src/ZF/qpair.ML

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(*  Title: 	ZF/qpair.ML
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

For qpair.thy.  

Quine-inspired ordered pairs and disjoint sums, for non-well-founded data
structures in ZF.  Does not precisely follow Quine's construction.  Thanks
to Thomas Forster for suggesting this approach!

W. V. Quine, On Ordered Pairs and Relations, in Selected Logic Papers,
1966.

Many proofs are borrowed from pair.ML and sum.ML

Do we EVER have rank(a) < rank(<a;b>) ?  Perhaps if the latter rank
    is not a limit ordinal? 
*)


open QPair;

(**** Quine ordered pairing ****)

(** Lemmas for showing that <a;b> uniquely determines a and b **)

val QPair_iff = prove_goalw QPair.thy [QPair_def]
    "<a;b> = <c;d> <-> a=c & b=d"
 (fn _=> [rtac sum_equal_iff 1]);

val QPair_inject = standard (QPair_iff RS iffD1 RS conjE);

val QPair_inject1 = prove_goal QPair.thy "<a;b> = <c;d> ==> a=c"
 (fn [major]=>
  [ (rtac (major RS QPair_inject) 1), (assume_tac 1) ]);

val QPair_inject2 = prove_goal QPair.thy "<a;b> = <c;d> ==> b=d"
 (fn [major]=>
  [ (rtac (major RS QPair_inject) 1), (assume_tac 1) ]);


(*** QSigma: Disjoint union of a family of sets
     Generalizes Cartesian product ***)

val QSigmaI = prove_goalw QPair.thy [QSigma_def]
    "[| a:A;  b:B(a) |] ==> <a;b> : QSigma(A,B)"
 (fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]);

(*The general elimination rule*)
val QSigmaE = prove_goalw QPair.thy [QSigma_def]
    "[| c: QSigma(A,B);  \
\       !!x y.[| x:A;  y:B(x);  c=<x;y> |] ==> P \
\    |] ==> P"
 (fn major::prems=>
  [ (cut_facts_tac [major] 1),
    (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]);

(** Elimination rules for <a;b>:A*B -- introducing no eigenvariables **)

val QSigmaE2 = 
  rule_by_tactic (REPEAT_FIRST (etac QPair_inject ORELSE' bound_hyp_subst_tac)
		  THEN prune_params_tac)
      (read_instantiate [("c","<a;b>")] QSigmaE);  

val QSigmaD1 = prove_goal QPair.thy "<a;b> : QSigma(A,B) ==> a : A"
 (fn [major]=>
  [ (rtac (major RS QSigmaE2) 1), (assume_tac 1) ]);

val QSigmaD2 = prove_goal QPair.thy "<a;b> : QSigma(A,B) ==> b : B(a)"
 (fn [major]=>
  [ (rtac (major RS QSigmaE2) 1), (assume_tac 1) ]);

val QSigma_cong = prove_goalw QPair.thy [QSigma_def]
    "[| A=A';  !!x. x:A' ==> B(x)=B'(x) |] ==> \
\    QSigma(A,B) = QSigma(A',B')"
 (fn prems=> [ (simp_tac (ZF_ss addsimps prems) 1) ]);

val QSigma_empty1 = prove_goal QPair.thy "QSigma(0,B) = 0"
 (fn _ => [ (fast_tac (ZF_cs addIs [equalityI] addSEs [QSigmaE]) 1) ]);

val QSigma_empty2 = prove_goal QPair.thy "A <*> 0 = 0"
 (fn _ => [ (fast_tac (ZF_cs addIs [equalityI] addSEs [QSigmaE]) 1) ]);


(*** Eliminator - qsplit ***)

val qsplit = prove_goalw QPair.thy [qsplit_def]
    "qsplit(%x y.c(x,y), <a;b>) = c(a,b)"
 (fn _ => [ (fast_tac (ZF_cs addIs [the_equality] addEs [QPair_inject]) 1) ]);

val qsplit_type = prove_goal QPair.thy
    "[|  p:QSigma(A,B);   \
\        !!x y.[| x:A; y:B(x) |] ==> c(x,y):C(<x;y>) \
\    |] ==> qsplit(%x y.c(x,y), p) : C(p)"
 (fn major::prems=>
  [ (rtac (major RS QSigmaE) 1),
    (etac ssubst 1),
    (REPEAT (ares_tac (prems @ [qsplit RS ssubst]) 1)) ]);


val qpair_cs = ZF_cs addSIs [QSigmaI] addSEs [QSigmaE2, QSigmaE, QPair_inject];

