 109f1d...
Proof:
Let z be given.
Assume Hz: SNo z.
Let p be given.
Assume H1.
We prove the intermediate claim LLz: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
Set L to be the term {x ∈ SNoS_ (SNoLev z)|x < z}.
Set R to be the term {y ∈ SNoS_ (SNoLev z)|z < y}.
We prove the intermediate claim L1: z = SNoCut L R.
An exact proof term for the current goal is SNo_eta z Hz.
We prove the intermediate claim LL: ∀x, x ∈ L → SNo x ∧ SNoLev x ∈ SNoLev z ∧ x < z.
Let x be given.
Assume Hx: x ∈ L.
Apply SepE (SNoS_ (SNoLev z)) (λx ⇒ x < z) x Hx to the current goal.
Assume Hx1: x ∈ SNoS_ (SNoLev z).
Assume Hx2: x < z.
Apply SNoS_E (SNoLev z) LLz x Hx1 to the current goal.
Let beta be given.
Assume Hx3.
Apply Hx3 to the current goal.
Assume Hb: beta ∈ SNoLev z.
Assume Hx3: SNo_ beta x.
We prove the intermediate claim Lb: ordinal beta.
An exact proof term for the current goal is ordinal_Hered (SNoLev z) LLz beta Hb.
We prove the intermediate claim Lx1: SNo x.
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha x.
We use beta to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Lb.
An exact proof term for the current goal is Hx3.
We prove the intermediate claim Lx2: SNoLev x = beta.
Apply SNoLev_prop x Lx1 to the current goal.
Assume Hx4: ordinal (SNoLev x).
Assume Hx5: SNo_ (SNoLev x) x.
Apply SNoLev_uniq x to the current goal.
An exact proof term for the current goal is Hx4.
An exact proof term for the current goal is Lb.
Apply Hx5 to the current goal.
An exact proof term for the current goal is Hx3.
We prove the intermediate claim Lx3: SNoLev x ∈ SNoLev z.
rewrite the current goal using Lx2 (from left to right).
An exact proof term for the current goal is Hb.
Apply and3I to the current goal.
We will prove SNo x.
An exact proof term for the current goal is Lx1.
We will prove SNoLev x ∈ SNoLev z.
An exact proof term for the current goal is Lx3.
We will prove x < z.
An exact proof term for the current goal is Hx2.
We prove the intermediate claim LR: ∀y, y ∈ R → SNo y ∧ SNoLev y ∈ SNoLev z ∧ z < y.
Let y be given.
Assume Hy: y ∈ R.
Apply SepE (SNoS_ (SNoLev z)) (λy ⇒ z < y) y Hy to the current goal.
Assume Hy1: y ∈ SNoS_ (SNoLev z).
Assume Hy2: z < y.
Apply SNoS_E (SNoLev z) LLz y Hy1 to the current goal.
Let beta be given.
Assume Hy3.
Apply Hy3 to the current goal.
Assume Hb: beta ∈ SNoLev z.
Assume Hy3: SNo_ beta y.
We prove the intermediate claim Lb: ordinal beta.
An exact proof term for the current goal is ordinal_Hered (SNoLev z) LLz beta Hb.
We prove the intermediate claim Ly1: SNo y.
We will prove ∃alpha, ordinal alpha ∧ SNo_ alpha y.
We use beta to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Lb.
An exact proof term for the current goal is Hy3.
We prove the intermediate claim Ly2: SNoLev y = beta.
Apply SNoLev_prop y Ly1 to the current goal.
Assume Hy4: ordinal (SNoLev y).
Assume Hy5: SNo_ (SNoLev y) y.
Apply SNoLev_uniq y to the current goal.
An exact proof term for the current goal is Hy4.
An exact proof term for the current goal is Lb.
Apply Hy5 to the current goal.
An exact proof term for the current goal is Hy3.
We prove the intermediate claim Ly3: SNoLev y ∈ SNoLev z.
rewrite the current goal using Ly2 (from left to right).
An exact proof term for the current goal is Hb.
Apply and3I to the current goal.
We will prove SNo y.
An exact proof term for the current goal is Ly1.
We will prove SNoLev y ∈ SNoLev z.
An exact proof term for the current goal is Ly3.
We will prove z < y.
An exact proof term for the current goal is Hy2.
Apply H1 L R to the current goal.
We will prove SNoCutP L R.
We will prove (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y).
Apply and3I to the current goal.
Let x be given.
Assume Hx.
Apply LL x Hx to the current goal.
Assume H _.
Apply H to the current goal.
Assume H _.
An exact proof term for the current goal is H.
Let y be given.
Assume Hy.
Apply LR y Hy to the current goal.
Assume H _.
Apply H to the current goal.
Assume H _.
An exact proof term for the current goal is H.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
Apply LL x Hx to the current goal.
Assume H2.
Apply H2 to the current goal.
Assume H2: SNo x.
Assume H3: SNoLev x ∈ SNoLev z.
Assume H4: x < z.
Apply LR y Hy to the current goal.
Assume H5.
Apply H5 to the current goal.
Assume H5: SNo y.
Assume H6: SNoLev y ∈ SNoLev z.
Assume H7: z < y.
An exact proof term for the current goal is SNoLt_tra x z y H2 Hz H5 H4 H7.
We will prove ∀x ∈ L, SNoLev x ∈ SNoLev z.
Let x be given.
Assume Hx.
Apply LL x Hx to the current goal.
Assume H _.
Apply H to the current goal.
Assume _ H.
An exact proof term for the current goal is H.
We will prove ∀y ∈ R, SNoLev y ∈ SNoLev z.
Let y be given.
Assume Hy.
Apply LR y Hy to the current goal.
Assume H _.
Apply H to the current goal.
Assume _ H.
An exact proof term for the current goal is H.
We will prove z = SNoCut L R.
An exact proof term for the current goal is L1.
∎
Theorem. (SNo_ind) The following is provable:
∀P : set → prop, (∀L R, SNoCutP L R → (∀x ∈ L, P x) → (∀y ∈ R, P y) → P (SNoCut L R)) → ∀z, SNo z → P z
In Proofgold the corresponding term root is 5b4c39... and proposition id is 94cbe6...
Proof:
Let P be given.
Assume H1.
We prove the intermediate claim L1: ∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → P z.
Apply ordinal_ind to the current goal.
Let alpha be given.
Assume Ha: ordinal alpha.
Assume IH: ∀beta ∈ alpha, ∀z, SNo z → SNoLev z ∈ beta → P z.
Let z be given.
Assume Hz: SNo z.
Assume Hz2: SNoLev z ∈ alpha.
We will prove P z.
We prove the intermediate claim LLz: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
Apply SNo_etaE z Hz to the current goal.
Let L and R be given.
Assume H2: SNoCutP L R.
Assume H3: ∀x ∈ L, SNoLev x ∈ SNoLev z.
Assume H4: ∀y ∈ R, SNoLev y ∈ SNoLev z.
