[700] | 1 | include "Clight/Cexec.ma". |
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[226] | 2 | |
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[487] | 3 | definition yields ≝ λA.λa:res A.λv':A. |
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[399] | 4 | match a with [ OK v ⇒ v' = v | _ ⇒ False ]. |
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[239] | 5 | |
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[399] | 6 | (* This tells us that some execution of a results in v'. |
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[239] | 7 | (There may be many possible executions due to I/O, but we're trying to prove |
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| 8 | that one particular one exists corresponding to a derivation in the inductive |
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| 9 | semantics.) *) |
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[487] | 10 | let rec yieldsIO (A:Type[0]) (a:IO io_out io_in A) (v':A) on a : Prop ≝ |
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[399] | 11 | match a with [ Value v ⇒ v' = v | Interact _ k ⇒ ∃r.yieldsIO A (k r) v' | _ ⇒ False ]. |
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| 12 | |
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[487] | 13 | definition yields_sig : ∀A,P. res (Sig A P) → A → Prop ≝ |
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| 14 | λA,P,e,v. match e with [ OK v' ⇒ match v' with [ dp v'' _ ⇒ v = v'' ] | _ ⇒ False]. |
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[399] | 15 | |
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[487] | 16 | let rec yieldsIO_sig (A:Type[0]) (P:A → Prop) (e:IO io_out io_in (Sig A P)) (v:A) on e : Prop ≝ |
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[239] | 17 | match e with |
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[487] | 18 | [ Value v' ⇒ match v' with [ dp v'' _ ⇒ v = v'' ] |
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[399] | 19 | | Interact _ k ⇒ ∃r.yieldsIO_sig A P (k r) v |
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[239] | 20 | | _ ⇒ False]. |
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| 21 | |
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[487] | 22 | lemma remove_io_sig: ∀A. ∀P:A → Prop. ∀a,v',p. |
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[399] | 23 | yieldsIO A a v' → |
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[487] | 24 | yieldsIO_sig A P (io_inject io_out io_in A P (Some ? a) p) v'. |
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| 25 | #A #P #a elim a; |
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| 26 | [ #a #k #IH #v' #p #H whd in H ⊢ %; elim H; #r #H' %{ r} @IH @H' |
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| 27 | | #v #v' #p #H @H |
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| 28 | | #a #b *; |
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| 29 | ] qed. |
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[399] | 30 | |
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[487] | 31 | lemma yields_eq: ∀A,a,v'. yields A a v' → a = OK ? v'. |
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| 32 | #A #a #v' cases a; //; whd in ⊢ (% → ?); *; |
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| 33 | qed. |
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[399] | 34 | |
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[487] | 35 | lemma yields_sig_eq: ∀A,P,e,v. yields_sig A P e v → ∃p. e = OK ? (dp … v p). |
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| 36 | #A #P #e #v cases e; |
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| 37 | [ #vp cases vp; #v' #p #H whd in H; >H %{ p} @refl |
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| 38 | | *; |
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| 39 | ] qed. |
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[399] | 40 | |
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[499] | 41 | lemma is_pointer_compat_true: ∀b,sp. |
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| 42 | pointer_compat b sp → |
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| 43 | is_pointer_compat b sp = true. |
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| 44 | #b #sp #H whd in ⊢ (??%?); |
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| 45 | elim (pointer_compat_dec b sp); |
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[487] | 46 | [ // |
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| 47 | | #H' @False_ind @(absurd … H H') |
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| 48 | ] qed. |
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[226] | 49 | |
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[500] | 50 | lemma dec_true: ∀P:Prop.∀f:P + ¬P.∀p:P.∀Q:(P + ¬P) → Type[0]. (∀p'.Q (inl ?? p')) → Q f. |
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| 51 | #P #f #p #Q #H cases f; |
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| 52 | [ @H |
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| 53 | | #np cut False [ @(absurd ? p np) | * ] |
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[487] | 54 | ] qed. |
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[226] | 55 | |
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[500] | 56 | lemma dec_false: ∀P:Prop.∀f:P + ¬P.∀p:¬P.∀Q:(P + ¬P) → Type[0]. (∀p'.Q (inr ?? p')) → Q f. |
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| 57 | #P #f #p #Q #H cases f; |