(*** qconverse ***)

val qconverseI = prove_goalw QPair.thy [qconverse_def]
    "!!a b r. <a;b>:r ==> <b;a>:qconverse(r)"
 (fn _ => [ (fast_tac qpair_cs 1) ]);

val qconverseD = prove_goalw QPair.thy [qconverse_def]
    "!!a b r. <a;b> : qconverse(r) ==> <b;a> : r"
 (fn _ => [ (fast_tac qpair_cs 1) ]);

val qconverseE = prove_goalw QPair.thy [qconverse_def]
    "[| yx : qconverse(r);  \
\       !!x y. [| yx=<y;x>;  <x;y>:r |] ==> P \
\    |] ==> P"
 (fn [major,minor]=>
  [ (rtac (major RS ReplaceE) 1),
    (REPEAT (eresolve_tac [exE, conjE, minor] 1)),
    (hyp_subst_tac 1),
    (assume_tac 1) ]);

val qconverse_cs = qpair_cs addSIs [qconverseI] 
			    addSEs [qconverseD,qconverseE];

val qconverse_of_qconverse = prove_goal QPair.thy
    "!!A B r. r<=QSigma(A,B) ==> qconverse(qconverse(r)) = r"
 (fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]);

val qconverse_type = prove_goal QPair.thy
    "!!A B r. r <= A <*> B ==> qconverse(r) <= B <*> A"
 (fn _ => [ (fast_tac qconverse_cs 1) ]);

val qconverse_of_prod = prove_goal QPair.thy "qconverse(A <*> B) = B <*> A"
 (fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]);

val qconverse_empty = prove_goal QPair.thy "qconverse(0) = 0"
 (fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]);


(*** qsplit for predicates: result type o ***)

goalw QPair.thy [qfsplit_def] "!!R a b. R(a,b) ==> qfsplit(R, <a;b>)";
by (REPEAT (ares_tac [refl,exI,conjI] 1));
val qfsplitI = result();

val major::prems = goalw QPair.thy [qfsplit_def]
    "[| qfsplit(R,z);  !!x y. [| z = <x;y>;  R(x,y) |] ==> P |] ==> P";
by (cut_facts_tac [major] 1);
by (REPEAT (eresolve_tac (prems@[asm_rl,exE,conjE]) 1));
val qfsplitE = result();

goal QPair.thy "!!R a b. qfsplit(R,<a;b>) ==> R(a,b)";
by (REPEAT (eresolve_tac [asm_rl,qfsplitE,QPair_inject,ssubst] 1));
val qfsplitD = result();


(**** The Quine-inspired notion of disjoint sum ****)

val qsum_defs = [qsum_def,QInl_def,QInr_def,qcase_def];

(** Introduction rules for the injections **)

goalw QPair.thy qsum_defs "!!a A B. a : A ==> QInl(a) : A <+> B";
by (REPEAT (ares_tac [UnI1,QSigmaI,singletonI] 1));
val QInlI = result();

goalw QPair.thy qsum_defs "!!b A B. b : B ==> QInr(b) : A <+> B";
by (REPEAT (ares_tac [UnI2,QSigmaI,singletonI] 1));
val QInrI = result();

(** Elimination rules **)

val major::prems = goalw QPair.thy qsum_defs
    "[| u: A <+> B;  \
\       !!x. [| x:A;  u=QInl(x) |] ==> P; \
\       !!y. [| y:B;  u=QInr(y) |] ==> P \
\    |] ==> P";
by (rtac (major RS UnE) 1);
by (REPEAT (rtac refl 1
     ORELSE eresolve_tac (prems@[QSigmaE,singletonE,ssubst]) 1));
val qsumE = result();