Assume H5: z = SNoCut L R.
Apply H2 to the current goal.
Assume H6.
Apply H6 to the current goal.
Assume H6: ∀x ∈ L, SNo x.
Assume H7: ∀y ∈ R, SNo y.
Assume H8: ∀x ∈ L, ∀y ∈ R, x < y.
rewrite the current goal using H5 (from left to right).
We will prove P (SNoCut L R).
Apply H1 to the current goal.
We will prove SNoCutP L R.
An exact proof term for the current goal is H2.
We will prove ∀x, x ∈ L → P x.
Let x be given.
Assume Hx: x ∈ L.
Apply IH (SNoLev z) Hz2 x to the current goal.
We will prove SNo x.
An exact proof term for the current goal is H6 x Hx.
We will prove SNoLev x ∈ SNoLev z.
An exact proof term for the current goal is H3 x Hx.
We will prove ∀y, y ∈ R → P y.
Let y be given.
Assume Hy: y ∈ R.
Apply IH (SNoLev z) Hz2 y to the current goal.
We will prove SNo y.
An exact proof term for the current goal is H7 y Hy.
We will prove SNoLev y ∈ SNoLev z.
An exact proof term for the current goal is H4 y Hy.
Let z be given.
Assume Hz: SNo z.
We prove the intermediate claim L2: ordinal (ordsucc (SNoLev z)).
Apply ordinal_ordsucc to the current goal.
Apply SNoLev_ordinal to the current goal.
An exact proof term for the current goal is Hz.
Apply L1 (ordsucc (SNoLev z)) L2 z Hz to the current goal.
Apply ordsuccI2 to the current goal.
∎
Beginning of Section SurrealRecI
Variable F : set → (set → set) → set
Let default : set ≝ Eps_i (λ_ ⇒ True)
Let G : set → (set → set → set) → set → set ≝ λalpha g ⇒ If_ii (ordinal alpha) (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : set ⇒ default)
Definition. We define SNo_rec_i to be λz ⇒ In_rec_ii G (SNoLev z) z of type set → set.
In Proofgold the corresponding term root is be45df... and object id is 230ba8...
Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
Theorem. (SNo_rec_i_eq) The following is provable:
∀z, SNo z → SNo_rec_i z = F z SNo_rec_i
In Proofgold the corresponding term root is d5ffa8... and proposition id is f547c8...
Proof:
Let z be given.
Assume Hz: SNo z.
We will prove SNo_rec_i z = F z SNo_rec_i.
We will prove In_rec_ii G (SNoLev z) z = F z (λz ⇒ In_rec_ii G (SNoLev z) z).
We prove the intermediate claim L1: ∀alpha, ∀g h : set → set → set, (∀x ∈ alpha, g x = h x) → G alpha g = G alpha h.
Let alpha, g and h be given.
Assume Hgh: ∀x ∈ alpha, g x = h x.
We will prove G alpha g = G alpha h.
We will prove (If_ii (ordinal alpha) (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : set ⇒ default)) = (If_ii (ordinal alpha) (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ h (SNoLev w) w) else default) (λz : set ⇒ default)).
Apply xm (ordinal alpha) to the current goal.
Assume H1: ordinal alpha.
rewrite the current goal using If_ii_1 (ordinal alpha) (λz ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ h (SNoLev w) w) else default) (λz : set ⇒ default) H1 (from left to right).
rewrite the current goal using If_ii_1 (ordinal alpha) (λz ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : set ⇒ default) H1 (from left to right).
We will prove (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) = (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ h (SNoLev w) w) else default).
Apply func_ext set set to the current goal.
Let z be given.
We will prove (if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) = (if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ h (SNoLev w) w) else default).
Apply xm (z ∈ SNoS_ (ordsucc alpha)) to the current goal.
Assume Hz: z ∈ SNoS_ (ordsucc alpha).
rewrite the current goal using If_i_1 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default Hz (from left to right).
rewrite the current goal using If_i_1 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default Hz (from left to right).
We will prove F z (λw ⇒ g (SNoLev w) w) = F z (λw ⇒ h (SNoLev w) w).
We prove the intermediate claim Lsa: ordinal (ordsucc alpha).
An exact proof term for the current goal is ordinal_ordsucc alpha H1.
Apply SNoS_E2 (ordsucc alpha) Lsa z Hz to the current goal.
Assume Hz1: SNoLev z ∈ ordsucc alpha.
Assume Hz2: ordinal (SNoLev z).
Assume Hz3: SNo z.
Assume Hz4: SNo_ (SNoLev z) z.
We will prove F z (λw ⇒ g (SNoLev w) w) = F z (λw ⇒ h (SNoLev w) w).
Apply Fr to the current goal.
We will prove SNo z.
An exact proof term for the current goal is Hz3.
We will prove ∀w ∈ SNoS_ (SNoLev z), g (SNoLev w) w = h (SNoLev w) w.
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev z).
Apply SNoS_E2 (SNoLev z) Hz2 w Hw to the current goal.
Assume Hw1: SNoLev w ∈ SNoLev z.
Assume Hw2: ordinal (SNoLev w).
Assume Hw3: SNo w.
Assume Hw4: SNo_ (SNoLev w) w.
We prove the intermediate claim LLw: SNoLev w ∈ alpha.
Apply ordsuccE alpha (SNoLev z) Hz1 to the current goal.
Assume H2: SNoLev z ∈ alpha.
Apply H1 to the current goal.
Assume Ha1 _.
An exact proof term for the current goal is Ha1 (SNoLev z) H2 (SNoLev w) Hw1.
Assume H2: SNoLev z = alpha.
rewrite the current goal using H2 (from right to left).
An exact proof term for the current goal is Hw1.
rewrite the current goal using Hgh (SNoLev w) LLw (from left to right).
Use reflexivity.
Assume Hz: z ∉ SNoS_ (ordsucc alpha).
rewrite the current goal using If_i_0 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default Hz (from left to right).
An exact proof term for the current goal is If_i_0 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default Hz.
Assume H1: ¬ ordinal alpha.
rewrite the current goal using If_ii_0 (ordinal alpha) (λz ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ h (SNoLev w) w) else default) (λz : set ⇒ default) H1 (from left to right).
An exact proof term for the current goal is If_ii_0 (ordinal alpha) (λz ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : set ⇒ default) H1.
rewrite the current goal using In_rec_ii_eq G L1 (from left to right).
We will prove G (SNoLev z) (In_rec_ii G) z = F z (λz ⇒ In_rec_ii G (SNoLev z) z).
We will prove (If_ii (ordinal (SNoLev z)) (λu : set ⇒ if u ∈ SNoS_ (ordsucc (SNoLev z)) then F u (λw : set ⇒ In_rec_ii G (SNoLev w) w) else default) (λ_ : set ⇒ default)) z = F z (λz ⇒ In_rec_ii G (SNoLev z) z).
