--- a/src/ZF/mono.ML Fri Jun 28 20:01:09 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,191 +0,0 @@
-(* Title: ZF/mono
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1993 University of Cambridge
-
-Monotonicity of various operations (for lattice properties see subset.ML)
-*)
-
-(** Replacement, in its various formulations **)
-
-(*Not easy to express monotonicity in P, since any "bigger" predicate
- would have to be single-valued*)
-Goal "A<=B ==> Replace(A,P) <= Replace(B,P)";
-by (blast_tac (claset() addSEs [ReplaceE]) 1);
-qed "Replace_mono";
-
-Goal "A<=B ==> {f(x). x:A} <= {f(x). x:B}";
-by (Blast_tac 1);
-qed "RepFun_mono";
-
-Goal "A<=B ==> Pow(A) <= Pow(B)";
-by (Blast_tac 1);
-qed "Pow_mono";
-
-Goal "A<=B ==> Union(A) <= Union(B)";
-by (Blast_tac 1);
-qed "Union_mono";
-
-val prems = Goal
- "[| A<=C; !!x. x:A ==> B(x)<=D(x) \
-\ |] ==> (UN x:A. B(x)) <= (UN x:C. D(x))";
-by (blast_tac (claset() addIs (prems RL [subsetD])) 1);
-qed "UN_mono";
-
-(*Intersection is ANTI-monotonic. There are TWO premises! *)
-Goal "[| A<=B; a:A |] ==> Inter(B) <= Inter(A)";
-by (Blast_tac 1);
-qed "Inter_anti_mono";
-
-Goal "C<=D ==> cons(a,C) <= cons(a,D)";
-by (Blast_tac 1);
-qed "cons_mono";
-
-Goal "[| A<=C; B<=D |] ==> A Un B <= C Un D";
-by (Blast_tac 1);
-qed "Un_mono";
-
-Goal "[| A<=C; B<=D |] ==> A Int B <= C Int D";
-by (Blast_tac 1);
-qed "Int_mono";
-
-Goal "[| A<=C; D<=B |] ==> A-B <= C-D";
-by (Blast_tac 1);
-qed "Diff_mono";
-
-(** Standard products, sums and function spaces **)
-
-Goal "[| A<=C; ALL x:A. B(x) <= D(x) |] ==> Sigma(A,B) <= Sigma(C,D)";
-by (Blast_tac 1);
-qed "Sigma_mono_lemma";
-bind_thm ("Sigma_mono", ballI RSN (2,Sigma_mono_lemma));
-
-Goalw sum_defs "[| A<=C; B<=D |] ==> A+B <= C+D";
-by (REPEAT (ares_tac [subset_refl,Un_mono,Sigma_mono] 1));
-qed "sum_mono";
-
-(*Note that B->A and C->A are typically disjoint!*)
-Goal "B<=C ==> A->B <= A->C";
-by (blast_tac (claset() addIs [lam_type] addEs [Pi_lamE]) 1);
-qed "Pi_mono";
-
-Goalw [lam_def] "A<=B ==> Lambda(A,c) <= Lambda(B,c)";
-by (etac RepFun_mono 1);
-qed "lam_mono";
-
-(** Quine-inspired ordered pairs, products, injections and sums **)
-
-Goalw [QPair_def] "[| a<=c; b<=d |] ==> <a;b> <= <c;d>";
-by (REPEAT (ares_tac [sum_mono] 1));
-qed "QPair_mono";
-
-Goal "[| A<=C; ALL x:A. B(x) <= D(x) |] ==> \
-\ QSigma(A,B) <= QSigma(C,D)";
-by (Blast_tac 1);
-qed "QSigma_mono_lemma";
-bind_thm ("QSigma_mono", ballI RSN (2,QSigma_mono_lemma));
-
-Goalw [QInl_def] "a<=b ==> QInl(a) <= QInl(b)";
-by (REPEAT (ares_tac [subset_refl RS QPair_mono] 1));
-qed "QInl_mono";
-
-Goalw [QInr_def] "a<=b ==> QInr(a) <= QInr(b)";
