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
exists 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
exists 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
Union is a term of type
set → set .
Axiom. (
UnionEq ) We take the following as an axiom:
Primitive . The name
Power is a term of type
set → set .
Axiom. (
PowerEq ) We take the following as an axiom:
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:
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 Union X ∈ U of type
set → prop .
Definition. We define
Power_closed to be
λU : set ⇒ ∀X : set , X ∈ U Power 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
/:e 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:
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 /:e Y z /:e 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 ∈ Union X Axiom. (
UnionE ) We take the following as an axiom:
Axiom. (
UnionE_impred ) We take the following as an axiom:
∀X x : set , x ∈ Union X ∀p : prop , (∀Y : set , x ∈ Y Y ∈ X p ) → p Axiom. (
PowerI ) We take the following as an axiom:
Axiom. (
PowerE ) We take the following as an axiom:
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 } exists 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 Axiom. (
Sing_inj ) We take the following as an axiom:
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:
Axiom. (
binunionE' ) We take the following as an axiom:
∀X Y z, ∀p : prop , (z ∈ X p → (z ∈ Y p → (z ∈ X :\/: Y p 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 exists 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
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_Empty ) We take the following as an axiom:
Axiom. (
Sep_Subq ) We take the following as an axiom:
∀X : set , ∀P : set → prop { x ∈ X | P x } ⊆ 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 } exists 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 .
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 /:e Y → z ∈ X :\: Y Axiom. (
setminusE ) We take the following as an axiom:
Axiom. (
setminusE1 ) We take the following as an axiom:
∀X Y z, (z ∈ X :\: Y → z ∈ X Axiom. (
setminusE2 ) We take the following as an axiom:
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:
Axiom. (
In_1_3 ) We take the following as an axiom:
Axiom. (
In_2_3 ) We take the following as an axiom:
Axiom. (
In_1_4 ) We take the following as an axiom:
Axiom. (
In_2_4 ) We take the following as an axiom:
Axiom. (
In_3_4 ) We take the following as an axiom:
Axiom. (
In_1_5 ) We take the following as an axiom:
Axiom. (
In_2_5 ) We take the following as an axiom:
Axiom. (
In_3_5 ) We take the following as an axiom:
Axiom. (
In_4_5 ) We take the following as an axiom:
Axiom. (
In_1_6 ) We take the following as an axiom:
Axiom. (
In_1_7 ) We take the following as an axiom:
Axiom. (
In_1_8 ) 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_3 ) We take the following as an axiom:
Axiom. (
nat_4 ) We take the following as an axiom:
Axiom. (
nat_5 ) We take the following as an axiom:
Axiom. (
nat_6 ) We take the following as an axiom:
Axiom. (
nat_7 ) We take the following as an axiom:
Axiom. (
nat_8 ) 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. (
neq_1_3 ) We take the following as an axiom:
Axiom. (
neq_2_3 ) We take the following as an axiom:
Axiom. (
neq_2_4 ) We take the following as an axiom:
Axiom. (
neq_3_4 ) We take the following as an axiom:
Axiom. (
ZF_closed_E ) We take the following as an axiom:
Primitive . The name
omega 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 , exists u ∈ X , f u = w of type
set → set → (set → set → prop .
Axiom. (
bijI ) We take the following as an axiom:
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 , exists 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 ⇒ exists 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
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 (exists 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 (exists 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 (exists 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 (exists 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 : exists 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 : exists 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 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
Definition. We define
In_rec_i_G to be
λX Y ⇒ ∀R : set → set → prop (∀X : set , ∀f : set → set (∀ x ∈ X , R x (f x → R X (F X f ) → R X Y of type
set → set → prop .
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
Definition. We define
In_rec_G_ii to be
λX Y ⇒ ∀R : set → (set → set → prop (∀X : set , ∀f : set → (set → set (∀ x ∈ X , R x (f x → R X (F X f ) → R X Y of type
set → (set → set → prop .
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
Definition. We define
In_rec_G_iii to be
λX Y ⇒ ∀R : set → (set → set → set → prop (∀X : set , ∀f : set → (set → set → set (∀ x ∈ X , R x (f x → R X (F X f ) → R X Y of type
set → (set → set → set → prop .
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
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:
Axiom. (
add_nat_0L ) We take the following as an axiom:
Axiom. (
add_nat_SL ) We take the following as an axiom:
Axiom. (
add_nat_com ) 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:
Axiom. (
Inj1_setsum ) We take the following as an axiom:
Axiom. (
eq_1_Sing0 ) We take the following as an axiom:
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:
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
Sigma_ 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:
Axiom. (
proj0_Sigma ) We take the following as an axiom:
Axiom. (
proj1_Sigma ) We take the following as an axiom:
Axiom. (
pair_Sigma_E1 ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀x y : set , pair x y ∈ (Sigma_ x ∈ X , Y x y ∈ Y x Axiom. (
Sigma_E ) We take the following as an axiom:
Definition. We define
setprod to be
λX Y : set ⇒ Sigma_ 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 , exists 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:
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 ∈ (Sigma_ 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 ∈ (Sigma_ 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 ) Notation . We use
Pi_ 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:
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 ∈ (Pi_ x ∈ X , Y x Axiom. (
ap_Pi ) We take the following as an axiom:
∀X : set , ∀Y : set → set ∀f : set , ∀x : set , f ∈ (Pi_ x ∈ X , Y x x ∈ X f x ∈ Y x Definition. We define
setexp to be
λX Y : set ⇒ Pi_ 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
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 .
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 .
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. (
SNo_0_eq_0 ) 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
Notation . We use
:*: as an infix operator with priority 440 and which associates to the left corresponding to applying term
setprod .
Notation . We use
:^: as an infix operator with priority 430 and which associates to the left corresponding to applying term
setexp .
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 .
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 .
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 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:
Axiom. (
mul_SNoCutP_lem ) We take the following as an axiom:
∀Lx Rx Ly Ry x y, SNoCutP Lx Rx SNoCutP Ly Ry x = SNoCut Lx Rx y = SNoCut Ly Ry SNoCutP ({ w 0 * y + x * w 1 + - w 0 * w 1 | w ∈ Lx :*: Ly } :\/: { z 0 * y + x * z 1 + - z 0 * z 1 | z ∈ Rx :*: Ry } ({ w 0 * y + x * w 1 + - w 0 * w 1 | w ∈ Lx :*: Ry } :\/: { z 0 * y + x * z 1 + - z 0 * z 1 | z ∈ Rx :*: Ly } /\ x * y = SNoCut ({ w 0 * y + x * w 1 + - w 0 * w 1 | w ∈ Lx :*: Ly } :\/: { z 0 * y + x * z 1 + - z 0 * z 1 | z ∈ Rx :*: Ry } ({ w 0 * y + x * w 1 + - w 0 * w 1 | w ∈ Lx :*: Ry } :\/: { z 0 * y + x * z 1 + - z 0 * z 1 | z ∈ Rx :*: Ly } /\ ∀q : prop , (∀LxLy' RxRy' LxRy' RxLy', (∀ u ∈ LxLy' , ∀p : prop , (∀ w ∈ Lx , ∀ w' ∈ Ly , SNo w SNo w' w < x w' < y u = w * y + x * w' + - w * w' p → p → (∀ w ∈ Lx , ∀ w' ∈ Ly , w * y + x * w' + - w * w' ∈ LxLy' → (∀ u ∈ RxRy' , ∀p : prop , (∀ z ∈ Rx , ∀ z' ∈ Ry , SNo z SNo z' x < z y < z' u = z * y + x * z' + - z * z' p → p → (∀ z ∈ Rx , ∀ z' ∈ Ry , z * y + x * z' + - z * z' ∈ RxRy' → (∀ u ∈ LxRy' , ∀p : prop , (∀ w ∈ Lx , ∀ z ∈ Ry , SNo w SNo z w < x y < z u = w * y + x * z + - w * z p → p → (∀ w ∈ Lx , ∀ z ∈ Ry , w * y + x * z + - w * z ∈ LxRy' → (∀ u ∈ RxLy' , ∀p : prop , (∀ z ∈ Rx , ∀ w ∈ Ly , SNo z SNo w x < z w < y u = z * y + x * w + - z * w p → p → (∀ z ∈ Rx , ∀ w ∈ Ly , z * y + x * w + - z * w ∈ RxLy' → SNoCutP (LxLy' :\/: RxRy' (LxRy' :\/: RxLy' x * y = SNoCut (LxLy' :\/: RxRy' (LxRy' :\/: RxLy' q ) → q Axiom. (
mul_SNoCut_abs ) We take the following as an axiom:
∀Lx Rx Ly Ry x y, SNoCutP Lx Rx SNoCutP Ly Ry x = SNoCut Lx Rx y = SNoCut Ly Ry ∀q : prop , (∀LxLy' RxRy' LxRy' RxLy', (∀ u ∈ LxLy' , ∀p : prop , (∀ w ∈ Lx , ∀ w' ∈ Ly , SNo w SNo w' w < x w' < y u = w * y + x * w' + - w * w' p → p → (∀ w ∈ Lx , ∀ w' ∈ Ly , w * y + x * w' + - w * w' ∈ LxLy' → (∀ u ∈ RxRy' , ∀p : prop , (∀ z ∈ Rx , ∀ z' ∈ Ry , SNo z SNo z' x < z y < z' u = z * y + x * z' + - z * z' p → p → (∀ z ∈ Rx , ∀ z' ∈ Ry , z * y + x * z' + - z * z' ∈ RxRy' → (∀ u ∈ LxRy' , ∀p : prop , (∀ w ∈ Lx , ∀ z ∈ Ry , SNo w SNo z w < x y < z u = w * y + x * z + - w * z p → p → (∀ w ∈ Lx , ∀ z ∈ Ry , w * y + x * z + - w * z ∈ LxRy' → (∀ u ∈ RxLy' , ∀p : prop , (∀ z ∈ Rx , ∀ w ∈ Ly , SNo z SNo w x < z w < y u = z * y + x * w + - z * w p → p → (∀ z ∈ Rx , ∀ w ∈ Ly , z * y + x * w + - z * w ∈ RxLy' → SNoCutP (LxLy' :\/: RxRy' (LxRy' :\/: RxLy' x * y = SNoCut (LxLy' :\/: RxRy' (LxRy' :\/: RxLy' q ) → q 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 .