We prove the intermediate claim L2: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
rewrite the current goal using If_ii_1 (ordinal (SNoLev z)) (λu : set ⇒ if u ∈ SNoS_ (ordsucc (SNoLev z)) then F u (λw : set ⇒ In_rec_ii G (SNoLev w) w) else default) (λ_ : set ⇒ default) L2 (from left to right).
We will prove (if z ∈ SNoS_ (ordsucc (SNoLev z)) then F z (λw : set ⇒ In_rec_ii G (SNoLev w) w) else default) = F z (λz ⇒ In_rec_ii G (SNoLev z) z).
We prove the intermediate claim L3: z ∈ SNoS_ (ordsucc (SNoLev z)).
An exact proof term for the current goal is SNoS_SNoLev z Hz.
An exact proof term for the current goal is If_i_1 (z ∈ SNoS_ (ordsucc (SNoLev z))) (F z (λw : set ⇒ In_rec_ii G (SNoLev w) w)) default L3.
∎
End of Section SurrealRecI
Beginning of Section SurrealRecII
Variable F : set → (set → (set → set)) → (set → set)
Let default : (set → set) ≝ Descr_ii (λ_ ⇒ True)
Let G : set → (set → set → (set → set)) → set → (set → set) ≝ λalpha g ⇒ If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default)
Definition. We define SNo_rec_ii to be λz ⇒ In_rec_iii G (SNoLev z) z of type set → (set → set).
In Proofgold the corresponding term root is e148e6... and object id is c3f218...
Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
Theorem. (SNo_rec_ii_eq) The following is provable:
∀z, SNo z → SNo_rec_ii z = F z SNo_rec_ii
In Proofgold the corresponding term root is 39e866... and proposition id is 1c5373...
Proof:
Let z be given.
Assume Hz: SNo z.
We will prove SNo_rec_ii z = F z SNo_rec_ii.
We will prove In_rec_iii G (SNoLev z) z = F z (λz ⇒ In_rec_iii G (SNoLev z) z).
We prove the intermediate claim L1: ∀alpha, ∀g h : set → set → (set → set), (∀x ∈ alpha, g x = h x) → G alpha g = G alpha h.
Let alpha, g and h be given.
Assume Hgh: ∀x ∈ alpha, g x = h x.
We will prove G alpha g = G alpha h.
We will prove (If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default)) = (If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default) (λz : set ⇒ default)).
Apply xm (ordinal alpha) to the current goal.
Assume H1: ordinal alpha.
rewrite the current goal using If_iii_1 (ordinal alpha) (λz ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default) (λz : set ⇒ default) H1 (from left to right).
rewrite the current goal using If_iii_1 (ordinal alpha) (λz ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default) H1 (from left to right).
We will prove (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) = (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default).
Apply func_ext set (set → set) to the current goal.
Let z be given.
We will prove (If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) = (If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default).
Apply xm (z ∈ SNoS_ (ordsucc alpha)) to the current goal.
Assume Hz: z ∈ SNoS_ (ordsucc alpha).
rewrite the current goal using If_ii_1 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default Hz (from left to right).
rewrite the current goal using If_ii_1 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default Hz (from left to right).
We will prove F z (λw ⇒ g (SNoLev w) w) = F z (λw ⇒ h (SNoLev w) w).
We prove the intermediate claim Lsa: ordinal (ordsucc alpha).
An exact proof term for the current goal is ordinal_ordsucc alpha H1.
Apply SNoS_E2 (ordsucc alpha) Lsa z Hz to the current goal.
Assume Hz1: SNoLev z ∈ ordsucc alpha.
Assume Hz2: ordinal (SNoLev z).
Assume Hz3: SNo z.
Assume Hz4: SNo_ (SNoLev z) z.
We will prove F z (λw ⇒ g (SNoLev w) w) = F z (λw ⇒ h (SNoLev w) w).
Apply Fr to the current goal.
We will prove SNo z.
An exact proof term for the current goal is Hz3.
We will prove ∀w ∈ SNoS_ (SNoLev z), g (SNoLev w) w = h (SNoLev w) w.
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev z).
Apply SNoS_E2 (SNoLev z) Hz2 w Hw to the current goal.
Assume Hw1: SNoLev w ∈ SNoLev z.
Assume Hw2: ordinal (SNoLev w).
Assume Hw3: SNo w.
Assume Hw4: SNo_ (SNoLev w) w.
We prove the intermediate claim LLw: SNoLev w ∈ alpha.
Apply ordsuccE alpha (SNoLev z) Hz1 to the current goal.
Assume H2: SNoLev z ∈ alpha.
Apply H1 to the current goal.
Assume Ha1 _.
An exact proof term for the current goal is Ha1 (SNoLev z) H2 (SNoLev w) Hw1.
Assume H2: SNoLev z = alpha.
rewrite the current goal using H2 (from right to left).
An exact proof term for the current goal is Hw1.
rewrite the current goal using Hgh (SNoLev w) LLw (from left to right).
Use reflexivity.
Assume Hz: z ∉ SNoS_ (ordsucc alpha).
rewrite the current goal using If_ii_0 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default Hz (from left to right).
An exact proof term for the current goal is If_ii_0 (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default Hz.
Assume H1: ¬ ordinal alpha.
rewrite the current goal using If_iii_0 (ordinal alpha) (λz ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ h (SNoLev w) w)) default) (λz : set ⇒ default) H1 (from left to right).
An exact proof term for the current goal is If_iii_0 (ordinal alpha) (λz ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default) H1.
rewrite the current goal using In_rec_iii_eq G L1 (from left to right).
We will prove G (SNoLev z) (In_rec_iii G) z = F z (λz ⇒ In_rec_iii G (SNoLev z) z).
We will prove (If_iii (ordinal (SNoLev z)) (λu : set ⇒ If_ii (u ∈ SNoS_ (ordsucc (SNoLev z))) (F u (λw : set ⇒ In_rec_iii G (SNoLev w) w)) default) (λ_ : set ⇒ default)) z = F z (λz ⇒ In_rec_iii G (SNoLev z) z).
We prove the intermediate claim L2: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
rewrite the current goal using If_iii_1 (ordinal (SNoLev z)) (λu : set ⇒ If_ii (u ∈ SNoS_ (ordsucc (SNoLev z))) (F u (λw : set ⇒ In_rec_iii G (SNoLev w) w)) default) (λ_ : set ⇒ default) L2 (from left to right).
We will prove (If_ii (z ∈ SNoS_ (ordsucc (SNoLev z))) (F z (λw : set ⇒ In_rec_iii G (SNoLev w) w)) default) = F z (λz ⇒ In_rec_iii G (SNoLev z) z).
We prove the intermediate claim L3: z ∈ SNoS_ (ordsucc (SNoLev z)).
An exact proof term for the current goal is SNoS_SNoLev z Hz.
An exact proof term for the current goal is If_ii_1 (z ∈ SNoS_ (ordsucc (SNoLev z))) (F z (λw : set ⇒ In_rec_iii G (SNoLev w) w)) default L3.
