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src/HOL/BCV/DFAimpl.ML
changeset 7626 5997f35954d7
child 7961 422ac6888c7f 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BCV/DFAimpl.ML	Tue Sep 28 16:36:12 1999 +0200
@@ -0,0 +1,336 @@
+(*  Title:      HOL/BCV/DFAimpl.thy
+    ID:         $Id$
+    Author:     Tobias Nipkow
+    Copyright   1999 TUM
+*)
+
+(** merges **)
+
+Goal "!sos. size(merges t ps sos) = size sos";
+by(induct_tac "ps" 1);
+by(Auto_tac);
+qed_spec_mp "length_merges";
+Addsimps [length_merges];
+
+Goal
+ "!xs. xs : listsn n (option A) --> x : A --> (!p : set ps. p<n) --> \
+\      semilat A --> merges x ps xs : listsn n (option A)";
+by(induct_tac "ps" 1);
+by(Auto_tac);
+qed_spec_mp "merges_preserves_type";
+Addsimps [merges_preserves_type];
+(*AddSIs [merges_preserves_type];*)
+
+Goal "semilat A ==> !xs. xs : listsn n (option A) --> x:A --> (!p:set ps. p<n) \
+\ --> xs <= merges x ps xs";
+by(induct_tac "ps" 1);
+ by(Simp_tac 1);
+by(Simp_tac 1);
+by(Clarify_tac 1);
+br order_trans 1;
+ be list_update_incr 1;
+   by(Blast_tac 2);
+  ba 2;
+ be (Some_in_option RS iffD2) 1;
+by(blast_tac (claset() addSIs [listsnE_set]
+          addIs [semilat_plus,listsnE_length RS nth_in]) 1);
+qed_spec_mp "merges_incr";
+
+(* is redundant but useful *)
+Goal "[| xs!i = Some x; xs : listsn n (option A); i < n |] ==> x:A";
+bd listsnE_nth_in 1;
+ba 1;
+by(Asm_full_simp_tac 1);
+qed "listsn_optionE_in";
+(*Addsimps [listsn_optionE_in];*)
+
+Goal
+ "[| semilat L |] ==> \
+\ (!xs. xs : listsn n (option L) --> x:L --> (!p:set ps. p<n) --> \
+\      (merges x ps xs = xs) = (!p:set ps. Some x <= xs!p))";
+by(induct_tac "ps" 1);
+ by(Asm_simp_tac 1);
+by(Clarsimp_tac 1);
+by(rename_tac "p ps xs" 1);
+br iffI 1;
+ br context_conjI 1;
+  by(subgoal_tac "xs[p := Some x + xs!p] <= xs" 1);
+   by(EVERY[etac subst 2, rtac merges_incr 2]);
+      ba 2;
+     by(Force_tac 2);
+    ba 2;
+   ba 2;
+  by(exhaust_tac "xs!p" 1);
+   by(asm_full_simp_tac (simpset() addsimps [list_update_le_conv]) 1);
+  by(asm_full_simp_tac (simpset() addsimps
+      [list_update_le_conv,listsn_optionE_in]) 1);
+ by(Clarify_tac 1);
+ by(rotate_tac ~3 1);
+ by(asm_full_simp_tac (simpset() addsimps
+      [le_iff_plus_unchanged RS iffD1,listsn_optionE_in,
+       list_update_same_conv RS iffD2]) 1);
+by(Clarify_tac 1);
+by(asm_simp_tac (simpset() addsimps
+      [le_iff_plus_unchanged RS iffD1,listsn_optionE_in,
+       list_update_same_conv RS iffD2]) 1);
+qed_spec_mp "merges_same_conv";
+
+
+Goalw [le_list,plus_option]
+ "xs <= ys --> xs : listsn n (option L) --> ys : listsn n (option L) --> \
+\ p < n --> ys!p = Some y --> x <= y --> x : L --> semilat L --> \
+\ xs[p := Some x + xs!p] <= ys";
+by(simp_tac (simpset()  addsimps [nth_list_update]
+                        addsplits [option.split]) 1);
