[3] | 1 | (* *********************************************************************) |
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| 2 | (* *) |
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| 3 | (* The Compcert verified compiler *) |
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| 4 | (* *) |
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| 5 | (* Xavier Leroy, INRIA Paris-Rocquencourt *) |
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| 6 | (* *) |
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| 7 | (* Copyright Institut National de Recherche en Informatique et en *) |
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| 8 | (* Automatique. All rights reserved. This file is distributed *) |
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| 9 | (* under the terms of the GNU General Public License as published by *) |
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| 10 | (* the Free Software Foundation, either version 2 of the License, or *) |
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| 11 | (* (at your option) any later version. This file is also distributed *) |
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| 12 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
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| 13 | (* *) |
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| 14 | (* *********************************************************************) |
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| 15 | |
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| 16 | (* * Dynamic semantics for the Clight language *) |
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| 17 | |
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[474] | 18 | (*include "Coqlib.ma".*) |
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| 19 | (*include "Errors.ma".*) |
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| 20 | (*include "Integers.ma".*) |
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| 21 | (*include "Floats.ma".*) |
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| 22 | (*include "Values.ma".*) |
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| 23 | (*include "AST.ma".*) |
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| 24 | (*include "Mem.ma".*) |
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[3] | 25 | include "Globalenvs.ma". |
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| 26 | include "Csyntax.ma". |
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| 27 | include "Maps.ma". |
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[474] | 28 | (*include "Events.ma".*) |
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[3] | 29 | include "Smallstep.ma". |
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| 30 | |
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| 31 | (* * * Semantics of type-dependent operations *) |
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| 32 | |
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| 33 | (* * Interpretation of values as truth values. |
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| 34 | Non-zero integers, non-zero floats and non-null pointers are |
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| 35 | considered as true. The integer zero (which also represents |
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| 36 | the null pointer) and the float 0.0 are false. *) |
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| 37 | |
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[487] | 38 | inductive is_false: val → type → Prop ≝ |
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[3] | 39 | | is_false_int: ∀sz,sg. |
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| 40 | is_false (Vint zero) (Tint sz sg) |
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[484] | 41 | | is_false_pointer: ∀r,r',t. |
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| 42 | is_false (Vnull r) (Tpointer r' t) |
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[3] | 43 | | is_false_float: ∀sz. |
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| 44 | is_false (Vfloat Fzero) (Tfloat sz). |
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| 45 | |
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[487] | 46 | inductive is_true: val → type → Prop ≝ |
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[3] | 47 | | is_true_int_int: ∀n,sz,sg. |
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| 48 | n ≠ zero → |
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| 49 | is_true (Vint n) (Tint sz sg) |
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[500] | 50 | | is_true_pointer_pointer: ∀r,b,pc,ofs,s,t. |
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| 51 | is_true (Vptr r b pc ofs) (Tpointer s t) |
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[3] | 52 | | is_true_float: ∀f,sz. |
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| 53 | f ≠ Fzero → |
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| 54 | is_true (Vfloat f) (Tfloat sz). |
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| 55 | |
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[487] | 56 | inductive bool_of_val : val → type → val → Prop ≝ |
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[3] | 57 | | bool_of_val_true: ∀v,ty. |
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| 58 | is_true v ty → |
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| 59 | bool_of_val v ty Vtrue |
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| 60 | | bool_of_val_false: ∀v,ty. |
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| 61 | is_false v ty → |
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| 62 | bool_of_val v ty Vfalse. |
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| 63 | |
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| 64 | (* * The following [sem_] functions compute the result of an operator |
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| 65 | application. Since operators are overloaded, the result depends |
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| 66 | both on the static types of the arguments and on their run-time values. |
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| 67 | Unlike in C, automatic conversions between integers and floats |
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| 68 | are not performed. For instance, [e1 + e2] is undefined if [e1] |
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| 69 | is a float and [e2] an integer. The Clight producer must have explicitly |
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| 70 | promoted [e2] to a float. *) |
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| 71 | |
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[487] | 72 | let rec sem_neg (v: val) (ty: type) : option val ≝ |
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[3] | 73 | match ty with |
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| 74 | [ Tint _ _ ⇒ |
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| 75 | match v with |
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| 76 | [ Vint n ⇒ Some ? (Vint (neg n)) |
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| 77 | | _ => None ? |
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| 78 | ] |
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| 79 | | Tfloat _ ⇒ |
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| 80 | match v with |
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| 81 | [ Vfloat f ⇒ Some ? (Vfloat (Fneg f)) |
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| 82 | | _ ⇒ None ? |
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| 83 | ] |
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| 84 | | _ ⇒ None ? |
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| 85 | ]. |
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| 86 | |
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[487] | 87 | let rec sem_notint (v: val) : option val ≝ |
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[3] | 88 | match v with |
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| 89 | [ Vint n ⇒ Some ? (Vint (xor n mone)) |
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| 90 | | _ ⇒ None ? |
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| 91 | ]. |
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| 92 | |
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[487] | 93 | let rec sem_notbool (v: val) (ty: type) : option val ≝ |
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[3] | 94 | match ty with |
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| 95 | [ Tint _ _ ⇒ |
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| 96 | match v with |
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| 97 | [ Vint n ⇒ Some ? (of_bool (eq n zero)) |
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| 98 | | _ ⇒ None ? |
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| 99 | ] |
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[124] | 100 | | Tpointer _ _ ⇒ |
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[3] | 101 | match v with |
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[502] | 102 | [ Vptr _ _ _ _ ⇒ Some ? Vfalse |
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| 103 | | Vnull _ ⇒ Some ? Vtrue |
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[3] | 104 | | _ ⇒ None ? |
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| 105 | ] |
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| 106 | | Tfloat _ ⇒ |
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| 107 | match v with |
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| 108 | [ Vfloat f ⇒ Some ? (of_bool (Fcmp Ceq f Fzero)) |
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| 109 | | _ ⇒ None ? |
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| 110 | ] |
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| 111 | | _ ⇒ None ? |
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| 112 | ]. |
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| 113 | |
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[487] | 114 | let rec sem_add (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 115 | match classify_add t1 t2 with |
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| 116 | [ add_case_ii ⇒ (**r integer addition *) |
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| 117 | match v1 with |
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| 118 | [ Vint n1 ⇒ match v2 with |
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| 119 | [ Vint n2 ⇒ Some ? (Vint (add n1 n2)) |
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| 120 | | _ ⇒ None ? ] |
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| 121 | | _ ⇒ None ? ] |
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| 122 | | add_case_ff ⇒ (**r float addition *) |
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| 123 | match v1 with |
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| 124 | [ Vfloat n1 ⇒ match v2 with |
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| 125 | [ Vfloat n2 ⇒ Some ? (Vfloat (Fadd n1 n2)) |
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| 126 | | _ ⇒ None ? ] |
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| 127 | | _ ⇒ None ? ] |
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| 128 | | add_case_pi ty ⇒ (**r pointer plus integer *) |
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| 129 | match v1 with |
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[500] | 130 | [ Vptr r1 b1 p1 ofs1 ⇒ match v2 with |
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| 131 | [ Vint n2 ⇒ Some ? (Vptr r1 b1 p1 (add ofs1 (mul (repr (sizeof ty)) n2))) |
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[3] | 132 | | _ ⇒ None ? ] |
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[484] | 133 | | Vnull r ⇒ match v2 with |
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| 134 | [ Vint n2 ⇒ if eq n2 zero then Some ? (Vnull r) else None ? |
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| 135 | | _ ⇒ None ? ] |
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[3] | 136 | | _ ⇒ None ? ] |
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| 137 | | add_case_ip ty ⇒ (**r integer plus pointer *) |
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| 138 | match v1 with |
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| 139 | [ Vint n1 ⇒ match v2 with |
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[500] | 140 | [ Vptr r2 b2 p2 ofs2 ⇒ Some ? (Vptr r2 b2 p2 (add ofs2 (mul (repr (sizeof ty)) n1))) |
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[3] | 141 | | _ ⇒ None ? ] |
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| 142 | | _ ⇒ None ? ] |
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| 143 | | add_default ⇒ None ? |
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| 144 | ]. |
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| 145 | |
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[487] | 146 | let rec sem_sub (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 147 | match classify_sub t1 t2 with |
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| 148 | [ sub_case_ii ⇒ (**r integer subtraction *) |
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| 149 | match v1 with |
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| 150 | [ Vint n1 ⇒ match v2 with |
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| 151 | [ Vint n2 ⇒ Some ? (Vint (sub n1 n2)) |
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| 152 | | _ ⇒ None ? ] |
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| 153 | | _ ⇒ None ? ] |
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| 154 | | sub_case_ff ⇒ (**r float subtraction *) |
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| 155 | match v1 with |
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| 156 | [ Vfloat f1 ⇒ match v2 with |
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| 157 | [ Vfloat f2 ⇒ Some ? (Vfloat (Fsub f1 f2)) |
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| 158 | | _ ⇒ None ? ] |
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| 159 | | _ ⇒ None ? ] |
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| 160 | | sub_case_pi ty ⇒ (**r pointer minus integer *) |
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| 161 | match v1 with |
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[500] | 162 | [ Vptr r1 b1 p1 ofs1 ⇒ match v2 with |
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| 163 | [ Vint n2 ⇒ Some ? (Vptr r1 b1 p1 (sub ofs1 (mul (repr (sizeof ty)) n2))) |
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[3] | 164 | | _ ⇒ None ? ] |
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| 165 | | _ ⇒ None ? ] |
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| 166 | | sub_case_pp ty ⇒ (**r pointer minus pointer *) |
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| 167 | match v1 with |
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[500] | 168 | [ Vptr r1 b1 p1 ofs1 ⇒ match v2 with |
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| 169 | [ Vptr r2 b2 p2 ofs2 ⇒ |
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[496] | 170 | if eq_block b1 b2 then |
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[3] | 171 | if eq (repr (sizeof ty)) zero then None ? |
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| 172 | else Some ? (Vint (divu (sub ofs1 ofs2) (repr (sizeof ty)))) |
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| 173 | else None ? |
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| 174 | | _ ⇒ None ? ] |
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[484] | 175 | | Vnull r ⇒ match v2 with [ Vnull r' ⇒ Some ? (Vint zero) | _ ⇒ None ? ] |
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[3] | 176 | | _ ⇒ None ? ] |
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| 177 | | sub_default ⇒ None ? |
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| 178 | ]. |
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[124] | 179 | |
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[487] | 180 | let rec sem_mul (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 181 | match classify_mul t1 t2 with |
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| 182 | [ mul_case_ii ⇒ |
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| 183 | match v1 with |
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| 184 | [ Vint n1 ⇒ match v2 with |
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| 185 | [ Vint n2 ⇒ Some ? (Vint (mul n1 n2)) |
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| 186 | | _ ⇒ None ? ] |
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| 187 | | _ ⇒ None ? ] |
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| 188 | | mul_case_ff ⇒ |