∎
End of Section SurrealRecII
Beginning of Section SurrealRec2
Variable F : set → set → (set → set → set) → set
Let G : set → (set → set → set) → set → (set → set) → set ≝ λw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
Let H : set → (set → set → set) → set → set ≝ λw f z ⇒ if SNo z then SNo_rec_i (G w f) z else Empty
Definition. We define SNo_rec2 to be SNo_rec_ii H of type set → set → set.
In Proofgold the corresponding term root is 7d10ab... and object id is 951110...
Hypothesis Fr : ∀w, SNo w → ∀z, SNo z → ∀g h : set → set → set, (∀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
Theorem. (SNo_rec2_G_prop) The following is provable:
∀w, SNo w → ∀f k : set → set → set, (∀x ∈ SNoS_ (SNoLev w), f x = k x) → ∀z, SNo z → ∀g h : set → set, (∀u ∈ SNoS_ (SNoLev z), g u = h u) → G w f z g = G w k z h
In Proofgold the corresponding term root is 0f875d... and proposition id is de8737...
Proof:
Let w be given.
Assume Hw: SNo w.
Let f and k be given.
Assume Hfk: ∀x ∈ SNoS_ (SNoLev w), f x = k x.
Let z be given.
Assume Hz: SNo z.
Let g and h be given.
Assume Hgh: ∀u ∈ SNoS_ (SNoLev z), g u = h u.
We will prove G w f z g = G w k z h.
We will prove F w z (λx y ⇒ if x = w then g y else f x y) = F w z (λx y ⇒ if x = w then h y else k x y).
Apply Fr to the current goal.
We will prove SNo w.
An exact proof term for the current goal is Hw.
We will prove SNo z.
An exact proof term for the current goal is Hz.
Let x be given.
Assume Hx: x ∈ SNoS_ (SNoLev w).
Let y be given.
Assume Hy: SNo y.
We will prove (if x = w then g y else f x y) = (if x = w then h y else k x y).
We prove the intermediate claim Lxw: x ≠ w.
An exact proof term for the current goal is SNoS_In_neq w Hw x Hx.
rewrite the current goal using If_i_0 (x = w) (g y) (f x y) Lxw (from left to right).
rewrite the current goal using If_i_0 (x = w) (h y) (k x y) Lxw (from left to right).
We will prove f x y = k x y.
rewrite the current goal using Hfk x Hx (from left to right).
Use reflexivity.
Let y be given.
Assume Hy: y ∈ SNoS_ (SNoLev z).
We will prove (if w = w then g y else f w y) = (if w = w then h y else k w y).
rewrite the current goal using If_i_1 (w = w) (g y) (f w y) (λq H ⇒ H) (from left to right).
rewrite the current goal using If_i_1 (w = w) (h y) (k w y) (λq H ⇒ H) (from left to right).
We will prove g y = h y.
An exact proof term for the current goal is Hgh y Hy.
∎
Theorem. (SNo_rec2_eq_1) The following is provable:
∀w, SNo w → ∀f : set → set → set, ∀z, SNo z → SNo_rec_i (G w f) z = G w f z (SNo_rec_i (G w f))
In Proofgold the corresponding term root is 12a51a... and proposition id is 0dc50e...
Proof:
Let w be given.
Assume Hw: SNo w.
Let f be given.
We prove the intermediate claim L1: ∀z, SNo z → ∀g h : set → set, (∀u ∈ SNoS_ (SNoLev z), g u = h u) → G w f z g = G w f z h.
Let z be given.
Assume Hz.
Let g and h be given.
Assume Hgh.
We prove the intermediate claim L1a: ∀x ∈ SNoS_ (SNoLev w), f x = f x.
Let x be given.
Assume Hx.
Use reflexivity.
An exact proof term for the current goal is SNo_rec2_G_prop w Hw f f L1a z Hz g h Hgh.
An exact proof term for the current goal is SNo_rec_i_eq (G w f) L1.
∎
Theorem. (SNo_rec2_eq) The following is provable:
∀w, SNo w → ∀z, SNo z → SNo_rec2 w z = F w z SNo_rec2
In Proofgold the corresponding term root is e9ea41... and proposition id is ac9f2b...
Proof:
Let w be given.
Assume Hw: SNo w.
Let z be given.
Assume Hz: SNo z.
We will prove SNo_rec2 w z = F w z SNo_rec2.
We will prove SNo_rec_ii H w z = F w z SNo_rec2.
We prove the intermediate claim L1: ∀w, SNo w → ∀g h : set → set → set, (∀x ∈ SNoS_ (SNoLev w), g x = h x) → H w g = H w h.
Let w be given.
Assume Hw: SNo w.
Let g and h be given.
Assume Hgh: ∀x ∈ SNoS_ (SNoLev w), g x = h x.
We will prove H w g = H w h.
We will prove (λz : set ⇒ if SNo z then SNo_rec_i (G w g) z else Empty) = (λz : set ⇒ if SNo z then SNo_rec_i (G w h) z else Empty).
Apply func_ext set set to the current goal.
Let z be given.
We will prove (if SNo z then SNo_rec_i (G w g) z else Empty) = (if SNo z then SNo_rec_i (G w h) z else Empty).
Apply xm (SNo z) to the current goal.
Assume Hz: SNo z.
rewrite the current goal using If_i_1 (SNo z) (SNo_rec_i (G w g) z) Empty Hz (from left to right).
rewrite the current goal using If_i_1 (SNo z) (SNo_rec_i (G w h) z) Empty Hz (from left to right).
We will prove SNo_rec_i (G w g) z = SNo_rec_i (G w h) z.
We prove the intermediate claim L1a: ∀alpha, ordinal alpha → ∀z ∈ SNoS_ alpha, SNo_rec_i (G w g) z = SNo_rec_i (G w h) z.
Apply ordinal_ind to the current goal.
Let alpha be given.
Assume Ha: ordinal alpha.
Assume IH: ∀beta ∈ alpha, ∀z ∈ SNoS_ beta, SNo_rec_i (G w g) z = SNo_rec_i (G w h) z.
Let z be given.
Assume Hz: z ∈ SNoS_ alpha.
Apply SNoS_E2 alpha Ha z Hz to the current goal.
Assume Hz1: SNoLev z ∈ alpha.
Assume Hz2: ordinal (SNoLev z).
Assume Hz3: SNo z.
Assume Hz4: SNo_ (SNoLev z) z.
We will prove SNo_rec_i (G w g) z = SNo_rec_i (G w h) z.
rewrite the current goal using SNo_rec2_eq_1 w Hw g z Hz3 (from left to right).
rewrite the current goal using SNo_rec2_eq_1 w Hw h z Hz3 (from left to right).
We will prove G w g z (SNo_rec_i (G w g)) = G w h z (SNo_rec_i (G w h)).
Apply SNo_rec2_G_prop w Hw g h Hgh z Hz3 (SNo_rec_i (G w g)) (SNo_rec_i (G w h)) to the current goal.
