[20] | 1 | |
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| 2 | include "Csem.ma". |
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| 3 | |
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| 4 | include "extralib.ma". |
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[24] | 5 | include "IOMonad.ma". |
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[20] | 6 | |
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| 7 | include "Plogic/russell_support.ma". |
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| 8 | |
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| 9 | ndefinition P_to_P_option_res : ∀A:Type[0].∀P:A → CProp[0].option (res A) → CProp[0] ≝ |
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| 10 | λA,P,a.match a with [ None ⇒ False | Some y ⇒ match y return λ_.CProp[0] with [ Error ⇒ True | OK z ⇒ P z ]]. |
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| 11 | |
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| 12 | ndefinition err_inject : ∀A.∀P:A → Prop.∀a:option (res A).∀p:P_to_P_option_res A P a.res (sigma A P) ≝ |
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| 13 | λA.λP:A → Prop.λa:option (res A).λp:P_to_P_option_res A P a. |
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| 14 | (match a return λa'.a=a' → res (sigma A P) with |
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| 15 | [ None ⇒ λe1.? |
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| 16 | | Some b ⇒ λe1.(match b return λb'.b=b' → ? with |
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| 17 | [ Error ⇒ λ_. Error ? |
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| 18 | | OK c ⇒ λe2. OK ? (sig_intro A P c ?) |
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| 19 | ]) (refl ? b) |
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| 20 | ]) (refl ? a). |
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| 21 | ##[ nrewrite > e1 in p; nnormalize; *; |
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| 22 | ##| nrewrite > e1 in p; nrewrite > e2; nnormalize; // |
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| 23 | ##] nqed. |
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| 24 | |
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| 25 | ndefinition err_eject : ∀A.∀P: A → Prop. res (sigma A P) → res A ≝ |
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| 26 | λA,P,a.match a with [ Error ⇒ Error ? | OK b ⇒ |
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| 27 | match b with [ sig_intro w p ⇒ OK ? w] ]. |
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| 28 | |
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| 29 | ndefinition sig_eject : ∀A.∀P: A → Prop. sigma A P → A ≝ |
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| 30 | λA,P,a.match a with [ sig_intro w p ⇒ w]. |
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| 31 | |
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| 32 | ncoercion err_inject : |
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| 33 | ∀A.∀P:A → Prop.∀a.∀p:P_to_P_option_res ? P a.res (sigma A P) ≝ err_inject |
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| 34 | on a:option (res ?) to res (sigma ? ?). |
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| 35 | ncoercion err_eject : ∀A.∀P:A → Prop.∀c:res (sigma A P).res A ≝ err_eject |
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| 36 | on _c:res (sigma ? ?) to res ?. |
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| 37 | ncoercion sig_eject : ∀A.∀P:A → Prop.∀c:sigma A P.A ≝ sig_eject |
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| 38 | on _c:sigma ? ? to ?. |
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| 39 | |
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[250] | 40 | ndefinition P_res: ∀A.∀P:A → Prop.res A → Prop ≝ |
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| 41 | λA,P,a. match a with [ Error ⇒ True | OK v ⇒ P v ]. |
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| 42 | |
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| 43 | ndefinition exec_bool_of_val : ∀v:val. ∀ty:type. res bool ≝ |
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[20] | 44 | λv,ty. match v in val with |
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| 45 | [ Vint i ⇒ match ty with |
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[250] | 46 | [ Tint _ _ ⇒ OK ? (¬eq i zero) |
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| 47 | | Tpointer _ _ ⇒ OK ? (¬eq i zero) |
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| 48 | | _ ⇒ Error ? |
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[20] | 49 | ] |
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| 50 | | Vfloat f ⇒ match ty with |
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[250] | 51 | [ Tfloat _ ⇒ OK ? (¬Fcmp Ceq f Fzero) |
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| 52 | | _ ⇒ Error ? |
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[20] | 53 | ] |
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[124] | 54 | | Vptr _ _ _ ⇒ match ty with |
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[250] | 55 | [ Tint _ _ ⇒ OK ? true |
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| 56 | | Tpointer _ _ ⇒ OK ? true |
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| 57 | | _ ⇒ Error ? |
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[20] | 58 | ] |
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[250] | 59 | | _ ⇒ Error ? |
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| 60 | ]. |
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| 61 | |
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| 62 | nlemma exec_bool_of_val_sound: ∀v,ty,r. |
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| 63 | exec_bool_of_val v ty = OK ? r → bool_of_val v ty (of_bool r). |
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| 64 | #v ty r; |
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| 65 | ncases v; ##[ ##2: #i ##| ##3: #f ##| ##4: #sp b of ##] |
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| 66 | nwhd; ##[ ##4: #H; nwhd in H:(??%?); ndestruct ##] |
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| 67 | ncases ty; ##[ ##2,11,20: #sz sg ##| ##3,12,21: #sz ##| ##4,13,22: #sp ty ##| ##5,14,23: #sp ty n ##| ##6,15,24: #args rty ##| ##7,8,16,17,25,26: #id fs ##| ##9,18,27: #id ##] |
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| 68 | #H; nwhd in H:(??%?); ndestruct; |
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| 69 | ##[ ##1,4: nlapply (eq_spec i zero); nelim (eq i zero); |
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[20] | 70 | ##[ ##1,3: #e; nrewrite > e; napply bool_of_val_false; //; |
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| 71 | ##| ##2,4: #ne; napply bool_of_val_true; /2/; |
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| 72 | ##] |
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[250] | 73 | ##| ##3: nelim (eq_dec f Fzero); |
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[20] | 74 | ##[ #e; nrewrite > e; nrewrite > (Feq_zero_true …); napply bool_of_val_false; //; |
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| 75 | ##| #ne; nrewrite > (Feq_zero_false …); //; napply bool_of_val_true; /2/; |
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| 76 | ##] |
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[250] | 77 | ##| ##2,5: napply bool_of_val_true; // |
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[20] | 78 | ##] nqed. |
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| 79 | |
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[250] | 80 | nlemma bool_of_val_complete : ∀v,ty,r. bool_of_val v ty r → ∃b. r = of_bool b ∧ exec_bool_of_val v ty = OK ? b. |
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[20] | 81 | #v ty r H; nelim H; #v t H'; nelim H'; |
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[250] | 82 | ##[ #i is s ne; @ true; @; //; nwhd in ⊢ (??%?); nrewrite > (eq_false … ne); //; |
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[124] | 83 | ##| #p b i i0 s; @ true; @; // |
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[250] | 84 | ##| #i p t ne; @ true; @; //; nwhd in ⊢ (??%?); nrewrite > (eq_false … ne); //; |
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[124] | 85 | ##| #p b i p0 t0; @ true; @; // |
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[250] | 86 | ##| #f s ne; @ true; @; //; nwhd in ⊢ (??%?); nrewrite > (Feq_zero_false … ne); //; |
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| 87 | ##| #i s; @ false; @; //; |
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| 88 | ##| #p t; @ false; @; //; |
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| 89 | ##| #s; @ false; @; //; nwhd in ⊢ (??%?); nrewrite > (Feq_zero_true …); //; |
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[20] | 90 | ##] |
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| 91 | nqed. |
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| 92 | |
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| 93 | (* Prove a few minor results to make proof obligations easy. *) |
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| 94 | |
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| 95 | nlemma bind_assoc_r: ∀A,B,C,e,f,g. |
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| 96 | bind B C (bind A B e f) g = bind A C e (λx.bind B C (f x) g). |
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| 97 | #A B C e f g; ncases e; nnormalize; //; nqed. |
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| 98 | |
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| 99 | nlemma bind_OK: ∀A,B,P,e,f. |
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| 100 | (∀v. e = OK A v → match f v with [ Error ⇒ True | OK v' ⇒ P v' ]) → |
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| 101 | match bind A B e f with [ Error ⇒ True | OK v ⇒ P v ]. |
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| 102 | #A B P e f; nelim e; /2/; nqed. |
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| 103 | |
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| 104 | nlemma sig_bind_OK: ∀A,B. ∀P:A → Prop. ∀P':B → Prop. ∀e:res (sigma A P). ∀f:sigma A P → res B. |
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[124] | 105 | (∀v:A. ∀p:P v. match f (sig_intro A P v p) with [ Error ⇒ True | OK v' ⇒ P' v'] ) → |
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[20] | 106 | match bind (sigma A P) B e f with [ Error ⇒ True | OK v' ⇒ P' v' ]. |
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[124] | 107 | #A B P P' e f; nelim e; |
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| 108 | ##[ #v0; nelim v0; #v Hv IH; napply IH; |
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| 109 | ##| #_; napply I; |
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| 110 | ##] nqed. |
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[20] | 111 | |
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| 112 | nlemma bind2_OK: ∀A,B,C,P,e,f. |
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| 113 | (∀v1,v2. e = OK ? 〈v1,v2〉 → match f v1 v2 with [ Error ⇒ True | OK v' ⇒ P v' ]) → |
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| 114 | match bind2 A B C e f with [ Error ⇒ True | OK v ⇒ P v ]. |
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| 115 | #A B C P e f; nelim e; //; #v; ncases v; /2/; nqed. |
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| 116 | |
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| 117 | nlemma sig_bind2_OK: ∀A,B,C. ∀P:A×B → Prop. ∀P':C → Prop. ∀e:res (sigma (A×B) P). ∀f:A → B → res C. |
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| 118 | (∀v1:A.∀v2:B. P 〈v1,v2〉 → match f v1 v2 with [ Error ⇒ True | OK v' ⇒ P' v'] ) → |
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| 119 | match bind2 A B C e f with [ Error ⇒ True | OK v' ⇒ P' v' ]. |
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| 120 | #A B C P P' e f; nelim e; //; |
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| 121 | #v0; nelim v0; #v; nelim v; #v1 v2 Hv IH; napply IH; //; nqed. |
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| 122 | |
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| 123 | nlemma bool_val_distinct: Vtrue ≠ Vfalse. |
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| 124 | @; #H; nwhd in H:(??%%); ndestruct; napply (absurd ? e0 one_not_zero); |
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| 125 | nqed. |
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| 126 | |
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| 127 | nlemma bool_of: ∀v,ty,b. bool_of_val v ty (of_bool b) → |
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| 128 | if b then is_true v ty else is_false v ty. |
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| 129 | #v ty b; ncases b; #H; ninversion H; #v' ty' H' ev et ev; //; |
