Surreal Numbers as Sets

From Parts 1 and 2

Object. 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)
Definition. We define True to be ∀p : prop, p → p of type prop.
Definition. We define False to be ∀p : prop, p of type prop.
Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.
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.
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.
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.
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
Object. 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.
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
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
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.
Object. The name Empty is a term of type set.
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x ∈ Empty
Object. 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
Object. 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
Object. 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
Definition. We define TransSet to be λU : set ⇒ ∀x ∈ U, x ⊆ U of type set → prop.
Definition. We define Union_closed to be λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type set → prop.
Definition. We define Power_closed to be λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type set → prop.
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.
Definition. We define ZF_closed to be λU : set ⇒ Union_closed U ∧ Power_closed U ∧ Repl_closed U of type set → prop.
Object. 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
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U
Axiom. (FalseE) We take the following as an axiom:
False → ∀p : prop, p
Axiom. (TrueI) We take the following as an axiom:
True
Axiom. (andI) We take the following as an axiom:
∀A B : prop, A → B → A ∧ B
Axiom. (andEL) We take the following as an axiom:
∀A B : prop, A ∧ B → A
Axiom. (andER) We take the following as an axiom:
∀A B : prop, A ∧ B → B
Axiom. (orIL) We take the following as an axiom:
∀A B : prop, A → A ∨ B
Axiom. (orIR) We take the following as an axiom:
∀A B : prop, B → A ∨ B
Beginning of Section PropN
Variable P1 P2 P3 : prop
Axiom. (and3I) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
Axiom. (and3E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
Axiom. (or3I1) We take the following as an axiom:
P1 → P1 ∨ P2 ∨ P3
Axiom. (or3I2) We take the following as an axiom:
P2 → P1 ∨ P2 ∨ P3
Axiom. (or3I3) We take the following as an axiom:
P3 → P1 ∨ P2 ∨ P3
Axiom. (or3E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
Variable P4 : prop
Axiom. (and4I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
Variable P5 : prop
Axiom. (and5I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
End of Section PropN
Axiom. (not_or_and_demorgan) We take the following as an axiom:
∀A B : prop, ¬ (A ∨ B) → ¬ A ∧ ¬ B
Axiom. (not_ex_all_demorgan_i) We take the following as an axiom:
∀P : set → prop, (¬ ∃x, P x) → ∀x, ¬ P x
Axiom. (iffI) We take the following as an axiom:
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
Axiom. (iffEL) We take the following as an axiom:
∀A B : prop, (A ↔ B) → A → B
Axiom. (iffER) We take the following as an axiom:
∀A B : prop, (A ↔ B) → B → A
Axiom. (iff_refl) We take the following as an axiom:
∀A : prop, A ↔ A
Axiom. (iff_sym) We take the following as an axiom:
∀A B : prop, (A ↔ B) → (B ↔ A)
Axiom. (iff_trans) We take the following as an axiom:
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
Axiom. (eq_i_tra) We take the following as an axiom:
∀x y z, x = y → y = z → x = z
Axiom. (f_eq_i) We take the following as an axiom:
∀f : set → set, ∀x y, x = y → f x = f y
Axiom. (neq_i_sym) We take the following as an axiom:
∀x y, x ≠ y → y ≠ x
Definition. We define nIn to be λx X ⇒ ¬ In x X of type set → set → prop.
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)
Axiom. (pred_ext) We take the following as an axiom:
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
Axiom. (prop_ext_2) We take the following as an axiom:
∀p q : prop, (p → q) → (q → p) → p = q
Axiom. (Subq_ref) We take the following as an axiom:
∀X : set, X ⊆ X
Axiom. (Subq_tra) We take the following as an axiom:
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
Axiom. (Subq_contra) We take the following as an axiom:
∀X Y z : set, X ⊆ Y → z ∉ Y → z ∉ X
Axiom. (EmptyE) We take the following as an axiom:
∀x : set, x ∉ Empty
Axiom. (Subq_Empty) We take the following as an axiom:
∀X : set, Empty ⊆ X
Axiom. (Empty_Subq_eq) We take the following as an axiom:
∀X : set, X ⊆ Empty → X = Empty
Axiom. (Empty_eq) We take the following as an axiom:
∀X : set, (∀x, x ∉ X) → X = Empty
Axiom. (UnionI) We take the following as an axiom:
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
Axiom. (UnionE) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
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
Axiom. (PowerI) We take the following as an axiom:
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
Axiom. (PowerE) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
Axiom. (Empty_In_Power) We take the following as an axiom:
∀X : set, Empty ∈ 𝒫 X
Axiom. (Self_In_Power) We take the following as an axiom:
∀X : set, X ∈ 𝒫 X
Axiom. (xm) We take the following as an axiom:
∀P : prop, P ∨ ¬ P
Axiom. (dneg) We take the following as an axiom:
∀P : prop, ¬ ¬ P → P
Axiom. (not_all_ex_demorgan_i) We take the following as an axiom:
∀P : set → prop, ¬ (∀x, P x) → ∃x, ¬ P x
Axiom. (eq_or_nand) We take the following as an axiom:
or = (λx y : prop ⇒ ¬ (¬ x ∧ ¬ y))
Object. The name exactly1of2 is a term of type prop → prop → prop.
Axiom. (exactly1of2_I1) We take the following as an axiom:
∀A B : prop, A → ¬ B → exactly1of2 A B
Axiom. (exactly1of2_I2) We take the following as an axiom:
∀A B : prop, ¬ A → B → exactly1of2 A B
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
Axiom. (exactly1of2_or) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → A ∨ B
Axiom. (ReplI) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
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
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
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
Axiom. (Repl_Empty) We take the following as an axiom:
∀F : set → set, {F x|x ∈ Empty} = Empty
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}
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}
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
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
Object. The name If_i is a term of type prop → set → set → set.
