Primitive . The name
Eps_i is a term of type
(set → prop → set .
Axiom. (
Eps_i_ax ) We take the following as an axiom:
∀P : set → prop ∀x : set , P x P (Eps_i P Definition. We define
True to be
∀p : prop , p → p of type
prop .
Definition. We define
False to be
∀p : prop , p of type
prop .
Definition. We define
not to be
λA : prop ⇒ A → False of type
prop → prop .
Notation . We use
¬ as a prefix operator with priority 700 corresponding to applying term
not .
Definition. We define
and to be
λA B : prop ⇒ ∀p : prop , (A → B → p → p of type
prop → prop → prop .
Notation . We use
∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term
and .
Definition. We define
or to be
λA B : prop ⇒ ∀p : prop , (A → p → (B → p → p of type
prop → prop → prop .
Notation . We use
∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term
or .
Definition. We define
iff to be
λA B : prop ⇒ and (A → B (B → A of type
prop → prop → prop .
Notation . We use
↔ as an infix operator with priority 805 and no associativity corresponding to applying term
iff .
Beginning of Section Eq
Variable A : SType
Definition. We define
eq to be
λx y : A ⇒ ∀Q : A → A → prop Q x y Q y x of type
A → A → prop .
Definition. We define
neq to be
λx y : A ⇒ ¬ eq x y of type
A → A → prop .
End of Section Eq
Notation . We use
= as an infix operator with priority 502 and no associativity corresponding to applying term
eq .
Notation . We use
≠ as an infix operator with priority 502 and no associativity corresponding to applying term
neq .
Beginning of Section FE
Variable A B : SType
Axiom. (
func_ext ) We take the following as an axiom:
∀f g : A → B (∀x : A , f x = g x ) → f = g End of Section FE
Beginning of Section Ex
Variable A : SType
Definition. We define
ex to be
λQ : A → prop ∀P : prop , (∀x : A , Q x P ) → P of type
(A → prop → prop .
End of Section Ex
Notation . We use
∃ x ...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
ex .
Axiom. (
prop_ext ) We take the following as an axiom:
∀p q : prop , iff p q p = q Primitive . The name
In is a term of type
set → set → prop .
Notation . We use
∈ as an infix operator with priority 500 and no associativity corresponding to applying term
In . Furthermore, we may write
∀ x ∈ A , B to mean
∀ x : set, x ∈ A → B .
Definition. We define
Subq to be
λA B ⇒ ∀ x ∈ A , x ∈ B of type
set → set → prop .
Notation . We use
⊆ as an infix operator with priority 500 and no associativity corresponding to applying term
Subq . Furthermore, we may write
∀ x ⊆ A , B to mean
∀ x : set, x ⊆ A → B .
Axiom. (
set_ext ) We take the following as an axiom:
∀X Y : set , X ⊆ Y Y ⊆ X X = Y Axiom. (
In_ind ) We take the following as an axiom:
∀P : set → prop (∀X : set , (∀ x ∈ X , P x → P X ) → ∀X : set , P X Notation . We use
∃ x ...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
ex and handling ∈ or ⊆ ascriptions using
and .
Primitive . The name
Empty is a term of type
set .
Axiom. (
EmptyAx ) We take the following as an axiom:
Primitive . The name
⋃ is a term of type
set → set .
Axiom. (
UnionEq ) We take the following as an axiom:
Primitive . The name
𝒫 is a term of type
set → set .
Axiom. (
PowerEq ) We take the following as an axiom:
∀X Y : set , Y ∈ 𝒫 X ↔ Y ⊆ X Primitive . The name
Repl is a term of type
set → (set → set → set .
Notation .
{B | x ∈ A } is notation for
Repl A (λ x . B ).
Axiom. (
ReplEq ) We take the following as an axiom:
∀A : set , ∀F : set → set ∀y : set , y ∈ { F x | x ∈ A } ↔ ∃ x ∈ A , y = F x Definition. We define
TransSet to be
λU : set ⇒ ∀ x ∈ U , x ⊆ U of type
set → prop .
Definition. We define
Union_closed to be
λU : set ⇒ ∀X : set , X ∈ U ⋃ X ∈ U of type
set → prop .
Definition. We define
Power_closed to be
λU : set ⇒ ∀X : set , X ∈ U 𝒫 X ∈ U of type
set → prop .
Definition. We define
Repl_closed to be
λU : set ⇒ ∀X : set , X ∈ U ∀F : set → set (∀x : set , x ∈ X F x ∈ U ) → { F x | x ∈ X } ∈ U of type
set → prop .
Primitive . The name
UnivOf is a term of type
set → set .
Axiom. (
UnivOf_In ) We take the following as an axiom:
Axiom. (
UnivOf_Min ) We take the following as an axiom:
Axiom. (
FalseE ) We take the following as an axiom:
Axiom. (
TrueI ) We take the following as an axiom:
Axiom. (
andI ) We take the following as an axiom:
∀A B : prop , A → B → A ∧ B Axiom. (
andEL ) We take the following as an axiom:
Axiom. (
andER ) We take the following as an axiom:
Axiom. (
orIL ) We take the following as an axiom:
Axiom. (
orIR ) We take the following as an axiom:
Beginning of Section PropN
Variable P1 P2 P3 : prop
Axiom. (
and3I ) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3 Axiom. (
and3E ) We take the following as an axiom:
P1 ∧ P2 ∧ P3 (∀p : prop , (P1 → P2 → P3 → p → p ) Axiom. (
or3I1 ) We take the following as an axiom:
Axiom. (
or3I2 ) We take the following as an axiom:
Axiom. (
or3I3 ) We take the following as an axiom:
Axiom. (
or3E ) We take the following as an axiom:
P1 ∨ P2 ∨ P3 (∀p : prop , (P1 → p → (P2 → p → (P3 → p → p ) Variable P4 : prop
Axiom. (
and4I ) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4 Variable P5 : prop
Axiom. (
and5I ) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 End of Section PropN
Axiom. (
iffI ) We take the following as an axiom:
∀A B : prop , (A → B → (B → A → (A ↔ B Axiom. (
iffEL ) We take the following as an axiom:
∀A B : prop , (A ↔ B → A → B Axiom. (
iffER ) We take the following as an axiom:
∀A B : prop , (A ↔ B → B → A Axiom. (
iff_refl ) We take the following as an axiom:
Axiom. (
iff_sym ) We take the following as an axiom:
∀A B : prop , (A ↔ B → (B ↔ A Axiom. (
iff_trans ) We take the following as an axiom:
∀A B C : prop , (A ↔ B → (B ↔ C → (A ↔ C Axiom. (
eq_i_tra ) We take the following as an axiom:
∀x y z, x = y y = z x = z Axiom. (
f_eq_i ) We take the following as an axiom:
∀f : set → set ∀x y, x = y f x = f y Axiom. (
neq_i_sym ) We take the following as an axiom:
Definition. We define
nIn to be
λx X ⇒ ¬ In x X of type
set → set → prop .
Notation . We use
∉ as an infix operator with priority 502 and no associativity corresponding to applying term
nIn .
Axiom. (
Eps_i_ex ) We take the following as an axiom:
∀P : set → prop (∃ x, P x → P (Eps_i P Axiom. (
pred_ext ) We take the following as an axiom:
∀P Q : set → prop (∀x, P x ↔ Q x ) → P = Q Axiom. (
prop_ext_2 ) We take the following as an axiom:
∀p q : prop , (p → q → (q → p → p = q Axiom. (
Subq_ref ) We take the following as an axiom:
Axiom. (
Subq_tra ) We take the following as an axiom:
∀X Y Z : set , X ⊆ Y Y ⊆ Z X ⊆ Z Axiom. (
Subq_contra ) We take the following as an axiom:
∀X Y z : set , X ⊆ Y z ∉ Y z ∉ X Axiom. (
EmptyE ) We take the following as an axiom:
Axiom. (
Subq_Empty ) We take the following as an axiom:
Axiom. (
Empty_eq ) We take the following as an axiom:
Axiom. (
UnionI ) We take the following as an axiom:
∀X x Y : set , x ∈ Y Y ∈ X x ∈ ⋃ X Axiom. (
UnionE ) We take the following as an axiom:
∀X x : set , x ∈ ⋃ X ∃ Y : set , x ∈ Y ∧ Y ∈ X Axiom. (
UnionE_impred ) We take the following as an axiom:
∀X x : set , x ∈ ⋃ X ∀p : prop , (∀Y : set , x ∈ Y Y ∈ X p ) → p Axiom. (
PowerI ) We take the following as an axiom:
∀X Y : set , Y ⊆ X Y ∈ 𝒫 X Axiom. (
PowerE ) We take the following as an axiom:
∀X Y : set , Y ∈ 𝒫 X Y ⊆ X Axiom. (
xm ) We take the following as an axiom:
Axiom. (
dneg ) We take the following as an axiom:
Axiom. (
eq_or_nand ) We take the following as an axiom:
Primitive . The name
exactly1of2 is a term of type
prop → prop → prop .
Axiom. (
exactly1of2_E ) We take the following as an axiom:
∀A B : prop , exactly1of2 A B ∀p : prop , (A → ¬ B p → (¬ A B → p → p Axiom. (
ReplI ) We take the following as an axiom:
∀A : set , ∀F : set → set ∀x : set , x ∈ A F x ∈ { F x | x ∈ A } Axiom. (
ReplE ) We take the following as an axiom:
∀A : set , ∀F : set → set ∀y : set , y ∈ { F x | x ∈ A } ∃ x ∈ A , y = F x Axiom. (
ReplE_impred ) We take the following as an axiom:
∀A : set , ∀F : set → set ∀y : set , y ∈ { F x | x ∈ A } ∀p : prop , (∀x : set , x ∈ A y = F x p ) → p Axiom. (
ReplE' ) We take the following as an axiom:
∀X, ∀f : set → set ∀p : set → prop (∀ x ∈ X , p (f x → ∀ y ∈ { f x | x ∈ X } , p y Axiom. (
Repl_Empty ) We take the following as an axiom:
Axiom. (
ReplEq_ext ) We take the following as an axiom:
∀X, ∀F G : set → set (∀ x ∈ X , F x = G x → { F x | x ∈ X } = { G x | x ∈ X } Axiom. (
Repl_inv_eq ) We take the following as an axiom:
∀P : set → prop ∀f g : set → set (∀x, P x g (f x = x ) → ∀X, (∀ x ∈ X , P x → { g y | y ∈ { f x | x ∈ X } } = X Axiom. (
Repl_invol_eq ) We take the following as an axiom:
∀P : set → prop ∀f : set → set (∀x, P x f (f x = x ) → ∀X, (∀ x ∈ X , P x → { f y | y ∈ { f x | x ∈ X } } = X 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 .
Axiom. (
If_i_0 ) We take the following as an axiom:
Axiom. (
If_i_1 ) We take the following as an axiom:
Axiom. (
If_i_or ) 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 .
Axiom. (
UPairE ) We take the following as an axiom:
∀x y z : set , x ∈ { y , z } x = y ∨ x = z Axiom. (
UPairI1 ) We take the following as an axiom:
Axiom. (
UPairI2 ) 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 .
Axiom. (
SingI ) We take the following as an axiom:
Axiom. (
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 .