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| 189 | match v1 with |
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| 190 | [ Vfloat f1 ⇒ match v2 with |
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| 191 | [ Vfloat f2 ⇒ Some ? (Vfloat (Fmul f1 f2)) |
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| 192 | | _ ⇒ None ? ] |
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| 193 | | _ ⇒ None ? ] |
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| 194 | | mul_default ⇒ |
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| 195 | None ? |
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| 196 | ]. |
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| 197 | |
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[487] | 198 | let rec sem_div (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 199 | match classify_div t1 t2 with |
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| 200 | [ div_case_I32unsi ⇒ |
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| 201 | match v1 with |
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| 202 | [ Vint n1 ⇒ match v2 with |
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| 203 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (divu n1 n2)) |
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| 204 | | _ ⇒ None ? ] |
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| 205 | | _ ⇒ None ? ] |
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| 206 | | div_case_ii ⇒ |
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| 207 | match v1 with |
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| 208 | [ Vint n1 ⇒ match v2 with |
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| 209 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint(divs n1 n2)) |
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| 210 | | _ ⇒ None ? ] |
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| 211 | | _ ⇒ None ? ] |
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| 212 | | div_case_ff ⇒ |
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| 213 | match v1 with |
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| 214 | [ Vfloat f1 ⇒ match v2 with |
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| 215 | [ Vfloat f2 ⇒ Some ? (Vfloat(Fdiv f1 f2)) |
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| 216 | | _ ⇒ None ? ] |
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| 217 | | _ ⇒ None ? ] |
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| 218 | | div_default ⇒ |
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| 219 | None ? |
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| 220 | ]. |
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| 221 | |
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[487] | 222 | let rec sem_mod (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 223 | match classify_mod t1 t2 with |
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| 224 | [ mod_case_I32unsi ⇒ |
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| 225 | match v1 with |
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| 226 | [ Vint n1 ⇒ match v2 with |
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| 227 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (modu n1 n2)) |
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| 228 | | _ ⇒ None ? ] |
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| 229 | | _ ⇒ None ? ] |
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| 230 | | mod_case_ii ⇒ |
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| 231 | match v1 with |
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| 232 | [ Vint n1 ⇒ match v2 with |
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| 233 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (mods n1 n2)) |
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| 234 | | _ ⇒ None ? ] |
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| 235 | | _ ⇒ None ? ] |
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| 236 | | mod_default ⇒ |
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| 237 | None ? |
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| 238 | ]. |
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| 239 | |
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[487] | 240 | let rec sem_and (v1,v2: val) : option val ≝ |
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[3] | 241 | match v1 with |
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| 242 | [ Vint n1 ⇒ match v2 with |
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| 243 | [ Vint n2 ⇒ Some ? (Vint(i_and n1 n2)) |
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| 244 | | _ ⇒ None ? ] |
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| 245 | | _ ⇒ None ? |
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| 246 | ]. |
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| 247 | |
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[487] | 248 | let rec sem_or (v1,v2: val) : option val ≝ |
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[3] | 249 | match v1 with |
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| 250 | [ Vint n1 ⇒ match v2 with |
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| 251 | [ Vint n2 ⇒ Some ? (Vint(or n1 n2)) |
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| 252 | | _ ⇒ None ? ] |
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| 253 | | _ ⇒ None ? |
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| 254 | ]. |
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| 255 | |
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[487] | 256 | let rec sem_xor (v1,v2: val) : option val ≝ |
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[3] | 257 | match v1 with |
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| 258 | [ Vint n1 ⇒ match v2 with |
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| 259 | [ Vint n2 ⇒ Some ? (Vint(xor n1 n2)) |
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| 260 | | _ ⇒ None ? ] |
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| 261 | | _ ⇒ None ? |
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| 262 | ]. |
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| 263 | |
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[487] | 264 | let rec sem_shl (v1,v2: val): option val ≝ |
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[3] | 265 | match v1 with |
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| 266 | [ Vint n1 ⇒ match v2 with |
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| 267 | [ Vint n2 ⇒ |
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| 268 | if ltu n2 iwordsize then Some ? (Vint(shl n1 n2)) else None ? |
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| 269 | | _ ⇒ None ? ] |
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| 270 | | _ ⇒ None ? ]. |
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| 271 | |
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[487] | 272 | let rec sem_shr (v1: val) (t1: type) (v2: val) (t2: type): option val ≝ |
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[3] | 273 | match classify_shr t1 t2 with |
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| 274 | [ shr_case_I32unsi ⇒ |
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| 275 | match v1 with |
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| 276 | [ Vint n1 ⇒ match v2 with |
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| 277 | [ Vint n2 ⇒ |
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| 278 | if ltu n2 iwordsize then Some ? (Vint (shru n1 n2)) else None ? |
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| 279 | | _ ⇒ None ? ] |
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| 280 | | _ ⇒ None ? ] |
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| 281 | | shr_case_ii => |
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| 282 | match v1 with |
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| 283 | [ Vint n1 ⇒ match v2 with |
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| 284 | [ Vint n2 ⇒ |
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| 285 | if ltu n2 iwordsize then Some ? (Vint (shr n1 n2)) else None ? |
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| 286 | | _ ⇒ None ? ] |
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| 287 | | _ ⇒ None ? ] |
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| 288 | | shr_default ⇒ |
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| 289 | None ? |
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| 290 | ]. |
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| 291 | |
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[487] | 292 | let rec sem_cmp_mismatch (c: comparison): option val ≝ |
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[3] | 293 | match c with |
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| 294 | [ Ceq => Some ? Vfalse |
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| 295 | | Cne => Some ? Vtrue |
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| 296 | | _ => None ? |
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| 297 | ]. |
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| 298 | |
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[487] | 299 | let rec sem_cmp_match (c: comparison): option val ≝ |
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[484] | 300 | match c with |
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| 301 | [ Ceq => Some ? Vtrue |
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| 302 | | Cne => Some ? Vfalse |
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| 303 | | _ => None ? |
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| 304 | ]. |
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| 305 | |
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[487] | 306 | let rec sem_cmp (c:comparison) |
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[3] | 307 | (v1: val) (t1: type) (v2: val) (t2: type) |
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| 308 | (m: mem): option val ≝ |
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| 309 | match classify_cmp t1 t2 with |
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| 310 | [ cmp_case_I32unsi ⇒ |
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| 311 | match v1 with |
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| 312 | [ Vint n1 ⇒ match v2 with |
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| 313 | [ Vint n2 ⇒ Some ? (of_bool (cmpu c n1 n2)) |
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| 314 | | _ ⇒ None ? ] |
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| 315 | | _ ⇒ None ? ] |
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| 316 | | cmp_case_ipip ⇒ |
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| 317 | match v1 with |
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| 318 | [ Vint n1 ⇒ match v2 with |
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| 319 | [ Vint n2 ⇒ Some ? (of_bool (cmp c n1 n2)) |
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| 320 | | _ ⇒ None ? |
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| 321 | ] |
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[500] | 322 | | Vptr r1 b1 p1 ofs1 ⇒ |
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[3] | 323 | match v2 with |
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[500] | 324 | [ Vptr r2 b2 p2 ofs2 ⇒ |
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[484] | 325 | if valid_pointer m r1 b1 (signed ofs1) |
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| 326 | ∧ valid_pointer m r2 b2 (signed ofs2) then |
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[496] | 327 | if eq_block b1 b2 |
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[3] | 328 | then Some ? (of_bool (cmp c ofs1 ofs2)) |
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| 329 | else sem_cmp_mismatch c |
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| 330 | else None ? |
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[484] | 331 | | Vnull r2 ⇒ sem_cmp_mismatch c |
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[3] | 332 | | _ ⇒ None ? ] |
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[484] | 333 | | Vnull r1 ⇒ |
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| 334 | match v2 with |
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[500] | 335 | [ Vptr r2 b2 p2 ofs2 ⇒ sem_cmp_mismatch c |
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[484] | 336 | | Vnull r2 ⇒ sem_cmp_match c |
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| 337 | | _ ⇒ None ? |
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| 338 | ] |
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[3] | 339 | | _ ⇒ None ? ] |
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| 340 | | cmp_case_ff ⇒ |
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| 341 | match v1 with |
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| 342 | [ Vfloat f1 ⇒ |
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| 343 | match v2 with |
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| 344 | [ Vfloat f2 ⇒ Some ? (of_bool (Fcmp c f1 f2)) |
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| 345 | | _ ⇒ None ? ] |
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| 346 | | _ ⇒ None ? ] |
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| 347 | | cmp_default ⇒ None ? |
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| 348 | ]. |
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| 349 | |
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[487] | 350 | definition sem_unary_operation |
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[3] | 351 | : unary_operation → val → type → option val ≝ |
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| 352 | λop,v,ty. |
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| 353 | match op with |
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| 354 | [ Onotbool => sem_notbool v ty |
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| 355 | | Onotint => sem_notint v |
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| 356 | | Oneg => sem_neg v ty |
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| 357 | ]. |
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| 358 | |
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[487] | 359 | let rec sem_binary_operation |
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[3] | 360 | (op: binary_operation) |
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| 361 | (v1: val) (t1: type) (v2: val) (t2:type) |
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| 362 | (m: mem): option val ≝ |
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| 363 | match op with |
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| 364 | [ Oadd ⇒ sem_add v1 t1 v2 t2 |
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| 365 | | Osub ⇒ sem_sub v1 t1 v2 t2 |
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| 366 | | Omul ⇒ sem_mul v1 t1 v2 t2 |
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| 367 | | Omod ⇒ sem_mod v1 t1 v2 t2 |
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| 368 | | Odiv ⇒ sem_div v1 t1 v2 t2 |
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| 369 | | Oand ⇒ sem_and v1 v2 |
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| 370 | | Oor ⇒ sem_or v1 v2 |
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| 371 | | Oxor ⇒ sem_xor v1 v2 |
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| 372 | | Oshl ⇒ sem_shl v1 v2 |
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| 373 | | Oshr ⇒ sem_shr v1 t1 v2 t2 |
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| 374 | | Oeq ⇒ sem_cmp Ceq v1 t1 v2 t2 m |
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| 375 | | One ⇒ sem_cmp Cne v1 t1 v2 t2 m |
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| 376 | | Olt ⇒ sem_cmp Clt v1 t1 v2 t2 m |
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| 377 | | Ogt ⇒ sem_cmp Cgt v1 t1 v2 t2 m |
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| 378 | | Ole ⇒ sem_cmp Cle v1 t1 v2 t2 m |
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| 379 | | Oge ⇒ sem_cmp Cge v1 t1 v2 t2 m |
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| 380 | ]. |
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| 381 | |
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| 382 | (* * Semantic of casts. [cast v1 t1 t2 v2] holds if value [v1], |
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| 383 | viewed with static type [t1], can be cast to type [t2], |
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| 384 | resulting in value [v2]. *) |
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| 385 | |