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| 58 | [ #np cut False [ @(absurd ? np p) | * ] |
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| 59 | | @H |
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| 60 | ] qed. |
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| 61 | |
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[487] | 62 | theorem is_det: ∀p,s,s'. |
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[226] | 63 | initial_state p s → initial_state p s' → s = s'. |
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[487] | 64 | #p #s #s' #H1 #H2 |
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| 65 | inversion H1; #b1 #f1 #ge1 #m1 #e11 #e12 #e13 #e14 #e15 |
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| 66 | inversion H2; #b2 #f2 #ge2 #m2 #e21 #e22 #e23 #e24 #e25 |
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| 67 | >e11 in e21 #e1 >(?:ge1 = ge2) in e13 e14 |
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| 68 | [ 2: destruct (e1) skip (e11); @refl ] |
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| 69 | #e13 #e14 |
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[485] | 70 | |
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[487] | 71 | >e12 in e22 #e2 destruct (e2) skip (e12); |
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[485] | 72 | |
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[487] | 73 | >e13 in e23 #e3 >(?:b1 = b2) in e14 |
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| 74 | [ >e24 #e4 >(?:f2 = f1) |
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| 75 | [ //; |
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| 76 | | destruct (e4) skip (e24 e25); //; |
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| 77 | ] |
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| 78 | | destruct (e3) skip (e13); // |
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| 79 | ] qed. |
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[226] | 80 | |
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[487] | 81 | lemma remove_res_sig: ∀A. ∀P:A → Prop. ∀a,v',p. |
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[399] | 82 | yields A a v' → |
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[487] | 83 | yields_sig A P (err_inject A P (Some ? a) p) v'. |
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| 84 | #A #P #a cases a; |
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| 85 | [ #v #v' #p #H @H |
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| 86 | | #a #b *; |
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| 87 | ] qed. |
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[226] | 88 | |
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[385] | 89 | |
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[487] | 90 | theorem the_initial_state: |
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[485] | 91 | ∀p,s. initial_state p s → ∃ge. yields ? (make_initial_state p) 〈ge,s〉. |
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[487] | 92 | #p #s cases p; #fns #main #globs #H |
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| 93 | inversion H; |
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| 94 | #b #f #ge #m #e1 #e2 #e3 #e4 #e5 %{ge} |
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| 95 | whd in ⊢ (??%?); |
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| 96 | >e1 |
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| 97 | whd in ⊢ (??%?); |
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| 98 | >e2 |
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| 99 | whd in ⊢ (??%?); |
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| 100 | >e3 |
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| 101 | whd in ⊢ (??%?); |
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| 102 | >e4 |
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| 103 | whd; @refl |
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| 104 | qed. |
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[385] | 105 | |
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[487] | 106 | lemma cast_complete: ∀m,v,ty,ty',v'. |
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[399] | 107 | cast m v ty ty' v' → yields ? (exec_cast m v ty ty') v'. |
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[487] | 108 | #m #v #ty #ty' #v' #H |
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| 109 | elim H; |
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| 110 | [ #m #i #sz1 #sz2 #sg1 #sg2 @refl |
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| 111 | | #m #f #sz #szi #sg @refl |
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| 112 | | #m #i #sz #sz' #sg @refl |
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| 113 | | #m #f #sz #sz' @refl |
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[500] | 114 | | #m #r #r' #ty #ty' #b #pc #ofs #H1 #H2 #pc' |
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| 115 | elim H1 in pc ⊢ % [ #r1 #ty1 #pc | #r1 #ty1 #n1 #pc | #tys1 #ty1 #pc letin r1 ≝ Code ] |
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| 116 | whd in ⊢ (??%?) |
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| 117 | [ 1,2: @(dec_true ? (eq_region_dec r1 r1) (refl ??) …) #H0 whd in ⊢ (??%?) ] |
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| 118 | elim H2 in pc' ⊢ %; [ 1,4,7: #sp2 #ty2 | 2,5,8: #sp2 #ty2 #n2 | 3,6,9: #tys2 #ty2 letin sp2 ≝ Code ] |