Axiom. (
binunionI1 ) We take the following as an axiom:
∀X Y z : set , z ∈ X z ∈ X ∪ Y Axiom. (
binunionI2 ) We take the following as an axiom:
∀X Y z : set , z ∈ Y z ∈ X ∪ Y Axiom. (
binunionE ) We take the following as an axiom:
∀X Y z : set , z ∈ X ∪ Y z ∈ X ∨ z ∈ Y Axiom. (
binunionE' ) We take the following as an axiom:
∀X Y z, ∀p : prop , (z ∈ X p → (z ∈ Y p → (z ∈ X ∪ Y p Axiom. (
binunion_asso ) We take the following as an axiom:
∀X Y Z : set , X ∪ (Y ∪ Z = (X ∪ Y ∪ Z Axiom. (
binunion_com ) We take the following as an axiom:
∀X Y : set , X ∪ Y = Y ∪ X Definition. We define
SetAdjoin to be
λX y ⇒ X ∪ { y } of type
set → set → set .
Notation . We now use the set enumeration notation
{...,...,...} in general. If 0 elements are given, then
Empty is used to form the corresponding term. If 1 element is given, then
Sing is used to form the corresponding term. If 2 elements are given, then
UPair is used to form the corresponding term. If more than elements are given, then
SetAdjoin is used to reduce to the case with one fewer elements.
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 .
Axiom. (
famunionI ) We take the following as an axiom:
∀X : set , ∀F : (set → set , ∀x y : set , x ∈ X y ∈ F x y ∈ ⋃ x ∈ X F x Axiom. (
famunionE ) We take the following as an axiom:
∀X : set , ∀F : (set → set , ∀y : set , y ∈ (⋃ x ∈ X F x ∃ x ∈ X , y ∈ F x Axiom. (
famunionE_impred ) We take the following as an axiom:
∀X : set , ∀F : (set → set , ∀y : set , y ∈ (⋃ x ∈ X F x ∀p : prop , (∀x, x ∈ X y ∈ F x p ) → p
Beginning of Section SepSec
Variable X : set
Variable P : set → prop
Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z ) Primitive . The name
Sep is a term of type
set .
End of Section SepSec
Notation .
{x ∈ A | B } is notation for
Sep A (λ x . B ).
Axiom. (
SepI ) We take the following as an axiom:
∀X : set , ∀P : (set → prop , ∀x : set , x ∈ X P x x ∈ { x ∈ X | P x } Axiom. (
SepE ) We take the following as an axiom:
∀X : set , ∀P : (set → prop , ∀x : set , x ∈ { x ∈ X | P x } x ∈ X ∧ P x Axiom. (
SepE1 ) We take the following as an axiom:
∀X : set , ∀P : (set → prop , ∀x : set , x ∈ { x ∈ X | P x } x ∈ X Axiom. (
SepE2 ) We take the following as an axiom:
∀X : set , ∀P : (set → prop , ∀x : set , x ∈ { x ∈ X | P x } P x Axiom. (
Sep_Subq ) We take the following as an axiom:
∀X : set , ∀P : set → prop { x ∈ X | P x } ⊆ X Axiom. (
Sep_In_Power ) We take the following as an axiom:
∀X : set , ∀P : set → prop { x ∈ X | P x } ∈ 𝒫 X 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 ).
Axiom. (
ReplSepI ) We take the following as an axiom:
∀X : set , ∀P : set → prop ∀F : set → set ∀x : set , x ∈ X P x F x ∈ { F x | x ∈ X , P x } Axiom. (
ReplSepE ) We take the following as an axiom:
∀X : set , ∀P : set → prop ∀F : set → set ∀y : set , y ∈ { F x | x ∈ X , P x } ∃ x : set , x ∈ X ∧ P x ∧ y = F x Axiom. (
ReplSepE_impred ) We take the following as an axiom:
∀X : set , ∀P : set → prop ∀F : set → set ∀y : set , y ∈ { F x | x ∈ X , P x } ∀p : prop , (∀ x ∈ X , P x y = F x p → p 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 .
Axiom. (
binintersectI ) We take the following as an axiom:
∀X Y z, z ∈ X z ∈ Y z ∈ X ∩ Y Axiom. (
binintersectE ) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y z ∈ X ∧ z ∈ Y 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 .
Axiom. (
setminusI ) We take the following as an axiom:
∀X Y z, (z ∈ X → (z ∉ Y → z ∈ X ∖ Y Axiom. (
setminusE ) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y → z ∈ X ∧ z ∉ Y Axiom. (
setminusE1 ) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y → z ∈ X Axiom. (
In_irref ) We take the following as an axiom:
Axiom. (
In_no2cycle ) We take the following as an axiom:
Primitive . The name
ordsucc is a term of type
set → set .
Axiom. (
ordsuccI1 ) We take the following as an axiom:
Axiom. (
ordsuccI2 ) We take the following as an axiom:
Axiom. (
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 .
Axiom. (
ordsucc_inj ) We take the following as an axiom:
Axiom. (
In_0_1 ) We take the following as an axiom:
Axiom. (
In_0_2 ) We take the following as an axiom:
Axiom. (
In_1_2 ) We take the following as an axiom:
Definition. We define
nat_p to be
λn : set ⇒ ∀p : set → prop p 0 (∀x : set , p x p (ordsucc x ) → p n of type
set → prop .
Axiom. (
nat_0 ) We take the following as an axiom:
Axiom. (
nat_ordsucc ) We take the following as an axiom:
Axiom. (
nat_1 ) We take the following as an axiom:
Axiom. (
nat_2 ) We take the following as an axiom:
Axiom. (
nat_ind ) We take the following as an axiom:
Axiom. (
nat_inv ) We take the following as an axiom:
Axiom. (
nat_p_trans ) We take the following as an axiom:
Axiom. (
nat_trans ) We take the following as an axiom:
Axiom. (
cases_1 ) We take the following as an axiom:
∀ i ∈ 1 , ∀p : set → prop p 0 p i Axiom. (
cases_2 ) We take the following as an axiom:
∀ i ∈ 2 , ∀p : set → prop p 0 p 1 p i Axiom. (
cases_3 ) We take the following as an axiom:
∀ i ∈ 3 , ∀p : set → prop p 0 p 1 p 2 p i Axiom. (
neq_0_1 ) We take the following as an axiom:
Axiom. (
neq_1_0 ) We take the following as an axiom:
Axiom. (
neq_0_2 ) We take the following as an axiom:
Axiom. (
neq_2_0 ) We take the following as an axiom:
Axiom. (
neq_1_2 ) We take the following as an axiom:
Axiom. (
ZF_closed_E ) We take the following as an axiom:
Primitive . The name
ω is a term of type
set .
Axiom. (
omega_nat_p ) We take the following as an axiom:
Axiom. (
nat_p_omega ) We take the following as an axiom:
Axiom. (
ordinal_1 ) We take the following as an axiom:
Axiom. (
ordinal_2 ) We take the following as an axiom:
Axiom. (
ordinal_ind ) We take the following as an axiom:
∀p : set → prop (∀alpha, ordinal alpha (∀ beta ∈ alpha , p beta → p alpha ) → ∀alpha, ordinal alpha p alpha Definition. We define
inj to be
λX Y f ⇒ (∀ u ∈ X , f u ∈ Y ∧ (∀ u v ∈ X , f u = f v u = v of type
set → set → (set → set → prop .
Definition. We define
bij to be
λX Y f ⇒ (∀ u ∈ X , f u ∈ Y ∧ (∀ u v ∈ X , f u = f v u = v ∧ (∀ w ∈ Y , ∃ u ∈ X , f u = w of type
set → set → (set → set → prop .
Axiom. (
bijI ) We take the following as an axiom:
∀X Y, ∀f : set → set (∀ u ∈ X , f u ∈ Y → (∀ u v ∈ X , f u = f v u = v → (∀ w ∈ Y , ∃ u ∈ X , f u = w → bij X Y f Axiom. (
bijE ) We take the following as an axiom:
∀X Y, ∀f : set → set bij X Y f ∀p : prop , ((∀ u ∈ X , f u ∈ Y → (∀ u v ∈ X , f u = f v u = v → (∀ w ∈ Y , ∃ u ∈ X , f u = w → p → p Primitive . The name
inv is a term of type
set → (set → set → set → set .
Axiom. (
surj_rinv ) We take the following as an axiom:
Axiom. (
inj_linv ) We take the following as an axiom:
∀X, ∀f : set → set (∀ u v ∈ X , f u = f v u = v → ∀ x ∈ X , inv X f (f x = x Axiom. (
bij_inv ) We take the following as an axiom:
∀X Y, ∀f : set → set bij X Y f bij Y X (inv X f Axiom. (
bij_id ) We take the following as an axiom:
Axiom. (
bij_comp ) We take the following as an axiom:
∀X Y Z : set , ∀f g : set → set bij X Y f bij Y Z g bij X Z (λx ⇒ g (f x ) Definition. We define
equip to be
λX Y : set ⇒ ∃ f : set → set , bij X Y f of type
set → set → prop .
Axiom. (
equip_ref ) We take the following as an axiom:
Axiom. (
equip_sym ) We take the following as an axiom:
Axiom. (
equip_tra ) We take the following as an axiom:
Beginning of Section SchroederBernstein
End of Section SchroederBernstein
Beginning of Section PigeonHole
End of Section PigeonHole
Definition. We define
finite to be
λX ⇒ ∃ n ∈ ω , equip X n of type
set → prop .
Axiom. (
finite_ind ) We take the following as an axiom:
Axiom. (
Subq_finite ) We take the following as an axiom:
Axiom. (
exandE_i ) We take the following as an axiom:
∀P Q : set → prop (∃ x, P x ∧ Q x → ∀r : prop , (∀x, P x Q x r ) → r Axiom. (
exandE_ii ) We take the following as an axiom:
∀P Q : (set → set → prop (∃ x : set → set , P x ∧ Q x → ∀p : prop , (∀x : set → set P x Q x p ) → p Axiom. (
exandE_iii ) We take the following as an axiom:
∀P Q : (set → set → set → prop (∃ x : set → set → set , P x ∧ Q x → ∀p : prop , (∀x : set → set → set P x Q x p ) → p Axiom. (
exandE_iiii ) We take the following as an axiom:
∀P Q : (set → set → set → set → prop (∃ x : set → set → set → set , P x ∧ Q x → ∀p : prop , (∀x : set → set → set → set P x Q x p ) → p
Beginning of Section Descr_ii
Variable P : (set → set → prop
Primitive . The name
Descr_ii is a term of type
set → set .
Hypothesis Pex : ∃ f : set → set , P f Hypothesis Puniq : ∀f g : set → set P f P g f = g End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (set → set → set → prop
Primitive . The name
Descr_iii is a term of type
set → set → set .
Hypothesis Pex : ∃ f : set → set → set , P f Hypothesis Puniq : ∀f g : set → set → set P f P g f = g End of Section Descr_iii
Beginning of Section Descr_Vo1
Primitive . The name
Descr_Vo1 is a term of type
Vo 1 .
Hypothesis Pex : ∃ f : Vo 1 , P f Hypothesis Puniq : ∀f g : Vo 1 P f P g f = g End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : set → set
Primitive . The name
If_ii is a term of type
set → set .
Axiom. (
If_ii_1 ) We take the following as an axiom:
Axiom. (
If_ii_0 ) We take the following as an axiom:
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : set → set → set
Primitive . The name
If_iii is a term of type
set → set → set .
Axiom. (
If_iii_1 ) We take the following as an axiom:
Axiom. (
If_iii_0 ) We take the following as an axiom:
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set → (set → set → set
Primitive . The name
In_rec_i is a term of type
set → set .
Hypothesis Fr : ∀X : set , ∀g h : set → set (∀ x ∈ X , g x = h x → F X g = F X h Axiom. (
In_rec_i_eq ) We take the following as an axiom:
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set → (set → (set → set → (set → set
Primitive . The name
In_rec_ii is a term of type
set → (set → set .
