(* Title: HOL/MicroJava/J/Eval.thy ID: $Id: Eval.thy,v 1.25 2005/06/17 14:13:09 haftmann Exp $ Author: David von Oheimb Copyright 1999 Technische Universitaet Muenchen *) header {* \isaheader{Operational Evaluation (big step) Semantics} *} theory Eval imports State WellType begin -- "Auxiliary notions" constdefs fits :: "java_mb prog => state => val => ty => bool" ("_,_\<turnstile>_ fits _"[61,61,61,61]60) "G,s\<turnstile>a' fits T ≡ case T of PrimT T' => False | RefT T' => a'=Null ∨ G\<turnstile>obj_ty(lookup_obj s a')\<preceq>T" constdefs catch ::"java_mb prog => xstate => cname => bool" ("_,_\<turnstile>catch _"[61,61,61]60) "G,s\<turnstile>catch C≡ case abrupt s of None => False | Some a => G,store s\<turnstile> a fits Class C" constdefs lupd :: "vname => val => state => state" ("lupd'(_\<mapsto>_')"[10,10]1000) "lupd vn v ≡ λ (hp,loc). (hp, (loc(vn\<mapsto>v)))" constdefs new_xcpt_var :: "vname => xstate => xstate" "new_xcpt_var vn ≡ λ(x,s). Norm (lupd(vn\<mapsto>the x) s)" -- "Evaluation relations" consts eval :: "java_mb prog => (xstate × expr × val × xstate) set" evals :: "java_mb prog => (xstate × expr list × val list × xstate) set" exec :: "java_mb prog => (xstate × stmt × xstate) set" syntax (xsymbols) eval :: "[java_mb prog,xstate,expr,val,xstate] => bool " ("_ \<turnstile> _ -_\<succ>_-> _" [51,82,60,82,82] 81) evals:: "[java_mb prog,xstate,expr list, val list,xstate] => bool " ("_ \<turnstile> _ -_[\<succ>]_-> _" [51,82,60,51,82] 81) exec :: "[java_mb prog,xstate,stmt, xstate] => bool " ("_ \<turnstile> _ -_-> _" [51,82,60,82] 81) syntax eval :: "[java_mb prog,xstate,expr,val,xstate] => bool " ("_ |- _ -_>_-> _" [51,82,60,82,82] 81) evals:: "[java_mb prog,xstate,expr list, val list,xstate] => bool " ("_ |- _ -_[>]_-> _" [51,82,60,51,82] 81) exec :: "[java_mb prog,xstate,stmt, xstate] => bool " ("_ |- _ -_-> _" [51,82,60,82] 81) translations "G\<turnstile>s -e \<succ> v-> (x,s')" <= "(s, e, v, x, s') ∈ eval G" "G\<turnstile>s -e \<succ> v-> s' " == "(s, e, v, s') ∈ eval G" "G\<turnstile>s -e[\<succ>]v-> (x,s')" <= "(s, e, v, x, s') ∈ evals G" "G\<turnstile>s -e[\<succ>]v-> s' " == "(s, e, v, s') ∈ evals G" "G\<turnstile>s -c -> (x,s')" <= "(s, c, x, s') ∈ exec G" "G\<turnstile>s -c -> s' " == "(s, c, s') ∈ exec G" inductive "eval G" "evals G" "exec G" intros (* evaluation of expressions *) XcptE:"G\<turnstile>(Some xc,s) -e\<succ>arbitrary-> (Some xc,s)" -- "cf. 15.5" -- "cf. 15.8.1" NewC: "[| h = heap s; (a,x) = new_Addr h; h'= h(a\<mapsto>(C,init_vars (fields (G,C)))) |] ==> G\<turnstile>Norm s -NewC C\<succ>Addr a-> c_hupd h' (x,s)" -- "cf. 15.15" Cast: "[| G\<turnstile>Norm s0 -e\<succ>v-> (x1,s1); x2 = raise_if (¬ cast_ok G C (heap s1) v) ClassCast x1 |] ==> G\<turnstile>Norm s0 -Cast C