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| 119 | #pc' whd in ⊢ (??%?) |
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| 120 | @(dec_true ? (pointer_compat_dec b sp2) pc') // |
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[487] | 121 | | #m #sz #si #ty'' #r #H cases H; //; |
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| 122 | | #m #t #t' #r #r' #H #H' cases H; try #a try #b try #c cases H'; try #d try #e try #f |
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[500] | 123 | whd in ⊢ (??%?); try @refl @(dec_true ? (eq_region_dec a a) (refl ??)) #H0 @refl |
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[487] | 124 | ] qed. |
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[226] | 125 | |
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| 126 | (* Use to narrow down the choice of expression to just the lvalues. *) |
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[498] | 127 | lemma lvalue_expr: ∀ge,env,m,e,ty,l,ofs,tr. ∀P:expr_descr → Prop. |
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| 128 | eval_lvalue ge env m (Expr e ty) l ofs tr → |
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[226] | 129 | (∀id. P (Evar id)) → (∀e'. P (Ederef e')) → (∀e',id. P (Efield e' id)) → |
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| 130 | P e. |
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[498] | 131 | #ge #env #m #e #ty #l #ofs #tr #P #H @(eval_lvalue_inv_ind … H) |
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[487] | 132 | [ #id #l #ty #e1 #e2 #e3 #e4 #e5 #e6 destruct; // |
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| 133 | | #id #sp #l #ty #e1 #e2 #e3 #e4 #e5 #e6 #e7 destruct; // |
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| 134 | | #e #ty #sp #l #ofs #tr #H #e1 #e2 #e3 #e4 #e5 destruct; // |
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[498] | 135 | | #e #id #ty #l #ofs #id' #fs #d #tr #H #e1 #e2 (* bogus? *) #_ #e3 #e4 #e5 #e6 #e7 destruct; // |
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| 136 | | #e #id #ty #l #ofs #id' #fs #tr #H #e1 (* bogus? *) #_ #e2 #e3 #e4 #e5 #e6 destruct; // |
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[487] | 137 | ] qed. |
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[226] | 138 | |
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[487] | 139 | lemma bool_of_val_3_complete : ∀v,ty,r. bool_of_val v ty r → ∃b. r = of_bool b ∧ yields ? (exec_bool_of_val v ty) b. |
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| 140 | #v #ty #r #H elim H; #v #t #H' elim H'; |
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| 141 | [ #i #is #s #ne %{ true} % //; whd; >(eq_false … ne) //; |
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[500] | 142 | | #r #b #pc #i #i0 #s %{ true} % // |
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[487] | 143 | | #f #s #ne %{ true} % //; whd; >(Feq_zero_false … ne) //; |
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| 144 | | #i #s %{ false} % //; |
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| 145 | | #r #r' #t %{ false} % //; |
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| 146 | | #s %{ false} % //; whd; >(Feq_zero_true …) //; |
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| 147 | ] |
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| 148 | qed. |
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[226] | 149 | |
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[487] | 150 | lemma bool_of_true: ∀v,ty. is_true v ty → yields ? (exec_bool_of_val v ty) true. |
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| 151 | #v #ty #H elim H; |
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| 152 | [ #i #is #s #ne whd; >(eq_false … ne) //; |
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| 153 | | #p #b #i #i0 #s // |
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| 154 | | #f #s #ne whd; >(Feq_zero_false … ne) //; |
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| 155 | ] |
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| 156 | qed. |
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[226] | 157 | |
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[487] | 158 | lemma bool_of_false: ∀v,ty. is_false v ty → yields ? (exec_bool_of_val v ty) false. |
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| 159 | #v #ty #H elim H; |
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| 160 | [ #i #s //; |
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| 161 | | #r #r' #t //; |
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| 162 | | #s whd; >(Feq_zero_true …) //; |
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| 163 | ] |
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| 164 | qed. |
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[226] | 165 | |
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[487] | 166 | lemma expr_lvalue_complete: ∀ge,env,m. |
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[399] | 167 | (∀e,v,tr. eval_expr ge env m e v tr → yields ? (exec_expr ge env m e) (〈v,tr〉)) ∧ |
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[498] | 168 | (∀e,l,off,tr. eval_lvalue ge env m e l off tr → yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉)). |
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[487] | 169 | #ge #env #m |
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| 170 | @(combined_expr_lvalue_ind ge env m |
