[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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[700] | 25 | include "common/Globalenvs.ma". |
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| 26 | include "Clight/Csyntax.ma". |
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[474] | 27 | (*include "Events.ma".*) |
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[700] | 28 | include "common/Smallstep.ma". |
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[3] | 29 | |
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| 30 | (* * * Semantics of type-dependent operations *) |
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| 31 | |
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| 32 | (* * Interpretation of values as truth values. |
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| 33 | Non-zero integers, non-zero floats and non-null pointers are |
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| 34 | considered as true. The integer zero (which also represents |
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| 35 | the null pointer) and the float 0.0 are false. *) |
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| 36 | |
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[487] | 37 | inductive is_false: val → type → Prop ≝ |
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[3] | 38 | | is_false_int: ∀sz,sg. |
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[961] | 39 | is_false (Vint sz (zero ?)) (Tint sz sg) |
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[484] | 40 | | is_false_pointer: ∀r,r',t. |
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| 41 | is_false (Vnull r) (Tpointer r' t) |
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[3] | 42 | | is_false_float: ∀sz. |
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| 43 | is_false (Vfloat Fzero) (Tfloat sz). |
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| 44 | |
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[487] | 45 | inductive is_true: val → type → Prop ≝ |
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[961] | 46 | | is_true_int_int: ∀sz,sg,n. |
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| 47 | n ≠ (zero ?) → |
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| 48 | is_true (Vint sz n) (Tint sz sg) |
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[1545] | 49 | | is_true_pointer_pointer: ∀ptr,s,t. |
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| 50 | is_true (Vptr ptr) (Tpointer s t) |
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[3] | 51 | | is_true_float: ∀f,sz. |
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| 52 | f ≠ Fzero → |
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| 53 | is_true (Vfloat f) (Tfloat sz). |
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| 54 | |
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[487] | 55 | inductive bool_of_val : val → type → val → Prop ≝ |
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[3] | 56 | | bool_of_val_true: ∀v,ty. |
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| 57 | is_true v ty → |
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| 58 | bool_of_val v ty Vtrue |
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| 59 | | bool_of_val_false: ∀v,ty. |
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| 60 | is_false v ty → |
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| 61 | bool_of_val v ty Vfalse. |
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| 62 | |
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| 63 | (* * The following [sem_] functions compute the result of an operator |
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| 64 | application. Since operators are overloaded, the result depends |
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| 65 | both on the static types of the arguments and on their run-time values. |
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| 66 | Unlike in C, automatic conversions between integers and floats |
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| 67 | are not performed. For instance, [e1 + e2] is undefined if [e1] |
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| 68 | is a float and [e2] an integer. The Clight producer must have explicitly |
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| 69 | promoted [e2] to a float. *) |
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| 70 | |
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[487] | 71 | let rec sem_neg (v: val) (ty: type) : option val ≝ |
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[3] | 72 | match ty with |
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[961] | 73 | [ Tint sz _ ⇒ |
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[3] | 74 | match v with |
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[961] | 75 | [ Vint sz' n ⇒ if eq_intsize sz sz' |
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| 76 | then Some ? (Vint ? (two_complement_negation ? n)) |
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| 77 | else None ? |
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[648] | 78 | | _ ⇒ None ? |
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[3] | 79 | ] |
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| 80 | | Tfloat _ ⇒ |
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| 81 | match v with |
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| 82 | [ Vfloat f ⇒ Some ? (Vfloat (Fneg f)) |
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| 83 | | _ ⇒ None ? |
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| 84 | ] |
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| 85 | | _ ⇒ None ? |
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| 86 | ]. |
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| 87 | |
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[487] | 88 | let rec sem_notint (v: val) : option val ≝ |
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[3] | 89 | match v with |
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[961] | 90 | [ Vint sz n ⇒ Some ? (Vint ? (exclusive_disjunction_bv ? n (mone ?))) (* XXX *) |
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[3] | 91 | | _ ⇒ None ? |
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| 92 | ]. |
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| 93 | |
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[487] | 94 | let rec sem_notbool (v: val) (ty: type) : option val ≝ |
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[3] | 95 | match ty with |
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[961] | 96 | [ Tint sz _ ⇒ |
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[3] | 97 | match v with |