Axiom. (
int_3_cases ) We take the following as an axiom:
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 .
Beginning of Section recip_SNo_pos
End of Section recip_SNo_pos
Axiom. (
recip_SNo_0 ) We take the following as an axiom:
Axiom. (
recip_SNo_1 ) We take the following as an axiom:
Axiom. (
recip_SNo_2 ) We take the following as an axiom:
Notation . We use
:/: as an infix operator with priority 353 and no associativity corresponding to applying term
div_SNo .
Axiom. (
SNo_div_SNo ) We take the following as an axiom:
Axiom. (
div_div_SNo ) We take the following as an axiom:
End of Section SurrealDiv
Beginning of Section SurrealSqrt
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 353 and no associativity corresponding to applying term
div_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 .
Beginning of Section sqrt_SNo_nonneg
End of Section sqrt_SNo_nonneg
End of Section SurrealSqrt
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 353 and no associativity corresponding to applying term
div_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 .
Axiom. (
real_I ) We take the following as an axiom:
Axiom. (
real_E ) We take the following as an axiom:
Axiom. (
real_SNo ) We take the following as an axiom:
Axiom. (
real_0 ) We take the following as an axiom:
Axiom. (
real_1 ) We take the following as an axiom:
Axiom. (
real_SNoCut ) We take the following as an axiom:
Axiom. (
SNo_approx_real_rep ) We take the following as an axiom:
∀ x ∈ real , ∀p : prop , (∀ f g ∈ SNoS_ omega :^: omega , (∀ n ∈ omega , f n < x → (∀ n ∈ omega , x < f n + eps_ n → (∀ n ∈ omega , ∀ i ∈ n , f i < f n → (∀ n ∈ omega , g n + - eps_ n < x → (∀ n ∈ omega , x < g n → (∀ n ∈ omega , ∀ i ∈ n , g n < g i → SNoCutP { f n | n ∈ omega } { g n | n ∈ omega } x = SNoCut { f n | n ∈ omega } { g n | n ∈ omega } p → p End of Section Reals
Axiom. (
mul_nat_1R ) We take the following as an axiom:
Axiom. (
mul_nat_com ) We take the following as an axiom:
Axiom. (
mul_nat_SL ) We take the following as an axiom:
Axiom. (
even_nat_0 ) We take the following as an axiom:
Axiom. (
odd_nat_1 ) We take the following as an axiom:
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 353 and no associativity corresponding to applying term
div_SNo .
Beginning of Section Alg
Variable extension_tag : set
Let ctag : set → set λalpha ⇒ SetAdjoin alpha extension_tag Notation . We use
'' as a postfix operator with priority 100 corresponding to applying term
ctag .
Variable F : set → prop
Hypothesis extension_tag_fresh : ∀x, F x ∀ u ∈ x , extension_tag /:e u Axiom. (
pair_tag_0 ) We take the following as an axiom:
Axiom. (
CD_carr_I ) We take the following as an axiom:
Axiom. (
CD_carr_E ) We take the following as an axiom:
Axiom. (
CD_proj0_1 ) We take the following as an axiom:
Axiom. (
CD_proj0_2 ) We take the following as an axiom:
∀x y, F x F y proj0 (pa x y = x Axiom. (
CD_proj1_1 ) We take the following as an axiom:
Axiom. (
CD_proj1_2 ) We take the following as an axiom:
∀x y, F x F y proj1 (pa x y = y Axiom. (
CD_proj0R ) We take the following as an axiom:
Axiom. (
CD_proj1R ) We take the following as an axiom:
Axiom. (
CD_proj0_F ) We take the following as an axiom:
Axiom. (
CD_proj1_F ) We take the following as an axiom:
Beginning of Section CD_minus_conj
Variable minus : set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus .
Variable conj : set → set
Definition. We define
CD_conj to be
λz ⇒ pa (conj (proj0 z (- proj1 z of type
set → set .
End of Section CD_minus_conj
Beginning of Section CD_add
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
End of Section CD_add
Beginning of Section CD_mul
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul .
End of Section CD_mul
Beginning of Section CD_minus_conj_clos
Variable minus : set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Hypothesis F_minus : ∀x, F x F (- x Axiom. (
CD_minus_CD ) We take the following as an axiom:
Variable conj : set → set
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Hypothesis F_conj : ∀x, F x F (conj x
Axiom. (
CD_conj_CD ) We take the following as an axiom:
End of Section CD_minus_conj_clos
Beginning of Section CD_add_clos
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_add : ∀x y, F x F y F (x + y Axiom. (
CD_add_CD ) We take the following as an axiom:
End of Section CD_add_clos
Beginning of Section CD_mul_clos
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y Axiom. (
CD_mul_CD ) We take the following as an axiom:
End of Section CD_mul_clos
Beginning of Section CD_minus_conj_F
Variable minus : set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Hypothesis F_minus_0 : - 0 = 0 Variable conj : set → set
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
End of Section CD_minus_conj_F
Beginning of Section CD_add_F
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_add_0_0 : 0 + 0 = 0 Axiom. (
CD_add_F_eq ) We take the following as an axiom:
∀x y, F x F y x :+: y = x + y End of Section CD_add_F
Beginning of Section CD_mul_F
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_mul : ∀x y, F x F y F (x * y Hypothesis F_minus_0 : - 0 = 0 Hypothesis F_add_0R : ∀x, F x x + 0 = x Hypothesis F_mul_0L : ∀x, F x 0 * x = 0 Hypothesis F_mul_0R : ∀x, F x x * 0 = 0 Axiom. (
CD_mul_F_eq ) We take the following as an axiom:
∀x y, F x F y x :*: y = x * y End of Section CD_mul_F
Beginning of Section CD_minus_invol
Variable minus : set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_minus_invol : ∀x, F x - - x = x End of Section CD_minus_invol
Beginning of Section CD_conj_invol
Variable minus : set → set
Variable conj : set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_minus_invol : ∀x, F x - - x = x Hypothesis F_conj_invol : ∀x, F x conj (conj x = x End of Section CD_conj_invol
Beginning of Section CD_conj_minus
Variable minus : set → set
Variable conj : set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_conj_minus : ∀x, F x conj (- x = - conj x End of Section CD_conj_minus
Beginning of Section CD_minus_add
Variable minus : set → set
Variable add : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y End of Section CD_minus_add
Beginning of Section CD_conj_add
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y Hypothesis F_conj_add : ∀x y, F x F y conj (x + y = conj x + conj y Axiom. (
CD_conj_add ) We take the following as an axiom:
End of Section CD_conj_add
Beginning of Section CD_add_com
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_add_com : ∀x y, F x F y x + y = y + x Axiom. (
CD_add_com ) We take the following as an axiom:
End of Section CD_add_com
Beginning of Section CD_add_assoc
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_add_assoc : ∀x y z, F x F y F z (x + y + z = x + (y + z End of Section CD_add_assoc
Beginning of Section CD_add_0R
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_add_0R : ∀x, F x x + 0 = x Axiom. (
CD_add_0R ) We take the following as an axiom:
End of Section CD_add_0R
Beginning of Section CD_add_0L
Variable add : set → set → set
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_add_0L : ∀x, F x 0 + x = x Axiom. (
CD_add_0L ) We take the following as an axiom:
End of Section CD_add_0L
Beginning of Section CD_add_minus_linv
Variable minus : set → set
Variable add : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_add_minus_linv : ∀x, F x - x + x = 0 End of Section CD_add_minus_linv
Beginning of Section CD_add_minus_rinv
Variable minus : set → set
Variable add : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_add_minus_rinv : ∀x, F x x + - x = 0 End of Section CD_add_minus_rinv
Beginning of Section CD_mul_0R
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus_0 : - 0 = 0 Hypothesis F_conj_0 : conj 0 = 0 Hypothesis F_add_0_0 : 0 + 0 = 0 Hypothesis F_mul_0L : ∀x, F x 0 * x = 0 Hypothesis F_mul_0R : ∀x, F x x * 0 = 0 Axiom. (
CD_mul_0R ) We take the following as an axiom:
End of Section CD_mul_0R
Beginning of Section CD_mul_0L
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_minus_0 : - 0 = 0 Hypothesis F_add_0_0 : 0 + 0 = 0 Hypothesis F_mul_0L : ∀x, F x 0 * x = 0 Hypothesis F_mul_0R : ∀x, F x x * 0 = 0 Axiom. (
CD_mul_0L ) We take the following as an axiom:
End of Section CD_mul_0L
Beginning of Section CD_mul_1R
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus_0 : - 0 = 0 Hypothesis F_conj_0 : conj 0 = 0 Hypothesis F_conj_1 : conj 1 = 1 Hypothesis F_add_0L : ∀x, F x 0 + x = x Hypothesis F_add_0R : ∀x, F x x + 0 = x Hypothesis F_mul_0L : ∀x, F x 0 * x = 0 Hypothesis F_mul_1R : ∀x, F x x * 1 = x Axiom. (
CD_mul_1R ) We take the following as an axiom:
End of Section CD_mul_1R
Beginning of Section CD_mul_1L
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_minus_0 : - 0 = 0 Hypothesis F_add_0R : ∀x, F x x + 0 = x Hypothesis F_mul_0L : ∀x, F x 0 * x = 0 Hypothesis F_mul_0R : ∀x, F x x * 0 = 0 Hypothesis F_mul_1L : ∀x, F x 1 * x = x Hypothesis F_mul_1R : ∀x, F x x * 1 = x Axiom. (
CD_mul_1L ) We take the following as an axiom:
End of Section CD_mul_1L
Beginning of Section CD_conj_mul
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y Hypothesis F_minus_invol : ∀x, F x - - x = x Hypothesis F_conj_invol : ∀x, F x conj (conj x = x Hypothesis F_conj_minus : ∀x, F x conj (- x = - conj x Hypothesis F_conj_add : ∀x y, F x F y conj (x + y = conj x + conj y Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y Hypothesis F_add_com : ∀x y, F x F y x + y = y + x Hypothesis F_conj_mul : ∀x y, F x F y conj (x * y = conj y * conj x Hypothesis F_minus_mul_distrR : ∀x y, F x F y x * (- y = - (x * y Hypothesis F_minus_mul_distrL : ∀x y, F x F y (- x * y = - (x * y Axiom. (
CD_conj_mul ) We take the following as an axiom:
End of Section CD_conj_mul
Beginning of Section CD_add_mul_distrR
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y Hypothesis F_add_assoc : ∀x y z, F x F y F z (x + y + z = x + (y + z Hypothesis F_add_com : ∀x y, F x F y x + y = y + x Hypothesis F_add_mul_distrL : ∀x y z, F x F y F z x * (y + z = x * y + x * z Hypothesis F_add_mul_distrR : ∀x y z, F x F y F z (x + y * z = x * z + y * z End of Section CD_add_mul_distrR
Beginning of Section CD_add_mul_distrL
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y Hypothesis F_conj_add : ∀x y, F x F y conj (x + y = conj x + conj y Hypothesis F_add_assoc : ∀x y z, F x F y F z (x + y + z = x + (y + z Hypothesis F_add_com : ∀x y, F x F y x + y = y + x Hypothesis F_add_mul_distrL : ∀x y z, F x F y F z x * (y + z = x * y + x * z Hypothesis F_add_mul_distrR : ∀x y z, F x F y F z (x + y * z = x * z + y * z End of Section CD_add_mul_distrL
Beginning of Section CD_minus_mul_distrR
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y Hypothesis F_conj_minus : ∀x, F x conj (- x = - conj x Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y Hypothesis F_minus_mul_distrR : ∀x y, F x F y x * (- y = - (x * y Hypothesis F_minus_mul_distrL : ∀x y, F x F y (- x * y = - (x * y End of Section CD_minus_mul_distrR
Beginning of Section CD_minus_mul_distrL
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y Hypothesis F_minus_add : ∀x y, F x F y - (x + y = - x + - y Hypothesis F_minus_mul_distrR : ∀x y, F x F y x * (- y = - (x * y Hypothesis F_minus_mul_distrL : ∀x y, F x F y (- x * y = - (x * y End of Section CD_minus_mul_distrL
Beginning of Section CD_exp_nat
Variable minus : set → set
Variable conj : set → set
Variable add : set → set → set
Variable mul : set → set → set
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
conj .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul .
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
CD_minus minus .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
CD_conj minus conj .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
CD_add add .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
CD_mul minus conj add mul .
Notation . We use
^ as an infix operator with priority 342 and no associativity corresponding to applying term
CD_exp_nat minus conj add mul .
Beginning of Section CD_exp_nat_1_2
Hypothesis F_minus_0 : - 0 = 0 Hypothesis F_conj_0 : conj 0 = 0 Hypothesis F_conj_1 : conj 1 = 1 Hypothesis F_add_0L : ∀x, F x 0 + x = x Hypothesis F_add_0R : ∀x, F x x + 0 = x Hypothesis F_mul_0L : ∀x, F x 0 * x = 0 Hypothesis F_mul_1R : ∀x, F x x * 1 = x End of Section CD_exp_nat_1_2
Hypothesis F_minus : ∀x, F x F (- x Hypothesis F_conj : ∀x, F x F (conj x
Hypothesis F_add : ∀x y, F x F y F (x + y Hypothesis F_mul : ∀x y, F x F y F (x * y End of Section CD_exp_nat
End of Section Alg
Beginning of Section Tags
Variable tagn : set
Hypothesis tagn_nat : nat_p tagn Hypothesis tagn_1 : 1 ∈ tagn End of Section Tags
Beginning of Section ExtendedSNo
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
tag .
End of Section ExtendedSNo
Beginning of Section SurComplex
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
tag .
Axiom. (
SNo_pair_0 ) We take the following as an axiom:
Axiom. (
CSNo_I ) We take the following as an axiom:
Axiom. (
CSNo_E ) We take the following as an axiom:
Axiom. (
SNo_CSNo ) We take the following as an axiom:
Axiom. (
CSNo_0 ) We take the following as an axiom:
Axiom. (
CSNo_1 ) We take the following as an axiom:
Notation . We use
'' as a postfix operator with priority 100 corresponding to applying term
ctag .