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[399] | 171 | (λe,v,tr,H. yields ? (exec_expr ge env m e) (〈v,tr〉)) |
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[498] | 172 | (λe,l,off,tr,H. yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉))); |
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[487] | 173 | [ #i #ty @refl |
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| 174 | | #f #ty @refl |
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[498] | 175 | | #e #ty #l #off #v #tr #H1 #H2 @(lvalue_expr … H1) |
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[487] | 176 | [ #id | #e' | #e' #id ] #H3 |
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| 177 | whd in ⊢ (??%?); |
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| 178 | >(yields_eq ??? H3) |
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| 179 | whd in ⊢ (??%?); >H2 @refl |
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[500] | 180 | | #e #ty #r #l #pc #off #tr #H1 #H2 whd in ⊢ (??%?); |
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| 181 | >(yields_eq ??? H2) whd in ⊢ (??%?) |
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| 182 | @(dec_true ? (pointer_compat_dec l r) pc) #pc' whd |
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[487] | 183 | @refl |
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| 184 | | #ty' #ty @refl |
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| 185 | | #op #e #ty #v1 #v #tr #H1 #H2 #H3 whd in ⊢ (??%?); |
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| 186 | >(yields_eq ??? H3) |
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| 187 | whd in ⊢ (??%?); >H2 @refl |
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| 188 | | #op #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #e3 #H4 #H5 whd in ⊢ (??%?); |
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| 189 | >(yields_eq ??? H4) whd in ⊢ (??%?); |
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| 190 | >(yields_eq ??? H5) whd in ⊢ (??%?); |
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| 191 | >e3 @refl |
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| 192 | | #e1 #e2 #e3 #ty #v1 #v2 #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?); |
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| 193 | >(yields_eq ??? H4) whd in ⊢ (??%?); |
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| 194 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 195 | >(yields_eq ??? H5) |
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| 196 | @refl |
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| 197 | | #e1 #e2 #e3 #ty #v1 #v2 #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?); |
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| 198 | >(yields_eq ??? H4) whd in ⊢ (??%?); |
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| 199 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 200 | >(yields_eq ??? H5) |
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| 201 | @refl |
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| 202 | | #e1 #e2 #ty #v1 #tr #H1 #H2 #H3 whd in ⊢ (??%?); |
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| 203 | >(yields_eq ??? H3) whd in ⊢ (??%?); |
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| 204 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 205 | @refl |
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| 206 | | #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 #H6 whd in ⊢ (??%?); |
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| 207 | >(yields_eq ??? H5) whd in ⊢ (??%?); |
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| 208 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 209 | >(yields_eq ??? H6) whd in ⊢ (??%?); |
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| 210 | elim (bool_of_val_3_complete … H4); #b *; #evb #Hb |
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| 211 | >(yields_eq ??? Hb) whd in ⊢ (??%?); <evb |
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| 212 | @refl |
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| 213 | | #e1 #e2 #ty #v1 #tr #H1 #H2 #H3 whd in ⊢ (??%?); |
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| 214 | >(yields_eq ??? H3) whd in ⊢ (??%?); |
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| 215 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 216 | @refl |
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| 217 | | #e1 #e2 #ty #v1 #v2 #v #tr1 #tr2 #H1 #H2 #H3 #H4 #H5 #H6 whd in ⊢ (??%?); |
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| 218 | >(yields_eq ??? H5) whd in ⊢ (??%?); |
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| 219 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 220 | >(yields_eq ??? H6) whd in ⊢ (??%?); |
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| 221 | elim (bool_of_val_3_complete … H4); #b *; #evb #Hb |
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| 222 | >(yields_eq ??? Hb) whd in ⊢ (??%?); <evb |
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| 223 | @refl |
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| 224 | | #e #ty #ty' #v1 #v #tr #H1 #H2 #H3 whd in ⊢ (??%?); |
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| 225 | >(yields_eq ??? H3) whd in ⊢ (??%?); |
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| 226 | >(yields_eq ??? (cast_complete … H2)) |
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| 227 | @refl |
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| 228 | | #e #ty #v #l #tr #H1 #H2 whd in ⊢ (??%?); |
