Theory "pair"

Parents relation

Signature

Type	Arity
prod	2
Constant	Type
##	:(α -> γ) -> (β -> δ) -> α # β -> γ # δ
,	:α -> β -> α # β
ABS_prod	:(α -> β -> bool) -> α # β
CURRY	:(α # β -> γ) -> α -> β -> γ
FST	:α # β -> α
LEX	:α reln -> β reln -> (α # β) reln
PROD_ALL	:(α -> bool) -> (β -> bool) -> α # β -> bool
REP_prod	:α # β -> α -> β -> bool
RPROD	:α reln -> β reln -> (α # β) reln
SND	:α # β -> β
SWAP	:β # α -> α # β
UNCURRY	:(α -> β -> γ) -> α # β -> γ
pair_CASE	:β # γ -> (β -> γ -> α) -> α

Definitions

prod_TY_DEF

⊢ ∃rep. TYPE_DEFINITION (λp. ∃x y. p = (λa b. a = x ∧ b = y)) rep

ABS_REP_prod

⊢ (∀a. ABS_prod (REP_prod a) = a) ∧
  ∀r. (λp. ∃x y. p = (λa b. a = x ∧ b = y)) r ⇔ REP_prod (ABS_prod r) = r

COMMA_DEF

⊢ ∀x y. (x,y) = ABS_prod (λa b. a = x ∧ b = y)

PAIR

⊢ ∀x. (FST x,SND x) = x

SWAP_def

⊢ ∀a. SWAP a = (SND a,FST a)

CURRY_DEF

⊢ ∀f x y. CURRY f x y = f (x,y)

UNCURRY

⊢ ∀f v. UNCURRY f v = f (FST v) (SND v)

PROD_ALL_def

⊢ ∀P Q p. PROD_ALL P Q p ⇔ P (FST p) ∧ Q (SND p)

PAIR_MAP

⊢ ∀f g p. (f ## g) p = (f (FST p),g (SND p))

pair_CASE_def

⊢ ∀p f. pair_CASE p f = f (FST p) (SND p)

LEX_DEF

⊢ ∀R1 R2. R1 LEX R2 = (λ(s,t) (u,v). R1 s u ∨ s = u ∧ R2 t v)

RPROD_DEF

⊢ ∀R1 R2. RPROD R1 R2 = (λ(s,t) (u,v). R1 s u ∧ R2 t v)

Theorems

PAIR_EQ

⊢ (x,y) = (a,b) ⇔ x = a ∧ y = b

CLOSED_PAIR_EQ

⊢ ∀x y a b. (x,y) = (a,b) ⇔ x = a ∧ y = b

ABS_PAIR_THM

⊢ ∀x. ∃q r. x = (q,r)

pair_CASES

⊢ ∀x. ∃q r. x = (q,r)

FST

⊢ ∀x y. FST (x,y) = x

SND

⊢ ∀x y. SND (x,y) = y

PAIR_FST_SND_EQ

⊢ ∀p q. p = q ⇔ FST p = FST q ∧ SND p = SND q

UNCURRY_VAR

⊢ ∀f v. UNCURRY f v = f (FST v) (SND v)

ELIM_UNCURRY

⊢ ∀f. UNCURRY f = (λx. f (FST x) (SND x))

UNCURRY_DEF

⊢ ∀f x y. UNCURRY f (x,y) = f x y

CURRY_UNCURRY_THM

⊢ ∀f. CURRY (UNCURRY f) = f

UNCURRY_CURRY_THM

⊢ ∀f. UNCURRY (CURRY f) = f

CURRY_ONE_ONE_THM

⊢ CURRY f = CURRY g ⇔ f = g

UNCURRY_ONE_ONE_THM

⊢ UNCURRY f = UNCURRY g ⇔ f = g

pair_Axiom

⊢ ∀f. ∃fn. ∀x y. fn (x,y) = f x y

UNCURRY_CONG

⊢ ∀f' f M' M.
      M = M' ∧ (∀x y. M' = (x,y) ⇒ f x y = f' x y) ⇒
      UNCURRY f M = UNCURRY f' M'

LAMBDA_PROD

⊢ ∀P. (λp. P p) = (λ(p1,p2). P (p1,p2))

EXISTS_PROD

⊢ (∃p. P p) ⇔ ∃p_1 p_2. P (p_1,p_2)

FORALL_PROD

⊢ (∀p. P p) ⇔ ∀p_1 p_2. P (p_1,p_2)

pair_induction

⊢ (∀p_1 p_2. P (p_1,p_2)) ⇒ ∀p. P p

PROD_ALL_THM

⊢ PROD_ALL P Q (x,y) ⇔ P x ∧ Q y

PROD_ALL_MONO

⊢ (∀x. P x ⇒ P' x) ∧ (∀y. Q y ⇒ Q' y) ⇒ PROD_ALL P Q p ⇒ PROD_ALL P' Q' p

PROD_ALL_CONG

⊢ ∀p p' P P' Q Q'.
      p = p' ∧ (∀x y. p' = (x,y) ⇒ (P x ⇔ P' x)) ∧
      (∀x y. p' = (x,y) ⇒ (Q y ⇔ Q' y)) ⇒
      (PROD_ALL P Q p ⇔ PROD_ALL P' Q' p')

