Primitive. The name
Eps_i is a term of type
(set → prop) → set.
L2Axiom. (
Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
L4Definition. We define
True to be
∀p : prop, p → p of type
prop.
L6Definition. We define
False to be
∀p : prop, p of type
prop.
L7Definition. 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.
L12Definition. 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.
L17Definition. 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.
L22Definition. 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
L30Definition. We define
eq to be
λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type
A → A → prop.
L31Definition. 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
L40Axiom. (
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
L45Definition. 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.
L50Axiom. (
prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
Primitive. The name
In is a term of type
set → set → prop.
Notation. We use
∈ as an infix operator with priority 500 and no associativity corresponding to applying term
In. Furthermore, we may write
∀ x ∈ A, B to mean
∀ x : set, x ∈ A → B.
L54Definition. 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.
L56Axiom. (
set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
L58Axiom. (
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.
Primitive. The name
Empty is a term of type
set.
L64Axiom. (
EmptyAx) We take the following as an axiom:
Primitive. The name
⋃ is a term of type
set → set.
L68Axiom. (
UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
Primitive. The name
𝒫 is a term of type
set → set.
L73Axiom. (
PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
Primitive. The name
Repl is a term of type
set → (set → set) → set.
Notation.
{B| x ∈ A} is notation for
Repl A (λ x . B).
L78Axiom. (
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
L80Definition. We define
TransSet to be
λU : set ⇒ ∀x ∈ U, x ⊆ U of type
set → prop.
L82Definition. We define
Union_closed to be
λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type
set → prop.
L84Definition. We define
Power_closed to be
λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type
set → prop.
L85Definition. 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.
Primitive. The name
UnivOf is a term of type
set → set.
L93Axiom. (
UnivOf_In) We take the following as an axiom:
L99Axiom. (
UnivOf_Min) We take the following as an axiom:
L104Axiom. (
FalseE) We take the following as an axiom:
L106Axiom. (
TrueI) We take the following as an axiom:
L108Axiom. (
notI) We take the following as an axiom:
L110Axiom. (
notE) We take the following as an axiom:
L112Axiom. (
andI) We take the following as an axiom:
∀A B : prop, A → B → A ∧ B
L114Axiom. (
andEL) We take the following as an axiom:
L116Axiom. (
andER) We take the following as an axiom:
L118Axiom. (
orIL) We take the following as an axiom:
L120Axiom. (
orIR) We take the following as an axiom:
L122Axiom. (
orE) We take the following as an axiom:
∀A B C : prop, (A → C) → (B → C) → A ∨ B → C
Beginning of Section PropN
L126Variable P1 P2 P3 : prop
L128Axiom. (
and3I) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
L130Axiom. (
and3E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
L131Axiom. (
or3I1) We take the following as an axiom:
L132Axiom. (
or3I2) We take the following as an axiom:
L133Axiom. (
or3I3) We take the following as an axiom:
L134Axiom. (
or3E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
L137Axiom. (
and4I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
L139Axiom. (
and4E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 ∧ P4 → (∀p : prop, (P1 → P2 → P3 → P4 → p) → p)
L140Axiom. (
or4I1) We take the following as an axiom:
L141Axiom. (
or4I2) We take the following as an axiom:
L142Axiom. (
or4I3) We take the following as an axiom:
L143Axiom. (
or4I4) We take the following as an axiom:
L144Axiom. (
or4E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 ∨ P4 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → (P4 → p) → p)
L147Axiom. (
and5I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
L149Axiom. (
and5E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 → (∀p : prop, (P1 → P2 → P3 → P4 → P5 → p) → p)
L150Axiom. (
or5I1) We take the following as an axiom:
P1 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5
L151Axiom. (
or5I2) We take the following as an axiom:
P2 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5
L152Axiom. (
or5I3) We take the following as an axiom:
P3 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5
L153Axiom. (
or5I4) We take the following as an axiom:
P4 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5
L154Axiom. (
or5I5) We take the following as an axiom:
P5 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5
L155Axiom. (
or5E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → (P4 → p) → (P5 → p) → p)
L158Axiom. (
and6I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P6 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6
L160Axiom. (
and6E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 → (∀p : prop, (P1 → P2 → P3 → P4 → P5 → P6 → p) → p)
L163Axiom. (
and7I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P6 → P7 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7
L165Axiom. (
and7E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 → (∀p : prop, (P1 → P2 → P3 → P4 → P5 → P6 → P7 → p) → p)
End of Section PropN
L168Axiom. (
iffI) We take the following as an axiom:
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
L170Axiom. (
iffEL) We take the following as an axiom:
∀A B : prop, (A ↔ B) → A → B
L171Axiom. (
iffER) We take the following as an axiom:
∀A B : prop, (A ↔ B) → B → A
L172Axiom. (
iff_ref) We take the following as an axiom:
L173Axiom. (
neq_i_sym) We take the following as an axiom:
L175Definition. 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.
L181Axiom. (
Eps_i_ex) We take the following as an axiom:
∀P : set → prop, (∃x, P x) → P (Eps_i P)
L183Axiom. (
pred_ext) We take the following as an axiom:
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
L185Axiom. (
prop_ext_2) We take the following as an axiom:
∀p q : prop, (p → q) → (q → p) → p = q
L186Axiom. (
pred_ext_2) We take the following as an axiom:
∀P Q : set → prop, P ⊆ Q → Q ⊆ P → P = Q
L187Axiom. (
Subq_ref) We take the following as an axiom:
L189Axiom. (
Subq_tra) We take the following as an axiom:
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
L190Axiom. (
Subq_contra) We take the following as an axiom:
∀X Y z : set, X ⊆ Y → z ∉ Y → z ∉ X
L191Axiom. (
EmptyE) We take the following as an axiom:
L193Axiom. (
Subq_Empty) We take the following as an axiom:
L195Axiom. (
Empty_eq) We take the following as an axiom:
L196Axiom. (
UnionI) We take the following as an axiom:
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
L198Axiom. (
UnionE) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
L199Axiom. (
UnionE_impred) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
L200Axiom. (
Union_Empty) We take the following as an axiom:
L202Axiom. (
PowerI) We take the following as an axiom:
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
L204Axiom. (
PowerE) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
L205Axiom. (
Power_Subq) We take the following as an axiom:
∀X Y : set, X ⊆ Y → 𝒫 X ⊆ 𝒫 Y
L210Axiom. (
xm) We take the following as an axiom:
L212Axiom. (
dneg) We take the following as an axiom:
L213Axiom. (
imp_not_or) We take the following as an axiom:
∀p q : prop, (p → q) → ¬ p ∨ q
Primitive. The name
exactly1of2 is a term of type
prop → prop → prop.
L223Axiom. (
exactly1of2_E) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
Primitive. The name
exactly1of3 is a term of type
prop → prop → prop → prop.
L245Axiom. (
exactly1of3_E) We take the following as an axiom:
∀A B C : prop, exactly1of3 A B C → ∀p : prop, (A → ¬ B → ¬ C → p) → (¬ A → B → ¬ C → p) → (¬ A → ¬ B → C → p) → p
L263Axiom. (
ReplI) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
L265Axiom. (
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
L267Axiom. (
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
L269Axiom. (
Repl_Empty) We take the following as an axiom:
L271Axiom. (
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}
L273Axiom. (
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}
Primitive. 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.
L283Axiom. (
If_i_0) We take the following as an axiom:
L286Axiom. (
If_i_1) We take the following as an axiom:
L289Axiom. (
If_i_or) We take the following as an axiom:
L291Axiom. (
If_i_eta) We take the following as an axiom:
Primitive. The name
UPair is a term of type
set → set → set.
Notation.
{x,y} is notation for
UPair x y.
L298Axiom. (
UPairE) We take the following as an axiom:
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
L301Axiom. (
UPairI1) We take the following as an axiom:
L303Axiom. (
UPairI2) We take the following as an axiom:
L305Axiom. (
UPair_com) We take the following as an axiom:
Primitive. The name
Sing is a term of type
set → set.
Notation.
{x} is notation for
Sing x.
L311Axiom. (
SingI) We take the following as an axiom:
L313Axiom. (
SingE) We take the following as an axiom:
∀x y : set, y ∈ {x} → y = x
Primitive. 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.
L320Axiom. (
binunionI1) We take the following as an axiom:
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
L322Axiom. (
binunionI2) We take the following as an axiom:
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
L324Axiom. (
binunionE) We take the following as an axiom:
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
L326Definition. 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.
L332Axiom. (
Repl_UPair) We take the following as an axiom:
∀F : set → set, ∀x y : set, {F z|z ∈ {x,y}} = {F x,F y}
L334Axiom. (
Repl_Sing) We take the following as an axiom:
∀F : set → set, ∀x : set, {F z|z ∈ {x}} = {F x}
L336Axiom. (
Repl_restr) We take the following as an axiom:
∀X : set, ∀F G : set → set, (∀x : set, x ∈ X → F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}
Primitive. 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.
L345Axiom. (
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
L347Axiom. (
famunionE) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
L349Axiom. (
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
L355Axiom. (
Power_Sing) We take the following as an axiom:
Primitive. The name
Sep is a term of type
set → (set → prop) → set.
Notation.
{x ∈ A | B} is notation for
Sep A (λ x . B).
L363Axiom. (
SepI) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
L365Axiom. (
SepE) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
L366Axiom. (
SepE1) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
L367Axiom. (
SepE2) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
L368Axiom. (
Sep_Subq) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
L370Axiom. (
Sep_In_Power) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
Primitive. 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).
L376Axiom. (
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}
L378Axiom. (
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
L380Axiom. (
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
Primitive. The name
ReplSep2 is a term of type
set → (set → set) → (set → set → prop) → (set → set → set) → set.
L385Axiom. (
ReplSep2I) We take the following as an axiom:
∀A, ∀B : set → set, ∀P : set → set → prop, ∀F : set → set → set, ∀x ∈ A, ∀y ∈ B x, P x y → F x y ∈ ReplSep2 A B P F
L387Axiom. (
ReplSep2E_impred) We take the following as an axiom:
∀A, ∀B : set → set, ∀P : set → set → prop, ∀F : set → set → set, ∀r ∈ ReplSep2 A B P F, ∀p : prop, (∀x ∈ A, ∀y ∈ B x, P x y → r = F x y → p) → p
L389Axiom. (
ReplSep2E) We take the following as an axiom:
∀A, ∀B : set → set, ∀P : set → set → prop, ∀F : set → set → set, ∀r ∈ ReplSep2 A B P F, ∃x ∈ A, ∃y ∈ B x, P x y ∧ r = F x y
L391Axiom. (
binunion_asso) We take the following as an axiom:
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
L393Axiom. (
binunion_com) We take the following as an axiom:
∀X Y : set, X ∪ Y = Y ∪ X
L401Axiom. (
binunion_nIn_I) We take the following as an axiom:
∀X Y z : set, z ∉ X → z ∉ Y → z ∉ X ∪ Y
L402Axiom. (
binunion_nIn_E) We take the following as an axiom:
∀X Y z : set, z ∉ X ∪ Y → z ∉ X ∧ z ∉ Y
Primitive. 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.
L409Axiom. (
binintersectI) We take the following as an axiom:
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
L411Axiom. (
binintersectE) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
Primitive. 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.
L435Axiom. (
setminusI) We take the following as an axiom:
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
L437Axiom. (
setminusE) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
L438Axiom. (
setminusE1) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
L439Axiom. (
setminusE2) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∉ Y
L453Axiom. (
In_irref) We take the following as an axiom:
L455Axiom. (
In_no2cycle) We take the following as an axiom:
L456Axiom. (
In_no3cycle) We take the following as an axiom:
L457Axiom. (
In_no4cycle) We take the following as an axiom:
∀x y z w, x ∈ y → y ∈ z → z ∈ w → w ∈ x → False
Primitive. The name
ordsucc is a term of type
set → set.
L461Axiom. (
ordsuccI1) We take the following as an axiom:
L463Axiom. (
ordsuccI2) We take the following as an axiom:
L464Axiom. (
ordsuccE) We take the following as an axiom:
Notation. Natural numbers 0,1,2,... are notation for the terms formed using
Empty as 0 and forming successors with
ordsucc.
L470Axiom. (
ordsucc_inj) We take the following as an axiom:
L473Axiom. (
In_0_1) We take the following as an axiom:
L475Axiom. (
In_0_2) We take the following as an axiom:
L476Axiom. (
In_1_2) We take the following as an axiom:
L477Definition. 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.