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 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 .
Axiom. (
CSNo_Re1 ) We take the following as an axiom:
Axiom. (
CSNo_Re2 ) We take the following as an axiom:
∀x y, SNo x SNo y Re (pa x y = x Axiom. (
CSNo_Im1 ) We take the following as an axiom:
Axiom. (
CSNo_Im2 ) We take the following as an axiom:
∀x y, SNo x SNo y Im (pa x y = y Axiom. (
CSNo_ReR ) We take the following as an axiom:
Axiom. (
CSNo_ImR ) We take the following as an axiom:
Axiom. (
CSNo_ReIm ) We take the following as an axiom:
∀z, CSNo z z = pa (Re z (Im z Let conj : set → set λx ⇒ x
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
minus_CSNo .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
conj_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 355 and which associates to the right corresponding to applying term
exp_CSNo_nat .
Axiom. (
mul_CSNo_CRe ) We take the following as an axiom:
∀z w, CSNo z CSNo w Re (mult' z w = Re z * Re w + - (Im w * Im z Axiom. (
mul_CSNo_CIm ) We take the following as an axiom:
∀z w, CSNo z CSNo w Im (mult' z w = Im w * Re z + Im z * Re w Axiom. (
SNo_Re ) We take the following as an axiom:
Axiom. (
SNo_Im ) We take the following as an axiom:
Axiom. (
Re_0 ) We take the following as an axiom:
Axiom. (
Im_0 ) We take the following as an axiom:
Axiom. (
Re_1 ) We take the following as an axiom:
Axiom. (
Im_1 ) We take the following as an axiom:
Axiom. (
Re_i ) We take the following as an axiom:
Axiom. (
Im_i ) We take the following as an axiom:
Axiom. (
conj_CSNo_0 ) We take the following as an axiom:
Axiom. (
conj_CSNo_1 ) We take the following as an axiom:
Axiom. (
conj_CSNo_i ) We take the following as an axiom:
Axiom. (
add_CSNo_0L ) We take the following as an axiom:
Axiom. (
add_CSNo_0R ) We take the following as an axiom:
Axiom. (
mul_CSNo_0R ) We take the following as an axiom:
Axiom. (
mul_CSNo_0L ) We take the following as an axiom:
Axiom. (
mul_CSNo_1R ) We take the following as an axiom:
Axiom. (
mul_CSNo_1L ) We take the following as an axiom:
End of Section SurComplex
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 .
Axiom. (
complex_I ) We take the following as an axiom:
Axiom. (
complex_E ) We take the following as an axiom:
Axiom. (
complex_0 ) We take the following as an axiom:
Axiom. (
complex_1 ) We take the following as an axiom:
Axiom. (
complex_i ) We take the following as an axiom:
Axiom. (
real_Re_eq ) We take the following as an axiom:
Axiom. (
real_Im_eq ) We take the following as an axiom:
Axiom. (
complex_eta ) We take the following as an axiom:
Beginning of Section ComplexDiv
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 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 ComplexDiv
End of Section Complex
Beginning of Section SurQuaternions
Proof: Let z be given.
Assume Hz .
∎
Theorem. (
HSNo_I ) The following is provable:
Theorem. (
HSNo_E ) The following is provable:
Theorem. (
CSNo_HSNo ) The following is provable:
Theorem. (
HSNo_0 ) The following is provable:
Proof:
An
exact proof term for the current goal is
CSNo_0 .
∎
Theorem. (
HSNo_1 ) The following is provable:
Proof:
An
exact proof term for the current goal is
CSNo_1 .
∎
Notation . We use
'' as a postfix operator with priority 100 corresponding to applying term
ctag .
Proof: Let z be given.
Assume Hz .
Apply HSNo_E z Hz to the current goal.
Let x and y be given.
Let v be given.
Let u be given.
We prove the intermediate
claim L1 :
v ∈ Union x .
An
exact proof term for the current goal is
UnionI x v u H1 H3 .
Let w be given.
We prove the intermediate
claim L2 :
v ∈ w '' .
rewrite the current goal using H4
An exact proof term for the current goal is H1
We prove the intermediate
claim L3 :
v ∈ Union y .
An
exact proof term for the current goal is
UnionI y v w H5 Hw .
Apply orIR to the current goal.
We use
3 to
witness the existential quantifier.
Apply andI to the current goal.
An
exact proof term for the current goal is
In_3_4 .
An
exact proof term for the current goal is
SingE { 3 } v H5 .
∎
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 .
Notation . We use
' as a postfix operator with priority 100 corresponding to applying term
conj_CSNo .
Proof: Let z be given.
Assume Hz .
Apply HSNo_E z Hz to the current goal.
Let x and y be given.
Assume Hx Hy .
rewrite the current goal using
HSNo_proj0_2 x y Hx Hy (from left to right).
rewrite the current goal using
HSNo_proj1_2 x y Hx Hy (from left to right).
∎
Notation . We use
:-: as a prefix operator with priority 358 corresponding to applying term
minus_HSNo .
Notation . We use
'' as a postfix operator with priority 100 corresponding to applying term
conj_HSNo .
Notation . We use
:+: as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_HSNo .
Notation . We use
:*: as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_HSNo .
Notation . We use
:^: as an infix operator with priority 355 and which associates to the right corresponding to applying term
exp_HSNo_nat .
Proof:
Apply HSNo_I to the current goal.
An
exact proof term for the current goal is
CSNo_0 .
An
exact proof term for the current goal is
CSNo_1 .
∎
Proof: We will
prove HSNo (pa 0 i .
Apply HSNo_I to the current goal.
An
exact proof term for the current goal is
CSNo_0 .
∎
Theorem. (
HSNo_p0_0 ) The following is provable:
Theorem. (
HSNo_p1_0 ) The following is provable:
Theorem. (
HSNo_p0_1 ) The following is provable:
Theorem. (
HSNo_p1_1 ) The following is provable:
Theorem. (
HSNo_p0_i ) The following is provable:
Theorem. (
HSNo_p1_i ) The following is provable:
Theorem. (
HSNo_p0_j ) The following is provable:
Theorem. (
HSNo_p1_j ) The following is provable:
Theorem. (
HSNo_p0_k ) The following is provable:
Theorem. (
HSNo_p1_k ) The following is provable:
Proof: Let x be given.
Assume Hx .
∎
Proof:
An
exact proof term for the current goal is
conj_CSNo_0 .
∎
Proof:
An
exact proof term for the current goal is
conj_CSNo_1 .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz .
We will
prove (x + y + z = (x + z + y .
rewrite the current goal using
add_CSNo_assoc x y z Hx Hy Hz (from left to right).
We will
prove x + y + z = (x + z + y .
rewrite the current goal using
add_CSNo_assoc x z y Hx Hz Hy (from left to right).
We will
prove x + y + z = x + z + y .
Use f_equal.
An
exact proof term for the current goal is
add_CSNo_com y z Hy Hz .
∎
Proof: Let x, y, z and w be given.
Assume Hx Hy Hz Hw .
We will
prove ((x + y + z + w = (x + z + (y + w .
We will
prove ((x + z + y + w = (x + z + (y + w .
∎
Proof: Let x, y, z and w be given.
Assume Hx Hy Hz Hw .
rewrite the current goal using
add_CSNo_com z w Hz Hw (from left to right).
We will
prove (x + y + (w + z = (x + w + (y + z .
∎
Proof:
We will
prove i * i = - 1 .
∎
Proof: We prove the intermediate
claim Ljj :
HSNo (j :*: j .
We prove the intermediate
claim Lm1 :
HSNo (:-: 1 .
We will
prove p0 (j :*: j = p0 (:-: 1 .
We will
prove p0 j * p0 j + - p1 j ' * p1 j = - p0 1 .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
conj_CSNo_1 (from left to right).
We will
prove p1 (j :*: j = p1 (:-: 1 .
We will
prove p1 j * p0 j + p1 j * p0 j ' = - p1 1 .
rewrite the current goal using
HSNo_p1_1 (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
We will
prove p1 j * p0 j + p1 j * p0 j ' = 0 .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
conj_CSNo_0 (from left to right).