+by(Clarify_tac 1);
+by(rotate_tac 3 1);
+by(force_tac (claset() addEs [listsn_optionE_in] addIs [semilat_lub],
+              simpset()) 1);
+qed_spec_mp "list_update_le_listI";
+
+Goal
+ "[| semilat(L); t:L; tos : listsn n (option L); \
+\    !p. p:set ps --> Some t <= tos!p; \
+\    !p. p:set ps --> p<n |] ==> \
+\ set qs <= set ps  --> \
+\ (!sos. sos : listsn n (option L) & sos <= tos --> merges t qs sos <= tos)";
+by(induct_tac "qs" 1);
+ by(Asm_simp_tac 1);
+by(force_tac (claset(),simpset() addsimps [list_update_le_listI]) 1);
+val lemma = result();
+
+Goal
+ "[| semilat(L); t:L; \
+\    !p. p:set ps --> Some t <= tos!p; \
+\    !p. p:set ps --> p<n; \
+\    sos <= tos; sos : listsn n (option L); tos : listsn n (option L) |] \
+\ ==> merges t ps sos <= tos";
+by(blast_tac (claset() addDs [lemma]) 1);
+qed "merges_pres_le_ub";
+
+
+(** next **)
+
+Goalw [is_next_def]
+ "[| is_next next; next step succs sos = None; succs_bounded succs n; \
+\    sos : listsn n S |] ==> \
+\ ? p<n. ? s. sos!p = Some s & step p s = None";
+by(subgoal_tac "n=size sos" 1);
+by(Blast_tac 1);
+by(Asm_simp_tac 1);
+qed "next_None";
+
+Goalw [is_next_def]
+ "[| is_next next; succs_bounded succs n; sos : listsn n S |] ==> \
+\ next step succs sos = Some sos --> \
+\ (!p<n. !s. sos!p = Some s --> (? t. \
+\         step p s = Some(t) & merges t (succs p) sos = sos))";
+by(subgoal_tac "n=size sos" 1);
+by(Blast_tac 1);
+by(Asm_simp_tac 1);
+qed "next_Some1";
+
+Goalw [is_next_def]
+ "[| is_next next; next step succs sos = Some sos'; sos' ~= sos; \
+\    succs_bounded succs n; sos : listsn n S |] ==> \
+\ ? p<n. ? s. sos!p = Some s & (? t. \
+\     step p s = Some(t) & merges t (succs p) sos = sos')";
+by(subgoal_tac "n=size sos" 1);
+by(Blast_tac 1);
+by(Asm_simp_tac 1);
+qed "next_Some2";
+
+Goal
+ "[| is_next next; succs_bounded succs n; sos : listsn n S |] ==> \
+\ (next step succs sos = Some sos) = \
+\ (!p<n. !s. sos!p = Some s --> (? t. \
+\         step p s = Some(t) & merges t (succs p) sos = sos))";
+br iffI 1;
+ by(asm_simp_tac (simpset() addsimps [next_Some1]) 1);
+by(exhaust_tac "next step succs sos" 1);
+ by(best_tac (claset() addSDs [next_None] addss simpset()) 1);
+by(rename_tac "sos'" 1);
+by(case_tac "sos' = sos" 1);
+ by(Blast_tac 1);
+by(best_tac (claset() addSDs [next_Some2] addss simpset()) 1);
+qed "next_Some1_eq";
+
+Addsimps [next_Some1_eq];
+
+Goalw [step_pres_type_def]
+ "[| step_pres_type step n L; s:L; p<n; step p s = Some(t) |] ==> t:L";
+by(Blast_tac 1);
+qed "step_pres_typeD";
+
+Goalw [succs_bounded_def]
+ "[| succs_bounded succs n; p < n; q : set(succs p) |] ==> q < n";
+by(Blast_tac 1);
+qed "succs_boundedD";
+
+Goal
+ "[| is_next next; semilat A; \
+\    step_pres_type step n A; succs_bounded succs n;\
+\    sos : listsn n (option A) |] ==> \
