author | paulson |
Wed, 16 Jun 2004 14:56:39 +0200 | |
changeset 14952 | 47455995693d |
parent 14045 | a34d89ce6097 |
child 14981 | e73f8140af78 |
permissions | -rw-r--r-- |
13673 | 1 |
(* Title: HOL/MicroJava/Comp/LemmasComp.thy |
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ID: $Id$ |
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Author: Martin Strecker |
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Copyright GPL 2002 |
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*) |
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(* Lemmas for compiler correctness proof *) |
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theory LemmasComp = TranslComp: |
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declare split_paired_All [simp del] |
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declare split_paired_Ex [simp del] |
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(**********************************************************************) |
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(* misc lemmas *) |
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lemma split_pairs: "(\<lambda>(a,b). (F a b)) (ab) = F (fst ab) (snd ab)" |
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proof - |
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have "(\<lambda>(a,b). (F a b)) (fst ab,snd ab) = F (fst ab) (snd ab)" |
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by (simp add: split_def) |
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then show ?thesis by simp |
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qed |
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lemma c_hupd_conv: |
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"c_hupd h' (xo, (h,l)) = (xo, (if xo = None then h' else h),l)" |
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by (simp add: c_hupd_def) |
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lemma gl_c_hupd [simp]: "(gl (c_hupd h xs)) = (gl xs)" |
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by (simp add: gl_def c_hupd_def split_pairs) |
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lemma c_hupd_xcpt_invariant [simp]: "gx (c_hupd h' (xo, st)) = xo" |
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by (case_tac st, simp only: c_hupd_conv gx_conv) |
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(* not added to simpset because of interference with c_hupd_conv *) |
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lemma c_hupd_hp_invariant: "gh (c_hupd hp (None, st)) = hp" |
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by (case_tac st, simp add: c_hupd_conv gh_def) |
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lemma unique_map_fst [rule_format]: "(\<forall> x \<in> set xs. (fst x = fst (f x) )) \<longrightarrow> |
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unique (map f xs) = unique xs" |
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proof (induct xs) |
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case Nil show ?case by simp |
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next |
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case (Cons a list) |
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show ?case |
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proof |
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assume fst_eq: "\<forall>x\<in>set (a # list). fst x = fst (f x)" |
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have fst_set: "(fst a \<in> fst ` set list) = (fst (f a) \<in> fst ` f ` set list)" |
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proof |
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assume as: "fst a \<in> fst ` set list" |
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then obtain x where fst_a_x: "x\<in>set list \<and> fst a = fst x" |
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by (auto simp add: image_iff) |
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then have "fst (f a) = fst (f x)" by (simp add: fst_eq) |
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with as show "(fst (f a) \<in> fst ` f ` set list)" by (simp add: fst_a_x) |
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next |
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assume as: "fst (f a) \<in> fst ` f ` set list" |
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then obtain x where fst_a_x: "x\<in>set list \<and> fst (f a) = fst (f x)" |
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by (auto simp add: image_iff) |
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then have "fst a = fst x" by (simp add: fst_eq) |
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with as show "fst a \<in> fst ` set list" by (simp add: fst_a_x) |
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qed |
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with fst_eq Cons |
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show "unique (map f (a # list)) = unique (a # list)" |
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by (simp add: unique_def fst_set) |
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qed |
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qed |
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lemma comp_unique: "unique (comp G) = unique G" |
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apply (simp add: comp_def) |
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apply (rule unique_map_fst) |
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apply (simp add: compClass_def split_beta) |
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done |
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(**********************************************************************) |
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(* invariance of properties under compilation *) |
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lemma comp_class_imp: |
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"(class G C = Some(D, fs, ms)) \<Longrightarrow> |