We will prove ∀y ∈ SNoS_ (SNoLev z), SNo_rec_i (G w g) y = SNo_rec_i (G w h) y.
Let y be given.
Assume Hy: y ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is IH (SNoLev z) Hz1 y Hy.
An exact proof term for the current goal is L1a (ordsucc (SNoLev z)) (ordinal_ordsucc (SNoLev z) (SNoLev_ordinal z Hz)) z (SNoS_SNoLev z Hz).
Assume Hz: ¬ SNo z.
rewrite the current goal using If_i_0 (SNo z) (SNo_rec_i (G w h) z) Empty Hz (from left to right).
An exact proof term for the current goal is If_i_0 (SNo z) (SNo_rec_i (G w g) z) Empty Hz.
We prove the intermediate claim L2: H w (SNo_rec_ii H) z = F w z SNo_rec2.
We will prove (if SNo z then SNo_rec_i (G w (SNo_rec_ii H)) z else Empty) = F w z SNo_rec2.
rewrite the current goal using If_i_1 (SNo z) (SNo_rec_i (G w (SNo_rec_ii H)) z) Empty Hz (from left to right).
We will prove SNo_rec_i (G w (SNo_rec_ii H)) z = F w z SNo_rec2.
We will prove SNo_rec_i (G w (SNo_rec_ii H)) z = F w z (SNo_rec_ii H).
rewrite the current goal using SNo_rec2_eq_1 w Hw (SNo_rec_ii H) z Hz (from left to right).
We will prove G w (SNo_rec_ii H) z (SNo_rec_i (G w (SNo_rec_ii H))) = F w z (SNo_rec_ii H).
We will prove F w z (λx y ⇒ if x = w then SNo_rec_i (G w (SNo_rec_ii H)) y else SNo_rec_ii H x y) = F w z (SNo_rec_ii H).
We prove the intermediate claim L2a: ∀x ∈ SNoS_ (SNoLev w), ∀y, SNo y → (if x = w then SNo_rec_i (G w (SNo_rec_ii H)) y else SNo_rec_ii H x y) = SNo_rec_ii H x y.
Let x be given.
Assume Hx: x ∈ SNoS_ (SNoLev w).
Let y be given.
Assume Hy: SNo y.
We prove the intermediate claim Lxw: x ≠ w.
An exact proof term for the current goal is SNoS_In_neq w Hw x Hx.
An exact proof term for the current goal is If_i_0 (x = w) (SNo_rec_i (G w (SNo_rec_ii H)) y) (SNo_rec_ii H x y) Lxw.
We prove the intermediate claim L2b: ∀y ∈ SNoS_ (SNoLev z), (if w = w then SNo_rec_i (G w (SNo_rec_ii H)) y else SNo_rec_ii H w y) = SNo_rec_ii H w y.
Let y be given.
Assume Hy: y ∈ SNoS_ (SNoLev z).
rewrite the current goal using If_i_1 (w = w) (SNo_rec_i (G w (SNo_rec_ii H)) y) (SNo_rec_ii H w y) (λq H ⇒ H) (from left to right).
We prove the intermediate claim Ly: SNo y.
Apply SNoS_E2 (SNoLev z) (SNoLev_ordinal z Hz) y Hy to the current goal.
Assume Hy1: SNoLev y ∈ SNoLev z.
Assume Hy2: ordinal (SNoLev y).
Assume Hy3: SNo y.
Assume Hy4: SNo_ (SNoLev y) y.
An exact proof term for the current goal is Hy3.
We will prove SNo_rec_i (G w (SNo_rec_ii H)) y = SNo_rec_ii H w y.
Apply eq_i_tra (SNo_rec_i (G w (SNo_rec_ii H)) y) (H w (SNo_rec_ii H) y) (SNo_rec_ii H w y) to the current goal.
We will prove SNo_rec_i (G w (SNo_rec_ii H)) y = H w (SNo_rec_ii H) y.
We will prove SNo_rec_i (G w (SNo_rec_ii H)) y = if SNo y then (SNo_rec_i (G w (SNo_rec_ii H)) y) else Empty.
Use symmetry.
An exact proof term for the current goal is If_i_1 (SNo y) (SNo_rec_i (G w (SNo_rec_ii H)) y) Empty Ly.
We will prove H w (SNo_rec_ii H) y = SNo_rec_ii H w y.
rewrite the current goal using SNo_rec_ii_eq H L1 w Hw (from left to right).
We will prove H w (SNo_rec_ii H) y = H w (SNo_rec_ii H) y.
An exact proof term for the current goal is (λq H ⇒ H).
An exact proof term for the current goal is Fr w Hw z Hz (λx y ⇒ if x = w then SNo_rec_i (G w (SNo_rec_ii H)) y else SNo_rec_ii H x y) (SNo_rec_ii H) L2a L2b.
We will prove SNo_rec_ii H w z = F w z SNo_rec2.
rewrite the current goal using SNo_rec_ii_eq H L1 w Hw (from left to right).
We will prove H w (SNo_rec_ii H) z = F w z SNo_rec2.
An exact proof term for the current goal is L2.
∎
End of Section SurrealRec2
Theorem. (SNo_ordinal_ind) The following is provable:
∀P : set → prop, (∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, P x) → (∀x, SNo x → P x)
In Proofgold the corresponding term root is 5b75fb... and proposition id is dc2c6d...
Proof:
Let P be given.
Assume H1.
Let x be given.
Assume Hx: SNo x.
We prove the intermediate claim LLx: ordinal (SNoLev x).
An exact proof term for the current goal is SNoLev_ordinal x Hx.
We prove the intermediate claim LsLx: ordinal (ordsucc (SNoLev x)).
An exact proof term for the current goal is ordinal_ordsucc (SNoLev x) LLx.
We prove the intermediate claim LxsLx: x ∈ SNoS_ (ordsucc (SNoLev x)).
An exact proof term for the current goal is SNoS_SNoLev x Hx.
An exact proof term for the current goal is H1 (ordsucc (SNoLev x)) LsLx x LxsLx.
∎
Theorem. (SNo_ordinal_ind2) The following is provable:
∀P : set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, P x y) → (∀x y, SNo x → SNo y → P x y)
In Proofgold the corresponding term root is ba3315... and proposition id is 4cd5ac...
Proof:
Let P be given.
Assume H1.
Let x and y be given.
Assume Hx: SNo x.
Assume Hy: SNo y.
We prove the intermediate claim LLx: ordinal (SNoLev x).
An exact proof term for the current goal is SNoLev_ordinal x Hx.
We prove the intermediate claim LsLx: ordinal (ordsucc (SNoLev x)).
An exact proof term for the current goal is ordinal_ordsucc (SNoLev x) LLx.
We prove the intermediate claim LxsLx: x ∈ SNoS_ (ordsucc (SNoLev x)).
An exact proof term for the current goal is SNoS_SNoLev x Hx.