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[487] | 386 | let rec cast_int_int (sz: intsize) (sg: signedness) (i: int) : int ≝ |
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[3] | 387 | match sz with |
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| 388 | [ I8 ⇒ match sg with [ Signed ⇒ sign_ext 8 i | Unsigned ⇒ zero_ext 8 i ] |
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| 389 | | I16 ⇒ match sg with [ Signed => sign_ext 16 i | Unsigned ⇒ zero_ext 16 i ] |
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| 390 | | I32 ⇒ i |
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| 391 | ]. |
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| 392 | |
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[487] | 393 | let rec cast_int_float (si : signedness) (i: int) : float ≝ |
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[3] | 394 | match si with |
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| 395 | [ Signed ⇒ floatofint i |
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| 396 | | Unsigned ⇒ floatofintu i |
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| 397 | ]. |
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| 398 | |
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[487] | 399 | let rec cast_float_int (si : signedness) (f: float) : int ≝ |
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[3] | 400 | match si with |
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| 401 | [ Signed ⇒ intoffloat f |
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| 402 | | Unsigned ⇒ intuoffloat f |
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| 403 | ]. |
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| 404 | |
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[487] | 405 | let rec cast_float_float (sz: floatsize) (f: float) : float ≝ |
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[3] | 406 | match sz with |
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| 407 | [ F32 ⇒ singleoffloat f |
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| 408 | | F64 ⇒ f |
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| 409 | ]. |
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| 410 | |
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[487] | 411 | inductive type_region : type → region → Prop ≝ |
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[484] | 412 | | type_rgn_pointer : ∀s,t. type_region (Tpointer s t) s |
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| 413 | | type_rgn_array : ∀s,t,n. type_region (Tarray s t n) s |
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[155] | 414 | (* XXX Is the following necessary? *) |
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[484] | 415 | | type_rgn_code : ∀tys,ty. type_region (Tfunction tys ty) Code. |
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[124] | 416 | |
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[487] | 417 | inductive cast : mem → val → type → type → val → Prop ≝ |
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[124] | 418 | | cast_ii: ∀m,i,sz2,sz1,si1,si2. (**r int to int *) |
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| 419 | cast m (Vint i) (Tint sz1 si1) (Tint sz2 si2) |
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[3] | 420 | (Vint (cast_int_int sz2 si2 i)) |
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[124] | 421 | | cast_fi: ∀m,f,sz1,sz2,si2. (**r float to int *) |
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| 422 | cast m (Vfloat f) (Tfloat sz1) (Tint sz2 si2) |
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[3] | 423 | (Vint (cast_int_int sz2 si2 (cast_float_int si2 f))) |
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[124] | 424 | | cast_if: ∀m,i,sz1,sz2,si1. (**r int to float *) |
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| 425 | cast m (Vint i) (Tint sz1 si1) (Tfloat sz2) |
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[3] | 426 | (Vfloat (cast_float_float sz2 (cast_int_float si1 i))) |
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[124] | 427 | | cast_ff: ∀m,f,sz1,sz2. (**r float to float *) |
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| 428 | cast m (Vfloat f) (Tfloat sz1) (Tfloat sz2) |
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[3] | 429 | (Vfloat (cast_float_float sz2 f)) |
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[500] | 430 | | cast_pp: ∀m,r,r',ty,ty',b,pc,ofs. |
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[484] | 431 | type_region ty r → |
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| 432 | type_region ty' r' → |
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[500] | 433 | ∀pc':pointer_compat b r'. |
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| 434 | cast m (Vptr r b pc ofs) ty ty' (Vptr r' b pc' ofs) |
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[484] | 435 | | cast_ip_z: ∀m,sz,sg,ty',r. |
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| 436 | type_region ty' r → |
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| 437 | cast m (Vint zero) (Tint sz sg) ty' (Vnull r) |
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| 438 | | cast_pp_z: ∀m,ty,ty',r,r'. |
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| 439 | type_region ty r → |
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| 440 | type_region ty' r' → |
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| 441 | cast m (Vnull r) ty ty' (Vnull r'). |
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[127] | 442 | |
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[3] | 443 | (* * * Operational semantics *) |
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| 444 | |
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| 445 | (* * The semantics uses two environments. The global environment |
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| 446 | maps names of functions and global variables to memory block references, |
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| 447 | and function pointers to their definitions. (See module [Globalenvs].) *) |
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| 448 | |
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[487] | 449 | definition genv ≝ (genv_t Genv) fundef. |
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[3] | 450 | |
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| 451 | (* * The local environment maps local variables to block references. |
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| 452 | The current value of the variable is stored in the associated memory |
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| 453 | block. *) |
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| 454 | |
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[487] | 455 | definition env ≝ (tree_t ? PTree) block. (* map variable -> location *) |
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[3] | 456 | |
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[487] | 457 | definition empty_env: env ≝ (empty …). |
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[3] | 458 | |
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| 459 | (* * [load_value_of_type ty m b ofs] computes the value of a datum |
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| 460 | of type [ty] residing in memory [m] at block [b], offset [ofs]. |
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| 461 | If the type [ty] indicates an access by value, the corresponding |
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| 462 | memory load is performed. If the type [ty] indicates an access by |
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| 463 | reference, the pointer [Vptr b ofs] is returned. *) |
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| 464 | |
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[498] | 465 | let rec load_value_of_type (ty: type) (m: mem) (b: block) (ofs: int) : option val ≝ |
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[3] | 466 | match access_mode ty with |
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[500] | 467 | [ By_value chunk ⇒ loadv chunk m (Vptr Any b ? ofs) |
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| 468 | | By_reference r ⇒ |
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| 469 | match pointer_compat_dec b r with |
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| 470 | [ inl p ⇒ Some ? (Vptr r b p ofs) |
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| 471 | | inr _ ⇒ None ? |
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| 472 | ] |
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[3] | 473 | | By_nothing ⇒ None ? |
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| 474 | ]. |
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[500] | 475 | cases b // |
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| 476 | qed. |
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[3] | 477 | |
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| 478 | (* * Symmetrically, [store_value_of_type ty m b ofs v] returns the |
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| 479 | memory state after storing the value [v] in the datum |
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| 480 | of type [ty] residing in memory [m] at block [b], offset [ofs]. |
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| 481 | This is allowed only if [ty] indicates an access by value. *) |
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| 482 | |
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[498] | 483 | let rec store_value_of_type (ty_dest: type) (m: mem) (loc: block) (ofs: int) (v: val) : option mem ≝ |
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[3] | 484 | match access_mode ty_dest with |
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[500] | 485 | [ By_value chunk ⇒ storev chunk m (Vptr Any loc ? ofs) v |
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[498] | 486 | | By_reference _ ⇒ None ? |
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[3] | 487 | | By_nothing ⇒ None ? |
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| 488 | ]. |
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[500] | 489 | cases loc // |
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| 490 | qed. |
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[3] | 491 | |
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| 492 | (* * Allocation of function-local variables. |
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| 493 | [alloc_variables e1 m1 vars e2 m2] allocates one memory block |
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| 494 | for each variable declared in [vars], and associates the variable |
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| 495 | name with this block. [e1] and [m1] are the initial local environment |
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| 496 | and memory state. [e2] and [m2] are the final local environment |
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| 497 | and memory state. *) |
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| 498 | |
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[487] | 499 | inductive alloc_variables: env → mem → |
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[3] | 500 | list (ident × type) → |
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| 501 | env → mem → Prop ≝ |
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| 502 | | alloc_variables_nil: |
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| 503 | ∀e,m. |
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| 504 | alloc_variables e m (nil ?) e m |
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| 505 | | alloc_variables_cons: |
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| 506 | ∀e,m,id,ty,vars,m1,b1,m2,e2. |
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[125] | 507 | alloc m 0 (sizeof ty) Any = 〈m1, b1〉 → |
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| 508 | alloc_variables (set … id b1 e) m1 vars e2 m2 → |
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[3] | 509 | alloc_variables e m (〈id, ty〉 :: vars) e2 m2. |
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| 510 | |
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| 511 | (* * Initialization of local variables that are parameters to a function. |
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| 512 | [bind_parameters e m1 params args m2] stores the values [args] |
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| 513 | in the memory blocks corresponding to the variables [params]. |
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| 514 | [m1] is the initial memory state and [m2] the final memory state. *) |
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| 515 | |
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[487] | 516 | inductive bind_parameters: env → |
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[3] | 517 | mem → list (ident × type) → list val → |
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| 518 | mem → Prop ≝ |
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| 519 | | bind_parameters_nil: |
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| 520 | ∀e,m. |
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| 521 | bind_parameters e m (nil ?) (nil ?) m |
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| 522 | | bind_parameters_cons: |
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[125] | 523 | ∀e,m,id,ty,params,v1,vl,b,m1,m2. |
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| 524 | get ??? id e = Some ? b → |
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[498] | 525 | store_value_of_type ty m b zero v1 = Some ? m1 → |
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[3] | 526 | bind_parameters e m1 params vl m2 → |
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| 527 | bind_parameters e m (〈id, ty〉 :: params) (v1 :: vl) m2. |
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| 528 | |
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| 529 | (* * Return the list of blocks in the codomain of [e]. *) |
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| 530 | |
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[487] | 531 | definition blocks_of_env : env → list block ≝ λe. |
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[125] | 532 | map ?? (λx. snd ?? x) (elements ??? e). |
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[3] | 533 | |
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| 534 | (* * Selection of the appropriate case of a [switch], given the value [n] |
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| 535 | of the selector expression. *) |
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| 536 | |
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[487] | 537 | let rec select_switch (n: int) (sl: labeled_statements) |
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[3] | 538 | on sl : labeled_statements ≝ |
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| 539 | match sl with |
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| 540 | [ LSdefault _ ⇒ sl |
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| 541 | | LScase c s sl' ⇒ if eq c n then sl else select_switch n sl' |
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| 542 | ]. |
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| 543 | |
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| 544 | (* * Turn a labeled statement into a sequence *) |
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| 545 | |
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[487] | 546 | let rec seq_of_labeled_statement (sl: labeled_statements) : statement ≝ |
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[3] | 547 | match sl with |
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| 548 | [ LSdefault s ⇒ s |
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| 549 | | LScase c s sl' ⇒ Ssequence s (seq_of_labeled_statement sl') |
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| 550 | ]. |
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| 551 | |
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| 552 | (* |
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| 553 | Section SEMANTICS. |
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| 554 | |
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| 555 | Variable ge: genv. |
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| 556 | |
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| 557 | (** ** Evaluation of expressions *) |
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| 558 | |
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| 559 | Section EXPR. |
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| 560 | |
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| 561 | Variable e: env. |