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(class (comp G) C = Some(D, fs, map (compMethod G C) ms))" |
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apply (simp add: class_def comp_def compClass_def) |
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apply (rule HOL.trans) |
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apply (rule map_of_map2) |
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apply auto |
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done |
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lemma comp_class_None: |
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"(class G C = None) = (class (comp G) C = None)" |
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apply (simp add: class_def comp_def compClass_def) |
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apply (simp add: map_of_in_set) |
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apply (simp add: image_compose [THEN sym] o_def split_beta del: image_compose) |
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done |
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lemma comp_is_class: "is_class (comp G) C = is_class G C" |
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by (case_tac "class G C", auto simp: is_class_def comp_class_None dest: comp_class_imp) |
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lemma comp_is_type: "is_type (comp G) T = is_type G T" |
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by ((cases T),simp,(induct G)) ((simp),(simp only: comp_is_class),(simp add: comp_is_class),(simp only: comp_is_class)) |
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lemma comp_classname: "is_class G C |
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\<Longrightarrow> fst (the (class G C)) = fst (the (class (comp G) C))" |
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by (case_tac "class G C", auto simp: is_class_def dest: comp_class_imp) |
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lemma comp_subcls1: "subcls1 (comp G) = subcls1 G" |
14952
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removal of x-symbol syntax <Sigma> for dependent products
paulson
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by (auto simp add: subcls1_def2 comp_classname comp_is_class) |
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lemma comp_widen: "((ty1,ty2) \<in> widen (comp G)) = ((ty1,ty2) \<in> widen G)" |
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apply rule |
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apply (cases "(ty1,ty2)" "comp G" rule: widen.cases) |
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apply (simp_all add: comp_subcls1 widen.null) |
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apply (cases "(ty1,ty2)" G rule: widen.cases) |
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apply (simp_all add: comp_subcls1 widen.null) |
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done |
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lemma comp_cast: "((ty1,ty2) \<in> cast (comp G)) = ((ty1,ty2) \<in> cast G)" |
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apply rule |
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apply (cases "(ty1,ty2)" "comp G" rule: cast.cases) |
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apply (simp_all add: comp_subcls1 cast.widen cast.subcls) |
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apply (rule cast.widen) apply (simp add: comp_widen) |
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apply (cases "(ty1,ty2)" G rule: cast.cases) |
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apply (simp_all add: comp_subcls1 cast.widen cast.subcls) |
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apply (rule cast.widen) apply (simp add: comp_widen) |
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done |
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lemma comp_cast_ok: "cast_ok (comp G) = cast_ok G" |
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by (simp add: expand_fun_eq cast_ok_def comp_widen) |
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lemma compClass_fst [simp]: "(fst (compClass G C)) = (fst C)" |
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by (simp add: compClass_def split_beta) |
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lemma compClass_fst_snd [simp]: "(fst (snd (compClass G C))) = (fst (snd C))" |
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by (simp add: compClass_def split_beta) |
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lemma compClass_fst_snd_snd [simp]: "(fst (snd (snd (compClass G C)))) = (fst (snd (snd C)))" |
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by (simp add: compClass_def split_beta) |
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lemma comp_wf_fdecl [simp]: "wf_fdecl (comp G) fd = wf_fdecl G fd" |
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by (case_tac fd, simp add: wf_fdecl_def comp_is_type) |
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lemma compClass_forall [simp]: " |
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(\<forall>x\<in>set (snd (snd (snd (compClass G C)))). P (fst x) (fst (snd x))) = |
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(\<forall>x\<in>set (snd (snd (snd C))). P (fst x) (fst (snd x)))" |
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by (simp add: compClass_def compMethod_def split_beta) |
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lemma comp_wf_mhead: "wf_mhead (comp G) S rT = wf_mhead G S rT" |
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by (simp add: wf_mhead_def split_beta comp_is_type) |
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lemma comp_ws_cdecl: " |
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ws_cdecl (TranslComp.comp G) (compClass G C) = ws_cdecl G C" |
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apply (simp add: ws_cdecl_def split_beta comp_is_class comp_subcls1) |
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apply (simp (no_asm_simp) add: comp_wf_mhead) |