We prove the intermediate claim LLy: ordinal (SNoLev y).
An exact proof term for the current goal is SNoLev_ordinal y Hy.
We prove the intermediate claim LsLy: ordinal (ordsucc (SNoLev y)).
An exact proof term for the current goal is ordinal_ordsucc (SNoLev y) LLy.
We prove the intermediate claim LysLy: y ∈ SNoS_ (ordsucc (SNoLev y)).
An exact proof term for the current goal is SNoS_SNoLev y Hy.
An exact proof term for the current goal is H1 (ordsucc (SNoLev x)) LsLx (ordsucc (SNoLev y)) LsLy x LxsLx y LysLy.
∎
Theorem. (SNo_ordinal_ind3) The following is provable:
∀P : set → set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma, ordinal gamma → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ gamma, P x y z) → (∀x y z, SNo x → SNo y → SNo z → P x y z)
In Proofgold the corresponding term root is fccdb6... and proposition id is dece8c...
Proof:
Let P be given.
Assume H1.
Let x, y and z be given.
Assume Hx: SNo x.
Assume Hy: SNo y.
Assume Hz: SNo z.
We prove the intermediate claim LLx: ordinal (SNoLev x).
An exact proof term for the current goal is SNoLev_ordinal x Hx.
We prove the intermediate claim LsLx: ordinal (ordsucc (SNoLev x)).
An exact proof term for the current goal is ordinal_ordsucc (SNoLev x) LLx.
We prove the intermediate claim LxsLx: x ∈ SNoS_ (ordsucc (SNoLev x)).
An exact proof term for the current goal is SNoS_SNoLev x Hx.
We prove the intermediate claim LLy: ordinal (SNoLev y).
An exact proof term for the current goal is SNoLev_ordinal y Hy.
We prove the intermediate claim LsLy: ordinal (ordsucc (SNoLev y)).
An exact proof term for the current goal is ordinal_ordsucc (SNoLev y) LLy.
We prove the intermediate claim LysLy: y ∈ SNoS_ (ordsucc (SNoLev y)).
An exact proof term for the current goal is SNoS_SNoLev y Hy.
We prove the intermediate claim LLz: ordinal (SNoLev z).
An exact proof term for the current goal is SNoLev_ordinal z Hz.
We prove the intermediate claim LsLz: ordinal (ordsucc (SNoLev z)).
An exact proof term for the current goal is ordinal_ordsucc (SNoLev z) LLz.
We prove the intermediate claim LzsLz: z ∈ SNoS_ (ordsucc (SNoLev z)).
An exact proof term for the current goal is SNoS_SNoLev z Hz.
An exact proof term for the current goal is H1 (ordsucc (SNoLev x)) LsLx (ordsucc (SNoLev y)) LsLy (ordsucc (SNoLev z)) LsLz x LxsLx y LysLy z LzsLz.
∎
Theorem. (SNoLev_ind) The following is provable:
∀P : set → prop, (∀x, SNo x → (∀w ∈ SNoS_ (SNoLev x), P w) → P x) → (∀x, SNo x → P x)
In Proofgold the corresponding term root is 8b2b42... and proposition id is 851a88...
Proof:
Let P be given.
Assume H1: ∀x, SNo x → (∀w ∈ SNoS_ (SNoLev x), P w) → P x.
We prove the intermediate claim L1: ∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, P x.
Apply ordinal_ind to the current goal.
Let alpha be given.
Assume Ha: ordinal alpha.
Assume IH: ∀beta ∈ alpha, ∀x ∈ SNoS_ beta, P x.
Let x be given.
Assume Hx: x ∈ SNoS_ alpha.
Apply SNoS_E2 alpha Ha x Hx to the current goal.
Assume Hx1: SNoLev x ∈ alpha.
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
Apply H1 x Hx3 to the current goal.
We will prove ∀w ∈ SNoS_ (SNoLev x), P w.
An exact proof term for the current goal is IH (SNoLev x) Hx1.
An exact proof term for the current goal is SNo_ordinal_ind P L1.
∎
Theorem. (SNoLev_ind2) The following is provable:
∀P : set → set → prop, (∀x y, 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) → ∀x y, SNo x → SNo y → P x y
In Proofgold the corresponding term root is 87eb9a... and proposition id is 800d4c...
Proof:
Let P be given.
Assume H1: ∀x y, 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.
We prove the intermediate claim L1: ∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, P x y.
Apply ordinal_ind to the current goal.
Let alpha be given.
Assume Ha: ordinal alpha.
Assume IHa: ∀gamma ∈ alpha, ∀beta, ordinal beta → ∀x ∈ SNoS_ gamma, ∀y ∈ SNoS_ beta, P x y.
Apply ordinal_ind to the current goal.
Let beta be given.
Assume Hb: ordinal beta.
Assume IHb: ∀delta ∈ beta, ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ delta, P x y.
Let x be given.
Assume Hx: x ∈ SNoS_ alpha.
Let y be given.
Assume Hy: y ∈ SNoS_ beta.
Apply SNoS_E2 alpha Ha x Hx to the current goal.
Assume Hx1: SNoLev x ∈ alpha.
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
Apply SNoS_E2 beta Hb y Hy to the current goal.
Assume Hy1: SNoLev y ∈ beta.
Assume Hy2: ordinal (SNoLev y).
Assume Hy3: SNo y.
Assume Hy4: SNo_ (SNoLev y) y.
Apply H1 x y Hx3 Hy3 to the current goal.
We will prove ∀w ∈ SNoS_ (SNoLev x), P w y.
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev x).
An exact proof term for the current goal is IHa (SNoLev x) Hx1 beta Hb w Hw y Hy.
We will prove ∀z ∈ SNoS_ (SNoLev y), P x z.
An exact proof term for the current goal is IHb (SNoLev y) Hy1 x Hx.
We will prove ∀w ∈ SNoS_ (SNoLev x), ∀z ∈ SNoS_ (SNoLev y), P w z.
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev x).
Let z be given.
Assume Hz: z ∈ SNoS_ (SNoLev y).
An exact proof term for the current goal is IHa (SNoLev x) Hx1 (SNoLev y) Hy2 w Hw z Hz.
An exact proof term for the current goal is SNo_ordinal_ind2 P L1.
∎
Theorem. (SNoLev_ind3) The following is provable:
∀P : set → set → set → prop, (∀x y z, 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) → ∀x y z, SNo x → SNo y → SNo z → P x y z
In Proofgold the corresponding term root is f1d779... and proposition id is ee16b9...
Proof:
Let P be given.
Assume H1: ∀x y z, 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.
We prove the intermediate claim L1: ∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma, ordinal gamma → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ gamma, P x y z.
Apply ordinal_ind to the current goal.
Let alpha be given.
Assume Ha: ordinal alpha.
Assume IHa: ∀delta ∈ alpha, ∀beta, ordinal beta → ∀gamma, ordinal gamma → ∀x ∈ SNoS_ delta, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ gamma, P x y z.
