(* Title: HOL/Bali/Evaln.thy ID: $Id: Evaln.thy,v 1.25 2007/09/26 17:19:38 wenzelm Exp $ Author: David von Oheimb and Norbert Schirmer *) header {* Operational evaluation (big-step) semantics of Java expressions and statements *} theory Evaln imports TypeSafe begin text {* Variant of @{term eval} relation with counter for bounded recursive depth. In principal @{term evaln} could replace @{term eval}. Validity of the axiomatic semantics builds on @{term evaln}. For recursive method calls the axiomatic semantics rule assumes the method ok to derive a proof for the body. To prove the method rule sound we need to perform induction on the recursion depth. For the completeness proof of the axiomatic semantics the notion of the most general formula is used. The most general formula right now builds on the ordinary evaluation relation @{term eval}. So sometimes we have to switch between @{term evaln} and @{term eval} and vice versa. To make this switch easy @{term evaln} also does all the technical accessibility tests @{term check_field_access} and @{term check_method_access} like @{term eval}. If it would omit them @{term evaln} and @{term eval} would only be equivalent for welltyped, and definitely assigned terms. *} inductive evaln :: "[prog, state, term, nat, vals, state] => bool" ("_\<turnstile>_ \<midarrow>_\<succ>\<midarrow>_-> '(_, _')" [61,61,80,61,0,0] 60) and evarn :: "[prog, state, var, vvar, nat, state] => bool" ("_\<turnstile>_ \<midarrow>_=\<succ>_\<midarrow>_-> _" [61,61,90,61,61,61] 60) and eval_n:: "[prog, state, expr, val, nat, state] => bool" ("_\<turnstile>_ \<midarrow>_-\<succ>_\<midarrow>_-> _" [61,61,80,61,61,61] 60) and evalsn :: "[prog, state, expr list, val list, nat, state] => bool" ("_\<turnstile>_ \<midarrow>_\<doteq>\<succ>_\<midarrow>_-> _" [61,61,61,61,61,61] 60) and execn :: "[prog, state, stmt, nat, state] => bool" ("_\<turnstile>_ \<midarrow>_\<midarrow>_-> _" [61,61,65, 61,61] 60) for G :: prog where "G\<turnstile>s \<midarrow>c \<midarrow>n-> s' ≡ G\<turnstile>s \<midarrow>In1r c\<succ>\<midarrow>n-> (♦ , s')" | "G\<turnstile>s \<midarrow>e-\<succ>v \<midarrow>n-> s' ≡ G\<turnstile>s \<midarrow>In1l e\<succ>\<midarrow>n-> (In1 v , s')" | "G\<turnstile>s \<midarrow>e=\<succ>vf \<midarrow>n-> s' ≡ G\<turnstile>s \<midarrow>In2 e\<succ>\<midarrow>n-> (In2 vf, s')" | "G\<turnstile>s \<midarrow>e\<doteq>\<succ>v \<midarrow>n-> s' ≡ G\<turnstile>s \<midarrow>In3 e\<succ>\<midarrow>n-> (In3 v , s')" --{* propagation of abrupt completion *} | Abrupt: "G\<turnstile>(Some xc,s) \<midarrow>t\<succ>\<midarrow>n-> (arbitrary3 t,(Some xc,s))" --{* evaluation of variables *} | LVar: "G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<midarrow>n-> Norm s" | FVar: "[|G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n-> s2; (v,s2') = fvar statDeclC stat fn a s2; s3 = check_field_access G accC statDeclC fn stat a s2'|] ==> G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<midarrow>n-> s3" | AVar: "[|G\<turnstile> Norm s0 \<midarrow>e1-\<succ>a\<midarrow>n-> s1 ; G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n-> s2; (v,s2') = avar G i a s2|] ==> G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<midarrow>n-> s2'" --{* evaluation of expressions *} | NewC: "[|G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n-> s1; G\<turnstile> s1 \<midarrow>halloc (CInst C)\<succ>a-> s2|] ==> G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<midarrow>n-> s2" | NewA: "[|G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>e-\<succ>i'\<midarrow>n-> s2; G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc (Arr T (the_Intg i'))\<succ>a-> s3|] ==> G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<midarrow>n-> s3" | Cast: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1; s2 = abupd (raise_if (¬G,snd s1\<turnstile>v fits T) ClassCast) s1|] ==> G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<midarrow>n-> s2" | Inst: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1; b = (v≠Null ∧ G,store s1\<turnstile>v fits RefT T)|] ==> G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<midarrow>n-> s1" | Lit: "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<midarrow>n-> Norm s" | UnOp: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1|] ==> G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>(eval_unop unop v)\<midarrow>n-> s1" | BinOp: "[|G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then (In1l e2) else (In1r Skip)) \<succ>\<midarrow>n-> (In1 v2,s2)|] ==> G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>(eval_binop binop v1 v2)\<midarrow>n-> s2" | Super: "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<midarrow>n-> Norm s" | Acc: "[|G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v,f)\<midarrow>n-> s1|] ==> G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<midarrow>n-> s1" | Ass: "[|G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w,f)\<midarrow>n-> s1; G\<turnstile> s1 \<midarrow>e-\<succ>v \<midarrow>n-> s2|] ==> G\<turnstile>Norm s0 \<midarrow>va:=e-\<succ>v\<midarrow>n-> assign f v s2" | Cond: "[|G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<midarrow>n-> s1; G\<turnstile> s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n-> s2|] ==> G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>n-> s2" | Call: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n-> s2; D = invocation_declclass G mode (store s2) a' statT (|name=mn,parTs=pTs|)),; s3=init_lvars G D (|name=mn,parTs=pTs|)), mode a' vs s2; s3' = check_method_access G accC statT mode (|name=mn,parTs=pTs|)), a' s3; G\<turnstile>s3'\<midarrow>Methd D (|name=mn,parTs=pTs|)),-\<succ>v\<midarrow>n-> s4 |] ==> G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e·mn({pTs}args)-\<succ>v\<midarrow>n-> (restore_lvars s2 s4)" | Methd:"[|G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<midarrow>n-> s1|] ==> G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<midarrow>Suc n-> s1" | Body: "[|G\<turnstile>Norm s0\<midarrow>Init D\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>c\<midarrow>n-> s2; s3 = (if (∃ l. abrupt s2 = Some (Jump (Break l)) ∨ abrupt s2 = Some (Jump (Cont l))) then abupd (λ x. Some (Error CrossMethodJump)) s2 else s2)|]==> G\<turnstile>Norm s0 \<midarrow>Body D c -\<succ>the (locals (store s2) Result)\<midarrow>n->abupd (absorb Ret) s3" --{* evaluation of expression lists *} | Nil: "G\<turnstile>Norm s0 \<midarrow>[]\<doteq>\<succ>[]\<midarrow>n-> Norm s0" | Cons: "[|G\<turnstile>Norm s0 \<midarrow>e -\<succ> v \<midarrow>n-> s1; G\<turnstile> s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n-> s2|] ==> G\<turnstile>Norm s0 \<midarrow>e#es\<doteq>\<succ>v#vs\<midarrow>n-> s2" --{* execution of statements *} | Skip: "G\<turnstile>Norm s \<midarrow>Skip\<midarrow>n-> Norm s" | Expr: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1|] ==> G\<turnstile>Norm s0 \<midarrow>Expr e\<midarrow>n-> s1" | Lab: "[|G\<turnstile>Norm s0 \<midarrow>c \<midarrow>n-> s1|] ==> G\<turnstile>Norm s0 \<midarrow>l• c\<midarrow>n-> abupd (absorb l) s1" | Comp: "[|G\<turnstile>Norm s0 \<midarrow>c1 \<midarrow>n-> s1; G\<turnstile> s1 \<midarrow>c2 \<midarrow>n-> s2|] ==> G\<turnstile>Norm s0 \<midarrow>c1;; c2\<midarrow>n-> s2" | If: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n-> s1; G\<turnstile> s1\<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n-> s2|] ==> G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2 \<midarrow>n-> s2" | Loop: "[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n-> s1; if the_Bool b then (G\<turnstile>s1 \<midarrow>c\<midarrow>n-> s2 ∧ G\<turnstile>(abupd (absorb (Cont l)) s2) \<midarrow>l• While(e) c\<midarrow>n-> s3) else s3 = s1|] ==> G\<turnstile>Norm s0 \<midarrow>l• While(e) c\<midarrow>n-> s3" | Jmp: "G\<turnstile>Norm s \<midarrow>Jmp j\<midarrow>n-> (Some (Jump j), s)" | Throw:"[|G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n-> s1|] ==> G\<turnstile>Norm s0 \<midarrow>Throw e\<midarrow>n-> abupd (throw a') s1" | Try: "[|G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>sxalloc-> s2; if G,s2\<turnstile>catch tn then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n-> s3 else s3 = s2|] ==> G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(tn vn) c2\<midarrow>n-> s3" | Fin: "[|G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n-> (x1,s1); G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n-> s2; s3=(if (∃ err. x1=Some (Error err)) then (x1,s1) else abupd (abrupt_if (x1≠None) x1) s2)|] ==> G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>n-> s3" | Init: "[|the (class G C) = c; if inited C (globs s0) then s3 = Norm s0 else (G\<turnstile>Norm (init_class_obj G C s0) \<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n-> s1 ∧ G\<turnstile>set_lvars empty s1 \<midarrow>init c\<midarrow>n-> s2 ∧ s3 = restore_lvars s1 s2)|] ==> G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n-> s3" monos if_bool_eq_conj declare split_if [split del] split_if_asm [split del] option.split [split del] option.split_asm [split del] not_None_eq [simp del] split_paired_All [simp del] split_paired_Ex [simp del] declaration {* K (Simplifier.map_ss (fn ss => ss delloop "split_all_tac")) *} inductive_cases evaln_cases: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (v, s')" inductive_cases evaln_elim_cases: "G\<turnstile>(Some xc, s) \<midarrow>t \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1r Skip \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (Jmp j) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (l• c) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In3 ([]) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In3 (e#es) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (Lit w) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (UnOp unop e) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (BinOp binop e1 e2) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In2 (LVar vn) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (Cast T e) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (e InstOf T) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (Super) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (Acc va) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1r (Expr e) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (c1;; c2) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1l (Methd C sig) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1l (Body D c) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1l (e0 ? e1 : e2) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1r (If(e) c1 Else c2) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (l• While(e) c) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (c1 Finally c2) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (Throw e) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1l (NewC C) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (New T[e]) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l (Ass va e) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1r (Try c1 Catch(tn vn) c2) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In2 ({accC,statDeclC,stat}e..fn) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In2 (e1.[e2]) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1l ({accC,statT,mode}e·mn({pT}p)) \<succ>\<midarrow>n-> (v, s')" "G\<turnstile>Norm s \<midarrow>In1r (Init C) \<succ>\<midarrow>n-> (x, s')" "G\<turnstile>Norm s \<midarrow>In1r (Init C) \<succ>\<midarrow>n-> (x, s')" declare split_if [split] split_if_asm [split] option.split [split] option.split_asm [split] not_None_eq [simp] split_paired_All [simp] split_paired_Ex [simp] declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *} lemma evaln_Inj_elim: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (w,s') ==> case t of In1 ec => (case ec of Inl e => (∃v. w = In1 v) | Inr c => w = ♦) | In2 e => (∃v. w = In2 v) | In3 e => (∃v. w = In3 v)" apply (erule evaln_cases , auto) apply (induct_tac "t") apply (induct_tac "a") apply auto done text {* The following simplification procedures set up the proper injections of terms and their corresponding values in the evaluation relation: E.g. an expression (injection @{term In1l} into terms) always evaluates to ordinary values (injection @{term In1} into generalised values @{term vals}). *} lemma evaln_expr_eq: "G\<turnstile>s \<midarrow>In1l t\<succ>\<midarrow>n-> (w, s') = (∃v. w=In1 v ∧ G\<turnstile>s \<midarrow>t-\<succ>v \<midarrow>n-> s')" by (auto, frule evaln_Inj_elim, auto) lemma evaln_var_eq: "G\<turnstile>s \<midarrow>In2 t\<succ>\<midarrow>n-> (w, s') = (∃vf. w=In2 vf ∧ G\<turnstile>s \<midarrow>t=\<succ>vf\<midarrow>n-> s')" by (auto, frule evaln_Inj_elim, auto) lemma evaln_exprs_eq: "G\<turnstile>s \<midarrow>In3 t\<succ>\<midarrow>n-> (w, s') = (∃vs. w=In3 vs ∧ G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<midarrow>n-> s')" by (auto, frule evaln_Inj_elim, auto) lemma evaln_stmt_eq: "G\<turnstile>s \<midarrow>In1r t\<succ>\<midarrow>n-> (w, s') = (w=♦ ∧ G\<turnstile>s \<midarrow>t \<midarrow>n-> s')" by (auto, frule evaln_Inj_elim, auto, frule evaln_Inj_elim, auto) simproc_setup evaln_expr ("G\<turnstile>s \<midarrow>In1l t\<succ>\<midarrow>n-> (w, s')") = {* fn _ => fn _ => fn ct => (case Thm.term_of ct of (_ $ _ $ _ $ _ $ _ $ (Const _ $ _) $ _) => NONE | _ => SOME (mk_meta_eq @{thm evaln_expr_eq})) *} simproc_setup evaln_var ("G\<turnstile>s \<midarrow>In2 t\<succ>\<midarrow>n-> (w, s')") = {* fn _ => fn _ => fn ct => (case Thm.term_of ct of (_ $ _ $ _ $ _ $ _ $ (Const _ $ _) $ _) => NONE | _ => SOME (mk_meta_eq @{thm evaln_var_eq})) *} simproc_setup evaln_exprs ("G\<turnstile>s \<midarrow>In3 t\<succ>\<midarrow>n-> (w, s')") = {* fn _ => fn _ => fn ct => (case Thm.term_of ct of (_ $ _ $ _ $ _ $ _ $ (Const _ $ _) $ _) => NONE | _ => SOME (mk_meta_eq @{thm evaln_exprs_eq})) *} simproc_setup evaln_stmt ("G\<turnstile>s \<midarrow>In1r t\<succ>\<midarrow>n-> (w, s')") = {* fn _ => fn _ => fn ct => (case Thm.term_of ct of (_ $ _ $ _ $ _ $ _ $ (Const _ $ _) $ _) => NONE | _ => SOME (mk_meta_eq @{thm evaln_stmt_eq})) *} ML_setup {* bind_thms ("evaln_AbruptIs", sum3_instantiate @{thm evaln.Abrupt}) *} declare evaln_AbruptIs [intro!] lemma evaln_Callee: "G\<turnstile>Norm s\<midarrow>In1l (Callee l e)\<succ>\<midarrow>n-> (v,s') = False" proof - { fix s t v s' assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (v,s')" and normal: "normal s" and callee: "t=In1l (Callee l e)" then have "False" by induct auto } then show ?thesis by (cases s') fastsimp qed lemma evaln_InsInitE: "G\<turnstile>Norm s\<midarrow>In1l (InsInitE c e)\<succ>\<midarrow>n-> (v,s') = False" proof - { fix s t v s' assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (v,s')" and normal: "normal s" and callee: "t=In1l (InsInitE c e)" then have "False" by induct auto } then show ?thesis by (cases s') fastsimp qed lemma evaln_InsInitV: "G\<turnstile>Norm s\<midarrow>In2 (InsInitV c w)\<succ>\<midarrow>n-> (v,s') = False" proof - { fix s t v s' assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (v,s')" and normal: "normal s" and callee: "t=In2 (InsInitV c w)" then have "False" by induct auto } then show ?thesis by (cases s') fastsimp qed lemma evaln_FinA: "G\<turnstile>Norm s\<midarrow>In1r (FinA a c)\<succ>\<midarrow>n-> (v,s') = False" proof - { fix s t v s' assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (v,s')" and normal: "normal s" and callee: "t=In1r (FinA a c)" then have "False" by induct auto } then show ?thesis by (cases s') fastsimp qed lemma evaln_abrupt_lemma: "G\<turnstile>s \<midarrow>e\<succ>\<midarrow>n-> (v,s') ==> fst s = Some xc --> s' = s ∧ v = arbitrary3 e" apply (erule evaln_cases , auto) done lemma evaln_abrupt: "!!s'. G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n-> (w,s') = (s' = (Some xc,s) ∧ w=arbitrary3 e ∧ G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n-> (arbitrary3 e,(Some xc,s)))" apply auto apply (frule evaln_abrupt_lemma, auto)+ done simproc_setup evaln_abrupt ("G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n-> (w,s')") = {* fn _ => fn _ => fn ct => (case Thm.term_of ct of (_ $ _ $ _ $ _ $ _ $ _ $ (Const (@{const_name Pair}, _) $ (Const (@{const_name Some},_) $ _)$ _)) => NONE | _ => SOME (mk_meta_eq @{thm evaln_abrupt})) *} lemma evaln_LitI: "G\<turnstile>s \<midarrow>Lit v-\<succ>(if normal s then v else arbitrary)\<midarrow>n-> s" apply (case_tac "s", case_tac "a = None") by (auto intro!: evaln.Lit) lemma CondI: "!!s1. [|G\<turnstile>s \<midarrow>e-\<succ>b\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n-> s2|] ==> G\<turnstile>s \<midarrow>e ? e1 : e2-\<succ>(if normal s1 then v else arbitrary)\<midarrow>n-> s2" apply (case_tac "s", case_tac "a = None") by (auto intro!: evaln.Cond) lemma evaln_SkipI [intro!]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n-> s" apply (case_tac "s", case_tac "a = None") by (auto intro!: evaln.Skip) lemma evaln_ExprI: "G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n-> s' ==> G\<turnstile>s \<midarrow>Expr e\<midarrow>n-> s'" apply (case_tac "s", case_tac "a = None") by (auto intro!: evaln.Expr) lemma evaln_CompI: "[|G\<turnstile>s \<midarrow>c1\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>c2\<midarrow>n-> s2|] ==> G\<turnstile>s \<midarrow>c1;; c2\<midarrow>n-> s2" apply (case_tac "s", case_tac "a = None") by (auto intro!: evaln.Comp) lemma evaln_IfI: "[|G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n-> s1; G\<turnstile>s1 \<midarrow>(if the_Bool v then c1 else c2)\<midarrow>n-> s2|] ==> G\<turnstile>s \<midarrow>If(e) c1 Else c2\<midarrow>n-> s2" apply (case_tac "s", case_tac "a = None") by (auto intro!: evaln.If) lemma evaln_SkipD [dest!]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n-> s' ==> s' = s" by (erule evaln_cases, auto) lemma evaln_Skip_eq [simp]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n-> s' = (s = s')" apply auto done section {* evaln implies eval *} lemma evaln_eval: assumes evaln: "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n-> (v,s1)" shows "G\<turnstile>s0 \<midarrow>t\<succ>-> (v,s1)" using evaln proof (induct) case (Loop s0 e b n s1 c s2 l s3) note `G\<turnstile>Norm s0 \<midarrow>e-\<succ>b-> s1` moreover have "if the_Bool b then (G\<turnstile>s1 \<midarrow>c-> s2) ∧ G\<turnstile>abupd (absorb (Cont l)) s2 \<midarrow>l• While(e) c-> s3 else s3 = s1" using Loop.hyps by simp ultimately show ?case by (rule eval.Loop) next case (Try s0 c1 n s1 s2 C vn c2 s3) note `G\<turnstile>Norm s0 \<midarrow>c1-> s1` moreover note `G\<turnstile>s1 \<midarrow>sxalloc-> s2` moreover have "if G,s2\<turnstile>catch C then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2-> s3 else s3 = s2" using Try.hyps by simp ultimately show ?case by (rule eval.Try) next case (Init C c s0 s3 n s1 s2) note `the (class G C) = c` moreover have "if inited C (globs s0) then s3 = Norm s0 else G\<turnstile>Norm ((init_class_obj G C) s0) \<midarrow>(if C = Object then Skip else Init (super c))-> s1 ∧ G\<turnstile>(set_lvars empty) s1 \<midarrow>init c-> s2 ∧ s3 = (set_lvars (locals (store s1))) s2" using Init.hyps by simp ultimately show ?case by (rule eval.Init) qed (rule eval.intros,(assumption+ | assumption?))