(** Injection and freeness equivalences, for rewriting **)

goalw QPair.thy qsum_defs "QInl(a)=QInl(b) <-> a=b";
by (simp_tac (ZF_ss addsimps [QPair_iff]) 1);
val QInl_iff = result();

goalw QPair.thy qsum_defs "QInr(a)=QInr(b) <-> a=b";
by (simp_tac (ZF_ss addsimps [QPair_iff]) 1);
val QInr_iff = result();

goalw QPair.thy qsum_defs "QInl(a)=QInr(b) <-> False";
by (simp_tac (ZF_ss addsimps [QPair_iff, one_not_0 RS not_sym]) 1);
val QInl_QInr_iff = result();

goalw QPair.thy qsum_defs "QInr(b)=QInl(a) <-> False";
by (simp_tac (ZF_ss addsimps [QPair_iff, one_not_0]) 1);
val QInr_QInl_iff = result();

(*Injection and freeness rules*)

val QInl_inject = standard (QInl_iff RS iffD1);
val QInr_inject = standard (QInr_iff RS iffD1);
val QInl_neq_QInr = standard (QInl_QInr_iff RS iffD1 RS FalseE);
val QInr_neq_QInl = standard (QInr_QInl_iff RS iffD1 RS FalseE);

val qsum_cs = 
    ZF_cs addIs [QInlI,QInrI] addSEs [qsumE,QInl_neq_QInr,QInr_neq_QInl]
          addSDs [QInl_inject,QInr_inject];

goal QPair.thy "!!A B. QInl(a): A<+>B ==> a: A";
by (fast_tac qsum_cs 1);
val QInlD = result();

goal QPair.thy "!!A B. QInr(b): A<+>B ==> b: B";
by (fast_tac qsum_cs 1);
val QInrD = result();

(** <+> is itself injective... who cares?? **)

goal QPair.thy
    "u: A <+> B <-> (EX x. x:A & u=QInl(x)) | (EX y. y:B & u=QInr(y))";
by (fast_tac qsum_cs 1);
val qsum_iff = result();

goal QPair.thy "A <+> B <= C <+> D <-> A<=C & B<=D";
by (fast_tac qsum_cs 1);
val qsum_subset_iff = result();

goal QPair.thy "A <+> B = C <+> D <-> A=C & B=D";
by (simp_tac (ZF_ss addsimps [extension,qsum_subset_iff]) 1);
by (fast_tac ZF_cs 1);
val qsum_equal_iff = result();

(*** Eliminator -- qcase ***)

goalw QPair.thy qsum_defs "qcase(c, d, QInl(a)) = c(a)";
by (rtac (qsplit RS trans) 1);
by (rtac cond_0 1);
val qcase_QInl = result();

goalw QPair.thy qsum_defs "qcase(c, d, QInr(b)) = d(b)";
by (rtac (qsplit RS trans) 1);
by (rtac cond_1 1);
val qcase_QInr = result();

val major::prems = goal QPair.thy
    "[| u: A <+> B; \
\       !!x. x: A ==> c(x): C(QInl(x));   \
\       !!y. y: B ==> d(y): C(QInr(y)) \
\    |] ==> qcase(c,d,u) : C(u)";
by (rtac (major RS qsumE) 1);
by (ALLGOALS (etac ssubst));
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps
			    (prems@[qcase_QInl,qcase_QInr]))));
val qcase_type = result();

(** Rules for the Part primitive **)

goal QPair.thy "Part(A <+> B,QInl) = {QInl(x). x: A}";
by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1);
val Part_QInl = result();

goal QPair.thy "Part(A <+> B,QInr) = {QInr(y). y: B}";
by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1);
val Part_QInr = result();

goal QPair.thy "Part(A <+> B, %x.QInr(h(x))) = {QInr(y). y: Part(B,h)}";
by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1);
val Part_QInr2 = result();

goal QPair.thy "!!A B C. C <= A <+> B ==> Part(C,QInl) Un Part(C,QInr) = C";
by (rtac equalityI 1);
by (rtac Un_least 1);
by (rtac Part_subset 1);
by (rtac Part_subset 1);
by (fast_tac (ZF_cs addIs [PartI] addSEs [qsumE]) 1);
val Part_qsum_equality = result();