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| 130 | napply False_ind; napply (absurd ? ev ?); |
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| 131 | ##[ ##2: napply sym_neq ##] napply bool_val_distinct; |
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| 132 | nqed. |
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| 133 | |
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| 134 | ndefinition opt_to_res ≝ λA.λv:option A. match v with [ None ⇒ Error A | Some v ⇒ OK A v ]. |
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| 135 | nlemma opt_OK: ∀A,P,e. |
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| 136 | (∀v. e = Some ? v → P v) → |
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| 137 | match opt_to_res A e with [ Error ⇒ True | OK v ⇒ P v ]. |
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| 138 | #A P e; nelim e; /2/; |
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| 139 | nqed. |
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| 140 | |
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| 141 | nlemma opt_bind_OK: ∀A,B,P,e,f. |
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| 142 | (∀v. e = Some A v → match f v with [ Error ⇒ True | OK v' ⇒ P v' ]) → |
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| 143 | match bind A B (opt_to_res A e) f with [ Error ⇒ True | OK v ⇒ P v ]. |
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| 144 | #A B P e f; nelim e; nnormalize; /2/; nqed. |
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| 145 | |
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| 146 | nlemma extract_subset_pair: ∀A,B,C,P. ∀e:{e:A×B | P e}. ∀Q:A→B→res C. ∀R:C→Prop. |
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| 147 | (∀a,b. eject ?? e = 〈a,b〉 → P 〈a,b〉 → match Q a b with [ OK v ⇒ R v | Error ⇒ True]) → |
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| 148 | match match eject ?? e with [ mk_pair a b ⇒ Q a b ] with [ OK v ⇒ R v | Error ⇒ True ]. |
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| 149 | #A B C P e Q R; ncases e; #e'; ncases e'; nnormalize; |
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| 150 | ##[ #H; napply (False_ind … H); |
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| 151 | ##| #e''; ncases e''; #a b Pab H; nnormalize; /2/; |
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| 152 | ##] nqed. |
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| 153 | |
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| 154 | (* |
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| 155 | nremark err_later: ∀A,B. ∀e:res A. match e with [ Error ⇒ Error B | OK v ⇒ Error B ] = Error B. |
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| 156 | #A B e; ncases e; //; nqed. |
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| 157 | *) |
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| 158 | |
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[250] | 159 | ndefinition try_cast_null : ∀m:mem. ∀i:int. ∀ty:type. ∀ty':type. res val ≝ |
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| 160 | λm:mem. λi:int. λty:type. λty':type. |
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[124] | 161 | match eq i zero with |
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| 162 | [ true ⇒ |
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| 163 | match ty with |
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[225] | 164 | [ Tint _ _ ⇒ |
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[124] | 165 | match ty' with |
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[250] | 166 | [ Tpointer _ _ ⇒ OK ? (Vint i) |
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| 167 | | Tarray _ _ _ ⇒ OK ? (Vint i) |
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| 168 | | Tfunction _ _ ⇒ OK ? (Vint i) |
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| 169 | | _ ⇒ Error ? |
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[124] | 170 | ] |
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[225] | 171 | | Tpointer _ _ ⇒ |
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| 172 | match ty' with |
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[250] | 173 | [ Tpointer _ _ ⇒ OK ? (Vint i) |
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| 174 | | Tarray _ _ _ ⇒ OK ? (Vint i) |
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| 175 | | Tfunction _ _ ⇒ OK ? (Vint i) |
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| 176 | | _ ⇒ Error ? |
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[225] | 177 | ] |
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[124] | 178 | | Tarray _ _ _ ⇒ |
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| 179 | match ty' with |
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[250] | 180 | [ Tpointer _ _ ⇒ OK ? (Vint i) |
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| 181 | | Tarray _ _ _ ⇒ OK ? (Vint i) |
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| 182 | | Tfunction _ _ ⇒ OK ? (Vint i) |
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| 183 | | _ ⇒ Error ? |
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[124] | 184 | ] |
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| 185 | | Tfunction _ _ ⇒ |
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| 186 | match ty' with |
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[250] | 187 | [ Tpointer _ _ ⇒ OK ? (Vint i) |
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| 188 | | Tarray _ _ _ ⇒ OK ? (Vint i) |
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| 189 | | Tfunction _ _ ⇒ OK ? (Vint i) |
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| 190 | | _ ⇒ Error ? |
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[124] | 191 | ] |
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[250] | 192 | | _ ⇒ Error ? |
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[124] | 193 | ] |
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[250] | 194 | | false ⇒ Error ? |
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| 195 | ]. |
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[124] | 196 | |
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[250] | 197 | nlemma try_cast_null_sound: ∀m,i,ty,ty',v'. try_cast_null m i ty ty' = OK ? v' → cast m (Vint i) ty ty' v'. |
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| 198 | #m i ty ty' v'; |
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| 199 | nwhd in ⊢ (??%? → ?); |
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| 200 | nlapply (eq_spec i zero); ncases (eq i zero); |
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| 201 | ##[ #e; nrewrite > e; |
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| 202 | ncases ty; ##[ ##| #sz sg ##| #fs ##| #sp ty ##| #sp ty n ##| #args rty ##| #id fs ##| #id fs ##| #id ##] |
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| 203 | nwhd in ⊢ (??%? → ?); ##[ ##1,3,7,8,9: #H; ndestruct ##] |
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| 204 | ncases ty'; ##[ ##2,11,20,29: #sz' sg' ##| ##3,12,21,30: #sz' ##| ##4,13,22,31: #sp' ty' ##| ##5,14,23,32: #sp' ty' n' ##| ##6,15,24,33: #args' res' ##| ##7,8,16,17,25,26,34,35: #id' fs' ##| ##9,18,27,36: #id' ##] |
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| 205 | nwhd in ⊢ (??%? → ?); #H; ndestruct (H); |
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| 206 | ##[ ##1,5,9: napply cast_ip_z ##| ##*: napply cast_pp_z ##] //; |
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| 207 | ##| #_; nwhd in ⊢ (??%? → ?); #H; ndestruct |
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| 208 | ##] |
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| 209 | nqed. |
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| 210 | |
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[155] | 211 | ndefinition ms_eq_dec : ∀s1,s2:memory_space. (s1 = s2) + (s1 ≠ s2). |
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| 212 | #s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed. |
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| 213 | |
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[250] | 214 | ndefinition exec_cast : ∀m:mem. ∀v:val. ∀ty:type. ∀ty':type. res val ≝ |
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[189] | 215 | λm:mem. λv:val. λty:type. λty':type. |
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[20] | 216 | match v with |
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| 217 | [ Vint i ⇒ |
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| 218 | match ty with |
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| 219 | [ Tint sz1 si1 ⇒ |
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| 220 | match ty' with |
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[250] | 221 | [ Tint sz2 si2 ⇒ OK ? (Vint (cast_int_int sz2 si2 i)) |
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| 222 | | Tfloat sz2 ⇒ OK ? (Vfloat (cast_float_float sz2 (cast_int_float si1 i))) |
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| 223 | | Tpointer _ _ ⇒ do r ← try_cast_null m i ty ty'; OK val r |
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| 224 | | Tarray _ _ _ ⇒ do r ← try_cast_null m i ty ty'; OK val r |
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| 225 | | Tfunction _ _ ⇒ do r ← try_cast_null m i ty ty'; OK val r |
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| 226 | | _ ⇒ Error ? |
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[20] | 227 | ] |
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[250] | 228 | | Tpointer _ _ ⇒ do r ← try_cast_null m i ty ty'; OK val r |
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| 229 | | Tarray _ _ _ ⇒ do r ← try_cast_null m i ty ty'; OK val r |
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| 230 | | Tfunction _ _ ⇒ do r ← try_cast_null m i ty ty'; OK val r |
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| 231 | | _ ⇒ Error ? |
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[20] | 232 | ] |
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| 233 | | Vfloat f ⇒ |
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| 234 | match ty with |
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| 235 | [ Tfloat sz ⇒ |
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| 236 | match ty' with |
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[250] | 237 | [ Tint sz' si' ⇒ OK ? (Vint (cast_int_int sz' si' (cast_float_int si' f))) |
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| 238 | | Tfloat sz' ⇒ OK ? (Vfloat (cast_float_float sz' f)) |
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| 239 | | _ ⇒ Error ? |
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[20] | 240 | ] |
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[250] | 241 | | _ ⇒ Error ? |
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[20] | 242 | ] |
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[124] | 243 | | Vptr p b ofs ⇒ |
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[189] | 244 | do s ← match ty with [ Tpointer s _ ⇒ OK ? s | Tarray s _ _ ⇒ OK ? s | Tfunction _ _ ⇒ OK ? Code | _ ⇒ Error ? ]; |
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| 245 | do u ← match ms_eq_dec p s with [ inl _ ⇒ OK ? something | inr _ ⇒ Error ? ]; |
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| 246 | do s' ← match ty' with |
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[127] | 247 | [ Tpointer s _ ⇒ OK ? s | Tarray s _ _ ⇒ OK ? s | Tfunction _ _ ⇒ OK ? Code |
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[189] | 248 | | _ ⇒ Error ? ]; |
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[125] | 249 | if is_pointer_compat (block_space m b) s' |
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| 250 | then OK ? (Vptr s' b ofs) |
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[250] | 251 | else Error ? |
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| 252 | | _ ⇒ Error ? |
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| 253 | ]. |
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| 254 | |
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| 255 | ndefinition exec_cast_sound : ∀m:mem. ∀v:val. ∀ty:type. ∀ty':type. ∀v':val. exec_cast m v ty ty' = OK ? v' → cast m v ty ty' v'. |
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| 256 | #m v ty ty' v'; |
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| 257 | ncases v; |
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| 258 | ##[ #H; nwhd in H:(??%?); ndestruct; |
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| 259 | ##| #i; ncases ty; |
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| 260 | ##[ #H; nwhd in H:(??%?); ndestruct; |
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| 261 | ##| ##3,9: #a; #H; nwhd in H:(??%?); ndestruct; |
