Theory "pair"

Signature

Definitions

prod_TY_DEF

|- ?rep. TYPE_DEFINITION (\p. ?x y. p = (\a b. (a = x) /\ (b = y))) rep

ABS_REP_prod

|- (!a. ABS_prod (REP_prod a) = a) /\
   !r.
     (\p. ?x y. p = (\a b. (a = x) /\ (b = y))) r =
     (REP_prod (ABS_prod r) = r)

COMMA_DEF

|- !x y. (x,y) = ABS_prod (\a b. (a = x) /\ (b = y))

PAIR

|- !x. (FST x,SND x) = x

CURRY_DEF

|- !f x y. CURRY f x y = f (x,y)

UNCURRY

|- !f v. UNCURRY f v = f (FST v) (SND v)

PAIR_MAP

|- !f g p. (f ## g) p = (f (FST p),g (SND p))

pair_case_def

|- pair_case = UNCURRY

LEX_DEF

|- !R1 R2. R1 LEX R2 = (\(s,t) (u,v). R1 s u \/ (s = u) /\ R2 t v)

RPROD_DEF

|- !R1 R2. RPROD R1 R2 = (\(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 = (\x. 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

|- !M M' f.
     (M = M') /\ (!x y. (M' = (x,y)) ==> (f x y = f' x y)) ==>
     (UNCURRY f M = UNCURRY f' M')

LAMBDA_PROD

|- !P. (\p. P p) = (\(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

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

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 o UNCURRY g = UNCURRY ($o f o g)

C_UNCURRY_L

|- combin$C (UNCURRY f) x = UNCURRY (combin$C (combin$C o f) x)

S_UNCURRY_R

|- S f (UNCURRY g) = UNCURRY (S (S o $o f o $,) g)

FORALL_UNCURRY

|- $! (UNCURRY f) = $! ($! o f)

PAIR_FUN_THM

|- !P. (?!f. P f) = ?!p. P (\a. (FST p a,SND p a))

pair_case_thm

|- pair_case f (x,y) = f x y

pair_case_cong

|- !M M' f.
     (M = M') /\ (!x y. (M' = (x,y)) ==> (f x y = f' x y)) ==>
     (pair_case f M = pair_case f' M')

LEX_DEF_THM

|- (R1 LEX R2) (a,b) (c,d) = R1 a c \/ (a = c) /\ R2 b d

WF_LEX

|- !R Q. WF R /\ WF Q ==> WF (R LEX Q)

WF_RPROD

|- !R Q. WF R /\ WF Q ==> WF (RPROD R Q)