Theory "pair"

Parents     relation

Signature

Type Arity
prod 2
Constant Type
, :'a -> 'b -> 'a # 'b
## :('a -> 'c) -> ('b -> 'd) -> 'a # 'b -> 'c # 'd
CURRY :('a # 'b -> 'c) -> 'a -> 'b -> 'c
UNCURRY :('a -> 'b -> 'c) -> 'a # 'b -> 'c
REP_prod :'a # 'b -> 'a -> 'b -> bool
ABS_prod :('a -> 'b -> bool) -> 'a # 'b
IS_PAIR :('a -> 'b -> bool) -> bool
MK_PAIR :'a -> 'b -> 'a -> 'b -> bool
FST :'a # 'b -> 'a
LEX :('a -> 'a -> bool) -> ('b -> 'b -> bool) -> 'a # 'b -> 'a # 'b -> bool
pair_case :('a -> 'b -> 'c) -> 'a # 'b -> 'c
SND :'a # 'b -> 'b
RPROD :('a -> 'a -> bool) -> ('b -> 'b -> bool) -> 'a # 'b -> 'a # 'b -> bool

Definitions

MK_PAIR_DEF
|- !x y. MK_PAIR x y = (\a b. (a = x) /\ (b = y))
IS_PAIR_DEF
|- !P. IS_PAIR P = ?x y. P = MK_PAIR x y
prod_TY_DEF
|- ?rep. TYPE_DEFINITION IS_PAIR rep
ABS_REP_prod
|- (!a. ABS_prod (REP_prod a) = a) /\
   !r. IS_PAIR r = (REP_prod (ABS_prod r) = r)
COMMA_DEF
|- !x y. (x,y) = ABS_prod (MK_PAIR x 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
|- 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)
FST
|- !x y. FST (x,y) = x
SND
|- !x y. SND (x,y) = y
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)
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)
pair_case_thm
|- 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)) ==>
     (case f M = case f' M')
WF_LEX
|- !R Q. WF R /\ WF Q ==> WF (R LEX Q)
WF_RPROD
|- !R Q. WF R /\ WF Q ==> WF (RPROD R Q)