-by (REPEAT (ares_tac [subset_refl RS QPair_mono] 1));
-qed "QInr_mono";
-
-Goal "[| A<=C; B<=D |] ==> A <+> B <= C <+> D";
-by (Blast_tac 1);
-qed "qsum_mono";
-
-
-(** Converse, domain, range, field **)
-
-Goal "r<=s ==> converse(r) <= converse(s)";
-by (Blast_tac 1);
-qed "converse_mono";
-
-Goal "r<=s ==> domain(r)<=domain(s)";
-by (Blast_tac 1);
-qed "domain_mono";
-
-bind_thm ("domain_rel_subset", [domain_mono, domain_subset] MRS subset_trans);
-
-Goal "r<=s ==> range(r)<=range(s)";
-by (Blast_tac 1);
-qed "range_mono";
-
-bind_thm ("range_rel_subset", [range_mono, range_subset] MRS subset_trans);
-
-Goal "r<=s ==> field(r)<=field(s)";
-by (Blast_tac 1);
-qed "field_mono";
-
-Goal "r <= A*A ==> field(r) <= A";
-by (etac (field_mono RS subset_trans) 1);
-by (Blast_tac 1);
-qed "field_rel_subset";
-
-
-(** Images **)
-
-val [prem1,prem2] = Goal
- "[| !! x y. <x,y>:r ==> <x,y>:s; A<=B |] ==> r``A <= s``B";
-by (blast_tac (claset() addIs [prem1, prem2 RS subsetD]) 1);
-qed "image_pair_mono";
-
-val [prem1,prem2] = Goal
- "[| !! x y. <x,y>:r ==> <x,y>:s; A<=B |] ==> r-``A <= s-``B";
-by (blast_tac (claset() addIs [prem1, prem2 RS subsetD]) 1);
-qed "vimage_pair_mono";
-
-Goal "[| r<=s; A<=B |] ==> r``A <= s``B";
-by (Blast_tac 1);
-qed "image_mono";
-
-Goal "[| r<=s; A<=B |] ==> r-``A <= s-``B";
-by (Blast_tac 1);
-qed "vimage_mono";
-
-val [sub,PQimp] = Goal
- "[| A<=B; !!x. x:A ==> P(x) --> Q(x) |] ==> Collect(A,P) <= Collect(B,Q)";
-by (blast_tac (claset() addIs [sub RS subsetD, PQimp RS mp]) 1);
-qed "Collect_mono";
-
-(** Monotonicity of implications -- some could go to FOL **)
-
-Goal "A<=B ==> x:A --> x:B";
-by (Blast_tac 1);
-qed "in_mono";
-
-goal IFOL.thy "!!P1 P2 Q1 Q2. [| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)";
-by (IntPr.fast_tac 1);
-qed "conj_mono";
-
-goal IFOL.thy "!!P1 P2 Q1 Q2. [| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)";
-by (IntPr.fast_tac 1);
-qed "disj_mono";
-
-goal IFOL.thy "!!P1 P2 Q1 Q2.[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)";
-by (IntPr.fast_tac 1);
-qed "imp_mono";
-
-goal IFOL.thy "P-->P";
-by (rtac impI 1);
-by (assume_tac 1);
-qed "imp_refl";
-
-val [PQimp] = goal IFOL.thy
- "[| !!x. P(x) --> Q(x) |] ==> (EX x. P(x)) --> (EX x. Q(x))";
-by IntPr.safe_tac;
-by (etac (PQimp RS mp RS exI) 1);
-qed "ex_mono";
-
-val [PQimp] = goal IFOL.thy
- "[| !!x. P(x) --> Q(x) |] ==> (ALL x. P(x)) --> (ALL x. Q(x))";
-by IntPr.safe_tac;
-by (etac (spec RS (PQimp RS mp)) 1);
-qed "all_mono";
-
-(*Used in intr_elim.ML and in individual datatype definitions*)
-bind_thms ("basic_monos",
- [subset_refl, imp_refl, disj_mono, conj_mono, ex_mono, Collect_mono, Part_mono, in_mono]);