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| 562 | Variable m: mem. |
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| 563 | *) |
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| 564 | (* * [eval_expr ge e m a v] defines the evaluation of expression [a] |
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| 565 | in r-value position. [v] is the value of the expression. |
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| 566 | [e] is the current environment and [m] is the current memory state. *) |
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| 567 | |
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[487] | 568 | inductive eval_expr (ge:genv) (e:env) (m:mem) : expr → val → trace → Prop ≝ |
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[3] | 569 | | eval_Econst_int: ∀i,ty. |
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[175] | 570 | eval_expr ge e m (Expr (Econst_int i) ty) (Vint i) E0 |
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[3] | 571 | | eval_Econst_float: ∀f,ty. |
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[175] | 572 | eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f) E0 |
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[498] | 573 | | eval_Elvalue: ∀a,ty,loc,ofs,v,tr. |
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| 574 | eval_lvalue ge e m (Expr a ty) loc ofs tr → |
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| 575 | load_value_of_type ty m loc ofs = Some ? v → |
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[175] | 576 | eval_expr ge e m (Expr a ty) v tr |
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[496] | 577 | | eval_Eaddrof: ∀a,ty,r,loc,ofs,tr. |
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[498] | 578 | eval_lvalue ge e m a loc ofs tr → |
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[500] | 579 | ∀pc:pointer_compat loc r. |
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| 580 | eval_expr ge e m (Expr (Eaddrof a) (Tpointer r ty)) (Vptr r loc pc ofs) tr |
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[3] | 581 | | eval_Esizeof: ∀ty',ty. |
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[175] | 582 | eval_expr ge e m (Expr (Esizeof ty') ty) (Vint (repr (sizeof ty'))) E0 |
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| 583 | | eval_Eunop: ∀op,a,ty,v1,v,tr. |
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| 584 | eval_expr ge e m a v1 tr → |
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| 585 | sem_unary_operation op v1 (typeof a) = Some ? v → |
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| 586 | eval_expr ge e m (Expr (Eunop op a) ty) v tr |
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| 587 | | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v,tr1,tr2. |
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| 588 | eval_expr ge e m a1 v1 tr1 → |
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| 589 | eval_expr ge e m a2 v2 tr2 → |
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| 590 | sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some ? v → |
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| 591 | eval_expr ge e m (Expr (Ebinop op a1 a2) ty) v (tr1⧺tr2) |
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| 592 | | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2,tr1,tr2. |
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| 593 | eval_expr ge e m a1 v1 tr1 → |
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| 594 | is_true v1 (typeof a1) → |
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| 595 | eval_expr ge e m a2 v2 tr2 → |
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| 596 | eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2 (tr1⧺tr2) |
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| 597 | | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3,tr1,tr2. |
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| 598 | eval_expr ge e m a1 v1 tr1 → |
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| 599 | is_false v1 (typeof a1) → |
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| 600 | eval_expr ge e m a3 v3 tr2 → |
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| 601 | eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3 (tr1⧺tr2) |
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| 602 | | eval_Eorbool_1: ∀a1,a2,ty,v1,tr. |
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| 603 | eval_expr ge e m a1 v1 tr → |
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| 604 | is_true v1 (typeof a1) → |
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| 605 | eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue tr |
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| 606 | | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2. |
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| 607 | eval_expr ge e m a1 v1 tr1 → |
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| 608 | is_false v1 (typeof a1) → |
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| 609 | eval_expr ge e m a2 v2 tr2 → |
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| 610 | bool_of_val v2 (typeof a2) v → |
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| 611 | eval_expr ge e m (Expr (Eorbool a1 a2) ty) v (tr1⧺tr2) |
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| 612 | | eval_Eandbool_1: ∀a1,a2,ty,v1,tr. |
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| 613 | eval_expr ge e m a1 v1 tr → |
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| 614 | is_false v1 (typeof a1) → |
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| 615 | eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse tr |
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| 616 | | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2. |
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| 617 | eval_expr ge e m a1 v1 tr1 → |
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| 618 | is_true v1 (typeof a1) → |
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| 619 | eval_expr ge e m a2 v2 tr2 → |
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| 620 | bool_of_val v2 (typeof a2) v → |
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| 621 | eval_expr ge e m (Expr (Eandbool a1 a2) ty) v (tr1⧺tr2) |
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| 622 | | eval_Ecast: ∀a,ty,ty',v1,v,tr. |
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| 623 | eval_expr ge e m a v1 tr → |
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| 624 | cast m v1 (typeof a) ty v → |
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| 625 | eval_expr ge e m (Expr (Ecast ty a) ty') v tr |
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| 626 | | eval_Ecost: ∀a,ty,v,l,tr. |
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| 627 | eval_expr ge e m a v tr → |
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| 628 | eval_expr ge e m (Expr (Ecost l a) ty) v (tr⧺Echarge l) |
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[3] | 629 | |
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[496] | 630 | (* * [eval_lvalue ge e m a r b ofs] defines the evaluation of expression [a] |
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[3] | 631 | in l-value position. The result is the memory location [b, ofs] |
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[496] | 632 | that contains the value of the expression [a]. The memory location should |
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| 633 | be representable in a pointer of region r. *) |
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[3] | 634 | |
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[498] | 635 | with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → block → int → trace → Prop ≝ |
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[125] | 636 | | eval_Evar_local: ∀id,l,ty. |
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| 637 | (* XXX notation? e!id*) get ??? id e = Some ? l → |
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[498] | 638 | eval_lvalue ge e m (Expr (Evar id) ty) l zero E0 |
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| 639 | | eval_Evar_global: ∀id,l,ty. |
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[175] | 640 | (* XXX e!id *) get ??? id e = None ? → |
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[498] | 641 | find_symbol ?? ge id = Some ? l → |
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| 642 | eval_lvalue ge e m (Expr (Evar id) ty) l zero E0 |
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[500] | 643 | | eval_Ederef: ∀a,ty,r,l,p,ofs,tr. |
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| 644 | eval_expr ge e m a (Vptr r l p ofs) tr → |
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[498] | 645 | eval_lvalue ge e m (Expr (Ederef a) ty) l ofs tr |
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| 646 | (* Aside: note that each block of memory is entirely contained within one |
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| 647 | memory region; hence adding a field offset will not produce a location |
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| 648 | outside of the original location's region. *) |
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| 649 | | eval_Efield_struct: ∀a,i,ty,l,ofs,id,fList,delta,tr. |
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| 650 | eval_lvalue ge e m a l ofs tr → |
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[175] | 651 | typeof a = Tstruct id fList → |
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| 652 | field_offset i fList = OK ? delta → |
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[498] | 653 | eval_lvalue ge e m (Expr (Efield a i) ty) l (add ofs (repr delta)) tr |
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| 654 | | eval_Efield_union: ∀a,i,ty,l,ofs,id,fList,tr. |
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| 655 | eval_lvalue ge e m a l ofs tr → |
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[175] | 656 | typeof a = Tunion id fList → |
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[498] | 657 | eval_lvalue ge e m (Expr (Efield a i) ty) l ofs tr. |
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[3] | 658 | |
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[487] | 659 | let rec eval_expr_ind (ge:genv) (e:env) (m:mem) |
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[226] | 660 | (P:∀a,v,tr. eval_expr ge e m a v tr → Prop) |
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| 661 | (eci:∀i,ty. P ??? (eval_Econst_int ge e m i ty)) |
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| 662 | (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty)) |
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[498] | 663 | (elv:∀a,ty,loc,ofs,v,tr,H1,H2. P ??? (eval_Elvalue ge e m a ty loc ofs v tr H1 H2)) |
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[500] | 664 | (ead:∀a,ty,r,loc,pc,ofs,tr,H. P ??? (eval_Eaddrof ge e m a ty r loc pc ofs tr H)) |
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[226] | 665 | (esz:∀ty',ty. P ??? (eval_Esizeof ge e m ty' ty)) |
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| 666 | (eun:∀op,a,ty,v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Eunop ge e m op a ty v1 v tr H1 H2)) |
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| 667 | (ebi:∀op,a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H2 → P ??? (eval_Ebinop ge e m op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3)) |
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| 668 | (ect:∀a1,a2,a3,ty,v1,v2,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Econdition_true ge e m a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3)) |
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| 669 | (ecf:∀a1,a2,a3,ty,v1,v3,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a3 v3 tr2 H3 → P ??? (eval_Econdition_false ge e m a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3)) |
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| 670 | (eo1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eorbool_1 ge e m a1 a2 ty v1 tr H1 H2)) |
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| 671 | (eo2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eorbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4)) |
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| 672 | (ea1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eandbool_1 ge e m a1 a2 ty v1 tr H1 H2)) |
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| 673 | (ea2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eandbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4)) |
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| 674 | (ecs:∀a,ty,ty',v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Ecast ge e m a ty ty' v1 v tr H1 H2)) |
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| 675 | (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H)) |
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| 676 | (a:expr) (v:val) (tr:trace) (ev:eval_expr ge e m a v tr) on ev : P a v tr ev ≝ |
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| 677 | match ev with |
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| 678 | [ eval_Econst_int i ty ⇒ eci i ty |
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| 679 | | eval_Econst_float f ty ⇒ ecF f ty |
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[498] | 680 | | eval_Elvalue a ty loc ofs v tr H1 H2 ⇒ elv a ty loc ofs v tr H1 H2 |
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[500] | 681 | | eval_Eaddrof a ty r loc pc ofs tr H ⇒ ead a ty r loc pc ofs tr H |
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[226] | 682 | | eval_Esizeof ty' ty ⇒ esz ty' ty |
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| 683 | | eval_Eunop op a ty v1 v tr H1 H2 ⇒ eun op a ty v1 v tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a v1 tr H1) |
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| 684 | | eval_Ebinop op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 ⇒ ebi op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H2) |
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| 685 | | eval_Econdition_true a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 ⇒ ect a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H3) |
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| 686 | | eval_Econdition_false a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 ⇒ ecf a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a3 v3 tr2 H3) |
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| 687 | | eval_Eorbool_1 a1 a2 ty v1 tr H1 H2 ⇒ eo1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr H1) |
---|
| 688 | | eval_Eorbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ eo2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H3) |
---|
| 689 | | eval_Eandbool_1 a1 a2 ty v1 tr H1 H2 ⇒ ea1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr H1) |
---|
| 690 | | eval_Eandbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ ea2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H3) |
---|
| 691 | | eval_Ecast a ty ty' v1 v tr H1 H2 ⇒ ecs a ty ty' v1 v tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a v1 tr H1) |
---|
| 692 | | eval_Ecost a ty v l tr H ⇒ eco a ty v l tr H (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a v tr H) |
---|
| 693 | ]. |
---|
| 694 | |
---|
[487] | 695 | inverter eval_expr_inv_ind for eval_expr : Prop. |
---|
[226] | 696 | |
---|
[487] | 697 | let rec eval_lvalue_ind (ge:genv) (e:env) (m:mem) |
---|
[498] | 698 | (P:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop) |
---|
| 699 | (lvl:∀id,l,ty,H. P ???? (eval_Evar_local ge e m id l ty H)) |
---|
| 700 | (lvg:∀id,l,ty,H1,H2. P ???? (eval_Evar_global ge e m id l ty H1 H2)) |
---|
[500] | 701 | (lde:∀a,ty,r,l,pc,ofs,tr,H. P ???? (eval_Ederef ge e m a ty r l pc ofs tr H)) |
---|
[498] | 702 | (lfs:∀a,i,ty,l,ofs,id,fList,delta,tr,H1,H2,H3. P a l ofs tr H1 → P ???? (eval_Efield_struct ge e m a i ty l ofs id fList delta tr H1 H2 H3)) |
---|
| 703 | (lfu:∀a,i,ty,l,ofs,id,fList,tr,H1,H2. P a l ofs tr H1 → P ???? (eval_Efield_union ge e m a i ty l ofs id fList tr H1 H2)) |
---|
| 704 | (a:expr) (loc:block) (ofs:int) (tr:trace) (ev:eval_lvalue ge e m a loc ofs tr) on ev : P a loc ofs tr ev ≝ |
---|
[226] | 705 | match ev with |
---|
| 706 | [ eval_Evar_local id l ty H ⇒ lvl id l ty H |
---|
[498] | 707 | | eval_Evar_global id l ty H1 H2 ⇒ lvg id l ty H1 H2 |
---|
[500] | 708 | | eval_Ederef a ty r l pc ofs tr H ⇒ lde a ty r l pc ofs tr H |
---|
[498] | 709 | | eval_Efield_struct a i ty l ofs id fList delta tr H1 H2 H3 ⇒ lfs a i ty l ofs id fList delta tr H1 H2 H3 (eval_lvalue_ind ge e m P lvl lvg lde lfs lfu a l ofs tr H1) |
---|
| 710 | | eval_Efield_union a i ty l ofs id fList tr H1 H2 ⇒ lfu a i ty l ofs id fList tr H1 H2 (eval_lvalue_ind ge e m P lvl lvg lde lfs lfu a l ofs tr H1) |