e\<succ>v-> (x2,s1)" -- "cf. 15.7.1" Lit: "G\<turnstile>Norm s -Lit v\<succ>v-> Norm s" BinOp:"[| G\<turnstile>Norm s -e1\<succ>v1-> s1; G\<turnstile>s1 -e2\<succ>v2-> s2; v = (case bop of Eq => Bool (v1 = v2) | Add => Intg (the_Intg v1 + the_Intg v2)) |] ==> G\<turnstile>Norm s -BinOp bop e1 e2\<succ>v-> s2" -- "cf. 15.13.1, 15.2" LAcc: "G\<turnstile>Norm s -LAcc v\<succ>the (locals s v)-> Norm s" -- "cf. 15.25.1" LAss: "[| G\<turnstile>Norm s -e\<succ>v-> (x,(h,l)); l' = (if x = None then l(va\<mapsto>v) else l) |] ==> G\<turnstile>Norm s -va::=e\<succ>v-> (x,(h,l'))" -- "cf. 15.10.1, 15.2" FAcc: "[| G\<turnstile>Norm s0 -e\<succ>a'-> (x1,s1); v = the (snd (the (heap s1 (the_Addr a'))) (fn,T)) |] ==> G\<turnstile>Norm s0 -{T}e..fn\<succ>v-> (np a' x1,s1)" -- "cf. 15.25.1" FAss: "[| G\<turnstile> Norm s0 -e1\<succ>a'-> (x1,s1); a = the_Addr a'; G\<turnstile>(np a' x1,s1) -e2\<succ>v -> (x2,s2); h = heap s2; (c,fs) = the (h a); h' = h(a\<mapsto>(c,(fs((fn,T)\<mapsto>v)))) |] ==> G\<turnstile>Norm s0 -{T}e1..fn:=e2\<succ>v-> c_hupd h' (x2,s2)" -- "cf. 15.11.4.1, 15.11.4.2, 15.11.4.4, 15.11.4.5, 14.15" Call: "[| G\<turnstile>Norm s0 -e\<succ>a'-> s1; a = the_Addr a'; G\<turnstile>s1 -ps[\<succ>]pvs-> (x,(h,l)); dynT = fst (the (h a)); (md,rT,pns,lvars,blk,res) = the (method (G,dynT) (mn,pTs)); G\<turnstile>(np a' x,(h,(init_vars lvars)(pns[\<mapsto>]pvs)(This\<mapsto>a'))) -blk-> s3; G\<turnstile> s3 -res\<succ>v -> (x4,s4) |] ==> G\<turnstile>Norm s0 -{C}e..mn({pTs}ps)\<succ>v-> (x4,(heap s4,l))" -- "evaluation of expression lists" -- "cf. 15.5" XcptEs:"G\<turnstile>(Some xc,s) -e[\<succ>]arbitrary-> (Some xc,s)" -- "cf. 15.11.???" Nil: "G\<turnstile>Norm s0 -[][\<succ>][]-> Norm s0" -- "cf. 15.6.4" Cons: "[| G\<turnstile>Norm s0 -e \<succ> v -> s1; G\<turnstile> s1 -es[\<succ>]vs-> s2 |] ==> G\<turnstile>Norm s0 -e#es[\<succ>]v#vs-> s2" -- "execution of statements" -- "cf. 14.1" XcptS:"G\<turnstile>(Some xc,s) -c-> (Some xc,s)" -- "cf. 14.5" Skip: "G\<turnstile>Norm s -Skip-> Norm s" -- "cf. 14.7" Expr: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1 |] ==> G\<turnstile>Norm s0 -Expr e-> s1" -- "cf. 14.2" Comp: "[| G\<turnstile>Norm s0 -c1-> s1; G\<turnstile> s1 -c2-> s2|] ==> G\<turnstile>Norm s0 -c1;; c2-> s2" -- "cf. 14.8.2" Cond: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1; G\<turnstile> s1 -(if the_Bool v then c1 else c2)-> s2|] ==> G\<turnstile>Norm s0 -If(e) c1 Else c2-> s2" -- "cf. 14.10, 14.10.1" LoopF:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; ¬the_Bool v |] ==> G\<turnstile>Norm s0 -While(e) c-> s1" LoopT:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; the_Bool v; G\<turnstile>s1 -c-> s2; G\<turnstile>s2 -While(e) c-> s3 |] ==> G\<turnstile>Norm s0 -While(e) c-> s3" lemmas