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(* Title: HOL/While
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 2000 TU Muenchen
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*)
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goalw_cterm [] (cterm_of (sign_of thy)
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(HOLogic.mk_Trueprop (hd while_aux.tcs)));
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br wf_same_fstI 1;
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br wf_same_fstI 1;
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by (asm_full_simp_tac (simpset() addsimps [wf_iff_no_infinite_down_chain]) 1);
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by(Blast_tac 1);
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val while_aux_tc = result();
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Goal
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"while_aux(b,c,s) = (if ? f. f 0 = s & (!i. b(f i) & c(f i) = f(i+1)) \
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\ then arbitrary \
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\ else (if b s then while_aux(b,c,c s) else s))";
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by(rtac (while_aux_tc RS (hd while_aux.simps) RS trans) 1);
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by(simp_tac (simpset() addsimps [same_fst_def]) 1);
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br refl 1;
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qed "while_aux_unfold";
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(*** The recursion equation for while: directly executable! ***)
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Goalw [while_def] "while b c s = (if b s then while b c (c s) else s)";
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by(rtac (while_aux_unfold RS trans) 1);
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by (Auto_tac);
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by(stac while_aux_unfold 1);
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by(Asm_full_simp_tac 1);
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by(Clarify_tac 1);
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by(eres_inst_tac [("x","%i. f(Suc i)")] allE 1);
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by(Blast_tac 1);
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qed "while_unfold";
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(*** The proof rule for while; P is the invariant ***)
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val [prem1,prem2,prem3] = Goal
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"[| !!s. [| P s; b s |] ==> P(c s); \
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\ !!s. [| P s; ~b s |] ==> Q s; \
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\ wf{(t,s). P s & b s & t = c s} |] \
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\ ==> P s --> Q(while b c s)";
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9747
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by(induct_thm_tac (prem3 RS wf_induct) "s" 1);
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9448
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by(Asm_full_simp_tac 1);
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by(Clarify_tac 1);
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by(stac while_unfold 1);
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by(asm_full_simp_tac (simpset() addsimps [prem1,prem2]) 1);
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qed_spec_mp "while_rule";
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(*** An application: computation of the lfp on finite sets via iteration ***)
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Goal
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"[| mono f; finite U; f U = U |] \
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\ ==> lfp f = fst(while (%(A,fA). A~=fA) (%(A,fA). (fA, f fA)) ({},f{}))";
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by(res_inst_tac [("P","%(A,B).(A <= U & B = f A & A <= B & B <= lfp f)")]
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while_rule 1);
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10186
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by(stac lfp_unfold 1);
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9448
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ba 1;
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by(Clarsimp_tac 1);
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by(blast_tac (claset() addDs [monoD]) 1);
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by(fast_tac (claset() addSIs [lfp_lowerbound] addss simpset()) 1);
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by(res_inst_tac [("r","((Pow U <*> UNIV) <*> (Pow U <*> UNIV)) Int \
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\ inv_image finite_psubset (op - U o fst)")]
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wf_subset 1);
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by(blast_tac (claset() addIs
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[wf_finite_psubset,Int_lower2 RSN (2,wf_subset)]) 1);
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by(clarsimp_tac (claset(),simpset() addsimps
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[inv_image_def,finite_psubset_def,order_less_le]) 1);
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by(blast_tac (claset() addSIs [finite_Diff] addDs [monoD]) 1);
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10186
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by(stac lfp_unfold 1);
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9448
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ba 1;
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by(asm_simp_tac (simpset() addsimps [monoD]) 1);
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qed "lfp_conv_while";
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(*** An example; requires integers
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Goal "{f n|n. A n | B n} = {f n|n. A n} Un {f n|n. B n}";
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by(Blast_tac 1);
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qed "lemma";
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Goal "P(lfp (%N::int set. {#0} Un {(n + #2) mod #6 |n. n:N})) = P{#0,#4,#2}";
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by(stac (read_instantiate [("U","{#0,#1,#2,#3,#4,#5}")] lfp_conv_while) 1);
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br monoI 1;
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by(Blast_tac 1);
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by(Simp_tac 1);
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by(simp_tac (simpset() addsimps [lemma,set_eq_subset]) 1);
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(* The fixpoint computation is performed purely by rewriting: *)
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by(simp_tac (simpset() addsimps [while_unfold,lemma,set_eq_subset]
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delsimps [subset_empty]) 1);
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result();
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***)
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