---|
[226] | 711 | ]. |
---|
| 712 | |
---|
[3] | 713 | (* |
---|
[226] | 714 | ninverter eval_lvalue_inv_ind for eval_lvalue : Prop. |
---|
| 715 | *) |
---|
| 716 | |
---|
[487] | 717 | definition eval_lvalue_inv_ind : |
---|
[226] | 718 | ∀x1: genv. |
---|
| 719 | ∀x2: env. |
---|
| 720 | ∀x3: mem. |
---|
| 721 | ∀x4: expr. |
---|
| 722 | ∀x6: block. |
---|
| 723 | ∀x7: int. |
---|
| 724 | ∀x8: trace. |
---|
| 725 | ∀P: |
---|
| 726 | ∀_z1430: expr. |
---|
| 727 | ∀_z1428: block. ∀_z1427: int. ∀_z1426: trace. Prop. |
---|
| 728 | ∀_H1: ?. |
---|
| 729 | ∀_H2: ?. |
---|
| 730 | ∀_H3: ?. |
---|
| 731 | ∀_H4: ?. |
---|
| 732 | ∀_H5: ?. |
---|
[498] | 733 | ∀_Hterm: eval_lvalue x1 x2 x3 x4 x6 x7 x8. |
---|
| 734 | P x4 x6 x7 x8 |
---|
[226] | 735 | := |
---|
| 736 | (λx1:genv. |
---|
| 737 | (λx2:env. |
---|
| 738 | (λx3:mem. |
---|
| 739 | (λx4:expr. |
---|
| 740 | (λx6:block. |
---|
| 741 | (λx7:int. |
---|
| 742 | (λx8:trace. |
---|
| 743 | (λP:∀_z1430: expr. |
---|
| 744 | ∀_z1428: block. |
---|
| 745 | ∀_z1427: int. ∀_z1426: trace. Prop. |
---|
| 746 | (λH1:?. |
---|
| 747 | (λH2:?. |
---|
| 748 | (λH3:?. |
---|
| 749 | (λH4:?. |
---|
| 750 | (λH5:?. |
---|
[498] | 751 | (λHterm:eval_lvalue x1 x2 x3 x4 x6 x7 x8. |
---|
[226] | 752 | ((λHcut:∀z1435: eq expr x4 x4. |
---|
| 753 | ∀z1433: eq block x6 x6. |
---|
| 754 | ∀z1432: eq int x7 x7. |
---|
| 755 | ∀z1431: eq trace x8 x8. |
---|
[498] | 756 | P x4 x6 x7 x8. |
---|
[226] | 757 | (Hcut (refl expr x4) |
---|
[498] | 758 | (refl block x6) |
---|
[226] | 759 | (refl int x7) (refl trace x8))) |
---|
[498] | 760 | ?))))))))))))))). |
---|
| 761 | [ @(eval_lvalue_ind x1 x2 x3 (λa,loc,ofs,tr,e. ∀e1:eq ? x4 a. ∀e3:eq ? x6 loc. ∀e4:eq ? x7 ofs. ∀e5:eq ? x8 tr. P a loc ofs tr) … Hterm) |
---|
[487] | 762 | [ @H1 | @H2 | @H3 | @H4 | @H5 ] |
---|
| 763 | | *: skip |
---|
| 764 | ] qed. |
---|
[226] | 765 | |
---|
[487] | 766 | let rec eval_expr_ind2 (ge:genv) (e:env) (m:mem) |
---|
[226] | 767 | (P:∀a,v,tr. eval_expr ge e m a v tr → Prop) |
---|
[498] | 768 | (Q:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop) |
---|
[226] | 769 | (eci:∀i,ty. P ??? (eval_Econst_int ge e m i ty)) |
---|
| 770 | (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty)) |
---|
[498] | 771 | (elv:∀a,ty,loc,ofs,v,tr,H1,H2. Q (Expr a ty) loc ofs tr H1 → P ??? (eval_Elvalue ge e m a ty loc ofs v tr H1 H2)) |
---|
[500] | 772 | (ead:∀a,ty,r,loc,pc,ofs,tr,H. Q a loc ofs tr H → P ??? (eval_Eaddrof ge e m a ty r loc ofs tr H pc)) |
---|
[226] | 773 | (esz:∀ty',ty. P ??? (eval_Esizeof ge e m ty' ty)) |
---|
| 774 | (eun:∀op,a,ty,v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Eunop ge e m op a ty v1 v tr H1 H2)) |
---|
| 775 | (ebi:∀op,a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H2 → P ??? (eval_Ebinop ge e m op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3)) |
---|
| 776 | (ect:∀a1,a2,a3,ty,v1,v2,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Econdition_true ge e m a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3)) |
---|
| 777 | (ecf:∀a1,a2,a3,ty,v1,v3,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a3 v3 tr2 H3 → P ??? (eval_Econdition_false ge e m a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3)) |
---|
| 778 | (eo1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eorbool_1 ge e m a1 a2 ty v1 tr H1 H2)) |
---|
| 779 | (eo2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eorbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4)) |
---|
| 780 | (ea1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eandbool_1 ge e m a1 a2 ty v1 tr H1 H2)) |
---|
| 781 | (ea2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eandbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4)) |
---|
| 782 | (ecs:∀a,ty,ty',v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Ecast ge e m a ty ty' v1 v tr H1 H2)) |
---|
| 783 | (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H)) |
---|
[498] | 784 | (lvl:∀id,l,ty,H. Q ???? (eval_Evar_local ge e m id l ty H)) |
---|
| 785 | (lvg:∀id,l,ty,H1,H2. Q ???? (eval_Evar_global ge e m id l ty H1 H2)) |
---|
[500] | 786 | (lde:∀a,ty,r,l,pc,ofs,tr,H. P a (Vptr r l pc ofs) tr H → Q ???? (eval_Ederef ge e m a ty r l pc ofs tr H)) |
---|
[498] | 787 | (lfs:∀a,i,ty,l,ofs,id,fList,delta,tr,H1,H2,H3. Q a l ofs tr H1 → Q ???? (eval_Efield_struct ge e m a i ty l ofs id fList delta tr H1 H2 H3)) |
---|
| 788 | (lfu:∀a,i,ty,l,ofs,id,fList,tr,H1,H2. Q a l ofs tr H1 → Q ???? (eval_Efield_union ge e m a i ty l ofs id fList tr H1 H2)) |
---|
[226] | 789 | |
---|
| 790 | (a:expr) (v:val) (tr:trace) (ev:eval_expr ge e m a v tr) on ev : P a v tr ev ≝ |
---|
| 791 | match ev with |
---|
| 792 | [ eval_Econst_int i ty ⇒ eci i ty |
---|
| 793 | | eval_Econst_float f ty ⇒ ecF f ty |
---|
[498] | 794 | | eval_Elvalue a ty loc ofs v tr H1 H2 ⇒ elv a ty loc ofs v tr H1 H2 (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu (Expr a ty) loc ofs tr H1) |
---|
[500] | 795 | | eval_Eaddrof a ty r loc ofs tr H pc ⇒ ead a ty r loc pc ofs tr H (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a loc ofs tr H) |
---|
[226] | 796 | | eval_Esizeof ty' ty ⇒ esz ty' ty |
---|
| 797 | | eval_Eunop op a ty v1 v tr H1 H2 ⇒ eun op a ty v1 v tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a v1 tr H1) |
---|
| 798 | | eval_Ebinop op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 ⇒ ebi op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H2) |
---|
| 799 | | eval_Econdition_true a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 ⇒ ect a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H3) |
---|
| 800 | | eval_Econdition_false a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 ⇒ ecf a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a3 v3 tr2 H3) |
---|
| 801 | | eval_Eorbool_1 a1 a2 ty v1 tr H1 H2 ⇒ eo1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr H1) |
---|
| 802 | | eval_Eorbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ eo2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H3) |
---|
| 803 | | eval_Eandbool_1 a1 a2 ty v1 tr H1 H2 ⇒ ea1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr H1) |
---|
| 804 | | eval_Eandbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ ea2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H3) |
---|
| 805 | | eval_Ecast a ty ty' v1 v tr H1 H2 ⇒ ecs a ty ty' v1 v tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a v1 tr H1) |
---|
| 806 | | eval_Ecost a ty v l tr H ⇒ eco a ty v l tr H (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a v tr H) |
---|
| 807 | ] |
---|
| 808 | and eval_lvalue_ind2 (ge:genv) (e:env) (m:mem) |
---|
| 809 | (P:∀a,v,tr. eval_expr ge e m a v tr → Prop) |
---|
[498] | 810 | (Q:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop) |
---|
[226] | 811 | (eci:∀i,ty. P ??? (eval_Econst_int ge e m i ty)) |
---|
| 812 | (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty)) |
---|
[498] | 813 | (elv:∀a,ty,loc,ofs,v,tr,H1,H2. Q (Expr a ty) loc ofs tr H1 → P ??? (eval_Elvalue ge e m a ty loc ofs v tr H1 H2)) |
---|
[500] | 814 | (ead:∀a,ty,r,loc,pc,ofs,tr,H. Q a loc ofs tr H → P ??? (eval_Eaddrof ge e m a ty r loc ofs tr H pc)) |
---|
[226] | 815 | (esz:∀ty',ty. P ??? (eval_Esizeof ge e m ty' ty)) |
---|
| 816 | (eun:∀op,a,ty,v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Eunop ge e m op a ty v1 v tr H1 H2)) |
---|
| 817 | (ebi:∀op,a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H2 → P ??? (eval_Ebinop ge e m op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3)) |
---|
| 818 | (ect:∀a1,a2,a3,ty,v1,v2,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Econdition_true ge e m a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3)) |
---|
| 819 | (ecf:∀a1,a2,a3,ty,v1,v3,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a3 v3 tr2 H3 → P ??? (eval_Econdition_false ge e m a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3)) |
---|
| 820 | (eo1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eorbool_1 ge e m a1 a2 ty v1 tr H1 H2)) |
---|
| 821 | (eo2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eorbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4)) |
---|
| 822 | (ea1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eandbool_1 ge e m a1 a2 ty v1 tr H1 H2)) |
---|
| 823 | (ea2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eandbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4)) |
---|
| 824 | (ecs:∀a,ty,ty',v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Ecast ge e m a ty ty' v1 v tr H1 H2)) |
---|
| 825 | (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H)) |
---|
[498] | 826 | (lvl:∀id,l,ty,H. Q ???? (eval_Evar_local ge e m id l ty H)) |
---|
| 827 | (lvg:∀id,l,ty,H1,H2. Q ???? (eval_Evar_global ge e m id l ty H1 H2)) |
---|
[500] | 828 | (lde:∀a,ty,r,l,pc,ofs,tr,H. P a (Vptr r l pc ofs) tr H → Q ???? (eval_Ederef ge e m a ty r l pc ofs tr H)) |
---|
[498] | 829 | (lfs:∀a,i,ty,l,ofs,id,fList,delta,tr,H1,H2,H3. Q a l ofs tr H1 → Q ???? (eval_Efield_struct ge e m a i ty l ofs id fList delta tr H1 H2 H3)) |
---|
| 830 | (lfu:∀a,i,ty,l,ofs,id,fList,tr,H1,H2. Q a l ofs tr H1 → Q ???? (eval_Efield_union ge e m a i ty l ofs id fList tr H1 H2)) |
---|
| 831 | (a:expr) (loc:block) (ofs:int) (tr:trace) (ev:eval_lvalue ge e m a loc ofs tr) on ev : Q a loc ofs tr ev ≝ |
---|
[226] | 832 | match ev with |
---|
| 833 | [ eval_Evar_local id l ty H ⇒ lvl id l ty H |
---|
[498] | 834 | | eval_Evar_global id l ty H1 H2 ⇒ lvg id l ty H1 H2 |
---|
[500] | 835 | | eval_Ederef a ty r l pc ofs tr H ⇒ lde a ty r l pc ofs tr H (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a (Vptr r l pc ofs) tr H) |
---|
[498] | 836 | | eval_Efield_struct a i ty l ofs id fList delta tr H1 H2 H3 ⇒ lfs a i ty l ofs id fList delta tr H1 H2 H3 (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a l ofs tr H1) |
---|
| 837 | | eval_Efield_union a i ty l ofs id fList tr H1 H2 ⇒ lfu a i ty l ofs id fList tr H1 H2 (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a l ofs tr H1) |
---|
[226] | 838 | ]. |
---|
| 839 | |
---|
[487] | 840 | definition combined_expr_lvalue_ind ≝ |
---|
[226] | 841 | λge,e,m,P,Q,eci,ecF,elv,ead,esz,eun,ebi,ect,ecf,eo1,eo2,ea1,ea2,ecs,eco,lvl,lvg,lde,lfs,lfu. |
---|
| 842 | conj ?? |
---|
| 843 | (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu) |
---|
| 844 | (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu). |
---|
| 845 | |
---|
| 846 | (* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a] |
---|
| 847 | in l-value position. The result is the memory location [b, ofs] |
---|
| 848 | that contains the value of the expression [a]. *) |
---|
| 849 | |
---|
| 850 | (* |
---|
| 851 | Scheme eval_expr_ind22 := Minimality for eval_expr Sort Prop |
---|
[3] | 852 | with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop. |
---|
| 853 | *) |
---|
| 854 | |
---|
| 855 | (* * [eval_exprlist ge e m al vl] evaluates a list of r-value |
---|
| 856 | expressions [al] to their values [vl]. *) |
---|
| 857 | |
---|
[487] | 858 | inductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr → list val → trace → Prop ≝ |
---|
[3] | 859 | | eval_Enil: |
---|
[175] | 860 | eval_exprlist ge e m (nil ?) (nil ?) E0 |
---|
| 861 | | eval_Econs: ∀a,bl,v,vl,tr1,tr2. |
---|
| 862 | eval_expr ge e m a v tr1 → |
---|
| 863 | eval_exprlist ge e m bl vl tr2 → |
---|
| 864 | eval_exprlist ge e m (a :: bl) (v :: vl) (tr1⧺tr2). |
---|
[3] | 865 | |
---|
| 866 | (*End EXPR.*) |
---|
| 867 | |
---|
| 868 | (* * ** Transition semantics for statements and functions *) |
---|
| 869 | |
---|
| 870 | (* * Continuations *) |
---|
| 871 | |
---|
[487] | 872 | inductive cont: Type[0] := |
---|
[3] | 873 | | Kstop: cont |
---|
| 874 | | Kseq: statement -> cont -> cont |
---|
| 875 | (**r [Kseq s2 k] = after [s1] in [s1;s2] *) |
---|
| 876 | | Kwhile: expr -> statement -> cont -> cont |
---|
| 877 | (**r [Kwhile e s k] = after [s] in [while (e) s] *) |
---|
| 878 | | Kdowhile: expr -> statement -> cont -> cont |
---|
| 879 | (**r [Kdowhile e s k] = after [s] in [do s while (e)] *) |
---|
| 880 | | Kfor2: expr -> statement -> statement -> cont -> cont |
---|
| 881 | (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *) |
---|
| 882 | | Kfor3: expr -> statement -> statement -> cont -> cont |
---|
| 883 | (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *) |
---|
| 884 | | Kswitch: cont -> cont |
---|
| 885 | (**r catches [break] statements arising out of [switch] *) |
---|
[498] | 886 | | Kcall: option (block × int × type) -> (**r where to store result *) |
---|
[3] | 887 | function -> (**r calling function *) |
---|
| 888 | env -> (**r local env of calling function *) |
---|
| 889 | cont -> cont. |
---|
| 890 | |
---|
| 891 | (* * Pop continuation until a call or stop *) |
---|
| 892 | |
---|
[487] | 893 | let rec call_cont (k: cont) : cont := |
---|
[3] | 894 | match k with |
---|
| 895 | [ Kseq s k => call_cont k |
---|
| 896 | | Kwhile e s k => call_cont k |
---|
| 897 | | Kdowhile e s k => call_cont k |
---|
| 898 | | Kfor2 e2 e3 s k => call_cont k |
---|
| 899 | | Kfor3 e2 e3 s k => call_cont k |
---|
| 900 | | Kswitch k => call_cont k |
---|
| 901 | | _ => k |
---|
| 902 | ]. |
---|
| 903 | |
---|
[487] | 904 | definition is_call_cont : cont → Prop ≝ λk. |
---|
[3] | 905 | match k with |
---|
| 906 | [ Kstop => True |
---|
| 907 | | Kcall _ _ _ _ => True |
---|
| 908 | | _ => False |
---|
| 909 | ]. |
---|
| 910 | |
---|
| 911 | (* * States *) |
---|
| 912 | |
---|
[487] | 913 | inductive state: Type[0] := |
---|
[3] | 914 | | State: |
---|
| 915 | ∀f: function. |
---|
| 916 | ∀s: statement. |
---|
| 917 | ∀k: cont. |
---|
| 918 | ∀e: env. |
---|
| 919 | ∀m: mem. state |
---|
| 920 | | Callstate: |
---|
| 921 | ∀fd: fundef. |
---|
| 922 | ∀args: list val. |
---|
| 923 | ∀k: cont. |
---|
| 924 | ∀m: mem. state |
---|
| 925 | | Returnstate: |
---|
| 926 | ∀res: val. |
---|
| 927 | ∀k: cont. |
---|
| 928 | ∀m: mem. state. |
---|
| 929 | |
---|
| 930 | (* * Find the statement and manufacture the continuation |
---|
| 931 | corresponding to a label *) |
---|
| 932 | |
---|
[487] | 933 | let rec find_label (lbl: label) (s: statement) (k: cont) |
---|
[3] | 934 | on s: option (statement × cont) := |
---|
| 935 | match s with |
---|
| 936 | [ Ssequence s1 s2 => |
---|
| 937 | match find_label lbl s1 (Kseq s2 k) with |
---|
| 938 | [ Some sk => Some ? sk |
---|
| 939 | | None => find_label lbl s2 k |
---|
| 940 | ] |
---|
| 941 | | Sifthenelse a s1 s2 => |
---|
| 942 | match find_label lbl s1 k with |
---|
| 943 | [ Some sk => Some ? sk |
---|
| 944 | | None => find_label lbl s2 k |
---|
| 945 | ] |
---|
| 946 | | Swhile a s1 => |
---|
| 947 | find_label lbl s1 (Kwhile a s1 k) |
---|
| 948 | | Sdowhile a s1 => |
---|
| 949 | find_label lbl s1 (Kdowhile a s1 k) |
---|
| 950 | | Sfor a1 a2 a3 s1 => |
---|
| 951 | match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with |
---|
| 952 | [ Some sk => Some ? sk |
---|
| 953 | | None => |
---|
| 954 | match find_label lbl s1 (Kfor2 a2 a3 s1 k) with |
---|
| 955 | [ Some sk => Some ? sk |
---|
| 956 | | None => find_label lbl a3 (Kfor3 a2 a3 s1 k) |
---|
| 957 | ] |
---|
| 958 | ] |
---|
| 959 | | Sswitch e sl => |
---|
| 960 | find_label_ls lbl sl (Kswitch k) |
---|
| 961 | | Slabel lbl' s' => |
---|
| 962 | match ident_eq lbl lbl' with |
---|
| 963 | [ inl _ ⇒ Some ? 〈s', k〉 |
---|
| 964 | | inr _ ⇒ find_label lbl s' k |
---|
| 965 | ] |
---|
| 966 | | _ => None ? |
---|
| 967 | ] |
---|
| 968 | |
---|
| 969 | and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont) |
---|
| 970 | on sl: option (statement × cont) := |
---|
| 971 | match sl with |
---|
| 972 | [ LSdefault s => find_label lbl s k |
---|
| 973 | | LScase _ s sl' => |
---|
| 974 | match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with |
---|
| 975 | [ Some sk => Some ? sk |
---|
| 976 | | None => find_label_ls lbl sl' k |
---|
| 977 | ] |
---|
| 978 | ]. |
---|
| 979 | |
---|
| 980 | (* * Transition relation *) |
---|
| 981 | |
---|
[457] | 982 | (* Strip off outer pointer for use when comparing function types. *) |
---|
[487] | 983 | definition fun_typeof ≝ |
---|
[457] | 984 | λe. match typeof e with |
---|
| 985 | [ Tvoid ⇒ Tvoid |
---|
| 986 | | Tint a b ⇒ Tint a b |
---|
| 987 | | Tfloat a ⇒ Tfloat a |
---|
| 988 | | Tpointer _ ty ⇒ ty |
---|
| 989 | | Tarray a b c ⇒ Tarray a b c |
---|
| 990 | | Tfunction a b ⇒ Tfunction a b |
---|
| 991 | | Tstruct a b ⇒ Tstruct a b |
---|
| 992 | | Tunion a b ⇒ Tunion a b |
---|
[481] | 993 | | Tcomp_ptr a b ⇒ Tcomp_ptr a b |
---|
[457] | 994 | ]. |
---|
| 995 | |
---|
[175] | 996 | (* XXX: note that cost labels in exprs expose a particular eval order. *) |
---|
[3] | 997 | |
---|
[487] | 998 | inductive step (ge:genv) : state → trace → state → Prop ≝ |
---|
[175] | 999 | |
---|
[498] | 1000 | | step_assign: ∀f,a1,a2,k,e,m,loc,ofs,v2,m',tr1,tr2. |
---|