ELIM_PEXISTS

⊢ (∃p. P (FST p) (SND p)) ⇔ ∃p1 p2. P p1 p2

ELIM_PFORALL

⊢ (∀p. P (FST p) (SND p)) ⇔ ∀p1 p2. P p1 p2

PFORALL_THM

⊢ ∀P. (∀x y. P x y) ⇔ ∀(x,y). P x y

PEXISTS_THM

⊢ ∀P. (∃x y. P x y) ⇔ ∃(x,y). P x y

ELIM_PEXISTS_EVAL

⊢ $? (UNCURRY (λx. P x)) ⇔ ∃x. $? (P x)

ELIM_PFORALL_EVAL

⊢ $! (UNCURRY (λx. P x)) ⇔ ∀x. $! (P x)

PAIR_MAP_THM

⊢ ∀f g x y. (f ## g) (x,y) = (f x,g y)

FST_PAIR_MAP

⊢ ∀p f g. FST ((f ## g) p) = f (FST p)

SND_PAIR_MAP

⊢ ∀p f g. SND ((f ## g) p) = g (SND p)

LET2_RAND

⊢ ∀P M N. P (let (x,y) = M in N x y) = (let (x,y) = M in P (N x y))

LET2_RATOR

⊢ ∀M N b. (let (x,y) = M in N x y) b = (let (x,y) = M in N x y b)

o_UNCURRY_R

⊢ f ∘ UNCURRY g = UNCURRY ($o f ∘ g)

C_UNCURRY_L

⊢ combin$C (UNCURRY f) x = UNCURRY (combin$C (combin$C ∘ f) x)

S_UNCURRY_R

⊢ S f (UNCURRY g) = UNCURRY (S (S ∘ $o f ∘ $,) g)

FORALL_UNCURRY

⊢ $! (UNCURRY f) ⇔ $! ($! ∘ f)

PAIR_FUN_THM

⊢ ∀P. (∃!f. P f) ⇔ ∃!p. P (λa. (FST p a,SND p a))

pair_case_thm

⊢ pair_CASE (x,y) f = f x y

pair_case_def

⊢ pair_CASE (x,y) f = f x y

pair_case_cong

⊢ ∀M M' f.
      M = M' ∧ (∀x y. M' = (x,y) ⇒ f x y = f' x y) ⇒
      pair_CASE M f = pair_CASE M' f'

pair_case_eq

⊢ pair_CASE p f = v ⇔ ∃x y. p = (x,y) ∧ f x y = v

datatype_pair

⊢ DATATYPE (pair $,)

LEX_DEF_THM

⊢ (R1 LEX R2) (a,b) (c,d) ⇔ R1 a c ∨ a = c ∧ R2 b d

LEX_MONO

⊢ (∀x y. R1 x y ⇒ R2 x y) ∧ (∀x y. R3 x y ⇒ R4 x y) ⇒
  (R1 LEX R3) x y ⇒
  (R2 LEX R4) x y

WF_LEX

⊢ ∀R Q. WF R ∧ WF Q ⇒ WF (R LEX Q)

WF_RPROD

⊢ ∀R Q. WF R ∧ WF Q ⇒ WF (RPROD R Q)

total_LEX

⊢ total R1 ∧ total R2 ⇒ total (R1 LEX R2)

transitive_LEX

⊢ transitive R1 ∧ transitive R2 ⇒ transitive (R1 LEX R2)

reflexive_LEX

⊢ reflexive (R1 LEX R2) ⇔ reflexive R1 ∨ reflexive R2

symmetric_LEX

⊢ symmetric R1 ∧ symmetric R2 ⇒ symmetric (R1 LEX R2)

LEX_CONG

⊢ ∀R1 R2 v1 v2 R1' R2' v1' v2'.
      v1 = v1' ∧ v2 = v2' ∧
      (∀a b c d. v1' = (a,b) ∧ v2' = (c,d) ⇒ (R1 a c ⇔ R1' a c)) ∧
      (∀a b c d. v1' = (a,b) ∧ v2' = (c,d) ∧ a = c ⇒ (R2 b d ⇔ R2' b d)) ⇒
      ((R1 LEX R2) v1 v2 ⇔ (R1' LEX R2') v1' v2')

FST_EQ_EQUIV

⊢ FST p = x ⇔ ∃y. p = (x,y)

SND_EQ_EQUIV

⊢ SND p = y ⇔ ∃x. p = (x,y)