Structure pairTheory

signature pairTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val ABS_REP_prod : thm
    val COMMA_DEF : thm
    val CURRY_DEF : thm
    val LEX_DEF : thm
    val PAIR : thm
    val PAIR_MAP : thm
    val RPROD_DEF : thm
    val UNCURRY : thm
    val pair_case_def : thm
    val prod_TY_DEF : thm
  
  (*  Theorems  *)
    val ABS_PAIR_THM : thm
    val CLOSED_PAIR_EQ : thm
    val CURRY_ONE_ONE_THM : thm
    val CURRY_UNCURRY_THM : thm
    val C_UNCURRY_L : thm
    val ELIM_PEXISTS : thm
    val ELIM_PFORALL : thm
    val ELIM_UNCURRY : thm
    val EXISTS_PROD : thm
    val FORALL_PROD : thm
    val FORALL_UNCURRY : thm
    val FST : thm
    val FST_PAIR_MAP : thm
    val LAMBDA_PROD : thm
    val LET2_RAND : thm
    val LET2_RATOR : thm
    val LEX_DEF_THM : thm
    val PAIR_EQ : thm
    val PAIR_FST_SND_EQ : thm
    val PAIR_FUN_THM : thm
    val PAIR_MAP_THM : thm
    val PEXISTS_THM : thm
    val PFORALL_THM : thm
    val SND : thm
    val SND_PAIR_MAP : thm
    val S_UNCURRY_R : thm
    val UNCURRY_CONG : thm
    val UNCURRY_CURRY_THM : thm
    val UNCURRY_DEF : thm
    val UNCURRY_ONE_ONE_THM : thm
    val UNCURRY_VAR : thm
    val WF_LEX : thm
    val WF_RPROD : thm
    val o_UNCURRY_R : thm
    val pair_Axiom : thm
    val pair_CASES : thm
    val pair_case_cong : thm
    val pair_case_thm : thm
    val pair_induction : thm
  
  val pair_grammars : type_grammar.grammar * term_grammar.grammar
  
  val pair_rwts : simpLib.ssfrag
  
  val pair_rws : thm list
  
  
  type hol_type = Abbrev.hol_type
  type term     = Abbrev.term
  type conv     = Abbrev.conv
  
  val uncurry_tm       : term
  val comma_tm         : term
  val dest_pair        : term -> term * term
  val strip_pair       : term -> term list
  val spine_pair       : term -> term list
  val is_vstruct       : term -> bool
  val mk_pabs          : term * term -> term
  val PAIRED_BETA_CONV : conv
  
(*
   [relation] Parent theory of "pair"
   
   [ABS_REP_prod]  Definition
      
      |- (!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]  Definition
      
      |- !x y. (x,y) = ABS_prod (\a b. (a = x) /\ (b = y))
   
   [CURRY_DEF]  Definition
      
      |- !f x y. CURRY f x y = f (x,y)
   
   [LEX_DEF]  Definition
      
      |- !R1 R2. R1 LEX R2 = (\(s,t) (u,v). R1 s u \/ (s = u) /\ R2 t v)
   
   [PAIR]  Definition
      
      |- !x. (FST x,SND x) = x
   
   [PAIR_MAP]  Definition
      
      |- !f g p. (f ## g) p = (f (FST p),g (SND p))
   
   [RPROD_DEF]  Definition
      
      |- !R1 R2. RPROD R1 R2 = (\(s,t) (u,v). R1 s u /\ R2 t v)
   
   [UNCURRY]  Definition
      
      |- !f v. UNCURRY f v = f (FST v) (SND v)
   
   [pair_case_def]  Definition
      
      |- pair_case = UNCURRY
   
   [prod_TY_DEF]  Definition
      
      |- ?rep. TYPE_DEFINITION (\p. ?x y. p = (\a b. (a = x) /\ (b = y))) rep
   
   [ABS_PAIR_THM]  Theorem
      
      |- !x. ?q r. x = (q,r)
   
   [CLOSED_PAIR_EQ]  Theorem
      
      |- !x y a b. ((x,y) = (a,b)) = (x = a) /\ (y = b)
   
   [CURRY_ONE_ONE_THM]  Theorem
      
      |- (CURRY f = CURRY g) = (f = g)
   
   [CURRY_UNCURRY_THM]  Theorem
      
      |- !f. CURRY (UNCURRY f) = f
   
   [C_UNCURRY_L]  Theorem
      
      |- combin$C (UNCURRY f) x = UNCURRY (combin$C (combin$C o f) x)
   