Hypothesis 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
Variable F : set → (set → (set → set → set → (set → set → set
Primitive . The name
In_rec_iii is a term of type
set → (set → set → set .
Hypothesis 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 NatRec
Variable z : set
Variable f : set → set → set
Let F : set → (set → set → set λn g ⇒ if ⋃ n ∈ n then f (⋃ n (g (⋃ n else z Axiom. (
nat_primrec_r ) We take the following as an axiom:
∀X : set , ∀g h : set → set (∀ x ∈ X , g x = h x → F X g = F X h 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 .
Axiom. (
add_nat_0R ) We take the following as an axiom:
Axiom. (
add_nat_SR ) We take the following as an axiom:
Axiom. (
add_nat_p ) We take the following as an axiom:
Definition. We define
mul_nat to be
λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r ) m of type
set → set → set .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_nat .
Axiom. (
mul_nat_0R ) We take the following as an axiom:
Axiom. (
mul_nat_SR ) We take the following as an axiom:
Axiom. (
mul_nat_p ) We take the following as an axiom:
End of Section NatArith
Axiom. (
Inj1_eq ) We take the following as an axiom:
Axiom. (
Inj1I1 ) We take the following as an axiom:
Axiom. (
Inj1I2 ) We take the following as an axiom:
Axiom. (
Inj1E ) We take the following as an axiom:
Axiom. (
Inj1NE1 ) We take the following as an axiom:
Axiom. (
Inj1NE2 ) We take the following as an axiom:
Definition. We define
Inj0 to be
λX ⇒ { Inj1 x | x ∈ X } of type
set → set .
Axiom. (
Inj0I ) We take the following as an axiom:
Axiom. (
Inj0E ) We take the following as an axiom:
Axiom. (
Unj_eq ) We take the following as an axiom:
Axiom. (
Unj_Inj1_eq ) We take the following as an axiom:
Axiom. (
Inj1_inj ) We take the following as an axiom:
Axiom. (
Unj_Inj0_eq ) We take the following as an axiom:
Axiom. (
Inj0_inj ) We take the following as an axiom:
Axiom. (
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 .
Axiom. (
Inj0_setsum ) We take the following as an axiom:
∀X Y x : set , x ∈ X Inj0 x ∈ X + Y Axiom. (
Inj1_setsum ) We take the following as an axiom:
∀X Y y : set , y ∈ Y Inj1 y ∈ X + Y Axiom. (
setsum_0_0 ) We take the following as an axiom:
Beginning of Section pair_setsum
Axiom. (
pairI0 ) We take the following as an axiom:
∀X Y x, x ∈ X pair 0 x ∈ pair X Y Axiom. (
pairI1 ) We take the following as an axiom:
∀X Y y, y ∈ Y pair 1 y ∈ pair X Y Axiom. (
pairE ) We take the following as an axiom:
∀X Y z, z ∈ pair X Y (∃ x ∈ X , z = pair 0 x ∨ (∃ y ∈ Y , z = pair 1 y Axiom. (
pairE0 ) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y x ∈ X Axiom. (
pairE1 ) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y y ∈ Y Axiom. (
proj0I ) We take the following as an axiom:
Axiom. (
proj0E ) We take the following as an axiom:
Axiom. (
proj1I ) We take the following as an axiom:
Axiom. (
proj1E ) We take the following as an axiom:
Definition. We define
Sigma to be
λX Y ⇒ ⋃ x ∈ X { pair x y | y ∈ Y x } of type
set → (set → set → set .
Notation . We use
∑ x ...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Sigma .
Axiom. (
pair_Sigma ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀ x ∈ X , ∀ y ∈ Y x , pair x y ∈ ∑ x ∈ X , Y x Axiom. (
proj0_Sigma ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀z : set , z ∈ (∑ x ∈ X , Y x proj0 z ∈ X Axiom. (
proj1_Sigma ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀z : set , z ∈ (∑ x ∈ X , Y x proj1 z ∈ Y (proj0 z Axiom. (
pair_Sigma_E1 ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀x y : set , pair x y ∈ (∑ x ∈ X , Y x y ∈ Y x Axiom. (
Sigma_E ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀z : set , z ∈ (∑ x ∈ X , Y x ∃ x ∈ X , ∃ y ∈ Y x , z = pair x y Definition. We define
setprod to be
λX Y : set ⇒ ∑ x ∈ X , Y of type
set → set → set .
Notation . We use
⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term
setprod .
Let lam : set → (set → set → set Sigma Definition. We define
ap to be
λf x ⇒ { proj1 z | z ∈ f , ∃ y : set , z = pair x y } of type
set → set → set .
Notation . When
x is a set, a term
x y is notation for
ap x y .
Notation .
λ x ∈ A ⇒ B is notation for the set
Sigma A (λ x : set ⇒ B ).
Notation . We now use n-tuple notation (
a0 ,...,
an-1 ) for n ≥ 2 for λ i ∈
n .
if i = 0
then a0 else ... if i =
n-2 then an-2 else an-1 .
Axiom. (
lamI ) We take the following as an axiom:
∀X : set , ∀F : set → set ∀ x ∈ X , ∀ y ∈ F x , pair x y ∈ λ x ∈ X ⇒ F x Axiom. (
lamE ) We take the following as an axiom:
∀X : set , ∀F : set → set ∀z : set , z ∈ (λ x ∈ X ⇒ F x ∃ x ∈ X , ∃ y ∈ F x , z = pair x y Axiom. (
apI ) We take the following as an axiom:
∀f x y, pair x y ∈ f y ∈ f x Axiom. (
apE ) We take the following as an axiom:
∀f x y, y ∈ f x pair x y ∈ f Axiom. (
beta ) We take the following as an axiom:
∀X : set , ∀F : set → set ∀x : set , x ∈ X (λ x ∈ X ⇒ F x x = F x Axiom. (
proj0_ap_0 ) We take the following as an axiom:
Axiom. (
proj1_ap_1 ) We take the following as an axiom:
Axiom. (
pair_ap_0 ) We take the following as an axiom:
∀x y : set , (pair x y 0 = x Axiom. (
pair_ap_1 ) We take the following as an axiom:
∀x y : set , (pair x y 1 = y Axiom. (
ap0_Sigma ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀z : set , z ∈ (∑ x ∈ X , Y x (z 0 ∈ X Axiom. (
ap1_Sigma ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀z : set , z ∈ (∑ x ∈ X , Y x (z 1 ∈ (Y (z 0 Definition. We define
pair_p to be
λu : set ⇒ pair (u 0 (u 1 = u of type
set → prop .
Axiom. (
pair_p_I ) We take the following as an axiom:
Axiom. (
tuple_pair ) We take the following as an axiom:
∀x y : set , pair x y = ( x , y ) Definition. We define
Pi to be
λX Y ⇒ { f ∈ 𝒫 (∑ x ∈ X , ⋃ (Y x | ∀ x ∈ X , f x ∈ Y x } of type
set → (set → set → set .
Notation . We use
∏ x ...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Pi .
Axiom. (
PiI ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀f : set , (∀ u ∈ f , pair_p u ∧ u 0 ∈ X → (∀ x ∈ X , f x ∈ Y x → f ∈ ∏ x ∈ X , Y x Axiom. (
lam_Pi ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀F : set → set (∀ x ∈ X , F x ∈ Y x → (λ x ∈ X ⇒ F x ∈ (∏ x ∈ X , Y x Axiom. (
ap_Pi ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀f : set , ∀x : set , f ∈ (∏ x ∈ X , Y x x ∈ X f x ∈ Y x Definition. We define
setexp to be
λX Y : set ⇒ ∏ y ∈ Y , X of type
set → set → set .
Notation . We use
:^: as an infix operator with priority 430 and which associates to the left corresponding to applying term
setexp .
Axiom. (
lamI2 ) We take the following as an axiom:
Beginning of Section Tuples
Variable x0 x1 : set
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 .
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 .
Definition. We define
PNoEq_ to be
λalpha p q ⇒ ∀ beta ∈ alpha , p beta ↔ q beta of type
set → (set → prop → (set → prop → prop .
Axiom. (
PNoEq_ref_ ) We take the following as an axiom:
∀alpha, ∀p : set → prop PNoEq_ alpha p p Axiom. (
PNoEq_sym_ ) We take the following as an axiom:
∀alpha, ∀p q : set → prop PNoEq_ alpha p q PNoEq_ alpha q p Axiom. (
PNoEq_tra_ ) We take the following as an axiom:
Axiom. (
PNoLt_E_ ) We take the following as an axiom:
Axiom. (
PNoLt_irref_ ) We take the following as an axiom:
∀alpha, ∀p : set → prop ¬ PNoLt_ alpha p p Axiom. (
PNoLt_mon_ ) We take the following as an axiom:
Axiom. (
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 .
Axiom. (
PNoLtI1 ) We take the following as an axiom:
Axiom. (
PNoLtI2 ) We take the following as an axiom:
Axiom. (
PNoLtI3 ) We take the following as an axiom:
Axiom. (
PNoLtE ) We take the following as an axiom:
Axiom. (
PNoLt_irref ) We take the following as an axiom:
∀alpha, ∀p : set → prop ¬ PNoLt alpha p alpha p Axiom. (
PNoLtEq_tra ) We take the following as an axiom:
Axiom. (
PNoEqLt_tra ) We take the following as an axiom:
Axiom. (
PNoLt_tra ) We take the following as an axiom:
Definition. We define
PNoLe to be
λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type
set → (set → prop → set → (set → prop → prop .
Axiom. (
PNoLeI1 ) We take the following as an axiom:
Axiom. (
PNoLeI2 ) We take the following as an axiom:
∀alpha, ∀p q : set → prop PNoEq_ alpha p q PNoLe alpha p alpha q Axiom. (
PNoLe_ref ) We take the following as an axiom:
∀alpha, ∀p : set → prop PNoLe alpha p alpha p Axiom. (
PNoLtLe_tra ) We take the following as an axiom:
Axiom. (
PNoLeLt_tra ) We take the following as an axiom:
Axiom. (
PNoEqLe_tra ) We take the following as an axiom:
Axiom. (
PNoLe_tra ) We take the following as an axiom:
Axiom. (
PNoLe_downc ) We take the following as an axiom:
Axiom. (
PNo_downc_ref ) We take the following as an axiom:
∀L : set → (set → prop → prop ∀alpha, ordinal alpha ∀p : set → prop L alpha p PNo_downc L alpha p Axiom. (
PNo_upc_ref ) We take the following as an axiom:
∀R : set → (set → prop → prop ∀alpha, ordinal alpha ∀p : set → prop R alpha p PNo_upc R alpha p Axiom. (
PNoLe_upc ) We take the following as an axiom:
Definition. We define
PNoLt_pwise to be
λL R ⇒ ∀gamma, ordinal gamma ∀p : set → prop L gamma p ∀delta, ordinal delta ∀q : set → prop R delta q PNoLt gamma p delta q of type
(set → (set → prop → prop → (set → (set → prop → prop → prop .
Axiom. (
PNo_extend0_eq ) We take the following as an axiom:
∀alpha, ∀p : set → prop PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha ) Axiom. (
PNo_extend1_eq ) We take the following as an axiom:
∀alpha, ∀p : set → prop PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha ) Definition. We define
PNo_lenbdd to be
λalpha L ⇒ ∀beta, ∀p : set → prop L beta p beta ∈ alpha of type
set → (set → (set → prop → prop → prop .