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
Axiom. (If_i_0) We take the following as an axiom:
∀p : prop, ∀x y : set, ¬ p → (if p then x else y) = y
Axiom. (If_i_1) We take the following as an axiom:
∀p : prop, ∀x y : set, p → (if p then x else y) = x
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
Object. The name UPair is a term of type set → set → set.
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
Axiom. (UPairI1) We take the following as an axiom:
∀y z : set, y ∈ {y,z}
Axiom. (UPairI2) We take the following as an axiom:
∀y z : set, z ∈ {y,z}
Object. The name Sing is a term of type set → set.
Notation. {x} is notation for Sing x.
Axiom. (SingI) We take the following as an axiom:
∀x : set, x ∈ {x}
Axiom. (SingE) We take the following as an axiom:
∀x y : set, y ∈ {x} → y = x
Object. The name binunion is a term of type set → set → set.
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
Axiom. (binunionI2) We take the following as an axiom:
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
Axiom. (binunionE) We take the following as an axiom:
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
Axiom. (binunionE') We take the following as an axiom:
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
Axiom. (binunion_asso) We take the following as an axiom:
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
Axiom. (binunion_com_Subq) We take the following as an axiom:
∀X Y : set, X ∪ Y ⊆ Y ∪ X
Axiom. (binunion_com) We take the following as an axiom:
∀X Y : set, X ∪ Y = Y ∪ X
Axiom. (binunion_idl) We take the following as an axiom:
∀X : set, Empty ∪ X = X
Axiom. (binunion_idr) We take the following as an axiom:
∀X : set, X ∪ Empty = X
Axiom. (binunion_Subq_1) We take the following as an axiom:
∀X Y : set, X ⊆ X ∪ Y
Axiom. (binunion_Subq_2) We take the following as an axiom:
∀X Y : set, Y ⊆ X ∪ Y
Axiom. (binunion_Subq_min) We take the following as an axiom:
∀X Y Z : set, X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z
Axiom. (Subq_binunion_eq) We take the following as an axiom:
∀X Y, (X ⊆ Y) = (X ∪ Y = Y)
Definition. We define SetAdjoin to be λX y ⇒ X ∪ {y} of type set → set → set.
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.
Object. The name famunion is a term of type set → (set → set) → set.
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
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
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
Axiom. (famunion_Empty) We take the following as an axiom:
∀F : set → set, (⋃x ∈ EmptyF x) = Empty
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
Object. The name Sep is a term of type set.
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}
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
Axiom. (SepE1) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
Axiom. (SepE2) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
Axiom. (Sep_Subq) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
Axiom. (Sep_In_Power) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
Object. The name ReplSep is a term of type set → (set → prop) → (set → set) → set.
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}
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
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
Object. The name binintersect is a term of type set → set → set.
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
Axiom. (binintersectE) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
Axiom. (binintersectE1) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X
Axiom. (binintersectE2) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ Y
Axiom. (binintersect_Subq_1) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ X
Axiom. (binintersect_Subq_2) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ Y
Axiom. (binintersect_Subq_eq_1) We take the following as an axiom:
∀X Y, X ⊆ Y → X ∩ Y = X
Axiom. (binintersect_Subq_max) We take the following as an axiom:
∀X Y Z : set, Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y
Axiom. (binintersect_com_Subq) We take the following as an axiom:
∀X Y : set, X ∩ Y ⊆ Y ∩ X
Axiom. (binintersect_com) We take the following as an axiom:
∀X Y : set, X ∩ Y = Y ∩ X
Object. The name setminus is a term of type set → set → set.
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
Axiom. (setminusE) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
Axiom. (setminusE1) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
Axiom. (setminus_Subq) We take the following as an axiom:
∀X Y : set, X ∖ Y ⊆ X
Axiom. (setminus_Subq_contra) We take the following as an axiom:
∀X Y Z : set, Z ⊆ Y → X ∖ Y ⊆ X ∖ Z
Axiom. (setminus_In_Power) We take the following as an axiom:
∀A U, A ∖ U ∈ 𝒫 A
Axiom. (In_irref) We take the following as an axiom:
∀x, x ∉ x
Axiom. (In_no2cycle) We take the following as an axiom:
∀x y, x ∈ y → y ∈ x → False
Object. The name ordsucc is a term of type set → set.
Axiom. (ordsuccI1) We take the following as an axiom:
∀x : set, x ⊆ ordsucc x
Axiom. (ordsuccI2) We take the following as an axiom:
∀x : set, x ∈ ordsucc x
Axiom. (ordsuccE) We take the following as an axiom:
∀x y : set, y ∈ ordsucc x → y ∈ x ∨ y = x
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
Axiom. (neq_ordsucc_0) We take the following as an axiom:
∀a : set, ordsucc a ≠ 0
Axiom. (ordsucc_inj) We take the following as an axiom:
∀a b : set, ordsucc a = ordsucc b → a = b
Axiom. (ordsucc_inj_contra) We take the following as an axiom:
∀a b : set, a ≠ b → ordsucc a ≠ ordsucc b
Axiom. (In_0_1) We take the following as an axiom:
0 ∈ 1
Axiom. (In_0_2) We take the following as an axiom:
0 ∈ 2
Axiom. (In_1_2) We take the following as an axiom:
1 ∈ 2
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.