+ lemma Suc_le_D_lemma: "[|Suc n <= m'; (!!m. n <= m ==> P (Suc m)) |] ==> P m'" apply (frule Suc_le_D) apply fast done lemma evaln_nonstrict [rule_format (no_asm), elim]: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (w, s') ==> ∀m. n≤m --> G\<turnstile>s \<midarrow>t\<succ>\<midarrow>m-> (w, s')" apply (erule evaln.induct) apply (tactic {* ALLGOALS (EVERY'[strip_tac, TRY o etac (thm "Suc_le_D_lemma"), REPEAT o smp_tac 1, resolve_tac (thms "evaln.intros") THEN_ALL_NEW TRY o atac]) *}) (* 3 subgoals *) apply (auto split del: split_if) done lemmas evaln_nonstrict_Suc = evaln_nonstrict [OF _ le_refl [THEN le_SucI]] lemma evaln_max2: "[|G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1-> (w1, s1'); G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2-> (w2, s2')|] ==> G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max n1 n2-> (w1, s1') ∧ G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max n1 n2-> (w2, s2')" by (fast intro: le_maxI1 le_maxI2) corollary evaln_max2E [consumes 2]: "[|G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1-> (w1, s1'); G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2-> (w2, s2'); [|G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max n1 n2-> (w1, s1');G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max n1 n2-> (w2, s2') |] ==> P |] ==> P" by (drule (1) evaln_max2) simp lemma evaln_max3: "[|G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1-> (w1, s1'); G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2-> (w2, s2'); G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>n3-> (w3, s3')|] ==> G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max (max n1 n2) n3-> (w1, s1') ∧ G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max (max n1 n2) n3-> (w2, s2') ∧ G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>max (max n1 n2) n3-> (w3, s3')" apply (drule (1) evaln_max2, erule thin_rl) apply (fast intro!: le_maxI1 le_maxI2) done corollary evaln_max3E: "[|G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1-> (w1, s1'); G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2-> (w2, s2'); G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>n3-> (w3, s3'); [|G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max (max n1 n2) n3-> (w1, s1'); G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max (max n1 n2) n3-> (w2, s2'); G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>max (max n1 n2) n3-> (w3, s3') |] ==> P |] ==> P" by (drule (2) evaln_max3) simp lemma le_max3I1: "(n2::nat) ≤ max n1 (max n2 n3)" proof - have "n2 ≤ max n2 n3" by (rule le_maxI1) also have "max n2 n3 ≤ max n1 (max n2 n3)" by (rule le_maxI2) finally show ?thesis . qed lemma le_max3I2: "(n3::nat) ≤ max n1 (max n2 n3)" proof - have "n3 ≤ max n2 n3" by (rule le_maxI2) also have "max n2 n3 ≤ max n1 (max n2 n3)" by (rule le_maxI2) finally show ?thesis . qed declare [[simproc del: wt_expr wt_var wt_exprs wt_stmt]] section {* eval implies evaln *} lemma eval_evaln: assumes eval: "G\<turnstile>s0 \<midarrow>t\<succ>-> (v,s1)" shows "∃n. G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n-> (v,s1)" using eval proof (induct) case (Abrupt xc s t) obtain n where "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<midarrow>n-> (arbitrary3 t, (Some xc, s))" by (iprover intro: evaln.Abrupt) then show ?case .. next case Skip show ?case by (blast intro: evaln.Skip) next case (Expr s0 e v s1) then obtain n where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>Expr e\<midarrow>n-> s1" by (rule evaln.Expr) then show ?case .. next case (Lab s0 c s1 l) then obtain n where "G\<turnstile>Norm s0 \<midarrow>c\<midarrow>n-> s1" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>l• c\<midarrow>n-> abupd (absorb l) s1" by (rule evaln.Lab) then show ?case .. next case (Comp s0 c1 s1 c2 s2) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>c2\<midarrow>n2-> s2" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>c1;; c2\<midarrow>max n1 n2-> s2" by (blast intro: evaln.Comp dest: evaln_max2 ) then show ?case .. next case (If s0 e b s1 c1 c2 s2) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n2-> s2" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2\<midarrow>max n1 n2-> s2" by (blast intro: evaln.If dest: evaln_max2) then show ?case .. next case (Loop s0 e b s1 c s2 l s3) from Loop.hyps obtain n1 where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n1-> s1" by (iprover) moreover from Loop.hyps obtain