∎
Proof: We prove the intermediate
claim Lkk :
HSNo (k :*: k .
We prove the intermediate
claim Lm1 :
HSNo (:-: 1 .
We will
prove p0 (k :*: k = p0 (:-: 1 .
We will
prove p0 k * p0 k + - p1 k ' * p1 k = - p0 1 .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
conj_CSNo_i (from left to right).
We will
prove 0 + i * i = - 1 .
rewrite the current goal using
Complex_i_sqr (from left to right).
We will
prove p1 (k :*: k = p1 (:-: 1 .
We will
prove p1 k * p0 k + p1 k * p0 k ' = - p1 1 .
rewrite the current goal using
HSNo_p1_1 (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
We will
prove p1 k * p0 k + p1 k * p0 k ' = 0 .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
conj_CSNo_0 (from left to right).
∎
Proof: We prove the intermediate
claim Lij :
HSNo (i :*: j .
We will
prove p0 (i :*: j = p0 k .
rewrite the current goal using
HSNo_p0_k (from left to right).
We will
prove p0 i * p0 j + - (p1 j ' * p1 i = 0 .
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
We will
prove p1 (i :*: j = p1 k .
rewrite the current goal using
HSNo_p1_k (from left to right).
We will
prove p1 j * p0 i + p1 i * p0 j ' = i .
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
∎
Proof: We prove the intermediate
claim Ljk :
HSNo (j :*: k .
We will
prove p0 (j :*: k = p0 i .
rewrite the current goal using
HSNo_p0_i (from left to right).
We will
prove p0 j * p0 k + - (p1 k ' * p1 j = i .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
We will
prove 0 + - (i ' = i .
rewrite the current goal using
conj_CSNo_i (from left to right).
We will
prove 0 + - - i = i .
We will
prove p1 (j :*: k = p1 i .
rewrite the current goal using
HSNo_p1_i (from left to right).
We will
prove p1 k * p0 j + p1 j * p0 k ' = 0 .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
∎
Proof: We prove the intermediate
claim Lki :
HSNo (k :*: i .
We will
prove p0 (k :*: i = p0 j .
rewrite the current goal using
HSNo_p0_j (from left to right).
We will
prove p0 k * p0 i + - (p1 i ' * p1 k = 0 .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
We will
prove p1 (k :*: i = p1 j .
rewrite the current goal using
HSNo_p1_j (from left to right).
We will
prove p1 i * p0 k + p1 k * p0 i ' = 1 .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
conj_CSNo_i (from left to right).
We will
prove 0 + i * (- i = 1 .
We will
prove 0 + - i * i = 1 .
rewrite the current goal using
Complex_i_sqr (from left to right).
∎
Proof: We prove the intermediate
claim Lji :
HSNo (j :*: i .
We will
prove p0 (j :*: i = p0 (:-: k .
rewrite the current goal using
HSNo_p0_k (from left to right).
We will
prove p0 j * p0 i + - (p1 i ' * p1 j = - 0 .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
We will
prove p1 (j :*: i = p1 (:-: k .
rewrite the current goal using
HSNo_p1_k (from left to right).
We will
prove p1 i * p0 j + p1 j * p0 i ' = - i .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
conj_CSNo_i (from left to right).
We will
prove 0 + - i = - i .
∎
Proof: We prove the intermediate
claim Lkj :
HSNo (k :*: j .
We will
prove p0 (k :*: j = p0 (:-: i .
rewrite the current goal using
HSNo_p0_i (from left to right).
We will
prove p0 k * p0 j + - (p1 j ' * p1 k = - i .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
We will
prove 0 + - i = - i .
We will
prove p1 (k :*: j = p1 (:-: i .
rewrite the current goal using
HSNo_p1_i (from left to right).
We will
prove p1 j * p0 k + p1 k * p0 j ' = - 0 .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
HSNo_p1_j (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
∎
Proof: We prove the intermediate
claim Lik :
HSNo (i :*: k .
We will
prove p0 (i :*: k = p0 (:-: j .
rewrite the current goal using
HSNo_p0_j (from left to right).
We will
prove p0 i * p0 k + - (p1 k ' * p1 i = - 0 .
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
We will
prove p1 (i :*: k = p1 (:-: j .
rewrite the current goal using
HSNo_p1_j (from left to right).
We will
prove p1 k * p0 i + p1 i * p0 k ' = - 1 .
rewrite the current goal using
HSNo_p0_i (from left to right).
rewrite the current goal using
HSNo_p1_i (from left to right).
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
HSNo_p1_k (from left to right).
rewrite the current goal using
Complex_i_sqr (from left to right).
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz .
We will
prove x + (y + z = z + (x + y .
Use transitivity with
x + (z + y ,
(x + z + y , and
(z + x + y .
Use f_equal.
An
exact proof term for the current goal is
add_CSNo_com y z Hy Hz .
Use symmetry.
An
exact proof term for the current goal is
add_CSNo_assoc x z y Hx Hz Hy .
Use f_equal.
An
exact proof term for the current goal is
add_CSNo_com x z Hx Hz .
An
exact proof term for the current goal is
add_CSNo_assoc z x y Hz Hx Hy .
∎
Proof: Let x, y and z be given.
Assume Hx Hy Hz .
We will
prove x * (y * z = z * (x * y .
Use transitivity with
x * (z * y ,
(x * z * y , and
(z * x * y .
Use f_equal.
An
exact proof term for the current goal is
mul_CSNo_com y z Hy Hz .
An
exact proof term for the current goal is
mul_CSNo_assoc x z y Hx Hz Hy .
Use f_equal.
An
exact proof term for the current goal is
mul_CSNo_com x z Hx Hz .
Use symmetry.
An
exact proof term for the current goal is
mul_CSNo_assoc z x y Hz Hx Hy .
∎
Proof: Let z, w and v be given.
Assume Hz Hw Hv .
We prove the intermediate
claim Lwv :
HSNo (mult' w v .
An
exact proof term for the current goal is
HSNo_mul_HSNo w v Hw Hv .
We prove the intermediate
claim Lzw :
HSNo (mult' z w .
An
exact proof term for the current goal is
HSNo_mul_HSNo z w Hz Hw .
We prove the intermediate
claim Lzwv1 :
HSNo (mult' z (mult' w v .
An
exact proof term for the current goal is
HSNo_mul_HSNo z (mult' w v Hz Lwv .
We prove the intermediate
claim Lzwv2 :
HSNo (mult' (mult' z w v .
An
exact proof term for the current goal is
HSNo_mul_HSNo (mult' z w v Lzw Hv .
We prove the intermediate
claim Lp0zR :
CSNo (p0 z .
An
exact proof term for the current goal is
HSNo_proj0R z Hz .
We prove the intermediate
claim Lp0wR :
CSNo (p0 w .
An
exact proof term for the current goal is
HSNo_proj0R w Hw .
We prove the intermediate
claim Lp0vR :
CSNo (p0 v .
An
exact proof term for the current goal is
HSNo_proj0R v Hv .
We prove the intermediate
claim Lp1zR :
CSNo (p1 z .
An
exact proof term for the current goal is
HSNo_proj1R z Hz .
We prove the intermediate
claim Lp1wR :
CSNo (p1 w .
An
exact proof term for the current goal is
HSNo_proj1R w Hw .
We prove the intermediate
claim Lp1vR :
CSNo (p1 v .
An
exact proof term for the current goal is
HSNo_proj1R v Hv .
We prove the intermediate
claim L1 :
CSNo (p0 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L2 :
CSNo (p1 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L3 :
CSNo (p1 v ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L4 :
CSNo (p0 v ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L5 :
CSNo (p0 v ' ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L6 :
CSNo (p1 w ' ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L7 :
CSNo (p1 w * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L8 :
CSNo (p0 z * p0 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L9 :
CSNo (p0 w * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L10 :
CSNo (p0 w * p0 v .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L11 :
CSNo (p1 v * p0 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L12 :
CSNo ((p1 v * p0 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L13 :
CSNo ((p0 w * p0 v ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L14 :
CSNo (p1 z * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L15 :
CSNo (p0 z * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L16 :
CSNo (p0 v ' * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L17 :
CSNo (p1 v * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L18 :
CSNo (p1 v ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L19 :
CSNo (p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L20 :
CSNo (p1 v ' * p1 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L21 :
CSNo (p1 w * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L22 :
CSNo (p0 w ' * p1 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L23 :
CSNo (p0 v ' ' * p1 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L24 :
CSNo ((p1 w * p0 v ' ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L25 :
CSNo ((p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L26 :
CSNo (- p1 w ' * p1 z .