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| 229 | >(yields_eq ??? H2) whd in ⊢ (??%?); |
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| 230 | @refl |
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[253] | 231 | |
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[226] | 232 | (* lvalues *) |
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[487] | 233 | | #id #l #ty #e1 whd in ⊢ (??%?); >e1 @refl |
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[498] | 234 | | #id #l #ty #e1 #e2 whd in ⊢ (??%?); >e1 |
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[487] | 235 | >e2 @refl |
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[500] | 236 | | #e #ty #r #l #pc #ofs #tr #H1 #H2 whd in ⊢ (??%?); |
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[487] | 237 | >(yields_eq ??? H2) |
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| 238 | @refl |
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[498] | 239 | | #e #i #ty #l #ofs #id #fList #delta #tr #H1 #H2 #H3 #H4 cases e in H2 H4 ⊢ %; |
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[487] | 240 | #e' #ty' #H2 whd in H2:(??%?); >H2 #H4 whd in ⊢ (??%?); |
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| 241 | >(yields_eq ??? H4) whd in ⊢ (??%?); |
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| 242 | >H3 @refl |
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[498] | 243 | | #e #i #ty #l #ofs #id #fList #tr cases e; #e' #ty' #H1 #H2 |
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[487] | 244 | whd in H2:(??%?); >H2 #H3 whd in ⊢ (??%?); |
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| 245 | >(yields_eq ??? H3) @refl |
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| 246 | ] qed. |
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[226] | 247 | |
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[487] | 248 | theorem expr_complete: ∀ge,env,m. |
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[399] | 249 | ∀e,v,tr. eval_expr ge env m e v tr → yields ? (exec_expr ge env m e) (〈v,tr〉). |
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[487] | 250 | #ge #env #m elim (expr_lvalue_complete ge env m); /2/; qed. |
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[226] | 251 | |
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[487] | 252 | theorem exprlist_complete: ∀ge,env,m,es,vs,tr. |
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[399] | 253 | eval_exprlist ge env m es vs tr → yields ? (exec_exprlist ge env m es) (〈vs,tr〉). |
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[487] | 254 | #ge #env #m #es #vs #tr #H elim H; |
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| 255 | [ @refl |
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| 256 | | #e #et #v #vt #tr #trt #H1 #H2 #H3 whd in ⊢ (??%?); |
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| 257 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 258 | >(yields_eq ??? H3) |
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| 259 | @refl |
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| 260 | ] qed. |
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[239] | 261 | |
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[487] | 262 | theorem lvalue_complete: ∀ge,env,m. |
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[498] | 263 | ∀e,l,off,tr. eval_lvalue ge env m e l off tr → yields ? (exec_lvalue ge env m e) (〈〈l,off〉,tr〉). |
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[487] | 264 | #ge #env #m elim (expr_lvalue_complete ge env m); /2/; qed. |
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[239] | 265 | |
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[487] | 266 | let rec P_typelist (P:type → Prop) (l:typelist) on l : Prop ≝ |
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[239] | 267 | match l with |
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| 268 | [ Tnil ⇒ True |
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| 269 | | Tcons h t ⇒ P h ∧ P_typelist P t |
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| 270 | ]. |
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| 271 | |
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[487] | 272 | let rec type_ind2l |
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[239] | 273 | (P:type → Prop) (Q:typelist → Prop) |
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| 274 | (vo:P Tvoid) |
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| 275 | (it:∀i,s. P (Tint i s)) |
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| 276 | (fl:∀f. P (Tfloat f)) |
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| 277 | (pt:∀s,t. P t → P (Tpointer s t)) |
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| 278 | (ar:∀s,t,n. P t → P (Tarray s t n)) |
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| 279 | (fn:∀tl,t. Q tl → P t → P (Tfunction tl t)) |
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| 280 | (st:∀i,fl. P (Tstruct i fl)) |
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| 281 | (un:∀i,fl. P (Tunion i fl)) |
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[481] | 282 | (cp:∀r,i. P (Tcomp_ptr r i)) |
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[239] | 283 | (nl:Q Tnil) |
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| 284 | (cs:∀t,tl. P t → Q tl → Q (Tcons t tl)) |
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| 285 | (t:type) on t : P t ≝ |