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apply (simp add: compClass_def compMethod_def split_beta unique_map_fst) |
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done |
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lemma comp_wf_syscls: "wf_syscls (comp G) = wf_syscls G" |
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apply (simp add: wf_syscls_def comp_def compClass_def split_beta) |
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apply (simp only: image_compose [THEN sym]) |
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apply (subgoal_tac "(Fun.comp fst (\<lambda>(C, cno::cname, fdls::fdecl list, jmdls). |
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(C, cno, fdls, map (compMethod G C) jmdls))) = fst") |
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apply (simp del: image_compose) |
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apply (simp add: expand_fun_eq split_beta) |
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done |
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lemma comp_ws_prog: "ws_prog (comp G) = ws_prog G" |
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apply (rule sym) |
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apply (simp add: ws_prog_def comp_ws_cdecl comp_unique) |
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apply (simp add: comp_wf_syscls) |
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apply (auto simp add: comp_ws_cdecl [THEN sym] TranslComp.comp_def) |
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done |
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lemma comp_class_rec: " wf ((subcls1 G)^-1) \<Longrightarrow> |
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class_rec (comp G) C t f = |
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class_rec G C t (\<lambda> C' fs' ms' r'. f C' fs' (map (compMethod G C') ms') r')" |
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apply (rule_tac a = C in wf_induct) apply assumption |
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apply (subgoal_tac "wf ((subcls1 (comp G))\<inverse>)") |
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apply (subgoal_tac "(class G x = None) \<or> (\<exists> D fs ms. (class G x = Some (D, fs, ms)))") |
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apply (erule disjE) |
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(* case class G x = None *) |
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apply (simp (no_asm_simp) add: class_rec_def comp_subcls1 wfrec cut_apply) |
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apply (simp add: comp_class_None) |
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(* case \<exists> D fs ms. (class G x = Some (D, fs, ms)) *) |
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apply (erule exE)+ |
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apply (frule comp_class_imp) |
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apply (frule_tac G="comp G" and C=x and t=t and f=f in class_rec_lemma) |
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apply assumption |
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apply (frule_tac G=G and C=x and t=t |
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and f="(\<lambda>C' fs' ms'. f C' fs' (map (compMethod G C') ms'))" in class_rec_lemma) |
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apply assumption |
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apply (simp only:) |
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apply (case_tac "x = Object") |
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apply simp |
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apply (frule subcls1I, assumption) |
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apply (drule_tac x=D in spec, drule mp, simp) |
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apply simp |
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(* subgoals *) |
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apply (case_tac "class G x") |
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apply auto |
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apply (simp add: comp_subcls1) |
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done |
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lemma comp_fields: "wf ((subcls1 G)^-1) \<Longrightarrow> |
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fields (comp G,C) = fields (G,C)" |
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by (simp add: fields_def comp_class_rec) |
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lemma comp_field: "wf ((subcls1 G)^-1) \<Longrightarrow> |
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field (comp G,C) = field (G,C)" |
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by (simp add: field_def comp_fields) |
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lemma class_rec_relation [rule_format (no_asm)]: "\<lbrakk> ws_prog G; |
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\<forall> fs ms. R (f1 Object fs ms t1) (f2 Object fs ms t2); |
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\<forall> C fs ms r1 r2. (R r1 r2) \<longrightarrow> (R (f1 C fs ms r1) (f2 C fs ms r2)) \<rbrakk> |
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\<Longrightarrow> ((class G C) \<noteq> None) \<longrightarrow> |
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R (class_rec G C t1 f1) (class_rec G C t2 f2)" |
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apply (frule wf_subcls1) (* establish wf ((subcls1 G)^-1) *) |
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apply (rule_tac a = C in wf_induct) apply assumption |
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apply (intro strip) |
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apply (subgoal_tac "(\<exists>D rT mb. class G x = Some (D, rT, mb))") |
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apply (erule exE)+ |
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apply (frule_tac C=x and t=t1 and f=f1 in class_rec_lemma) |
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apply assumption |