n2 where "if the_Bool b then (G\<turnstile>s1 \<midarrow>c\<midarrow>n2-> s2 ∧ G\<turnstile>(abupd (absorb (Cont l)) s2)\<midarrow>l• While(e) c\<midarrow>n2-> s3) else s3 = s1" by simp (iprover intro: evaln_nonstrict le_maxI1 le_maxI2) ultimately have "G\<turnstile>Norm s0 \<midarrow>l• While(e) c\<midarrow>max n1 n2-> s3" apply - apply (rule evaln.Loop) apply (iprover intro: evaln_nonstrict intro: le_maxI1) apply (auto intro: evaln_nonstrict intro: le_maxI2) done then show ?case .. next case (Jmp s j) have "G\<turnstile>Norm s \<midarrow>Jmp j\<midarrow>n-> (Some (Jump j), s)" by (rule evaln.Jmp) then show ?case .. next case (Throw s0 e a s1) then obtain n where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a\<midarrow>n-> s1" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>Throw e\<midarrow>n-> abupd (throw a) s1" by (rule evaln.Throw) then show ?case .. next case (Try s0 c1 s1 s2 catchC vn c2 s3) from Try.hyps obtain n1 where "G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n1-> s1" by (iprover) moreover note sxalloc = `G\<turnstile>s1 \<midarrow>sxalloc-> s2` moreover from Try.hyps obtain n2 where "if G,s2\<turnstile>catch catchC then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n2-> s3 else s3 = s2" by fastsimp ultimately have "G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(catchC vn) c2\<midarrow>max n1 n2-> s3" by (auto intro!: evaln.Try le_maxI1 le_maxI2) then show ?case .. next case (Fin s0 c1 x1 s1 c2 s2 s3) from Fin obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n1-> (x1, s1)" "G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n2-> s2" by iprover moreover note s3 = `s3 = (if ∃err. x1 = Some (Error err) then (x1, s1) else abupd (abrupt_if (x1 ≠ None) x1) s2)` ultimately have "G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>max n1 n2-> s3" by (blast intro: evaln.Fin dest: evaln_max2) then show ?case .. next case (Init C c s0 s3 s1 s2) note cls = `the (class G C) = c` moreover from Init.hyps obtain n where "if inited C (globs s0) then s3 = Norm s0 else (G\<turnstile>Norm (init_class_obj G C s0) \<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n-> s1 ∧ G\<turnstile>set_lvars empty s1 \<midarrow>init c\<midarrow>n-> s2 ∧ s3 = restore_lvars s1 s2)" by (auto intro: evaln_nonstrict le_maxI1 le_maxI2) ultimately have "G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n-> s3" by (rule evaln.Init) then show ?case .. next case (NewC s0 C s1 a s2) then obtain n where "G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n-> s1" by (iprover) with NewC have "G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<midarrow>n-> s2" by (iprover intro: evaln.NewC) then show ?case .. next case (NewA s0 T s1 e i s2 a s3) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>e-\<succ>i\<midarrow>n2-> s2" by (iprover) moreover note `G\<turnstile>abupd (check_neg i) s2 \<midarrow>halloc Arr T (the_Intg i)\<succ>a-> s3` ultimately have "G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<midarrow>max n1 n2-> s3" by (blast intro: evaln.NewA dest: evaln_max2) then show ?case .. next case (Cast s0 e v s1 s2 castT) then obtain n where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1" by (iprover) moreover note `s2 = abupd (raise_if (¬ G,snd s1\<turnstile>v fits castT) ClassCast) s1` ultimately have "G\<turnstile>Norm s0 \<midarrow>Cast castT e-\<succ>v\<midarrow>n-> s2" by (rule evaln.Cast) then show ?case .. next case (Inst s0 e v s1 b T) then obtain n where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1" by (iprover) moreover note `b = (v ≠ Null ∧ G,snd s1\<turnstile>v fits RefT T)` ultimately have "G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<midarrow>n-> s1" by (rule evaln.Inst) then show ?case .. next case (Lit s v) have "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<midarrow>n-> Norm s" by (rule evaln.Lit) then show ?case .. next case (UnOp s0 e v s1 unop) then obtain n where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n-> s1" by (iprover) hence "G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>eval_unop unop v\<midarrow>n-> s1" by (rule evaln.UnOp) then show ?case .. next case (BinOp s0 e1 v1 s1 binop e2 v2 s2) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ>\<midarrow>n2-> (In1 v2, s2)" by (iprover) hence "G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>(eval_binop binop v1 v2)\<midarrow>max n1 n2 -> s2" by (blast intro!: evaln.BinOp dest: evaln_max2) then show ?case .. next case (Super s ) have "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<midarrow>n-> Norm s" by (rule evaln.Super) then show ?case .. next case (Acc s0 va v f s1) then obtain n where "G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v, f)\<midarrow>n-> s1" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<midarrow>n-> s1" by (rule evaln.Acc) then show ?case .. next case (Ass s0 var w f s1 e v s2) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>var=\<succ>(w, f)\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>e-\<succ>v\<midarrow>n2-> s2" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>var:=e-\<succ>v\<midarrow>max n1 n2-> assign f v s2" by (blast intro: evaln.Ass dest: evaln_max2) then show ?case .. next case (Cond s0 e0 b s1 e1 e2 v s2) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n2-> s2" by (iprover) then have "G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>max n1 n2-> s2" by (blast intro: evaln.Cond dest: evaln_max2) then show ?case .. next case (Call s0 e a' s1 args vs s2 invDeclC mode statT mn pTs' s3 s3' accC' v s4) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n2-> s2" by iprover moreover note `invDeclC = invocation_declclass G mode (store s2) a' statT (|name=mn,parTs=pTs'|)),` moreover note `s3 = init_lvars G invDeclC (|name=mn,parTs=pTs'|)), mode a' vs s2` moreover note `s3'=check_method_access G accC' statT mode (|name=mn,parTs=pTs'|)), a' s3` moreover from Call.hyps obtain m where "G\<turnstile>s3' \<midarrow>Methd invDeclC (|name=mn, parTs=pTs'|)),-\<succ>v\<midarrow>m-> s4" by iprover ultimately have "G\<turnstile>Norm s0 \<midarrow>{accC',statT,mode}e·mn( {pTs'}args)-\<succ>v\<midarrow>max n1 (max n2 m)-> (set_lvars (locals (store s2))) s4" by (auto intro!: evaln.Call le_maxI1 le_max3I1 le_max3I2) thus ?case .. next case (Methd s0 D sig v s1) then obtain n where "G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<midarrow>n-> s1" by iprover then have "G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<midarrow>Suc n-> s1" by (rule evaln.Methd) then show ?case .. next case (Body s0 D s1 c s2 s3) from Body.hyps obtain n1 n2 where evaln_init: "G\<turnstile>Norm s0 \<midarrow>Init D\<midarrow>n1-> s1" and evaln_c: "G\<turnstile>s1 \<midarrow>c\<midarrow>n2-> s2" by (iprover) moreover note `s3 = (if ∃l. fst s2 = Some (Jump (Break l)) ∨ fst s2 = Some (Jump (Cont l)) then abupd (λx. Some (Error CrossMethodJump)) s2 else s2)` ultimately have "G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>the (locals (store s2) Result)\<midarrow>max n1 n2 -> abupd (absorb Ret) s3" by (iprover intro: evaln.Body dest: evaln_max2) then show ?case .. next case (LVar s vn ) obtain n where "G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<midarrow>n-> Norm s" by (iprover intro: evaln.LVar) then show ?case .. next case (FVar s0 statDeclC s1 e a s2 v s2' stat fn s3 accC) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n2-> s2" by iprover moreover note `s3 = check_field_access G accC statDeclC fn stat a s2'` and `(v, s2') = fvar statDeclC stat fn a s2` ultimately have "G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<midarrow>max n1 n2-> s3" by (iprover intro: evaln.FVar dest: evaln_max2) then show ?case .. next case (AVar s0 e1 a s1 e2 i s2 v s2') then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>a\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n2-> s2" by iprover moreover note `(v, s2') = avar G i a s2` ultimately have "G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<midarrow>max n1 n2-> s2'" by (blast intro!: evaln.AVar dest: evaln_max2) then show ?case .. next case (Nil s0) show ?case by (iprover intro: evaln.Nil) next case (Cons s0 e v s1 es vs s2) then obtain n1 n2 where "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n1-> s1" "G\<turnstile>s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n2-> s2" by iprover then have "G\<turnstile>Norm s0 \<midarrow>e # es\<doteq>\<succ>v # vs\<midarrow>max n1 n2-> s2" by (blast intro!: evaln.Cons dest: evaln_max2) then show ?case .. qed end
lemma evaln_Inj_elim:
G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (w, s')
==> case t of In1l e => ∃v. w = In1 v | In1r c => w = dummy_res
| In2 e => ∃v. w = In2 v | In3 e => ∃v. w = In3 v
lemma evaln_expr_eq:
G\<turnstile>s \<midarrow>In1l t\<succ>\<midarrow>n-> (w, s') =
(∃v. w = In1 v ∧ G\<turnstile>s \<midarrow>t-\<succ>v\<midarrow>n-> s')
lemma evaln_var_eq:
G\<turnstile>s \<midarrow>In2 t\<succ>\<midarrow>n-> (w, s') =
(∃vf. w = In2 vf ∧ G\<turnstile>s \<midarrow>t=\<succ>vf\<midarrow>n-> s')
lemma evaln_exprs_eq:
G\<turnstile>s \<midarrow>In3 t\<succ>\<midarrow>n-> (w, s') =
(∃vs. w = In3 vs ∧
G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<midarrow>n-> s')
lemma evaln_stmt_eq:
G\<turnstile>s \<midarrow>In1r t\<succ>\<midarrow>n-> (w, s') =
(w = dummy_res ∧ G\<turnstile>s \<midarrow>t\<midarrow>n-> s')
theorem evaln_AbruptIs:
G\<turnstile>(Some xc,
s) \<midarrow>x-\<succ>arbitrary\<midarrow>n-> (Some xc, s)