Axiom. (nat_0) We take the following as an axiom:
nat_p 0
Axiom. (nat_ordsucc) We take the following as an axiom:
∀n : set, nat_p n → nat_p (ordsucc n)
Axiom. (nat_1) We take the following as an axiom:
nat_p 1
Axiom. (nat_2) We take the following as an axiom:
nat_p 2
Axiom. (nat_0_in_ordsucc) We take the following as an axiom:
∀n, nat_p n → 0 ∈ ordsucc n
Axiom. (nat_ordsucc_in_ordsucc) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, ordsucc m ∈ ordsucc n
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
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
Axiom. (nat_inv) We take the following as an axiom:
∀n, nat_p n → n = 0 ∨ ∃x, nat_p x ∧ n = ordsucc x
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
Axiom. (nat_p_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, nat_p m
Axiom. (nat_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ n, m ⊆ n
Axiom. (nat_ordsucc_trans) We take the following as an axiom:
∀n, nat_p n → ∀m ∈ ordsucc n, m ⊆ n
Axiom. (Union_ordsucc_eq) We take the following as an axiom:
∀n, nat_p n → ⋃ (ordsucc n) = n
Axiom. (cases_1) We take the following as an axiom:
∀i ∈ 1, ∀p : set → prop, p 0 → p i
Axiom. (cases_2) We take the following as an axiom:
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
Axiom. (cases_3) We take the following as an axiom:
∀i ∈ 3, ∀p : set → prop, p 0 → p 1 → p 2 → p i
Axiom. (neq_0_1) We take the following as an axiom:
0 ≠ 1
Axiom. (neq_1_0) We take the following as an axiom:
1 ≠ 0
Axiom. (neq_0_2) We take the following as an axiom:
0 ≠ 2
Axiom. (neq_2_0) We take the following as an axiom:
2 ≠ 0
Axiom. (neq_1_2) We take the following as an axiom:
1 ≠ 2
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
Axiom. (ZF_Union_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, ⋃ X ∈ U
Axiom. (ZF_Power_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X ∈ U, 𝒫 X ∈ U
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
Axiom. (ZF_UPair_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x y ∈ U, {x,y} ∈ U
Axiom. (ZF_Sing_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x ∈ U, {x} ∈ U
Axiom. (ZF_binunion_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀X Y ∈ U, (X ∪ Y) ∈ U
Axiom. (ZF_ordsucc_closed) We take the following as an axiom:
∀U, ZF_closed U → ∀x ∈ U, ordsucc x ∈ U
Axiom. (nat_p_UnivOf_Empty) We take the following as an axiom:
∀n : set, nat_p n → n ∈ UnivOf Empty
Object. The name ω is a term of type set.
Axiom. (omega_nat_p) We take the following as an axiom:
∀n ∈ ω, nat_p n
Axiom. (nat_p_omega) We take the following as an axiom:
∀n : set, nat_p n → n ∈ ω
Axiom. (omega_ordsucc) We take the following as an axiom:
∀n ∈ ω, ordsucc n ∈ ω
Definition. We define ordinal to be λalpha : set ⇒ TransSet alpha ∧ ∀beta ∈ alpha, TransSet beta of type set → prop.
Axiom. (ordinal_TransSet) We take the following as an axiom:
∀alpha : set, ordinal alpha → TransSet alpha
Axiom. (ordinal_Empty) We take the following as an axiom:
ordinal Empty
Axiom. (ordinal_Hered) We take the following as an axiom:
∀alpha : set, ordinal alpha → ∀beta ∈ alpha, ordinal beta
Axiom. (TransSet_ordsucc) We take the following as an axiom:
∀X : set, TransSet X → TransSet (ordsucc X)
Axiom. (ordinal_ordsucc) We take the following as an axiom:
∀alpha : set, ordinal alpha → ordinal (ordsucc alpha)
Axiom. (nat_p_ordinal) We take the following as an axiom:
∀n : set, nat_p n → ordinal n
Axiom. (ordinal_1) We take the following as an axiom:
ordinal 1
Axiom. (ordinal_2) We take the following as an axiom:
ordinal 2
Axiom. (omega_TransSet) We take the following as an axiom:
TransSet ω
Axiom. (omega_ordinal) We take the following as an axiom:
ordinal ω
Axiom. (ordsucc_omega_ordinal) We take the following as an axiom:
ordinal (ordsucc ω)
Axiom. (TransSet_ordsucc_In_Subq) We take the following as an axiom:
∀X : set, TransSet X → ∀x ∈ X, ordsucc x ⊆ X
Axiom. (ordinal_ordsucc_In_Subq) We take the following as an axiom:
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ⊆ alpha
Axiom. (ordinal_trichotomy_or) We take the following as an axiom:
∀alpha beta : set, ordinal alpha → ordinal beta → alpha ∈ beta ∨ alpha = beta ∨ beta ∈ alpha
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
Axiom. (ordinal_In_Or_Subq) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → alpha ∈ beta ∨ beta ⊆ alpha
Axiom. (ordinal_linear) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta ∨ beta ⊆ alpha
Axiom. (ordinal_ordsucc_In_eq) We take the following as an axiom:
∀alpha beta, ordinal alpha → beta ∈ alpha → ordsucc beta ∈ alpha ∨ alpha = ordsucc beta
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)
Axiom. (ordinal_ordsucc_In) We take the following as an axiom:
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ∈ ordsucc alpha
Axiom. (ordinal_famunion) We take the following as an axiom:
∀X, ∀F : set → set, (∀x ∈ X, ordinal (F x)) → ordinal (⋃x ∈ XF x)
Axiom. (ordinal_binintersect) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∩ beta)
Axiom. (ordinal_binunion) We take the following as an axiom:
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∪ beta)
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
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
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.
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.
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
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
Object. The name inv is a term of type set → (set → set) → set → set.
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
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
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)
Axiom. (bij_id) We take the following as an axiom:
∀X, bij X X (λx ⇒ x)
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))
Definition. We define equip to be λX Y : set ⇒ ∃f : set → set, bij X Y f of type set → set → prop.