eval_evals_exec_induct = eval_evals_exec.induct [split_format (complete)] lemma NewCI: "[|new_Addr (heap s) = (a,x); s' = c_hupd (heap s(a\<mapsto>(C,init_vars (fields (G,C))))) (x,s)|] ==> G\<turnstile>Norm s -NewC C\<succ>Addr a-> s'" apply (simp (no_asm_simp)) apply (rule eval_evals_exec.NewC) apply auto done lemma eval_evals_exec_no_xcpt: "!!s s'. (G\<turnstile>(x,s) -e \<succ> v -> (x',s') --> x'=None --> x=None) ∧ (G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x'=None --> x=None) ∧ (G\<turnstile>(x,s) -c -> (x',s') --> x'=None --> x=None)" apply(simp (no_asm_simp) only: split_tupled_all) apply(rule eval_evals_exec_induct) apply(unfold c_hupd_def) apply(simp_all) done lemma eval_no_xcpt: "G\<turnstile>(x,s) -e\<succ>v-> (None,s') ==> x=None" apply (drule eval_evals_exec_no_xcpt [THEN conjunct1, THEN mp]) apply (fast) done lemma evals_no_xcpt: "G\<turnstile>(x,s) -e[\<succ>]v-> (None,s') ==> x=None" apply (drule eval_evals_exec_no_xcpt [THEN conjunct2, THEN conjunct1, THEN mp]) apply (fast) done lemma exec_no_xcpt: "G \<turnstile> (x, s) -c-> (None, s') ==> x = None" apply (drule eval_evals_exec_no_xcpt [THEN conjunct2 [THEN conjunct2], rule_format]) apply simp+ done lemma eval_evals_exec_xcpt: "!!s s'. (G\<turnstile>(x,s) -e \<succ> v -> (x',s') --> x=Some xc --> x'=Some xc ∧ s'=s) ∧ (G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x=Some xc --> x'=Some xc ∧ s'=s) ∧ (G\<turnstile>(x,s) -c -> (x',s') --> x=Some xc --> x'=Some xc ∧ s'=s)" apply (simp (no_asm_simp) only: split_tupled_all) apply (rule eval_evals_exec_induct) apply (unfold c_hupd_def) apply (simp_all) done lemma eval_xcpt: "G\<turnstile>(Some xc,s) -e\<succ>v-> (x',s') ==> x'=Some xc ∧ s'=s" apply (drule eval_evals_exec_xcpt [THEN conjunct1, THEN mp]) apply (fast) done lemma exec_xcpt: "G\<turnstile>(Some xc,s) -s0-> (x',s') ==> x'=Some xc ∧ s'=s" apply (drule eval_evals_exec_xcpt [THEN conjunct2, THEN conjunct2, THEN mp]) apply (fast) done end
lemmas eval_evals_exec_induct:
[| !!e a b xc. P1.0 (Some xc) a b e arbitrary (Some xc) a b; !!C a h h' aa b x. [| h = fst (aa, b); (a, x) = new_Addr h; h' = h(a |-> (C, init_vars (fields (G, C)))) |] ==> (%(v, x, y). P1.0 None aa b (NewC C) (Addr a) v x y) (c_hupd h' (x, aa, b)); !!C e a b aa ba v x1 x2. [| (Norm (a, b), e, v, x1, aa, ba) ∈ eval G; P1.0 None a b e v x1 aa ba; x2 = raise_if (¬ cast_ok G C (fst (aa, ba)) v) ClassCast x1 |] ==> P1.0 None a b (Cast C e) v x2 aa ba; !!a b v. P1.0 None a b (Lit v) v None a b; !!bop e1 e2 a b aa ab ba ac ad bb v v1 v2. [| (Norm (a, b), e1, v1, aa, ab, ba) ∈ eval G; P1.0 None a b e1 v1 aa ab ba; ((aa, ab, ba), e2, v2, ac, ad, bb) ∈ eval G; P1.0 aa ab ba e2 v2 ac ad bb; v = (case bop of Eq => Bool (v1 = v2) | Add => Intg (the_Intg v1 + the_Intg