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| 286 | match t return λt'.P t' with |
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| 287 | [ Tvoid ⇒ vo |
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| 288 | | Tint i s ⇒ it i s |
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| 289 | | Tfloat s ⇒ fl s |
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| 290 | | Tpointer s t' ⇒ pt s t' (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t') |
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| 291 | | Tarray s t' n ⇒ ar s t' n (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t') |
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| 292 | | Tfunction tl t' ⇒ fn tl t' (typelist_ind2l P Q vo it fl pt ar fn st un cp nl cs tl) (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t') |
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| 293 | | Tstruct i fs ⇒ st i fs |
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| 294 | | Tunion i fs ⇒ un i fs |
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[481] | 295 | | Tcomp_ptr r i ⇒ cp r i |
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[239] | 296 | ] |
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| 297 | and typelist_ind2l |
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| 298 | (P:type → Prop) (Q:typelist → Prop) |
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| 299 | (vo:P Tvoid) |
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| 300 | (it:∀i,s. P (Tint i s)) |
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| 301 | (fl:∀f. P (Tfloat f)) |
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| 302 | (pt:∀s,t. P t → P (Tpointer s t)) |
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| 303 | (ar:∀s,t,n. P t → P (Tarray s t n)) |
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| 304 | (fn:∀tl,t. Q tl → P t → P (Tfunction tl t)) |
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| 305 | (st:∀i,fl. P (Tstruct i fl)) |
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| 306 | (un:∀i,fl. P (Tunion i fl)) |
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[481] | 307 | (cp:∀r,i. P (Tcomp_ptr r i)) |
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[239] | 308 | (nl:Q Tnil) |
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| 309 | (cs:∀t,tl. P t → Q tl → Q (Tcons t tl)) |
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| 310 | (ts:typelist) on ts : Q ts ≝ |
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| 311 | match ts return λts'.Q ts' with |
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| 312 | [ Tnil ⇒ nl |
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| 313 | | Tcons t tl ⇒ cs t tl (type_ind2l P Q vo it fl pt ar fn st un cp nl cs t) |
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| 314 | (typelist_ind2l P Q vo it fl pt ar fn st un cp nl cs tl) |
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| 315 | ]. |
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| 316 | |
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[487] | 317 | lemma assert_type_eq_true: ∀t. ∃p.assert_type_eq t t = OK ? p. |
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| 318 | #t whd in ⊢ (??(λ_.??%?)); cases (type_eq_dec t t); #E |
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| 319 | [ %{ E} // |
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| 320 | | @False_ind @(absurd ?? E) // |
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| 321 | ] qed. |
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[239] | 322 | |
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[487] | 323 | lemma alloc_vars_complete: ∀env,m,l,env',m'. |
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[239] | 324 | alloc_variables env m l env' m' → |
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[487] | 325 | exec_alloc_variables env m l = 〈env', m'〉. |
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| 326 | #env #m #l #env' #m' #H elim H; |
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| 327 | [ #env'' #m'' % |
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| 328 | | #env1 #m1 #id #ty #l1 #m2 #loc #m3 #env2 #H1 #H2 #H3 |
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| 329 | < H3 whd in H1:(??%?) ⊢ (??%?) |
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[726] | 330 | destruct (H1) @refl |
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[487] | 331 | ] qed. |
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[239] | 332 | |
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[487] | 333 | lemma bind_params_complete: ∀e,m,params,vs,m2. |
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[239] | 334 | bind_parameters e m params vs m2 → |
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[487] | 335 | yields ? (exec_bind_parameters e m params vs) m2. |
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| 336 | #e #m #params #vs #m2 #H elim H; |
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| 337 | [ //; |
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| 338 | | #env1 #m1 #id #ty #l #v #tl #loc #m2 #m3 #H1 #H2 #H3 #H4 |
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| 339 | whd in ⊢ (??%?) |
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| 340 | >H1 whd in ⊢ (??%?); |
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| 341 | >H2 whd in ⊢ (??%?); |
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| 342 | @H4 |