+\ next step succs sos : option (listsn n (option A))";
+by(exhaust_tac "next step succs sos" 1);
+ by(ALLGOALS Asm_simp_tac);
+by(rename_tac "sos'" 1);
+by(case_tac "sos' = sos" 1);
+ by(Asm_simp_tac 1);
+by(blast_tac (claset() addIs [step_pres_typeD,succs_boundedD,listsn_optionE_in] addSIs [merges_preserves_type] addDs [next_Some2]) 1);
+qed_spec_mp "next_preserves_type";
+
+Goal
+ "[| is_next next; semilat A; \
+\    step_pres_type step n A; succs_bounded succs n; \
+\    next step succs xs = Some ys; xs : listsn n (option A) |] ==> xs <= ys";
+by(case_tac "ys = xs" 1);
+ by(Asm_full_simp_tac 1);
+by(blast_tac (claset() addSIs [merges_incr] addIs [listsn_optionE_in]
+       addDs [step_pres_typeD,succs_boundedD,next_Some2]) 1);
+qed_spec_mp "next_incr";
+
+val lemma = (Unify.trace_bound, Unify.search_bound);
+Unify.trace_bound := 50;
+Unify.search_bound := 50;
+
+Goalw [is_next_def]
+ "is_next (%step succs sos. itnext (size sos) step succs sos)";
+by(Clarify_tac 1);
+by(etac thin_rl 1);
+by(res_inst_tac [("n","length sos")] nat_induct 1);
+ by(Asm_full_simp_tac 1);
+by(asm_full_simp_tac (simpset() addsimps [less_Suc_eq,Let_def]
+                                addsplits [option.split])1);
+by(Blast_tac 1);
+qed "is_next_itnext";
+
+Unify.trace_bound := !(fst lemma);
+Unify.search_bound := !(snd lemma);
+
+(** fix step **)
+
+Goalw [step_mono_None_def]
+ "[| step_mono_None step n L; s : L; p < n; s <= t; step p s = None |] ==> \
+\ step p t = None";
+by(Blast_tac 1);
+qed "step_mono_NoneD";
+
+Goalw [step_mono_def]
+ "[| step_mono step n L; s : L; p < n; s <= t; step p s = Some(u) |] ==> \
+\ !v. step p t = Some(v) --> u <= v";
+by(Blast_tac 1);
+qed "step_monoD";
+
+Goalw [stable_def]
+"[| is_next next; semilat L; sos : listsn n (option L); \
+\   step_pres_type step n L; succs_bounded succs n |] \
+\ ==> (next step succs sos = Some sos) = (!p<n. stable step succs p sos)";
+by(Asm_simp_tac 1);
+br iffI 1;
+ by(Clarify_tac 1);
+ by(etac allE 1 THEN mp_tac 1);
+ by(etac allE 1 THEN mp_tac 1);
+ by(Clarify_tac 1);
+ bd(merges_same_conv RS iffD1)1;
+     ba 4;
+    ba 1;
+   by(blast_tac (claset() addIs [step_pres_typeD,listsn_optionE_in]) 1);
+  by(blast_tac (claset() addIs [succs_boundedD,listsn_optionE_in]) 1);
+ by(Blast_tac 1);
+by(Clarify_tac 1);
+by(etac allE 1 THEN mp_tac 1);
+by(Asm_full_simp_tac 1);
+by(Clarify_tac 1);
+by(Asm_simp_tac 1);
+by(blast_tac (claset() addSIs [merges_same_conv RS iffD2]
+     addIs [step_pres_typeD,succs_boundedD,listsn_optionE_in]) 1);
+qed "fixpoint_next_iff_stable";
+
+Goal
+ "[| semilat L; is_next next; succs_bounded succs n; \
+\    step_pres_type step n L; step_mono_None step n L; step_mono step n L; \
+\    tos:listsn n (option L); next step succs tos = Some tos; \
+\    sos:listsn n (option L); sos <= tos |] \