v2)) |] ==> P1.0 None a b (BinOp bop e1 e2) v ac ad bb; !!a b v. P1.0 None a b (LAcc v) (the (snd (a, b) v)) None a b; !!e h l l' a b v va x. [| (Norm (a, b), e, v, x, h, l) ∈ eval G; P1.0 None a b e v x h l; l' = (if x = None then l(va |-> v) else l) |] ==> P1.0 None a b (va::=e) v x h l'; !!T a' e fn a b aa ba v x1. [| (Norm (a, b), e, a', x1, aa, ba) ∈ eval G; P1.0 None a b e a' x1 aa ba; v = the (snd (the (fst (aa, ba) (the_Addr a'))) (fn, T)) |] ==> P1.0 None a b ({T}e..fn) v (np a' x1) aa ba; !!T a a' c e1 e2 fn fs h h' aa b ab ba ac bb v x1 x2. [| (Norm (aa, b), e1, a', x1, ab, ba) ∈ eval G; P1.0 None aa b e1 a' x1 ab ba; a = the_Addr a'; ((np a' x1, ab, ba), e2, v, x2, ac, bb) ∈ eval G; P1.0 (np a' x1) ab ba e2 v x2 ac bb; h = fst (ac, bb); (c, fs) = the (h a); h' = h(a |-> (c, fs((fn, T) |-> v))) |] ==> (%(va, x, y). P1.0 None aa b ({T}e1..fn:=e2) v va x y) (c_hupd h' (x2, ac, bb)); !!C a a' blk dynT e h l lvars md mn pTs pns ps pvs rT res aa b ab ac ba ad ae bb af bc v x x4. [| (Norm (aa, b), e, a', ab, ac, ba) ∈ eval G; P1.0 None aa b e a' ab ac ba; a = the_Addr a'; ((ab, ac, ba), ps, pvs, x, h, l) ∈ evals G; P2.0 ab ac ba ps pvs x h l; dynT = fst (the (h a)); (md, rT, pns, lvars, blk, res) = the (method (G, dynT) (mn, pTs)); ((np a' x, h, init_vars lvars(pns [|->] pvs, This |-> a')), blk, ad, ae, bb) ∈ exec G; P3.0 (np a' x) h (init_vars lvars(pns [|->] pvs, This |-> a')) blk ad ae bb; ((ad, ae, bb), res, v, x4, af, bc) ∈ eval G; P1.0 ad ae bb res v x4 af bc |] ==> P1.0 None aa b ({C}e..mn( {pTs}ps)) v x4 (fst (af, bc)) l; !!e a b xc. P2.0 (Some xc) a b e arbitrary (Some xc) a b; !!a b. P2.0 None a b [] [] None a b; !!e es a b aa ab ba ac ad bb v vs. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; ((aa, ab, ba), es, vs, ac, ad, bb) ∈ evals G; P2.0 aa ab ba es vs ac ad bb |] ==> P2.0 None a b (e # es) (v # vs) ac ad bb; !!c a b xc. P3.0 (Some xc) a b c (Some xc) a b; !!a b. P3.0 None a b Skip None a b; !!e a b aa ab ba v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba |] ==> P3.0 None a b (Expr e) aa ab ba; !!c1 c2 a b aa ab ba ac ad bb. [| (Norm (a, b), c1, aa, ab, ba) ∈ exec G; P3.0 None a b c1 aa ab ba; ((aa, ab, ba), c2, ac, ad, bb) ∈ exec G; P3.0 aa ab ba c2 ac ad bb |] ==> P3.0 None a b (c1;; c2) ac ad bb; !!c1 c2 e a b aa ab ba ac ad bb v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; ((aa, ab, ba), if the_Bool v then c1 else c2, ac, ad, bb) ∈ exec G; P3.0 aa ab ba (if the_Bool v then c1 else c2) ac ad bb |] ==> P3.0 None a b (If (e) c1 Else c2) ac ad bb; !!c e a b aa ab ba v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; ¬ the_Bool v |] ==> P3.0 None a b (While (e) c) aa ab ba; !!c e a b aa ab ba ac ad bb ae af bc v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; the_Bool