Apply ordinal_ind to the current goal.
Let beta be given.
Assume Hb: ordinal beta.
Assume IHb: ∀delta ∈ beta, ∀gamma, ordinal gamma → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ delta, ∀z ∈ SNoS_ gamma, P x y z.
Apply ordinal_ind to the current goal.
Let gamma be given.
Assume Hc: ordinal gamma.
Assume IHc: ∀delta ∈ gamma, ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ delta, P x y z.
Let x be given.
Assume Hx: x ∈ SNoS_ alpha.
Let y be given.
Assume Hy: y ∈ SNoS_ beta.
Let z be given.
Assume Hz: z ∈ SNoS_ gamma.
Apply SNoS_E2 alpha Ha x Hx to the current goal.
Assume Hx1: SNoLev x ∈ alpha.
Assume Hx2: ordinal (SNoLev x).
Assume Hx3: SNo x.
Assume Hx4: SNo_ (SNoLev x) x.
Apply SNoS_E2 beta Hb y Hy to the current goal.
Assume Hy1: SNoLev y ∈ beta.
Assume Hy2: ordinal (SNoLev y).
Assume Hy3: SNo y.
Assume Hy4: SNo_ (SNoLev y) y.
Apply SNoS_E2 gamma Hc z Hz to the current goal.
Assume Hz1: SNoLev z ∈ gamma.
Assume Hz2: ordinal (SNoLev z).
Assume Hz3: SNo z.
Assume Hz4: SNo_ (SNoLev z) z.
Apply H1 x y z Hx3 Hy3 Hz3 to the current goal.
We will prove ∀u ∈ SNoS_ (SNoLev x), P u y z.
Let u be given.
Assume Hu: u ∈ SNoS_ (SNoLev x).
An exact proof term for the current goal is IHa (SNoLev x) Hx1 beta Hb gamma Hc u Hu y Hy z Hz.
We will prove ∀v ∈ SNoS_ (SNoLev y), P x v z.
Let v be given.
Assume Hv: v ∈ SNoS_ (SNoLev y).
An exact proof term for the current goal is IHb (SNoLev y) Hy1 gamma Hc x Hx v Hv z Hz.
We will prove ∀w ∈ SNoS_ (SNoLev z), P x y w.
An exact proof term for the current goal is IHc (SNoLev z) Hz1 x Hx y Hy.
We will prove ∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), P u v z.
Let u be given.
Assume Hu: u ∈ SNoS_ (SNoLev x).
Let v be given.
Assume Hv: v ∈ SNoS_ (SNoLev y).
An exact proof term for the current goal is IHa (SNoLev x) Hx1 (SNoLev y) Hy2 gamma Hc u Hu v Hv z Hz.
We will prove ∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), P u y w.
Let u be given.
Assume Hu: u ∈ SNoS_ (SNoLev x).
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is IHa (SNoLev x) Hx1 beta Hb (SNoLev z) Hz2 u Hu y Hy w Hw.
We will prove ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P x v w.
Let v be given.
Assume Hv: v ∈ SNoS_ (SNoLev y).
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is IHb (SNoLev y) Hy1 (SNoLev z) Hz2 x Hx v Hv w Hw.
We will prove ∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P u v w.
Let u be given.
Assume Hu: u ∈ SNoS_ (SNoLev x).
Let v be given.
Assume Hv: v ∈ SNoS_ (SNoLev y).
Let w be given.
Assume Hw: w ∈ SNoS_ (SNoLev z).
An exact proof term for the current goal is IHa (SNoLev x) Hx1 (SNoLev y) Hy2 (SNoLev z) Hz2 u Hu v Hv w Hw.
An exact proof term for the current goal is SNo_ordinal_ind3 P L1.
∎
Theorem. (SNo_omega) The following is provable:
SNo ω
In Proofgold the corresponding term root is f65848... and proposition id is db7b5f...
Proof:
An exact proof term for the current goal is ordinal_SNo ω omega_ordinal.
∎
Theorem. (SNoLt_0_1) The following is provable:
0 < 1
In Proofgold the corresponding term root is dd6047... and proposition id is cb6392...
Proof:
Apply ordinal_In_SNoLt 1 ordinal_1 0 to the current goal.
We will prove 0 ∈ 1.
An exact proof term for the current goal is In_0_1.
∎
Theorem. (SNoLt_0_2) The following is provable:
0 < 2
In Proofgold the corresponding term root is 7be36f... and proposition id is f46934...
Proof:
Apply ordinal_In_SNoLt 2 ordinal_2 0 to the current goal.
We will prove 0 ∈ 2.
An exact proof term for the current goal is In_0_2.
∎
Theorem. (SNoLt_1_2) The following is provable:
1 < 2
In Proofgold the corresponding term root is cc44e8... and proposition id is 19d5e1...
Proof:
Apply ordinal_In_SNoLt 2 ordinal_2 1 to the current goal.
We will prove 1 ∈ 2.
An exact proof term for the current goal is In_1_2.
∎
Theorem. (restr_SNo_) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo_ alpha (x ∩ SNoElts_ alpha)
In Proofgold the corresponding term root is d78444... and proposition id is e0eca8...
Proof:
Let x be given.
Assume Hx.
Let alpha be given.
Assume Ha.
We prove the intermediate claim Lx1: SNo_ (SNoLev x) x.
An exact proof term for the current goal is SNoLev_ x Hx.
Apply Lx1 to the current goal.
Assume Hx1: x ⊆ SNoElts_ (SNoLev x).
Assume Hx2: ∀beta ∈ SNoLev x, exactly1of2 (SetAdjoin beta {1} ∈ x) (beta ∈ x).
We will prove x ∩ SNoElts_ alpha ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (SetAdjoin beta {1} ∈ x ∩ SNoElts_ alpha) (beta ∈ x ∩ SNoElts_ alpha).
Apply andI to the current goal.
An exact proof term for the current goal is binintersect_Subq_2 x (SNoElts_ alpha).
Let beta be given.
Assume Hb: beta ∈ alpha.
We prove the intermediate claim Lb: beta ∈ SNoLev x.
An exact proof term for the current goal is ordinal_TransSet (SNoLev x) (SNoLev_ordinal x Hx) alpha Ha beta Hb.
Apply exactly1of2_E (SetAdjoin beta {1} ∈ x) (beta ∈ x) (Hx2 beta Lb) to the current goal.
Assume H1: SetAdjoin beta {1} ∈ x.
Assume H2: beta ∉ x.
Apply exactly1of2_I1 to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is H1.
We will prove SetAdjoin beta {1} ∈ SNoElts_ alpha.
We will prove SetAdjoin beta {1} ∈ alpha ∪ {SetAdjoin beta {1}|beta ∈ alpha}.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is ReplI alpha (λbeta ⇒ SetAdjoin beta {1}) beta Hb.
We will prove beta ∉ x ∩ SNoElts_ alpha.