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| 343 | ] qed. |
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[239] | 344 | |
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[487] | 345 | lemma eventval_match_complete': ∀ev,ty,v. |
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| 346 | eventval_match ev ty v → yields ? (check_eventval' v ty) ev. |
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| 347 | #ev #ty #v #H elim H; //; qed. |
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[239] | 348 | |
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[487] | 349 | lemma eventval_list_match_complete: ∀vs,tys,evs. |
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| 350 | eventval_list_match evs tys vs → yields ? (check_eventval_list vs tys) evs. |
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| 351 | #vs #tys #evs #H elim H; |
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| 352 | [ // |
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| 353 | | #e #etl #ty #tytl #v #vtl #H1 #H2 #H3 whd in ⊢ (??%?) |
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| 354 | >(yields_eq ??? (eventval_match_complete' … H1)) whd in ⊢ (??%?) |
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| 355 | >(yields_eq ??? H3) whd in ⊢ (??%?) // |
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| 356 | ] qed. |
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[239] | 357 | |
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[487] | 358 | theorem step_complete: ∀ge,s,tr,s'. |
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[399] | 359 | step ge s tr s' → yieldsIO ? (exec_step ge s) 〈tr,s'〉. |
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[487] | 360 | #ge #s #tr #s' #H elim H; |
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[498] | 361 | [ #f #e #e1 #k #e2 #m #loc #ofs #v #m' #tr1 #tr2 #H1 #H2 #H3 whd in ⊢ (??%?); |
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[487] | 362 | >(yields_eq ??? (lvalue_complete … H1)) whd in ⊢ (??%?); |
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| 363 | >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?); |
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| 364 | >H3 @refl |
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| 365 | | #f #e #eargs #k #ef #m #vf #vargs #f' #tr1 #tr2 #H1 #H2 #H3 #H4 whd in ⊢ (??%?); |
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| 366 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 367 | >(yields_eq ??? (exprlist_complete … H2)) whd in ⊢ (??%?); |
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| 368 | >H3 whd in ⊢ (??%?); |
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| 369 | >H4 elim (assert_type_eq_true (fun_typeof e)); #pty #ety >ety |
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| 370 | @refl |
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[498] | 371 | | #f #el #ef #eargs #k #env #m #loc #ofs #vf #vargs #f' #tr1 #tr2 #tr3 #H1 #H2 #H3 #H4 #H5 whd in ⊢ (??%?); |
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[487] | 372 | >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?); |
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| 373 | >(yields_eq ??? (exprlist_complete … H3)) whd in ⊢ (??%?); |
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| 374 | >H4 whd in ⊢ (??%?); |
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| 375 | >H5 elim (assert_type_eq_true (fun_typeof ef)); #pty #ety >ety |
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| 376 | whd in ⊢ (??%?); |
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| 377 | >(yields_eq ??? (lvalue_complete … H1)) whd in ⊢ (??%?); |
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| 378 | @refl |
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| 379 | | #f #s1 #s2 #k #env #m @refl |
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| 380 | | 5,6,7: #f #s #k #env #m @refl |
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| 381 | | #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 382 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 383 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 384 | @refl |
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| 385 | | #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 386 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 387 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 388 | @refl |
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| 389 | | #f #e #s #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 390 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 391 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 392 | @refl |
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| 393 | | #f #e #s #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 394 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 395 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 396 | @refl |
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| 397 | | #f #s1 #e #s2 #k #env #m #H cases H; #es1 >es1 @refl |
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| 398 | | 13,14: #f #e #s #k #env #m @refl |
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| 399 | | #f #s1 #e #s2 #k #env #m #v #tr *; #es1 >es1 #H1 #H2 whd in ⊢ (??%?); |
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| 400 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 401 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 402 | @refl |