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[961] | 98 | [ Vint sz' n ⇒ if eq_intsize sz sz' |
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| 99 | then Some ? (of_bool (eq_bv ? n (zero ?))) |
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| 100 | else None ? |
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[3] | 101 | | _ ⇒ None ? |
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| 102 | ] |
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[124] | 103 | | Tpointer _ _ ⇒ |
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[3] | 104 | match v with |
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[1545] | 105 | [ Vptr _ ⇒ Some ? Vfalse |
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[502] | 106 | | Vnull _ ⇒ Some ? Vtrue |
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[3] | 107 | | _ ⇒ None ? |
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| 108 | ] |
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| 109 | | Tfloat _ ⇒ |
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| 110 | match v with |
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| 111 | [ Vfloat f ⇒ Some ? (of_bool (Fcmp Ceq f Fzero)) |
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| 112 | | _ ⇒ None ? |
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| 113 | ] |
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| 114 | | _ ⇒ None ? |
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| 115 | ]. |
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| 116 | |
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[487] | 117 | let rec sem_add (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 118 | match classify_add t1 t2 with |
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| 119 | [ add_case_ii ⇒ (**r integer addition *) |
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| 120 | match v1 with |
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[961] | 121 | [ Vint sz1 n1 ⇒ match v2 with |
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| 122 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 123 | (λn1. Some ? (Vint ? (addition_n ? n1 n2))) (None ?) |
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[3] | 124 | | _ ⇒ None ? ] |
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| 125 | | _ ⇒ None ? ] |
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| 126 | | add_case_ff ⇒ (**r float addition *) |
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| 127 | match v1 with |
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| 128 | [ Vfloat n1 ⇒ match v2 with |
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| 129 | [ Vfloat n2 ⇒ Some ? (Vfloat (Fadd n1 n2)) |
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| 130 | | _ ⇒ None ? ] |
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| 131 | | _ ⇒ None ? ] |
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| 132 | | add_case_pi ty ⇒ (**r pointer plus integer *) |
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| 133 | match v1 with |
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[1545] | 134 | [ Vptr ptr1 ⇒ match v2 with |
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| 135 | [ Vint sz2 n2 ⇒ Some ? (Vptr (shift_pointer_n ? ptr1 (sizeof ty) n2)) |
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[3] | 136 | | _ ⇒ None ? ] |
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[484] | 137 | | Vnull r ⇒ match v2 with |
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[961] | 138 | [ Vint sz2 n2 ⇒ if eq_bv ? n2 (zero ?) then Some ? (Vnull r) else None ? |
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[484] | 139 | | _ ⇒ None ? ] |
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[3] | 140 | | _ ⇒ None ? ] |
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| 141 | | add_case_ip ty ⇒ (**r integer plus pointer *) |
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| 142 | match v1 with |
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[961] | 143 | [ Vint sz1 n1 ⇒ match v2 with |
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[1545] | 144 | [ Vptr ptr2 ⇒ Some ? (Vptr (shift_pointer_n ? ptr2 (sizeof ty) n1)) |
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[961] | 145 | | Vnull r ⇒ if eq_bv ? n1 (zero ?) then Some ? (Vnull r) else None ? |
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[3] | 146 | | _ ⇒ None ? ] |
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| 147 | | _ ⇒ None ? ] |
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| 148 | | add_default ⇒ None ? |
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| 149 | ]. |
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| 150 | |
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[487] | 151 | let rec sem_sub (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 152 | match classify_sub t1 t2 with |
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| 153 | [ sub_case_ii ⇒ (**r integer subtraction *) |
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| 154 | match v1 with |
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[961] | 155 | [ Vint sz1 n1 ⇒ match v2 with |
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| 156 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 157 | (λn1.Some ? (Vint sz2 (subtraction ? n1 n2))) (None ?) |
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[3] | 158 | | _ ⇒ None ? ] |
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| 159 | | _ ⇒ None ? ] |
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| 160 | | sub_case_ff ⇒ (**r float subtraction *) |
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| 161 | match v1 with |
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| 162 | [ Vfloat f1 ⇒ match v2 with |
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| 163 | [ Vfloat f2 ⇒ Some ? (Vfloat (Fsub f1 f2)) |
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| 164 | | _ ⇒ None ? ] |
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| 165 | | _ ⇒ None ? ] |
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| 166 | | sub_case_pi ty ⇒ (**r pointer minus integer *) |
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| 167 | match v1 with |
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[1545] | 168 | [ Vptr ptr1 ⇒ match v2 with |
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| 169 | [ Vint sz2 n2 ⇒ Some ? (Vptr (neg_shift_pointer_n ? ptr1 (sizeof ty) n2)) |
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[3] | 170 | | _ ⇒ None ? ] |
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[776] | 171 | | Vnull r ⇒ match v2 with |
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[961] | 172 | [ Vint sz2 n2 ⇒ if eq_bv ? n2 (zero ?) then Some ? (Vnull r) else None ? |
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[776] | 173 | | _ ⇒ None ? ] |
