--- a/src/ZF/qpair.ML Sat Apr 05 16:18:58 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,281 +0,0 @@
-(* 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();