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| 403 | | #f #s1 #e #s2 #k #env #m #v #tr *; #es1 >es1 #H1 #H2 whd in ⊢ (??%?); |
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| 404 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 405 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 406 | @refl |
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| 407 | | #f #e #s #k #env #m @refl |
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| 408 | | #f #s1 #e #s2 #s3 #k #env #m #nskip whd in ⊢ (??%?); cases (is_Sskip s1); |
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| 409 | [ #H @False_ind @(absurd ? H nskip) |
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| 410 | | #H whd in ⊢ (??%?); @refl ] |
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| 411 | | #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 412 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 413 | >(yields_eq ??? (bool_of_false ?? H2)) |
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| 414 | @refl |
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| 415 | | #f #e #s1 #s2 #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 416 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 417 | >(yields_eq ??? (bool_of_true ?? H2)) |
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| 418 | @refl |
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| 419 | | #f #s1 #e #s2 #s3 #k #env #m *; #es1 >es1 @refl |
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| 420 | | 22,23: #f #e #s1 #s2 #k #env #m @refl |
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| 421 | | #f #k #env #m #H whd in ⊢ (??%?); >H @refl |
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| 422 | | #f #e #k #env #m #v #tr #H1 #H2 whd in ⊢ (??%?); |
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| 423 | @(dec_false ? (type_eq_dec (fn_return f) Tvoid) H1) #pf' |
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| 424 | whd in ⊢ (??%?); |
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| 425 | >(yields_eq ??? (expr_complete … H2)) whd in ⊢ (??%?); |
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| 426 | @refl |
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| 427 | | #f #k #env #m cases k; |
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| 428 | [ #H1 #H2 whd in ⊢ (??%?); >H2 @refl |
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| 429 | | #s' #k' whd in ⊢ (% → ?); *; |
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| 430 | | 3,4: #e' #s' #k' whd in ⊢ (% → ?); *; |
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| 431 | | 5,6: #e' #s1' #s2' #k' whd in ⊢ (% → ?); *; |
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| 432 | | #k' whd in ⊢ (% → ?); *; |
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| 433 | | #r #f' #env' #k' #H1 #H2 whd in ⊢ (??%?); >H2 @refl |
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| 434 | ] |
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| 435 | | #f #e #s #k #env #m #i #tr #H1 whd in ⊢ (??%?); |
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| 436 | >(yields_eq ??? (expr_complete … H1)) whd in ⊢ (??%?); |
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| 437 | @refl |
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| 438 | | #f #s #k #env #m *; #es >es @refl |
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| 439 | | #f #k #env #m @refl |
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| 440 | | #f #l #s #k #env #m @refl |
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| 441 | | #f #l #k #env #m #s #k' #H1 whd in ⊢ (??%?); >H1 @refl |
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| 442 | | #f #args #k #m1 #env #m2 #m3 #H1 #H2 whd in ⊢ (??%?); |
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| 443 | >(alloc_vars_complete … H1) whd in ⊢ (??%?); |
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| 444 | >(yields_eq ??? (bind_params_complete … H2)) |
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| 445 | // |
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| 446 | | #id #tys #rty #args #k #m #rv #tr #H whd in ⊢ (??%?); |
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| 447 | inversion H; #f' #args' #rv' #eargs #erv #H1 #H2 #e1 #e2 #e3 #e4 <e1 in H1 H2 |
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| 448 | #H1 #H2 |
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| 449 | >(yields_eq ??? (eventval_list_match_complete … H1)) whd in ⊢ (??%?); |
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| 450 | whd; inversion H2; #x #e5 #e6 #e7 %{ x} whd in ⊢ (??%?); |
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| 451 | @refl |
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| 452 | | #v #f #env #k #m @refl |
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[498] | 453 | | #v #f #env #k #m1 #m2 #loc #ofs #ty #H whd in ⊢ (??%?); |
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[487] | 454 | >H @refl |
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| 455 | | #f #l #s #k #env #m @refl |
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| 456 | ] qed. |
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[708] | 457 | |
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| 458 | lemma is_final_complete : ∀s,r. final_state s r → is_final s = Some ? r. |
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| 459 | #s0 #r0 * #r #m @refl qed. |
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| 460 | |
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