| 1001 | eval_lvalue ge e m a1 loc ofs tr1 → |
---|
[175] | 1002 | eval_expr ge e m a2 v2 tr2 → |
---|
[498] | 1003 | store_value_of_type (typeof a1) m loc ofs v2 = Some ? m' → |
---|
[3] | 1004 | step ge (State f (Sassign a1 a2) k e m) |
---|
[175] | 1005 | (tr1⧺tr2) (State f Sskip k e m') |
---|
[3] | 1006 | |
---|
[175] | 1007 | | step_call_none: ∀f,a,al,k,e,m,vf,vargs,fd,tr1,tr2. |
---|
| 1008 | eval_expr ge e m a vf tr1 → |
---|
| 1009 | eval_exprlist ge e m al vargs tr2 → |
---|
| 1010 | find_funct ?? ge vf = Some ? fd → |
---|
[457] | 1011 | type_of_fundef fd = fun_typeof a → |
---|
[3] | 1012 | step ge (State f (Scall (None ?) a al) k e m) |
---|
[175] | 1013 | (tr1⧺tr2) (Callstate fd vargs (Kcall (None ?) f e k) m) |
---|
[3] | 1014 | |
---|
[498] | 1015 | | step_call_some: ∀f,lhs,a,al,k,e,m,loc,ofs,vf,vargs,fd,tr1,tr2,tr3. |
---|
| 1016 | eval_lvalue ge e m lhs loc ofs tr1 → |
---|
[175] | 1017 | eval_expr ge e m a vf tr2 → |
---|
| 1018 | eval_exprlist ge e m al vargs tr3 → |
---|
| 1019 | find_funct ?? ge vf = Some ? fd → |
---|
[457] | 1020 | type_of_fundef fd = fun_typeof a → |
---|
[3] | 1021 | step ge (State f (Scall (Some ? lhs) a al) k e m) |
---|
[498] | 1022 | (tr1⧺tr2⧺tr3) (Callstate fd vargs (Kcall (Some ? 〈〈loc, ofs〉, typeof lhs〉) f e k) m) |
---|
[3] | 1023 | |
---|
| 1024 | | step_seq: ∀f,s1,s2,k,e,m. |
---|
| 1025 | step ge (State f (Ssequence s1 s2) k e m) |
---|
| 1026 | E0 (State f s1 (Kseq s2 k) e m) |
---|
| 1027 | | step_skip_seq: ∀f,s,k,e,m. |
---|
| 1028 | step ge (State f Sskip (Kseq s k) e m) |
---|
| 1029 | E0 (State f s k e m) |
---|
| 1030 | | step_continue_seq: ∀f,s,k,e,m. |
---|
| 1031 | step ge (State f Scontinue (Kseq s k) e m) |
---|
| 1032 | E0 (State f Scontinue k e m) |
---|
| 1033 | | step_break_seq: ∀f,s,k,e,m. |
---|
| 1034 | step ge (State f Sbreak (Kseq s k) e m) |
---|
| 1035 | E0 (State f Sbreak k e m) |
---|
| 1036 | |
---|
[175] | 1037 | | step_ifthenelse_true: ∀f,a,s1,s2,k,e,m,v1,tr. |
---|
| 1038 | eval_expr ge e m a v1 tr → |
---|
| 1039 | is_true v1 (typeof a) → |
---|
[3] | 1040 | step ge (State f (Sifthenelse a s1 s2) k e m) |
---|
[175] | 1041 | tr (State f s1 k e m) |
---|
| 1042 | | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1,tr. |
---|
| 1043 | eval_expr ge e m a v1 tr → |
---|
| 1044 | is_false v1 (typeof a) → |
---|
[3] | 1045 | step ge (State f (Sifthenelse a s1 s2) k e m) |
---|
[175] | 1046 | tr (State f s2 k e m) |
---|
[3] | 1047 | |
---|
[175] | 1048 | | step_while_false: ∀f,a,s,k,e,m,v,tr. |
---|
| 1049 | eval_expr ge e m a v tr → |
---|
| 1050 | is_false v (typeof a) → |
---|
[3] | 1051 | step ge (State f (Swhile a s) k e m) |
---|
[175] | 1052 | tr (State f Sskip k e m) |
---|
| 1053 | | step_while_true: ∀f,a,s,k,e,m,v,tr. |
---|
| 1054 | eval_expr ge e m a v tr → |
---|
| 1055 | is_true v (typeof a) → |
---|
[3] | 1056 | step ge (State f (Swhile a s) k e m) |
---|
[175] | 1057 | tr (State f s (Kwhile a s k) e m) |
---|
[3] | 1058 | | step_skip_or_continue_while: ∀f,x,a,s,k,e,m. |
---|
[175] | 1059 | x = Sskip ∨ x = Scontinue → |
---|
[3] | 1060 | step ge (State f x (Kwhile a s k) e m) |
---|
| 1061 | E0 (State f (Swhile a s) k e m) |
---|
| 1062 | | step_break_while: ∀f,a,s,k,e,m. |
---|
| 1063 | step ge (State f Sbreak (Kwhile a s k) e m) |
---|
| 1064 | E0 (State f Sskip k e m) |
---|
| 1065 | |
---|
| 1066 | | step_dowhile: ∀f,a,s,k,e,m. |
---|
| 1067 | step ge (State f (Sdowhile a s) k e m) |
---|
| 1068 | E0 (State f s (Kdowhile a s k) e m) |
---|
[175] | 1069 | | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v,tr. |
---|
| 1070 | x = Sskip ∨ x = Scontinue → |
---|
| 1071 | eval_expr ge e m a v tr → |
---|
| 1072 | is_false v (typeof a) → |
---|
[3] | 1073 | step ge (State f x (Kdowhile a s k) e m) |
---|
[175] | 1074 | tr (State f Sskip k e m) |
---|
| 1075 | | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v,tr. |
---|
| 1076 | x = Sskip ∨ x = Scontinue → |
---|
| 1077 | eval_expr ge e m a v tr → |
---|
| 1078 | is_true v (typeof a) → |
---|
[3] | 1079 | step ge (State f x (Kdowhile a s k) e m) |
---|
[175] | 1080 | tr (State f (Sdowhile a s) k e m) |
---|
[3] | 1081 | | step_break_dowhile: ∀f,a,s,k,e,m. |
---|
| 1082 | step ge (State f Sbreak (Kdowhile a s k) e m) |
---|
| 1083 | E0 (State f Sskip k e m) |
---|
| 1084 | |
---|
| 1085 | | step_for_start: ∀f,a1,a2,a3,s,k,e,m. |
---|
[175] | 1086 | a1 ≠ Sskip → |
---|
[3] | 1087 | step ge (State f (Sfor a1 a2 a3 s) k e m) |
---|
| 1088 | E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m) |
---|
[175] | 1089 | | step_for_false: ∀f,a2,a3,s,k,e,m,v,tr. |
---|
| 1090 | eval_expr ge e m a2 v tr → |
---|
| 1091 | is_false v (typeof a2) → |
---|
[3] | 1092 | step ge (State f (Sfor Sskip a2 a3 s) k e m) |
---|
[175] | 1093 | tr (State f Sskip k e m) |
---|
| 1094 | | step_for_true: ∀f,a2,a3,s,k,e,m,v,tr. |
---|
| 1095 | eval_expr ge e m a2 v tr → |
---|
| 1096 | is_true v (typeof a2) → |
---|
[3] | 1097 | step ge (State f (Sfor Sskip a2 a3 s) k e m) |
---|
[175] | 1098 | tr (State f s (Kfor2 a2 a3 s k) e m) |
---|
[3] | 1099 | | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m. |
---|
[175] | 1100 | x = Sskip ∨ x = Scontinue → |
---|
[3] | 1101 | step ge (State f x (Kfor2 a2 a3 s k) e m) |
---|
| 1102 | E0 (State f a3 (Kfor3 a2 a3 s k) e m) |
---|
| 1103 | | step_break_for2: ∀f,a2,a3,s,k,e,m. |
---|
| 1104 | step ge (State f Sbreak (Kfor2 a2 a3 s k) e m) |
---|
| 1105 | E0 (State f Sskip k e m) |
---|
| 1106 | | step_skip_for3: ∀f,a2,a3,s,k,e,m. |
---|
| 1107 | step ge (State f Sskip (Kfor3 a2 a3 s k) e m) |
---|
| 1108 | E0 (State f (Sfor Sskip a2 a3 s) k e m) |
---|
| 1109 | |
---|
| 1110 | | step_return_0: ∀f,k,e,m. |
---|
[175] | 1111 | fn_return f = Tvoid → |
---|
[3] | 1112 | step ge (State f (Sreturn (None ?)) k e m) |
---|
| 1113 | E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e))) |
---|
[175] | 1114 | | step_return_1: ∀f,a,k,e,m,v,tr. |
---|
| 1115 | fn_return f ≠ Tvoid → |
---|
| 1116 | eval_expr ge e m a v tr → |
---|
[3] | 1117 | step ge (State f (Sreturn (Some ? a)) k e m) |
---|
[175] | 1118 | tr (Returnstate v (call_cont k) (free_list m (blocks_of_env e))) |
---|
[3] | 1119 | | step_skip_call: ∀f,k,e,m. |
---|
[175] | 1120 | is_call_cont k → |
---|
| 1121 | fn_return f = Tvoid → |
---|
[3] | 1122 | step ge (State f Sskip k e m) |
---|
| 1123 | E0 (Returnstate Vundef k (free_list m (blocks_of_env e))) |
---|
| 1124 | |
---|
[175] | 1125 | | step_switch: ∀f,a,sl,k,e,m,n,tr. |
---|
| 1126 | eval_expr ge e m a (Vint n) tr → |
---|
[3] | 1127 | step ge (State f (Sswitch a sl) k e m) |
---|
[175] | 1128 | tr (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m) |
---|
[3] | 1129 | | step_skip_break_switch: ∀f,x,k,e,m. |
---|
[175] | 1130 | x = Sskip ∨ x = Sbreak → |
---|
[3] | 1131 | step ge (State f x (Kswitch k) e m) |
---|
| 1132 | E0 (State f Sskip k e m) |
---|
| 1133 | | step_continue_switch: ∀f,k,e,m. |
---|
| 1134 | step ge (State f Scontinue (Kswitch k) e m) |
---|
| 1135 | E0 (State f Scontinue k e m) |
---|
| 1136 | |
---|
| 1137 | | step_label: ∀f,lbl,s,k,e,m. |
---|
| 1138 | step ge (State f (Slabel lbl s) k e m) |
---|
| 1139 | E0 (State f s k e m) |
---|
| 1140 | |
---|
| 1141 | | step_goto: ∀f,lbl,k,e,m,s',k'. |
---|
[175] | 1142 | find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 → |
---|
[3] | 1143 | step ge (State f (Sgoto lbl) k e m) |
---|
| 1144 | E0 (State f s' k' e m) |
---|
| 1145 | |
---|
| 1146 | | step_internal_function: ∀f,vargs,k,m,e,m1,m2. |
---|
[175] | 1147 | alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 → |
---|
| 1148 | bind_parameters e m1 (fn_params f) vargs m2 → |
---|
[3] | 1149 | step ge (Callstate (Internal f) vargs k m) |
---|
| 1150 | E0 (State f (fn_body f) k e m2) |
---|
| 1151 | |
---|
| 1152 | | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t. |
---|
[175] | 1153 | event_match (external_function id targs tres) vargs t vres → |
---|
[3] | 1154 | step ge (Callstate (External id targs tres) vargs k m) |
---|
| 1155 | t (Returnstate vres k m) |
---|
| 1156 | |
---|
| 1157 | | step_returnstate_0: ∀v,f,e,k,m. |
---|
| 1158 | step ge (Returnstate v (Kcall (None ?) f e k) m) |
---|
| 1159 | E0 (State f Sskip k e m) |
---|
| 1160 | |
---|
[498] | 1161 | | step_returnstate_1: ∀v,f,e,k,m,m',loc,ofs,ty. |
---|
| 1162 | store_value_of_type ty m loc ofs v = Some ? m' → |
---|
| 1163 | step ge (Returnstate v (Kcall (Some ? 〈〈loc, ofs〉, ty〉) f e k) m) |
---|
[175] | 1164 | E0 (State f Sskip k e m') |
---|
| 1165 | |
---|
| 1166 | | step_cost: ∀f,lbl,s,k,e,m. |
---|
| 1167 | step ge (State f (Scost lbl s) k e m) |
---|
| 1168 | (Echarge lbl) (State f s k e m). |
---|
[3] | 1169 | (* |
---|
| 1170 | (** * Alternate big-step semantics *) |
---|
| 1171 | |
---|
| 1172 | (** ** Big-step semantics for terminating statements and functions *) |
---|
| 1173 | |
---|
| 1174 | (** The execution of a statement produces an ``outcome'', indicating |
---|
| 1175 | how the execution terminated: either normally or prematurely |
---|
| 1176 | through the execution of a [break], [continue] or [return] statement. *) |
---|
| 1177 | |
---|
[487] | 1178 | inductive outcome: Type[0] := |
---|
[3] | 1179 | | Out_break: outcome (**r terminated by [break] *) |
---|
| 1180 | | Out_continue: outcome (**r terminated by [continue] *) |
---|
| 1181 | | Out_normal: outcome (**r terminated normally *) |
---|
| 1182 | | Out_return: option val -> outcome. (**r terminated by [return] *) |
---|
| 1183 | |
---|
[487] | 1184 | inductive out_normal_or_continue : outcome -> Prop := |
---|
[3] | 1185 | | Out_normal_or_continue_N: out_normal_or_continue Out_normal |
---|
| 1186 | | Out_normal_or_continue_C: out_normal_or_continue Out_continue. |
---|
| 1187 | |
---|
[487] | 1188 | inductive out_break_or_return : outcome -> outcome -> Prop := |
---|
[3] | 1189 | | Out_break_or_return_B: out_break_or_return Out_break Out_normal |
---|
| 1190 | | Out_break_or_return_R: ∀ov. |
---|
| 1191 | out_break_or_return (Out_return ov) (Out_return ov). |
---|
| 1192 | |
---|
| 1193 | Definition outcome_switch (out: outcome) : outcome := |
---|
| 1194 | match out with |
---|
| 1195 | | Out_break => Out_normal |
---|
| 1196 | | o => o |
---|
| 1197 | end. |
---|
| 1198 | |
---|
| 1199 | Definition outcome_result_value (out: outcome) (t: type) (v: val) : Prop := |
---|
| 1200 | match out, t with |
---|
| 1201 | | Out_normal, Tvoid => v = Vundef |
---|
| 1202 | | Out_return None, Tvoid => v = Vundef |
---|
| 1203 | | Out_return (Some v'), ty => ty <> Tvoid /\ v'=v |
---|
| 1204 | | _, _ => False |
---|
| 1205 | end. |
---|
| 1206 | |
---|
| 1207 | (** [exec_stmt ge e m1 s t m2 out] describes the execution of |
---|
| 1208 | the statement [s]. [out] is the outcome for this execution. |
---|
| 1209 | [m1] is the initial memory state, [m2] the final memory state. |
---|
| 1210 | [t] is the trace of input/output events performed during this |
---|
| 1211 | evaluation. *) |
---|
| 1212 | |
---|
[487] | 1213 | inductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop := |
---|
[3] | 1214 | | exec_Sskip: ∀e,m. |
---|
| 1215 | exec_stmt e m Sskip |
---|
| 1216 | E0 m Out_normal |
---|
| 1217 | | exec_Sassign: ∀e,m,a1,a2,loc,ofs,v2,m'. |
---|
| 1218 | eval_lvalue e m a1 loc ofs -> |
---|
| 1219 | eval_expr e m a2 v2 -> |
---|
| 1220 | store_value_of_type (typeof a1) m loc ofs v2 = Some m' -> |
---|
| 1221 | exec_stmt e m (Sassign a1 a2) |
---|
| 1222 | E0 m' Out_normal |
---|
| 1223 | | exec_Scall_none: ∀e,m,a,al,vf,vargs,f,t,m',vres. |
---|
| 1224 | eval_expr e m a vf -> |
---|
| 1225 | eval_exprlist e m al vargs -> |
---|
| 1226 | Genv.find_funct ge vf = Some f -> |
---|
| 1227 | type_of_fundef f = typeof a -> |
---|
| 1228 | eval_funcall m f vargs t m' vres -> |
---|
| 1229 | exec_stmt e m (Scall None a al) |
---|
| 1230 | t m' Out_normal |
---|
| 1231 | | exec_Scall_some: ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t,m',vres,m''. |
---|
| 1232 | eval_lvalue e m lhs loc ofs -> |
---|
| 1233 | eval_expr e m a vf -> |
---|
| 1234 | eval_exprlist e m al vargs -> |
---|
| 1235 | Genv.find_funct ge vf = Some f -> |
---|
| 1236 | type_of_fundef f = typeof a -> |
---|
| 1237 | eval_funcall m f vargs t m' vres -> |
---|
| 1238 | store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' -> |
---|
| 1239 | exec_stmt e m (Scall (Some lhs) a al) |
---|
| 1240 | t m'' Out_normal |
---|
| 1241 | | exec_Sseq_1: ∀e,m,s1,s2,t1,m1,t2,m2,out. |
---|
| 1242 | exec_stmt e m s1 t1 m1 Out_normal -> |
---|
| 1243 | exec_stmt e m1 s2 t2 m2 out -> |
---|
| 1244 | exec_stmt e m (Ssequence s1 s2) |
---|
| 1245 | (t1 ** t2) m2 out |
---|
| 1246 | | exec_Sseq_2: ∀e,m,s1,s2,t1,m1,out. |
---|
| 1247 | exec_stmt e m s1 t1 m1 out -> |
---|
| 1248 | out <> Out_normal -> |
---|
| 1249 | exec_stmt e m (Ssequence s1 s2) |
---|
| 1250 | t1 m1 out |
---|
| 1251 | | exec_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t,m',out. |
---|
| 1252 | eval_expr e m a v1 -> |
---|
| 1253 | is_true v1 (typeof a) -> |
---|
| 1254 | exec_stmt e m s1 t m' out -> |
---|
| 1255 | exec_stmt e m (Sifthenelse a s1 s2) |
---|
| 1256 | t m' out |
---|
| 1257 | | exec_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t,m',out. |
---|
| 1258 | eval_expr e m a v1 -> |
---|
| 1259 | is_false v1 (typeof a) -> |
---|
| 1260 | exec_stmt e m s2 t m' out -> |
---|
| 1261 | exec_stmt e m (Sifthenelse a s1 s2) |
---|
| 1262 | t m' out |
---|
| 1263 | | exec_Sreturn_none: ∀e,m. |
---|
| 1264 | exec_stmt e m (Sreturn None) |
---|
| 1265 | E0 m (Out_return None) |
---|
| 1266 | | exec_Sreturn_some: ∀e,m,a,v. |
---|
| 1267 | eval_expr e m a v -> |
---|
| 1268 | exec_stmt e m (Sreturn (Some a)) |
---|
| 1269 | E0 m (Out_return (Some v)) |
---|
| 1270 | | exec_Sbreak: ∀e,m. |
---|
| 1271 | exec_stmt e m Sbreak |
---|
| 1272 | E0 m Out_break |
---|
| 1273 | | exec_Scontinue: ∀e,m. |
---|
| 1274 | exec_stmt e m Scontinue |
---|
| 1275 | E0 m Out_continue |
---|
| 1276 | | exec_Swhile_false: ∀e,m,a,s,v. |
---|
| 1277 | eval_expr e m a v -> |
---|
| 1278 | is_false v (typeof a) -> |
---|
| 1279 | exec_stmt e m (Swhile a s) |
---|
| 1280 | E0 m Out_normal |
---|
| 1281 | | exec_Swhile_stop: ∀e,m,a,v,s,t,m',out',out. |
---|
| 1282 | eval_expr e m a v -> |
---|
| 1283 | is_true v (typeof a) -> |
---|
| 1284 | exec_stmt e m s t m' out' -> |
---|
| 1285 | out_break_or_return out' out -> |
---|
| 1286 | exec_stmt e m (Swhile a s) |
---|
| 1287 | t m' out |
---|
| 1288 | | exec_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2,m2,out. |
---|
| 1289 | eval_expr e m a v -> |
---|
| 1290 | is_true v (typeof a) -> |
---|
| 1291 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1292 | out_normal_or_continue out1 -> |
---|
| 1293 | exec_stmt e m1 (Swhile a s) t2 m2 out -> |
---|
| 1294 | exec_stmt e m (Swhile a s) |
---|
| 1295 | (t1 ** t2) m2 out |
---|
| 1296 | | exec_Sdowhile_false: ∀e,m,s,a,t,m1,out1,v. |
---|
| 1297 | exec_stmt e m s t m1 out1 -> |
---|
| 1298 | out_normal_or_continue out1 -> |
---|
| 1299 | eval_expr e m1 a v -> |
---|
| 1300 | is_false v (typeof a) -> |
---|
| 1301 | exec_stmt e m (Sdowhile a s) |
---|
| 1302 | t m1 Out_normal |
---|
| 1303 | | exec_Sdowhile_stop: ∀e,m,s,a,t,m1,out1,out. |
---|
| 1304 | exec_stmt e m s t m1 out1 -> |
---|
| 1305 | out_break_or_return out1 out -> |
---|
| 1306 | exec_stmt e m (Sdowhile a s) |
---|
| 1307 | t m1 out |
---|
| 1308 | | exec_Sdowhile_loop: ∀e,m,s,a,m1,m2,t1,t2,out,out1,v. |
---|
| 1309 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1310 | out_normal_or_continue out1 -> |
---|
| 1311 | eval_expr e m1 a v -> |
---|
| 1312 | is_true v (typeof a) -> |
---|
| 1313 | exec_stmt e m1 (Sdowhile a s) t2 m2 out -> |
---|
| 1314 | exec_stmt e m (Sdowhile a s) |
---|
| 1315 | (t1 ** t2) m2 out |
---|
| 1316 | | exec_Sfor_start: ∀e,m,s,a1,a2,a3,out,m1,m2,t1,t2. |
---|
| 1317 | a1 <> Sskip -> |
---|
| 1318 | exec_stmt e m a1 t1 m1 Out_normal -> |
---|
| 1319 | exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out -> |
---|
| 1320 | exec_stmt e m (Sfor a1 a2 a3 s) |
---|
| 1321 | (t1 ** t2) m2 out |
---|
| 1322 | | exec_Sfor_false: ∀e,m,s,a2,a3,v. |
---|
| 1323 | eval_expr e m a2 v -> |
---|
| 1324 | is_false v (typeof a2) -> |
---|
| 1325 | exec_stmt e m (Sfor Sskip a2 a3 s) |
---|
| 1326 | E0 m Out_normal |
---|
| 1327 | | exec_Sfor_stop: ∀e,m,s,a2,a3,v,m1,t,out1,out. |
---|
| 1328 | eval_expr e m a2 v -> |
---|
| 1329 | is_true v (typeof a2) -> |
---|
| 1330 | exec_stmt e m s t m1 out1 -> |
---|
| 1331 | out_break_or_return out1 out -> |
---|
| 1332 | exec_stmt e m (Sfor Sskip a2 a3 s) |
---|
| 1333 | t m1 out |
---|
| 1334 | | exec_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,m3,t1,t2,t3,out1,out. |
---|
| 1335 | eval_expr e m a2 v -> |
---|
| 1336 | is_true v (typeof a2) -> |
---|
| 1337 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1338 | out_normal_or_continue out1 -> |
---|
| 1339 | exec_stmt e m1 a3 t2 m2 Out_normal -> |
---|