Axiom. (equip_ref) We take the following as an axiom:
∀X, equip X X
Axiom. (equip_sym) We take the following as an axiom:
∀X Y, equip X Y → equip Y X
Axiom. (equip_tra) We take the following as an axiom:
∀X Y Z, equip X Y → equip Y Z → equip X Z
Axiom. (equip_0_Empty) We take the following as an axiom:
∀X, equip X 0 → X = 0
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
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
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}
Axiom. (setminus_antimonotone) We take the following as an axiom:
∀A U V, U ⊆ V → A ∖ V ⊆ A ∖ U
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
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)
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
End of Section PigeonHole
Definition. We define finite to be λX ⇒ ∃n ∈ ω, equip X n of type set → prop.
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
Axiom. (finite_Empty) We take the following as an axiom:
finite 0
Axiom. (adjoin_finite) We take the following as an axiom:
∀X y, finite X → finite (X ∪ {y})
Axiom. (binunion_finite) We take the following as an axiom:
∀X, finite X → ∀Y, finite Y → finite (X ∪ Y)
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)
Axiom. (Subq_finite) We take the following as an axiom:
∀X, finite X → ∀Y, Y ⊆ X → finite Y
Axiom. (TransSet_In_ordsucc_Subq) We take the following as an axiom:
∀x y, TransSet y → x ∈ ordsucc y → x ⊆ y
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
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
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
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
Beginning of Section Descr_ii
Variable P : (set → set) → prop
Object. The name Descr_ii is a term of type set → set.
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
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (set → set → set) → prop
Object. The name Descr_iii is a term of type set → set → set.
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
End of Section Descr_iii
Beginning of Section Descr_Vo1
Variable P : Vo 1 → prop
Object. The name Descr_Vo1 is a term of type Vo 1.
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
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : set → set
Object. The name If_ii is a term of type set → set.
Axiom. (If_ii_1) We take the following as an axiom:
p → If_ii = f
Axiom. (If_ii_0) We take the following as an axiom:
¬ p → If_ii = g
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : set → set → set
Object. The name If_iii is a term of type set → set → set.
Axiom. (If_iii_1) We take the following as an axiom:
p → If_iii = f
Axiom. (If_iii_0) We take the following as an axiom:
¬ p → If_iii = g
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set → (set → set) → set
Object. The name In_rec_i is a term of type set → set.
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
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set → (set → (set → set)) → (set → set)
Object. The name In_rec_ii is a term of type set → (set → set).
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
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set → (set → (set → set → set)) → (set → set → set)
Object. The name In_rec_iii is a term of type set → (set → set → set).
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
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.
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
Axiom. (nat_primrec_0) We take the following as an axiom:
nat_primrec 0 = z
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)
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.
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
Axiom. (add_nat_SR) We take the following as an axiom:
∀n m : set, nat_p m → n + ordsucc m = ordsucc (n + m)
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)
Axiom. (add_nat_1_1_2) We take the following as an axiom:
1 + 1 = 2
Definition. We define mul_nat to be λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type set → set → set.
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
Axiom. (mul_nat_SR) We take the following as an axiom:
∀n m : set, nat_p m → n * ordsucc m = n + n * m
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)
End of Section NatArith
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} ∪ {f x|x ∈ X}) of type set → set.
Axiom. (Inj1_eq) We take the following as an axiom:
∀X : set, Inj1 X = {0} ∪ {Inj1 x|x ∈ X}
Axiom. (Inj1I1) We take the following as an axiom:
∀X : set, 0 ∈ Inj1 X
Axiom. (Inj1I2) We take the following as an axiom:
∀X x : set, x ∈ X → Inj1 x ∈ Inj1 X
Axiom. (Inj1E) We take the following as an axiom:
∀X y : set, y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X, y = Inj1 x
Axiom. (Inj1NE1) We take the following as an axiom:
∀x : set, Inj1 x ≠ 0
Axiom. (Inj1NE2) We take the following as an axiom:
∀x : set, Inj1 x ∉ {0}
Definition. We define Inj0 to be λX ⇒ {Inj1 x|x ∈ X} of type set → set.
Axiom. (Inj0I) We take the following as an axiom:
∀X x : set, x ∈ X → Inj1 x ∈ Inj0 X
Axiom. (Inj0E) We take the following as an axiom:
∀X y : set, y ∈ Inj0 X → ∃x : set, x ∈ X ∧ y = Inj1 x
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|x ∈ X ∖ {0}}) of type set → set.
Axiom. (Unj_eq) We take the following as an axiom:
∀X : set, Unj X = {Unj x|x ∈ X ∖ {0}}
Axiom. (Unj_Inj1_eq) We take the following as an axiom:
∀X : set, Unj (Inj1 X) = X
Axiom. (Inj1_inj) We take the following as an axiom:
∀X Y : set, Inj1 X = Inj1 Y → X = Y
Axiom. (Unj_Inj0_eq) We take the following as an axiom:
∀X : set, Unj (Inj0 X) = X
Axiom. (Inj0_inj) We take the following as an axiom:
∀X Y : set, Inj0 X = Inj0 Y → X = Y
Axiom. (Inj0_0) We take the following as an axiom:
Inj0 0 = 0
Axiom. (Inj0_Inj1_neq) We take the following as an axiom:
∀X Y : set, Inj0 X ≠ Inj1 Y
Definition. We define setsum to be λX Y ⇒ {Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y} of type set → set → set.