Definition. We define
PNo_least_rep2 to be
λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta ¬ p x of type
(set → (set → prop → prop → (set → (set → prop → prop → set → (set → prop → prop .
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 .
Axiom. (
PNo_bd_pred ) We take the following as an axiom:
Axiom. (
PNo_bd_In ) We take the following as an axiom:
Beginning of Section TaggedSets
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
tag .
Definition. We define
SNoElts_ to be
λalpha ⇒ alpha ∪ { beta ' | beta ∈ alpha } of type
set → set .
Axiom. (
SNoElts_mon ) We take the following as an axiom:
Axiom. (
PNoEq_PSNo ) We take the following as an axiom:
Axiom. (
SNo_PSNo ) We take the following as an axiom:
Primitive . The name
SNo is a term of type
set → prop .
Axiom. (
SNo_SNo ) We take the following as an axiom:
Primitive . The name
SNoLev is a term of type
set → set .
Axiom. (
SNoLev_uniq ) We take the following as an axiom:
Axiom. (
SNoLev_prop ) We take the following as an axiom:
Axiom. (
SNoLev_ ) We take the following as an axiom:
Axiom. (
SNoLev_PSNo ) We take the following as an axiom:
Axiom. (
SNo_Subq ) We take the following as an axiom:
Definition. We define
SNoEq_ to be
λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x ) (λbeta ⇒ beta ∈ y ) of type
set → set → set → prop .
Axiom. (
SNoEq_I ) We take the following as an axiom:
Axiom. (
SNo_eq ) 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 .
Axiom. (
SNoLtLe ) We take the following as an axiom:
Axiom. (
SNoLeE ) We take the following as an axiom:
Axiom. (
SNoEq_sym_ ) We take the following as an axiom:
Axiom. (
SNoEq_tra_ ) We take the following as an axiom:
Axiom. (
SNoLtE ) We take the following as an axiom:
Axiom. (
SNoLtI2 ) We take the following as an axiom:
Axiom. (
SNoLtI3 ) We take the following as an axiom:
Axiom. (
SNoLt_irref ) We take the following as an axiom:
Axiom. (
SNoLt_tra ) We take the following as an axiom:
Axiom. (
SNoLe_ref ) We take the following as an axiom:
Axiom. (
SNoLtLe_tra ) We take the following as an axiom:
Axiom. (
SNoLeLt_tra ) We take the following as an axiom:
Axiom. (
SNoLe_tra ) We take the following as an axiom:
Axiom. (
SNoLtLe_or ) We take the following as an axiom:
Axiom. (
SNoCutP_L_0 ) We take the following as an axiom:
Axiom. (
SNoCutP_0_R ) We take the following as an axiom:
Axiom. (
SNoCutP_0_0 ) We take the following as an axiom:
Axiom. (
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 .
Axiom. (
SNoS_I ) We take the following as an axiom:
Axiom. (
SNoS_I2 ) We take the following as an axiom:
Axiom. (
SNoS_Subq ) We take the following as an axiom:
Axiom. (
SNoS_E2 ) We take the following as an axiom:
Axiom. (
SNoS_In_neq ) We take the following as an axiom:
Axiom. (
SNoS_SNoLev ) We take the following as an axiom:
Axiom. (
SNoL_E ) We take the following as an axiom:
Axiom. (
SNoR_E ) We take the following as an axiom:
Axiom. (
SNoL_SNoS_ ) We take the following as an axiom:
Axiom. (
SNoR_SNoS_ ) We take the following as an axiom:
Axiom. (
SNoL_SNoS ) We take the following as an axiom:
Axiom. (
SNoR_SNoS ) We take the following as an axiom:
Axiom. (
SNoL_I ) We take the following as an axiom:
Axiom. (
SNoR_I ) We take the following as an axiom:
Axiom. (
SNo_eta ) We take the following as an axiom:
Axiom. (
SNoCut_Le ) We take the following as an axiom:
Axiom. (
SNoCut_ext ) We take the following as an axiom:
Axiom. (
ordinal_SNo ) We take the following as an axiom:
Axiom. (
nat_p_SNo ) We take the following as an axiom:
Axiom. (
omega_SNo ) We take the following as an axiom:
Axiom. (
SNo_0 ) We take the following as an axiom:
Axiom. (
SNo_1 ) We take the following as an axiom:
Axiom. (
SNo_2 ) We take the following as an axiom:
Axiom. (
SNoLev_0 ) We take the following as an axiom:
Axiom. (
SNoCut_0_0 ) We take the following as an axiom:
Axiom. (
SNoL_0 ) We take the following as an axiom:
Axiom. (
SNoR_0 ) We take the following as an axiom:
Axiom. (
SNoL_1 ) We take the following as an axiom:
Axiom. (
SNoR_1 ) We take the following as an axiom:
Axiom. (
eps_0_1 ) We take the following as an axiom:
Axiom. (
SNo__eps_ ) We take the following as an axiom:
Axiom. (
SNo_eps_ ) We take the following as an axiom:
Axiom. (
SNo_eps_1 ) We take the following as an axiom:
Axiom. (
SNoLev_eps_ ) We take the following as an axiom:
Axiom. (
SNo_eps_pos ) We take the following as an axiom:
Axiom. (
eps_SNo_eq ) We take the following as an axiom:
Axiom. (
eps_SNoCutP ) We take the following as an axiom:
Axiom. (
eps_SNoCut ) We take the following as an axiom:
End of Section TaggedSets2
Axiom. (
SNo_etaE ) We take the following as an axiom:
Axiom. (
SNo_ind ) We take the following as an axiom:
Beginning of Section SurrealRecI
Variable F : set → (set → set → set
Primitive . The name
SNo_rec_i is a term of type
set → set .
Hypothesis Fr : ∀z, SNo z ∀g h : set → set (∀ w ∈ SNoS_ (SNoLev z , g w = h w → F z g = F z h End of Section SurrealRecI
Beginning of Section SurrealRecII
Variable F : set → (set → (set → set → (set → set
Let G : set → (set → set → (set → set → set → (set → set λalpha g ⇒ If_iii (ordinal alpha (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha (F z (λw ⇒ g (SNoLev w w ) default ) (λz : set ⇒ default ) Primitive . The name
SNo_rec_ii is a term of type
set → (set → set .
Hypothesis Fr : ∀z, SNo z ∀g h : set → (set → set (∀ w ∈ SNoS_ (SNoLev z , g w = h w → F z g = F z h End of Section SurrealRecII
Beginning of Section SurrealRec2
Variable F : set → set → (set → set → set → set
Let G : set → (set → set → set → set → (set → set → set λw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y ) Primitive . The name
SNo_rec2 is a term of type
set → set → set .
Axiom. (
SNo_rec2_eq ) We take the following as an axiom:
End of Section SurrealRec2
Axiom. (
SNoLev_ind ) We take the following as an axiom:
Axiom. (
SNoLev_ind2 ) We take the following as an axiom:
Axiom. (
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 Axiom. (
SNo_omega ) We take the following as an axiom:
Axiom. (
SNoLt_0_1 ) We take the following as an axiom:
Axiom. (
SNoLt_0_2 ) We take the following as an axiom:
Axiom. (
SNoLt_1_2 ) We take the following as an axiom:
Axiom. (
restr_SNo_ ) We take the following as an axiom:
Axiom. (
restr_SNo ) We take the following as an axiom:
Axiom. (
restr_SNoEq ) We take the following as an axiom:
Beginning of Section SurrealMinus
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 .
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo .
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 .
Axiom. (
add_SNo_eq ) We take the following as an axiom:
Axiom. (
SNo_add_SNo ) We take the following as an axiom:
Axiom. (
add_SNo_Lt1 ) We take the following as an axiom:
Axiom. (
add_SNo_Le1 ) We take the following as an axiom:
Axiom. (
add_SNo_Lt2 ) We take the following as an axiom:
Axiom. (
add_SNo_Le2 ) We take the following as an axiom:
Axiom. (
add_SNo_Lt3 ) We take the following as an axiom:
Axiom. (
add_SNo_Le3 ) We take the following as an axiom:
Axiom. (
add_SNo_com ) We take the following as an axiom:
Axiom. (
add_SNo_0L ) We take the following as an axiom:
Axiom. (
add_SNo_0R ) We take the following as an axiom:
Axiom. (
minus_SNo_0 ) We take the following as an axiom:
Axiom. (
add_SNo_Lt4 ) We take the following as an axiom:
End of Section SurrealAdd
Beginning of Section SurrealMul
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 .
Axiom. (
mul_SNo_eq ) We take the following as an axiom:
Axiom. (
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 Axiom. (
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 Axiom. (
SNo_mul_SNo ) We take the following as an axiom:
Axiom. (
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 Axiom. (
mul_SNo_Lt ) We take the following as an axiom:
Axiom. (
mul_SNo_Le ) We take the following as an axiom:
Axiom. (
mul_SNo_Subq_lem ) We take the following as an axiom:
∀x y X Y Z W, ∀U U', (∀u, u ∈ U (∀q : prop , (∀ w0 ∈ X , ∀ w1 ∈ Y , u = w0 * y + x * w1 + - w0 * w1 q → (∀ z0 ∈ Z , ∀ z1 ∈ W , u = z0 * y + x * z1 + - z0 * z1 q → q ) ) → (∀ w0 ∈ X , ∀ w1 ∈ Y , w0 * y + x * w1 + - w0 * w1 ∈ U' → (∀ w0 ∈ Z , ∀ w1 ∈ W , w0 * y + x * w1 + - w0 * w1 ∈ U' → U ⊆ U' Axiom. (
mul_SNo_com ) We take the following as an axiom:
Beginning of Section mul_SNo_assoc_lems
Variable M : set → set → set
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
M .
Hypothesis SNo_M : ∀x y, SNo x SNo y SNo (x * y Hypothesis DL : ∀x y z, SNo x SNo y SNo z x * (y + z = x * y + x * z Hypothesis DR : ∀x y z, SNo x SNo y SNo z (x + y * z = x * z + y * z Hypothesis M_Lt : ∀x y u v, SNo x SNo y SNo u SNo v u < x v < y u * y + x * v < x * y + u * v Hypothesis M_Le : ∀x y u v, SNo x SNo y SNo u SNo v u ≤ x v ≤ y u * y + x * v ≤ x * y + u * v Axiom. (
mul_SNo_assoc_lem1 ) We take the following as an axiom:
∀x y z, SNo x SNo y SNo z (∀ u ∈ SNoS_ (SNoLev x , u * (y * z = (u * y * z → (∀ v ∈ SNoS_ (SNoLev y , x * (v * z = (x * v * z → (∀ w ∈ SNoS_ (SNoLev z , x * (y * w = (x * y * w → (∀ u ∈ SNoS_ (SNoLev x , ∀ v ∈ SNoS_ (SNoLev y , u * (v * z = (u * v * z → (∀ u ∈ SNoS_ (SNoLev x , ∀ w ∈ SNoS_ (SNoLev z , u * (y * w = (u * y * w → (∀ v ∈ SNoS_ (SNoLev y , ∀ w ∈ SNoS_ (SNoLev z , x * (v * w = (x * v * w → (∀ u ∈ SNoS_ (SNoLev x , ∀ v ∈ SNoS_ (SNoLev y , ∀ w ∈ SNoS_ (SNoLev z , u * (v * w = (u * v * w → ∀L, (∀ u ∈ L , ∀q : prop , (∀ v ∈ SNoL x , ∀ w ∈ SNoL (y * z , u = v * (y * z + x * w + - v * w q → (∀ v ∈ SNoR x , ∀ w ∈ SNoR (y * z , u = v * (y * z + x * w + - v * w q → q → ∀ u ∈ L , u < (x * y * z Axiom. (
mul_SNo_assoc_lem2 ) We take the following as an axiom:
∀x y z, SNo x SNo y SNo z (∀ u ∈ SNoS_ (SNoLev x , u * (y * z = (u * y * z → (∀ v ∈ SNoS_ (SNoLev y , x * (v * z = (x * v * z → (∀ w ∈ SNoS_ (SNoLev z , x * (y * w = (x * y * w → (∀ u ∈ SNoS_ (SNoLev x , ∀ v ∈ SNoS_ (SNoLev y , u * (v * z = (u * v * z → (∀ u ∈ SNoS_ (SNoLev x , ∀ w ∈ SNoS_ (SNoLev z , u * (y * w = (u * y * w → (∀ v ∈ SNoS_ (SNoLev y , ∀ w ∈ SNoS_ (SNoLev z , x * (v * w = (x * v * w → (∀ u ∈ SNoS_ (SNoLev x , ∀ v ∈ SNoS_ (SNoLev y , ∀ w ∈ SNoS_ (SNoLev z , u * (v * w = (u * v * w → ∀R, (∀ u ∈ R , ∀q : prop , (∀ v ∈ SNoL x , ∀ w ∈ SNoR (y * z , u = v * (y * z + x * w + - v * w q → (∀ v ∈ SNoR x , ∀ w ∈ SNoL (y * z , u = v * (y * z + x * w + - v * w q → q → ∀ u ∈ R , (x * y * z < u End of Section mul_SNo_assoc_lems
Axiom. (
SNo_foil ) We take the following as an axiom:
Axiom. (
SNo_foil_mm ) We take the following as an axiom:
End of Section SurrealMul
Beginning of Section SurrealExp
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 342 and which associates to the right corresponding to applying term
exp_SNo_nat .