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| 262 | ##| ##7,8: #a b; #H; nwhd in H:(??%?); ndestruct; |
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| 263 | ##| #sz1 si1; ncases ty'; |
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| 264 | ##[ #H; nwhd in H:(??%?); ndestruct; |
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| 265 | ##| ##3,9: #a; #H; nwhd in H:(??%?); ndestruct; // |
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| 266 | ##| ##2,7,8: #a b; #H; nwhd in H:(??%?); ndestruct; // |
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| 267 | ##| ##4,5,6: ##[ #sp ty''; nletin t ≝ (Tpointer sp ty'') |
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| 268 | ##| #sp ty'' n; nletin t ≝ (Tarray sp ty'' n) |
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| 269 | ##| #args rty; nletin t ≝ (Tfunction args rty) ##] |
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| 270 | nwhd in ⊢ (??%? → ?); |
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| 271 | nlapply (try_cast_null_sound m i (Tint sz1 si1) t v'); |
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| 272 | ncases (try_cast_null m i (Tint sz1 si1) t); |
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| 273 | ##[ ##1,3,5: #v''; #H' e; napply H'; napply e; |
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| 274 | ##| ##*: #_; nwhd in ⊢ (??%? → ?); #H; ndestruct (H); |
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| 275 | ##] |
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| 276 | ##] |
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| 277 | ##| ##*: ##[ #sp ty''; nletin t ≝ (Tpointer sp ty'') |
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| 278 | ##| #sp ty'' n; nletin t ≝ (Tarray sp ty'' n) |
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| 279 | ##| #args rty; nletin t ≝ (Tfunction args rty) ##] |
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| 280 | nwhd in ⊢ (??%? → ?); |
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| 281 | nlapply (try_cast_null_sound m i t ty' v'); |
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| 282 | ncases (try_cast_null m i t ty'); |
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| 283 | ##[ ##1,3,5: #v''; #H' e; napply H'; napply e; |
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| 284 | ##| ##*: #_; nwhd in ⊢ (??%? → ?); #H; ndestruct (H); |
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| 285 | ##] |
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| 286 | ##] |
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| 287 | ##| #f; ncases ty; ##[ ##3,9: #x; ##| ##2,4,6,7,8: #x y; ##| ##5: #x y z; ##] |
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| 288 | ##[ ncases ty'; ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##] |
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| 289 | nwhd in e:(??%?); ndestruct; //; |
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| 290 | ##| ##*: #e; nwhd in e:(??%?); ndestruct |
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| 291 | ##] |
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| 292 | ##| #sp b of; nwhd in ⊢ (??%? → ?); #e; |
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| 293 | nelim (bind_inversion ????? e); #s; *; #es e'; |
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| 294 | nelim (bind_inversion ????? e'); #u; *; #eu e''; |
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| 295 | nelim (bind_inversion ????? e''); #s'; *; #es' e'''; |
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[155] | 296 | ncut (type_space ty s); |
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[250] | 297 | ##[ ncases ty in es:(??%?) ⊢ %; ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##] |
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| 298 | nwhd in e:(??%?); ndestruct; //; |
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[155] | 299 | ##| #Hty; |
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| 300 | ncut (type_space ty' s'); |
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| 301 | ##[ ncases ty' in es' ⊢ %; ##[ #e; ##| ##3,9: #a e; ##| ##2,4,6,7,8: #a b e; ##| #a b c e; ##] |
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| 302 | nwhd in e:(??%?); ndestruct; //; |
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| 303 | ##| #Hty'; |
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[250] | 304 | ncut (s = sp). nelim (ms_eq_dec sp s) in eu; //; nnormalize; #_; #e; ndestruct. |
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[155] | 305 | #e; nrewrite < e; |
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[250] | 306 | nwhd in match (is_pointer_compat ??) in e'''; |
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| 307 | ncases (pointer_compat_dec (block_space m b) s') in e'''; #Hcompat e'''; |
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| 308 | nwhd in e''':(??%?); ndestruct (e'''); /2/ |
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[155] | 309 | ##] |
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[124] | 310 | ##] |
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[20] | 311 | ##] nqed. |
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| 312 | |
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[124] | 313 | ndefinition load_value_of_type' ≝ |
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[125] | 314 | λty,m,l. match l with [ mk_pair pl ofs ⇒ match pl with [ mk_pair psp loc ⇒ |
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| 315 | load_value_of_type ty m psp loc ofs ] ]. |
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[124] | 316 | |
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[20] | 317 | (* To make the evaluation of bare lvalue expressions invoke exec_lvalue with |
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| 318 | a structurally smaller value, we break out the surrounding Expr constructor |
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| 319 | and use exec_lvalue'. *) |
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| 320 | |
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[250] | 321 | nlet rec exec_expr (ge:genv) (en:env) (m:mem) (e:expr) on e : res (val×trace) ≝ |
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[20] | 322 | match e with |
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| 323 | [ Expr e' ty ⇒ |
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| 324 | match e' with |
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[250] | 325 | [ Econst_int i ⇒ OK ? 〈Vint i, E0〉 |
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| 326 | | Econst_float f ⇒ OK ? 〈Vfloat f, E0〉 |
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| 327 | | Evar _ ⇒ |
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[189] | 328 | do 〈l,tr〉 ← exec_lvalue' ge en m e' ty; |
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| 329 | do v ← opt_to_res ? (load_value_of_type' ty m l); |
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[250] | 330 | OK ? 〈v,tr〉 |
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| 331 | | Ederef _ ⇒ |
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[189] | 332 | do 〈l,tr〉 ← exec_lvalue' ge en m e' ty; |
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| 333 | do v ← opt_to_res ? (load_value_of_type' ty m l); |
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[250] | 334 | OK ? 〈v,tr〉 |
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| 335 | | Efield _ _ ⇒ |
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[189] | 336 | do 〈l,tr〉 ← exec_lvalue' ge en m e' ty; |
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| 337 | do v ← opt_to_res ? (load_value_of_type' ty m l); |
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[250] | 338 | OK ? 〈v,tr〉 |
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| 339 | | Eaddrof a ⇒ |
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[189] | 340 | do 〈plo,tr〉 ← exec_lvalue ge en m a; |
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[250] | 341 | OK ? 〈match plo with [ mk_pair pl ofs ⇒ match pl with [ mk_pair pcl loc ⇒ Vptr pcl loc ofs ] ], tr〉 |
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| 342 | | Esizeof ty' ⇒ OK ? 〈Vint (repr (sizeof ty')), E0〉 |
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| 343 | | Eunop op a ⇒ |
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[189] | 344 | do 〈v1,tr〉 ← exec_expr ge en m a; |
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| 345 | do v ← opt_to_res ? (sem_unary_operation op v1 (typeof a)); |
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[250] | 346 | OK ? 〈v,tr〉 |
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| 347 | | Ebinop op a1 a2 ⇒ |
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[189] | 348 | do 〈v1,tr1〉 ← exec_expr ge en m a1; |
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| 349 | do 〈v2,tr2〉 ← exec_expr ge en m a2; |
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| 350 | do v ← opt_to_res ? (sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m); |
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[250] | 351 | OK ? 〈v,tr1⧺tr2〉 |
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| 352 | | Econdition a1 a2 a3 ⇒ |
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[189] | 353 | do 〈v,tr1〉 ← exec_expr ge en m a1; |
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[250] | 354 | do b ← exec_bool_of_val v (typeof a1); |
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[189] | 355 | do 〈v',tr2〉 ← match b return λ_.res (val×trace) with |
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[175] | 356 | [ true ⇒ (exec_expr ge en m a2) |
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[189] | 357 | | false ⇒ (exec_expr ge en m a3) ]; |
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[250] | 358 | OK ? 〈v',tr1⧺tr2〉 |
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[20] | 359 | (* if b then exec_expr ge en m a2 else exec_expr ge en m a3)*) |
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[250] | 360 | | Eorbool a1 a2 ⇒ |
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[189] | 361 | do 〈v1,tr1〉 ← exec_expr ge en m a1; |
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[250] | 362 | do b1 ← exec_bool_of_val v1 (typeof a1); |
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[175] | 363 | match b1 return λ_.res (val×trace) with [ true ⇒ OK ? 〈Vtrue,tr1〉 | false ⇒ |
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[189] | 364 | do 〈v2,tr2〉 ← exec_expr ge en m a2; |
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[250] | 365 | do b2 ← exec_bool_of_val v2 (typeof a2); |
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| 366 | OK ? 〈of_bool b2, tr1⧺tr2〉 ] |
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| 367 | | Eandbool a1 a2 ⇒ |
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[189] | 368 | do 〈v1,tr1〉 ← exec_expr ge en m a1; |
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[250] | 369 | do b1 ← exec_bool_of_val v1 (typeof a1); |
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[175] | 370 | match b1 return λ_.res (val×trace) with [ true ⇒ |
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[189] | 371 | do 〈v2,tr2〉 ← exec_expr ge en m a2; |
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[250] | 372 | do b2 ← exec_bool_of_val v2 (typeof a2); |
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[175] | 373 | OK ? 〈of_bool b2, tr1⧺tr2〉 |
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[250] | 374 | | false ⇒ OK ? 〈Vfalse,tr1〉 ] |
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| 375 | | Ecast ty' a ⇒ |
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[189] | 376 | do 〈v,tr〉 ← exec_expr ge en m a; |
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| 377 | do v' ← exec_cast m v (typeof a) ty'; |
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[250] | 378 | OK ? 〈v',tr〉 |
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| 379 | | Ecost l a ⇒ |
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[189] | 380 | do 〈v,tr〉 ← exec_expr ge en m a; |
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[250] | 381 | OK ? 〈v,tr⧺(Echarge l)〉 |
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[20] | 382 | ] |
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| 383 | ] |
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[250] | 384 | and exec_lvalue' (ge:genv) (en:env) (m:mem) (e':expr_descr) (ty:type) on e' : res (memory_space × block × int × trace) ≝ |
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[20] | 385 | match e' with |
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| 386 | [ Evar id ⇒ |
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| 387 | match (get … id en) with |
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[250] | 388 | [ None ⇒ do 〈sp,l〉 ← opt_to_res ? (find_symbol ? ? ge id); OK ? 〈〈〈sp,l〉,zero〉,E0〉 (* global *) |