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[3] | 174 | | _ ⇒ None ? ] |
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| 175 | | sub_case_pp ty ⇒ (**r pointer minus pointer *) |
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| 176 | match v1 with |
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[1545] | 177 | [ Vptr ptr1 ⇒ match v2 with |
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| 178 | [ Vptr ptr2 ⇒ |
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| 179 | if eq_block (pblock ptr1) (pblock ptr2) then |
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[961] | 180 | if eqb (sizeof ty) 0 then None ? |
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[1545] | 181 | else match division_u ? (sub_offset ? (poff ptr1) (poff ptr2)) (repr (sizeof ty)) with |
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[744] | 182 | [ None ⇒ None ? |
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[961] | 183 | | Some v ⇒ Some ? (Vint I32 v) (* XXX choose size from result type? *) |
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[744] | 184 | ] |
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[3] | 185 | else None ? |
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| 186 | | _ ⇒ None ? ] |
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[961] | 187 | | Vnull r ⇒ match v2 with [ Vnull r' ⇒ Some ? (Vint I32 (*XXX*) (zero ?)) | _ ⇒ None ? ] |
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[3] | 188 | | _ ⇒ None ? ] |
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| 189 | | sub_default ⇒ None ? |
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| 190 | ]. |
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[124] | 191 | |
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[487] | 192 | let rec sem_mul (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 193 | match classify_mul t1 t2 with |
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| 194 | [ mul_case_ii ⇒ |
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| 195 | match v1 with |
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[961] | 196 | [ Vint sz1 n1 ⇒ match v2 with |
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| 197 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 198 | (λn1. Some ? (Vint sz2 (\snd (split ??? (multiplication ? n1 n2))))) (None ?) |
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[3] | 199 | | _ ⇒ None ? ] |
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| 200 | | _ ⇒ None ? ] |
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| 201 | | mul_case_ff ⇒ |
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| 202 | match v1 with |
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| 203 | [ Vfloat f1 ⇒ match v2 with |
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| 204 | [ Vfloat f2 ⇒ Some ? (Vfloat (Fmul f1 f2)) |
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| 205 | | _ ⇒ None ? ] |
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| 206 | | _ ⇒ None ? ] |
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| 207 | | mul_default ⇒ |
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| 208 | None ? |
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| 209 | ]. |
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| 210 | |
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[487] | 211 | let rec sem_div (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 212 | match classify_div t1 t2 with |
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| 213 | [ div_case_I32unsi ⇒ |
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| 214 | match v1 with |
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[961] | 215 | [ Vint sz1 n1 ⇒ match v2 with |
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| 216 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 217 | (λn1. option_map … (Vint ?) (division_u ? n1 n2)) (None ?) |
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[3] | 218 | | _ ⇒ None ? ] |
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| 219 | | _ ⇒ None ? ] |
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| 220 | | div_case_ii ⇒ |
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| 221 | match v1 with |
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[961] | 222 | [ Vint sz1 n1 ⇒ match v2 with |
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| 223 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 224 | (λn1. option_map … (Vint ?) (division_s ? n1 n2)) (None ?) |
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[3] | 225 | | _ ⇒ None ? ] |
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| 226 | | _ ⇒ None ? ] |
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| 227 | | div_case_ff ⇒ |
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| 228 | match v1 with |
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| 229 | [ Vfloat f1 ⇒ match v2 with |
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| 230 | [ Vfloat f2 ⇒ Some ? (Vfloat(Fdiv f1 f2)) |
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| 231 | | _ ⇒ None ? ] |
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| 232 | | _ ⇒ None ? ] |
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| 233 | | div_default ⇒ |
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| 234 | None ? |
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| 235 | ]. |
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| 236 | |
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[487] | 237 | let rec sem_mod (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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[3] | 238 | match classify_mod t1 t2 with |
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| 239 | [ mod_case_I32unsi ⇒ |
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| 240 | match v1 with |
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[961] | 241 | [ Vint sz1 n1 ⇒ match v2 with |
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| 242 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 243 | (λn1. option_map … (Vint ?) (modulus_u ? n1 n2)) (None ?) |
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[3] | 244 | | _ ⇒ None ? ] |
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| 245 | | _ ⇒ None ? ] |
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| 246 | | mod_case_ii ⇒ |
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| 247 | match v1 with |
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[961] | 248 | [ Vint sz1 n1 ⇒ match v2 with |
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| 249 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 250 | (λn1. option_map … (Vint ?) (modulus_s ? n1 n2)) (None ?) |
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[3] | 251 | | _ ⇒ None ? ] |