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apply (frule_tac C=x and t=t2 and f=f2 in class_rec_lemma) |
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apply assumption |
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apply (simp only:) |
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apply (case_tac "x = Object") |
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apply simp |
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apply (frule subcls1I, assumption) |
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apply (drule_tac x=D in spec, drule mp, simp) |
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apply simp |
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apply (subgoal_tac "(\<exists>D' rT' mb'. class G D = Some (D', rT', mb'))") |
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apply blast |
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(* subgoals *) |
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apply (frule class_wf_struct) apply assumption |
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apply (simp add: ws_cdecl_def is_class_def) |
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apply (simp add: subcls1_def2 is_class_def) |
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apply auto |
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done |
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syntax |
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mtd_mb :: "cname \<times> ty \<times> 'c \<Rightarrow> 'c" |
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translations |
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"mtd_mb" => "Fun.comp snd snd" |
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lemma map_of_map_fst: "\<lbrakk> inj f; |
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\<forall>x\<in>set xs. fst (f x) = fst x; \<forall>x\<in>set xs. fst (g x) = fst x \<rbrakk> |
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\<Longrightarrow> map_of (map g xs) k |
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= option_map (\<lambda> e. (snd (g ((inv f) (k, e))))) (map_of (map f xs) k)" |
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apply (induct xs) |
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apply simp |
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apply (simp del: split_paired_All) |
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apply (case_tac "k = fst a") |
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apply (simp del: split_paired_All) |
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apply (subgoal_tac "(inv f (fst a, snd (f a))) = a", simp) |
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apply (subgoal_tac "(fst a, snd (f a)) = f a", simp) |
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apply (erule conjE)+ |
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apply (drule_tac s ="fst (f a)" and t="fst a" in sym) |
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apply simp |
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apply (simp add: surjective_pairing) |
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done |
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lemma comp_method [rule_format (no_asm)]: "\<lbrakk> ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> |
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((method (comp G, C) S) = |
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option_map (\<lambda> (D,rT,b). (D, rT, mtd_mb (compMethod G D (S, rT, b)))) |
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(method (G, C) S))" |
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apply (simp add: method_def) |
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apply (frule wf_subcls1) |
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apply (simp add: comp_class_rec) |
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apply (simp add: map_compose [THEN sym] split_iter split_compose del: map_compose) |
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apply (rule_tac R="%x y. ((x S) = (option_map (\<lambda>(D, rT, b). |
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(D, rT, snd (snd (compMethod G D (S, rT, b))))) (y S)))" |
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in class_rec_relation) |
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apply assumption |
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13673 | 297 |
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apply (intro strip) |
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apply simp |
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apply (rule trans) |
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apply (rule_tac f="(\<lambda>(s, m). (s, Object, m))" and |
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g="(Fun.comp (\<lambda>(s, m). (s, Object, m)) (compMethod G Object))" |
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in map_of_map_fst) |
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defer (* inj \<dots> *) |
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apply (simp add: inj_on_def split_beta) |
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13673 | 308 |
apply (simp add: split_beta compMethod_def) |
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apply (simp add: map_of_map [THEN sym]) |
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apply (simp add: split_beta) |
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apply (simp add: map_compose [THEN sym] Fun.comp_def split_beta) |
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apply (subgoal_tac "(\<lambda>x\<Colon>(vname list \<times> fdecl list \<times> stmt \<times> expr) mdecl. |
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(fst x, Object, |
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snd (compMethod G Object |
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(inv (\<lambda>(s\<Colon>sig, m\<Colon>ty \<times> vname list \<times> fdecl list \<times> stmt \<times> expr). |
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(s, Object, m)) |