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| 389 | | Some loc ⇒ OK ? 〈〈〈Any,loc〉,zero〉,E0〉 (* local *) |
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[20] | 390 | ] |
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[250] | 391 | | Ederef a ⇒ |
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[189] | 392 | do 〈v,tr〉 ← exec_expr ge en m a; |
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[20] | 393 | match v with |
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[175] | 394 | [ Vptr sp l ofs ⇒ OK ? 〈〈〈sp,l〉,ofs〉,tr〉 |
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[20] | 395 | | _ ⇒ Error ? |
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[250] | 396 | ] |
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[20] | 397 | | Efield a i ⇒ |
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| 398 | match (typeof a) with |
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[250] | 399 | [ Tstruct id fList ⇒ |
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[189] | 400 | do 〈plofs,tr〉 ← exec_lvalue ge en m a; |
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| 401 | do delta ← field_offset i fList; |
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[250] | 402 | OK ? 〈〈\fst plofs,add (\snd plofs) (repr delta)〉,tr〉 |
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| 403 | | Tunion id fList ⇒ |
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[189] | 404 | do 〈plofs,tr〉 ← exec_lvalue ge en m a; |
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[250] | 405 | OK ? 〈plofs,tr〉 |
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| 406 | | _ ⇒ Error ? |
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[20] | 407 | ] |
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[250] | 408 | | _ ⇒ Error ? |
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[20] | 409 | ] |
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[250] | 410 | and exec_lvalue (ge:genv) (en:env) (m:mem) (e:expr) on e : res (memory_space × block × int × trace) ≝ |
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[20] | 411 | match e with [ Expr e' ty ⇒ exec_lvalue' ge en m e' ty ]. |
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[250] | 412 | |
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| 413 | nlemma P_res_to_P: ∀A,P,e,v. P_res A P e → e = OK A v → P v. |
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| 414 | #A P e v H1 H2; nrewrite > H2 in H1; nwhd in ⊢ (% → ?); //; nqed. |
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| 415 | |
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[251] | 416 | (* We define a special induction principle tailored to the recursive definition |
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| 417 | above. *) |
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| 418 | |
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| 419 | ndefinition is_not_lvalue: expr_descr → Prop ≝ |
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| 420 | λe. match e with [ Evar _ ⇒ False | Ederef _ ⇒ False | Efield _ _ ⇒ False | _ ⇒ True ]. |
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| 421 | |
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| 422 | ndefinition Plvalue ≝ λP:(expr → Prop).λe,ty. |
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| 423 | match e return λ_.Prop with [ Evar _ ⇒ P (Expr e ty) | Ederef _ ⇒ P (Expr e ty) | Efield _ _ ⇒ P (Expr e ty) | _ ⇒ True ]. |
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| 424 | |
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| 425 | nlet rec expr_lvalue_ind |
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| 426 | (P:expr → Prop) |
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| 427 | (Q:expr_descr → type → Prop) |
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| 428 | (ci:∀ty,i.P (Expr (Econst_int i) ty)) |
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| 429 | (cf:∀ty,f.P (Expr (Econst_float f) ty)) |
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| 430 | (lv:∀e,ty. Q e ty → Plvalue P e ty) |
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| 431 | (vr:∀v,ty.Q (Evar v) ty) |
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| 432 | (dr:∀e,ty.P e → Q (Ederef e) ty) |
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| 433 | (ao:∀ty,e,ty'.Q e ty' → P (Expr (Eaddrof (Expr e ty')) ty)) |
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| 434 | (uo:∀ty,op,e.P e → P (Expr (Eunop op e) ty)) |
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| 435 | (bo:∀ty,op,e1,e2.P e1 → P e2 → P (Expr (Ebinop op e1 e2) ty)) |
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| 436 | (ca:∀ty,ty',e.P e → P (Expr (Ecast ty' e) ty)) |
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| 437 | (cd:∀ty,e1,e2,e3.P e1 → P e2 → P e3 → P (Expr (Econdition e1 e2 e3) ty)) |
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| 438 | (ab:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eandbool e1 e2) ty)) |
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| 439 | (ob:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eorbool e1 e2) ty)) |
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| 440 | (sz:∀ty,ty'. P (Expr (Esizeof ty') ty)) |
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| 441 | (fl:∀ty,e,ty',i. Q e ty' → Q (Efield (Expr e ty') i) ty) |
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| 442 | (co:∀ty,l,e. P e → P (Expr (Ecost l e) ty)) |
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| 443 | (xx:∀e,ty. is_not_lvalue e → Q e ty) |
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| 444 | (e:expr) on e : P e ≝ |
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| 445 | match e with |
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| 446 | [ Expr e' ty ⇒ |
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| 447 | match e' with |
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| 448 | [ Econst_int i ⇒ ci ty i |
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| 449 | | Econst_float f ⇒ cf ty f |
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| 450 | | Evar v ⇒ lv (Evar v) ty (vr v ty) |
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| 451 | | Ederef e'' ⇒ lv (Ederef e'') ty (dr e'' ty (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'')) |
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| 452 | | Eaddrof e'' ⇒ match e'' with [ Expr e0 ty0 ⇒ ao ty e0 ty0 (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e0 ty0) ] |
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| 453 | | Eunop op e'' ⇒ uo ty op e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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| 454 | | Ebinop op e1 e2 ⇒ bo ty op e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) |
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| 455 | | Ecast ty' e'' ⇒ ca ty ty' e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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| 456 | | Econdition e1 e2 e3 ⇒ cd ty e1 e2 e3 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e3) |
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| 457 | | Eandbool e1 e2 ⇒ ab ty e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) |
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| 458 | | Eorbool e1 e2 ⇒ ob ty e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) |
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| 459 | | Esizeof ty' ⇒ sz ty ty' |
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| 460 | | Efield e'' i ⇒ match e'' with [ Expr ef tyf ⇒ lv (Efield (Expr ef tyf) i) ty (fl ty ef tyf i (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx ef tyf)) ] |
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| 461 | | Ecost l e'' ⇒ co ty l e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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| 462 | ] |
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| 463 | ] |
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| 464 | and lvalue_expr_ind |
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| 465 | (P:expr → Prop) |
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| 466 | (Q:expr_descr → type → Prop) |
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| 467 | (ci:∀ty,i.P (Expr (Econst_int i) ty)) |
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| 468 | (cf:∀ty,f.P (Expr (Econst_float f) ty)) |
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| 469 | (lv:∀e,ty. Q e ty → Plvalue P e ty) |
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| 470 | (vr:∀v,ty.Q (Evar v) ty) |
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| 471 | (dr:∀e,ty.P e → Q (Ederef e) ty) |
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| 472 | (ao:∀ty,e,ty'.Q e ty' → P (Expr (Eaddrof (Expr e ty')) ty)) |
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| 473 | (uo:∀ty,op,e.P e → P (Expr (Eunop op e) ty)) |
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| 474 | (bo:∀ty,op,e1,e2.P e1 → P e2 → P (Expr (Ebinop op e1 e2) ty)) |
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| 475 | (ca:∀ty,ty',e.P e → P (Expr (Ecast ty' e) ty)) |
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| 476 | (cd:∀ty,e1,e2,e3.P e1 → P e2 → P e3 → P (Expr (Econdition e1 e2 e3) ty)) |
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| 477 | (ab:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eandbool e1 e2) ty)) |
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| 478 | (ob:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eorbool e1 e2) ty)) |
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| 479 | (sz:∀ty,ty'. P (Expr (Esizeof ty') ty)) |
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| 480 | (fl:∀ty,e,ty',i. Q e ty' → Q (Efield (Expr e ty') i) ty) |
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| 481 | (co:∀ty,l,e. P e → P (Expr (Ecost l e) ty)) |
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| 482 | (xx:∀e,ty. is_not_lvalue e → Q e ty) |
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| 483 | (e:expr_descr) (ty:type) on e : Q e ty ≝ |
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| 484 | match e return λe0. Q e0 ty with |
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| 485 | [ Evar v ⇒ vr v ty |
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| 486 | | Ederef e'' ⇒ dr e'' ty (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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| 487 | | Efield e' i ⇒ match e' return λe1.Q (Efield e1 i) ty with [ Expr e'' ty'' ⇒ fl ty e'' ty'' i (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'' ty'') ] |
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| 488 | | _ ⇒ xx ? ty ? |
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| 489 | ]. nwhd; napply I; nqed. |
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| 490 | |
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| 491 | |
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| 492 | ntheorem exec_expr_sound: ∀ge:genv. ∀en:env. ∀m:mem. ∀e:expr. |
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| 493 | (P_res ? (λx.eval_expr ge en m e (\fst x) (\snd x)) (exec_expr ge en m e)). |
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| 494 | #ge en m e; napply (expr_lvalue_ind ? (λe',ty.P_res ? (λr.eval_lvalue ge en m (Expr e' ty) (\fst (\fst (\fst r))) (\snd (\fst (\fst r))) (\snd (\fst r)) (\snd r)) (exec_lvalue' ge en m e' ty)) … e); |
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| 495 | ##[ ##1,2: #ty c; nwhd; //; |
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| 496 | (* expressions that are lvalues *) |
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| 497 | ##| #e' ty; ncases e'; //; ##[ #i He' ##| #e He' ##| #e i He' ##] nwhd in He' ⊢ %; |
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[250] | 498 | napply bind2_OK; #x; ncases x; #y; ncases y; #sp loc ofs tr H; |
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[251] | 499 | napply opt_bind_OK; #vl evl; nwhd in evl:(??%?); napply (eval_Elvalue … evl); |
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| 500 | nrewrite > H in He'; #He'; napply He'; |
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| 501 | ##| #v ty; |
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| 502 | nwhd in ⊢ (???%); |
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| 503 | nlapply (refl ? (get ident PTree block v en)); |
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| 504 | ncases (get ident PTree block v en) in ⊢ (???% → %); |
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| 505 | ##[ #eget; napply opt_bind_OK; #sploc; ncases sploc; #sp loc efind; |
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| 506 | nwhd; napply (eval_Evar_global … eget efind); |