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| 252 | | _ ⇒ None ? ] |
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| 253 | | mod_default ⇒ |
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| 254 | None ? |
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| 255 | ]. |
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| 256 | |
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[487] | 257 | let rec sem_and (v1,v2: val) : option val ≝ |
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[3] | 258 | match v1 with |
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[961] | 259 | [ Vint sz1 n1 ⇒ match v2 with |
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| 260 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 261 | (λn1. Some ? (Vint ? (conjunction_bv ? n1 n2))) (None ?) |
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[3] | 262 | | _ ⇒ None ? ] |
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| 263 | | _ ⇒ None ? |
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| 264 | ]. |
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| 265 | |
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[487] | 266 | let rec sem_or (v1,v2: val) : option val ≝ |
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[3] | 267 | match v1 with |
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[961] | 268 | [ Vint sz1 n1 ⇒ match v2 with |
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| 269 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 270 | (λn1. Some ? (Vint ? (inclusive_disjunction_bv ? n1 n2))) (None ?) |
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[3] | 271 | | _ ⇒ None ? ] |
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| 272 | | _ ⇒ None ? |
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| 273 | ]. |
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| 274 | |
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[487] | 275 | let rec sem_xor (v1,v2: val) : option val ≝ |
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[3] | 276 | match v1 with |
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[961] | 277 | [ Vint sz1 n1 ⇒ match v2 with |
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| 278 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 279 | (λn1. Some ? (Vint ? (exclusive_disjunction_bv ? n1 n2))) (None ?) |
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[3] | 280 | | _ ⇒ None ? ] |
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| 281 | | _ ⇒ None ? |
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| 282 | ]. |
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| 283 | |
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[487] | 284 | let rec sem_shl (v1,v2: val): option val ≝ |
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[3] | 285 | match v1 with |
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[961] | 286 | [ Vint sz1 n1 ⇒ match v2 with |
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| 287 | [ Vint sz2 n2 ⇒ |
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| 288 | if lt_u ? n2 (bitvector_of_nat … (bitsize_of_intsize sz1)) |
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| 289 | then Some ? (Vint sz1 (shift_left ?? (nat_of_bitvector … n2) n1 false)) |
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| 290 | else None ? |
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[3] | 291 | | _ ⇒ None ? ] |
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| 292 | | _ ⇒ None ? ]. |
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| 293 | |
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[487] | 294 | let rec sem_shr (v1: val) (t1: type) (v2: val) (t2: type): option val ≝ |
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[3] | 295 | match classify_shr t1 t2 with |
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| 296 | [ shr_case_I32unsi ⇒ |
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| 297 | match v1 with |
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[961] | 298 | [ Vint sz1 n1 ⇒ match v2 with |
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| 299 | [ Vint sz2 n2 ⇒ |
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| 300 | if lt_u ? n2 (bitvector_of_nat … (bitsize_of_intsize sz1)) |
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| 301 | then Some ? (Vint ? (shift_right ?? (nat_of_bitvector … n2) n1 false)) |
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| 302 | else None ? |
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[3] | 303 | | _ ⇒ None ? ] |
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| 304 | | _ ⇒ None ? ] |
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| 305 | | shr_case_ii => |
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| 306 | match v1 with |
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[961] | 307 | [ Vint sz1 n1 ⇒ match v2 with |
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| 308 | [ Vint sz2 n2 ⇒ |
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| 309 | if lt_u ? n2 (bitvector_of_nat … (bitsize_of_intsize sz1)) |
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| 310 | then Some ? (Vint ? (shift_right ?? (nat_of_bitvector … n2) n1 (head' … n1))) |
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| 311 | else None ? |
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[3] | 312 | | _ ⇒ None ? ] |
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| 313 | | _ ⇒ None ? ] |
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| 314 | | shr_default ⇒ |
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| 315 | None ? |
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| 316 | ]. |
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| 317 | |
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[487] | 318 | let rec sem_cmp_mismatch (c: comparison): option val ≝ |
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[3] | 319 | match c with |
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| 320 | [ Ceq => Some ? Vfalse |
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| 321 | | Cne => Some ? Vtrue |
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| 322 | | _ => None ? |
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| 323 | ]. |
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| 324 | |
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[487] | 325 | let rec sem_cmp_match (c: comparison): option val ≝ |
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[484] | 326 | match c with |
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| 327 | [ Ceq => Some ? Vtrue |
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| 328 | | Cne => Some ? Vfalse |
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| 329 | | _ => None ? |
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| 330 | ]. |