L479Axiom. (
nat_0) We take the following as an axiom:
L481Axiom. (
nat_ordsucc) We take the following as an axiom:
L482Axiom. (
nat_1) We take the following as an axiom:
L483Axiom. (
nat_2) We take the following as an axiom:
L484Axiom. (
nat_3) We take the following as an axiom:
L485Axiom. (
nat_4) We take the following as an axiom:
L486Axiom. (
nat_5) We take the following as an axiom:
L487Axiom. (
nat_6) We take the following as an axiom:
L488Axiom. (
nat_7) We take the following as an axiom:
L489Axiom. (
nat_8) We take the following as an axiom:
L490Axiom. (
nat_9) We take the following as an axiom:
L491Axiom. (
nat_10) We take the following as an axiom:
L492Axiom. (
nat_11) We take the following as an axiom:
L493Axiom. (
nat_12) We take the following as an axiom:
L494Axiom. (
nat_13) We take the following as an axiom:
L495Axiom. (
nat_14) We take the following as an axiom:
L496Axiom. (
nat_15) We take the following as an axiom:
L497Axiom. (
nat_16) We take the following as an axiom:
L498Axiom. (
nat_17) We take the following as an axiom:
L502Axiom. (
nat_ind) We take the following as an axiom:
L503Axiom. (
nat_inv) We take the following as an axiom:
L505Axiom. (
nat_p_trans) We take the following as an axiom:
L506Axiom. (
nat_trans) We take the following as an axiom:
L510Axiom. (
In_0_3) We take the following as an axiom:
L512Axiom. (
In_1_3) We take the following as an axiom:
L513Axiom. (
In_2_3) We take the following as an axiom:
L514Axiom. (
In_0_4) We take the following as an axiom:
L515Axiom. (
In_1_4) We take the following as an axiom:
L516Axiom. (
In_2_4) We take the following as an axiom:
L517Axiom. (
In_3_4) We take the following as an axiom:
L518Axiom. (
In_0_5) We take the following as an axiom:
L519Axiom. (
In_1_5) We take the following as an axiom:
L520Axiom. (
In_2_5) We take the following as an axiom:
L521Axiom. (
In_3_5) We take the following as an axiom:
L522Axiom. (
In_4_5) We take the following as an axiom:
L523Axiom. (
In_0_6) We take the following as an axiom:
L524Axiom. (
In_1_6) We take the following as an axiom:
L525Axiom. (
In_2_6) We take the following as an axiom:
L526Axiom. (
In_3_6) We take the following as an axiom:
L527Axiom. (
In_4_6) We take the following as an axiom:
L528Axiom. (
In_5_6) We take the following as an axiom:
L529Axiom. (
cases_1) We take the following as an axiom:
∀i ∈ 1, ∀p : set → prop, p 0 → p i
L531Axiom. (
cases_2) We take the following as an axiom:
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
L532Axiom. (
cases_3) We take the following as an axiom:
∀i ∈ 3, ∀p : set → prop, p 0 → p 1 → p 2 → p i
L533Axiom. (
cases_4) We take the following as an axiom:
∀i ∈ 4, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p i
L534Axiom. (
cases_5) We take the following as an axiom:
∀i ∈ 5, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p i
L535Axiom. (
cases_6) We take the following as an axiom:
∀i ∈ 6, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p i
L536Axiom. (
neq_0_1) We take the following as an axiom:
L538Axiom. (
neq_0_2) We take the following as an axiom:
L539Axiom. (
neq_1_2) We take the following as an axiom:
L540Axiom. (
neq_1_0) We take the following as an axiom:
L541Axiom. (
neq_2_0) We take the following as an axiom:
L542Axiom. (
neq_2_1) We take the following as an axiom:
L543Axiom. (
neq_3_0) We take the following as an axiom:
L544Axiom. (
neq_3_1) We take the following as an axiom:
L545Axiom. (
neq_3_2) We take the following as an axiom:
L546Axiom. (
neq_4_0) We take the following as an axiom:
L547Axiom. (
neq_4_1) We take the following as an axiom:
L548Axiom. (
neq_4_2) We take the following as an axiom:
L549Axiom. (
neq_4_3) We take the following as an axiom:
L550Axiom. (
neq_5_0) We take the following as an axiom:
L551Axiom. (
neq_5_1) We take the following as an axiom:
L552Axiom. (
neq_5_2) We take the following as an axiom:
L553Axiom. (
neq_5_3) We take the following as an axiom:
L554Axiom. (
neq_5_4) We take the following as an axiom:
L555Axiom. (
ZF_closed_I) We take the following as an axiom:
L561Axiom. (
ZF_closed_E) We take the following as an axiom:
Primitive. The name
ω is a term of type
set.
L595Axiom. (
omega_nat_p) We take the following as an axiom:
L597Axiom. (
nat_p_omega) We take the following as an axiom:
L617Axiom. (
ordinal_1) We take the following as an axiom:
L619Axiom. (
ordinal_2) We take the following as an axiom:
L651Axiom. (
ordinal_Sep) We take the following as an axiom:
∀alpha, ordinal alpha → ∀p : set → prop, (∀beta ∈ alpha, ∀gamma ∈ beta, p beta → p gamma) → ordinal {beta ∈ alpha|p beta}
L653Definition. 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.
L659Definition. We define
surj to be
λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type
set → set → (set → set) → prop.
L665Definition. 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.
L673Axiom. (
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
L679Axiom. (
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
Primitive. The name
inv is a term of type
set → (set → set) → set → set.
L691Axiom. (
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
L693Axiom. (
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
L695Axiom. (
bij_inv) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
L697Axiom. (
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))
L699Axiom. (
bij_id) We take the following as an axiom:
L701Axiom. (
bij_inj) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → inj X Y f
L703Axiom. (
bij_surj) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → surj X Y f
L705Axiom. (
inj_surj_bij) We take the following as an axiom:
∀X Y, ∀f : set → set, inj X Y f → surj X Y f → bij X Y f
L707Axiom. (
surj_inv_inj) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀y ∈ Y, ∃x ∈ X, f x = y) → inj Y X (inv X f)
L709Definition. We define
atleastp to be
λX Y : set ⇒ ∃f : set → set, inj X Y f of type
set → set → prop.
L712Definition. We define
equip to be
λX Y : set ⇒ ∃f : set → set, bij X Y f of type
set → set → prop.
L715Axiom. (
equip_ref) We take the following as an axiom:
L717Axiom. (
equip_sym) We take the following as an axiom:
L718Axiom. (
equip_tra) We take the following as an axiom:
L719Definition. We define
finite to be
λX ⇒ ∃n ∈ ω, equip X n of type
set → prop.
L722Axiom. (
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
L727Axiom. (
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
L732Axiom. (
f_eq_i) We take the following as an axiom:
∀f : set → set, ∀x y, x = y → f x = f y
L734Axiom. (
f_eq_i_i) We take the following as an axiom:
∀f : set → set → set, ∀x y z w, x = y → z = w → f x z = f y w
L735Axiom. (
eq_i_tra) We take the following as an axiom:
∀x y z, x = y → y = z → x = z
L736Definition. We define
nSubq to be
λX Y ⇒ ¬ Subq X Y of type
set → set → prop.
Notation. We use
⊈ as an infix operator with priority 502 and no associativity corresponding to applying term
nSubq.
L742Axiom. (
Sing_inv) We take the following as an axiom:
∀x Y, {x} = Y → x ∈ Y ∧ ∀y ∈ Y, y = x
L746Axiom. (
inv_Repl_eq) We take the following as an axiom:
∀X, ∀f g : set → set, (∀x ∈ X, f (g x) = x) → {f y|y ∈ {g x|x ∈ X}} = X
L747Axiom. (
invol_Repl_eq) We take the following as an axiom:
∀X, ∀f : set → set, (∀x ∈ X, f (f x) = x) → {f y|y ∈ {f x|x ∈ X}} = X
L748Axiom. (
Eps_i_set_R) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀x ∈ X, P x → Eps_i (λx ⇒ x ∈ X ∧ P x) ∈ X ∧ P (Eps_i (λx ⇒ x ∈ X ∧ P x))
L750Axiom. (
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
L752Axiom. (
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
L754Axiom. (
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
L756Axiom. (
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
L758Axiom. (
exandE_iio) We take the following as an axiom:
∀P Q : (set → set → prop) → prop, (∃x : set → set → prop, P x ∧ Q x) → ∀p : prop, (∀x : set → set → prop, P x → Q x → p) → p
L760Axiom. (
exandE_iiio) We take the following as an axiom:
∀P Q : (set → set → set → prop) → prop, (∃x : set → set → set → prop, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → prop, P x → Q x → p) → p
Beginning of Section Descr_ii
L764Variable P : (set → set) → prop
Primitive. The name
Descr_ii is a term of type
set → set.
L769Hypothesis Pex : ∃f : set → set, P f
L771Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
End of Section Descr_ii
Beginning of Section Descr_iii
L778Variable P : (set → set → set) → prop
Primitive. The name
Descr_iii is a term of type
set → set → set.
L783Hypothesis Pex : ∃f : set → set → set, P f
L785Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
End of Section Descr_iii
Beginning of Section Descr_iio
L792Variable P : (set → set → prop) → prop
Primitive. The name
Descr_iio is a term of type
set → set → prop.
L797Hypothesis Pex : ∃f : set → set → prop, P f
L799Hypothesis Puniq : ∀f g : set → set → prop, P f → P g → f = g
End of Section Descr_iio
Beginning of Section Descr_Vo1
Primitive. The name
Descr_Vo1 is a term of type
Vo 1.
L811Hypothesis Pex : ∃f : Vo 1, P f
L813Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
End of Section Descr_Vo1
Beginning of Section Descr_Vo2
Primitive. The name
Descr_Vo2 is a term of type
Vo 2.
L825Hypothesis Pex : ∃f : Vo 2, P f
L827Hypothesis Puniq : ∀f g : Vo 2, P f → P g → f = g
End of Section Descr_Vo2
Beginning of Section If_ii
L836Variable f g : set → set
Primitive. The name
If_ii is a term of type
set → set.
L840Axiom. (
If_ii_1) We take the following as an axiom:
L842Axiom. (
If_ii_0) We take the following as an axiom:
End of Section If_ii
Beginning of Section If_iii
L850Variable f g : set → set → set
Primitive. The name
If_iii is a term of type
set → set → set.
L854Axiom. (
If_iii_1) We take the following as an axiom:
L856Axiom. (
If_iii_0) We take the following as an axiom:
End of Section If_iii
Beginning of Section If_Vo1
Primitive. The name
If_Vo1 is a term of type
Vo 1.
L868Axiom. (
If_Vo1_1) We take the following as an axiom:
L870Axiom. (
If_Vo1_0) We take the following as an axiom:
End of Section If_Vo1
Beginning of Section If_iio
L878Variable f g : set → set → prop
Primitive. The name
If_iio is a term of type
set → set → prop.
L882Axiom. (
If_iio_1) We take the following as an axiom:
L884Axiom. (
If_iio_0) We take the following as an axiom:
End of Section If_iio
Beginning of Section If_Vo2
Primitive. The name
If_Vo2 is a term of type
Vo 2.
L896Axiom. (
If_Vo2_1) We take the following as an axiom:
L898Axiom. (
If_Vo2_0) We take the following as an axiom:
End of Section If_Vo2
Beginning of Section EpsilonRec_i
L904Variable F : set → (set → set) → set
Primitive. The name
In_rec_i is a term of type
set → set.
L909Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
L911Axiom. (
In_rec_i_eq) We take the following as an axiom:
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
L917Variable F : set → (set → (set → set)) → (set → set)
Primitive. The name
In_rec_ii is a term of type
set → (set → set).
L922Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
L930Variable F : set → (set → (set → set → set)) → (set → set → set)
Primitive. The name
In_rec_iii is a term of type
set → (set → set → set).
L935Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_iii
Beginning of Section EpsilonRec_iio
L943Variable F : set → (set → (set → set → prop)) → (set → set → prop)
Primitive. The name
In_rec_iio is a term of type
set → (set → set → prop).
L948Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → prop), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_iio
Beginning of Section EpsilonRec_Vo1
L956Variable F : set → (set → Vo 1) → Vo 1
Primitive. The name
In_rec_Vo1 is a term of type
set → Vo 1.
L961Hypothesis Fr : ∀X : set, ∀g h : set → Vo 1, (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_Vo1
Beginning of Section EpsilonRec_Vo2
L969Variable F : set → (set → Vo 2) → Vo 2
Primitive. The name
In_rec_Vo2 is a term of type
set → Vo 2.
L974Hypothesis Fr : ∀X : set, ∀g h : set → Vo 2, (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_Vo2
Beginning of Section If_Vo3
Primitive. The name
If_Vo3 is a term of type
Vo 3.
L988Axiom. (
If_Vo3_1) We take the following as an axiom:
L990Axiom. (
If_Vo3_0) We take the following as an axiom:
End of Section If_Vo3
Beginning of Section Descr_Vo3
Primitive. The name
Descr_Vo3 is a term of type
Vo 3.
L1001Hypothesis Pex : ∃f : Vo 3, P f
L1003Hypothesis Puniq : ∀f g : Vo 3, P f → P g → f = g
End of Section Descr_Vo3
Beginning of Section EpsilonRec_Vo3
L1010Variable F : set → (set → Vo 3) → Vo 3
Primitive. The name
In_rec_Vo3 is a term of type
set → Vo 3.
L1015Hypothesis Fr : ∀X : set, ∀g h : set → Vo 3, (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_Vo3
Beginning of Section If_Vo4
Primitive. The name
If_Vo4 is a term of type
Vo 4.
L1029Axiom. (
If_Vo4_1) We take the following as an axiom:
L1031Axiom. (
If_Vo4_0) We take the following as an axiom:
End of Section If_Vo4
Beginning of Section Descr_Vo4
Primitive. The name
Descr_Vo4 is a term of type
Vo 4.
L1041Hypothesis Pex : ∃f : Vo 4, P f
L1043Hypothesis Puniq : ∀f g : Vo 4, P f → P g → f = g
End of Section Descr_Vo4
Beginning of Section EpsilonRec_Vo4
L1050Variable F : set → (set → Vo 4) → Vo 4
Primitive. The name
In_rec_Vo4 is a term of type
set → Vo 4.