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| 507 | ##| #loc eget; napply (eval_Evar_local … eget); |
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| 508 | ##] |
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| 509 | ##| #ty e He; nwhd in ⊢ (???%); |
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| 510 | napply bind2_OK; #v; ncases v; //; #sp l ofs tr Hv; nwhd; |
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| 511 | napply eval_Ederef; nrewrite > Hv in He; #He; napply He; |
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| 512 | ##| #ty e'' ty'' He''; napply bind2_OK; #x; ncases x; #y; ncases y; #sp loc ofs tr H; |
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| 513 | nwhd; napply eval_Eaddrof; nrewrite > H in He''; #He''; napply He''; |
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| 514 | ##| #ty op e1 He1; napply bind2_OK; #v1 tr Hv1; |
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[175] | 515 | napply opt_bind_OK; #v ev; |
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[250] | 516 | napply (eval_Eunop … ev); |
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[251] | 517 | nrewrite > Hv1 in He1; #He1; napply He1; |
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| 518 | ##| #ty op e1 e2 He1 He2; |
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| 519 | napply bind2_OK; #v1 tr1 ev1; nrewrite > ev1 in He1; #He1; |
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| 520 | napply bind2_OK; #v2 tr2 ev2; nrewrite > ev2 in He2; #He2; |
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| 521 | napply opt_bind_OK; #v ev; nwhd in He1 He2; nwhd; |
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| 522 | napply (eval_Ebinop … He1 He2 ev); |
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| 523 | ##| #ty ty' e' He'; |
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| 524 | napply bind2_OK; #v tr Hv; nrewrite > Hv in He'; #He'; |
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[250] | 525 | napply bind_OK; #v' ev'; |
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[251] | 526 | napply (eval_Ecast … He' ?); |
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[250] | 527 | /2/; |
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[251] | 528 | ##| #ty e1 e2 e3 He1 He2 He3; |
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| 529 | napply bind2_OK; #vb tr1 Hvb; nrewrite > Hvb in He1; #He1; |
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[250] | 530 | napply bind_OK; #b; |
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| 531 | ncases b; #eb; nlapply (exec_bool_of_val_sound … eb); #Hb; |
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[251] | 532 | napply bind2_OK; #v tr Hv; |
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| 533 | ##[ nrewrite > Hv in He2; #He2; nwhd in He2 Hv:(??%?) ⊢%; |
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| 534 | napply (eval_Econdition_true … He1 ? He2); napply (bool_of ??? Hb); |
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| 535 | ##| nrewrite > Hv in He3; #He3; nwhd in He3 Hv:(??%?) ⊢%; |
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| 536 | napply (eval_Econdition_false … He1 ? He3); napply (bool_of ??? Hb); |
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[20] | 537 | ##] |
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[251] | 538 | ##| #ty e1 e2 He1 He2; |
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| 539 | napply bind2_OK; #v1 tr1 Hv1; nrewrite > Hv1 in He1; #He1; |
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[250] | 540 | napply bind_OK; #b1; ncases b1; #eb1; nlapply (exec_bool_of_val_sound … eb1); #Hb1; |
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[251] | 541 | ##[ napply bind2_OK; #v2 tr2 Hv2; nrewrite > Hv2 in He2; #He2; |
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[250] | 542 | napply bind_OK; #b2 eb2; |
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[251] | 543 | napply (eval_Eandbool_2 … He1 … He2); |
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[250] | 544 | ##[ napply (bool_of … Hb1); ##| napply (exec_bool_of_val_sound … eb2); ##] |
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[251] | 545 | ##| napply (eval_Eandbool_1 … He1); napply (bool_of … Hb1); |
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[20] | 546 | ##] |
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[251] | 547 | ##| #ty e1 e2 He1 He2; |
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| 548 | napply bind2_OK; #v1 tr1 Hv1; nrewrite > Hv1 in He1; #He1; |
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[250] | 549 | napply bind_OK; #b1; ncases b1; #eb1; nlapply (exec_bool_of_val_sound … eb1); #Hb1; |
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[251] | 550 | ##[ napply (eval_Eorbool_1 … He1); napply (bool_of … Hb1); |
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| 551 | ##| napply bind2_OK; #v2 tr2 Hv2; nrewrite > Hv2 in He2; #He2; |
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[250] | 552 | napply bind_OK; #b2 eb2; |
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[251] | 553 | napply (eval_Eorbool_2 … He1 … He2); |
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[250] | 554 | ##[ napply (bool_of … Hb1); ##| napply (exec_bool_of_val_sound … eb2); ##] |
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[20] | 555 | ##] |
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[251] | 556 | ##| #ty ty'; nwhd; // |
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| 557 | ##| #ty e' ty' i; ncases ty'; //; |
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| 558 | ##[ #id fs He'; napply bind2_OK; #x; ncases x; #sp l ofs H; |
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| 559 | napply bind_OK; #delta Hdelta; nrewrite > H in He'; #He'; |
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| 560 | napply (eval_Efield_struct … He' (refl ??) Hdelta); |
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| 561 | ##| #id fs He'; napply bind2_OK; #x; ncases x; #sp l ofs H; |
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| 562 | nrewrite > H in He'; #He'; |
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| 563 | napply (eval_Efield_union … He' (refl ??)); |
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| 564 | ##] |
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| 565 | ##| #ty l e' He'; napply bind2_OK; #v tr1 H; nrewrite > H in He'; #He'; |
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| 566 | napply (eval_Ecost … He'); |
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| 567 | (* exec_lvalue fails on non-lvalues. *) |
---|
| 568 | ##| #e' ty; ncases e'; |
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| 569 | ##[ ##1,2,5,12: #a H ##| ##3,4: #a; * ##| ##13,14: #a b; * ##| ##6,8,10,11: #a b H ##| ##7,9: #a b c H ##] |
---|
| 570 | napply I; |
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[20] | 571 | ##] nqed. |
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| 572 | |
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[251] | 573 | nlemma addrof_eval_lvalue: ∀ge,en,m,e,sp,loc,off,tr,ty. |
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| 574 | eval_expr ge en m (Expr (Eaddrof e) ty) (Vptr sp loc off) tr → |
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| 575 | eval_lvalue ge en m e sp loc off tr. |
---|
| 576 | #ge en m e sp loc off tr ty H; ninversion H; |
---|
| 577 | ##[ ##1,2,5: #a b H; napply False_ind; ndestruct (H); |
---|
| 578 | ##| #a b c d e f g H1 i H2; nrewrite < H2 in H1; #H1; napply False_ind; |
---|
| 579 | napply (eval_lvalue_inv_ind … H1); |
---|
| 580 | ##[ #a b c d bad; ndestruct (bad); |
---|
| 581 | ##| #a b c d e f bad; ndestruct (bad); |
---|
| 582 | ##| #a b c d e f g bad; ndestruct (bad); |
---|
| 583 | ##| #a b c d e f g h i j k l m n bad; napply False_ind; ndestruct (bad); |
---|
| 584 | ##| #a b c d e f g h i j k l bad; ndestruct (bad); |
---|
| 585 | ##] |
---|
| 586 | ##| #e' ty' sp' loc' ofs' tr' H e1 e2 e3; ndestruct (e1 e2 e3); napply H; |
---|
| 587 | ##| #a b c d e f g h i bad; ndestruct (bad); |
---|
| 588 | ##| #a b c d e f g h i j k l k l bad; ndestruct (bad); |
---|
| 589 | ##| #a b c d e f g h i j k l m bad; ndestruct (bad); |
---|
| 590 | ##| #a b c d e f g h i j k l m bad; ndestruct (bad); |
---|
| 591 | ##| #a b c d e f g h bad; ndestruct (bad); |
---|
| 592 | ##| #a b c d e f g h i j k l m n bad; ndestruct (bad); |
---|
| 593 | ##| #a b c d e f g h bad; ndestruct (bad); |
---|
| 594 | ##| #a b c d e f g h i j k l m n bad; ndestruct (bad); |
---|
| 595 | ##| #a b c d e f g h i bad; ndestruct (bad); |
---|
| 596 | ##| #a b c d e f g bad; ndestruct (bad); |
---|
| 597 | ##] nqed. |
---|
| 598 | |
---|
| 599 | nlemma addrof_exec_lvalue: ∀ge,en,m,e,sp,loc,off,tr,ty. |
---|
| 600 | exec_lvalue ge en m e = OK ? 〈〈〈sp,loc〉,off〉,tr〉 → |
---|
| 601 | exec_expr ge en m (Expr (Eaddrof e) ty) = OK ? 〈Vptr sp loc off, tr〉. |
---|
| 602 | #ge en m e sp loc off tr ty H; nwhd in ⊢ (??%?); |
---|
| 603 | nrewrite > H; //; |
---|
| 604 | nqed. |
---|
| 605 | |
---|
| 606 | ntheorem exec_lvalue_sound: ∀ge,en,m,e. |
---|
| 607 | P_res ? (λr.eval_lvalue ge en m e (\fst (\fst (\fst r))) (\snd (\fst (\fst r))) (\snd (\fst r)) (\snd r)) (exec_lvalue ge en m e). |
---|
| 608 | #ge en m e; nlapply (refl ? (exec_lvalue ge en m e)); |
---|
| 609 | ncases (exec_lvalue ge en m e) in ⊢ (???% → %); |
---|
| 610 | ##[ #x; ncases x; #y; ncases y; #z; ncases z; #sp loc off tr H; nwhd; |
---|
| 611 | napply (addrof_eval_lvalue … Tvoid); |
---|
| 612 | nlapply (addrof_exec_lvalue … Tvoid H); #H'; |
---|
| 613 | nlapply (exec_expr_sound ge en m (Expr (Eaddrof e) Tvoid)); |
---|
| 614 | nrewrite > H'; #H''; napply H''; |
---|
| 615 | ##| #_; nwhd; napply I; |
---|
| 616 | ##] nqed. |
---|
| 617 | |
---|
| 618 | (* Plain equality versions of the above *) |
---|
| 619 | |
---|
| 620 | ndefinition exec_expr_sound' ≝ λge,en,m,e,v. |
---|
| 621 | λH:exec_expr ge en m e = OK ? v. |
---|
| 622 | P_res_to_P ???? (exec_expr_sound ge en m e) H. |
---|
| 623 | |
---|
| 624 | ndefinition exec_lvalue_sound' ≝ λge,en,m,e,sp,loc,off,tr. |
---|
| 625 | λH:exec_lvalue ge en m e = OK ? 〈〈〈sp,loc〉,off〉,tr〉. |
---|
| 626 | P_res_to_P ???? (exec_lvalue_sound ge en m e) H. |
---|
| 627 | |
---|
[20] | 628 | (* TODO: Can we do this sensibly with a map combinator? *) |
---|
[251] | 629 | nlet rec exec_exprlist (ge:genv) (e:env) (m:mem) (l:list expr) on l : res (list val×trace) ≝ |
---|
[20] | 630 | match l with |
---|
[251] | 631 | [ nil ⇒ OK ? 〈nil val, E0〉 |
---|
| 632 | | cons e1 es ⇒ |
---|
[189] | 633 | do 〈v,tr1〉 ← exec_expr ge e m e1; |
---|
| 634 | do 〈vs,tr2〉 ← exec_exprlist ge e m es; |
---|
[251] | 635 | OK ? 〈cons val v vs, tr1⧺tr2〉 |
---|
| 636 | ]. |
---|
[20] | 637 | |
---|
[251] | 638 | nlemma exec_exprlist_sound: ∀ge,e,m,l. P_res ? (λvltr:list val×trace. eval_exprlist ge e m l (\fst vltr) (\snd vltr)) (exec_exprlist ge e m l). |
---|
| 639 | #ge e m l; nelim l; |
---|
| 640 | nwhd; //; |
---|
| 641 | #e1 es; #IH; |
---|
| 642 | napply bind2_OK; #v tr1 Hv; |
---|
| 643 | napply bind2_OK; #vs tr2 Hvs; |
---|
| 644 | nwhd; napply eval_Econs; |
---|
| 645 | ##[ napply (P_res_to_P … (exec_expr_sound ge e m e1) Hv); |
---|
| 646 | ##| napply (P_res_to_P … IH Hvs); |
---|
| 647 | ##] nqed. |
---|
| 648 | |
---|
[20] | 649 | (* Don't really want to use subset rather than sigma here, but can't be bothered |
---|
| 650 | with *another* set of coercions. XXX: why do I have to get the recursive |
---|
| 651 | call's property manually? *) |
---|
| 652 | |
---|
| 653 | nlet rec exec_alloc_variables (en:env) (m:mem) (l:list (ident × type)) on l : { r:env × mem | alloc_variables en m l (\fst r) (\snd r) } ≝ |
---|
| 654 | match l with |
---|
| 655 | [ nil ⇒ Some ? 〈en, m〉 |
---|
| 656 | | cons h vars ⇒ |
---|
| 657 | match h with [ mk_pair id ty ⇒ |
---|
[125] | 658 | match alloc m 0 (sizeof ty) Any with [ mk_pair m1 b1 ⇒ |
---|
| 659 | match exec_alloc_variables (set … id b1 en) m1 vars with |
---|
[20] | 660 | [ sig_intro r p ⇒ r ] |
---|
[125] | 661 | ]]]. nwhd; |
---|
| 662 | ##[ //; |
---|
| 663 | ##| nelim (exec_alloc_variables (set ident ? ? c3 c7 en) c6 c1); |
---|
| 664 | #H; nelim H; //; #H0; nelim H0; nnormalize; #en' m' IH; |
---|
[20] | 665 | napply (alloc_variables_cons … IH); /2/; |
---|
| 666 | nqed. |
---|
| 667 | |
---|
| 668 | (* TODO: can we establish that length params = length vs in advance? *) |
---|
| 669 | nlet rec exec_bind_parameters (e:env) (m:mem) (params:list (ident × type)) (vs:list val) on params : res (Σm2:mem. bind_parameters e m params vs m2) ≝ |
---|
| 670 | match params with |
---|
| 671 | [ nil ⇒ match vs with [ nil ⇒ Some ? (OK ? m) | cons _ _ ⇒ Some ? (Error ?) ] |
---|
| 672 | | cons idty params' ⇒ match idty with [ mk_pair id ty ⇒ |
---|
| 673 | match vs with |
---|
| 674 | [ nil ⇒ Some ? (Error ?) |
---|
| 675 | | cons v1 vl ⇒ Some ? ( |
