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(* Title: ZF/mono


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ID: $Id$


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Author: Lawrence C Paulson, Cambridge University Computer Laboratory


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Copyright 1993 University of Cambridge


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Monotonicity of various operations (for lattice properties see subset.ML)


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*)


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(** Replacement, in its various formulations **)


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(*Not easy to express monotonicity in P, since any "bigger" predicate


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would have to be singlevalued*)


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goal ZF.thy "!!A B. A<=B ==> Replace(A,P) <= Replace(B,P)";


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by (fast_tac ZF_cs 1);


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val Replace_mono = result();


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goal ZF.thy "!!A B. A<=B ==> {f(x). x:A} <= {f(x). x:B}";


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by (fast_tac ZF_cs 1);


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val RepFun_mono = result();


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goal ZF.thy "!!A B. A<=B ==> Pow(A) <= Pow(B)";


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by (fast_tac ZF_cs 1);


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val Pow_mono = result();


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goal ZF.thy "!!A B. A<=B ==> Union(A) <= Union(B)";


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by (fast_tac ZF_cs 1);


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val Union_mono = result();


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val prems = goal ZF.thy


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"[ A<=C; !!x. x:A ==> B(x)<=D(x) \


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\ ] ==> (UN x:A. B(x)) <= (UN x:C. D(x))";


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by (fast_tac (ZF_cs addIs (prems RL [subsetD])) 1);


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val UN_mono = result();


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(*Intersection is ANTImonotonic. There are TWO premises! *)


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goal ZF.thy "!!A B. [ A<=B; a:A ] ==> Inter(B) <= Inter(A)";


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by (fast_tac ZF_cs 1);


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val Inter_anti_mono = result();


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goal ZF.thy "!!C D. C<=D ==> cons(a,C) <= cons(a,D)";


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by (fast_tac ZF_cs 1);


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val cons_mono = result();


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goal ZF.thy "!!A B C D. [ A<=C; B<=D ] ==> A Un B <= C Un D";


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by (fast_tac ZF_cs 1);


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val Un_mono = result();


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goal ZF.thy "!!A B C D. [ A<=C; B<=D ] ==> A Int B <= C Int D";


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by (fast_tac ZF_cs 1);


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val Int_mono = result();


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goal ZF.thy "!!A B C D. [ A<=C; D<=B ] ==> AB <= CD";


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by (fast_tac ZF_cs 1);


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val Diff_mono = result();


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(** Standard products, sums and function spaces **)


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goal ZF.thy "!!A B C D. [ A<=C; ALL x:A. B(x) <= D(x) ] ==> \


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\ Sigma(A,B) <= Sigma(C,D)";


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by (fast_tac ZF_cs 1);


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val Sigma_mono_lemma = result();


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val Sigma_mono = ballI RSN (2,Sigma_mono_lemma);


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goalw Sum.thy sum_defs "!!A B C D. [ A<=C; B<=D ] ==> A+B <= C+D";


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by (REPEAT (ares_tac [subset_refl,Un_mono,Sigma_mono] 1));


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val sum_mono = result();


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(*Note that B>A and C>A are typically disjoint!*)


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goal ZF.thy "!!A B C. B<=C ==> A>B <= A>C";


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by (fast_tac (ZF_cs addIs [lam_type] addEs [Pi_lamE]) 1);


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val Pi_mono = result();


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goalw ZF.thy [lam_def] "!!A B. A<=B ==> Lambda(A,c) <= Lambda(B,c)";


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by (etac RepFun_mono 1);


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val lam_mono = result();


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(** Quineinspired ordered pairs, products, injections and sums **)


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goalw QPair.thy [QPair_def] "!!a b c d. [ a<=c; b<=d ] ==> <a;b> <= <c;d>";


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by (REPEAT (ares_tac [sum_mono] 1));


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val QPair_mono = result();


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goal QPair.thy "!!A B C D. [ A<=C; ALL x:A. B(x) <= D(x) ] ==> \


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\ QSigma(A,B) <= QSigma(C,D)";


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by (fast_tac (ZF_cs addIs [QSigmaI] addSEs [QSigmaE]) 1);


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val QSigma_mono_lemma = result();


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val QSigma_mono = ballI RSN (2,QSigma_mono_lemma);


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goalw QPair.thy [QInl_def] "!!a b. a<=b ==> QInl(a) <= QInl(b)";


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by (REPEAT (ares_tac [subset_refl RS QPair_mono] 1));


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val QInl_mono = result();


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goalw QPair.thy [QInr_def] "!!a b. a<=b ==> QInr(a) <= QInr(b)";