Assume H3.
An exact proof term for the current goal is H2 (binintersectE1 x (SNoElts_ alpha) beta H3).
Assume H1: SetAdjoin beta {1} ∉ x.
Assume H2: beta ∈ x.
Apply exactly1of2_I2 to the current goal.
Assume H3: SetAdjoin beta {1} ∈ x ∩ SNoElts_ alpha.
An exact proof term for the current goal is H1 (binintersectE1 x (SNoElts_ alpha) (SetAdjoin beta {1}) H3).
Apply binintersectI to the current goal.
An exact proof term for the current goal is H2.
We will prove beta ∈ SNoElts_ alpha.
We will prove beta ∈ alpha ∪ {SetAdjoin beta {1}|beta ∈ alpha}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is Hb.
∎
Theorem. (restr_SNo) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo (x ∩ SNoElts_ alpha)
In Proofgold the corresponding term root is 930b56... and proposition id is d141ed...
Proof:
Let x be given.
Assume Hx.
Let alpha be given.
Assume Ha.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is ordinal_Hered (SNoLev x) (SNoLev_ordinal x Hx) alpha Ha.
We prove the intermediate claim L1: SNo_ alpha (x ∩ SNoElts_ alpha).
An exact proof term for the current goal is restr_SNo_ x Hx alpha Ha.
An exact proof term for the current goal is SNo_SNo alpha La (x ∩ SNoElts_ alpha) L1.
∎
Theorem. (restr_SNoLev) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoLev (x ∩ SNoElts_ alpha) = alpha
In Proofgold the corresponding term root is cae8b5... and proposition id is 8b1853...
Proof:
Let x be given.
Assume Hx.
Let alpha be given.
Assume Ha.
We prove the intermediate claim La: ordinal alpha.
An exact proof term for the current goal is ordinal_Hered (SNoLev x) (SNoLev_ordinal x Hx) alpha Ha.
We prove the intermediate claim L1: SNo_ alpha (x ∩ SNoElts_ alpha).
An exact proof term for the current goal is restr_SNo_ x Hx alpha Ha.
An exact proof term for the current goal is SNoLev_uniq2 alpha La (x ∩ SNoElts_ alpha) L1.
∎
Theorem. (restr_SNoEq) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoEq_ alpha (x ∩ SNoElts_ alpha) x
In Proofgold the corresponding term root is 829f8b... and proposition id is 704f5a...
Proof:
Let x be given.
Assume Hx.
Let alpha be given.
Assume Ha.
Apply SNoEq_I to the current goal.
Let beta be given.
Assume Hb: beta ∈ alpha.
We will prove beta ∈ x ∩ SNoElts_ alpha ↔ beta ∈ x.
Apply iffI to the current goal.
Assume H1: beta ∈ x ∩ SNoElts_ alpha.
An exact proof term for the current goal is binintersectE1 x (SNoElts_ alpha) beta H1.
Assume H1: beta ∈ x.
Apply binintersectI to the current goal.
An exact proof term for the current goal is H1.
We will prove beta ∈ alpha ∪ {SetAdjoin beta {1}|beta ∈ alpha}.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is Hb.
∎
Theorem. (SNo_extend0_restr_eq) The following is provable:
∀x, SNo x → x = SNo_extend0 x ∩ SNoElts_ (SNoLev x)
In Proofgold the corresponding term root is 2c202e... and proposition id is 9827a9...
Proof:
Let x be given.
Assume Hx.
We prove the intermediate claim L1: SNo (SNo_extend0 x).
An exact proof term for the current goal is SNo_extend0_SNo x Hx.
We prove the intermediate claim L2: SNoLev x ∈ SNoLev (SNo_extend0 x).
rewrite the current goal using SNo_extend0_SNoLev x Hx (from left to right).
Apply ordsuccI2 to the current goal.
Apply SNo_eq to the current goal.
An exact proof term for the current goal is Hx.
Apply restr_SNo to the current goal.
An exact proof term for the current goal is L1.
An exact proof term for the current goal is L2.
We will prove SNoLev x = SNoLev (SNo_extend0 x ∩ SNoElts_ (SNoLev x)).
Use symmetry.
Apply restr_SNoLev to the current goal.
An exact proof term for the current goal is L1.
An exact proof term for the current goal is L2.
We will prove SNoEq_ (SNoLev x) x (SNo_extend0 x ∩ SNoElts_ (SNoLev x)).
Apply SNoEq_sym_ to the current goal.
Apply SNoEq_tra_ (SNoLev x) (SNo_extend0 x ∩ SNoElts_ (SNoLev x)) (SNo_extend0 x) x to the current goal.
We will prove SNoEq_ (SNoLev x) (SNo_extend0 x ∩ SNoElts_ (SNoLev x)) (SNo_extend0 x).
An exact proof term for the current goal is restr_SNoEq (SNo_extend0 x) L1 (SNoLev x) L2.
We will prove SNoEq_ (SNoLev x) (SNo_extend0 x) x.
An exact proof term for the current goal is SNo_extend0_SNoEq x Hx.
∎
Theorem. (SNo_extend1_restr_eq) The following is provable:
∀x, SNo x → x = SNo_extend1 x ∩ SNoElts_ (SNoLev x)
In Proofgold the corresponding term root is 75bbe8... and proposition id is 275196...
Proof:
Let x be given.
Assume Hx.
We prove the intermediate claim L1: SNo (SNo_extend1 x).
An exact proof term for the current goal is SNo_extend1_SNo x Hx.
We prove the intermediate claim L2: SNoLev x ∈ SNoLev (SNo_extend1 x).
rewrite the current goal using SNo_extend1_SNoLev x Hx (from left to right).
Apply ordsuccI2 to the current goal.
Apply SNo_eq to the current goal.
An exact proof term for the current goal is Hx.
Apply restr_SNo to the current goal.
An exact proof term for the current goal is L1.
An exact proof term for the current goal is L2.
We will prove SNoLev x = SNoLev (SNo_extend1 x ∩ SNoElts_ (SNoLev x)).
Use symmetry.
Apply restr_SNoLev to the current goal.
An exact proof term for the current goal is L1.
An exact proof term for the current goal is L2.
We will prove SNoEq_ (SNoLev x) x (SNo_extend1 x ∩ SNoElts_ (SNoLev x)).
Apply SNoEq_sym_ to the current goal.
Apply SNoEq_tra_ (SNoLev x) (SNo_extend1 x ∩ SNoElts_ (SNoLev x)) (SNo_extend1 x) x to the current goal.
We will prove SNoEq_ (SNoLev x) (SNo_extend1 x ∩ SNoElts_ (SNoLev x)) (SNo_extend1 x).
An exact proof term for the current goal is restr_SNoEq (SNo_extend1 x) L1 (SNoLev x) L2.
We will prove SNoEq_ (SNoLev x) (SNo_extend1 x) x.
An exact proof term for the current goal is SNo_extend1_SNoEq x Hx.
∎