---|
[189] | 676 | do b ← opt_to_res ? (get … id e); |
---|
| 677 | do m1 ← opt_to_res ? (store_value_of_type ty m Any b zero v1); |
---|
[20] | 678 | err_eject ?? (exec_bind_parameters e m1 params' vl)) (* FIXME: don't want to have to eject here *) |
---|
| 679 | ] |
---|
| 680 | ] ]. |
---|
| 681 | nwhd; //; |
---|
[125] | 682 | napply opt_bind_OK; #b eb; |
---|
[20] | 683 | napply opt_bind_OK; #m1 em1; |
---|
[250] | 684 | napply sig_bind_OK; #m2 Hm2; |
---|
[20] | 685 | napply (bind_parameters_cons … eb em1 Hm2); |
---|
| 686 | nqed. |
---|
| 687 | |
---|
| 688 | ndefinition is_not_void : ∀t:type. res (Σu:unit. t ≠ Tvoid) ≝ |
---|
| 689 | λt. match t with |
---|
| 690 | [ Tvoid ⇒ Some ? (Error ?) |
---|
| 691 | | _ ⇒ Some ? (OK ??) |
---|
| 692 | ]. nwhd; //; @; #H; ndestruct; nqed. |
---|
| 693 | |
---|
| 694 | ninductive decide : Type ≝ |
---|
| 695 | | dy : decide | dn : decide. |
---|
| 696 | |
---|
| 697 | ndefinition dodecide : ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ ¬P ]).P + ¬P. |
---|
| 698 | #P d;ncases d;/2/; nqed. |
---|
| 699 | |
---|
| 700 | ncoercion decide_inject : |
---|
| 701 | ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ ¬P ]).P + ¬P ≝ dodecide |
---|
| 702 | on d:decide to ? + (¬?). |
---|
| 703 | |
---|
| 704 | ndefinition dodecide2 : ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ True ]).res P. |
---|
| 705 | #P d; ncases d; nnormalize; #p; ##[ napply (OK ? p); ##| napply Error ##] nqed. |
---|
| 706 | |
---|
| 707 | ncoercion decide_inject2 : |
---|
| 708 | ∀P:Prop.∀d.∀p:(match d with [ dy ⇒ P | dn ⇒ True ]).res P ≝ dodecide2 |
---|
| 709 | on d:decide to res ?. |
---|
| 710 | |
---|
| 711 | alias id "Tint" = "cic:/matita/c-semantics/Csyntax/type.con(0,2,0)". |
---|
| 712 | alias id "Tfloat" = "cic:/matita/c-semantics/Csyntax/type.con(0,3,0)". |
---|
| 713 | ndefinition sz_eq_dec : ∀s1,s2:intsize. (s1 = s2) + (s1 ≠ s2). |
---|
| 714 | #s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed. |
---|
| 715 | ndefinition sg_eq_dec : ∀s1,s2:signedness. (s1 = s2) + (s1 ≠ s2). |
---|
| 716 | #s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed. |
---|
| 717 | ndefinition fs_eq_dec : ∀s1,s2:floatsize. (s1 = s2) + (s1 ≠ s2). |
---|
| 718 | #s1; ncases s1; #s2; ncases s2; /2/; @2; @; #H; ndestruct; nqed. |
---|
| 719 | |
---|
| 720 | nlet rec assert_type_eq (t1,t2:type) : res (t1 = t2) ≝ |
---|
| 721 | match t1 with |
---|
| 722 | [ Tvoid ⇒ match t2 with [ Tvoid ⇒ dy | _ ⇒ dn ] |
---|
| 723 | | Tint sz sg ⇒ match t2 with [ Tint sz' sg' ⇒ match sz_eq_dec sz sz' with [ inl _ ⇒ match sg_eq_dec sg sg' with [ inl _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] |
---|
| 724 | | Tfloat f ⇒ match t2 with [ Tfloat f' ⇒ match fs_eq_dec f f' with [ inl _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] |
---|
[124] | 725 | | Tpointer s t ⇒ match t2 with [ Tpointer s' t' ⇒ |
---|
| 726 | match ms_eq_dec s s' with [ inl _ ⇒ |
---|
| 727 | match assert_type_eq t t' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] |
---|
| 728 | | Tarray s t n ⇒ match t2 with [ Tarray s' t' n' ⇒ |
---|
| 729 | match ms_eq_dec s s' with [ inl _ ⇒ |
---|
| 730 | match assert_type_eq t t' with [ OK _ ⇒ |
---|
| 731 | match decidable_eq_Z_Type n n' with [ inl _ ⇒ dy | inr _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] |
---|
[20] | 732 | | Tfunction tl t ⇒ match t2 with [ Tfunction tl' t' ⇒ match assert_typelist_eq tl tl' with [ OK _ ⇒ |
---|
| 733 | match assert_type_eq t t' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] |
---|
| 734 | | Tstruct i fl ⇒ |
---|
| 735 | match t2 with [ Tstruct i' fl' ⇒ match ident_eq i i' with [ inl _ ⇒ |
---|
| 736 | match assert_fieldlist_eq fl fl' with [ OK _ ⇒ dy | _ ⇒ dn ] | inr _ ⇒ dn ] | _ ⇒ dn ] |
---|
| 737 | | Tunion i fl ⇒ |
---|
| 738 | match t2 with [ Tunion i' fl' ⇒ match ident_eq i i' with [ inl _ ⇒ |
---|
| 739 | match assert_fieldlist_eq fl fl' with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] | _ ⇒ dn ] |
---|
| 740 | | Tcomp_ptr i ⇒ match t2 with [ Tcomp_ptr i' ⇒ match ident_eq i i' with [ inl _ ⇒ dy | inr _ ⇒ dn ] | _ ⇒ dn ] |
---|
| 741 | ] |
---|
| 742 | and assert_typelist_eq (tl1,tl2:typelist) : res (tl1 = tl2) ≝ |
---|
| 743 | match tl1 with |
---|
| 744 | [ Tnil ⇒ match tl2 with [ Tnil ⇒ dy | _ ⇒ dn ] |
---|
| 745 | | Tcons t1 ts1 ⇒ match tl2 with [ Tnil ⇒ dn | Tcons t2 ts2 ⇒ |
---|
| 746 | match assert_type_eq t1 t2 with [ OK _ ⇒ |
---|
| 747 | match assert_typelist_eq ts1 ts2 with [ OK _ ⇒ dy | _ ⇒ dn ] | _ ⇒ dn ] ] |
---|
| 748 | ] |
---|
| 749 | and assert_fieldlist_eq (fl1,fl2:fieldlist) : res (fl1 = fl2) ≝ |
---|
| 750 | match fl1 with |
---|
| 751 | [ Fnil ⇒ match fl2 with [ Fnil ⇒ dy | _ ⇒ dn ] |
---|
| 752 | | Fcons i1 t1 fs1 ⇒ match fl2 with [ Fnil ⇒ dn | Fcons i2 t2 fs2 ⇒ |
---|
| 753 | match ident_eq i1 i2 with [ inl _ ⇒ |
---|
| 754 | match assert_type_eq t1 t2 with [ OK _ ⇒ |
---|
| 755 | match assert_fieldlist_eq fs1 fs2 with [ OK _ ⇒ dy | _ ⇒ dn ] |
---|
| 756 | | _ ⇒ dn ] | _ ⇒ dn ] ] |
---|
| 757 | ]. |
---|
| 758 | (* A poor man's clear, otherwise automation picks up recursive calls without |
---|
| 759 | checking that the argument is smaller. *) |
---|
| 760 | ngeneralize in assert_type_eq; |
---|
| 761 | ngeneralize in assert_typelist_eq; |
---|
| 762 | ngeneralize in assert_fieldlist_eq; #avoid1; #_; #avoid2; #_; #avoid3; #_; nwhd; //; |
---|
| 763 | (* XXX: I have no idea why the first // didn't catch these. *) |
---|
| 764 | //; //; //; //; //; //; //; //; //; |
---|
| 765 | nqed. |
---|
| 766 | |
---|
| 767 | nlet rec is_is_call_cont (k:cont) : (is_call_cont k) + (¬is_call_cont k) ≝ |
---|
| 768 | match k with |
---|
| 769 | [ Kstop ⇒ dy |
---|
| 770 | | Kcall _ _ _ _ ⇒ dy |
---|
| 771 | | _ ⇒ dn |
---|
| 772 | ]. nwhd; //; @; #H; nelim H; nqed. |
---|
| 773 | |
---|
| 774 | nlet rec is_Sskip (s:statement) : (s = Sskip) + (s ≠ Sskip) ≝ |
---|
| 775 | match s with |
---|
| 776 | [ Sskip ⇒ dy |
---|
| 777 | | _ ⇒ dn |
---|
| 778 | ]. |
---|
| 779 | ##[ //; |
---|
| 780 | ##| ##*: @; #H; ndestruct; |
---|
| 781 | ##] nqed. |
---|
| 782 | |
---|
| 783 | (* IO monad *) |
---|
| 784 | |
---|
[24] | 785 | (* Interactions are function calls that return a value and do not change |
---|
| 786 | the rest of the Clight program's state. *) |
---|
[25] | 787 | ndefinition io_out ≝ (ident × (list eventval)). |
---|
[20] | 788 | |
---|
[25] | 789 | ndefinition do_io : ident → list eventval → IO eventval io_out eventval ≝ |
---|
| 790 | λfn,args. Interact ?? eventval 〈fn,args〉 (λres. Value ?? eventval res). |
---|
[20] | 791 | |
---|
[25] | 792 | ndefinition ret: ∀T. T → (IO eventval io_out T) ≝ |
---|
| 793 | λT,x.(Value ?? T x). |
---|
[20] | 794 | |
---|
[24] | 795 | (* Checking types of values given to / received from an external function call. *) |
---|
[20] | 796 | |
---|
| 797 | ndefinition check_eventval : ∀ev:eventval. ∀ty:typ. res (Σv:val. eventval_match ev ty v) ≝ |
---|
| 798 | λev,ty. |
---|
| 799 | match ty with |
---|
| 800 | [ Tint ⇒ match ev with [ EVint i ⇒ Some ? (OK ? (Vint i)) | _ ⇒ Some ? (Error ?) ] |
---|
| 801 | | Tfloat ⇒ match ev with [ EVfloat f ⇒ Some ? (OK ? (Vfloat f)) | _ ⇒ Some ? (Error ?) ] |
---|
| 802 | | _ ⇒ Some ? (Error ?) |
---|
| 803 | ]. nwhd; //; nqed. |
---|
| 804 | |
---|
| 805 | ndefinition check_eventval' : ∀v:val. ∀ty:typ. res (Σev:eventval. eventval_match ev ty v) ≝ |
---|
| 806 | λv,ty. |
---|
| 807 | match ty with |
---|
| 808 | [ Tint ⇒ match v with [ Vint i ⇒ Some ? (OK ? (EVint i)) | _ ⇒ Some ? (Error ?) ] |
---|
| 809 | | Tfloat ⇒ match v with [ Vfloat f ⇒ Some ? (OK ? (EVfloat f)) | _ ⇒ Some ? (Error ?) ] |
---|
| 810 | | _ ⇒ Some ? (Error ?) |
---|
| 811 | ]. nwhd; //; nqed. |
---|
| 812 | |
---|
| 813 | nlet rec check_eventval_list (vs: list val) (tys: list typ) : res (Σevs:list eventval. eventval_list_match evs tys vs) ≝ |
---|
| 814 | match vs with |
---|
| 815 | [ nil ⇒ match tys with [ nil ⇒ Some ? (OK ? (nil ?)) | _ ⇒ Some ? (Error ?) ] |
---|
| 816 | | cons v vt ⇒ |
---|
| 817 | match tys with |
---|
| 818 | [ nil ⇒ Some ? (Error ?) |
---|
| 819 | | cons ty tyt ⇒ Some ? ( |
---|
[189] | 820 | do ev ← check_eventval' v ty; |
---|
| 821 | do evt ← check_eventval_list vt tyt; |
---|
[20] | 822 | OK ? ((sig_eject ?? ev)::evt)) |
---|
| 823 | ] |
---|
| 824 | ]. nwhd; //; |
---|
[124] | 825 | napply sig_bind_OK; #ev Hev; |
---|
| 826 | napply sig_bind_OK; #evt Hevt; |
---|
[20] | 827 | nnormalize; /2/; |
---|
| 828 | nqed. |
---|
| 829 | |
---|
| 830 | (* execution *) |
---|
| 831 | |
---|
[124] | 832 | ndefinition store_value_of_type' ≝ |
---|
| 833 | λty,m,l,v. |
---|
| 834 | match l with [ mk_pair pl ofs ⇒ |
---|
| 835 | match pl with [ mk_pair pcl loc ⇒ |
---|
| 836 | store_value_of_type ty m pcl loc ofs v ] ]. |
---|
| 837 | |
---|
[25] | 838 | nlet rec exec_step (ge:genv) (st:state) on st : (IO eventval io_out (Σr:trace × state. step ge st (\fst r) (\snd r))) ≝ |
---|
[20] | 839 | match st with |
---|
| 840 | [ State f s k e m ⇒ |
---|
| 841 | match s with |
---|
| 842 | [ Sassign a1 a2 ⇒ Some ? ( |
---|
[208] | 843 | ! 〈l,tr1〉 ← exec_lvalue ge e m a1; |
---|
| 844 | ! 〈v2,tr2〉 ← exec_expr ge e m a2; |
---|
| 845 | ! m' ← store_value_of_type' (typeof a1) m l v2; |
---|
[175] | 846 | ret ? 〈tr1⧺tr2, State f Sskip k e m'〉) |
---|
[20] | 847 | | Scall lhs a al ⇒ Some ? ( |
---|
[208] | 848 | ! 〈vf,tr2〉 ← exec_expr ge e m a; |
---|
| 849 | ! 〈vargs,tr3〉 ← exec_exprlist ge e m al; |
---|
| 850 | ! fd ← find_funct ? ? ge vf; |
---|
| 851 | ! p ← err_to_io … (assert_type_eq (type_of_fundef fd) (typeof a)); |
---|
[20] | 852 | (* |
---|
| 853 | ! k' ← match lhs with |
---|
| 854 | [ None ⇒ ret ? (Kcall (None ?) f e k) |
---|
| 855 | | Some lhs' ⇒ |
---|
[208] | 856 | ! locofs ← exec_lvalue ge e m lhs'; |
---|
[20] | 857 | ret ? (Kcall (Some ? 〈sig_eject ?? locofs, typeof lhs'〉) f e k) |
---|
[208] | 858 | ]; |
---|
[20] | 859 | ret ? 〈E0, Callstate fd vargs k' m〉) |
---|
| 860 | *) |
---|
| 861 | match lhs with |
---|
[175] | 862 | [ None ⇒ ret ? 〈tr2⧺tr3, Callstate fd vargs (Kcall (None ?) f e k) m〉 |
---|
[20] | 863 | | Some lhs' ⇒ |
---|
[208] | 864 | ! 〈locofs,tr1〉 ← exec_lvalue ge e m lhs'; |
---|
[175] | 865 | ret ? 〈tr1⧺tr2⧺tr3, Callstate fd vargs (Kcall (Some ? 〈locofs, typeof lhs'〉) f e k) m〉 |
---|
[20] | 866 | ]) |
---|
| 867 | | Ssequence s1 s2 ⇒ Some ? (ret ? 〈E0, State f s1 (Kseq s2 k) e m〉) |
---|
| 868 | | Sskip ⇒ |
---|
| 869 | match k with |
---|
| 870 | [ Kseq s k' ⇒ Some ? (ret ? 〈E0, State f s k' e m〉) |
---|
| 871 | | Kstop ⇒ |
---|
| 872 | match fn_return f with |
---|
| 873 | [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef k (free_list m (blocks_of_env e))〉) |
---|
[25] | 874 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 875 | ] |
---|
| 876 | | Kcall _ _ _ _ ⇒ |
---|
| 877 | match fn_return f with |
---|
| 878 | [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef k (free_list m (blocks_of_env e))〉) |
---|
[25] | 879 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 880 | ] |
---|
| 881 | | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f (Swhile a s') k' e m〉) |
---|
| 882 | | Kdowhile a s' k' ⇒ Some ? ( |
---|
[208] | 883 | ! 〈v,tr〉 ← exec_expr ge e m a; |
---|
[250] | 884 | ! b ← err_to_io … (exec_bool_of_val v (typeof a)); |
---|
[20] | 885 | match b with |
---|
[175] | 886 | [ true ⇒ ret ? 〈tr, State f (Sdowhile a s') k' e m〉 |
---|
| 887 | | false ⇒ ret ? 〈tr, State f Sskip k' e m〉 |
---|
[20] | 888 | ]) |
---|
| 889 | | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f a3 (Kfor3 a2 a3 s' k') e m〉) |
---|
| 890 | | Kfor3 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f (Sfor Sskip a2 a3 s') k' e m〉) |
---|
| 891 | | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉) |
---|
[25] | 892 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 893 | ] |
---|
| 894 | | Scontinue ⇒ |
---|
| 895 | match k with |
---|
| 896 | [ Kseq s' k' ⇒ Some ? (ret ? 〈E0, State f Scontinue k' e m〉) |
---|
| 897 | | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f (Swhile a s') k' e m〉) |
---|
| 898 | | Kdowhile a s' k' ⇒ Some ? ( |
---|
[208] | 899 | ! 〈v,tr〉 ← exec_expr ge e m a; |
---|
[250] | 900 | ! b ← err_to_io … (exec_bool_of_val v (typeof a)); |
---|
[20] | 901 | match b with |
---|
[175] | 902 | [ true ⇒ ret ? 〈tr, State f (Sdowhile a s') k' e m〉 |
---|
| 903 | | false ⇒ ret ? 〈tr, State f Sskip k' e m〉 |
---|
[20] | 904 | ]) |
---|
| 905 | | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f a3 (Kfor3 a2 a3 s' k') e m〉) |
---|
| 906 | | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Scontinue k' e m〉) |
---|
[25] | 907 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 908 | ] |
---|
| 909 | | Sbreak ⇒ |
---|
| 910 | match k with |
---|
| 911 | [ Kseq s' k' ⇒ Some ? (ret ? 〈E0, State f Sbreak k' e m〉) |
---|
| 912 | | Kwhile a s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉) |
---|