L1055Hypothesis Fr : ∀X : set, ∀g h : set → Vo 4, (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_Vo4
L1061Definition. We define
bigintersect to be
λ(D : (set → prop) → prop)(x : set) ⇒ ∀P : set → prop, D P → P x.
L1063Definition. We define
reflexive to be
λR ⇒ ∀x : set, R x x of type
(set → set → prop) → prop.
L1065Definition. We define
irreflexive to be
λR ⇒ ∀x : set, ¬ R x x of type
(set → set → prop) → prop.
L1066Definition. We define
symmetric to be
λR ⇒ ∀x y : set, R x y → R y x of type
(set → set → prop) → prop.
L1067Definition. We define
antisymmetric to be
λR ⇒ ∀x y : set, R x y → R y x → x = y of type
(set → set → prop) → prop.
L1068Definition. We define
transitive to be
λR ⇒ ∀x y z : set, R x y → R y z → R x z of type
(set → set → prop) → prop.
L1071Definition. We define
linear to be
λR ⇒ ∀x y : set, R x y ∨ R y x of type
(set → set → prop) → prop.
L1072Definition. We define
trichotomous_or to be
λR ⇒ ∀x y : set, R x y ∨ x = y ∨ R y x of type
(set → set → prop) → prop.
L1077Axiom. (
per_sym) We take the following as an axiom:
L1079Axiom. (
per_tra) We take the following as an axiom:
L1081Axiom. (
per_stra1) We take the following as an axiom:
∀R : set → set → prop, per R → ∀x y z : set, R y x → R y z → R x z
L1083Axiom. (
per_stra2) We take the following as an axiom:
∀R : set → set → prop, per R → ∀x y z : set, R x y → R z y → R x z
L1085Axiom. (
per_stra3) We take the following as an axiom:
∀R : set → set → prop, per R → ∀x y z : set, R y x → R z y → R x z
L1087Axiom. (
per_ref1) We take the following as an axiom:
∀R : set → set → prop, per R → ∀x y : set, R x y → R x x
L1089Axiom. (
per_ref2) We take the following as an axiom:
∀R : set → set → prop, per R → ∀x y : set, R x y → R y y
L1094Definition. We define
reflclos to be
λR x y ⇒ R x y ∨ x = y of type
(set → set → prop) → (set → set → prop).
Beginning of Section Zermelo1908
Primitive. The name
ZermeloWO is a term of type
set → set → prop.
L1118Axiom. (
ZermeloWO_wo) We take the following as an axiom:
∀p : set → prop, (∃x : set, p x) → ∃x : set, p x ∧ ∀y : set, p y → ZermeloWO x y
L1125Axiom. (
Zermelo_WO) We take the following as an axiom:
∃r : set → set → prop, totalorder r ∧ (∀p : set → prop, (∃x : set, p x) → ∃x : set, p x ∧ ∀y : set, p y → r x y)
End of Section Zermelo1908
L1135Axiom. (
eq_imp_or) We take the following as an axiom:
(λx y : prop ⇒ (x → y)) = (λx y : prop ⇒ (¬ x ∨ y))
L1139Axiom. (
Empty_or_ex) We take the following as an axiom:
L1141Axiom. (
nIn_0_0) We take the following as an axiom:
L1143Axiom. (
nIn_1_0) We take the following as an axiom:
L1144Axiom. (
nIn_2_0) We take the following as an axiom:
L1145Axiom. (
nIn_1_1) We take the following as an axiom:
L1146Axiom. (
nIn_2_2) We take the following as an axiom:
L1147Axiom. (
Subq_0_0) We take the following as an axiom:
L1148Axiom. (
Subq_0_1) We take the following as an axiom:
L1149Axiom. (
Subq_0_2) We take the following as an axiom:
L1150Axiom. (
nSubq_1_0) We take the following as an axiom:
L1151Axiom. (
Subq_1_1) We take the following as an axiom:
L1152Axiom. (
Subq_1_2) We take the following as an axiom:
L1153Axiom. (
nSubq_2_0) We take the following as an axiom:
L1154Axiom. (
nSubq_2_1) We take the following as an axiom:
L1155Axiom. (
Subq_2_2) We take the following as an axiom:
L1156Axiom. (
In_0_7) We take the following as an axiom:
L1157Axiom. (
In_1_7) We take the following as an axiom:
L1158Axiom. (
In_2_7) We take the following as an axiom:
L1159Axiom. (
In_3_7) We take the following as an axiom:
L1160Axiom. (
In_4_7) We take the following as an axiom:
L1161Axiom. (
In_5_7) We take the following as an axiom:
L1162Axiom. (
In_6_7) We take the following as an axiom:
L1163Axiom. (
In_0_8) We take the following as an axiom:
L1164Axiom. (
In_1_8) We take the following as an axiom:
L1165Axiom. (
In_2_8) We take the following as an axiom:
L1166Axiom. (
In_3_8) We take the following as an axiom:
L1167Axiom. (
In_4_8) We take the following as an axiom:
L1168Axiom. (
In_5_8) We take the following as an axiom:
L1169Axiom. (
In_6_8) We take the following as an axiom:
L1170Axiom. (
In_7_8) We take the following as an axiom:
L1171Axiom. (
In_0_9) We take the following as an axiom:
L1172Axiom. (
In_1_9) We take the following as an axiom:
L1173Axiom. (
In_2_9) We take the following as an axiom:
L1174Axiom. (
In_3_9) We take the following as an axiom:
L1175Axiom. (
In_4_9) We take the following as an axiom:
L1176Axiom. (
In_5_9) We take the following as an axiom:
L1177Axiom. (
In_6_9) We take the following as an axiom:
L1178Axiom. (
In_7_9) We take the following as an axiom:
L1179Axiom. (
In_8_9) We take the following as an axiom:
Beginning of Section NatRec
L1184Variable f : set → set → set
L1185Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
L1188Axiom. (
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
End of Section NatRec
Beginning of Section NatArith
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_nat.
L1202Axiom. (
add_nat_0R) We take the following as an axiom:
L1204Axiom. (
add_nat_SR) We take the following as an axiom:
L1206Axiom. (
add_nat_p) We take the following as an axiom:
L1210Axiom. (
add_nat_0L) We take the following as an axiom:
L1212Axiom. (
add_nat_SL) We take the following as an axiom:
L1214Axiom. (
add_nat_com) We take the following as an axiom:
L1216Definition. 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.
L1220Axiom. (
mul_nat_0R) We take the following as an axiom:
L1222Axiom. (
mul_nat_SR) We take the following as an axiom:
L1224Axiom. (
mul_nat_p) We take the following as an axiom:
L1226Axiom. (
mul_nat_0L) We take the following as an axiom:
L1228Axiom. (
mul_nat_SL) We take the following as an axiom:
L1230Axiom. (
mul_nat_com) We take the following as an axiom:
L1240Definition. We define
divides_nat to be
λm n ⇒ m ∈ ω ∧ n ∈ ω ∧ ∃k ∈ ω, m * k = n of type
set → set → prop.
L1248Definition. We define
exp_nat to be
λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type
set → set → set.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_nat.
L1252Definition. We define
even_nat to be
λn ⇒ n ∈ ω ∧ ∃m ∈ ω, n = 2 * m of type
set → prop.
L1254Definition. We define
odd_nat to be
λn ⇒ n ∈ ω ∧ ∀m ∈ ω, n ≠ 2 * m of type
set → prop.
End of Section NatArith
L1263Axiom. (
cases_7) We take the following as an axiom:
∀i ∈ 7, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p i
L1265Axiom. (
cases_8) We take the following as an axiom:
∀i ∈ 8, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p 7 → p i
L1266Axiom. (
cases_9) We take the following as an axiom:
∀i ∈ 9, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p 7 → p 8 → p i
L1267Axiom. (
nIn_2_1) We take the following as an axiom:
L1269Axiom. (
neq_6_0) We take the following as an axiom:
L1270Axiom. (
neq_6_1) We take the following as an axiom:
L1271Axiom. (
neq_6_2) We take the following as an axiom:
L1272Axiom. (
neq_6_3) We take the following as an axiom:
L1273Axiom. (
neq_6_4) We take the following as an axiom:
L1274Axiom. (
neq_6_5) We take the following as an axiom:
L1275Axiom. (
neq_7_0) We take the following as an axiom:
L1276Axiom. (
neq_7_1) We take the following as an axiom:
L1277Axiom. (
neq_7_2) We take the following as an axiom:
L1278Axiom. (
neq_7_3) We take the following as an axiom:
L1279Axiom. (
neq_7_4) We take the following as an axiom:
L1280Axiom. (
neq_7_5) We take the following as an axiom:
L1281Axiom. (
neq_7_6) We take the following as an axiom:
L1282Axiom. (
neq_8_0) We take the following as an axiom:
L1283Axiom. (
neq_8_1) We take the following as an axiom:
L1284Axiom. (
neq_8_2) We take the following as an axiom:
L1285Axiom. (
neq_8_3) We take the following as an axiom:
L1286Axiom. (
neq_8_4) We take the following as an axiom:
L1287Axiom. (
neq_8_5) We take the following as an axiom:
L1288Axiom. (
neq_8_6) We take the following as an axiom:
L1289Axiom. (
neq_8_7) We take the following as an axiom:
L1290Axiom. (
neq_9_0) We take the following as an axiom:
L1291Axiom. (
neq_9_1) We take the following as an axiom:
L1292Axiom. (
neq_9_2) We take the following as an axiom:
L1293Axiom. (
neq_9_3) We take the following as an axiom:
L1294Axiom. (
neq_9_4) We take the following as an axiom:
L1295Axiom. (
neq_9_5) We take the following as an axiom:
L1296Axiom. (
neq_9_6) We take the following as an axiom:
L1297Axiom. (
neq_9_7) We take the following as an axiom:
L1298Axiom. (
neq_9_8) We take the following as an axiom:
L1301Axiom. (
eq_1_Sing0) We take the following as an axiom:
L1305Axiom. (
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
L1319Axiom. (
Inj1_eq) We take the following as an axiom:
L1321Axiom. (
Inj1I1) We take the following as an axiom:
L1322Axiom. (
Inj1I2) We take the following as an axiom:
L1323Axiom. (
Inj1E) We take the following as an axiom:
L1324Axiom. (
Inj1NE1) We take the following as an axiom:
L1325Axiom. (
Inj1NE2) We take the following as an axiom:
L1326Definition. We define
Inj0 to be
λX ⇒ {Inj1 x|x ∈ X} of type
set → set.
L1329Axiom. (
Inj0I) We take the following as an axiom:
L1331Axiom. (
Inj0E) We take the following as an axiom:
L1335Axiom. (
Unj_eq) We take the following as an axiom:
L1337Axiom. (
Unj_Inj1_eq) We take the following as an axiom:
L1338Axiom. (
Inj1_inj) We take the following as an axiom:
L1339Axiom. (
Unj_Inj0_eq) We take the following as an axiom:
L1340Axiom. (
Inj0_inj) We take the following as an axiom:
L1341Axiom. (
Inj0_0) We take the following as an axiom:
Notation. We use
+ as an infix operator with priority 450 and which associates to the left corresponding to applying term
setsum.
L1349Axiom. (
Inj0_setsum) We take the following as an axiom:
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
L1351Axiom. (
Inj1_setsum) We take the following as an axiom:
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
L1357Axiom. (
setsum_0_0) We take the following as an axiom:
L1360Axiom. (
setsum_mon) We take the following as an axiom:
∀X Y W Z, X ⊆ W → Y ⊆ Z → X + Y ⊆ W + Z
Beginning of Section pair_setsum
L1375Axiom. (
pair_0_0) We take the following as an axiom:
L1377Axiom. (
pair_1_0_1) We take the following as an axiom:
L1378Axiom. (
pair_1_1_2) We take the following as an axiom:
L1386Axiom. (
pairI0) We take the following as an axiom:
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
L1387Axiom. (
pairI1) We take the following as an axiom:
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
L1388Axiom. (
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)
L1389Axiom. (
pairE0) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
L1390Axiom. (
pairE1) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
L1391Axiom. (
pairEq) 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)
L1392Axiom. (
pairSubq) We take the following as an axiom:
∀X Y W Z, X ⊆ W → Y ⊆ Z → pair X Y ⊆ pair W Z
L1393Axiom. (
proj0I) We take the following as an axiom:
L1394Axiom. (
proj0E) We take the following as an axiom:
L1395Axiom. (
proj1I) We take the following as an axiom:
L1396Axiom. (
proj1E) We take the following as an axiom:
L1399Axiom. (
pair_inj) We take the following as an axiom:
∀x y w z : set, pair x y = pair w z → x = w ∧ y = z
L1401Definition. 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.