v; ((aa, ab, ba), c, ac, ad, bb) ∈ exec G; P3.0 aa ab ba c ac ad bb; ((ac, ad, bb), While (e) c, ae, af, bc) ∈ exec G; P3.0 ac ad bb (While (e) c) ae af bc |] ==> P3.0 None a b (While (e) c) ae af bc |] ==> (((xka, xkb, xkc), xja, xia, xha, xhb, xhc) ∈ eval G --> P1.0 xka xkb xkc xja xia xha xhb xhc) ∧ (((xga, xgb, xgc), xfa, xea, xda, xdb, xdc) ∈ evals G --> P2.0 xga xgb xgc xfa xea xda xdb xdc) ∧ (((xca, xcb, xcc), xba, xaa, xab, xac) ∈ exec G --> P3.0 xca xcb xcc xba xaa xab xac)
lemmas eval_evals_exec_induct:
[| !!e a b xc. P1.0 (Some xc) a b e arbitrary (Some xc) a b; !!C a h h' aa b x. [| h = fst (aa, b); (a, x) = new_Addr h; h' = h(a |-> (C, init_vars (fields (G, C)))) |] ==> (%(v, x, y). P1.0 None aa b (NewC C) (Addr a) v x y) (c_hupd h' (x, aa, b)); !!C e a b aa ba v x1 x2. [| (Norm (a, b), e, v, x1, aa, ba) ∈ eval G; P1.0 None a b e v x1 aa ba; x2 = raise_if (¬ cast_ok G C (fst (aa, ba)) v) ClassCast x1 |] ==> P1.0 None a b (Cast C e) v x2 aa ba; !!a b v. P1.0 None a b (Lit v) v None a b; !!bop e1 e2 a b aa ab ba ac ad bb v v1 v2. [| (Norm (a, b), e1, v1, aa, ab, ba) ∈ eval G; P1.0 None a b e1 v1 aa ab ba; ((aa, ab, ba), e2, v2, ac, ad, bb) ∈ eval G; P1.0 aa ab ba e2 v2 ac ad bb; v = (case bop of Eq => Bool (v1 = v2) | Add => Intg (the_Intg v1 + the_Intg v2)) |] ==> P1.0 None a b (BinOp bop e1 e2) v ac ad bb; !!a b v. P1.0 None a b (LAcc v) (the (snd (a, b) v)) None a b; !!e h l l' a b v va x. [| (Norm (a, b), e, v, x, h, l) ∈ eval G; P1.0 None a b e v x h l; l' = (if x = None then l(va |-> v) else l) |] ==> P1.0 None a b (va::=e) v x h l'; !!T a' e fn a b aa ba v x1. [| (Norm (a, b), e, a', x1, aa, ba) ∈ eval G; P1.0 None a b e a' x1 aa ba; v = the (snd (the (fst (aa, ba) (the_Addr a'))) (fn, T)) |] ==> P1.0 None a b ({T}e..fn) v (np a' x1) aa ba; !!T a a' c e1 e2 fn fs h h' aa b ab ba ac bb v x1 x2. [| (Norm (aa, b), e1, a', x1, ab, ba) ∈ eval G; P1.0 None aa b e1 a' x1 ab ba; a = the_Addr a'; ((np a' x1, ab, ba), e2, v, x2, ac, bb) ∈ eval G; P1.0 (np a' x1) ab ba e2 v x2 ac bb; h = fst (ac, bb); (c, fs) = the (h a); h' = h(a |-> (c, fs((fn, T) |-> v))) |] ==> (%(va, x, y). P1.0 None aa b ({T}e1..fn:=e2) v va x y) (c_hupd h' (x2, ac, bb)); !!C a a' blk dynT e h l lvars md mn pTs pns ps pvs rT res aa b ab ac ba ad ae bb af bc v x x4. [| (Norm (aa, b), e, a', ab, ac, ba) ∈ eval G; P1.0 None aa b e a' ab ac ba; a = the_Addr a'; ((ab, ac, ba), ps, pvs, x, h, l) ∈ evals G; P2.0 ab ac ba ps pvs x h l; dynT = fst (the (h a)); (md, rT, pns, lvars, blk, res) = the (method (G, dynT) (mn, pTs)); ((np a' x, h, init_vars lvars(pns [|->] pvs, This |-> a')), blk, ad, ae, bb) ∈ exec G; P3.0 (np a' x) h (init_vars lvars(pns [|->] pvs, This |-> a')) blk ad ae bb; ((ad, ae, bb), res, v, x4, af, bc) ∈ eval G; P1.0 ad ae bb res v x4 af bc |] ==> P1.0 None aa b ({C}e..mn( {pTs}ps)) v x4 (fst (af, bc)) l; !!e a b xc. P2.0 (Some xc) a b e arbitrary (Some xc) a b; !!a b. P2.0 None a b [] [] None a b; !!e es a b aa ab ba ac ad bb v vs. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; ((aa, ab, ba), es, vs, ac, ad, bb) ∈ evals G; P2.0 aa ab ba es vs ac ad bb |] ==> P2.0 None a b (e # es) (v # vs) ac ad bb; !!c a b xc. P3.0 (Some xc) a b c (Some xc) a b; !!a b. P3.0 None a b Skip None a b; !!e a b aa ab ba v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba |] ==> P3.0 None a b (Expr e) aa ab ba; !!c1 c2 a b aa ab ba ac ad bb. [| (Norm (a, b), c1, aa, ab, ba) ∈ exec G; P3.0 None a b c1 aa ab ba; ((aa, ab, ba), c2, ac, ad, bb) ∈ exec G; P3.0 aa ab ba c2 ac ad bb |] ==> P3.0 None a b (c1;; c2) ac ad bb; !!c1 c2 e a b aa ab ba ac ad bb v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; ((aa, ab, ba), if the_Bool v then c1 else c2, ac, ad, bb) ∈ exec G; P3.0 aa ab ba (if the_Bool v then c1 else c2) ac ad bb |] ==> P3.0 None a b (If (e) c1 Else c2) ac ad bb; !!c e a b aa ab ba v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; ¬ the_Bool v |] ==> P3.0 None a b (While (e) c) aa ab ba; !!c e a b aa ab ba ac ad bb ae af bc v. [| (Norm (a, b), e, v, aa, ab, ba) ∈ eval G; P1.0 None a b e v aa ab ba; the_Bool v; ((aa, ab, ba), c, ac, ad, bb) ∈ exec G; P3.0 aa ab ba c ac ad bb; ((ac, ad, bb), While (e) c, ae, af, bc) ∈ exec G; P3.0 ac ad bb (While (e) c) ae af bc |] ==> P3.0 None a b (While (e) c) ae af bc |] ==> (((xka, xkb, xkc), xja, xia, xha, xhb, xhc) ∈ eval G --> P1.0 xka xkb xkc xja xia xha xhb xhc) ∧ (((xga, xgb, xgc), xfa, xea, xda, xdb, xdc) ∈ evals G --> P2.0 xga xgb xgc xfa xea xda xdb xdc) ∧ (((xca, xcb, xcc), xba, xaa, xab, xac) ∈ exec G --> P3.0 xca xcb xcc xba xaa xab xac)
lemma NewCI:
[| new_Addr (fst s) = (a, x); s' = c_hupd (fst s(a |-> (C, init_vars (fields (G, C))))) (x, s) |] ==> G |- Norm s -NewC C>Addr a-> s'
lemma eval_evals_exec_no_xcpt:
(G |- (x, s) -e>v-> (x', s') --> x' = None --> x = None) ∧ (G |- (x, s) -es[>]vs-> (x', s') --> x' = None --> x = None) ∧ (G |- (x, s) -c-> (x', s') --> x' = None --> x = None)
lemma eval_no_xcpt:
G |- (x, s) -e>v-> Norm s' ==> x = None
lemma evals_no_xcpt:
G |- (x, s) -e[>]v-> Norm s' ==> x = None
lemma exec_no_xcpt:
G |- (x, s) -c-> Norm s' ==> x = None
lemma eval_evals_exec_xcpt:
(G |- (x, s) -e>v-> (x', s') --> x = Some xc --> x' = Some xc ∧ s' = s) ∧ (G |- (x, s) -es[>]vs-> (x', s') --> x = Some xc --> x' = Some xc ∧ s' = s) ∧ (G |- (x, s) -c-> (x', s') --> x = Some xc --> x' = Some xc ∧ s' = s)
lemma eval_xcpt:
G |- (Some xc, s) -e>v-> (x', s') ==> x' = Some xc ∧ s' = s
lemma exec_xcpt:
G |- (Some xc, s) -s0.0-> (x', s') ==> x' = Some xc ∧ s' = s