Axiom. (
eps_bd_1 ) We take the following as an axiom:
Axiom. (
SNoS_finite ) We take the following as an axiom:
End of Section SurrealExp
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 .
Primitive . The name
int is a term of type
set .
Axiom. (
int_SNo ) We take the following as an axiom:
Axiom. (
int_add_SNo ) We take the following as an axiom:
Axiom. (
int_mul_SNo ) We take the following as an axiom:
End of Section Int
Beginning of Section SurrealAbs
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 .
Axiom. (
abs_SNo_0 ) We take the following as an axiom:
Axiom. (
pos_abs_SNo ) We take the following as an axiom:
Axiom. (
neg_abs_SNo ) We take the following as an axiom:
Axiom. (
SNo_abs_SNo ) We take the following as an axiom:
Axiom. (
abs_SNo_Lev ) We take the following as an axiom:
End of Section SurrealAbs
Beginning of Section SNoMaxMin
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 342 and which associates to the right corresponding to applying term
exp_SNo_nat .
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 .
End of Section SNoMaxMin
Beginning of Section DiadicRationals
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 .
End of Section DiadicRationals
Beginning of Section SurrealDiv
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 .
Proof: Let x and xi be given.
Assume Hx Hxi Hxpos Hxxi .
Assume H1 .
An exact proof term for the current goal is H1
rewrite the current goal using Hxxi
We will
prove 0 = x * xi .
rewrite the current goal using H1
Use symmetry.
rewrite the current goal using Hxxi
We will
prove x * xi < 0 .
An
exact proof term for the current goal is
mul_SNo_pos_neg x xi Hx Hxi Hxpos H1 .
∎
Proof: Let x, x', x'i, y and y' be given.
Assume Hx Hxpos Hx' Hx'i Hx'x'i Hy Hxy1 Hy' Hy'y .
Apply SepE (SNoL x (λw ⇒ 0 < w ) x' Hx' to the current goal.
Apply SNoL_E x Hx x' Hx'L to the current goal.
We prove the intermediate
claim Lxy :
SNo (x * y .
An
exact proof term for the current goal is
SNo_mul_SNo x y Hx Hy .
We prove the intermediate
claim Lxy' :
SNo (x * y' .
An
exact proof term for the current goal is
SNo_mul_SNo x y' Hx Hy' .
We prove the intermediate
claim L1 :
0 < 1 + - x * y .
We will
prove 0 + x * y < 1 .
rewrite the current goal using
add_SNo_0L (x * y Lxy (from left to right).
An exact proof term for the current goal is Hxy1
We prove the intermediate
claim L2 :
(x' + - x * x'i < 0 .
We will
prove x' + - x < 0 .
We will
prove x' < 0 + x .
rewrite the current goal using
add_SNo_0L x Hx (from left to right).
An exact proof term for the current goal is Hx'3
An
exact proof term for the current goal is
SNo_recip_pos_pos x' x'i Hx'1 Hx'i Hx'pos Hx'x'i .
We prove the intermediate
claim L3 :
1 + - x * y' < 0 .
rewrite the current goal using Hy'y
We will
prove (1 + - x * y * (x' + - x * x'i < 0 .
An exact proof term for the current goal is L1
An exact proof term for the current goal is L2
We will
prove 1 < x * y' .
rewrite the current goal using
add_SNo_0L (x * y' Lxy' (from right to left).
We will
prove 1 < 0 + x * y' .
∎
Proof: Let x, x', x'i, y and y' be given.
Assume Hx Hxpos Hx' Hx'i Hx'x'i Hy Hxy1 Hy' Hy'y .
Apply SepE (SNoL x (λw ⇒ 0 < w ) x' Hx' to the current goal.
Apply SNoL_E x Hx x' Hx'L to the current goal.
We prove the intermediate
claim Lxy :
SNo (x * y .
An
exact proof term for the current goal is
SNo_mul_SNo x y Hx Hy .
We prove the intermediate
claim Lxy' :
SNo (x * y' .
An
exact proof term for the current goal is
SNo_mul_SNo x y' Hx Hy' .
We prove the intermediate
claim L1 :
1 + - x * y < 0 .
We will
prove 1 < 0 + x * y .
rewrite the current goal using
add_SNo_0L (x * y Lxy (from left to right).
An exact proof term for the current goal is Hxy1
We prove the intermediate
claim L2 :
(x' + - x * x'i < 0 .
We will
prove x' + - x < 0 .
We will
prove x' < 0 + x .
rewrite the current goal using
add_SNo_0L x Hx (from left to right).
An exact proof term for the current goal is Hx'3
An
exact proof term for the current goal is
SNo_recip_pos_pos x' x'i Hx'1 Hx'i Hx'pos Hx'x'i .
We prove the intermediate
claim L3 :
0 < 1 + - x * y' .
rewrite the current goal using Hy'y
We will
prove 0 < (1 + - x * y * (x' + - x * x'i .
An exact proof term for the current goal is L1
An exact proof term for the current goal is L2
We will
prove x * y' < 1 .
rewrite the current goal using
add_SNo_0L (x * y' Lxy' (from right to left).
We will
prove 0 + x * y' < 1 .
∎
Proof: Let x, x', x'i, y and y' be given.
Assume Hx Hxpos Hx' Hx'i Hx'x'i Hy Hxy1 Hy' Hy'y .
Apply SNoR_E x Hx x' Hx' to the current goal.
We prove the intermediate
claim Lxy :
SNo (x * y .
An
exact proof term for the current goal is
SNo_mul_SNo x y Hx Hy .
We prove the intermediate
claim Lxy' :
SNo (x * y' .
An
exact proof term for the current goal is
SNo_mul_SNo x y' Hx Hy' .
We prove the intermediate
claim Lx'pos :
0 < x' .
An
exact proof term for the current goal is
SNoLt_tra 0 x x' SNo_0 Hx Hx'1 Hxpos Hx'3 .
We prove the intermediate
claim L1 :
0 < 1 + - x * y .
We will
prove 0 + x * y < 1 .
rewrite the current goal using
add_SNo_0L (x * y Lxy (from left to right).
An exact proof term for the current goal is Hxy1
We prove the intermediate
claim L2 :
0 < (x' + - x * x'i .
We will
prove 0 < x' + - x .
We will
prove 0 + x < x' .
rewrite the current goal using
add_SNo_0L x Hx (from left to right).
An exact proof term for the current goal is Hx'3
An
exact proof term for the current goal is
SNo_recip_pos_pos x' x'i Hx'1 Hx'i Lx'pos Hx'x'i .
We prove the intermediate
claim L3 :
0 < 1 + - x * y' .
rewrite the current goal using Hy'y
We will
prove 0 < (1 + - x * y * (x' + - x * x'i .
An exact proof term for the current goal is L1
An exact proof term for the current goal is L2
We will
prove x * y' < 1 .
rewrite the current goal using
add_SNo_0L (x * y' Lxy' (from right to left).
We will
prove 0 + x * y' < 1 .
∎
Proof: Let x, x', x'i, y and y' be given.
Assume Hx Hxpos Hx' Hx'i Hx'x'i Hy Hxy1 Hy' Hy'y .
Apply SNoR_E x Hx x' Hx' to the current goal.
We prove the intermediate
claim Lxy :
SNo (x * y .
An
exact proof term for the current goal is
SNo_mul_SNo x y Hx Hy .
We prove the intermediate
claim Lxy' :
SNo (x * y' .
An
exact proof term for the current goal is
SNo_mul_SNo x y' Hx Hy' .
We prove the intermediate
claim Lx'pos :
0 < x' .
An
exact proof term for the current goal is
SNoLt_tra 0 x x' SNo_0 Hx Hx'1 Hxpos Hx'3 .
We prove the intermediate
claim L1 :
1 + - x * y < 0 .
We will
prove 1 < 0 + x * y .
rewrite the current goal using
add_SNo_0L (x * y Lxy (from left to right).
An exact proof term for the current goal is Hxy1
We prove the intermediate
claim L2 :
0 < (x' + - x * x'i .
We will
prove 0 < x' + - x .
We will
prove 0 + x < x' .
rewrite the current goal using
add_SNo_0L x Hx (from left to right).
An exact proof term for the current goal is Hx'3
An
exact proof term for the current goal is
SNo_recip_pos_pos x' x'i Hx'1 Hx'i Lx'pos Hx'x'i .
We prove the intermediate
claim L3 :
1 + - x * y' < 0 .
rewrite the current goal using Hy'y
We will
prove (1 + - x * y * (x' + - x * x'i < 0 .
An exact proof term for the current goal is L1
An exact proof term for the current goal is L2
We will
prove 1 < x * y' .
rewrite the current goal using
add_SNo_0L (x * y' Lxy' (from right to left).
We will
prove 1 < 0 + x * y' .
∎
Definition. We define
SNo_recipauxset to be
λY x X g ⇒ ⋃ y ∈ Y { (1 + (x' + - x * y * g x' | x' ∈ X } of type
set → set → set → (set → set → set .
Proof: Let Y, x, X, g and y be given.
Assume Hy .
Let x' be given.
Assume Hx' .
An
exact proof term for the current goal is
ReplI X (λx' ⇒ (1 + (x' + - x * y * g x' ) x' Hx' .
∎
Proof: Let Y, x, X, g and z be given.
Assume Hz .
Let p be given.
Assume H1 .
Let y be given.
Assume Hy .
Let x' be given.
An exact proof term for the current goal is H1 y Hy x' Hx' H3
∎
Proof: Let Y, x, X, g and h be given.
Assume Hgh .
Let y be given.
Assume Hy .
Apply ReplEq_ext X (λx' ⇒ (1 + (x' + - x * y * g x' ) (λx' ⇒ (1 + (x' + - x * y * h x' ) to the current goal.
Let x' be given.
Assume Hx' .