An exact proof term for the current goal is ??
We prove the intermediate
claim L27 :
CSNo (p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L28 :
CSNo (p0 v ' * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L29 :
CSNo (- p1 v ' * p1 w .
An exact proof term for the current goal is ??
We prove the intermediate
claim L30 :
CSNo (p1 w ' * p1 w ' ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L31 :
CSNo ((- p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L32 :
CSNo (p1 v * p0 z * p0 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L33 :
CSNo (p1 v * p0 w * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L34 :
CSNo (p0 z * p0 w * p0 v .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L35 :
CSNo (- (p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L36 :
CSNo (p1 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L37 :
CSNo (p1 w * p0 z * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L38 :
CSNo (p1 w * p0 v ' * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L39 :
CSNo (p1 z * p1 v * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L40 :
CSNo (p0 z * p1 v ' * p1 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L41 :
CSNo (p0 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L42 :
CSNo (- p1 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
We prove the intermediate
claim L43 :
CSNo (p1 z * p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L44 :
CSNo (p1 z * p0 v ' * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L45 :
CSNo (- p1 z * p1 v * p1 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L46 :
CSNo (- p0 z * p1 v ' * p1 w .
An exact proof term for the current goal is ??
We prove the intermediate
claim L47 :
CSNo (p0 w ' * p1 v ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L48 :
CSNo (- p0 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
We prove the intermediate
claim L49 :
CSNo (- p0 w ' * p1 v ' * p1 z .
An exact proof term for the current goal is ??
We prove the intermediate
claim L50 :
CSNo (p1 w * p0 z + p1 z * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L51 :
CSNo (p1 v * p0 w + p1 w * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L52 :
CSNo (p0 z * p0 w + - p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L53 :
CSNo (p0 w * p0 v + - p1 v ' * p1 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L54 :
CSNo ((p1 v * p0 w + p1 w * p0 v ' ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L55 :
CSNo ((p0 w * p0 v + - p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L56 :
CSNo (p0 w ' * p1 v ' + p0 v ' ' * p1 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L57 :
CSNo ((p1 v * p0 w ' + (p1 w * p0 v ' ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L58 :
CSNo ((p1 v * p0 w + p1 w * p0 v ' * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L59 :
CSNo (p0 v ' * p0 w ' + - (p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L60 :
CSNo ((p0 w * p0 v ' + (- p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L61 :
CSNo (p1 v * (p0 z * p0 w + - p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L62 :
CSNo ((p1 w * p0 z + p1 z * p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L63 :
CSNo (p0 z * (p0 w * p0 v + - p1 v ' * p1 w .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L64 :
CSNo ((p1 v * p0 w + p1 w * p0 v ' ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L65 :
CSNo (p1 z * (p0 w * p0 v + - p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L66 :
CSNo (- (p1 v * p0 w + p1 w * p0 v ' ' * p1 z .
An exact proof term for the current goal is ??
We prove the intermediate
claim L67 :
CSNo (p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L68 :
CSNo (p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L69 :
CSNo (- p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L70 :
CSNo (- p1 v * p1 w ' * p1 z + p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L71 :
CSNo (p1 w * p0 v ' * p0 z + p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L72 :
CSNo (- p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L73 :
CSNo (p1 v * (p0 z * p0 w + - p1 w ' * p1 z + (p1 w * p0 z + p1 z * p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L74 :
CSNo ((p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 w * p0 v + - p1 v ' * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L75 :
CSNo (p0 z * (p0 w * p0 v + - p1 v ' * p1 w + - (p1 v * p0 w + p1 w * p0 v ' ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L76 :
CSNo (p1 v * p0 z * p0 w + - p1 v * p1 w ' * p1 z + p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L77 :
CSNo (p1 v * p0 w * p0 z + p1 w * p0 v ' * p0 z + p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L78 :
CSNo (p0 z * p0 w * p0 v + - p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L79 :
CSNo (p0 v * p1 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L80 :
CSNo (p1 z * p0 v .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L81 :
CSNo (p1 w ' * p1 z * p0 v .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L82 :
CSNo (- p1 w ' * p1 z * p0 v .
An exact proof term for the current goal is ??
We prove the intermediate
claim L83 :
CSNo (p1 v ' * p1 w * p0 z .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L84 :
CSNo (p1 v ' * p1 z * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L85 :
CSNo (- p1 v ' * p1 w * p0 z .
An exact proof term for the current goal is ??
We prove the intermediate
claim L86 :
CSNo (- p1 v ' * p1 z * p0 w ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L87 :
CSNo (- p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L88 :
CSNo (- p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
We prove the intermediate
claim L89 :
CSNo (- p1 v ' .
An exact proof term for the current goal is ??
We prove the intermediate
claim L90 :
CSNo (- p1 v * p1 w ' .
An exact proof term for the current goal is ??
We will
prove p0 (mult' z (mult' w v = p0 (mult' (mult' z w v .
Use transitivity with
p0 z * p0 (mult' w v + - p1 (mult' w v ' * p1 z ,
p0 z * (p0 w * p0 v + - p1 v ' * p1 w + - (p1 v * p0 w + p1 w * p0 v ' ' * p1 z ,
p0 z * p0 w * p0 v + - p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z ,
p0 z * p0 w * p0 v + - p1 w ' * p1 z * p0 v + - p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' ,
(p0 z * p0 w + - p1 w ' * p1 z * p0 v + - p1 v ' * (p1 w * p0 z + p1 z * p0 w ' , and
p0 (mult' z w * p0 v + - p1 v ' * p1 (mult' z w .
An
exact proof term for the current goal is
mul_HSNo_proj0 z (mult' w v Hz Lwv .
We will
prove p0 z * p0 (mult' w v + - p1 (mult' w v ' * p1 z = p0 z * (p0 w * p0 v + - p1 v ' * p1 w + - (p1 v * p0 w + p1 w * p0 v ' ' * p1 z .
Use f_equal.
Use f_equal.
Use f_equal.
Use f_equal.
We will
prove p0 z * (p0 w * p0 v + - p1 v ' * p1 w + - (p1 v * p0 w + p1 w * p0 v ' ' * p1 z = p0 z * p0 w * p0 v + - p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z .
rewrite the current goal using
add_CSNo_assoc (p0 z * p0 w * p0 v (- p0 z * p1 v ' * p1 w (- p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z ?? ?? ?? (from right to left).
Use f_equal.
We will
prove p0 z * (p0 w * p0 v + - p1 v ' * p1 w = p0 z * p0 w * p0 v + - p0 z * p1 v ' * p1 w .
rewrite the current goal using
mul_CSNo_distrL (p0 z (p0 w * p0 v (- p1 v ' * p1 w ?? ?? ?? (from left to right).
We will
prove p0 z * p0 w * p0 v + p0 z * (- p1 v ' * p1 w = p0 z * p0 w * p0 v + - p0 z * p1 v ' * p1 w .
Use f_equal.
We will
prove p0 z * (- p1 v ' * p1 w = - p0 z * p1 v ' * p1 w .
We will
prove - (p1 v * p0 w + p1 w * p0 v ' ' * p1 z = - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z .
Use transitivity with and
- (p0 w ' * p1 v ' * p1 z + p0 v * p1 w ' * p1 z .
Use f_equal.
We will
prove (p1 v * p0 w + p1 w * p0 v ' ' * p1 z = p0 w ' * p1 v ' * p1 z + p0 v * p1 w ' * p1 z .
Use transitivity with and
(p0 w ' * p1 v ' + p0 v * p1 w ' * p1 z .
Use f_equal.
We will
prove (p1 v * p0 w + p1 w * p0 v ' ' = p0 w ' * p1 v ' + p0 v * p1 w ' .
