author  clasohm 
Tue, 30 Jan 1996 13:42:57 +0100  
changeset 1461  6bcb44e4d6e5 
parent 803  4c8333ab3eae 
child 1956  589af052bcd4 
permissions  rwrr 
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(* Title: ZF/Finite.ML 
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ID: $Id$ 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1994 University of Cambridge 
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Finite powerset operator; finite function space 
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prove X:Fin(A) ==> X < nat 

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prove: b: Fin(A) ==> inj(b,b)<=surj(b,b) 

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*) 

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open Finite; 

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(*** Finite powerset operator ***) 
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goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)"; 
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by (rtac lfp_mono 1); 

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by (REPEAT (rtac Fin.bnd_mono 1)); 

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by (REPEAT (ares_tac (Pow_mono::basic_monos) 1)); 

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qed "Fin_mono"; 
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(* A : Fin(B) ==> A <= B *) 

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val FinD = Fin.dom_subset RS subsetD RS PowD; 

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(** Induction on finite sets **) 

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(*Discharging x~:y entails extra work*) 

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val major::prems = goal Finite.thy 

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"[ b: Fin(A); \ 

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\ P(0); \ 

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\ !!x y. [ x: A; y: Fin(A); x~:y; P(y) ] ==> P(cons(x,y)) \ 

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\ ] ==> P(b)"; 

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by (rtac (major RS Fin.induct) 1); 

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by (excluded_middle_tac "a:b" 2); 

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by (etac (cons_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*) 
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by (REPEAT (ares_tac prems 1)); 
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qed "Fin_induct"; 
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(** Simplification for Fin **) 

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val Fin_ss = arith_ss addsimps Fin.intrs; 

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(*The union of two finite sets is finite.*) 

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val major::prems = goal Finite.thy 

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"[ b: Fin(A); c: Fin(A) ] ==> b Un c : Fin(A)"; 

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by (rtac (major RS Fin_induct) 1); 

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by (ALLGOALS (asm_simp_tac (Fin_ss addsimps (prems@[Un_0, Un_cons])))); 

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qed "Fin_UnI"; 
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(*The union of a set of finite sets is finite.*) 

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val [major] = goal Finite.thy "C : Fin(Fin(A)) ==> Union(C) : Fin(A)"; 

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by (rtac (major RS Fin_induct) 1); 

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by (ALLGOALS (asm_simp_tac (Fin_ss addsimps [Union_0, Union_cons, Fin_UnI]))); 

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qed "Fin_UnionI"; 
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(*Every subset of a finite set is finite.*) 

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goal Finite.thy "!!b A. b: Fin(A) ==> ALL z. z<=b > z: Fin(A)"; 

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by (etac Fin_induct 1); 

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by (simp_tac (Fin_ss addsimps [subset_empty_iff]) 1); 

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by (asm_simp_tac (ZF_ss addsimps subset_cons_iff::distrib_rews) 1); 
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by (safe_tac ZF_cs); 

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by (eres_inst_tac [("b","z")] (cons_Diff RS subst) 1); 

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by (asm_simp_tac Fin_ss 1); 

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qed "Fin_subset_lemma"; 
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goal Finite.thy "!!c b A. [ c<=b; b: Fin(A) ] ==> c: Fin(A)"; 

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by (REPEAT (ares_tac [Fin_subset_lemma RS spec RS mp] 1)); 

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qed "Fin_subset"; 
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val major::prems = goal Finite.thy 

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"[ c: Fin(A); b: Fin(A); \ 
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\ P(b); \ 

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\ !!x y. [ x: A; y: Fin(A); x:y; P(y) ] ==> P(y{x}) \ 
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\ ] ==> c<=b > P(bc)"; 

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by (rtac (major RS Fin_induct) 1); 

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by (rtac (Diff_cons RS ssubst) 2); 

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by (ALLGOALS (asm_simp_tac (Fin_ss addsimps (prems@[Diff_0, cons_subset_iff, 

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Diff_subset RS Fin_subset])))); 
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qed "Fin_0_induct_lemma"; 
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val prems = goal Finite.thy 

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"[ b: Fin(A); \ 
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\ P(b); \ 

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\ !!x y. [ x: A; y: Fin(A); x:y; P(y) ] ==> P(y{x}) \ 
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\ ] ==> P(0)"; 

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by (rtac (Diff_cancel RS subst) 1); 

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by (rtac (Fin_0_induct_lemma RS mp) 1); 

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by (REPEAT (ares_tac (subset_refl::prems) 1)); 

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qed "Fin_0_induct"; 
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(*Functions from a finite ordinal*) 

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val prems = goal Finite.thy "n: nat ==> n>A <= Fin(nat*A)"; 

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by (nat_ind_tac "n" prems 1); 

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by (simp_tac (ZF_ss addsimps [Pi_empty1, Fin.emptyI, subset_iff, cons_iff]) 1); 

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by (asm_simp_tac (ZF_ss addsimps [succ_def, mem_not_refl RS cons_fun_eq]) 1); 

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by (fast_tac (ZF_cs addSIs [Fin.consI]) 1); 

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qed "nat_fun_subset_Fin"; 
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(*** Finite function space ***) 

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goalw Finite.thy FiniteFun.defs 

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"!!A B C D. [ A<=C; B<=D ] ==> A > B <= C > D"; 

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by (rtac lfp_mono 1); 

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by (REPEAT (rtac FiniteFun.bnd_mono 1)); 

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by (REPEAT (ares_tac (Fin_mono::Sigma_mono::basic_monos) 1)); 

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qed "FiniteFun_mono"; 
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goal Finite.thy "!!A B. A<=B ==> A > A <= B > B"; 

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

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qed "FiniteFun_mono1"; 
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goal Finite.thy "!!h. h: A >B ==> h: domain(h) > B"; 

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by (etac FiniteFun.induct 1); 

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by (simp_tac (ZF_ss addsimps [empty_fun, domain_0]) 1); 

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by (asm_simp_tac (ZF_ss addsimps [fun_extend3, domain_cons]) 1); 

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qed "FiniteFun_is_fun"; 
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goal Finite.thy "!!h. h: A >B ==> domain(h) : Fin(A)"; 

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by (etac FiniteFun.induct 1); 

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by (simp_tac (Fin_ss addsimps [domain_0]) 1); 

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by (asm_simp_tac (Fin_ss addsimps [domain_cons]) 1); 

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qed "FiniteFun_domain_Fin"; 
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changed useless "qed" calls for lemmas back to uses of "result",
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diff
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bind_thm ("FiniteFun_apply_type", FiniteFun_is_fun RS apply_type); 
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(*Every subset of a finite function is a finite function.*) 

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goal Finite.thy "!!b A. b: A>B ==> ALL z. z<=b > z: A>B"; 

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by (etac FiniteFun.induct 1); 

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by (simp_tac (ZF_ss addsimps subset_empty_iff::FiniteFun.intrs) 1); 

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by (asm_simp_tac (ZF_ss addsimps subset_cons_iff::distrib_rews) 1); 

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by (safe_tac ZF_cs); 

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by (eres_inst_tac [("b","z")] (cons_Diff RS subst) 1); 

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by (dtac (spec RS mp) 1 THEN assume_tac 1); 

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by (fast_tac (ZF_cs addSIs FiniteFun.intrs) 1); 

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qed "FiniteFun_subset_lemma"; 
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goal Finite.thy "!!c b A. [ c<=b; b: A>B ] ==> c: A>B"; 

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by (REPEAT (ares_tac [FiniteFun_subset_lemma RS spec RS mp] 1)); 

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qed "FiniteFun_subset"; 
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