+\ ==> ? sos'. next step succs sos = Some sos' & sos' <= tos";
+by(subgoal_tac
+   "!p<n. !s. sos!p = Some s --> (? u. \
+\             step p s = Some(u) & (!q:set(succs p). Some u<=tos!q))" 1);
+ by(exhaust_tac "next step succs sos" 1);
+  bd next_None 1;
+     ba 1;
+    ba 1;
+   ba 1;
+  by(Force_tac 1);
+ by(rename_tac "sos'" 1);
+ by(case_tac "sos' = sos" 1);
+  by(Blast_tac 1);
+ bd next_Some2 1;
+    by(EVERY1[atac, atac, atac, atac]);
+ by(Clarify_tac 1);
+ by(etac allE 1 THEN mp_tac 1);
+ by(etac allE 1 THEN mp_tac 1);
+ by(Clarify_tac 1);
+ by(EVERY1[rtac exI, rtac conjI, atac]);
+ br merges_pres_le_ub 1;
+       ba 1;
+      by(blast_tac (claset() addIs [step_pres_typeD,listsn_optionE_in]) 1);
+     by(Asm_full_simp_tac 1);
+    by(blast_tac (claset() addIs [succs_boundedD,listsn_optionE_in]) 1);
+   by(REPEAT(atac 1));
+by(Clarify_tac 1);
+by(exhaust_tac "tos!p" 1);
+ by(force_tac (claset() addDs [le_listD],simpset()) 1);
+by(rename_tac "t" 1);
+by(subgoal_tac "s <= t" 1);
+ by(force_tac (claset() addDs [le_listD],simpset()) 2);
+by(exhaust_tac "step p s" 1);
+ bd step_mono_NoneD 1;
+     ba 4;
+    by(blast_tac (claset() addIs [listsn_optionE_in]) 1);
+   ba 1;
+  ba 1;
+ by(Force_tac 1);
+bd step_monoD 1;
+    ba 4;
+   by(blast_tac (claset() addIs [listsn_optionE_in]) 1);
+  ba 1;
+ ba 1;
+by(Asm_full_simp_tac 1);
+by(etac allE 1 THEN mp_tac 1);
+by(etac allE 1 THEN mp_tac 1);
+by(Clarify_tac 1);
+by(Asm_full_simp_tac 1);
+by(forward_tac[merges_same_conv RS iffD1]1);
+    ba 4;
+   ba 1;
+  by(blast_tac (claset() addIs [step_pres_typeD,listsn_optionE_in]) 1);
+ by(blast_tac (claset() addIs [succs_boundedD,listsn_optionE_in]) 1);
+by(blast_tac (claset() addIs [order_trans]) 1);
+val lemma = result();
+
+Goalw [is_dfa_def]
+ "[| semilat L; acc L; is_next next; \
+\    step_pres_type step n L; succs_bounded succs n; \
+\    step_mono_None step n L; step_mono step n L |] ==> \
+\ is_dfa (%sos. fix(next step succs, sos)) step succs n L";
+by(Clarify_tac 1);
+by(stac fix_iff_has_fixpoint 1);
+     by(etac (acc_option RS acc_listsn) 1);
+    by(blast_tac (claset() addIs [lemma]) 1);
+   by(blast_tac (claset() addIs [next_preserves_type]) 1);
+  by(blast_tac (claset() addIs [next_incr]) 1);
+ ba 1;
+by(asm_simp_tac (simpset() delsimps [next_Some1_eq] addsimps [fixpoint_next_iff_stable]) 1);
+qed "is_dfa_fix_next";
+
+Goal
+ "[| semilat L; acc L; is_next next; \
+\    step_pres_type step n L; succs_bounded succs n; \
+\    step_mono_None step n L; step_mono step n L; \
+\    wti_is_fix_step step wti succs n L; \
+\    sos : listsn n (option L) |] ==> \
+\ fix(next step succs, sos) = \
+\ (? tos:listsn n (option L). sos<=tos & welltyping wti tos)";
+by(blast_tac (claset() addSDs [dfa_iff_welltyping] addSIs [is_dfa_fix_next]) 1);
+qed "fix_next_iff_welltyping";
isabelle