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
Axiom. (Inj1_setsum) We take the following as an axiom:
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
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)
Axiom. (Inj0_setsum_0L) We take the following as an axiom:
∀X : set, 0 + X = Inj0 X
Axiom. (Subq_1_Sing0) We take the following as an axiom:
1 ⊆ {0}
Axiom. (Inj1_setsum_1L) We take the following as an axiom:
∀X : set, 1 + X = Inj1 X
Axiom. (nat_setsum1_ordsucc) We take the following as an axiom:
∀n : set, nat_p n → 1 + n = ordsucc n
Axiom. (setsum_0_0) We take the following as an axiom:
0 + 0 = 0
Axiom. (setsum_1_0_1) We take the following as an axiom:
1 + 0 = 1
Axiom. (setsum_1_1_2) We take the following as an axiom:
1 + 1 = 2
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.
Definition. We define proj1 to be λZ ⇒ {Unj z|z ∈ Z, ∃y : set, Inj1 y = z} of type set → set.
Axiom. (Inj0_pair_0_eq) We take the following as an axiom:
Inj0 = pair 0
Axiom. (Inj1_pair_1_eq) We take the following as an axiom:
Inj1 = pair 1
Axiom. (pairI0) We take the following as an axiom:
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
Axiom. (pairI1) We take the following as an axiom:
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
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)
Axiom. (pairE0) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
Axiom. (pairE1) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
Axiom. (proj0I) We take the following as an axiom:
∀w u : set, pair 0 u ∈ w → u ∈ proj0 w
Axiom. (proj0E) We take the following as an axiom:
∀w u : set, u ∈ proj0 w → pair 0 u ∈ w
Axiom. (proj1I) We take the following as an axiom:
∀w u : set, pair 1 u ∈ w → u ∈ proj1 w
Axiom. (proj1E) We take the following as an axiom:
∀w u : set, u ∈ proj1 w → pair 1 u ∈ w
Axiom. (proj0_pair_eq) We take the following as an axiom:
∀X Y : set, proj0 (pair X Y) = X
Axiom. (proj1_pair_eq) We take the following as an axiom:
∀X Y : set, proj1 (pair X Y) = Y
Definition. We define Sigma to be λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type set → (set → set) → set.
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
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)
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
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
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)
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
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
Definition. We define setprod to be λX Y : set ⇒ ∑x ∈ X, Y of type set → set → set.
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.
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).
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
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
Axiom. (apI) We take the following as an axiom:
∀f x y, pair x y ∈ f → y ∈ f x
Axiom. (apE) We take the following as an axiom:
∀f x y, y ∈ f x → pair x y ∈ f
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
Axiom. (proj0_ap_0) We take the following as an axiom:
∀u, proj0 u = u 0
Axiom. (proj1_ap_1) We take the following as an axiom:
∀u, proj1 u = u 1
Axiom. (pair_ap_0) We take the following as an axiom:
∀x y : set, (pair x y) 0 = x
Axiom. (pair_ap_1) We take the following as an axiom:
∀x y : set, (pair x y) 1 = y
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
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))
Definition. We define pair_p to be λu : set ⇒ pair (u 0) (u 1) = u of type set → prop.
Axiom. (pair_p_I) We take the following as an axiom:
∀x y, pair_p (pair x y)
Axiom. (Subq_2_UPair01) We take the following as an axiom:
2 ⊆ {0,1}
Axiom. (tuple_pair) We take the following as an axiom:
∀x y : set, pair x y = (x,y)
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.
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
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)
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
Definition. We define setexp to be λX Y : set ⇒ ∏y ∈ Y, X of type set → set → set.
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))
Axiom. (lamI2) We take the following as an axiom:
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
Beginning of Section Tuples
Variable x0 x1 : set
Axiom. (tuple_2_0_eq) We take the following as an axiom:
(x0,x1) 0 = x0
Axiom. (tuple_2_1_eq) We take the following as an axiom:
(x0,x1) 1 = x1
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}
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
Axiom. (tuple_2_setprod) We take the following as an axiom:
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
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.
Object. The name DescrR_i_io_1 is a term of type (set → (set → prop) → prop) → set.
Object. The name DescrR_i_io_2 is a term of type (set → (set → prop) → prop) → set → prop.
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)
Definition. We define PNoEq_ to be λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type set → (set → prop) → (set → prop) → prop.
Axiom. (PNoEq_ref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
Axiom. (PNoEq_sym_) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
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
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
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.
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
Axiom. (PNoLt_irref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
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
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
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
Object. The name PNoLt is a term of type set → (set → prop) → set → (set → prop) → prop.
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
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
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
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
Axiom. (PNoLt_irref) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
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
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
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
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
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.
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
Axiom. (PNoLeI2) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
Axiom. (PNoLe_ref) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
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
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
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
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
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
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.
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.
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
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
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
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
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.
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)
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.
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.
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.
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
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
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
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
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
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
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.
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.
Axiom. (PNo_extend0_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
Axiom. (PNo_extend1_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
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)
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.
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)
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)
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
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
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.
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.
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.
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
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
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
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
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
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
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
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
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.
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.
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
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)
Object. The name PNo_bd is a term of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set.
Object. The name PNo_pred is a term of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.
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)
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)
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

Part 3

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}
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (not_ordinal_Sing1) The following is provable:
¬ ordinal {1}
Proof:
Assume H1.
Apply H1 to the current goal.
Assume H2 _.
An exact proof term for the current goal is not_TransSet_Sing1 H2.
∎
Try to prove it yourself. Main Goal
Theorem. (tagged_not_ordinal) The following is provable:
∀y, ¬ ordinal (y ')
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (tagged_notin_ordinal) The following is provable:
∀alpha y, ordinal alpha → (y ') ∉ alpha
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.
∎
Try to prove it yourself. Main Goal
Theorem. (tagged_eqE_Subq) The following is provable:
∀alpha beta, ordinal alpha → alpha ' = beta ' → alpha ⊆ beta
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (tagged_eqE_eq) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha ' = beta ' → alpha = beta
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))).