| 913 | | Kdowhile a s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉) |
---|
| 914 | | Kfor2 a2 a3 s' k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉) |
---|
| 915 | | Kswitch k' ⇒ Some ? (ret ? 〈E0, State f Sskip k' e m〉) |
---|
[25] | 916 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 917 | ] |
---|
| 918 | | Sifthenelse a s1 s2 ⇒ Some ? ( |
---|
[208] | 919 | ! 〈v,tr〉 ← exec_expr ge e m a; |
---|
[250] | 920 | ! b ← err_to_io … (exec_bool_of_val v (typeof a)); |
---|
[175] | 921 | ret ? 〈tr, State f (if b then s1 else s2) k e m〉) |
---|
[20] | 922 | | Swhile a s' ⇒ Some ? ( |
---|
[208] | 923 | ! 〈v,tr〉 ← exec_expr ge e m a; |
---|
[250] | 924 | ! b ← err_to_io … (exec_bool_of_val v (typeof a)); |
---|
[175] | 925 | ret ? 〈tr, if b then State f s' (Kwhile a s' k) e m |
---|
| 926 | else State f Sskip k e m〉) |
---|
[20] | 927 | | Sdowhile a s' ⇒ Some ? (ret ? 〈E0, State f s' (Kdowhile a s' k) e m〉) |
---|
| 928 | | Sfor a1 a2 a3 s' ⇒ |
---|
| 929 | match is_Sskip a1 with |
---|
| 930 | [ inl _ ⇒ Some ? ( |
---|
[208] | 931 | ! 〈v,tr〉 ← exec_expr ge e m a2; |
---|
[250] | 932 | ! b ← err_to_io … (exec_bool_of_val v (typeof a2)); |
---|
[175] | 933 | ret ? 〈tr, State f (if b then s' else Sskip) (if b then (Kfor2 a2 a3 s' k) else k) e m〉) |
---|
[20] | 934 | | inr _ ⇒ Some ? (ret ? 〈E0, State f a1 (Kseq (Sfor Sskip a2 a3 s') k) e m〉) |
---|
| 935 | ] |
---|
| 936 | | Sreturn a_opt ⇒ |
---|
| 937 | match a_opt with |
---|
| 938 | [ None ⇒ match fn_return f with |
---|
| 939 | [ Tvoid ⇒ Some ? (ret ? 〈E0, Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e))〉) |
---|
[25] | 940 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 941 | ] |
---|
| 942 | | Some a ⇒ Some ? ( |
---|
[251] | 943 | ! u ← err_to_io_sig … (is_not_void (fn_return f)); |
---|
[208] | 944 | ! 〈v,tr〉 ← exec_expr ge e m a; |
---|
[175] | 945 | ret ? 〈tr, Returnstate v (call_cont k) (free_list m (blocks_of_env e))〉) |
---|
[20] | 946 | ] |
---|
| 947 | | Sswitch a sl ⇒ Some ? ( |
---|
[208] | 948 | ! 〈v,tr〉 ← exec_expr ge e m a; |
---|
[175] | 949 | match v with [ Vint n ⇒ ret ? 〈tr, State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m〉 |
---|
[25] | 950 | | _ ⇒ Wrong ??? ]) |
---|
[20] | 951 | | Slabel lbl s' ⇒ Some ? (ret ? 〈E0, State f s' k e m〉) |
---|
| 952 | | Sgoto lbl ⇒ |
---|
| 953 | match find_label lbl (fn_body f) (call_cont k) with |
---|
| 954 | [ Some sk' ⇒ match sk' with [ mk_pair s' k' ⇒ Some ? (ret ? 〈E0, State f s' k' e m〉) ] |
---|
[25] | 955 | | None ⇒ Some ? (Wrong ???) |
---|
[20] | 956 | ] |
---|
[175] | 957 | | Scost lbl s' ⇒ Some ? (ret ? 〈Echarge lbl, State f s' k e m〉) |
---|
[20] | 958 | ] |
---|
| 959 | | Callstate f0 vargs k m ⇒ |
---|
| 960 | match f0 with |
---|
| 961 | [ Internal f ⇒ Some ? ( |
---|
| 962 | match exec_alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) with [ mk_pair e m1 ⇒ |
---|
[251] | 963 | ! m2 ← err_to_io_sig … (exec_bind_parameters e m1 (fn_params f) vargs); |
---|
[20] | 964 | ret ? 〈E0, State f (fn_body f) k e m2〉 |
---|
| 965 | ]) |
---|
| 966 | | External f argtys retty ⇒ Some ? ( |
---|
[251] | 967 | ! evargs ← err_to_io_sig … (check_eventval_list vargs (typlist_of_typelist argtys)); |
---|
[208] | 968 | ! evres ← do_io f evargs; |
---|
[251] | 969 | ! vres ← err_to_io_sig … (check_eventval evres (proj_sig_res (signature_of_type argtys retty))); |
---|
[20] | 970 | ret ? 〈(Eextcall f evargs evres), Returnstate vres k m〉) |
---|
| 971 | ] |
---|
| 972 | | Returnstate res k m ⇒ |
---|
| 973 | match k with |
---|
| 974 | [ Kcall r f e k' ⇒ |
---|
| 975 | match r with |
---|
| 976 | [ None ⇒ |
---|
| 977 | match res with |
---|
| 978 | [ Vundef ⇒ Some ? (ret ? 〈E0, (State f Sskip k' e m)〉) |
---|
[25] | 979 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 980 | ] |
---|
| 981 | | Some r' ⇒ |
---|
[124] | 982 | match r' with [ mk_pair l ty ⇒ |
---|
| 983 | Some ? ( |
---|
[208] | 984 | ! m' ← store_value_of_type' ty m l res; |
---|
[20] | 985 | ret ? 〈E0, (State f Sskip k' e m')〉) |
---|
| 986 | ] |
---|
| 987 | ] |
---|
[25] | 988 | | _ ⇒ Some ? (Wrong ???) |
---|
[20] | 989 | ] |
---|
| 990 | ]. nwhd; //; |
---|
| 991 | ##[ nrewrite > c7; napply step_skip_call; //; napply c8; |
---|
| 992 | ##| napply step_skip_or_continue_while; @; //; |
---|
[251] | 993 | ##| napply res_bindIO2_OK; #v tr Hv; |
---|
[250] | 994 | nletin bexpr ≝ (exec_bool_of_val v (typeof c7)); |
---|
| 995 | nlapply (refl ? bexpr); |
---|
| 996 | ncases bexpr in ⊢ (???% → %); |
---|
| 997 | ##[ #b; ncases b; #eb; nlapply (exec_bool_of_val_sound … eb); #Hb; |
---|
| 998 | nwhd in ⊢ (?????%); |
---|
[251] | 999 | ##[ napply (step_skip_or_continue_dowhile_true … (exec_expr_sound' … Hv)); |
---|
[250] | 1000 | ##[ @; // ##| napply (bool_of … Hb); ##] |
---|
[251] | 1001 | ##| napply (step_skip_or_continue_dowhile_false … (exec_expr_sound' … Hv)); |
---|
[250] | 1002 | ##[ @; // ##| napply (bool_of … Hb); ##] |
---|
| 1003 | ##] |
---|
| 1004 | ##| #_; //; |
---|
[20] | 1005 | ##] |
---|
| 1006 | ##| napply step_skip_or_continue_for2; @; //; |
---|
| 1007 | ##| napply step_skip_break_switch; @; //; |
---|
| 1008 | ##| nrewrite > c11; napply step_skip_call; //; napply c12; |
---|
[251] | 1009 | ##| napply res_bindIO2_OK; #x; ncases x; #y; ncases y; #pcl loc ofs tr1 Hlval; |
---|
| 1010 | napply res_bindIO2_OK; #v2 tr2 Hv2; |
---|
[20] | 1011 | napply opt_bindIO_OK; #m' em'; |
---|
[251] | 1012 | nwhd; napply (step_assign … (exec_lvalue_sound' … Hlval) (exec_expr_sound' … Hv2) em'); |
---|
| 1013 | ##| napply res_bindIO2_OK; #vf tr1 Hvf0; nlapply (exec_expr_sound' … Hvf0); #Hvf; |
---|
| 1014 | napply res_bindIO2_OK; #vargs tr2 Hvargs0; nlapply (P_res_to_P ???? (exec_exprlist_sound …) Hvargs0); #Hvargs; |
---|
[20] | 1015 | napply opt_bindIO_OK; #fd efd; |
---|
| 1016 | napply bindIO_OK; #ety; |
---|
| 1017 | ncases c6; nwhd; |
---|
| 1018 | ##[ napply (step_call_none … Hvf Hvargs efd ety); |
---|
| 1019 | ##| #lhs'; |
---|
[251] | 1020 | napply res_bindIO2_OK; #x; ncases x; #y; ncases y; #pcl loc ofs tr3 Hlocofs; |
---|
| 1021 | nwhd; napply (step_call_some … (exec_lvalue_sound' … Hlocofs) Hvf Hvargs efd ety); |
---|
[20] | 1022 | ##] |
---|
[251] | 1023 | ##| napply res_bindIO2_OK; #v tr Hv; |
---|
[250] | 1024 | nletin bexpr ≝ (exec_bool_of_val v (typeof c6)); |
---|
| 1025 | nlapply (refl ? bexpr); ncases bexpr in ⊢ (???% → %); //; |
---|
| 1026 | #b; ncases b; #eb; nlapply (exec_bool_of_val_sound … eb); #Hb; |
---|
[251] | 1027 | ##[ napply (step_ifthenelse_true … (exec_expr_sound' … Hv)); napply (bool_of … Hb); |
---|
| 1028 | ##| napply (step_ifthenelse_false … (exec_expr_sound' … Hv)); napply (bool_of … Hb) |
---|
[20] | 1029 | ##] |
---|
[251] | 1030 | ##| napply res_bindIO2_OK; #v tr Hv; |
---|
[250] | 1031 | nletin bexpr ≝ (exec_bool_of_val v (typeof c6)); |
---|
| 1032 | nlapply (refl ? bexpr); ncases bexpr in ⊢ (???% → %); //; |
---|
| 1033 | #b; ncases b; #eb; nlapply (exec_bool_of_val_sound … eb); #Hb; |
---|
[251] | 1034 | ##[ napply (step_while_true … (exec_expr_sound' … Hv)); napply (bool_of … Hb); |
---|
| 1035 | ##| napply (step_while_false … (exec_expr_sound' … Hv)); napply (bool_of … Hb); |
---|
[20] | 1036 | ##] |
---|
| 1037 | ##| nrewrite > c11; |
---|
[251] | 1038 | napply res_bindIO2_OK; #v tr Hv; |
---|
[250] | 1039 | nletin bexpr ≝ (exec_bool_of_val v (typeof c7)); |
---|
| 1040 | nlapply (refl ? bexpr); ncases bexpr in ⊢ (???% → %); //; |
---|
| 1041 | #b; ncases b; #eb; nlapply (exec_bool_of_val_sound … eb); #Hb; |
---|
[251] | 1042 | ##[ napply (step_for_true … (exec_expr_sound' … Hv)); napply (bool_of … Hb); |
---|
| 1043 | ##| napply (step_for_false … (exec_expr_sound' … Hv)); napply (bool_of … Hb); |
---|
[20] | 1044 | ##] |
---|
| 1045 | ##| napply step_for_start; //; |
---|
| 1046 | ##| napply step_skip_break_switch; @2; //; |
---|
| 1047 | ##| napply step_skip_or_continue_while; @2; //; |
---|
[251] | 1048 | ##| napply res_bindIO2_OK; #v tr Hv; |
---|
[250] | 1049 | nletin bexpr ≝ (exec_bool_of_val v (typeof c7)); |
---|
| 1050 | nlapply (refl ? bexpr); ncases bexpr in ⊢ (???% → %); //; |
---|
| 1051 | #b; ncases b; #eb; nlapply (exec_bool_of_val_sound … eb); #Hb; |
---|
[251] | 1052 | ##[ napply (step_skip_or_continue_dowhile_true … (exec_expr_sound' … Hv)); |
---|
[20] | 1053 | ##[ @2; // ##| napply (bool_of … Hb); ##] |
---|
[251] | 1054 | ##| napply (step_skip_or_continue_dowhile_false … (exec_expr_sound' … Hv)); |
---|
[20] | 1055 | ##[ @2; // ##| napply (bool_of … Hb); ##] |
---|
| 1056 | ##] |
---|
| 1057 | ##| napply step_skip_or_continue_for2; @2; // |
---|
| 1058 | ##| napply step_return_0; napply c9; |
---|
| 1059 | ##| napply sig_bindIO_OK; #u Hnotvoid; |
---|
[251] | 1060 | napply res_bindIO2_OK; #v tr Hv; |
---|
| 1061 | nwhd; napply (step_return_1 … Hnotvoid (exec_expr_sound' … Hv)); |
---|
| 1062 | ##| napply res_bindIO2_OK; #v; ncases v; //; #n tr Hv; |
---|
| 1063 | napply step_switch; napply (exec_expr_sound' … Hv); |
---|
[20] | 1064 | ##| napply step_goto; nrewrite < c12; napply c9; |
---|
| 1065 | ##| napply extract_subset_pair_io; #e m1 ealloc Halloc; |
---|
| 1066 | napply sig_bindIO_OK; #m2 Hbind; |
---|
| 1067 | nwhd; napply (step_internal_function … Halloc Hbind); |
---|
| 1068 | ##| napply sig_bindIO_OK; #evs Hevs; |
---|
| 1069 | napply bindIO_OK; #eres; |
---|
| 1070 | napply sig_bindIO_OK; #res Hres; |
---|
| 1071 | nwhd; napply step_external_function; @; ##[ napply Hevs; ##| napply Hres; ##] |
---|
[124] | 1072 | ##| ncases c11; #x; ncases x; #pcl b ofs; |
---|
| 1073 | napply opt_bindIO_OK; #m' em'; napply step_returnstate_1; nwhd in em':(??%?); //; |
---|
[20] | 1074 | ##] |
---|
| 1075 | nqed. |
---|
| 1076 | |
---|
[250] | 1077 | nlet rec make_initial_state (p:program) : IO eventval io_out state ≝ |
---|
[20] | 1078 | let ge ≝ globalenv Genv ?? p in |
---|
| 1079 | let m0 ≝ init_mem Genv ?? p in |
---|
[208] | 1080 | ! 〈sp,b〉 ← find_symbol ? ? ge (prog_main ?? p); |
---|
| 1081 | ! u ← opt_to_io … (match ms_eq_dec sp Code with [ inl _ ⇒ Some ? something | inr _ ⇒ None ? ]); |
---|
| 1082 | ! f ← find_funct_ptr ? ? ge b; |
---|
[250] | 1083 | ret ? (Callstate f (nil ?) Kstop m0). |
---|
| 1084 | |
---|
| 1085 | nlemma make_initial_state_sound : ∀p. P_io ??? (λs.initial_state p s) (make_initial_state p). |
---|
| 1086 | #p; ncases p; #fns main vars; |
---|
| 1087 | nwhd in ⊢ (?????%); |
---|
[125] | 1088 | napply opt_bindIO2_OK; #sp b esb; |
---|
| 1089 | napply opt_bindIO_OK; #u ecode; |
---|
[20] | 1090 | napply opt_bindIO_OK; #f ef; |
---|
[156] | 1091 | ncases sp in esb ecode; #esb ecode; nwhd in ecode:(??%%); ##[ ##1,2,3,4,5: ndestruct (ecode); ##] |
---|
[125] | 1092 | nwhd; napply (initial_state_intro … esb ef); |
---|
[20] | 1093 | nqed. |
---|
| 1094 | |
---|
[239] | 1095 | ndefinition is_final_state : ∀st:state. (Σr. final_state st r) + (¬∃r. final_state st r). |
---|
[20] | 1096 | #st; nelim st; |
---|
| 1097 | ##[ #f s k e m; @2; @;*; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1098 | ##| #f l k m; @2; @;*; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1099 | ##| #v k m; ncases k; |
---|
| 1100 | ##[ ncases v; |
---|
| 1101 | ##[ ##2: #i; @1; @ i; //; |
---|
| 1102 | ##| ##1: @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1103 | ##| #f; @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
[124] | 1104 | ##| #pcl b of; @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
[20] | 1105 | ##] |
---|
| 1106 | ##| #a b; @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1107 | ##| ##3,4: #a b c; @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1108 | ##| ##5,6,8: #a b c d; @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1109 | ##| #a; @2; @; *; #r H; ninversion H; #i m e; ndestruct; |
---|
| 1110 | ##] |
---|
| 1111 | ##] nqed. |
---|
| 1112 | |
---|
| 1113 | nlet rec exec_steps (n:nat) (ge:genv) (s:state) : |
---|
[25] | 1114 | IO eventval io_out (Σts:trace×state. star (mk_transrel ?? step) ge s (\fst ts) (\snd ts)) ≝ |
---|
[20] | 1115 | match is_final_state s with |
---|
| 1116 | [ inl _ ⇒ Some ? (ret ? 〈E0, s〉) |
---|
| 1117 | | inr _ ⇒ |
---|
| 1118 | match n with |
---|
| 1119 | [ O ⇒ Some ? (ret ? 〈E0, s〉) |
---|
| 1120 | | S n' ⇒ Some ? ( |
---|
[208] | 1121 | ! 〈t,s'〉 ← exec_step ge s; |
---|
[152] | 1122 | (* ! 〈t,s'〉 ← match s with |
---|
| 1123 | [ State f s k e m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (State f s k e (mk_mem c n p)) ] |
---|
| 1124 | | Callstate fd args k m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (Callstate fd args k (mk_mem c n p)) ] |
---|
| 1125 | | Returnstate r k m ⇒ match m with [ mk_mem c n p ⇒ exec_step ge (Returnstate r k (mk_mem c n p)) ] |
---|
[208] | 1126 | ] ;*) |
---|
[152] | 1127 | ! 〈t',s''〉 ← match s' with |
---|
| 1128 | [ State f s k e m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (State f s k e (mk_mem c n p)) ] |
---|