Use transitivity with and
(p1 v * p0 w ' + (p1 w * p0 v ' ' .
An
exact proof term for the current goal is
conj_add_CSNo (p1 v * p0 w (p1 w * p0 v ' ?? ?? .
We will
prove (p1 v * p0 w ' + (p1 w * p0 v ' ' = p0 w ' * p1 v ' + p0 v * p1 w ' .
Use f_equal.
An
exact proof term for the current goal is
conj_mul_CSNo (p1 v (p0 w ?? ?? .
rewrite the current goal using
conj_mul_CSNo (p1 w (p0 v ' ?? ?? (from left to right).
We will
prove p0 v ' ' * p1 w ' = p0 v * p1 w ' .
Use f_equal.
We will
prove (p0 w ' * p1 v ' + p0 v * p1 w ' * p1 z = p0 w ' * p1 v ' * p1 z + p0 v * p1 w ' * p1 z .
rewrite the current goal using
mul_CSNo_distrR (p0 w ' * p1 v ' (p0 v * p1 w ' (p1 z ?? ?? ?? (from left to right).
Use f_equal.
Use symmetry.
An
exact proof term for the current goal is
mul_CSNo_assoc (p0 w ' (p1 v ' (p1 z ?? ?? ?? .
Use symmetry.
An
exact proof term for the current goal is
mul_CSNo_assoc (p0 v (p1 w ' (p1 z ?? ?? ?? .
We will
prove - (p0 w ' * p1 v ' * p1 z + p0 v * p1 w ' * p1 z = - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z .
An
exact proof term for the current goal is
minus_add_CSNo (p0 w ' * p1 v ' * p1 z (p0 v * p1 w ' * p1 z ?? ?? .
We will
prove p0 z * p0 w * p0 v + - p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z = p0 z * p0 w * p0 v + - p1 w ' * p1 z * p0 v + - p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' .
Use f_equal.
We will
prove - p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z + - p0 v * p1 w ' * p1 z = - p1 w ' * p1 z * p0 v + - p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' .
We will
prove - p0 v * p1 w ' * p1 z + - p0 z * p1 v ' * p1 w + - p0 w ' * p1 v ' * p1 z = - p1 w ' * p1 z * p0 v + - p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' .
Use f_equal.
We will
prove - p0 v * p1 w ' * p1 z = - p1 w ' * p1 z * p0 v .
Use f_equal.
Use symmetry.
Use f_equal.
We will
prove - p0 z * p1 v ' * p1 w = - p1 v ' * p1 w * p0 z .
Use f_equal.
Use symmetry.
We will
prove - p0 w ' * p1 v ' * p1 z = - p1 v ' * p1 z * p0 w ' .
Use f_equal.
Use symmetry.
We will
prove p0 z * p0 w * p0 v + - p1 w ' * p1 z * p0 v + - p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' = (p0 z * p0 w + - p1 w ' * p1 z * p0 v + - p1 v ' * (p1 w * p0 z + p1 z * p0 w ' .
rewrite the current goal using
add_CSNo_assoc (p0 z * p0 w * p0 v (- p1 w ' * p1 z * p0 v (- p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' ?? ?? ?? (from right to left).
Use f_equal.
We will
prove p0 z * p0 w * p0 v + - p1 w ' * p1 z * p0 v = (p0 z * p0 w + - p1 w ' * p1 z * p0 v .
rewrite the current goal using
mul_CSNo_assoc (p0 z (p0 w (p0 v ?? ?? ?? (from left to right).
We will
prove (p0 z * p0 w * p0 v + - p1 w ' * p1 z * p0 v = (p0 z * p0 w + - p1 w ' * p1 z * p0 v .
rewrite the current goal using
mul_CSNo_assoc (p1 w ' (p1 z (p0 v ?? ?? ?? (from left to right).
We will
prove (p0 z * p0 w * p0 v + (- p1 w ' * p1 z * p0 v = (p0 z * p0 w + - p1 w ' * p1 z * p0 v .
Use symmetry.
An
exact proof term for the current goal is
mul_CSNo_distrR (p0 z * p0 w (- p1 w ' * p1 z (p0 v ?? ?? ?? .
We will
prove - p1 v ' * p1 w * p0 z + - p1 v ' * p1 z * p0 w ' = - p1 v ' * (p1 w * p0 z + p1 z * p0 w ' .
rewrite the current goal using
mul_CSNo_distrL (- p1 v ' (p1 w * p0 z (p1 z * p0 w ' ?? ?? ?? (from right to left).
We will
prove (p0 z * p0 w + - p1 w ' * p1 z * p0 v + - p1 v ' * (p1 w * p0 z + p1 z * p0 w ' = p0 (mult' z w * p0 v + - p1 v ' * p1 (mult' z w .
Use f_equal.
Use f_equal.
Use symmetry.
Use f_equal.
Use f_equal.
Use symmetry.
Use symmetry.
An
exact proof term for the current goal is
mul_HSNo_proj0 (mult' z w v Lzw Hv .
We will
prove p1 (mult' z (mult' w v = p1 (mult' (mult' z w v .
Use transitivity with
p1 (mult' w v * p0 z + p1 z * p0 (mult' w v ' ,
(p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 w * p0 v + - (p1 v ' * p1 w ' ,
(p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 v ' * p0 w ' + - p1 v * p1 w ' ,
p1 v * p0 w * p0 z + p1 w * p0 v ' * p0 z + p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' ,
p1 v * p0 z * p0 w + - p1 v * p1 w ' * p1 z + p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' ,
p1 v * (p0 z * p0 w + - p1 w ' * p1 z + (p1 w * p0 z + p1 z * p0 w ' * p0 v ' , and
p1 v * p0 (mult' z w + p1 (mult' z w * p0 v ' .
An
exact proof term for the current goal is
mul_HSNo_proj1 z (mult' w v Hz Lwv .
We will
prove p1 (mult' w v * p0 z + p1 z * p0 (mult' w v ' = (p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 w * p0 v + - (p1 v ' * p1 w ' .
Use f_equal.
Use f_equal.
Use f_equal.
We will
prove (p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 w * p0 v + - (p1 v ' * p1 w ' = (p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 v ' * p0 w ' + - p1 v * p1 w ' .
Use f_equal.
Use f_equal.
We will
prove (p0 w * p0 v + - (p1 v ' * p1 w ' = p0 v ' * p0 w ' + - p1 v * p1 w ' .
rewrite the current goal using
conj_add_CSNo (p0 w * p0 v (- (p1 v ' * p1 w ?? ?? (from left to right).
rewrite the current goal using
conj_minus_CSNo (p1 v ' * p1 w ?? (from left to right).
We will
prove (p0 w * p0 v ' + - (p1 v ' * p1 w ' = p0 v ' * p0 w ' + - p1 v * p1 w ' .
Use f_equal.
An
exact proof term for the current goal is
conj_mul_CSNo (p0 w (p0 v ?? ?? .
Use f_equal.
rewrite the current goal using
conj_mul_CSNo (p1 v ' (p1 w ?? ?? (from left to right).
We will
prove p1 w ' * p1 v ' ' = p1 v * p1 w ' .
rewrite the current goal using
conj_CSNo_invol (p1 v ?? (from left to right).
We will
prove p1 w ' * p1 v = p1 v * p1 w ' .
An
exact proof term for the current goal is
mul_CSNo_com (p1 w ' (p1 v ?? ?? .
We will
prove (p1 v * p0 w + p1 w * p0 v ' * p0 z + p1 z * (p0 v ' * p0 w ' + - p1 v * p1 w ' = p1 v * p0 w * p0 z + p1 w * p0 v ' * p0 z + p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' .
rewrite the current goal using
add_CSNo_assoc (p1 v * p0 w * p0 z (p1 w * p0 v ' * p0 z (p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' ?? ?? ?? (from right to left).
Use f_equal.
We will
prove (p1 v * p0 w + p1 w * p0 v ' * p0 z = p1 v * p0 w * p0 z + p1 w * p0 v ' * p0 z .
rewrite the current goal using
mul_CSNo_distrR (p1 v * p0 w (p1 w * p0 v ' (p0 z ?? ?? ?? (from left to right).
rewrite the current goal using
mul_CSNo_assoc (p1 v (p0 w (p0 z ?? ?? ?? (from left to right).
rewrite the current goal using
mul_CSNo_assoc (p1 w (p0 v ' (p0 z ?? ?? ?? (from left to right).