L1408Axiom. (
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
L1414Axiom. (
proj0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
L1416Axiom. (
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)
L1418Axiom. (
pair_Sigma_E0) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → x ∈ X
L1420Axiom. (
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
L1422Axiom. (
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
L1424Axiom. (
Sigma_Eq) 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
L1426Axiom. (
Sigma_mon) We take the following as an axiom:
∀X Y : set, X ⊆ Y → ∀Z W : set → set, (∀x ∈ X, Z x ⊆ W x) → (∑x ∈ X, Z x) ⊆ ∑y ∈ Y, W y
L1429Axiom. (
Sigma_mon0) We take the following as an axiom:
∀X Y : set, X ⊆ Y → ∀Z : set → set, (∑x ∈ X, Z x) ⊆ ∑y ∈ Y, Z y
L1431Axiom. (
Sigma_mon1) We take the following as an axiom:
∀X : set, ∀Z W : set → set, (∀x, x ∈ X → Z x ⊆ W x) → (∑x ∈ X, Z x) ⊆ ∑x ∈ X, W x
L1433Axiom. (
Sigma_Power_1) We take the following as an axiom:
∀X : set, X ∈ 𝒫 1 → ∀Y : set → set, (∀x ∈ X, Y x ∈ 𝒫 1) → (∑x ∈ X, Y x) ∈ 𝒫 1
L1435Definition. 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.
L1440Axiom. (
pair_setprod) We take the following as an axiom:
∀X Y : set, ∀(x ∈ X)(y ∈ Y), pair x y ∈ X ⨯ Y
L1446Axiom. (
pair_setprod_E0) We take the following as an axiom:
∀X Y x y : set, pair x y ∈ X ⨯ Y → x ∈ X
L1448Axiom. (
pair_setprod_E1) We take the following as an axiom:
∀X Y x y : set, pair x y ∈ X ⨯ Y → y ∈ Y
L1450Let lam : set → (set → set) → set ≝ Sigma
L1453Definition. 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).
Notation. We now use n-tuple notation (
a0,...,
an-1) for n ≥ 2 for λ i ∈
n .
if i = 0
then a0 else ... if i =
n-2 then an-2 else an-1.
L1459Axiom. (
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
L1461Axiom. (
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
L1463Axiom. (
lamEq) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀z, z ∈ (λx ∈ X ⇒ F x) ↔ ∃x ∈ X, ∃y ∈ F x, z = pair x y
L1465Axiom. (
apI) We take the following as an axiom:
∀f x y, pair x y ∈ f → y ∈ f x
L1467Axiom. (
apE) We take the following as an axiom:
∀f x y, y ∈ f x → pair x y ∈ f
L1469Axiom. (
apEq) We take the following as an axiom:
∀f x y, y ∈ f x ↔ pair x y ∈ f
L1471Axiom. (
beta) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
L1473Axiom. (
beta0) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x : set, x ∉ X → (λx ∈ X ⇒ F x) x = 0
L1475Axiom. (
proj0_ap_0) We take the following as an axiom:
L1477Axiom. (
proj1_ap_1) We take the following as an axiom:
L1479Axiom. (
pair_ap_0) We take the following as an axiom:
∀x y : set, (pair x y) 0 = x
L1481Axiom. (
pair_ap_1) We take the following as an axiom:
∀x y : set, (pair x y) 1 = y
L1483Axiom. (
pair_ap_n2) We take the following as an axiom:
∀x y i : set, i ∉ 2 → (pair x y) i = 0
L1485Axiom. (
ap0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
L1487Axiom. (
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))
L1489Definition. We define
pair_p to be
λu : set ⇒ pair (u 0) (u 1) = u of type
set → prop.
L1492Axiom. (
pair_p_I) We take the following as an axiom:
L1494Axiom. (
pair_p_I2) We take the following as an axiom:
L1498Definition. We define
tuple_p to be
λn u ⇒ ∀z ∈ u, ∃i ∈ n, ∃x : set, z = pair i x of type
set → set → prop.
L1505Axiom. (
tuple_pair) We take the following as an axiom:
∀x y : set, pair x y = (x,y)
L1507Definition. 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.
L1512Axiom. (
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
L1515Axiom. (
PiE) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, f ∈ (∏x ∈ X, Y x) → (∀u ∈ f, pair_p u ∧ u 0 ∈ X) ∧ (∀x ∈ X, f x ∈ Y x)
L1518Axiom. (
PiEq) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, f ∈ Pi X Y ↔ (∀u ∈ f, pair_p u ∧ u 0 ∈ X) ∧ (∀x ∈ X, f x ∈ Y x)
L1521Axiom. (
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)
L1524Axiom. (
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
L1526Axiom. (
Pi_ext_Subq) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f g ∈ (∏x ∈ X, Y x), (∀x ∈ X, f x ⊆ g x) → f ⊆ g
L1528Axiom. (
Pi_ext) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f g ∈ (∏x ∈ X, Y x), (∀x ∈ X, f x = g x) → f = g
L1530Axiom. (
Pi_eta) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, f ∈ (∏x ∈ X, Y x) → (λx ∈ X ⇒ f x) = f
L1532Definition. 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.
L1539Axiom. (
lamI2) We take the following as an axiom:
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
L1541Axiom. (
lamE2) We take the following as an axiom:
∀X, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = (x,y)
L1543Axiom. (
tuple_2_inj) We take the following as an axiom:
∀x y w z : set, (x,y) = (w,z) → x = w ∧ y = z
Beginning of Section Tuples
L1547Variable x0 x1 : set
End of Section Tuples
L1554Definition. We define
Sep2 to be
λX Y R ⇒ {u ∈ ∑x ∈ X, Y x|R (u 0) (u 1)} of type
set → (set → set) → (set → set → prop) → set.
L1557Axiom. (
Sep2I) We take the following as an axiom:
∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x ∈ X, ∀y ∈ Y x, R x y → (x,y) ∈ Sep2 X Y R
L1560Axiom. (
Sep2E) We take the following as an axiom:
∀X, ∀Y : set → set, ∀R : set → set → prop, ∀u ∈ Sep2 X Y R, ∃x ∈ X, ∃y ∈ Y x, u = (x,y) ∧ R x y
L1563Axiom. (
Sep2E') We take the following as an axiom:
∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → x ∈ X ∧ y ∈ Y x ∧ R x y
L1566Axiom. (
Sep2E'1) We take the following as an axiom:
∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → x ∈ X
L1569Axiom. (
Sep2E'2) We take the following as an axiom:
∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → y ∈ Y x
L1572Axiom. (
Sep2E'3) We take the following as an axiom:
∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → R x y
L1575Definition. We define
set_of_pairs to be
λX ⇒ ∀x ∈ X, ∃y z, x = (y,z) of type
set → prop.
L1585Axiom. (
Sep2_ext) We take the following as an axiom:
∀X, ∀Y : set → set, ∀R R' : set → set → prop, (∀x ∈ X, ∀y ∈ Y x, R x y ↔ R' x y) → Sep2 X Y R = Sep2 X Y R'
L1589Axiom. (
lam_ext_sub) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → (λx ∈ X ⇒ F x) ⊆ (λx ∈ X ⇒ G x)
L1591Axiom. (
lam_ext) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → (λx ∈ X ⇒ F x) = (λx ∈ X ⇒ G x)
L1593Axiom. (
lam_eta) We take the following as an axiom:
∀X, ∀F : set → set, (λx ∈ X ⇒ (λx ∈ X ⇒ F x) x) = (λx ∈ X ⇒ F x)
L1595Axiom. (
tuple_2_eta) We take the following as an axiom:
L1597Definition. We define
lam2 to be
λX Y F ⇒ λx ∈ X ⇒ λy ∈ Y x ⇒ F x y of type
set → (set → set) → (set → set → set) → set.
L1600Axiom. (
beta2) We take the following as an axiom:
∀X, ∀Y : set → set, ∀F : set → set → set, ∀x ∈ X, ∀y ∈ Y x, lam2 X Y F x y = F x y
L1602Axiom. (
lam2_ext) We take the following as an axiom:
∀X, ∀Y : set → set, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y x, F x y = G x y) → lam2 X Y F = lam2 X Y G
L1606Definition. We define
encode_u to be
lam of type
set → (set → set) → set.
L1608Definition. We define
decode_u to be
ap of type
set → set → set.
L1609Definition. We define
encode_b to be
λX F ⇒ lam2 X (λ_ ⇒ X) F of type
set → (set → set → set) → set.
L1611Definition. We define
decode_b to be
λF x y ⇒ F x y of type
set → set → set → set.
L1612Definition. We define
encode_p to be
λX P ⇒ Sep X P of type
set → (set → prop) → set.
L1614Definition. We define
decode_p to be
λP x ⇒ x ∈ P of type
set → set → prop.
L1615Definition. We define
encode_r to be
λX R ⇒ Sep2 X (λ_ ⇒ X) R of type
set → (set → set → prop) → set.
L1617Definition. We define
decode_r to be
λR x y ⇒ (x,y) ∈ R of type
set → set → set → prop.
L1618Definition. We define
encode_c to be
λX C ⇒ Sep (𝒫 X) (λU ⇒ (C (λx ⇒ x ∈ U))) of type
set → ((set → prop) → prop) → set.
L1620Definition. We define
decode_c to be
λC U ⇒ ∃V, (∀x, U x ↔ x ∈ V) ∧ V ∈ C of type
set → (set → prop) → prop.
L1639Axiom. (
encode_c_ext) We take the following as an axiom:
∀X, ∀C C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → encode_c X C = encode_c X C'
L1641Axiom. (
setprod_mon) We take the following as an axiom:
∀X Y : set, X ⊆ Y → ∀Z W : set, Z ⊆ W → X ⨯ Z ⊆ Y ⨯ W
L1643Axiom. (
setprod_mon0) We take the following as an axiom:
∀X Y : set, X ⊆ Y → ∀Z : set, X ⨯ Z ⊆ Y ⨯ Z
L1645Axiom. (
setprod_mon1) We take the following as an axiom:
∀X : set, ∀Z W : set, Z ⊆ W → X ⨯ Z ⊆ X ⨯ W
L1649Axiom. (
Sigma_eta) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (z 0) (z 1) = z
L1651Axiom. (
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}
L1657Axiom. (
Pi_Power_1) We take the following as an axiom:
∀X : set, ∀Y : set → set, (∀x ∈ X, Y x ∈ 𝒫 1) → (∏x ∈ X, Y x) ∈ 𝒫 1
L1659Axiom. (
Pi_0_dom_mon) We take the following as an axiom:
∀X Y : set, ∀A : set → set, X ⊆ Y → (∀y ∈ Y, y ∉ X → 0 ∈ A y) → (∏x ∈ X, A x) ⊆ ∏y ∈ Y, A y
L1662Axiom. (
Pi_cod_mon) We take the following as an axiom:
∀X : set, ∀A B : set → set, (∀x ∈ X, A x ⊆ B x) → (∏x ∈ X, A x) ⊆ ∏x ∈ X, B x
L1664Axiom. (
Pi_0_mon) We take the following as an axiom:
∀X Y : set, ∀A B : set → set, (∀x ∈ X, A x ⊆ B x) → X ⊆ Y → (∀y ∈ Y, y ∉ X → 0 ∈ B y) → (∏x ∈ X, A x) ⊆ ∏y ∈ Y, B y
L1668Axiom. (
setexp_2_eq) We take the following as an axiom:
L1670Axiom. (
setexp_0_dom_mon) We take the following as an axiom:
∀A : set, 0 ∈ A → ∀X Y : set, X ⊆ Y → AX ⊆ AY
L1672Axiom. (
setexp_0_mon) We take the following as an axiom:
∀X Y A B : set, 0 ∈ B → A ⊆ B → X ⊆ Y → AX ⊆ BY
L1676Axiom. (
tupleI0) We take the following as an axiom:
L1678Axiom. (
tupleI1) We take the following as an axiom:
L1680Axiom. (
tupleE) We take the following as an axiom:
L1682Axiom. (
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
L1684Axiom. (
tuple_2_setprod) We take the following as an axiom:
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
L1686Axiom. (
tuple_Sigma_eta) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), (z 0,z 1) = z
L1688Axiom. (
apI2) We take the following as an axiom:
∀f x y, (x,y) ∈ f → y ∈ f x
L1690Axiom. (
apE2) We take the following as an axiom:
∀f x y, y ∈ f x → (x,y) ∈ f
L1692Axiom. (
ap_const_0) We take the following as an axiom:
L1698Axiom. (
tuple_3_eta) We take the following as an axiom:
L1700Axiom. (
tuple_4_eta) We take the following as an axiom:
∀x y z w, (λi ∈ 4 ⇒ (x,y,z,w) i) = (x,y,z,w)
Beginning of Section Tuples
L1704Variable x0 x1 x2 : set
L1735Axiom. (
tuple_6_0_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5) 0 = x0
L1736Axiom. (
tuple_6_1_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5) 1 = x1
L1738Axiom. (
tuple_6_2_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5) 2 = x2
L1740Axiom. (
tuple_6_3_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5) 3 = x3
L1742Axiom. (
tuple_6_4_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5) 4 = x4
L1744Axiom. (
tuple_6_5_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5) 5 = x5
L1748Axiom. (
tuple_7_0_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 0 = x0
L1749Axiom. (
tuple_7_1_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 1 = x1
L1751Axiom. (
tuple_7_2_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 2 = x2
L1753Axiom. (
tuple_7_3_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 3 = x3
L1755Axiom. (
tuple_7_4_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 4 = x4
L1757Axiom. (
tuple_7_5_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 5 = x5
L1759Axiom. (
tuple_7_6_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6) 6 = x6
L1763Axiom. (
tuple_8_0_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 0 = x0
L1765Axiom. (
tuple_8_1_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 1 = x1
L1767Axiom. (
tuple_8_2_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 2 = x2
L1769Axiom. (
tuple_8_3_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 3 = x3
L1771Axiom. (
tuple_8_4_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 4 = x4
L1773Axiom. (
tuple_8_5_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 5 = x5
L1775Axiom. (
tuple_8_6_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 6 = x6
L1777Axiom. (
tuple_8_7_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7) 7 = x7
L1781Axiom. (
tuple_9_0_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 0 = x0
L1782Axiom. (
tuple_9_1_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 1 = x1
L1784Axiom. (
tuple_9_2_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 2 = x2
L1786Axiom. (
tuple_9_3_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 3 = x3
L1788Axiom. (
tuple_9_4_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 4 = x4
L1790Axiom. (
tuple_9_5_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 5 = x5
L1792Axiom. (
tuple_9_6_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 6 = x6
L1794Axiom. (
tuple_9_7_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 7 = x7
L1796Axiom. (
tuple_9_8_eq) We take the following as an axiom:
(x0,x1,x2,x3,x4,x5,x6,x7,x8) 8 = x8
End of Section Tuples
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.