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(S, Object, snd x))))) |
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= (\<lambda>x. (fst x, Object, fst (snd x), |
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snd (snd (compMethod G Object (S, snd x)))))") |
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apply (simp only:) |
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apply (simp add: expand_fun_eq) |
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apply (intro strip) |
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apply (subgoal_tac "(inv (\<lambda>(s, m). (s, Object, m)) (S, Object, snd x)) = (S, snd x)") |
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apply (simp only:) |
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apply (simp add: compMethod_def split_beta) |
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apply (rule inv_f_eq) |
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defer |
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defer |
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13673 | 329 |
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apply (intro strip) |
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apply (simp add: map_add_Some_iff map_of_map del: split_paired_Ex) |
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apply (simp add: map_add_def) |
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apply (subgoal_tac "inj (\<lambda>(s, m). (s, Ca, m))") |
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apply (drule_tac g="(Fun.comp (\<lambda>(s, m). (s, Ca, m)) (compMethod G Ca))" and xs=ms |
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and k=S in map_of_map_fst) |
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apply (simp add: split_beta) |
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apply (simp add: compMethod_def split_beta) |
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apply (case_tac "(map_of (map (\<lambda>(s, m). (s, Ca, m)) ms) S)") |
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apply simp |
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apply simp apply (simp add: split_beta map_of_map) apply (erule exE) apply (erule conjE)+ |
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apply (drule_tac t=a in sym) |
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apply (subgoal_tac "(inv (\<lambda>(s, m). (s, Ca, m)) (S, a)) = (S, snd a)") |
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apply simp |
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apply (subgoal_tac "\<forall>x\<in>set ms. fst ((Fun.comp (\<lambda>(s, m). (s, Ca, m)) (compMethod G Ca)) x) = fst x") |
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prefer 2 apply (simp (no_asm_simp) add: compMethod_def split_beta) |
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apply (simp add: map_of_map2) |
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apply (simp (no_asm_simp) add: compMethod_def split_beta) |
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14045 | 349 |
-- "remaining subgoals" |
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apply (auto intro: inv_f_eq simp add: inj_on_def is_class_def) |
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13673 | 351 |
done |
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lemma comp_wf_mrT: "\<lbrakk> ws_prog G; is_class G D\<rbrakk> \<Longrightarrow> |
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wf_mrT (TranslComp.comp G) (C, D, fs, map (compMethod G a) ms) = |
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wf_mrT G (C, D, fs, ms)" |
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apply (simp add: wf_mrT_def compMethod_def split_beta) |
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apply (simp add: comp_widen) |
|
360 |
apply (rule iffI) |
|
13673 | 361 |
apply (intro strip) |
362 |
apply simp |
|
14045 | 363 |
apply (drule bspec) apply assumption |
364 |
apply (drule_tac x=D' in spec) apply (drule_tac x=rT' in spec) apply (drule mp) |
|
365 |
prefer 2 apply assumption |
|
366 |
apply (simp add: comp_method [of G D]) |
|
367 |
apply (erule exE)+ |
|
368 |
apply (subgoal_tac "z =(fst z, fst (snd z), snd (snd z))") |
|
369 |
apply (rule exI) |
|
370 |
apply (simp) |
|
371 |
apply (simp add: split_paired_all) |
|
372 |
apply (intro strip) |
|
373 |
apply (simp add: comp_method) |
|
374 |
apply auto |
|
13673 | 375 |
done |
376 |
||
377 |
||
378 |
(**********************************************************************) |
|
379 |
(* DIVERSE OTHER LEMMAS *) |
|
380 |
(**********************************************************************) |
|
381 |
||
382 |
lemma max_spec_preserves_length: |
|
383 |
"max_spec G C (mn, pTs) = {((md,rT),pTs')} |
|
384 |
\<Longrightarrow> length pTs = length pTs'" |
|
385 |
apply (frule max_spec2mheads) |
|
386 |
apply (erule exE)+ |
|
387 |
apply (simp add: list_all2_def) |
|
388 |
done |
|
389 |
||
390 |
||
391 |
lemma ty_exprs_length [simp]: "(E\<turnstile>es[::]Ts \<longrightarrow> length es = length Ts)" |
|
392 |
apply (subgoal_tac "(E\<turnstile>e::T \<longrightarrow> True) \<and> (E\<turnstile>es[::]Ts \<longrightarrow> length es = length Ts) \<and> (E\<turnstile>s\<surd> \<longrightarrow> True)") |
|
393 |
apply blast |
|
394 |
apply (rule ty_expr_ty_exprs_wt_stmt.induct) |
|
395 |
apply auto |
|
396 |
done |
|
397 |
||
398 |
||
399 |
lemma max_spec_preserves_method_rT [simp]: |
|
400 |
"max_spec G C (mn, pTs) = {((md,rT),pTs')} |
|
401 |
\<Longrightarrow> method_rT (the (method (G, C) (mn, pTs'))) = rT" |
|
402 |
apply (frule max_spec2mheads) |
|
403 |
apply (erule exE)+ |
|
404 |
apply (simp add: method_rT_def) |
|
405 |
done |
|
406 |
||
14045 | 407 |
(**********************************************************************************) |
408 |
||
409 |
declare compClass_fst [simp del] |
|
410 |
declare compClass_fst_snd [simp del] |
|
411 |
declare compClass_fst_snd_snd [simp del] |
|
412 |
||
413 |
declare split_paired_All [simp add] |
|
414 |
declare split_paired_Ex [simp add] |
|
13673 | 415 |
|
416 |
end |