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by (REPEAT (ares_tac [subset_refl RS QPair_mono] 1));


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val QInr_mono = result();


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goal QPair.thy "!!A B C D. [ A<=C; B<=D ] ==> A <+> B <= C <+> D";


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by (fast_tac qsum_cs 1);


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val qsum_mono = result();


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(** Converse, domain, range, field **)


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goal ZF.thy "!!r s. r<=s ==> converse(r) <= converse(s)";


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by (fast_tac ZF_cs 1);


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val converse_mono = result();


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goal ZF.thy "!!r s. r<=s ==> domain(r)<=domain(s)";


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by (fast_tac ZF_cs 1);


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val domain_mono = result();


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val [prem] = goal ZF.thy "r <= Sigma(A,B) ==> domain(r) <= A";


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by (rtac (domain_subset RS (prem RS domain_mono RS subset_trans)) 1);


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val domain_rel_subset = result();


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goal ZF.thy "!!r s. r<=s ==> range(r)<=range(s)";


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by (fast_tac ZF_cs 1);


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val range_mono = result();


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val [prem] = goal ZF.thy "r <= A*B ==> range(r) <= B";


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by (rtac (range_subset RS (prem RS range_mono RS subset_trans)) 1);


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val range_rel_subset = result();


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goal ZF.thy "!!r s. r<=s ==> field(r)<=field(s)";


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by (fast_tac ZF_cs 1);


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val field_mono = result();


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goal ZF.thy "!!r A. r <= A*A ==> field(r) <= A";


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by (etac (field_mono RS subset_trans) 1);


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by (fast_tac ZF_cs 1);


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val field_rel_subset = result();


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(** Images **)


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val [prem1,prem2] = goal ZF.thy


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"[ !! x y. <x,y>:r ==> <x,y>:s; A<=B ] ==> r``A <= s``B";


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by (fast_tac (ZF_cs addIs [prem1, prem2 RS subsetD]) 1);


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val image_pair_mono = result();


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val [prem1,prem2] = goal ZF.thy


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"[ !! x y. <x,y>:r ==> <x,y>:s; A<=B ] ==> r``A <= s``B";


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by (fast_tac (ZF_cs addIs [prem1, prem2 RS subsetD]) 1);


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val vimage_pair_mono = result();


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goal ZF.thy "!!r s. [ r<=s; A<=B ] ==> r``A <= s``B";


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by (fast_tac ZF_cs 1);


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val image_mono = result();


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goal ZF.thy "!!r s. [ r<=s; A<=B ] ==> r``A <= s``B";


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by (fast_tac ZF_cs 1);


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val vimage_mono = result();


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val [sub,PQimp] = goal ZF.thy


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"[ A<=B; !!x. x:A ==> P(x) > Q(x) ] ==> Collect(A,P) <= Collect(B,Q)";


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by (fast_tac (ZF_cs addIs [sub RS subsetD, PQimp RS mp]) 1);


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val Collect_mono = result();


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(** Monotonicity of implications  some could go to FOL **)


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goal ZF.thy "!!A B x. A<=B ==> x:A > x:B";


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by (rtac impI 1);


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by (etac subsetD 1);


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by (assume_tac 1);


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val in_mono = result();


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goal IFOL.thy "!!P1 P2 Q1 Q2. [ P1>Q1; P2>Q2 ] ==> (P1&P2) > (Q1&Q2)";


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by (Int.fast_tac 1);


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val conj_mono = result();


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goal IFOL.thy "!!P1 P2 Q1 Q2. [ P1>Q1; P2>Q2 ] ==> (P1P2) > (Q1Q2)";


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by (Int.fast_tac 1);


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val disj_mono = result();


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goal IFOL.thy "!!P1 P2 Q1 Q2.[ Q1>P1; P2>Q2 ] ==> (P1>P2)>(Q1>Q2)";


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by (Int.fast_tac 1);


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val imp_mono = result();


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goal IFOL.thy "P>P";


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by (rtac impI 1);


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by (assume_tac 1);


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val imp_refl = result();


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val [PQimp] = goal IFOL.thy


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"[ !!x. P(x) > Q(x) ] ==> (EX x.P(x)) > (EX x.Q(x))";


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by (fast_tac (FOL_cs addIs [PQimp RS mp]) 1);


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val ex_mono = result();


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val [PQimp] = goal IFOL.thy


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"[ !!x. P(x) > Q(x) ] ==> (ALL x.P(x)) > (ALL x.Q(x))";


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by (fast_tac (FOL_cs addIs [PQimp RS mp]) 1);


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val all_mono = result();