L1814Axiom. (
tuple_3_in_A_3) We take the following as an axiom:
∀x y z A, x ∈ A → y ∈ A → z ∈ A → (x,y,z) ∈ A3
L1816Axiom. (
tuple_3_bij_3) We take the following as an axiom:
∀x y z, x ∈ 3 → y ∈ 3 → z ∈ 3 → x ≠ y → x ≠ z → y ≠ z → bij 3 3 (λi ⇒ (x,y,z) i)
L1818Axiom. (
tuple_4_in_A_4) We take the following as an axiom:
∀x y z w A, x ∈ A → y ∈ A → z ∈ A → w ∈ A → (x,y,z,w) ∈ A4
L1820Axiom. (
tuple_4_bij_4) We take the following as an axiom:
∀x y z w, x ∈ 4 → y ∈ 4 → z ∈ 4 → w ∈ 4 → x ≠ y → x ≠ z → x ≠ w → y ≠ z → y ≠ w → z ≠ w → bij 4 4 (λi ⇒ (x,y,z,w) i)
L1822Axiom. (
iff_refl) We take the following as an axiom:
L1824Axiom. (
iff_sym) We take the following as an axiom:
∀A B : prop, (A ↔ B) → (B ↔ A)
L1826Axiom. (
iff_trans) We take the following as an axiom:
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
L1836Axiom. (
eq_or_nand) We take the following as an axiom:
Primitive. The name
EpsR_i_i_1 is a term of type
(set → set → prop) → set.
Primitive. The name
EpsR_i_i_2 is a term of type
(set → set → prop) → set.
L1844Axiom. (
EpsR_i_i_12) We take the following as an axiom:
Primitive. The name
DescrR_i_io_1 is a term of type
(set → (set → prop) → prop) → set.
Primitive. The name
DescrR_i_io_2 is a term of type
(set → (set → prop) → prop) → set → prop.
L1854Definition. We define
PNoEq_ to be
λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type
set → (set → prop) → (set → prop) → prop.
L1861Axiom. (
PNoEq_ref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
L1863Axiom. (
PNoEq_sym_) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
L1865Axiom. (
PNoEq_tra_) We take the following as an axiom:
L1869Definition. 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.
L1872Axiom. (
PNoLt_E_) We take the following as an axiom:
L1875Axiom. (
PNoLt_irref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
L1877Axiom. (
PNoLt_mon_) We take the following as an axiom:
L1882Axiom. (
PNoLt_tra_) We take the following as an axiom:
Primitive. The name
PNoLt is a term of type
set → (set → prop) → set → (set → prop) → prop.
L1887Axiom. (
PNoLtI1) We take the following as an axiom:
L1890Axiom. (
PNoLtI2) We take the following as an axiom:
L1893Axiom. (
PNoLtI3) We take the following as an axiom:
L1896Axiom. (
PNoLtE) We take the following as an axiom:
L1904Axiom. (
PNoLtE2) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoLt alpha p alpha q → PNoLt_ alpha p q
L1907Axiom. (
PNoLt_irref) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
L1913Axiom. (
PNoLtEq_tra) We take the following as an axiom:
L1915Axiom. (
PNoEqLt_tra) We take the following as an axiom:
L1917Axiom. (
PNoLt_tra) We take the following as an axiom:
L1919Definition. 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.
L1922Axiom. (
PNoLeI1) We take the following as an axiom:
L1925Axiom. (
PNoLeI2) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
L1928Axiom. (
PNoLe_ref) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
L1934Axiom. (
PNoLtLe_tra) We take the following as an axiom:
L1936Axiom. (
PNoLeLt_tra) We take the following as an axiom:
L1938Axiom. (
PNoEqLe_tra) We take the following as an axiom:
L1940Axiom. (
PNoLeEq_tra) We take the following as an axiom:
L1942Axiom. (
PNoLe_tra) We take the following as an axiom:
L1944Definition. 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.
L1947Definition. 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.
L1950Axiom. (
PNoLe_downc) We take the following as an axiom:
L1954Axiom. (
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
L1956Axiom. (
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
L1958Axiom. (
PNoLe_upc) We take the following as an axiom:
L1962Definition. 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.
L2005Axiom. (
PNo_extend0_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
L2007Axiom. (
PNo_extend1_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
L2015Definition. 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.
L2107Definition. 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.
Primitive. The name
PNo_bd is a term of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → set.
Primitive. The name
PNo_pred is a term of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.
L2139Axiom. (
PNo_bd_pred) We take the following as an axiom:
L2146Axiom. (
PNo_bd_ord) We take the following as an axiom:
L2168Axiom. (
PNo_bd_In) We take the following as an axiom:
L2175Definition. We define
PNoCutL to be
λalpha p beta q ⇒ beta ∈ alpha ∧ PNoLt beta q alpha p of type
set → (set → prop) → set → (set → prop) → prop.
L2178Definition. We define
PNoCutR to be
λalpha p beta q ⇒ beta ∈ alpha ∧ PNoLt alpha p beta q of type
set → (set → prop) → set → (set → prop) → prop.
Beginning of Section TaggedSets
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
L2213Definition. We define
SNoElts_ to be
λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type
set → set.
L2215Axiom. (
SNoElts_mon) We take the following as an axiom:
L2224Axiom. (
PNoEq_PSNo) We take the following as an axiom:
L2226Axiom. (
SNo_PSNo) We take the following as an axiom:
Primitive. The name
SNo is a term of type
set → prop.
L2233Axiom. (
SNo_SNo) We take the following as an axiom:
Primitive. The name
SNoLev is a term of type
set → set.
L2240Axiom. (
SNoLev_uniq) We take the following as an axiom:
L2242Axiom. (
SNoLev_prop) We take the following as an axiom:
L2246Axiom. (
SNoLev_) We take the following as an axiom:
L2250Axiom. (
SNoLev_PSNo) We take the following as an axiom:
L2252Axiom. (
SNo_Subq) We take the following as an axiom:
L2254Definition. We define
SNoEq_ to be
λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type
set → set → set → prop.
L2257Axiom. (
SNoEq_I) We take the following as an axiom:
L2259Axiom. (
SNoEq_E) We take the following as an axiom:
L2261Axiom. (
SNoEq_E1) We take the following as an axiom:
L2263Axiom. (
SNoEq_E2) We take the following as an axiom:
L2267Axiom. (
SNo_eq) We take the following as an axiom:
Notation. We use
'' as a postfix operator with priority 100 corresponding to applying term
ctag.
L2280Definition. We define
SNo_pair to be
λx y ⇒ x ∪ {u ''|u ∈ y} of type
set → set → set.
L2286Axiom. (
SNo_pair_0) We take the following as an axiom:
End of Section TaggedSets
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
L2300Axiom. (
SNoLtLe) We take the following as an axiom:
L2302Axiom. (
SNoLeE) We take the following as an axiom:
L2304Axiom. (
SNoEq_ref_) We take the following as an axiom:
L2306Axiom. (
SNoEq_sym_) We take the following as an axiom:
L2308Axiom. (
SNoEq_tra_) We take the following as an axiom:
L2310Axiom. (
SNoLtE) We take the following as an axiom:
L2317Axiom. (
SNoLtI2) We take the following as an axiom:
L2325Axiom. (
SNoLtI3) We take the following as an axiom:
L2331Axiom. (
SNoLt_irref) We take the following as an axiom:
L2341Axiom. (
SNoLt_tra) We take the following as an axiom:
L2343Axiom. (
SNoLe_ref) We take the following as an axiom:
L2347Axiom. (
SNoLtLe_tra) We take the following as an axiom:
L2349Axiom. (
SNoLeLt_tra) We take the following as an axiom:
L2351Axiom. (
SNoLe_tra) We take the following as an axiom:
L2353Axiom. (
SNoLtLe_or) We take the following as an axiom:
L2366Definition. 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.
L2389Axiom. (
SNoCutP_L_0) We take the following as an axiom:
L2391Axiom. (
SNoCutP_0_R) We take the following as an axiom:
L2392Axiom. (
SNoCutP_0_0) We take the following as an axiom:
L2393Axiom. (
SNoCut_0_0) We take the following as an axiom:
L2400Axiom. (
SNoS_E) We take the following as an axiom:
Beginning of Section TaggedSets2
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
L2407Axiom. (
SNoS_I) We take the following as an axiom:
L2409Axiom. (
SNoS_I2) We take the following as an axiom:
L2411Axiom. (
SNoS_Subq) We take the following as an axiom:
L2415Axiom. (
SNoS_E2) We take the following as an axiom:
L2420Axiom. (
SNoS_In_neq) We take the following as an axiom:
L2422Axiom. (
SNoS_SNoLev) We take the following as an axiom:
L2429Axiom. (
SNoL_E) We take the following as an axiom:
L2434Axiom. (
SNoR_E) We take the following as an axiom:
L2439Axiom. (
SNoL_SNoS) We take the following as an axiom:
L2441Axiom. (
SNoR_SNoS) We take the following as an axiom:
L2442Axiom. (
SNoL_SNoS_) We take the following as an axiom:
L2443Axiom. (
SNoR_SNoS_) We take the following as an axiom:
L2444Axiom. (
SNoL_I) We take the following as an axiom:
L2446Axiom. (
SNoR_I) We take the following as an axiom:
L2448Axiom. (
SNo_eta) We take the following as an axiom:
L2463Axiom. (
SNoCut_Le) We take the following as an axiom:
L2468Axiom. (
SNoCut_ext) We take the following as an axiom:
L2500Axiom. (
SNo_0) We take the following as an axiom:
L2502Axiom. (
SNoLev_0) We take the following as an axiom:
L2503Axiom. (
SNoL_0) We take the following as an axiom:
L2504Axiom. (
SNoR_0) We take the following as an axiom:
L2505Axiom. (
SNoL_1) We take the following as an axiom:
L2506Axiom. (
SNoR_1) We take the following as an axiom:
L2536Axiom. (
nat_p_SNo) We take the following as an axiom:
L2538Axiom. (
omega_SNo) We take the following as an axiom:
L2544Axiom. (
eps_0_1) We take the following as an axiom:
L2545Axiom. (
SNo__eps_) We take the following as an axiom:
L2546Axiom. (
SNo_eps_) We take the following as an axiom:
L2547Axiom. (
SNoLev_eps_) We take the following as an axiom:
L2550Axiom. (
SNo_eps_pos) We take the following as an axiom:
End of Section TaggedSets2
L2555Axiom. (
ordinal_SNo) We take the following as an axiom:
L2567Axiom. (
SNo_etaE) We take the following as an axiom:
L2576Axiom. (
SNo_ind) We take the following as an axiom:
∀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
Beginning of Section SurrealRecI
L2587Variable F : set → (set → set) → set
Primitive. The name
SNo_rec_i is a term of type
set → set.
L2592Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
End of Section SurrealRecI
Beginning of Section SurrealRecII
L2602Variable F : set → (set → (set → set)) → (set → set)
Primitive. The name
SNo_rec_ii is a term of type
set → (set → set).
L2607Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
End of Section SurrealRecII
Beginning of Section SurrealRec2
L2617Variable F : set → set → (set → set → set) → set
Primitive. The name
SNo_rec2 is a term of type
set → set → set.
L2628Axiom. (
SNo_rec2_eq) We take the following as an axiom:
End of Section SurrealRec2
L2653Axiom. (
SNoLev_ind) We take the following as an axiom:
L2658Axiom. (
SNoLev_ind2) We take the following as an axiom:
L2666Axiom. (
SNoLev_ind3) We take the following as an axiom:
∀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
L2678Axiom. (
SNo_1) We take the following as an axiom:
L2680Axiom. (
SNo_2) We take the following as an axiom:
L2681Axiom. (
SNo_omega) We take the following as an axiom:
L2682Axiom. (
SNoLt_0_1) We take the following as an axiom:
L2683Axiom. (
SNoLt_0_2) We take the following as an axiom:
L2684Axiom. (
SNoLt_1_2) We take the following as an axiom:
L2687Axiom. (
restr_SNo_) We take the following as an axiom:
L2688Axiom. (
restr_SNo) We take the following as an axiom:
L2690Axiom. (
restr_SNoEq) We take the following as an axiom:
Primitive. The name
pack_e is a term of type
set → set → set.
L2699Axiom. (
pack_e_0_eq) We take the following as an axiom:
L2703Axiom. (
pack_e_1_eq) We take the following as an axiom:
L2707Axiom. (
pack_e_inj) We take the following as an axiom:
L2709Definition. We define
struct_e to be
λS ⇒ ∀q : set → prop, (∀X : set, ∀c : set, c ∈ X → q (pack_e X c)) → q S of type
set → prop.