∎
Try to prove it yourself. Main Goal
Theorem. (tagged_ReplE) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → beta ' ∈ {gamma '|gamma ∈ alpha} → beta ∈ alpha
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (ordinal_notin_tagged_Repl) The following is provable:
∀alpha Y, ordinal alpha → alpha ∉ {y '|y ∈ Y}
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.
∎
Try to prove it yourself. Main Goal
Definition. We define SNoElts_ to be λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type set → set.
Theorem. (SNoElts_mon) The following is provable:
∀alpha beta, alpha ⊆ beta → SNoElts_ alpha ⊆ SNoElts_ beta
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).
∎
Try to prove it yourself. Main Goal
Definition. We define SNo_ to be λalpha x ⇒ x ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ x) (beta ∈ x) of type set → set → prop.
Definition. We define PSNo to be λalpha p ⇒ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta} of type set → (set → prop) → set.
Theorem. (PNoEq_PSNo) The following is provable:
∀alpha, ordinal alpha → ∀p : set → prop, PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo alpha p) p
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_PSNo) The following is provable:
∀alpha, ordinal alpha → ∀p : set → prop, SNo_ alpha (PSNo alpha p)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
Theorem. (SNo_PSNo_eta_) The following is provable:
∀alpha x, ordinal alpha → SNo_ alpha x → x = PSNo alpha (λbeta ⇒ beta ∈ x)
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Definition. We define SNo to be λx ⇒ ∃alpha, ordinal alpha ∧ SNo_ alpha x of type set → prop.
Theorem. (SNo_SNo) The following is provable:
∀alpha, ordinal alpha → ∀z, SNo_ alpha z → SNo z
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.
∎
Try to prove it yourself. Main Goal
Definition. We define SNoLev to be λx ⇒ Eps_i (λalpha ⇒ ordinal alpha ∧ SNo_ alpha x) of type set → set.
Theorem. (SNoLev_uniq_Subq) The following is provable:
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha ⊆ beta
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Main Goal
Theorem. (SNoLev_uniq) The following is provable:
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha = beta
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_prop) The following is provable:
∀x, SNo x → ordinal (SNoLev x) ∧ SNo_ (SNoLev x) x
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_ordinal) The following is provable:
∀x, SNo x → ordinal (SNoLev x)
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_) The following is provable:
∀x, SNo x → SNo_ (SNoLev x) x
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_PSNo_eta) The following is provable:
∀x, SNo x → x = PSNo (SNoLev x) (λbeta ⇒ beta ∈ x)
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_PSNo) The following is provable:
∀alpha, ordinal alpha → ∀p : set → prop, SNoLev (PSNo alpha p) = alpha
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type set → set → set → prop.
Theorem. (SNoEq_I) The following is provable:
∀alpha x y, (∀beta ∈ alpha, beta ∈ x ↔ beta ∈ y) → SNoEq_ alpha x y
Proof:
Let alpha, x and y be given.
Assume Hxy.
An exact proof term for the current goal is Hxy.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_eq) The following is provable:
∀x y, SNo x → SNo y → SNoLev x = SNoLev y → SNoEq_ (SNoLev x) x y → x = y
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.
∎
Try to prove it yourself. Main Goal
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.
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.
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
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLeE) The following is provable:
∀x y, SNo x → SNo y → x ≤ y → x < y ∨ x = y
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoEq_sym_) The following is provable:
∀alpha x y, SNoEq_ alpha x y → SNoEq_ alpha y x
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).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoEq_tra_) The following is provable:
∀alpha x y z, SNoEq_ alpha x y → SNoEq_ alpha y z → SNoEq_ alpha x z
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).
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Subgoal 8 Main Goal
Theorem. (SNoLtI2) The following is provable:
∀x y, SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → x < y
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLtI3) The following is provable:
∀x y, SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → x < y
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLt_irref) The following is provable:
∀x, ¬ SNoLt x x
Proof:
Let x be given.
An exact proof term for the current goal is PNoLt_irref (SNoLev x) (λbeta ⇒ beta ∈ x).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLt_trichotomy_or) The following is provable:
∀x y, SNo x → SNo y → x < y ∨ x = y ∨ y < x
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLt_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x < y → y < z → x < z
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLe_ref) The following is provable:
∀x, SNoLe x x
Proof:
Let x be given.
An exact proof term for the current goal is PNoLe_ref (SNoLev x) (λbeta ⇒ beta ∈ x).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLe_antisym) The following is provable:
∀x y, SNo x → SNo y → x ≤ y → y ≤ x → x = y
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLtLe_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x < y → y ≤ z → x < z
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLeLt_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y < z → x < z
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLe_tra) The following is provable:
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y ≤ z → x ≤ z
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLtLe_or) The following is provable:
∀x y, SNo x → SNo y → x < y ∨ y ≤ x
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
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.
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.
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)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Subgoal 8 Subgoal 9 Subgoal 10 Subgoal 11 Subgoal 12 Subgoal 13 Subgoal 14 Subgoal 15 Subgoal 16 Subgoal 17 Subgoal 18 Subgoal 19 Subgoal 20 Subgoal 21 Subgoal 22 Subgoal 23 Subgoal 24 Subgoal 25 Subgoal 26 Subgoal 27 Subgoal 28 Subgoal 29 Subgoal 30 Subgoal 31 Subgoal 32 Subgoal 33 Subgoal 34 Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoCutP_L_0) The following is provable:
∀L, (∀x ∈ L, SNo x) → SNoCutP L 0
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoCutP_0_R) The following is provable:
∀R, (∀x ∈ R, SNo x) → SNoCutP 0 R
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoCutP_0_0) The following is provable:
SNoCutP 0 0
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.