| 1129 | | Callstate fd args k m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (Callstate fd args k (mk_mem c n p)) ] |
---|
| 1130 | | Returnstate r k m ⇒ match m with [ mk_mem c n p ⇒ exec_steps n' ge (Returnstate r k (mk_mem c n p)) ] |
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[208] | 1131 | ] ; |
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| 1132 | (* ! 〈t',s''〉 ← exec_steps n' ge s';*) |
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[20] | 1133 | ret ? 〈t ⧺ t',s''〉) |
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| 1134 | ] |
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| 1135 | ]. nwhd; /2/; |
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[152] | 1136 | napply sig_bindIO2_OK; #t s'; ncases s'; ##[ #f st k e m; ##| #fd args k m; ##| #r k m; ##] |
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| 1137 | nwhd in ⊢ (? → ?????(??????%?)); |
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| 1138 | ncases m; #mc mn mp; #H1; |
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| 1139 | nwhd in ⊢ (?????(??????%?)); |
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[20] | 1140 | napply sig_bindIO2_OK; #t' s'' IH; |
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| 1141 | nwhd; napply (star_step … IH); //; |
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| 1142 | nqed. |
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| 1143 | (* |
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| 1144 | nlet rec exec_steps_without_proof (n:nat) (ge:genv) (s:state) : |
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| 1145 | res (trace×state) ≝ |
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| 1146 | match is_final_state s with |
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| 1147 | [ inl _ ⇒ OK ? 〈E0, s〉 |
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| 1148 | | inr _ ⇒ |
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| 1149 | match n with |
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| 1150 | [ O ⇒ OK ? 〈E0, s〉 |
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| 1151 | | S n' ⇒ |
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[208] | 1152 | 〈t,s'〉 ← exec_step ge s; |
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| 1153 | 〈t',s''〉 ← exec_steps_without_proof n' ge s'; |
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[20] | 1154 | OK ? 〈t ⧺ t',s''〉 |
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| 1155 | ] |
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| 1156 | ]. |
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| 1157 | *) |
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[174] | 1158 | |
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| 1159 | (* A (possibly non-terminating) execution. *) |
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| 1160 | ncoinductive execution : Type ≝ |
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[239] | 1161 | | e_stop : trace → int → execution |
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[174] | 1162 | | e_step : trace → state → execution → execution |
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| 1163 | | e_wrong : execution |
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| 1164 | | e_interact : io_out → (eventval → execution) → execution. |
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| 1165 | |
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| 1166 | nlet corec exec_inf_aux (ge:genv) (s:IO eventval io_out (trace×state)) : execution ≝ |
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| 1167 | match s with |
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| 1168 | [ Wrong ⇒ e_wrong |
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| 1169 | | Value v ⇒ match v with [ mk_pair t s' ⇒ |
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| 1170 | match is_final_state s' with |
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[239] | 1171 | [ inl r ⇒ e_stop t (sig_eject … r) |
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[174] | 1172 | | inr _ ⇒ e_step t s' (exec_inf_aux ge (exec_step ge s')) ] ] |
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| 1173 | | Interact out k' ⇒ e_interact out (λv. exec_inf_aux ge (k' v)) |
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| 1174 | ]. |
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| 1175 | |
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[239] | 1176 | |
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[174] | 1177 | ndefinition exec_inf : program → execution ≝ |
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[250] | 1178 | λp. exec_inf_aux (globalenv Genv ?? p) (! s ← make_initial_state p; ret ? 〈E0,s〉). |
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[174] | 1179 | |
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[239] | 1180 | nremark execution_cases: ∀e. |
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| 1181 | e = match e with [ e_stop tr r ⇒ e_stop tr r | e_step tr s e ⇒ e_step tr s e |
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| 1182 | | e_wrong ⇒ e_wrong | e_interact o k ⇒ e_interact o k ]. |
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| 1183 | #e; ncases e; //; nqed. |
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| 1184 | |
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| 1185 | nlemma exec_inf_aux_unfold: ∀ge,s. exec_inf_aux ge s = |
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| 1186 | match s with |
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| 1187 | [ Wrong ⇒ e_wrong |
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| 1188 | | Value v ⇒ match v with [ mk_pair t s' ⇒ |
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| 1189 | match is_final_state s' with |
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| 1190 | [ inl r ⇒ e_stop t (sig_eject … r) |
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| 1191 | | inr _ ⇒ e_step t s' (exec_inf_aux ge (exec_step ge s')) ] ] |
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| 1192 | | Interact out k' ⇒ e_interact out (λv. exec_inf_aux ge (k' v)) |
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| 1193 | ]. |
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[245] | 1194 | #ge s; nrewrite > (execution_cases (exec_inf_aux …)); ncases s; |
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| 1195 | ##[ #o k |
---|
| 1196 | ##| #x; ncases x; #tr s'; ncases s'; |
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| 1197 | ##[ #fn st k env m |
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| 1198 | ##| #fd args k mem |
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| 1199 | ##| #v k mem; (* for final state check *) ncases k; |
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| 1200 | ##[ ncases v; ##[ ##2,3: #v' ##| ##4: #sp loc off ##] |
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| 1201 | ##| #s' k' ##| ##3,4: #e s' k' ##| ##5,6: #e s' s'' k' ##| #k' ##| #a b c d ##] |
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| 1202 | ##] |
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| 1203 | ##| ##] |
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| 1204 | nwhd in ⊢ (??%%); //; |
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| 1205 | nqed. |
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[239] | 1206 | |
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| 1207 | (* Finite number of steps without interaction. *) |
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| 1208 | ninductive execution_steps : trace → execution → execution → Prop ≝ |
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| 1209 | | steps_none : ∀e. execution_steps E0 e e |
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| 1210 | | steps_one : ∀e,e',tr,tr',s. execution_steps tr' e e' → execution_steps (tr⧺tr') (e_step tr s e) e'. |
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| 1211 | |
---|
| 1212 | (* Finite number of steps allowing interaction. *) |
---|
| 1213 | ninductive execution_isteps : trace → execution → execution → Prop ≝ |
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| 1214 | | isteps_none : ∀e. execution_isteps E0 e e |
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| 1215 | | isteps_step : ∀e,e',tr,tr',s. execution_isteps tr e e' → execution_isteps (tr⧺tr') (e_step tr s e) e' |
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| 1216 | | isteps_interact : ∀k,o,i,e',tr. execution_isteps tr (k i) e' → execution_isteps tr (e_interact o k) e'. |
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| 1217 | |
---|
| 1218 | ninductive execution_terminates : trace → int → execution → Prop ≝ |
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| 1219 | | terminates : ∀tr,tr',r,e. execution_isteps tr e (e_stop tr' r) → execution_terminates (tr⧺tr') r e. |
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| 1220 | |
---|
| 1221 | ncoinductive execution_diverging : execution → Prop ≝ |
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| 1222 | | diverging_step : ∀s,e. execution_diverging e → execution_diverging (e_step E0 s e). |
---|
| 1223 | |
---|
| 1224 | (* Makes a finite number of interactions (including cost labels) before diverging. *) |
---|
| 1225 | ninductive execution_diverges : trace → execution → Prop ≝ |
---|
| 1226 | | diverges_diverging: ∀tr,e,e'. |
---|
| 1227 | execution_isteps tr e e' → |
---|
| 1228 | execution_diverging e' → |
---|
| 1229 | execution_diverges tr e. |
---|
| 1230 | |
---|
| 1231 | (* NB: "reacting" includes hitting a cost label. *) |
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| 1232 | ncoinductive execution_reacts : traceinf → execution → Prop ≝ |
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| 1233 | | reacting: ∀tr,tr',e,e'. execution_reacts tr' e' → execution_isteps tr e e' → tr ≠ E0 → execution_reacts (tr⧻tr') e. |
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| 1234 | |
---|
| 1235 | ninductive execution_goes_wrong: trace → execution → Prop ≝ |
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| 1236 | | go_wrong: ∀tr,e. execution_isteps tr e e_wrong → execution_goes_wrong tr e. |
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| 1237 | |
---|
| 1238 | ninductive execution_matches_behavior: execution → program_behavior → Prop ≝ |
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| 1239 | | emb_terminates: ∀e,tr,r. |
---|
| 1240 | execution_terminates tr r e → |
---|
| 1241 | execution_matches_behavior e (Terminates tr r) |
---|
| 1242 | | emb_diverges: ∀e,tr. |
---|
| 1243 | execution_diverges tr e → |
---|
| 1244 | execution_matches_behavior e (Diverges tr) |
---|
| 1245 | | emb_reacts: ∀e,tr. |
---|
| 1246 | execution_reacts tr e → |
---|
| 1247 | execution_matches_behavior e (Reacts tr) |
---|
| 1248 | | emb_wrong: ∀e,tr. |
---|
| 1249 | execution_goes_wrong tr e → |
---|
| 1250 | execution_matches_behavior e (Goes_wrong tr). |
---|
| 1251 | |
---|
[245] | 1252 | (* We don't morally need the cut, but the proof I tried without it failed the |
---|
| 1253 | guarded-by-constructors check and it wasn't apparent why. *) |
---|
| 1254 | |
---|
| 1255 | nlet corec silent_sound ge s e |
---|
| 1256 | (H0:execution_diverging e) |
---|
| 1257 | (EXEC:exec_inf_aux ge (Value ??? 〈E0,s〉) = e) |
---|
| 1258 | : forever_silent (mk_transrel ?? step) … ge s ≝ ?. |
---|
| 1259 | ncut (∃s2.∃e2.execution_diverging e2 ∧ exec_inf_aux ge (exec_step ge s) = e_step E0 s2 e2); |
---|
| 1260 | ##[ ncases H0 in EXEC ⊢ %; #s1 e1 H1; |
---|
| 1261 | nrewrite > (exec_inf_aux_unfold …); nwhd in ⊢ (??%? → ?); |
---|
| 1262 | ncases (is_final_state s); #p EXEC; nwhd in EXEC:(??%?); ndestruct; |
---|
| 1263 | ninversion H1; #s2 e2 H2 EXEC2; |
---|
| 1264 | @ s2; @ e2; @; //; |
---|
| 1265 | ##| *; #s2; *; #e2; *; #H2 EXEC2; |
---|
| 1266 | napply (forever_silent_intro (mk_transrel ?? step) … ge s s2 ? (silent_sound ge s2 (e_step E0 s2 e2) ??)); |
---|
| 1267 | ncut (∃p.exec_step ge s = Value ??? (sig_intro ?? 〈E0,s2〉 p)); |
---|
| 1268 | ##[ ##1,3,5: nrewrite > (exec_inf_aux_unfold …) in EXEC2; nelim (exec_step ge s); |
---|
| 1269 | ##[ ##1,4,7: #o k IH EXEC2; nwhd in EXEC2:(??%?); ndestruct; |
---|
| 1270 | ##| ##2,5,8: #x; ncases x; #y; ncases y; #tr' s' p; nwhd in ⊢ (??%? → ?); |
---|
| 1271 | ncases (is_final_state s'); #p' EXEC2; nwhd in EXEC2:(??%?); ndestruct; |
---|
| 1272 | @ p; napply refl; |
---|
| 1273 | ##| ##3,6,9: #EXEC2; nwhd in EXEC2:(??%?); ndestruct |
---|
| 1274 | ##] |
---|
| 1275 | ##| *; #p EXEC1; napply p |
---|
| 1276 | ##| *; #p EXEC1; nrewrite > EXEC1 in EXEC2; #EXEC2; nwhd in EXEC2:(??(??%)?); napply EXEC2; |
---|
| 1277 | ##| *; #p EXEC1; @; napply H2; |
---|
| 1278 | ##] |
---|
| 1279 | ##] |
---|
| 1280 | nqed. |
---|
| 1281 | |
---|
[239] | 1282 | (* |
---|
| 1283 | ntheorem exec_inf_sound: ∀p. ∃b.execution_matches_behavior (exec_inf p) b ∧ exec_program p b. |
---|
[245] | 1284 | *) |
---|