Primitive. The name
unpack_e_i is a term of type
set → (set → set → set) → set.
Primitive. The name
unpack_e_o is a term of type
set → (set → set → prop) → prop.
Primitive. The name
pack_u is a term of type
set → (set → set) → set.
L2732Axiom. (
pack_u_0_eq) We take the following as an axiom:
∀S X, ∀F : set → set, S = pack_u X F → X = S 0
L2736Axiom. (
pack_u_1_eq) We take the following as an axiom:
L2740Axiom. (
pack_u_inj) We take the following as an axiom:
L2742Axiom. (
pack_u_ext) We take the following as an axiom:
L2746Definition. We define
struct_u to be
λS ⇒ ∀q : set → prop, (∀X, ∀F : set → set, (∀x ∈ X, F x ∈ X) → q (pack_u X F)) → q S of type
set → prop.
Primitive. The name
unpack_u_i is a term of type
set → (set → (set → set) → set) → set.
L2757Axiom. (
unpack_u_i_eq) We take the following as an axiom:
∀Phi : set → (set → set) → set, ∀X, ∀F : set → set, (∀F' : set → set, (∀x ∈ X, F x = F' x) → Phi X F' = Phi X F) → unpack_u_i (pack_u X F) Phi = Phi X F
Primitive. The name
unpack_u_o is a term of type
set → (set → (set → set) → prop) → prop.
L2766Axiom. (
unpack_u_o_eq) We take the following as an axiom:
∀Phi : set → (set → set) → prop, ∀X, ∀F : set → set, (∀F' : set → set, (∀x ∈ X, F x = F' x) → Phi X F' = Phi X F) → unpack_u_o (pack_u X F) Phi = Phi X F
Primitive. The name
pack_b is a term of type
set → (set → set → set) → set.
L2775Axiom. (
pack_b_0_eq) We take the following as an axiom:
∀S X, ∀F : set → set → set, S = pack_b X F → X = S 0
L2779Axiom. (
pack_b_1_eq) We take the following as an axiom:
L2783Axiom. (
pack_b_inj) We take the following as an axiom:
∀X X', ∀F F' : set → set → set, pack_b X F = pack_b X' F' → X = X' ∧ ∀x y ∈ X, F x y = F' x y
L2785Axiom. (
pack_b_ext) We take the following as an axiom:
∀X, ∀F F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → pack_b X F = pack_b X F'
L2789Definition. We define
struct_b to be
λS ⇒ ∀q : set → prop, (∀X : set, ∀F : set → set → set, (∀x y ∈ X, F x y ∈ X) → q (pack_b X F)) → q S of type
set → prop.
Primitive. The name
unpack_b_i is a term of type
set → (set → (set → set → set) → set) → set.
L2800Axiom. (
unpack_b_i_eq) We take the following as an axiom:
∀Phi : set → (set → set → set) → set, ∀X, ∀F : set → set → set, (∀F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → Phi X F' = Phi X F) → unpack_b_i (pack_b X F) Phi = Phi X F
Primitive. The name
unpack_b_o is a term of type
set → (set → (set → set → set) → prop) → prop.
L2809Axiom. (
unpack_b_o_eq) We take the following as an axiom:
∀Phi : set → (set → set → set) → prop, ∀X, ∀F : set → set → set, (∀F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → Phi X F' = Phi X F) → unpack_b_o (pack_b X F) Phi = Phi X F
Primitive. The name
pack_p is a term of type
set → (set → prop) → set.
L2818Axiom. (
pack_p_0_eq) We take the following as an axiom:
∀S X, ∀P : set → prop, S = pack_p X P → X = S 0
L2822Axiom. (
pack_p_1_eq) We take the following as an axiom:
L2826Axiom. (
pack_p_inj) We take the following as an axiom:
L2828Axiom. (
pack_p_ext) We take the following as an axiom:
L2832Definition. We define
struct_p to be
λS ⇒ ∀q : set → prop, (∀X : set, ∀P : set → prop, q (pack_p X P)) → q S of type
set → prop.
Primitive. The name
unpack_p_i is a term of type
set → (set → (set → prop) → set) → set.
L2841Axiom. (
unpack_p_i_eq) We take the following as an axiom:
∀Phi : set → (set → prop) → set, ∀X, ∀P : set → prop, (∀P' : set → prop, (∀x ∈ X, P x ↔ P' x) → Phi X P' = Phi X P) → unpack_p_i (pack_p X P) Phi = Phi X P
Primitive. The name
unpack_p_o is a term of type
set → (set → (set → prop) → prop) → prop.
L2850Axiom. (
unpack_p_o_eq) We take the following as an axiom:
∀Phi : set → (set → prop) → prop, ∀X, ∀P : set → prop, (∀P' : set → prop, (∀x ∈ X, P x ↔ P' x) → Phi X P' = Phi X P) → unpack_p_o (pack_p X P) Phi = Phi X P
Primitive. The name
pack_r is a term of type
set → (set → set → prop) → set.
L2859Axiom. (
pack_r_0_eq) We take the following as an axiom:
∀S X, ∀R : set → set → prop, S = pack_r X R → X = S 0
L2861Axiom. (
pack_r_0_eq2) We take the following as an axiom:
∀X, ∀R : set → set → prop, X = pack_r X R 0
L2863Axiom. (
pack_r_1_eq) We take the following as an axiom:
L2867Axiom. (
pack_r_inj) We take the following as an axiom:
∀X X', ∀R R' : set → set → prop, pack_r X R = pack_r X' R' → X = X' ∧ ∀x y ∈ X, R x y = R' x y
L2869Axiom. (
pack_r_ext) We take the following as an axiom:
∀X, ∀R R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → pack_r X R = pack_r X R'
L2873Definition. We define
struct_r to be
λS ⇒ ∀q : set → prop, (∀X : set, ∀R : set → set → prop, q (pack_r X R)) → q S of type
set → prop.
Primitive. The name
unpack_r_i is a term of type
set → (set → (set → set → prop) → set) → set.
L2882Axiom. (
unpack_r_i_eq) We take the following as an axiom:
∀Phi : set → (set → set → prop) → set, ∀X, ∀R : set → set → prop, (∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X R' = Phi X R) → unpack_r_i (pack_r X R) Phi = Phi X R
Primitive. The name
unpack_r_o is a term of type
set → (set → (set → set → prop) → prop) → prop.
L2891Axiom. (
unpack_r_o_eq) We take the following as an axiom:
∀Phi : set → (set → set → prop) → prop, ∀X, ∀R : set → set → prop, (∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X R' = Phi X R) → unpack_r_o (pack_r X R) Phi = Phi X R
Primitive. The name
pack_c is a term of type
set → ((set → prop) → prop) → set.
L2900Axiom. (
pack_c_0_eq) We take the following as an axiom:
∀S X, ∀C : (set → prop) → prop, S = pack_c X C → X = S 0
L2902Axiom. (
pack_c_0_eq2) We take the following as an axiom:
∀X, ∀C : (set → prop) → prop, X = pack_c X C 0
L2904Axiom. (
pack_c_1_eq) We take the following as an axiom:
∀S X, ∀C : (set → prop) → prop, S = pack_c X C → ∀U : set → prop, (∀x, U x → x ∈ X) → C U = decode_c (S 1) U
L2906Axiom. (
pack_c_1_eq2) We take the following as an axiom:
∀X, ∀C : (set → prop) → prop, ∀U : set → prop, (∀x, U x → x ∈ X) → C U = decode_c (pack_c X C 1) U
L2908Axiom. (
pack_c_inj) We take the following as an axiom:
∀X X', ∀C C' : (set → prop) → prop, pack_c X C = pack_c X' C' → X = X' ∧ ∀U : set → prop, (∀x, U x → x ∈ X) → C U = C' U
L2910Axiom. (
pack_c_ext) We take the following as an axiom:
∀X, ∀C C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → pack_c X C = pack_c X C'
L2914Definition. We define
struct_c to be
λS ⇒ ∀q : set → prop, (∀X : set, ∀C : (set → prop) → prop, q (pack_c X C)) → q S of type
set → prop.
Primitive. The name
unpack_c_i is a term of type
set → (set → ((set → prop) → prop) → set) → set.
L2923Axiom. (
unpack_c_i_eq) We take the following as an axiom:
∀Phi : set → ((set → prop) → prop) → set, ∀X, ∀C : (set → prop) → prop, (∀C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → Phi X C' = Phi X C) → unpack_c_i (pack_c X C) Phi = Phi X C
Primitive. The name
unpack_c_o is a term of type
set → (set → ((set → prop) → prop) → prop) → prop.
L2932Axiom. (
unpack_c_o_eq) We take the following as an axiom:
∀Phi : set → ((set → prop) → prop) → prop, ∀X, ∀C : (set → prop) → prop, (∀C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → Phi X C' = Phi X C) → unpack_c_o (pack_c X C) Phi = Phi X C
Primitive. The name
canonical_elt is a term of type
(set → set → prop) → set → set.
Primitive. The name
quotient is a term of type
(set → set → prop) → set → prop.
L2947Axiom. (
quotient_prop1) We take the following as an axiom:
∀R : set → set → prop, ∀x : set, quotient R x → R x x
Primitive. The name
canonical_elt_def is a term of type
(set → set → prop) → (set → set) → set → set.
Primitive. The name
quotient_def is a term of type
(set → set → prop) → (set → set) → set → prop.
Beginning of Section explicit_Nats
L2988Variable S : set → set
L2992Axiom. (
explicit_Nats_I) We take the following as an axiom:
(base ∈ N) → (∀m ∈ N, S m ∈ N) → (∀m ∈ N, S m ≠ base) → (∀m n ∈ N, S m = S n → m = n) → (∀p : set → prop, p base → (∀m, p m → p (S m)) → (∀m ∈ N, p m)) → explicit_Nats
L3000Axiom. (
explicit_Nats_E) We take the following as an axiom:
∀q : prop, (explicit_Nats → (base ∈ N) → (∀m ∈ N, S m ∈ N) → (∀m ∈ N, S m ≠ base) → (∀m n ∈ N, S m = S n → m = n) → (∀p : set → prop, p base → (∀m, p m → p (S m)) → (∀m ∈ N, p m)) → q) → explicit_Nats → q
End of Section explicit_Nats
Beginning of Section explicit_Nats_zero
L3040Variable S : set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
explicit_Nats_zero_plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
explicit_Nats_zero_mult.
End of Section explicit_Nats_zero
Beginning of Section explicit_Nats_one
L3067Variable S : set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
explicit_Nats_one_plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
explicit_Nats_one_mult.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
explicit_Nats_one_exp.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
explicit_Nats_one_lt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
explicit_Nats_one_le.
End of Section explicit_Nats_one
Beginning of Section explicit_Nats_transfer
L3110Variable S : set → set
L3112Variable base' : set
L3113Variable S' : set → set
L3114Variable f : set → set
End of Section explicit_Nats_transfer
Beginning of Section AssocComm
L3124Variable plus : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
L3127Axiom. (
AssocComm_identities) We take the following as an axiom:
(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → ∀p : prop, ((∀x y z ∈ R, x + y + z = y + x + z) → (∀x y z ∈ R, x + y + z = z + x + y) → (∀x y z w ∈ R, (x + y) + (z + w) = (x + z) + (y + w)) → (∀x y z w ∈ R, x + y + z + w = w + x + y + z) → (∀x y z w ∈ R, x + y + z + w = z + w + x + y) → p) → p
End of Section AssocComm
Beginning of Section Group1
Beginning of Section Group1Explicit
L3148Variable op : set → set → set
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
op.
L3151Definition. We define
explicit_Group to be
(∀a b ∈ G, a * b ∈ G) ∧ (∀a b c ∈ G, a * (b * c) = (a * b) * c) ∧ ∃e ∈ G, (∀a ∈ G, e * a = a ∧ a * e = a) ∧ (∀a ∈ G, ∃b ∈ G, a * b = e ∧ b * a = e) of type
prop.
Notation. We use
- as a postfix operator with priority 340 corresponding to applying term
explicit_Group_inverse.
End of Section Group1Explicit
Beginning of Section Group1Explicit2
L3204Variable op : set → set → set
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
op.
Beginning of Section Group1Explicit2RepIndep
L3209Variable op' : set → set → set
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
op'.
L3212Hypothesis Hopop' : ∀a b ∈ G, a * b = a ⨯ b
End of Section Group1Explicit2RepIndep
End of Section Group1Explicit2
Beginning of Section Group1Explicit3RepIndep
L3238Variable op : set → set → set
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
op.
L3241Variable op' : set → set → set
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
op'.
L3244Hypothesis Hopop' : ∀a b ∈ G, a * b = a ⨯ b
End of Section Group1Explicit3RepIndep
End of Section Group1
L3263Axiom. (
GroupI) We take the following as an axiom:
L3265Axiom. (
GroupE) We take the following as an axiom:
Beginning of Section Group2
L3276Variable op : set → set → set
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
op.
Notation. We use
- as a postfix operator with priority 340 corresponding to applying term
explicit_Group_inverse G op.
End of Section Group2
Beginning of Section Group3
L3306Variable op op' : set → set → set
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
op.
Notation. We use
- as a postfix operator with priority 340 corresponding to applying term
explicit_Group_inverse G op.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
op'.