We will
prove (1 + (x' + - x * y * g x' = (1 + (x' + - x * y * h x' .
Use f_equal.
An exact proof term for the current goal is Hgh x' Hx'
∎
Proof: Let x and g be given.
∎
Proof: Let x, g and n be given.
Assume Hn .
∎
Proof: Let x be given.
Assume Hx Hxpos .
Let g be given.
Assume Hg .
Let x' be given.
Assume Hx' Hx'pos .
Let y and y' be given.
Assume Hy Hy' .
Assume _ _ Hx'1 _ .
rewrite the current goal using Hy'
An
exact proof term for the current goal is
SNo_1 .
An exact proof term for the current goal is Hx'1
An exact proof term for the current goal is Hx
An exact proof term for the current goal is Hy
Apply Hg x' Hx' Hx'pos
Assume H _ .
An exact proof term for the current goal is H
Let x' be given.
Assume Hx' Hx'pos .
Let y and y' be given.
Assume Hy Hy' .
Assume _ _ Hx'1 _ .
Apply Hg x' Hx' Hx'pos
Assume Hgx'1 Hgx'2 .
We will
prove 1 + - x * y' = (1 + - x * y * (x' * g x' + (- x * g x' .
rewrite the current goal using Hgx'2
Use f_equal.
We will
prove - x * y' = (- x * g x' + - x * y + (- x * y * (- x * g x' .
rewrite the current goal using Hy'
We will
prove - x * ((1 + (x' + - x * y * g x' = (- x * g x' + - x * y + (- x * y * (- x * g x' .
We will
prove - x * (1 * g x' + ((x' + - x * y * g x' = (- x * g x' + - x * y + (- x * y * (- x * g x' .
rewrite the current goal using
mul_SNo_oneL (g x' Hgx'1 (from left to right).
We will
prove - x * (g x' + ((x' + - x * y * g x' = (- x * g x' + - x * y + (- x * y * (- x * g x' .
We will
prove (- x * (g x' + ((x' + - x * y * g x' = (- x * g x' + - x * y + (- x * y * (- x * g x' .
We will
prove (- x * g x' + (- x * (((x' + - x * y * g x' = (- x * g x' + - x * y + (- x * y * (- x * g x' .
Use f_equal.
We will
prove (- x * (((x' + - x * y * g x' = - x * y + (- x * y * (- x * g x' .
We will
prove (- x * ((x' * y + (- x * y * g x' = - x * y + (- x * y * (- x * g x' .
We will
prove (- x * ((x' * y * g x' + ((- x * y * g x' = - x * y + (- x * y * (- x * g x' .
rewrite the current goal using
mul_SNo_com x' y Hx'1 Hy (from left to right).
rewrite the current goal using
mul_SNo_assoc y x' (g x' Hy Hx'1 Hgx'1 (from right to left).
rewrite the current goal using Hgx'2
rewrite the current goal using
mul_SNo_oneR y Hy (from left to right).
We will
prove (- x * (y + ((- x * y * g x' = - x * y + (- x * y * (- x * g x' .
We will
prove (- x * y + (- x * ((- x * y * g x' = - x * y + (- x * y * (- x * g x' .
We will
prove - x * y + (- x * ((- x * y * g x' = - x * y + (- x * y * (- x * g x' .
Use f_equal.
We will
prove (- x * ((- x * y * g x' = (- x * y * (- x * g x' .
Use f_equal.
We will
prove (- x * ((- x * y = (- x * y * (- x .
We will
prove (- x * (y * (- x = (- x * y * (- x .
We will
prove (- x * (y * (- x = ((- x * y * (- x .
Apply andI to the current goal.
Let y be given.
rewrite the current goal using
tuple_2_0_eq (from left to right).
rewrite the current goal using
SingE 0 y H1 (from left to right).
Apply andI to the current goal.
An
exact proof term for the current goal is
SNo_0 .
rewrite the current goal using
mul_SNo_zeroR x Hx (from left to right).
An
exact proof term for the current goal is
SNoLt_0_1 .
Let y be given.
rewrite the current goal using
tuple_2_1_eq (from left to right).
An
exact proof term for the current goal is
EmptyE y H1 .
Let k be given.
Assume IH .
Apply IH
Apply andI to the current goal.
Let y' be given.
rewrite the current goal using
tuple_2_0_eq (from left to right).
Assume H1 .
Assume H1 .
An exact proof term for the current goal is IH1 y'
Assume H1 .
Let y be given.
Let x' be given.
Apply SNoR_E x Hx x' Hx' to the current goal.
Assume Hx'1 Hx'2 Hx'3 .
Apply SNoS_I2 x' x Hx'1 Hx Hx'2 to the current goal.
We prove the intermediate
claim Lx'pos :
0 < x' .
An
exact proof term for the current goal is
SNoLt_tra 0 x x' SNo_0 Hx Hx'1 Hxpos Hx'3 .
Apply IH1 y Hy
Apply Hg x' Lx' Lx'pos
We prove the intermediate
claim Ly' :
SNo y' .
An exact proof term for the current goal is L1 x' Lx' Lx'pos y y' Hy1 H2
Apply andI to the current goal.
An exact proof term for the current goal is Ly'
Apply SNo_recip_lem3 x x' (g x' y y' Hx Hxpos Hx' Hgx'1 Hgx'2 Hy1 Hxy1 Ly' to the current goal.
An exact proof term for the current goal is L2 x' Lx' Lx'pos y y' Hy1 H2
Assume H1 .
Let y be given.
Let x' be given.
Apply SepE (SNoL x (λw ⇒ 0 < w ) x' Hx' to the current goal.
Apply SNoL_E x Hx x' Hx'0 to the current goal.
Assume Hx'1 Hx'2 Hx'3 .
Apply SNoS_I2 x' x Hx'1 Hx Hx'2 to the current goal.
Apply IH2 y Hy
Apply Hg x' Lx' Hx'pos
We prove the intermediate
claim Ly' :
SNo y' .
An exact proof term for the current goal is L1 x' Lx' Hx'pos y y' Hy1 H2
Apply andI to the current goal.
An exact proof term for the current goal is Ly'
Apply SNo_recip_lem2 x x' (g x' y y' Hx Hxpos Hx' Hgx'1 Hgx'2 Hy1 Hxy1 Ly' to the current goal.
An exact proof term for the current goal is L2 x' Lx' Hx'pos y y' Hy1 H2
An exact proof term for the current goal is Hk
Let y' be given.
rewrite the current goal using
tuple_2_1_eq (from left to right).
Assume H1 .
Assume H1 .
An exact proof term for the current goal is IH2 y'
Assume H1 .
Let y be given.
Let x' be given.
Apply SepE (SNoL x (λw ⇒ 0 < w ) x' Hx' to the current goal.
Apply SNoL_E x Hx x' Hx'0 to the current goal.
Assume Hx'1 Hx'2 Hx'3 .
Apply SNoS_I2 x' x Hx'1 Hx Hx'2 to the current goal.
Apply IH1 y Hy
Apply Hg x' Lx' Hx'pos
We prove the intermediate
claim Ly' :
SNo y' .
An exact proof term for the current goal is L1 x' Lx' Hx'pos y y' Hy1 H2
Apply andI to the current goal.
An exact proof term for the current goal is Ly'
Apply SNo_recip_lem1 x x' (g x' y y' Hx Hxpos Hx' Hgx'1 Hgx'2 Hy1 Hxy1 Ly' to the current goal.
An exact proof term for the current goal is L2 x' Lx' Hx'pos y y' Hy1 H2
Assume H1 .
Let y be given.
Let x' be given.
Apply SNoR_E x Hx x' Hx' to the current goal.
Assume Hx'1 Hx'2 Hx'3 .
Apply SNoS_I2 x' x Hx'1 Hx Hx'2 to the current goal.
We prove the intermediate
claim Lx'pos :
0 < x' .
An
exact proof term for the current goal is
SNoLt_tra 0 x x' SNo_0 Hx Hx'1 Hxpos Hx'3 .
Apply IH2 y Hy
Apply Hg x' Lx' Lx'pos
We prove the intermediate
claim Ly' :
SNo y' .
An exact proof term for the current goal is L1 x' Lx' Lx'pos y y' Hy1 H2
Apply andI to the current goal.
An exact proof term for the current goal is Ly'
Apply SNo_recip_lem4 x x' (g x' y y' Hx Hxpos Hx' Hgx'1 Hgx'2 Hy1 Hxy1 Ly' to the current goal.
An exact proof term for the current goal is L2 x' Lx' Lx'pos y y' Hy1 H2
An exact proof term for the current goal is Hk
∎
Proof: Let x be given.
Assume Hx Hxpos .
Let g be given.
Assume Hg .
Apply and3I to the current goal.
Let y be given.
Assume Hy .
Let k be given.
Assume Hk .
Assume H2 _ .
Apply H2 y H1
Assume H3 _ .
An exact proof term for the current goal is H3
Let y be given.
Assume Hy .
Let k be given.
Assume Hk .
Assume _ H2 .
Apply H2 y H1
Assume H3 _ .
An exact proof term for the current goal is H3
Let w be given.
Assume Hw .
Let z be given.
Assume Hz .
Let k be given.
Assume Hk .
Assume H2 _ .
Apply H2 w H1
Let k' be given.
Assume Hk' .
Assume _ H6 .
Apply H6 z H5
An exact proof term for the current goal is H9
We will
prove x * z ≤ x * w .
An exact proof term for the current goal is H4
∎
Proof: Let x be given.
Assume Hx .
Let g and h be given.
Assume Hgh .
rewrite the current goal using
SNo_recipaux_0 x h (from left to right).
Let k be given.
rewrite the current goal using
SNo_recipaux_S x g k Hk (from left to right).
rewrite the current goal using
SNo_recipaux_S x h k Hk (from left to right).
rewrite the current goal using IH
Use f_equal.
Use f_equal.
Let w be given.
Apply SNoR_E x Hx w Hw to the current goal.
Assume Hw1 Hw2 Hw3 .
Apply Hgh
An
exact proof term for the current goal is
SNoS_I2 w x Hw1 Hx Hw2 .
Let w be given.
Assume Hw1 Hw2 Hw3 .
Apply Hgh
An
exact proof term for the current goal is
SNoS_I2 w x Hw1 Hx Hw2 .
Use f_equal.
Use f_equal.
Let w be given.
Assume Hw1 Hw2 Hw3 .
Apply Hgh
An
exact proof term for the current goal is
SNoS_I2 w x Hw1 Hx Hw2 .
Let w be given.
Apply SNoR_E x Hx w Hw to the current goal.
Assume Hw1 Hw2 Hw3 .
Apply Hgh
An
exact proof term for the current goal is
SNoS_I2 w x Hw1 Hx Hw2 .
rewrite the current goal using L1
rewrite the current goal using L2
Use reflexivity.
∎
Beginning of Section recip_SNo_pos
Proof:
Let x be given.
Assume Hx .
Let g and h be given.
Assume Hgh .
Use f_equal.
Let k be given.
Assume Hk .
Use f_equal.
An exact proof term for the current goal is Hgh
An
exact proof term for the current goal is
omega_nat_p k Hk .
Let k be given.
Assume Hk .
Use f_equal.
An exact proof term for the current goal is Hgh
An
exact proof term for the current goal is
omega_nat_p k Hk .
∎
Proof:
Let x be given.
We prove the intermediate
claim L1L :
∀ w ∈ L , x * w < 1 .
Let w be given.
Assume Hw .
Let k be given.
Assume H2 _ .
Apply H2 w H1
Assume _ H3 .