   [ELIM_PEXISTS]  Theorem
      
      |- (?p. P (FST p) (SND p)) = ?p1 p2. P p1 p2
   
   [ELIM_PFORALL]  Theorem
      
      |- (!p. P (FST p) (SND p)) = !p1 p2. P p1 p2
   
   [ELIM_UNCURRY]  Theorem
      
      |- !f. UNCURRY f = (\x. f (FST x) (SND x))
   
   [EXISTS_PROD]  Theorem
      
      |- (?p. P p) = ?p_1 p_2. P (p_1,p_2)
   
   [FORALL_PROD]  Theorem
      
      |- (!p. P p) = !p_1 p_2. P (p_1,p_2)
   
   [FORALL_UNCURRY]  Theorem
      
      |- $! (UNCURRY f) = $! ($! o f)
   
   [FST]  Theorem
      
      |- !x y. FST (x,y) = x
   
   [FST_PAIR_MAP]  Theorem
      
      |- !p f g. FST ((f ## g) p) = f (FST p)
   
   [LAMBDA_PROD]  Theorem
      
      |- !P. (\p. P p) = (\(p1,p2). P (p1,p2))
   
   [LET2_RAND]  Theorem
      
      |- !P M N. P (let (x,y) = M in N x y) = (let (x,y) = M in P (N x y))
   
   [LET2_RATOR]  Theorem
      
      |- !M N b. (let (x,y) = M in N x y) b = (let (x,y) = M in N x y b)
   
   [LEX_DEF_THM]  Theorem
      
      |- (R1 LEX R2) (a,b) (c,d) = R1 a c \/ (a = c) /\ R2 b d
   
   [PAIR_EQ]  Theorem
      
      |- ((x,y) = (a,b)) = (x = a) /\ (y = b)
   
   [PAIR_FST_SND_EQ]  Theorem
      
      |- !p q. (p = q) = (FST p = FST q) /\ (SND p = SND q)
   
   [PAIR_FUN_THM]  Theorem
      
      |- !P. (?!f. P f) = ?!p. P (\a. (FST p a,SND p a))
   
   [PAIR_MAP_THM]  Theorem
      
      |- !f g x y. (f ## g) (x,y) = (f x,g y)
   
   [PEXISTS_THM]  Theorem
      
      |- !P. (?x y. P x y) = ?(x,y). P x y
   
   [PFORALL_THM]  Theorem
      
      |- !P. (!x y. P x y) = !(x,y). P x y
   
   [SND]  Theorem
      
      |- !x y. SND (x,y) = y
   
   [SND_PAIR_MAP]  Theorem
      
      |- !p f g. SND ((f ## g) p) = g (SND p)
   
   [S_UNCURRY_R]  Theorem
      
      |- S f (UNCURRY g) = UNCURRY (S (S o $o f o $,) g)
   
   [UNCURRY_CONG]  Theorem
      
      |- !M M' f.
           (M = M') /\ (!x y. (M' = (x,y)) ==> (f x y = f' x y)) ==>
           (UNCURRY f M = UNCURRY f' M')
   
   [UNCURRY_CURRY_THM]  Theorem
      
      |- !f. UNCURRY (CURRY f) = f
   
   [UNCURRY_DEF]  Theorem
      
      |- !f x y. UNCURRY f (x,y) = f x y
   
   [UNCURRY_ONE_ONE_THM]  Theorem
      
      |- (UNCURRY f = UNCURRY g) = (f = g)
   
   [UNCURRY_VAR]  Theorem
      
      |- !f v. UNCURRY f v = f (FST v) (SND v)
   
   [WF_LEX]  Theorem
      
      |- !R Q. WF R /\ WF Q ==> WF (R LEX Q)
   
   [WF_RPROD]  Theorem
      
      |- !R Q. WF R /\ WF Q ==> WF (RPROD R Q)
   
   [o_UNCURRY_R]  Theorem
      
      |- f o UNCURRY g = UNCURRY ($o f o g)
   
   [pair_Axiom]  Theorem
      
      |- !f. ?fn. !x y. fn (x,y) = f x y
   
   [pair_CASES]  Theorem
      
      |- !x. ?q r. x = (q,r)
   
   [pair_case_cong]  Theorem
      
      |- !M M' f.
           (M = M') /\ (!x y. (M' = (x,y)) ==> (f x y = f' x y)) ==>
           (pair_case f M = pair_case f' M')
   
   [pair_case_thm]  Theorem
      
      |- pair_case f (x,y) = f x y
   
   [pair_induction]  Theorem
      
      |- (!p_1 p_2. P (p_1,p_2)) ==> !p. P p
   
   
*)
end