∎
Try to prove it yourself. Main Goal
Definition. We define SNoS_ to be λalpha ⇒ {x ∈ 𝒫 (SNoElts_ alpha)|∃beta ∈ alpha, SNo_ beta x} of type set → set.
Theorem. (SNoS_E) The following is provable:
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∃beta ∈ alpha, SNo_ beta x
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoS_I2) The following is provable:
∀x y, SNo x → SNo y → SNoLev x ∈ SNoLev y → x ∈ SNoS_ (SNoLev y)
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).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoS_Subq) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → SNoS_ alpha ⊆ SNoS_ beta
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_uniq2) The following is provable:
∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNoLev x = alpha
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Main Goal
Theorem. (SNoS_In_neq) The following is provable:
∀w, SNo w → ∀x ∈ SNoS_ (SNoLev w), x ≠ w
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoS_SNoLev) The following is provable:
∀z, SNo z → z ∈ SNoS_ (ordsucc (SNoLev z))
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.
∎
Try to prove it yourself. Main Goal
Definition. We define SNoL to be λz ⇒ {x ∈ SNoS_ (SNoLev z)|x < z} of type set → set.
Definition. We define SNoR to be λz ⇒ {y ∈ SNoS_ (SNoLev z)|z < y} of type set → set.
Theorem. (SNoCutP_SNoL_SNoR) The following is provable:
∀z, SNo z → SNoCutP (SNoL z) (SNoR z)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoL_SNoS_) The following is provable:
∀z, SNoL z ⊆ SNoS_ (SNoLev z)
Proof:
Let z be given.
An exact proof term for the current goal is Sep_Subq (SNoS_ (SNoLev z)) (λx ⇒ x < z).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoR_SNoS_) The following is provable:
∀z, SNoR z ⊆ SNoS_ (SNoLev z)
Proof:
Let z be given.
An exact proof term for the current goal is Sep_Subq (SNoS_ (SNoLev z)) (λy ⇒ z < y).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoL_SNoS) The following is provable:
∀x, SNo x → ∀w ∈ SNoL x, w ∈ SNoS_ (SNoLev x)
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoR_SNoS) The following is provable:
∀x, SNo x → ∀z ∈ SNoR x, z ∈ SNoS_ (SNoLev x)
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoL_I) The following is provable:
∀z, SNo z → ∀x, SNo x → SNoLev x ∈ SNoLev z → x < z → x ∈ SNoL z
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoR_I) The following is provable:
∀z, SNo z → ∀y, SNo y → SNoLev y ∈ SNoLev z → z < y → y ∈ SNoR z
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_eta) The following is provable:
∀z, SNo z → z = SNoCut (SNoL z) (SNoR z)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Main Goal
Theorem. (SNoCutP_SNo_SNoCut) The following is provable:
∀L R, SNoCutP L R → SNo (SNoCut L R)
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoCutP_SNoCut_L) The following is provable:
∀L R, SNoCutP L R → ∀x ∈ L, x < SNoCut L R
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoCutP_SNoCut_R) The following is provable:
∀L R, SNoCutP L R → ∀y ∈ R, SNoCut L R < y
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
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
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.
∎
Try to prove it yourself. Main Goal
Theorem. (ordinal_SNo_) The following is provable:
∀alpha, ordinal alpha → SNo_ alpha alpha
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.
∎
Try to prove it yourself. Main Goal
Theorem. (ordinal_SNo) The following is provable:
∀alpha, ordinal alpha → SNo alpha
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.
∎
Try to prove it yourself. Main Goal
Theorem. (ordinal_SNoLev) The following is provable:
∀alpha, ordinal alpha → SNoLev alpha = alpha
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).
∎
Try to prove it yourself. Main Goal
Theorem. (ordinal_SNoLev_max) The following is provable:
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → z < alpha
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (ordinal_SNoL) The following is provable:
∀alpha, ordinal alpha → SNoL alpha = SNoS_ alpha
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (ordinal_SNoR) The following is provable:
∀alpha, ordinal alpha → SNoR alpha = Empty
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (nat_p_SNo) The following is provable:
∀n, nat_p n → SNo n
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.
∎
Try to prove it yourself. Main Goal
Theorem. (omega_SNo) The following is provable:
∀n ∈ ω, SNo n
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.
∎
Try to prove it yourself. Main Goal
Theorem. (omega_SNoS_omega) The following is provable:
ω ⊆ SNoS_ ω
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.
∎
Try to prove it yourself. Main Goal
Theorem. (ordinal_In_SNoLt) The following is provable:
∀alpha, ordinal alpha → ∀beta ∈ alpha, beta < alpha
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
Theorem. (ordinal_SNoLev_max_2) The following is provable:
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → z ≤ alpha
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Main Goal
Theorem. (ordinal_Subq_SNoLe) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → alpha ≤ beta
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Main Goal
Theorem. (ordinal_SNoLt_In) The following is provable:
∀alpha beta, ordinal alpha → ordinal beta → alpha < beta → alpha ∈ beta
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.
∎
Try to prove it yourself. Main Goal
Theorem. (omega_nonneg) The following is provable:
∀m ∈ ω, 0 ≤ m
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_0) The following is provable:
SNo 0
Proof:
An exact proof term for the current goal is ordinal_SNo 0 ordinal_Empty.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_1) The following is provable:
SNo 1
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_2) The following is provable:
SNo 2
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_0) The following is provable:
SNoLev 0 = 0
Proof:
An exact proof term for the current goal is ordinal_SNoLev 0 ordinal_Empty.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoCut_0_0) The following is provable:
SNoCut 0 0 = 0
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNoL_0) The following is provable:
SNoL 0 = 0
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNoR_0) The following is provable:
SNoR 0 = 0
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNoL_1) The following is provable:
SNoL 1 = 1
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNoR_1) The following is provable:
SNoR 1 = 0
Proof:
An exact proof term for the current goal is ordinal_SNoR 1 (nat_p_ordinal 1 nat_1).