Notation. We use
:-: as a postfix operator with priority 340 corresponding to applying term
explicit_Group_inverse G op'.
L3314Hypothesis Hopop' : ∀a b ∈ G, a * b = a ⨯ b
End of Section Group3
Notation. We use
≤ as an infix operator with priority 400 and no associativity corresponding to applying term
subgroup.
L3350Axiom. (
subgroup_E) We take the following as an axiom:
∀H G, H ≤ G → ∀q : set → set → prop, (∀H G, ∀op : set → set → set, (∀a b ∈ G, op a b ∈ G) → Group (pack_b H op) → H ⊆ G → q (pack_b H op) (pack_b G op)) → q H G
Beginning of Section Group4
L3366Let G ≝ {f ∈ AA|bij A A (λx ⇒ f x)}
L3368Let op ≝ λf g : set ⇒ λx ∈ A ⇒ g (f x)
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
op.
L3380Let H ≝ {f ∈ AA|bij A A (λx ⇒ f x) ∧ ∀x ∈ B, f x = x}
End of Section Group4
L3386Definition. We define
symgroup to be
λA ⇒ pack_b {f ∈ AA|bij A A (λx ⇒ f x)} (λf g ⇒ λx ∈ A ⇒ g (f x)) of type
set → set.
L3425Definition. We define
Group_carrier to be
λGs ⇒ Gs 0 of type
set → set.
L3427Definition. We define
Group_op to be
λGs ⇒ decode_b (Gs 1) of type
set → set → set → set.
Beginning of Section Group2
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
Group_op Gs.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
Group_op Gs'.
End of Section Group2
Beginning of Section explicit_Ring
L3457Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L3464Axiom. (
explicit_Ring_I) We take the following as an axiom:
(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → explicit_Ring
L3476Axiom. (
explicit_Ring_E) We take the following as an axiom:
∀q : prop, (explicit_Ring → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → q) → explicit_Ring → q
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
explicit_Ring_minus.
End of Section explicit_Ring
Beginning of Section explicit_Ring_with_id
L3516Variable zero one : set
L3518Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L3525Axiom. (
explicit_Ring_with_id_I) We take the following as an axiom:
(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x * one = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → explicit_Ring_with_id
L3541Axiom. (
explicit_Ring_with_id_E) We take the following as an axiom:
∀q : prop, (explicit_Ring_with_id → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x * one = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → q) → explicit_Ring_with_id → q
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
explicit_Ring_minus R zero plus mult.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
explicit_Ring_with_id_exp_nat.
End of Section explicit_Ring_with_id
Beginning of Section explicit_Ring_with_id_RepIndep2
L3605Variable zero one : set
L3607Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L3611Variable plus' mult' : set → set → set
Notation. We use
+ as an infix operator with priority 355 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L3615Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b
L3617Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b
End of Section explicit_Ring_with_id_RepIndep2
Beginning of Section explicit_CRing_with_id
L3626Variable zero one : set
L3628Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L3635Axiom. (
explicit_CRing_with_id_I) We take the following as an axiom:
(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → explicit_CRing_with_id
L3650Axiom. (
explicit_CRing_with_id_E) We take the following as an axiom:
∀q : prop, (explicit_CRing_with_id → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → q) → explicit_CRing_with_id → q
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
explicit_Ring_minus R zero plus mult.
End of Section explicit_CRing_with_id
Beginning of Section explicit_CRing_with_id_RepIndep2
L3705Variable zero one : set
L3707Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L3711Variable plus' mult' : set → set → set
Notation. We use
+ as an infix operator with priority 355 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L3715Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b
L3717Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b
End of Section explicit_CRing_with_id_RepIndep2
Primitive. The name
pack_b_b_e is a term of type
set → (set → set → set) → (set → set → set) → set → set.
Primitive. The name
struct_b_b_e is a term of type
set → prop.
Primitive. The name
unpack_b_b_e_i is a term of type
set → (set → (set → set → set) → (set → set → set) → set → set) → set.
Primitive. The name
unpack_b_b_e_o is a term of type
set → (set → (set → set → set) → (set → set → set) → set → prop) → prop.
Primitive. The name
pack_b_b_e_e is a term of type
set → (set → set → set) → (set → set → set) → set → set → set.
Primitive. The name
unpack_b_b_e_e_i is a term of type
set → (set → (set → set → set) → (set → set → set) → set → set → set) → set.
Primitive. The name
unpack_b_b_e_e_o is a term of type
set → (set → (set → set → set) → (set → set → set) → set → set → prop) → prop.
Beginning of Section explicit_Reals
L3779Variable zero one : set
L3781Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L3788Axiom. (
explicit_Field_I) We take the following as an axiom:
(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x ≠ zero → ∃y ∈ R, x * y = one) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → explicit_Field
L3804Axiom. (
explicit_Field_E) We take the following as an axiom:
∀q : prop, (explicit_Field → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x ≠ zero → ∃y ∈ R, x * y = one) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → q) → explicit_Field → q
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
explicit_Field_minus.
L3850Variable leq : set → set → prop
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
leq.
L3888Definition. We define
lt to be
λx y ⇒ x ≤ y ∧ x ≠ y of type
set → set → prop.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
lt.
L3896Let Npos ≝ {n ∈ N|n ≠ zero}
End of Section explicit_Reals
Beginning of Section CRing_with_id
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
CRing_with_id_plus Rs.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
CRing_with_id_mult Rs.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
CRing_with_id_omega_exp.
End of Section CRing_with_id
Beginning of Section explicit_Reals
L4025Variable zero one : set
L4027Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
explicit_Field_minus R zero one plus mult.
L4033Variable leq : set → set → prop
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
leq.
L4039Let Npos ≝ {n ∈ N|n ≠ zero}
Beginning of Section explicit_Reals_Q_min_props
L4073Let Npos' ≝ {n ∈ N'|n ≠ zero}
End of Section explicit_Reals_Q_min_props
End of Section explicit_Reals
Beginning of Section explicit_Field_transfer
L4100Variable zero one : set
L4102Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L4108Variable zero' one' : set
L4110Variable plus' mult' : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L4114Variable f : set → set
End of Section explicit_Field_transfer
Beginning of Section explicit_Field_RepIndep2
L4130Variable zero one : set
L4132Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L4136Variable plus' mult' : set → set → set
Notation. We use
+ as an infix operator with priority 355 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L4140Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b
L4142Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b
End of Section explicit_Field_RepIndep2
Beginning of Section explicit_OrderedField_transfer
L4159Variable zero one : set
L4161Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L4165Variable leq : set → set → prop
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
leq.
L4172Variable zero' one' : set
L4174Variable plus' mult' : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L4178Variable leq' : set → set → prop
L4180Variable f : set → set
End of Section explicit_OrderedField_transfer
L4193Definition. We define
Field_carrier to be
λFs ⇒ Fs 0 of type
set → set.
L4197Definition. We define
Field_zero to be
λFs ⇒ Fs 3 of type
set → set.
L4198Definition. We define
Field_one to be
λFs ⇒ Fs 4 of type
set → set.
Primitive. The name
Field_minus is a term of type
set → set → set.
Beginning of Section Field
L4206Hypothesis HFs : Field Fs
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
Field_plus Fs.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
Field_mult Fs.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
Field_minus Fs.
L4214Axiom. (
Field_eta) We take the following as an axiom:
L4228Axiom. (
Field_one_L) We take the following as an axiom:
Primitive. The name
Field_div is a term of type
set → set → set.
Notation. We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
Field_div.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
CRing_with_id_omega_exp Fs.
L4267Axiom. (
Field_dist_R) We take the following as an axiom:
∀x y z ∈ F, (x + y) * z = x * z + y * z
End of Section Field
Beginning of Section Field2
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
Field_plus Fs.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
Field_mult Fs.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
Field_plus Fs'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
Field_mult Fs'.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
Field_minus Fs.
Notation. We use
:-: as a prefix operator with priority 358 corresponding to applying term
Field_minus Fs'.
Primitive. The name
subfield is a term of type
prop.
L4297Axiom. (
subfield_I) We take the following as an axiom:
L4303Axiom. (
subfield_E) We take the following as an axiom:
subfield → ∀p : prop, (Field Fs → Field Fs' → F ⊆ F' → zero = zero' → one = one' → (∀a b ∈ F, a + b = a + b) → (∀a b ∈ F, a * b = a ⨯ b) → p) → p
Primitive. The name
Field_Hom is a term of type
set → prop.
L4313Axiom. (
Field_Hom_I) We take the following as an axiom:
∀g, Field Fs → Field Fs' → g ∈ F'F → g zero = zero' → g one = one' → (∀a b ∈ F, g (a + b) = g a + g b) → (∀a b ∈ F, g (a * b) = g a ⨯ g b) → Field_Hom g
L4322Axiom. (
Field_Hom_E) We take the following as an axiom:
End of Section Field2
L4368Axiom. (
radical_field_extension_E) We take the following as an axiom:
∀Fs Fs', radical_field_extension Fs Fs' → ∀p : prop, (Field Fs → Field Fs' → subfield Fs Fs' → ∀r ∈ ω, ∀Fseq, Fseq 0 = Fs → Fseq r = Fs' → (∀i ∈ ordsucc r, Field (Fseq i)) → (∀i ∈ ordsucc r, ∀j ∈ ordsucc i, subfield (Fseq j) (Fseq i)) → (∀i ∈ r, ∃a ∈ Field_carrier (Fseq (ordsucc i)), ∃n ∈ ω, CRing_with_id_omega_exp (Fseq (ordsucc i)) a n ∈ Field_carrier (Fseq i) ∧ Field_extension_by_1 (Fseq i) (Fseq (ordsucc i)) a) → p) → p
L4397Definition. We define
lam_comp to be
λA f g ⇒ λx ∈ A ⇒ f (g x) of type
set → set → set → set.
L4399Definition. We define
lam_id to be
λA ⇒ λx ∈ A ⇒ x of type
set → set.
Beginning of Section explicit_Reals_transfer
L4418Variable zero one : set
L4420Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L4424Variable leq : set → set → prop
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
leq.
L4431Variable zero' one' : set
L4433Variable plus' mult' : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L4437Variable leq' : set → set → prop
L4439Variable f : set → set
End of Section explicit_Reals_transfer
Beginning of Section explicit_Complex
L4456Variable Re Im : set → set
L4458Variable zero one i : set
L4459Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
End of Section explicit_Complex
Beginning of Section RealsToComplex
L4485Variable zero one : set
L4487Variable plus mult : set → set → set
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
explicit_Field_minus R zero one plus mult.
L4493Variable leq : set → set → prop
L4495Variable pa : set → set → set
L4499Let Re : set → set ≝ λz ⇒ Eps_i (λx ⇒ x ∈ R ∧ ∃y ∈ R, z = pa x y)
L4501Let Im : set → set ≝ λz ⇒ Eps_i (λy ⇒ y ∈ R ∧ z = pa (Re z) y)
L4502Let Re' : set → set ≝ λz ⇒ pa (Re z) zero
L4503Let Im' : set → set ≝ λz ⇒ pa (Im z) zero
L4504Let R' ≝ {z ∈ C|Re' z = z}
L4506Let zero' : set ≝ pa zero zero
L4508Let one' : set ≝ pa one zero
L4509Let i' : set ≝ pa zero one
L4510Let plus' : set → set → set ≝ λz w ⇒ pa (Re z + Re w) (Im z + Im w)
L4511Let mult' : set → set → set ≝ λz w ⇒ pa (Re z * Re w + - (Im z * Im w)) (Re z * Im w + Im z * Re w)
End of Section RealsToComplex
Beginning of Section SurrealArithmetic
Primitive. The name
minus_SNo is a term of type
set → set.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
L4559Axiom. (
minus_SNo_0) We take the following as an axiom:
L4560Axiom. (
SNo_momega) We take the following as an axiom:
Primitive. The name
add_SNo is a term 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_SNo.
L4574Axiom. (
add_SNo_eq) We take the following as an axiom:
L4585Axiom. (
SNo_add_SNo) We take the following as an axiom:
L4589Axiom. (
add_SNo_Lt1) We take the following as an axiom:
L4591Axiom. (
add_SNo_Le1) We take the following as an axiom:
L4593Axiom. (
add_SNo_Lt2) We take the following as an axiom:
L4595Axiom. (
add_SNo_Le2) We take the following as an axiom:
L4601Axiom. (
add_SNo_Lt3) We take the following as an axiom:
L4603Axiom. (
add_SNo_Le3) We take the following as an axiom:
L4611Axiom. (
add_SNo_com) We take the following as an axiom:
L4613Axiom. (
add_SNo_0L) We take the following as an axiom:
L4615Axiom. (
add_SNo_0R) We take the following as an axiom:
Primitive. The name
mul_SNo is a term 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_SNo.
L4723Axiom. (
mul_SNo_eq) We take the following as an axiom:
L4732Axiom. (
mul_SNo_eq_2) We take the following as an axiom:
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
L4753Axiom. (
mul_SNo_prop_1) We take the following as an axiom:
∀x, SNo x → ∀y, SNo y → ∀p : prop, (SNo (x * y) → (∀u ∈ SNoL x, ∀v ∈ SNoL y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoR x, ∀v ∈ SNoR y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoL x, ∀v ∈ SNoR y, x * y + u * v < u * y + x * v) → (∀u ∈ SNoR x, ∀v ∈ SNoL y, x * y + u * v < u * y + x * v) → p) → p
L4763Axiom. (
SNo_mul_SNo) We take the following as an axiom:
L4765Axiom. (
mul_SNo_eq_3) We take the following as an axiom:
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, SNoCutP L R → (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
L4786Axiom. (
mul_SNo_Lt) We take the following as an axiom:
L4789Axiom. (
mul_SNo_Le) We take the following as an axiom:
L4816Axiom. (
mul_SNo_com) We take the following as an axiom:
Primitive. The name
abs_SNo is a term of type
set → set.