Use reflexivity.
We will
prove p1 z * (p0 v ' * p0 w ' + - p1 v * p1 w ' = p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' .
rewrite the current goal using
mul_CSNo_distrL (p1 z (p0 v ' * p0 w ' (- p1 v * p1 w ' ?? ?? ?? (from left to right).
Use f_equal.
We will
prove p1 z * (- p1 v * p1 w ' = - p1 z * p1 v * p1 w ' .
We will
prove p1 v * p0 w * p0 z + p1 w * p0 v ' * p0 z + p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' = p1 v * p0 z * p0 w + - p1 v * p1 w ' * p1 z + p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' .
Use f_equal.
Use f_equal.
An
exact proof term for the current goal is
mul_CSNo_com (p0 w (p0 z ?? ?? .
We will
prove p1 w * p0 v ' * p0 z + p1 z * p0 v ' * p0 w ' + - p1 z * p1 v * p1 w ' = - p1 v * p1 w ' * p1 z + p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' .
rewrite the current goal using
add_CSNo_rotate_3_1 (p1 w * p0 v ' * p0 z (p1 z * p0 v ' * p0 w ' (- p1 z * p1 v * p1 w ' ?? ?? ?? (from left to right).
Use f_equal.
We will
prove - p1 z * p1 v * p1 w ' = - p1 v * p1 w ' * p1 z .
Use f_equal.
Use symmetry.
Use f_equal.
We will
prove p1 w * p0 v ' * p0 z = p1 w * p0 z * p0 v ' .
Use f_equal.
An
exact proof term for the current goal is
mul_CSNo_com (p0 v ' (p0 z ?? ?? .
We will
prove p1 z * p0 v ' * p0 w ' = p1 z * p0 w ' * p0 v ' .
Use f_equal.
An
exact proof term for the current goal is
mul_CSNo_com (p0 v ' (p0 w ' ?? ?? .
We will
prove p1 v * p0 z * p0 w + - p1 v * p1 w ' * p1 z + p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' = p1 v * (p0 z * p0 w + - p1 w ' * p1 z + (p1 w * p0 z + p1 z * p0 w ' * p0 v ' .
rewrite the current goal using
add_CSNo_assoc (p1 v * p0 z * p0 w (- p1 v * p1 w ' * p1 z (p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' ?? ?? ?? (from right to left).
Use f_equal.
We will
prove p1 v * p0 z * p0 w + - p1 v * p1 w ' * p1 z = p1 v * (p0 z * p0 w + - p1 w ' * p1 z .
rewrite the current goal using
mul_CSNo_distrL (p1 v (p0 z * p0 w (- p1 w ' * p1 z ?? ?? ?? (from left to right).
We will
prove p1 v * p0 z * p0 w + - p1 v * p1 w ' * p1 z = p1 v * p0 z * p0 w + p1 v * (- p1 w ' * p1 z .
Use f_equal.
We will
prove - p1 v * p1 w ' * p1 z = p1 v * (- p1 w ' * p1 z .
Use symmetry.
We will
prove p1 w * p0 z * p0 v ' + p1 z * p0 w ' * p0 v ' = (p1 w * p0 z + p1 z * p0 w ' * p0 v ' .
rewrite the current goal using
mul_CSNo_assoc (p1 w (p0 z (p0 v ' ?? ?? ?? (from left to right).
rewrite the current goal using
mul_CSNo_assoc (p1 z (p0 w ' (p0 v ' ?? ?? ?? (from left to right).
Use symmetry.
An
exact proof term for the current goal is
mul_CSNo_distrR (p1 w * p0 z (p1 z * p0 w ' (p0 v ' ?? ?? ?? .
We will
prove p1 v * (p0 z * p0 w + - p1 w ' * p1 z + (p1 w * p0 z + p1 z * p0 w ' * p0 v ' = p1 v * p0 (mult' z w + p1 (mult' z w * p0 v ' .
Use f_equal.
Use f_equal.
Use symmetry.
Use f_equal.
Use symmetry.
Use symmetry.
We will
prove p1 (mult' (mult' z w v = p1 v * p0 (mult' z w + p1 (mult' z w * p0 v ' .
An
exact proof term for the current goal is
mul_HSNo_proj1 (mult' z w v Lzw Hv .
∎
Proof:
An
exact proof term for the current goal is
conj_CSNo_i .
∎
Proof:
We will
prove p0 (j '' = p0 (:-: j .
We will
prove p0 j ' = - p0 j .
rewrite the current goal using
HSNo_p0_j (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
We will
prove p1 (j '' = p1 (:-: j .
We will
prove - p1 j = - p1 j .
Use reflexivity.
∎
Proof:
We will
prove p0 (k '' = p0 (:-: k .
We will
prove p0 k ' = - p0 k .
rewrite the current goal using
HSNo_p0_k (from left to right).
rewrite the current goal using
minus_CSNo_0 (from left to right).
We will
prove p1 (k '' = p1 (:-: k .
We will
prove - p1 k = - p1 k .
Use reflexivity.
∎
End of Section SurQuaternions
Beginning of Section Quaternions
Notation . We use
- as a prefix operator with priority 358 corresponding to applying term
minus_HSNo .
Notation . We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_HSNo .
Notation . We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_HSNo .
Proof: Let x be given.
Assume Hx .
Let y be given.
Assume Hy .
rewrite the current goal using
tuple_2_0_eq x y (from right to left).
rewrite the current goal using
tuple_2_1_eq x y (from right to left) at position 2.
∎
Proof: Let z be given.
Assume Hz .
Let p be given.
Assume Hp .
Let u be given.
∎
Proof: Let z be given.
Assume Hz .
Let x be given.
Assume Hx .
Let y be given.
Assume Hy .
rewrite the current goal using Hzxy
Apply HSNo_I to the current goal.
An
exact proof term for the current goal is
complex_CSNo x Hx .
An
exact proof term for the current goal is
complex_CSNo y Hy .
∎
Proof: Let x be given.
rewrite the current goal using
CSNo_pair_0 x (from right to left).
An exact proof term for the current goal is Hx
An
exact proof term for the current goal is
complex_0 .
∎
Proof:
An
exact proof term for the current goal is
complex_0 .
An
exact proof term for the current goal is
complex_1 .
∎
Proof:
An
exact proof term for the current goal is
complex_0 .
An
exact proof term for the current goal is
complex_i .
∎
Proof: Let x be given.
Assume Hx .
Let y be given.
Assume Hy .
∎
Proof: Let x be given.
Assume Hx .
Let y be given.
Assume Hy .
∎
Proof: Let z be given.
Assume Hz .
Let x be given.
Assume Hx .
Let y be given.
Assume Hy Hzxy .
rewrite the current goal using Hzxy
An exact proof term for the current goal is Hx
∎
Proof: Let z be given.
Assume Hz .
Let x be given.
Assume Hx .
Let y be given.
Assume Hy Hzxy .
rewrite the current goal using Hzxy
An exact proof term for the current goal is Hy
∎
Proof: Let z be given.
Assume Hz .
Let w be given.
Assume Hw .
∎
Proof: Let z be given.
Assume Hz .
∎
Proof: Let z be given.
Assume Hz .
∎
Proof: Let z be given.
Assume Hz .
Let w be given.
Assume Hw .
∎
Proof: Let z be given.
Assume Hz .
Let w be given.
Assume Hw .
We prove the intermediate
claim Lz0 :
p0 z ∈ complex .
We prove the intermediate
claim Lz1 :
p1 z ∈ complex .
We prove the intermediate
claim Lw0 :
p0 w ∈ complex .
We prove the intermediate
claim Lw1 :
p1 w ∈ complex .
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
An exact proof term for the current goal is ??
∎
Proof: Let x be given.
Assume Hx .
rewrite the current goal using
CSNo_pair_0 x (from right to left) at position 1.
∎
Proof: Let x be given.
Assume Hx .
rewrite the current goal using
CSNo_pair_0 x (from right to left).
∎
End of Section Quaternions