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_max_SNoLev) The following is provable:
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → SNoLev x = x
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
Theorem. (SNo_max_ordinal) The following is provable:
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → ordinal x
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.
∎
Try to prove it yourself. Main Goal
Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∧ delta ≠ SNoLev x) of type set → set.
Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∨ delta = SNoLev x) of type set → set.
Theorem. (SNo_extend0_SNo_) The following is provable:
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend0 x)
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).
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNo_extend1_SNo_) The following is provable:
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend1 x)
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).
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNo_extend0_SNo) The following is provable:
∀x, SNo x → SNo (SNo_extend0 x)
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).
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend1_SNo) The following is provable:
∀x, SNo x → SNo (SNo_extend1 x)
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).
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend0_SNoLev) The following is provable:
∀x, SNo x → SNoLev (SNo_extend0 x) = ordsucc (SNoLev x)
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).
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend1_SNoLev) The following is provable:
∀x, SNo x → SNoLev (SNo_extend1 x) = ordsucc (SNoLev x)
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).
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend0_nIn) The following is provable:
∀x, SNo x → SNoLev x ∉ SNo_extend0 x
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend1_In) The following is provable:
∀x, SNo x → SNoLev x ∈ SNo_extend1 x
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend0_SNoEq) The following is provable:
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend0 x) x
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNo_extend1_SNoEq) The following is provable:
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend1 x) x
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNoLev_0_eq_0) The following is provable:
∀x, SNo x → SNoLev x = 0 → x = 0
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.
∎
Try to prove it yourself. Main Goal
Definition. We define eps_ to be λn ⇒ {0} ∪ {(ordsucc m) '|m ∈ n} of type set → set.
Theorem. (eps_ordinal_In_eq_0) The following is provable:
∀n alpha, ordinal alpha → alpha ∈ eps_ n → alpha = 0
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.
∎
Try to prove it yourself. Main Goal
Theorem. (eps_0_1) The following is provable:
eps_ 0 = 1
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo__eps_) The following is provable:
∀n ∈ ω, SNo_ (ordsucc n) (eps_ n)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Main Goal
Theorem. (SNo_eps_) The following is provable:
∀n ∈ ω, SNo (eps_ n)
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).
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNo_eps_1) The following is provable:
SNo (eps_ 1)
Proof:
An exact proof term for the current goal is SNo_eps_ 1 (nat_p_omega 1 nat_1).
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLev_eps_) The following is provable:
∀n ∈ ω, SNoLev (eps_ n) = ordsucc n
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNo_eps_SNoS_omega) The following is provable:
∀n ∈ ω, eps_ n ∈ SNoS_ ω
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_eps_decr) The following is provable:
∀n ∈ ω, ∀m ∈ n, eps_ n < eps_ m
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Main Goal
Theorem. (SNo_eps_pos) The following is provable:
∀n ∈ ω, 0 < eps_ n
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_pos_eps_Lt) The following is provable:
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → eps_ n < x
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
Theorem. (SNo_pos_eps_Le) The following is provable:
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc (ordsucc n)), 0 < x → eps_ n ≤ x
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
Theorem. (eps_SNoCutP) The following is provable:
∀n ∈ ω, SNoCutP {0} {eps_ m|m ∈ n}
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (eps_SNoCut) The following is provable:
∀n ∈ ω, eps_ n = SNoCut {0} {eps_ m|m ∈ n}
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Subgoal 8 Subgoal 9 Subgoal 10 Subgoal 11 Subgoal 12 Subgoal 13 Subgoal 14 Subgoal 15 Subgoal 16 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Subgoal 8 Subgoal 9 Subgoal 10 Subgoal 11 Subgoal 12 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main 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.
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Main Goal
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).
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
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Main Goal
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.
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
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
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))
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (SNo_rec2_eq) The following is provable:
∀w, SNo w → ∀z, SNo z → SNo_rec2 w z = F w z SNo_rec2
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Main Goal
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)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Main Goal
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)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Main Goal
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)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Subgoal 3 Subgoal 4 Subgoal 5 Subgoal 6 Subgoal 7 Subgoal 8 Subgoal 9 Main Goal
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)
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
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
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.
∎
Try to prove it yourself. Subgoal 1 Main Goal
Theorem. (SNo_omega) The following is provable:
SNo ω
Proof:
An exact proof term for the current goal is ordinal_SNo ω omega_ordinal.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLt_0_1) The following is provable:
0 < 1
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLt_0_2) The following is provable:
0 < 2
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNoLt_1_2) The following is provable:
1 < 2
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.
∎
Try to prove it yourself. Main Goal
Theorem. (restr_SNo_) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo_ alpha (x ∩ SNoElts_ alpha)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (restr_SNo) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo (x ∩ SNoElts_ alpha)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (restr_SNoLev) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoLev (x ∩ SNoElts_ alpha) = alpha
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (restr_SNoEq) The following is provable:
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoEq_ alpha (x ∩ SNoElts_ alpha) x
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.
∎
Try to prove it yourself. Main Goal
Theorem. (SNo_extend0_restr_eq) The following is provable:
∀x, SNo x → x = SNo_extend0 x ∩ SNoElts_ (SNoLev x)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal
Theorem. (SNo_extend1_restr_eq) The following is provable:
∀x, SNo x → x = SNo_extend1 x ∩ SNoElts_ (SNoLev x)
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.
∎
Try to prove it yourself. Subgoal 1 Subgoal 2 Main Goal