| 1340 | exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out -> |
---|
| 1341 | exec_stmt e m (Sfor Sskip a2 a3 s) |
---|
| 1342 | (t1 ** t2 ** t3) m3 out |
---|
| 1343 | | exec_Sswitch: ∀e,m,a,t,n,sl,m1,out. |
---|
| 1344 | eval_expr e m a (Vint n) -> |
---|
| 1345 | exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out -> |
---|
| 1346 | exec_stmt e m (Sswitch a sl) |
---|
| 1347 | t m1 (outcome_switch out) |
---|
| 1348 | |
---|
| 1349 | (** [eval_funcall m1 fd args t m2 res] describes the invocation of |
---|
| 1350 | function [fd] with arguments [args]. [res] is the value returned |
---|
| 1351 | by the call. *) |
---|
| 1352 | |
---|
| 1353 | with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop := |
---|
| 1354 | | eval_funcall_internal: ∀m,f,vargs,t,e,m1,m2,m3,out,vres. |
---|
| 1355 | alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 -> |
---|
| 1356 | bind_parameters e m1 f.(fn_params) vargs m2 -> |
---|
| 1357 | exec_stmt e m2 f.(fn_body) t m3 out -> |
---|
| 1358 | outcome_result_value out f.(fn_return) vres -> |
---|
| 1359 | eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres |
---|
| 1360 | | eval_funcall_external: ∀m,id,targs,tres,vargs,t,vres. |
---|
| 1361 | event_match (external_function id targs tres) vargs t vres -> |
---|
| 1362 | eval_funcall m (External id targs tres) vargs t m vres. |
---|
| 1363 | |
---|
| 1364 | Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop |
---|
| 1365 | with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop. |
---|
| 1366 | |
---|
| 1367 | (** ** Big-step semantics for diverging statements and functions *) |
---|
| 1368 | |
---|
| 1369 | (** Coinductive semantics for divergence. |
---|
| 1370 | [execinf_stmt ge e m s t] holds if the execution of statement [s] |
---|
| 1371 | diverges, i.e. loops infinitely. [t] is the possibly infinite |
---|
| 1372 | trace of observable events performed during the execution. *) |
---|
| 1373 | |
---|
[487] | 1374 | Coinductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop := |
---|
[3] | 1375 | | execinf_Scall_none: ∀e,m,a,al,vf,vargs,f,t. |
---|
| 1376 | eval_expr e m a vf -> |
---|
| 1377 | eval_exprlist e m al vargs -> |
---|
| 1378 | Genv.find_funct ge vf = Some f -> |
---|
| 1379 | type_of_fundef f = typeof a -> |
---|
| 1380 | evalinf_funcall m f vargs t -> |
---|
| 1381 | execinf_stmt e m (Scall None a al) t |
---|
| 1382 | | execinf_Scall_some: ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t. |
---|
| 1383 | eval_lvalue e m lhs loc ofs -> |
---|
| 1384 | eval_expr e m a vf -> |
---|
| 1385 | eval_exprlist e m al vargs -> |
---|
| 1386 | Genv.find_funct ge vf = Some f -> |
---|
| 1387 | type_of_fundef f = typeof a -> |
---|
| 1388 | evalinf_funcall m f vargs t -> |
---|
| 1389 | execinf_stmt e m (Scall (Some lhs) a al) t |
---|
| 1390 | | execinf_Sseq_1: ∀e,m,s1,s2,t. |
---|
| 1391 | execinf_stmt e m s1 t -> |
---|
| 1392 | execinf_stmt e m (Ssequence s1 s2) t |
---|
| 1393 | | execinf_Sseq_2: ∀e,m,s1,s2,t1,m1,t2. |
---|
| 1394 | exec_stmt e m s1 t1 m1 Out_normal -> |
---|
| 1395 | execinf_stmt e m1 s2 t2 -> |
---|
| 1396 | execinf_stmt e m (Ssequence s1 s2) (t1 *** t2) |
---|
| 1397 | | execinf_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t. |
---|
| 1398 | eval_expr e m a v1 -> |
---|
| 1399 | is_true v1 (typeof a) -> |
---|
| 1400 | execinf_stmt e m s1 t -> |
---|
| 1401 | execinf_stmt e m (Sifthenelse a s1 s2) t |
---|
| 1402 | | execinf_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t. |
---|
| 1403 | eval_expr e m a v1 -> |
---|
| 1404 | is_false v1 (typeof a) -> |
---|
| 1405 | execinf_stmt e m s2 t -> |
---|
| 1406 | execinf_stmt e m (Sifthenelse a s1 s2) t |
---|
| 1407 | | execinf_Swhile_body: ∀e,m,a,v,s,t. |
---|
| 1408 | eval_expr e m a v -> |
---|
| 1409 | is_true v (typeof a) -> |
---|
| 1410 | execinf_stmt e m s t -> |
---|
| 1411 | execinf_stmt e m (Swhile a s) t |
---|
| 1412 | | execinf_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2. |
---|
| 1413 | eval_expr e m a v -> |
---|
| 1414 | is_true v (typeof a) -> |
---|
| 1415 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1416 | out_normal_or_continue out1 -> |
---|
| 1417 | execinf_stmt e m1 (Swhile a s) t2 -> |
---|
| 1418 | execinf_stmt e m (Swhile a s) (t1 *** t2) |
---|
| 1419 | | execinf_Sdowhile_body: ∀e,m,s,a,t. |
---|
| 1420 | execinf_stmt e m s t -> |
---|
| 1421 | execinf_stmt e m (Sdowhile a s) t |
---|
| 1422 | | execinf_Sdowhile_loop: ∀e,m,s,a,m1,t1,t2,out1,v. |
---|
| 1423 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1424 | out_normal_or_continue out1 -> |
---|
| 1425 | eval_expr e m1 a v -> |
---|
| 1426 | is_true v (typeof a) -> |
---|
| 1427 | execinf_stmt e m1 (Sdowhile a s) t2 -> |
---|
| 1428 | execinf_stmt e m (Sdowhile a s) (t1 *** t2) |
---|
| 1429 | | execinf_Sfor_start_1: ∀e,m,s,a1,a2,a3,t. |
---|
| 1430 | execinf_stmt e m a1 t -> |
---|
| 1431 | execinf_stmt e m (Sfor a1 a2 a3 s) t |
---|
| 1432 | | execinf_Sfor_start_2: ∀e,m,s,a1,a2,a3,m1,t1,t2. |
---|
| 1433 | a1 <> Sskip -> |
---|
| 1434 | exec_stmt e m a1 t1 m1 Out_normal -> |
---|
| 1435 | execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 -> |
---|
| 1436 | execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2) |
---|
| 1437 | | execinf_Sfor_body: ∀e,m,s,a2,a3,v,t. |
---|
| 1438 | eval_expr e m a2 v -> |
---|
| 1439 | is_true v (typeof a2) -> |
---|
| 1440 | execinf_stmt e m s t -> |
---|
| 1441 | execinf_stmt e m (Sfor Sskip a2 a3 s) t |
---|
| 1442 | | execinf_Sfor_next: ∀e,m,s,a2,a3,v,m1,t1,t2,out1. |
---|
| 1443 | eval_expr e m a2 v -> |
---|
| 1444 | is_true v (typeof a2) -> |
---|
| 1445 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1446 | out_normal_or_continue out1 -> |
---|
| 1447 | execinf_stmt e m1 a3 t2 -> |
---|
| 1448 | execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2) |
---|
| 1449 | | execinf_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,t1,t2,t3,out1. |
---|
| 1450 | eval_expr e m a2 v -> |
---|
| 1451 | is_true v (typeof a2) -> |
---|
| 1452 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1453 | out_normal_or_continue out1 -> |
---|
| 1454 | exec_stmt e m1 a3 t2 m2 Out_normal -> |
---|
| 1455 | execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 -> |
---|
| 1456 | execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3) |
---|
| 1457 | | execinf_Sswitch: ∀e,m,a,t,n,sl. |
---|
| 1458 | eval_expr e m a (Vint n) -> |
---|
| 1459 | execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t -> |
---|
| 1460 | execinf_stmt e m (Sswitch a sl) t |
---|
| 1461 | |
---|
| 1462 | (** [evalinf_funcall ge m fd args t] holds if the invocation of function |
---|
| 1463 | [fd] on arguments [args] diverges, with observable trace [t]. *) |
---|
| 1464 | |
---|
| 1465 | with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop := |
---|
| 1466 | | evalinf_funcall_internal: ∀m,f,vargs,t,e,m1,m2. |
---|
| 1467 | alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 -> |
---|
| 1468 | bind_parameters e m1 f.(fn_params) vargs m2 -> |
---|
| 1469 | execinf_stmt e m2 f.(fn_body) t -> |
---|
| 1470 | evalinf_funcall m (Internal f) vargs t. |
---|
| 1471 | |
---|
| 1472 | End SEMANTICS. |
---|
| 1473 | *) |
---|
| 1474 | (* * * Whole-program semantics *) |
---|
| 1475 | |
---|
| 1476 | (* * Execution of whole programs are described as sequences of transitions |
---|
| 1477 | from an initial state to a final state. An initial state is a [Callstate] |
---|
| 1478 | corresponding to the invocation of the ``main'' function of the program |
---|
| 1479 | without arguments and with an empty continuation. *) |
---|
| 1480 | |
---|
[487] | 1481 | inductive initial_state (p: clight_program): state -> Prop := |
---|
[485] | 1482 | | initial_state_intro: ∀b,f,ge,m0. |
---|
| 1483 | globalenv Genv ?? p = OK ? ge → |
---|
| 1484 | init_mem Genv ?? p = OK ? m0 → |
---|
[496] | 1485 | find_symbol ?? ge (prog_main ?? p) = Some ? b → |
---|
[485] | 1486 | find_funct_ptr ?? ge b = Some ? f → |
---|
[3] | 1487 | initial_state p (Callstate f (nil ?) Kstop m0). |
---|
| 1488 | |
---|
| 1489 | (* * A final state is a [Returnstate] with an empty continuation. *) |
---|
| 1490 | |
---|
[487] | 1491 | inductive final_state: state -> int -> Prop := |
---|
[3] | 1492 | | final_state_intro: ∀r,m. |
---|
| 1493 | final_state (Returnstate (Vint r) Kstop m) r. |
---|
| 1494 | |
---|
| 1495 | (* * Execution of a whole program: [exec_program p beh] |
---|
| 1496 | holds if the application of [p]'s main function to no arguments |
---|
| 1497 | in the initial memory state for [p] has [beh] as observable |
---|
| 1498 | behavior. *) |
---|
| 1499 | |
---|
[487] | 1500 | definition exec_program : clight_program → program_behavior → Prop ≝ λp,beh. |
---|
[485] | 1501 | ∀ge. globalenv ??? p = OK ? ge → |
---|
| 1502 | program_behaves (mk_transrel ?? step) (initial_state p) final_state ge beh. |
---|
[3] | 1503 | (* |
---|
| 1504 | (** Big-step execution of a whole program. *) |
---|
| 1505 | |
---|
[487] | 1506 | inductive bigstep_program_terminates (p: program): trace -> int -> Prop := |
---|
[3] | 1507 | | bigstep_program_terminates_intro: ∀b,f,m1,t,r. |
---|
| 1508 | let ge := Genv.globalenv p in |
---|
| 1509 | let m0 := Genv.init_mem p in |
---|
| 1510 | Genv.find_symbol ge p.(prog_main) = Some b -> |
---|
| 1511 | Genv.find_funct_ptr ge b = Some f -> |
---|
| 1512 | eval_funcall ge m0 f nil t m1 (Vint r) -> |
---|
| 1513 | bigstep_program_terminates p t r. |
---|
| 1514 | |
---|
[487] | 1515 | inductive bigstep_program_diverges (p: program): traceinf -> Prop := |
---|
[3] | 1516 | | bigstep_program_diverges_intro: ∀b,f,t. |
---|
| 1517 | let ge := Genv.globalenv p in |
---|
| 1518 | let m0 := Genv.init_mem p in |
---|
| 1519 | Genv.find_symbol ge p.(prog_main) = Some b -> |
---|
| 1520 | Genv.find_funct_ptr ge b = Some f -> |
---|
| 1521 | evalinf_funcall ge m0 f nil t -> |
---|
| 1522 | bigstep_program_diverges p t. |
---|
| 1523 | |
---|
| 1524 | (** * Implication from big-step semantics to transition semantics *) |
---|
| 1525 | |
---|
| 1526 | Section BIGSTEP_TO_TRANSITIONS. |
---|
| 1527 | |
---|
| 1528 | Variable prog: program. |
---|
| 1529 | Let ge : genv := Genv.globalenv prog. |
---|
| 1530 | |
---|
| 1531 | Definition exec_stmt_eval_funcall_ind |
---|
| 1532 | (PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop) |
---|
| 1533 | (PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) := |
---|
| 1534 | fun a b c d e f g h i j k l m n o p q r s t u v w x y => |
---|
| 1535 | conj (exec_stmt_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y) |
---|
| 1536 | (eval_funcall_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y). |
---|
| 1537 | |
---|
[487] | 1538 | inductive outcome_state_match |
---|
[3] | 1539 | (e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop := |
---|
| 1540 | | osm_normal: |
---|
| 1541 | outcome_state_match e m f k Out_normal (State f Sskip k e m) |
---|
| 1542 | | osm_break: |
---|
| 1543 | outcome_state_match e m f k Out_break (State f Sbreak k e m) |
---|
| 1544 | | osm_continue: |
---|
| 1545 | outcome_state_match e m f k Out_continue (State f Scontinue k e m) |
---|
| 1546 | | osm_return_none: ∀k'. |
---|
| 1547 | call_cont k' = call_cont k -> |
---|
| 1548 | outcome_state_match e m f k |
---|
| 1549 | (Out_return None) (State f (Sreturn None) k' e m) |
---|
| 1550 | | osm_return_some: ∀a,v,k'. |
---|
| 1551 | call_cont k' = call_cont k -> |
---|
| 1552 | eval_expr ge e m a v -> |
---|
| 1553 | outcome_state_match e m f k |
---|
| 1554 | (Out_return (Some v)) (State f (Sreturn (Some a)) k' e m). |
---|
| 1555 | |
---|
| 1556 | Lemma is_call_cont_call_cont: |
---|
| 1557 | ∀k. is_call_cont k -> call_cont k = k. |
---|
| 1558 | Proof. |
---|
| 1559 | destruct k; simpl; intros; contradiction || auto. |
---|
| 1560 | Qed. |
---|
| 1561 | |
---|
| 1562 | Lemma exec_stmt_eval_funcall_steps: |
---|
| 1563 | (∀e,m,s,t,m',out. |
---|
| 1564 | exec_stmt ge e m s t m' out -> |
---|
| 1565 | ∀f,k. exists S, |
---|
| 1566 | star step ge (State f s k e m) t S |
---|
| 1567 | /\ outcome_state_match e m' f k out S) |
---|
| 1568 | /\ |
---|
| 1569 | (∀m,fd,args,t,m',res. |
---|
| 1570 | eval_funcall ge m fd args t m' res -> |
---|
| 1571 | ∀k. |
---|
| 1572 | is_call_cont k -> |
---|
| 1573 | star step ge (Callstate fd args k m) t (Returnstate res k m')). |
---|
| 1574 | Proof. |
---|
| 1575 | apply exec_stmt_eval_funcall_ind; intros. |
---|
| 1576 | |
---|
| 1577 | (* skip *) |
---|
| 1578 | econstructor; split. apply star_refl. constructor. |
---|
| 1579 | |
---|
| 1580 | (* assign *) |
---|
| 1581 | econstructor; split. apply star_one. econstructor; eauto. constructor. |
---|
| 1582 | |
---|
| 1583 | (* call none *) |
---|
| 1584 | econstructor; split. |
---|
| 1585 | eapply star_left. econstructor; eauto. |
---|
| 1586 | eapply star_right. apply H4. simpl; auto. econstructor. reflexivity. traceEq. |
---|
| 1587 | constructor. |
---|
| 1588 | |
---|
| 1589 | (* call some *) |
---|
| 1590 | econstructor; split. |
---|
| 1591 | eapply star_left. econstructor; eauto. |
---|
| 1592 | eapply star_right. apply H5. simpl; auto. econstructor; eauto. reflexivity. traceEq. |
---|
| 1593 | constructor. |
---|
| 1594 | |
---|
| 1595 | (* sequence 2 *) |
---|
| 1596 | destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. inv B1. |
---|
| 1597 | destruct (H2 f k) as [S2 [A2 B2]]. |
---|
| 1598 | econstructor; split. |
---|
| 1599 | eapply star_left. econstructor. |
---|
| 1600 | eapply star_trans. eexact A1. |
---|
| 1601 | eapply star_left. constructor. eexact A2. |
---|
| 1602 | reflexivity. reflexivity. traceEq. |
---|
| 1603 | auto. |
---|
| 1604 | |
---|
| 1605 | (* sequence 1 *) |
---|
| 1606 | destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. |
---|
| 1607 | set (S2 := |
---|
| 1608 | match out with |
---|
| 1609 | | Out_break => State f Sbreak k e m1 |
---|
| 1610 | | Out_continue => State f Scontinue k e m1 |
---|
| 1611 | | _ => S1 |
---|
| 1612 | end). |
---|
| 1613 | exists S2; split. |
---|
| 1614 | eapply star_left. econstructor. |
---|
| 1615 | eapply star_trans. eexact A1. |
---|
| 1616 | unfold S2; inv B1. |
---|
| 1617 | congruence. |
---|
| 1618 | apply star_one. apply step_break_seq. |
---|
| 1619 | apply star_one. apply step_continue_seq. |
---|
| 1620 | apply star_refl. |
---|
| 1621 | apply star_refl. |
---|
| 1622 | reflexivity. traceEq. |
---|
| 1623 | unfold S2; inv B1; congruence || econstructor; eauto. |
---|
| 1624 | |
---|
| 1625 | (* ifthenelse true *) |
---|
| 1626 | destruct (H2 f k) as [S1 [A1 B1]]. |
---|
| 1627 | exists S1; split. |
---|
| 1628 | eapply star_left. eapply step_ifthenelse_true; eauto. eexact A1. traceEq. |
---|
| 1629 | auto. |
---|
| 1630 | |
---|
| 1631 | (* ifthenelse false *) |
---|
| 1632 | destruct (H2 f k) as [S1 [A1 B1]]. |
---|
| 1633 | exists S1; split. |
---|
| 1634 | eapply star_left. eapply step_ifthenelse_false; eauto. eexact A1. traceEq. |
---|
| 1635 | auto. |
---|
| 1636 | |
---|
| 1637 | (* return none *) |
---|
| 1638 | econstructor; split. apply star_refl. constructor. auto. |
---|
| 1639 | |
---|
| 1640 | (* return some *) |
---|
| 1641 | econstructor; split. apply star_refl. econstructor; eauto. |
---|
| 1642 | |
---|
| 1643 | (* break *) |
---|
| 1644 | econstructor; split. apply star_refl. constructor. |
---|
| 1645 | |
---|
| 1646 | (* continue *) |
---|
| 1647 | econstructor; split. apply star_refl. constructor. |
---|
| 1648 | |
---|
| 1649 | (* while false *) |
---|
| 1650 | econstructor; split. |
---|
| 1651 | apply star_one. eapply step_while_false; eauto. |
---|
| 1652 | constructor. |
---|
| 1653 | |
---|
| 1654 | (* while stop *) |
---|
| 1655 | destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]]. |
---|
| 1656 | set (S2 := |
---|
| 1657 | match out' with |
---|
| 1658 | | Out_break => State f Sskip k e m' |
---|
| 1659 | | _ => S1 |
---|
| 1660 | end). |
---|
| 1661 | exists S2; split. |
---|
| 1662 | eapply star_left. eapply step_while_true; eauto. |
---|
| 1663 | eapply star_trans. eexact A1. |
---|
| 1664 | unfold S2. inversion H3; subst. |
---|
| 1665 | inv B1. apply star_one. constructor. |
---|
| 1666 | apply star_refl. |
---|
| 1667 | reflexivity. traceEq. |
---|
| 1668 | unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto. |
---|
| 1669 | |
---|
| 1670 | (* while loop *) |
---|
| 1671 | destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]]. |
---|
| 1672 | destruct (H5 f k) as [S2 [A2 B2]]. |
---|
| 1673 | exists S2; split. |
---|
| 1674 | eapply star_left. eapply step_while_true; eauto. |
---|
| 1675 | eapply star_trans. eexact A1. |
---|
| 1676 | eapply star_left. |
---|
| 1677 | inv H3; inv B1; apply step_skip_or_continue_while; auto. |
---|
| 1678 | eexact A2. |
---|
| 1679 | reflexivity. reflexivity. traceEq. |
---|
| 1680 | auto. |
---|
| 1681 | |
---|
| 1682 | (* dowhile false *) |
---|
| 1683 | destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]]. |