An exact proof term for the current goal is H3
We prove the intermediate
claim L1R :
∀ z ∈ R , 1 < x * z .
Let z be given.
Assume Hz .
Let k be given.
Assume _ H2 .
Apply H2 z H1
Assume _ H3 .
An exact proof term for the current goal is H3
We prove the intermediate
claim L2 :
SNoCutP L R .
Apply L2
Assume HLHR .
Apply HLHR
Set y to be the term
SNoCut L R .
We prove the intermediate
claim L3 :
SNo (x * y .
An
exact proof term for the current goal is
SNo_mul_SNo x y Hx H1 .
We prove the intermediate
claim L4 :
0 < y .
Apply H3
An
exact proof term for the current goal is
nat_0 .
rewrite the current goal using
tuple_2_0_eq (from left to right).
Apply SingI to the current goal.
We prove the intermediate
claim L5 :
0 < x * y .
An
exact proof term for the current goal is
mul_SNo_pos_pos x y Hx H1 Hxpos L4 .
We prove the intermediate
claim L6 :
∀ w ∈ SNoL y , ∃ w' ∈ L , w ≤ w' .
We prove the intermediate
claim L7 :
∀ z ∈ SNoR y , ∃ z' ∈ R , z' ≤ z .
Apply andI to the current goal.
An exact proof term for the current goal is H1
Apply dneg to the current goal.
We prove the intermediate
claim L8 :
1 ∈ SNoR (x * y .
Apply SNoR_I to the current goal.
An exact proof term for the current goal is L3
An
exact proof term for the current goal is
SNo_1 .
An exact proof term for the current goal is H7
Apply HC
An
exact proof term for the current goal is
pos_low_eq_one (x * y L3 L5 H7 .
An exact proof term for the current goal is H6
Let v, w and w' be given.
Assume Hv1 Hw1 Hw' HvS Hvpos H7 Hw'' H8 .
We prove the intermediate
claim Lw''1 :
SNo w'' .
An exact proof term for the current goal is HL w'' Hw''
We will
prove 1 < 1 + v * (y + - w'' .
rewrite the current goal using
add_SNo_0R 1 SNo_1 (from right to left) at position 1.
We will
prove 0 < v * (y + - w'' .
We will
prove 0 < y + - w'' .
An exact proof term for the current goal is H3 w'' Hw''
We will
prove 1 + v * (y + - w'' ≤ 1 .
We will
prove 1 + v * y + v * (- w'' ≤ 1 .
We will
prove 1 + v * y + - v * w'' ≤ 1 .
Apply IH v HvS Hvpos
rewrite the current goal using Hrv2
We will
prove - (v + - x * w' + v * y ≤ 1 .
We will
prove (- (v + - x * w' + v * y ≤ 1 .
We will
prove (- v + x * w' + v * y ≤ 1 .
We will
prove (- v + x * w' + v * y ≤ (- v + x * w + v * y .
We will
prove (- v + x * w' ≤ (- v + x * w .
An exact proof term for the current goal is H8
We will
prove (- v + x * w + v * y ≤ 1 .
We will
prove ((- v * w + x * w + v * y ≤ 1 .
We will
prove (- v * w + x * w + v * y ≤ 1 .
We will
prove - v * w + x * w + v * y ≤ 1 .
We will
prove x * w + v * y ≤ 1 + v * w .
We will
prove v * y + x * w ≤ 1 + v * w .
An exact proof term for the current goal is H7
Let v be given.
Let w be given.
Apply SNoL_E x Hx v Hv to the current goal.
Apply SNoR_E y H1 w Hw to the current goal.
An
exact proof term for the current goal is
SNoS_I2 v x Hv1 Hx Hv2 .
We prove the intermediate
claim Lxw :
SNo (x * w .
An
exact proof term for the current goal is
SNo_mul_SNo x w Hx Hw1 .
Apply L7 w Hw
Let w' be given.
Assume Hw' .
Apply Hw'
We prove the intermediate
claim Lw' :
SNo w' .
An exact proof term for the current goal is HR w' Hw'1
We prove the intermediate
claim Lxw' :
SNo (x * w' .
An
exact proof term for the current goal is
SNo_mul_SNo x w' Hx Lw' .
We prove the intermediate
claim Lvpos :
0 < v .
An exact proof term for the current goal is H8
We will
prove 1 < x * w' .
An exact proof term for the current goal is L1R w' Hw'1
We will
prove x * w' ≤ 1 .
We will
prove x * w' ≤ x * w .
We will
prove x * w ≤ v * (y + - w + x * w .
rewrite the current goal using
add_SNo_0L (x * w Lxw (from right to left) at position 1.
We will
prove 0 + x * w ≤ v * (y + - w + x * w .
We will
prove 0 ≤ v * (y + - w .
We will
prove y + - w < 0 .
rewrite the current goal using
add_SNo_0L w Hw1 (from left to right).
An exact proof term for the current goal is Hw3
We will
prove v * (y + - w + x * w ≤ 1 .
We will
prove (v * y + v * (- w + x * w ≤ 1 .
We will
prove (v * y + x * w + v * (- w ≤ 1 .
We will
prove v * y + x * w ≤ 1 + - v * (- w .
An exact proof term for the current goal is H7
We prove the intermediate
claim Lw'' :
w'' ∈ L .
Let k be given.
rewrite the current goal using
tuple_2_0_eq (from left to right).
An exact proof term for the current goal is H9
Apply SepI to the current goal.
An exact proof term for the current goal is Hv
An exact proof term for the current goal is Lvpos
Apply L9 v w w' Hv1 Hw1 Lw' LvS Lvpos H7 Lw''
We will
prove (- v + x * w' ≤ (- v + x * w .
We will
prove 0 ≤ - v + x .
We will
prove 0 ≤ x + - v .
rewrite the current goal using
add_SNo_0L v Hv1 (from left to right).
An exact proof term for the current goal is Hv3
An exact proof term for the current goal is Lw'
An exact proof term for the current goal is Hw1
An exact proof term for the current goal is Hw'2
Let v be given.
Let w be given.
Apply SNoR_E x Hx v Hv to the current goal.
Apply SNoL_E y H1 w Hw to the current goal.
An
exact proof term for the current goal is
SNoS_I2 v x Hv1 Hx Hv2 .
We prove the intermediate
claim Lxw :
SNo (x * w .
An
exact proof term for the current goal is
SNo_mul_SNo x w Hx Hw1 .
Apply L6 w Hw
Let w' be given.
Assume Hw' .
Apply Hw'
We prove the intermediate
claim Lw' :
SNo w' .
An exact proof term for the current goal is HL w' Hw'1
We prove the intermediate
claim Lxw' :
SNo (x * w' .
An
exact proof term for the current goal is
SNo_mul_SNo x w' Hx Lw' .
We prove the intermediate
claim Lvpos :
0 < v .
An
exact proof term for the current goal is
SNoLt_tra 0 x v SNo_0 Hx Hv1 Hxpos Hv3 .
We prove the intermediate
claim Lw'' :
w'' ∈ L .
Let k be given.
rewrite the current goal using
tuple_2_0_eq (from left to right).
An exact proof term for the current goal is H9
We will
prove v ∈ SNoR x .
An exact proof term for the current goal is Hv
Apply L9 v w w' Hv1 Hw1 Lw' LvS Lvpos H7 Lw''
We will
prove (- v + x * w' ≤ (- v + x * w .
We will
prove - v + x ≤ 0 .
We will
prove x + - v ≤ 0 .
rewrite the current goal using
add_SNo_0L v Hv1 (from left to right).
An exact proof term for the current goal is Hv3
An exact proof term for the current goal is Lw'
An exact proof term for the current goal is Hw1
An exact proof term for the current goal is Hw'2
Apply HC
An exact proof term for the current goal is H6
We prove the intermediate
claim L10 :
1 ∈ SNoL (x * y .
Apply SNoL_I to the current goal.
An exact proof term for the current goal is L3
An
exact proof term for the current goal is
SNo_1 .
An exact proof term for the current goal is H7
Apply HC
An
exact proof term for the current goal is
pos_low_eq_one (x * y L3 L5 H7 .
An exact proof term for the current goal is H6
Let v, w and w' be given.
Assume Hv1 Hw1 Hw' HvS Hvpos H7 Hw'' H8 .
We prove the intermediate
claim Lw''1 :
SNo w'' .
An exact proof term for the current goal is HR w'' Hw''
We will
prove 1 ≤ 1 + v * (y + - w'' .
We will
prove 1 ≤ 1 + v * y + v * (- w'' .
We will
prove 1 ≤ 1 + v * y + - v * w'' .
Apply IH v HvS Hvpos
rewrite the current goal using Hrv2
We will
prove 1 ≤ - (v + - x * w' + v * y .
We will
prove 1 ≤ (- (v + - x * w' + v * y .
We will
prove 1 ≤ (- v + x * w' + v * y .
We will
prove 1 ≤ (- v + x * w + v * y .
We will
prove 1 ≤ ((- v * w + x * w + v * y .
We will
prove 1 ≤ (- v * w + x * w + v * y .
We will
prove 1 ≤ - v * w + x * w + v * y .
We will
prove 1 ≤ (x * w + v * y + - v * w .
We will
prove 1 + v * w ≤ x * w + v * y .
We will
prove 1 + v * w ≤ v * y + x * w .
An exact proof term for the current goal is H7
We will
prove (- v + x * w + v * y ≤ (- v + x * w' + v * y .
We will
prove (- v + x * w ≤ (- v + x * w' .
An exact proof term for the current goal is H8
We will
prove 1 + v * (y + - w'' < 1 .
rewrite the current goal using
add_SNo_0R 1 SNo_1 (from right to left) at position 4.
We will
prove v * (y + - w'' < 0 .
We will
prove y + - w'' < 0 .
We will
prove y < 0 + w'' .
rewrite the current goal using
add_SNo_0L w'' Lw''1 (from left to right).
An exact proof term for the current goal is H4 w'' Hw''
Let v be given.
Let w be given.
Apply SNoL_E x Hx v Hv to the current goal.
Apply SNoL_E y H1 w Hw to the current goal.
An
exact proof term for the current goal is
SNoS_I2 v x Hv1 Hx Hv2 .
We prove the intermediate
claim Lxw :
SNo (x * w .
An
exact proof term for the current goal is
SNo_mul_SNo x w Hx Hw1 .
Apply L6 w Hw
Let w' be given.
Assume Hw' .
Apply Hw'
We prove the intermediate
claim Lw' :
SNo w' .
An exact proof term for the current goal is HL w' Hw'1
We prove the intermediate
claim Lxw' :
SNo (x * w' .
An
exact proof term for the current goal is
SNo_mul_SNo x w' Hx Lw' .
We prove the intermediate
claim Lvpos :
0 < v .
An exact proof term for the current goal is H8
We will
prove 1 ≤ x * w' .
We will
prove 1 ≤ v * (y + - w + x * w .
We will
prove 1 ≤ (v * y + v * (- w + x * w .
We will
prove 1 ≤ (v * y + x * w + v * (- w .
We will
prove 1 ≤ (v * y + x * w + - v * w .
We will
prove 1 + v * w ≤ v * y + x * w .
An exact proof term for the current goal is H7
We will
prove v * (y + - w + x * w ≤ x * w .
rewrite the current goal using
add_SNo_0L (x * w Lxw (from right to left) at position 2.
We will
prove v * (y + - w + x * w ≤ 0 + x * w .
We will
prove v * (y + - w ≤ 0 .
We will
prove 0 < y + - w .
rewrite the current goal using
add_SNo_0L w Hw1 (from left to right).