L4843Axiom. (
abs_SNo_0) We take the following as an axiom:
L4844Axiom. (
pos_abs_SNo) We take the following as an axiom:
L4845Axiom. (
neg_abs_SNo) We take the following as an axiom:
L4846Axiom. (
SNo_abs_SNo) We take the following as an axiom:
L4847Axiom. (
abs_SNo_Lev) We take the following as an axiom:
L4852Axiom. (
eps_SNo_eq) We take the following as an axiom:
L4853Axiom. (
eps_SNoCutP) We take the following as an axiom:
L4854Axiom. (
eps_SNoCut) We take the following as an axiom:
L4856Axiom. (
SNo_eps_1) We take the following as an axiom:
L4866Definition. We define
SNo_ord_seq to be
λlambda s ⇒ ∀alpha ∈ lambda, SNo (s alpha) of type
set → (set → set) → prop.
Notation. We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
div_SNo.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
End of Section SurrealArithmetic
L4895Axiom. (
CSNo_I) We take the following as an axiom:
L4897Axiom. (
CSNo_E) We take the following as an axiom:
L4901Axiom. (
SNo_CSNo) We take the following as an axiom:
Beginning of Section Complex
L4917Axiom. (
CSNo_Re1) We take the following as an axiom:
L4919Axiom. (
CSNo_Re2) We take the following as an axiom:
∀x y, SNo x → SNo y → Re (pa x y) = x
L4920Axiom. (
CSNo_Im1) We take the following as an axiom:
∀z, CSNo z → SNo (Im z) ∧ z = pa (Re z) (Im z)
L4921Axiom. (
CSNo_Im2) We take the following as an axiom:
∀x y, SNo x → SNo y → Im (pa x y) = y
L4922Axiom. (
CSNo_ReR) We take the following as an axiom:
L4923Axiom. (
CSNo_ImR) We take the following as an axiom:
L4924Axiom. (
CSNo_ReIm) We take the following as an axiom:
∀z, CSNo z → z = pa (Re z) (Im z)
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
L4930Definition. We define
minus_CSNo to be
λz ⇒ pa (- Re z) (- Im z) of type
set → set.
L4932Definition. We define
add_CSNo to be
λz w ⇒ pa (Re z + Re w) (Im z + Im w) of type
set → set → set.
L4933Definition. We define
mul_CSNo to be
λz w ⇒ pa (Re z * Re w + - (Im z * Im w)) (Re z * Im w + Im z * Re w) of type
set → set → set.
L4937Axiom. (
SNo_Re) We take the following as an axiom:
L4939Axiom. (
SNo_Im) We take the following as an axiom:
L4941Axiom. (
Re_0) We take the following as an axiom:
L4943Axiom. (
Im_0) We take the following as an axiom:
L4945Axiom. (
Re_1) We take the following as an axiom:
L4947Axiom. (
Im_1) We take the following as an axiom:
L4949Axiom. (
Re_i) We take the following as an axiom:
L4951Axiom. (
Im_i) We take the following as an axiom:
L4957Axiom. (
add_CSNo_0L) We take the following as an axiom:
L4959Axiom. (
add_CSNo_0R) We take the following as an axiom:
End of Section Complex
Beginning of Section Complex
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_CSNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_CSNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_CSNo.
Notation. We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
div_CSNo.
End of Section Complex
Primitive. The name
pack_b_b_r_e_e is a term of type
set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set.
L4989Axiom. (
pack_b_b_r_e_e_0_eq) We take the following as an axiom:
∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → X = S 0
L4991Axiom. (
pack_b_b_r_e_e_0_eq2) We take the following as an axiom:
∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, X = pack_b_b_r_e_e X f g R c d 0
L4993Axiom. (
pack_b_b_r_e_e_1_eq) We take the following as an axiom:
∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → ∀x y ∈ X, f x y = decode_b (S 1) x y
L4997Axiom. (
pack_b_b_r_e_e_2_eq) We take the following as an axiom:
∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → ∀x y ∈ X, g x y = decode_b (S 2) x y
L5001Axiom. (
pack_b_b_r_e_e_3_eq) We take the following as an axiom:
∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → ∀x y ∈ X, R x y = decode_r (S 3) x y
L5005Axiom. (
pack_b_b_r_e_e_4_eq) We take the following as an axiom:
∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → c = S 4
L5007Axiom. (
pack_b_b_r_e_e_4_eq2) We take the following as an axiom:
∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, c = pack_b_b_r_e_e X f g R c d 4
L5009Axiom. (
pack_b_b_r_e_e_5_eq) We take the following as an axiom:
∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → d = S 5
L5011Axiom. (
pack_b_b_r_e_e_5_eq2) We take the following as an axiom:
∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, d = pack_b_b_r_e_e X f g R c d 5
L5013Axiom. (
pack_b_b_r_e_e_inj) We take the following as an axiom:
∀X X', ∀f f' : set → set → set, ∀g g' : set → set → set, ∀R R' : set → set → prop, ∀c c' : set, ∀d d' : set, pack_b_b_r_e_e X f g R c d = pack_b_b_r_e_e X' f' g' R' c' d' → X = X' ∧ (∀x y ∈ X, f x y = f' x y) ∧ (∀x y ∈ X, g x y = g' x y) ∧ (∀x y ∈ X, R x y = R' x y) ∧ c = c' ∧ d = d'
L5015Axiom. (
pack_b_b_r_e_e_ext) We take the following as an axiom:
∀X, ∀f f' : set → set → set, ∀g g' : set → set → set, ∀R R' : set → set → prop, ∀c, ∀d, (∀x y ∈ X, f x y = f' x y) → (∀x y ∈ X, g x y = g' x y) → (∀x y ∈ X, R x y ↔ R' x y) → pack_b_b_r_e_e X f g R c d = pack_b_b_r_e_e X f' g' R' c d
L5021Definition. We define
struct_b_b_r_e_e to be
λS ⇒ ∀q : set → prop, (∀X : set, ∀f : set → set → set, (∀x y ∈ X, f x y ∈ X) → ∀g : set → set → set, (∀x y ∈ X, g x y ∈ X) → ∀R : set → set → prop, ∀c : set, c ∈ X → ∀d : set, d ∈ X → q (pack_b_b_r_e_e X f g R c d)) → q S of type
set → prop.
Primitive. The name
unpack_b_b_r_e_e_i is a term of type
set → (set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set) → set.
L5038Axiom. (
unpack_b_b_r_e_e_i_eq) We take the following as an axiom:
∀Phi : set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set, ∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, (∀f' : set → set → set, (∀x y ∈ X, f x y = f' x y) → ∀g' : set → set → set, (∀x y ∈ X, g x y = g' x y) → ∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X f' g' R' c d = Phi X f g R c d) → unpack_b_b_r_e_e_i (pack_b_b_r_e_e X f g R c d) Phi = Phi X f g R c d
Primitive. The name
unpack_b_b_r_e_e_o is a term of type
set → (set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → prop) → prop.
L5047Axiom. (
unpack_b_b_r_e_e_o_eq) We take the following as an axiom:
∀Phi : set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → prop, ∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, (∀f' : set → set → set, (∀x y ∈ X, f x y = f' x y) → ∀g' : set → set → set, (∀x y ∈ X, g x y = g' x y) → ∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X f' g' R' c d = Phi X f g R c d) → unpack_b_b_r_e_e_o (pack_b_b_r_e_e X f g R c d) Phi = Phi X f g R c d
Beginning of Section explicit_OrderedField_RepIndep2
L5060Variable zero one : set
L5062Variable plus mult : set → set → set
L5063Variable leq : set → set → prop
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L5067Variable plus' mult' : set → set → set
L5069Variable leq' : set → set → prop
Notation. We use
+ as an infix operator with priority 355 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L5072Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b
L5074Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b
L5075Hypothesis Hll' : ∀a b ∈ R, leq a b ↔ leq' a b
End of Section explicit_OrderedField_RepIndep2
Beginning of Section explicit_Reals_RepIndep2
L5090Variable zero one : set
L5092Variable plus mult : set → set → set
L5093Variable leq : set → set → prop
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
plus.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult.
L5097Variable plus' mult' : set → set → set
L5099Variable leq' : set → set → prop
Notation. We use
+ as an infix operator with priority 355 and which associates to the right corresponding to applying term
plus'.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mult'.
L5102Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b
L5104Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b
L5105Hypothesis Hll' : ∀a b ∈ R, leq a b ↔ leq' a b
End of Section explicit_Reals_RepIndep2
Beginning of Section RealsStruct
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
RealsStruct_plus Rs.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
RealsStruct_mult Rs.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
RealsStruct_leq Rs.
L5155Definition. We define
RealsStruct_lt to be
λx y ⇒ x ≤ y ∧ x ≠ y of type
set → set → prop.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
RealsStruct_lt.
L5231Axiom. (
RealsStruct_Arch) We take the following as an axiom:
∀x y ∈ R, zero < x → zero ≤ y → ∃n ∈ N, y ≤ n * x
L5233Axiom. (
RealsStruct_Compl) We take the following as an axiom:
∀a b ∈ RN, (∀n ∈ N, a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n) → ∃x ∈ R, ∀n ∈ N, a n ≤ x ∧ x ≤ b n
L5268Axiom. (
RealsStruct_Z_props) We take the following as an axiom:
∀p : prop, ((∀n ∈ Npos, - n ∈ Z) → zero ∈ Z → Npos ⊆ Z → Z ⊆ R → (∀n ∈ Z, ∀q : prop, (- n ∈ Npos → q) → (n = zero → q) → (n ∈ Npos → q) → q) → one ∈ Z → - one ∈ Z → (∀m ∈ Z, - m ∈ Z) → (∀n m ∈ Z, n + m ∈ Z) → (∀n m ∈ Z, n * m ∈ Z) → p) → p
L5287Axiom. (
RealsStruct_Q_props) We take the following as an axiom:
∀p : prop, (Q ⊆ R → (∀x ∈ Q, ∀q : prop, (x ∈ R → ∀n ∈ Z, ∀m ∈ Npos, m * x = n → q) → q) → (∀x ∈ R, ∀n ∈ Z, ∀m ∈ Npos, m * x = n → x ∈ Q) → p) → p
Notation. We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
Field_div (Field_of_RealsStruct Rs).
End of Section RealsStruct
Beginning of Section Int
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
L5346Definition. We define
int to be
ω ∪ {- n|n ∈ ω} of type
set.
L5348Axiom. (
int_SNo_cases) We take the following as an axiom:
∀p : set → prop, (∀n ∈ ω, p n) → (∀n ∈ ω, p (- n)) → ∀x ∈ int, p x
L5353Axiom. (
int_SNo) We take the following as an axiom:
L5358Axiom. (
int_add_SNo) We take the following as an axiom:
L5360Axiom. (
int_mul_SNo) We take the following as an axiom:
L5361Axiom. (
add_SNo_4_3) We take the following as an axiom:
L5363Axiom. (
add_SNo_4_4) We take the following as an axiom:
L5364Axiom. (
nat_p_int) We take the following as an axiom:
End of Section Int
Beginning of Section IDL
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
End of Section IDL
Beginning of Section Reals
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
Primitive. The name
real is a term of type
set.
L5914Axiom. (
real_I) We take the following as an axiom:
L5920Axiom. (
real_E) We take the following as an axiom:
L5932Axiom. (
real_SNo) We take the following as an axiom:
L5938Axiom. (
real_0) We take the following as an axiom:
L5940Axiom. (
real_1) We take the following as an axiom:
L6007Axiom. (
SNo_approx_real_rep) We take the following as an axiom:
∀x ∈ real, ∀p : prop, (∀f g ∈ SNoS_ ωω, (∀n ∈ ω, f n < x) → (∀n ∈ ω, x < f n + eps_ n) → (∀n ∈ ω, ∀i ∈ n, f i < f n) → (∀n ∈ ω, g n + - eps_ n < x) → (∀n ∈ ω, x < g n) → (∀n ∈ ω, ∀i ∈ n, g n < g i) → SNoCutP {f n|n ∈ ω} {g n|n ∈ ω} → x = SNoCut {f n|n ∈ ω} {g n|n ∈ ω} → p) → p
End of Section Reals
Beginning of Section Complex
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_CSNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_CSNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_CSNo.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
L6068Axiom. (
complex_I) We take the following as an axiom:
L6070Axiom. (
complex_E) We take the following as an axiom:
L6075Axiom. (
complex_0) We take the following as an axiom:
L6076Axiom. (
complex_1) We take the following as an axiom:
L6077Axiom. (
complex_i) We take the following as an axiom:
L6086Axiom. (
real_Re_eq) We take the following as an axiom:
L6087Axiom. (
real_Im_eq) We take the following as an axiom:
L6091Axiom. (
complex_eta) We take the following as an axiom:
End of Section Complex
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_CSNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_CSNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_CSNo.
Notation. We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
div_CSNo.