G\<turnstile>(Some xc,
s) \<midarrow>x=\<succ>arbitrary\<midarrow>n-> (Some xc, s)
G\<turnstile>(Some xc,
s) \<midarrow>x\<doteq>\<succ>arbitrary\<midarrow>n-> (Some xc, s)
G\<turnstile>(Some xc, s) \<midarrow>x\<midarrow>n-> (Some xc, s)
lemma evaln_Callee:
G\<turnstile>Norm s \<midarrow>In1l (Callee l e)\<succ>\<midarrow>n-> (v, s') =
False
lemma evaln_InsInitE:
G\<turnstile>Norm s \<midarrow>In1l (InsInitE c
e)\<succ>\<midarrow>n-> (v, s') =
False
lemma evaln_InsInitV:
G\<turnstile>Norm s \<midarrow>In2 (InsInitV c w)\<succ>\<midarrow>n-> (v, s') =
False
lemma evaln_FinA:
G\<turnstile>Norm s \<midarrow>In1r (FinA a c)\<succ>\<midarrow>n-> (v, s') =
False
lemma evaln_abrupt_lemma:
G\<turnstile>s \<midarrow>e\<succ>\<midarrow>n-> (v, s')
==> fst s = Some xc --> s' = s ∧ v = arbitrary3 e
lemma evaln_abrupt:
G\<turnstile>(Some xc, s) \<midarrow>e\<succ>\<midarrow>n-> (w, s') =
(s' = (Some xc, s) ∧
w = arbitrary3 e ∧
G\<turnstile>(Some xc,
s) \<midarrow>e\<succ>\<midarrow>n-> (arbitrary3
e, (Some xc, s)))
lemma evaln_LitI:
G\<turnstile>s \<midarrow>Lit v-\<succ>(if normal s then v
else arbitrary)\<midarrow>n-> s
lemma CondI:
[| G\<turnstile>s \<midarrow>e-\<succ>b\<midarrow>n-> s1.0;
G\<turnstile>s1.0 \<midarrow>(if the_Bool b then e1.0
else e2.0)-\<succ>v\<midarrow>n-> s2.0 |]
==> G\<turnstile>s \<midarrow>e ? e1.0 : e2.0-\<succ>(if normal s1.0 then v
else arbitrary)\<midarrow>n-> s2.0
lemma evaln_SkipI:
G\<turnstile>s \<midarrow>Skip\<midarrow>n-> s
lemma evaln_ExprI:
G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n-> s'
==> G\<turnstile>s \<midarrow>Expr e\<midarrow>n-> s'
lemma evaln_CompI:
[| G\<turnstile>s \<midarrow>c1.0\<midarrow>n-> s1.0;
G\<turnstile>s1.0 \<midarrow>c2.0\<midarrow>n-> s2.0 |]
==> G\<turnstile>s \<midarrow>c1.0;; c2.0\<midarrow>n-> s2.0
lemma evaln_IfI:
[| G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n-> s1.0;
G\<turnstile>s1.0 \<midarrow>(if the_Bool v then c1.0
else c2.0)\<midarrow>n-> s2.0 |]
==> G\<turnstile>s \<midarrow>If(e) c1.0 Else c2.0\<midarrow>n-> s2.0
lemma evaln_SkipD:
G\<turnstile>s \<midarrow>Skip\<midarrow>n-> s' ==> s' = s
lemma evaln_Skip_eq:
G\<turnstile>s \<midarrow>Skip\<midarrow>n-> s' = (s = s')
lemma evaln_eval:
G\<turnstile>s0.0 \<midarrow>t\<succ>\<midarrow>n-> (v, s1.0)
==> G\<turnstile>s0.0 \<midarrow>t\<succ>-> (v, s1.0)
lemma Suc_le_D_lemma:
[| Suc n ≤ m'; !!m. n ≤ m ==> P (Suc m) |] ==> P m'
lemma evaln_nonstrict:
[| G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (w, s'); n ≤ m |]
==> G\<turnstile>s \<midarrow>t\<succ>\<midarrow>m-> (w, s')
lemma evaln_nonstrict_Suc:
G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n-> (w, s')
==> G\<turnstile>s \<midarrow>t\<succ>\<midarrow>Suc n-> (w, s')
lemma evaln_max2:
[| G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>n1.0-> (w1.0, s1');
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>n2.0-> (w2.0, s2') |]
==> G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>max n1.0
n2.0-> (w1.0, s1') ∧
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>max n1.0
n2.0-> (w2.0, s2')
corollary evaln_max2E:
[| G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>n1.0-> (w1.0, s1');
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>n2.0-> (w2.0, s2');
[| G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>max n1.0
n2.0-> (w1.0, s1');
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>max n1.0
n2.0-> (w2.0, s2') |]
==> P |]
==> P
lemma evaln_max3:
[| G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>n1.0-> (w1.0, s1');
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>n2.0-> (w2.0, s2');
G\<turnstile>s3.0 \<midarrow>t3.0\<succ>\<midarrow>n3.0-> (w3.0, s3') |]
==> G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>max (max n1.0 n2.0)
n3.0-> (w1.0, s1') ∧
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>max (max n1.0 n2.0)
n3.0-> (w2.0, s2') ∧
G\<turnstile>s3.0 \<midarrow>t3.0\<succ>\<midarrow>max (max n1.0 n2.0)
n3.0-> (w3.0, s3')
corollary evaln_max3E:
[| G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>n1.0-> (w1.0, s1');
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>n2.0-> (w2.0, s2');
G\<turnstile>s3.0 \<midarrow>t3.0\<succ>\<midarrow>n3.0-> (w3.0, s3');
[| G\<turnstile>s1.0 \<midarrow>t1.0\<succ>\<midarrow>max (max n1.0 n2.0)
n3.0-> (w1.0, s1');
G\<turnstile>s2.0 \<midarrow>t2.0\<succ>\<midarrow>max (max n1.0 n2.0)
n3.0-> (w2.0, s2');
G\<turnstile>s3.0 \<midarrow>t3.0\<succ>\<midarrow>max (max n1.0 n2.0)
n3.0-> (w3.0, s3') |]
==> P |]
==> P
lemma le_max3I1:
n2.0 ≤ max n1.0 (max n2.0 n3.0)
lemma le_max3I2:
n3.0 ≤ max n1.0 (max n2.0 n3.0)
lemma eval_evaln:
G\<turnstile>s0.0 \<midarrow>t\<succ>-> (v, s1.0)
==> ∃n. G\<turnstile>s0.0 \<midarrow>t\<succ>\<midarrow>n-> (v, s1.0)