---|
| 1684 | exists (State f Sskip k e m1); split. |
---|
| 1685 | eapply star_left. constructor. |
---|
| 1686 | eapply star_right. eexact A1. |
---|
| 1687 | inv H1; inv B1; eapply step_skip_or_continue_dowhile_false; eauto. |
---|
| 1688 | reflexivity. traceEq. |
---|
| 1689 | constructor. |
---|
| 1690 | |
---|
| 1691 | (* dowhile stop *) |
---|
| 1692 | destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]]. |
---|
| 1693 | set (S2 := |
---|
| 1694 | match out1 with |
---|
| 1695 | | Out_break => State f Sskip k e m1 |
---|
| 1696 | | _ => S1 |
---|
| 1697 | end). |
---|
| 1698 | exists S2; split. |
---|
| 1699 | eapply star_left. apply step_dowhile. |
---|
| 1700 | eapply star_trans. eexact A1. |
---|
| 1701 | unfold S2. inversion H1; subst. |
---|
| 1702 | inv B1. apply star_one. constructor. |
---|
| 1703 | apply star_refl. |
---|
| 1704 | reflexivity. traceEq. |
---|
| 1705 | unfold S2. inversion H1; subst. constructor. inv B1; econstructor; eauto. |
---|
| 1706 | |
---|
| 1707 | (* dowhile loop *) |
---|
| 1708 | destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]]. |
---|
| 1709 | destruct (H5 f k) as [S2 [A2 B2]]. |
---|
| 1710 | exists S2; split. |
---|
| 1711 | eapply star_left. apply step_dowhile. |
---|
| 1712 | eapply star_trans. eexact A1. |
---|
| 1713 | eapply star_left. |
---|
| 1714 | inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto. |
---|
| 1715 | eexact A2. |
---|
| 1716 | reflexivity. reflexivity. traceEq. |
---|
| 1717 | auto. |
---|
| 1718 | |
---|
| 1719 | (* for start *) |
---|
| 1720 | destruct (H1 f (Kseq (Sfor Sskip a2 a3 s) k)) as [S1 [A1 B1]]. inv B1. |
---|
| 1721 | destruct (H3 f k) as [S2 [A2 B2]]. |
---|
| 1722 | exists S2; split. |
---|
| 1723 | eapply star_left. apply step_for_start; auto. |
---|
| 1724 | eapply star_trans. eexact A1. |
---|
| 1725 | eapply star_left. constructor. eexact A2. |
---|
| 1726 | reflexivity. reflexivity. traceEq. |
---|
| 1727 | auto. |
---|
| 1728 | |
---|
| 1729 | (* for false *) |
---|
| 1730 | econstructor; split. |
---|
| 1731 | eapply star_one. eapply step_for_false; eauto. |
---|
| 1732 | constructor. |
---|
| 1733 | |
---|
| 1734 | (* for stop *) |
---|
| 1735 | destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]]. |
---|
| 1736 | set (S2 := |
---|
| 1737 | match out1 with |
---|
| 1738 | | Out_break => State f Sskip k e m1 |
---|
| 1739 | | _ => S1 |
---|
| 1740 | end). |
---|
| 1741 | exists S2; split. |
---|
| 1742 | eapply star_left. eapply step_for_true; eauto. |
---|
| 1743 | eapply star_trans. eexact A1. |
---|
| 1744 | unfold S2. inversion H3; subst. |
---|
| 1745 | inv B1. apply star_one. constructor. |
---|
| 1746 | apply star_refl. |
---|
| 1747 | reflexivity. traceEq. |
---|
| 1748 | unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto. |
---|
| 1749 | |
---|
| 1750 | (* for loop *) |
---|
| 1751 | destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]]. |
---|
| 1752 | destruct (H5 f (Kfor3 a2 a3 s k)) as [S2 [A2 B2]]. inv B2. |
---|
| 1753 | destruct (H7 f k) as [S3 [A3 B3]]. |
---|
| 1754 | exists S3; split. |
---|
| 1755 | eapply star_left. eapply step_for_true; eauto. |
---|
| 1756 | eapply star_trans. eexact A1. |
---|
| 1757 | eapply star_trans with (s2 := State f a3 (Kfor3 a2 a3 s k) e m1). |
---|
| 1758 | inv H3; inv B1. |
---|
| 1759 | apply star_one. constructor. auto. |
---|
| 1760 | apply star_one. constructor. auto. |
---|
| 1761 | eapply star_trans. eexact A2. |
---|
| 1762 | eapply star_left. constructor. |
---|
| 1763 | eexact A3. |
---|
| 1764 | reflexivity. reflexivity. reflexivity. reflexivity. traceEq. |
---|
| 1765 | auto. |
---|
| 1766 | |
---|
| 1767 | (* switch *) |
---|
| 1768 | destruct (H1 f (Kswitch k)) as [S1 [A1 B1]]. |
---|
| 1769 | set (S2 := |
---|
| 1770 | match out with |
---|
| 1771 | | Out_normal => State f Sskip k e m1 |
---|
| 1772 | | Out_break => State f Sskip k e m1 |
---|
| 1773 | | Out_continue => State f Scontinue k e m1 |
---|
| 1774 | | _ => S1 |
---|
| 1775 | end). |
---|
| 1776 | exists S2; split. |
---|
| 1777 | eapply star_left. eapply step_switch; eauto. |
---|
| 1778 | eapply star_trans. eexact A1. |
---|
| 1779 | unfold S2; inv B1. |
---|
| 1780 | apply star_one. constructor. auto. |
---|
| 1781 | apply star_one. constructor. auto. |
---|
| 1782 | apply star_one. constructor. |
---|
| 1783 | apply star_refl. |
---|
| 1784 | apply star_refl. |
---|
| 1785 | reflexivity. traceEq. |
---|
| 1786 | unfold S2. inv B1; simpl; econstructor; eauto. |
---|
| 1787 | |
---|
| 1788 | (* call internal *) |
---|
| 1789 | destruct (H2 f k) as [S1 [A1 B1]]. |
---|
| 1790 | eapply star_left. eapply step_internal_function; eauto. |
---|
| 1791 | eapply star_right. eexact A1. |
---|
| 1792 | inv B1; simpl in H3; try contradiction. |
---|
| 1793 | (* Out_normal *) |
---|
| 1794 | assert (fn_return f = Tvoid /\ vres = Vundef). |
---|
| 1795 | destruct (fn_return f); auto || contradiction. |
---|
| 1796 | destruct H5. subst vres. apply step_skip_call; auto. |
---|
| 1797 | (* Out_return None *) |
---|
| 1798 | assert (fn_return f = Tvoid /\ vres = Vundef). |
---|
| 1799 | destruct (fn_return f); auto || contradiction. |
---|
| 1800 | destruct H6. subst vres. |
---|
| 1801 | rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5. |
---|
| 1802 | apply step_return_0; auto. |
---|
| 1803 | (* Out_return Some *) |
---|
| 1804 | destruct H3. subst vres. |
---|
| 1805 | rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5. |
---|
| 1806 | eapply step_return_1; eauto. |
---|
| 1807 | reflexivity. traceEq. |
---|
| 1808 | |
---|
| 1809 | (* call external *) |
---|
| 1810 | apply star_one. apply step_external_function; auto. |
---|
| 1811 | Qed. |
---|
| 1812 | |
---|
| 1813 | Lemma exec_stmt_steps: |
---|
| 1814 | ∀e,m,s,t,m',out. |
---|
| 1815 | exec_stmt ge e m s t m' out -> |
---|
| 1816 | ∀f,k. exists S, |
---|
| 1817 | star step ge (State f s k e m) t S |
---|
| 1818 | /\ outcome_state_match e m' f k out S. |
---|
| 1819 | Proof (proj1 exec_stmt_eval_funcall_steps). |
---|
| 1820 | |
---|
| 1821 | Lemma eval_funcall_steps: |
---|
| 1822 | ∀m,fd,args,t,m',res. |
---|
| 1823 | eval_funcall ge m fd args t m' res -> |
---|
| 1824 | ∀k. |
---|
| 1825 | is_call_cont k -> |
---|
| 1826 | star step ge (Callstate fd args k m) t (Returnstate res k m'). |
---|
| 1827 | Proof (proj2 exec_stmt_eval_funcall_steps). |
---|
| 1828 | |
---|
| 1829 | Definition order (x y: unit) := False. |
---|
| 1830 | |
---|
| 1831 | Lemma evalinf_funcall_forever: |
---|
| 1832 | ∀m,fd,args,T,k. |
---|
| 1833 | evalinf_funcall ge m fd args T -> |
---|
| 1834 | forever_N step order ge tt (Callstate fd args k m) T. |
---|
| 1835 | Proof. |
---|
| 1836 | cofix CIH_FUN. |
---|
| 1837 | assert (∀e,m,s,T,f,k. |
---|
| 1838 | execinf_stmt ge e m s T -> |
---|
| 1839 | forever_N step order ge tt (State f s k e m) T). |
---|
| 1840 | cofix CIH_STMT. |
---|
| 1841 | intros. inv H. |
---|
| 1842 | |
---|
| 1843 | (* call none *) |
---|
| 1844 | eapply forever_N_plus. |
---|
| 1845 | apply plus_one. eapply step_call_none; eauto. |
---|
| 1846 | apply CIH_FUN. eauto. traceEq. |
---|
| 1847 | (* call some *) |
---|
| 1848 | eapply forever_N_plus. |
---|
| 1849 | apply plus_one. eapply step_call_some; eauto. |
---|
| 1850 | apply CIH_FUN. eauto. traceEq. |
---|
| 1851 | |
---|
| 1852 | (* seq 1 *) |
---|
| 1853 | eapply forever_N_plus. |
---|
| 1854 | apply plus_one. econstructor. |
---|
| 1855 | apply CIH_STMT; eauto. traceEq. |
---|
| 1856 | (* seq 2 *) |
---|
| 1857 | destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kseq s2 k)) as [S1 [A1 B1]]. |
---|
| 1858 | inv B1. |
---|
| 1859 | eapply forever_N_plus. |
---|
| 1860 | eapply plus_left. constructor. eapply star_trans. eexact A1. |
---|
| 1861 | apply star_one. constructor. reflexivity. reflexivity. |
---|
| 1862 | apply CIH_STMT; eauto. traceEq. |
---|
| 1863 | |
---|
| 1864 | (* ifthenelse true *) |
---|
| 1865 | eapply forever_N_plus. |
---|
| 1866 | apply plus_one. eapply step_ifthenelse_true; eauto. |
---|
| 1867 | apply CIH_STMT; eauto. traceEq. |
---|
| 1868 | (* ifthenelse false *) |
---|
| 1869 | eapply forever_N_plus. |
---|
| 1870 | apply plus_one. eapply step_ifthenelse_false; eauto. |
---|
| 1871 | apply CIH_STMT; eauto. traceEq. |
---|
| 1872 | |
---|
| 1873 | (* while body *) |
---|
| 1874 | eapply forever_N_plus. |
---|
| 1875 | eapply plus_one. eapply step_while_true; eauto. |
---|
| 1876 | apply CIH_STMT; eauto. traceEq. |
---|
| 1877 | (* while loop *) |
---|
| 1878 | destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kwhile a s0 k)) as [S1 [A1 B1]]. |
---|
| 1879 | eapply forever_N_plus with (s2 := State f (Swhile a s0) k e m1). |
---|
| 1880 | eapply plus_left. eapply step_while_true; eauto. |
---|
| 1881 | eapply star_right. eexact A1. |
---|
| 1882 | inv H3; inv B1; apply step_skip_or_continue_while; auto. |
---|
| 1883 | reflexivity. reflexivity. |
---|
| 1884 | apply CIH_STMT; eauto. traceEq. |
---|
| 1885 | |
---|
| 1886 | (* dowhile body *) |
---|
| 1887 | eapply forever_N_plus. |
---|
| 1888 | eapply plus_one. eapply step_dowhile. |
---|
| 1889 | apply CIH_STMT; eauto. |
---|
| 1890 | traceEq. |
---|
| 1891 | |
---|
| 1892 | (* dowhile loop *) |
---|
| 1893 | destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kdowhile a s0 k)) as [S1 [A1 B1]]. |
---|
| 1894 | eapply forever_N_plus with (s2 := State f (Sdowhile a s0) k e m1). |
---|
| 1895 | eapply plus_left. eapply step_dowhile. |
---|
| 1896 | eapply star_right. eexact A1. |
---|
| 1897 | inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto. |
---|
| 1898 | reflexivity. reflexivity. |
---|
| 1899 | apply CIH_STMT. eauto. |
---|
| 1900 | traceEq. |
---|
| 1901 | |
---|
| 1902 | (* for start 1 *) |
---|
| 1903 | assert (a1 <> Sskip). red; intros; subst. inv H0. |
---|
| 1904 | eapply forever_N_plus. |
---|
| 1905 | eapply plus_one. apply step_for_start; auto. |
---|
| 1906 | apply CIH_STMT; eauto. |
---|
| 1907 | traceEq. |
---|
| 1908 | |
---|
| 1909 | (* for start 2 *) |
---|
| 1910 | destruct (exec_stmt_steps _ _ _ _ _ _ H1 f (Kseq (Sfor Sskip a2 a3 s0) k)) as [S1 [A1 B1]]. |
---|
| 1911 | inv B1. |
---|
| 1912 | eapply forever_N_plus. |
---|
| 1913 | eapply plus_left. eapply step_for_start; eauto. |
---|
| 1914 | eapply star_right. eexact A1. |
---|
| 1915 | apply step_skip_seq. |
---|
| 1916 | reflexivity. reflexivity. |
---|
| 1917 | apply CIH_STMT; eauto. |
---|
| 1918 | traceEq. |
---|
| 1919 | |
---|
| 1920 | (* for body *) |
---|
| 1921 | eapply forever_N_plus. |
---|
| 1922 | apply plus_one. eapply step_for_true; eauto. |
---|
| 1923 | apply CIH_STMT; eauto. |
---|
| 1924 | traceEq. |
---|
| 1925 | |
---|
| 1926 | (* for next *) |
---|
| 1927 | destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]]. |
---|
| 1928 | eapply forever_N_plus. |
---|
| 1929 | eapply plus_left. eapply step_for_true; eauto. |
---|
| 1930 | eapply star_trans. eexact A1. |
---|
| 1931 | apply star_one. |
---|
| 1932 | inv H3; inv B1; apply step_skip_or_continue_for2; auto. |
---|
| 1933 | reflexivity. reflexivity. |
---|
| 1934 | apply CIH_STMT; eauto. |
---|
| 1935 | traceEq. |
---|
| 1936 | |
---|
| 1937 | (* for body *) |
---|
| 1938 | destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]]. |
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| 1939 | destruct (exec_stmt_steps _ _ _ _ _ _ H4 f (Kfor3 a2 a3 s0 k)) as [S2 [A2 B2]]. |
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| 1940 | inv B2. |
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| 1941 | eapply forever_N_plus. |
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| 1942 | eapply plus_left. eapply step_for_true; eauto. |
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| 1943 | eapply star_trans. eexact A1. |
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| 1944 | eapply star_left. inv H3; inv B1; apply step_skip_or_continue_for2; auto. |
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| 1945 | eapply star_right. eexact A2. |
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| 1946 | constructor. |
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| 1947 | reflexivity. reflexivity. reflexivity. reflexivity. |
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| 1948 | apply CIH_STMT; eauto. |
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| 1949 | traceEq. |
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| 1950 | |
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| 1951 | (* switch *) |
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| 1952 | eapply forever_N_plus. |
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| 1953 | eapply plus_one. eapply step_switch; eauto. |
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| 1954 | apply CIH_STMT; eauto. |
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| 1955 | traceEq. |
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| 1956 | |
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| 1957 | (* call internal *) |
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| 1958 | intros. inv H0. |
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| 1959 | eapply forever_N_plus. |
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| 1960 | eapply plus_one. econstructor; eauto. |
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| 1961 | apply H; eauto. |
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| 1962 | traceEq. |
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| 1963 | Qed. |
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| 1964 | |
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| 1965 | Theorem bigstep_program_terminates_exec: |
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| 1966 | ∀t,r. bigstep_program_terminates prog t r -> exec_program prog (Terminates t r). |
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| 1967 | Proof. |
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| 1968 | intros. inv H. unfold ge0, m0 in *. |
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| 1969 | econstructor. |
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| 1970 | econstructor. eauto. eauto. |
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| 1971 | apply eval_funcall_steps. eauto. red; auto. |
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| 1972 | econstructor. |
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| 1973 | Qed. |
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| 1974 | |
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| 1975 | Theorem bigstep_program_diverges_exec: |
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| 1976 | ∀T. bigstep_program_diverges prog T -> |
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| 1977 | exec_program prog (Reacts T) \/ |
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| 1978 | exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T. |
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| 1979 | Proof. |
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| 1980 | intros. inv H. |
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| 1981 | set (st := Callstate f nil Kstop m0). |
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| 1982 | assert (forever step ge0 st T). |
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| 1983 | eapply forever_N_forever with (order := order). |
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| 1984 | red; intros. constructor; intros. red in H. elim H. |
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| 1985 | eapply evalinf_funcall_forever; eauto. |
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| 1986 | destruct (forever_silent_or_reactive _ _ _ _ _ _ H) |
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| 1987 | as [A | [t [s' [T' [B [C D]]]]]]. |
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| 1988 | left. econstructor. econstructor. eauto. eauto. auto. |
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| 1989 | right. exists t. split. |
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| 1990 | econstructor. econstructor; eauto. eauto. auto. |
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| 1991 | subst T. rewrite <- (E0_right t) at 1. apply traceinf_prefix_app. constructor. |
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| 1992 | Qed. |
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| 1993 | |
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| 1994 | End BIGSTEP_TO_TRANSITIONS. |
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| 1995 | |
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| 1996 | |
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| 1997 | |
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| 1998 | *) |
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| 1999 | |
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| 2000 | |
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