An exact proof term for the current goal is Hw3
We will
prove x * w ≤ x * w' .
We will
prove x * w' < 1 .
An exact proof term for the current goal is L1L w' Hw'1
We prove the intermediate
claim Lw'' :
w'' ∈ R .
Let k be given.
rewrite the current goal using
tuple_2_1_eq (from left to right).
An exact proof term for the current goal is H9
Apply SepI to the current goal.
An exact proof term for the current goal is Hv
An exact proof term for the current goal is Lvpos
Apply L11 v w w' Hv1 Hw1 Lw' LvS Lvpos H7 Lw''
We will
prove (- v + x * w ≤ (- v + x * w' .
We will
prove 0 ≤ - v + x .
We will
prove 0 ≤ x + - v .
rewrite the current goal using
add_SNo_0L v Hv1 (from left to right).
An exact proof term for the current goal is Hv3
An exact proof term for the current goal is Hw1
An exact proof term for the current goal is Lw'
An exact proof term for the current goal is Hw'2
Let v be given.
Let w be given.
Apply SNoR_E x Hx v Hv to the current goal.
Apply SNoR_E y H1 w Hw to the current goal.
An
exact proof term for the current goal is
SNoS_I2 v x Hv1 Hx Hv2 .
We prove the intermediate
claim Lvpos :
0 < v .
An
exact proof term for the current goal is
SNoLt_tra 0 x v SNo_0 Hx Hv1 Hxpos Hv3 .
We prove the intermediate
claim Lxw :
SNo (x * w .
An
exact proof term for the current goal is
SNo_mul_SNo x w Hx Hw1 .
Apply L7 w Hw
Let w' be given.
Assume Hw' .
Apply Hw'
We prove the intermediate
claim Lw' :
SNo w' .
An exact proof term for the current goal is HR w' Hw'1
We prove the intermediate
claim Lxw' :
SNo (x * w' .
An
exact proof term for the current goal is
SNo_mul_SNo x w' Hx Lw' .
We prove the intermediate
claim Lw'' :
w'' ∈ R .
Let k be given.
rewrite the current goal using
tuple_2_1_eq (from left to right).
An exact proof term for the current goal is H9
We will
prove v ∈ SNoR x .
An exact proof term for the current goal is Hv
Apply L11 v w w' Hv1 Hw1 Lw' LvS Lvpos H7 Lw''
We will
prove (- v + x * w ≤ (- v + x * w' .
Apply L11 v w w' Hv1 Hw1 Lw' LvS Lvpos H7 Lw''
We will
prove (- v + x * w ≤ (- v + x * w' .
We will
prove - v + x ≤ 0 .
We will
prove x + - v ≤ 0 .
rewrite the current goal using
add_SNo_0L v Hv1 (from left to right).
An exact proof term for the current goal is Hv3
An exact proof term for the current goal is Hw1
An exact proof term for the current goal is Lw'
An exact proof term for the current goal is Hw'2
∎
Proof: Let x be given.
Assume Hx Hxpos .
Assume H _ .
An exact proof term for the current goal is H
∎
Proof: Let x be given.
Assume Hx Hxpos .
Assume _ H .
An exact proof term for the current goal is H
∎
Proof: Let x be given.
Assume Hx Hxpos .
rewrite the current goal using H1
Assume H1 .
An exact proof term for the current goal is H1
∎
Proof: Let x be given.
Assume Hx Hxpos .
rewrite the current goal using Hrx0
An exact proof term for the current goal is Lrxpos
∎
Proof: Let n be given.
Assume Hn .
We prove the intermediate
claim Len1 :
SNo (eps_ n .
We prove the intermediate
claim Len2 :
0 < eps_ n .
We prove the intermediate
claim Len3 :
eps_ n ≠ 0 .
rewrite the current goal using H1
An exact proof term for the current goal is Len2
∎
Proof: Let n be given.
Assume Hn .
∎
End of Section recip_SNo_pos
Proof: Let x be given.
Assume Hxpos .
∎
Proof: Let x be given.
Assume Hx Hxneg .
We prove the intermediate
claim L1 :
¬ (0 < x .
∎
Proof: Let x be given.
Assume Hx .
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is H1
rewrite the current goal using H1
rewrite the current goal using
recip_SNo_0 (from left to right).
An
exact proof term for the current goal is
SNo_0 .
∎
Proof: Let x be given.
Assume Hx Hx0 .
We prove the intermediate
claim L1 :
0 < - x .
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is H1
An exact proof term for the current goal is Hx0 H1
∎
Proof: Let x be given.
Assume Hx Hx0 .
∎
Proof: Let n be given.
Assume Hn .
∎
Proof: Let n be given.
Assume Hn .
We prove the intermediate
claim L1 :
0 < 2 ^ n .
∎
Proof: Let x be given.
Assume Hx .
We prove the intermediate
claim Lmx :
SNo (- x .
We prove the intermediate
claim Lmxpos :
0 < - x .
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is H1
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is L1
rewrite the current goal using H1
rewrite the current goal using
recip_SNo_0 (from left to right).
An
exact proof term for the current goal is
recip_SNo_0 .
∎
Proof: Let x be given.
Assume Hx Hxpos .
∎
Proof: Let x be given.
Assume Hx Hxneg .
We prove the intermediate
claim Lmx :
SNo (- x .
We prove the intermediate
claim Lmxpos :
0 < - x .
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is Hxneg
An exact proof term for the current goal is Hx
An exact proof term for the current goal is Lmxpos
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is Lmxpos
∎
Notation . We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
div_SNo .
Proof: Let x and y be given.
Assume Hx Hy .
∎
Proof: Let x be given.
Assume Hx .
∎
Proof: Let x be given.
Assume Hx .
rewrite the current goal using
recip_SNo_0 (from left to right).
∎
Proof: Let x and y be given.
Assume Hx Hy Hy0 .
rewrite the current goal using
recip_SNo_invL y Hy Hy0 (from left to right).
An
exact proof term for the current goal is
mul_SNo_oneR x Hx .
∎
Proof: Let x and y be given.
Assume Hx Hy Hy0 .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz .
Apply xm (y = 0 to the current goal.
rewrite the current goal using H1
We will
prove y * (x :/: y * z = x * z .
We will
prove (y * (x :/: y * z = x * z .
Use f_equal.
We will
prove y * (x :/: y = x .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz .
Use transitivity with
(x :/: y * z , and
(x * z :/: y .
An
exact proof term for the current goal is
mul_div_SNo_R x y z Hx Hy Hz .
Use f_equal.
An
exact proof term for the current goal is
mul_SNo_com x z Hx Hz .
∎
Proof: Let x and y be given.
Assume Hx Hy Hy0 .
rewrite the current goal using
mul_div_SNo_R x y y Hx Hy Hy (from right to left).
We will
prove (x :/: y * y = x .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz .
We prove the intermediate
claim Lxdy :
SNo (x :/: y .
An
exact proof term for the current goal is
SNo_div_SNo x y Hx Hy .
We prove the intermediate
claim Lxdydz :
SNo ((x :/: y :/: z .
An
exact proof term for the current goal is
SNo_div_SNo (x :/: y z Lxdy Hz .
We prove the intermediate
claim Lxy :
SNo (x * y .
An
exact proof term for the current goal is
SNo_mul_SNo x y Hx Hy .
We prove the intermediate
claim Lyz :
SNo (y * z .
An
exact proof term for the current goal is
SNo_mul_SNo y z Hy Hz .
We prove the intermediate
claim Lxdyz :
SNo (x :/: (y * z .
An
exact proof term for the current goal is
SNo_div_SNo x (y * z Hx Lyz .
Apply xm (y = 0 to the current goal.
rewrite the current goal using Hy0
rewrite the current goal using
mul_SNo_zeroL z Hz (from left to right).
Apply xm (z = 0 to the current goal.
rewrite the current goal using Hz0
rewrite the current goal using
mul_SNo_zeroR y Hy (from left to right).
We prove the intermediate
claim Lyz0 :
y * z ≠ 0 .
Apply Hz0
We will
prove y * z = y * 0 .
rewrite the current goal using
mul_SNo_zeroR y Hy (from left to right).
An exact proof term for the current goal is H1
rewrite the current goal using
mul_div_SNo_invR x y Hx Hy Hy0 (from left to right).
We will
prove x = y * z * (x :/: (y * z .
rewrite the current goal using
mul_SNo_assoc y z (x :/: (y * z Hy Hz Lxdyz (from left to right).
Use symmetry.
∎
Proof: Let x, y, z and w be given.
Assume Hx Hy Hz Hw .
rewrite the current goal using
mul_div_SNo_R x y z Hx Hy Hz (from left to right).
∎
Proof: Let x be given.
Assume Hx Hxpos .
An exact proof term for the current goal is H1
rewrite the current goal using
mul_SNo_zeroR x Hx (from right to left).
∎
Proof: Let x be given.
Assume Hx Hxneg .
We prove the intermediate
claim Lmx :
SNo (- x .
We prove the intermediate
claim L1 :
0 < - x .
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is Hxneg
rewrite the current goal using
minus_SNo_0 (from left to right).
An exact proof term for the current goal is L1
∎
Proof: Let x and y be given.
Assume Hx Hy Hxpos Hypos .
∎
Proof: Let x and y be given.
Assume Hx Hy Hxneg Hyneg .
An
exact proof term for the current goal is
recip_SNo_neg' y Hy Hyneg .
∎
Proof: Let x and y be given.
Assume Hx Hy Hxpos Hyneg .
An
exact proof term for the current goal is
recip_SNo_neg' y Hy Hyneg .
∎
Proof: Let x and y be given.
Assume Hx Hy Hxneg Hypos .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz Hypos .
We will
prove x :/: y < z .
rewrite the current goal using
mul_SNo_oneR z Hz (from right to left).
We will
prove x :/: y < z * 1 .
We prove the intermediate
claim Ly0 :
y ≠ 0 .
rewrite the current goal using H2
An exact proof term for the current goal is Hypos
rewrite the current goal using
recip_SNo_invR y Hy Ly0 (from right to left).
An exact proof term for the current goal is H1
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz Hypos .
We will
prove z < x :/: y .
rewrite the current goal using
mul_SNo_oneR z Hz (from right to left).
We will
prove z * 1 < x :/: y .
We prove the intermediate
claim Ly0 :
y ≠ 0 .
rewrite the current goal using H2
An exact proof term for the current goal is Hypos
rewrite the current goal using
recip_SNo_invR y Hy Ly0 (from right to left).
An exact proof term for the current goal is H1
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz Hypos .
rewrite the current goal using
mul_SNo_oneR x Hx (from right to left).
We will
prove x * 1 < z * y .
We prove the intermediate
claim Ly0 :
y ≠ 0 .
rewrite the current goal using H2
An exact proof term for the current goal is Hypos
rewrite the current goal using
recip_SNo_invL y Hy Ly0 (from right to left).
We will
prove (x :/: y * y < z * y .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz Hypos .
rewrite the current goal using
mul_SNo_oneR x Hx (from right to left).
We will
prove z * y < x * 1 .
We prove the intermediate
claim Ly0 :
y ≠ 0 .
rewrite the current goal using H2
An exact proof term for the current goal is Hypos
rewrite the current goal using
recip_SNo_invL y Hy Ly0 (from right to left).
We will
prove z * y < (x :/: y * y .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz Hy0 H1 .
We will
prove y * (x :/: y = y * z .
rewrite the current goal using
mul_div_SNo_invR x y Hx Hy Hy0 (from left to right).
An exact proof term for the current goal is H1
∎
End of Section SurrealDiv