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| 331 | |
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[487] | 332 | let rec sem_cmp (c:comparison) |
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[3] | 333 | (v1: val) (t1: type) (v2: val) (t2: type) |
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| 334 | (m: mem): option val ≝ |
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| 335 | match classify_cmp t1 t2 with |
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| 336 | [ cmp_case_I32unsi ⇒ |
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| 337 | match v1 with |
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[961] | 338 | [ Vint sz1 n1 ⇒ match v2 with |
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| 339 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 340 | (λn1. Some ? (of_bool (cmpu_int ? c n1 n2))) (None ?) |
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[3] | 341 | | _ ⇒ None ? ] |
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| 342 | | _ ⇒ None ? ] |
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[816] | 343 | | cmp_case_ii ⇒ |
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[3] | 344 | match v1 with |
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[961] | 345 | [ Vint sz1 n1 ⇒ match v2 with |
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| 346 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 347 | (λn1. Some ? (of_bool (cmp_int ? c n1 n2))) (None ?) |
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[3] | 348 | | _ ⇒ None ? |
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| 349 | ] |
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[816] | 350 | | _ ⇒ None ? |
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| 351 | ] |
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| 352 | | cmp_case_pp ⇒ |
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| 353 | match v1 with |
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[1545] | 354 | [ Vptr ptr1 ⇒ |
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[3] | 355 | match v2 with |
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[1545] | 356 | [ Vptr ptr2 ⇒ |
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| 357 | if valid_pointer m ptr1 |
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| 358 | ∧ valid_pointer m ptr2 then |
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| 359 | if eq_block (pblock ptr1) (pblock ptr2) |
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| 360 | then Some ? (of_bool (cmp_offset c (poff ptr1) (poff ptr2))) |
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[3] | 361 | else sem_cmp_mismatch c |
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| 362 | else None ? |
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[484] | 363 | | Vnull r2 ⇒ sem_cmp_mismatch c |
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[3] | 364 | | _ ⇒ None ? ] |
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[484] | 365 | | Vnull r1 ⇒ |
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| 366 | match v2 with |
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[1545] | 367 | [ Vptr ptr2 ⇒ sem_cmp_mismatch c |
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[484] | 368 | | Vnull r2 ⇒ sem_cmp_match c |
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| 369 | | _ ⇒ None ? |
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| 370 | ] |
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[3] | 371 | | _ ⇒ None ? ] |
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| 372 | | cmp_case_ff ⇒ |
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| 373 | match v1 with |
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| 374 | [ Vfloat f1 ⇒ |
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| 375 | match v2 with |
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| 376 | [ Vfloat f2 ⇒ Some ? (of_bool (Fcmp c f1 f2)) |
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| 377 | | _ ⇒ None ? ] |
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| 378 | | _ ⇒ None ? ] |
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| 379 | | cmp_default ⇒ None ? |
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| 380 | ]. |
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| 381 | |
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[487] | 382 | definition sem_unary_operation |
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[3] | 383 | : unary_operation → val → type → option val ≝ |
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| 384 | λop,v,ty. |
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| 385 | match op with |
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| 386 | [ Onotbool => sem_notbool v ty |
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| 387 | | Onotint => sem_notint v |
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| 388 | | Oneg => sem_neg v ty |
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| 389 | ]. |
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| 390 | |
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[487] | 391 | let rec sem_binary_operation |
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[3] | 392 | (op: binary_operation) |
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| 393 | (v1: val) (t1: type) (v2: val) (t2:type) |
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| 394 | (m: mem): option val ≝ |
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| 395 | match op with |
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| 396 | [ Oadd ⇒ sem_add v1 t1 v2 t2 |
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| 397 | | Osub ⇒ sem_sub v1 t1 v2 t2 |
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| 398 | | Omul ⇒ sem_mul v1 t1 v2 t2 |
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| 399 | | Omod ⇒ sem_mod v1 t1 v2 t2 |
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| 400 | | Odiv ⇒ sem_div v1 t1 v2 t2 |
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| 401 | | Oand ⇒ sem_and v1 v2 |
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| 402 | | Oor ⇒ sem_or v1 v2 |
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| 403 | | Oxor ⇒ sem_xor v1 v2 |
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| 404 | | Oshl ⇒ sem_shl v1 v2 |
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| 405 | | Oshr ⇒ sem_shr v1 t1 v2 t2 |
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| 406 | | Oeq ⇒ sem_cmp Ceq v1 t1 v2 t2 m |
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| 407 | | One ⇒ sem_cmp Cne v1 t1 v2 t2 m |
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| 408 | | Olt ⇒ sem_cmp Clt v1 t1 v2 t2 m |
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| 409 | | Ogt ⇒ sem_cmp Cgt v1 t1 v2 t2 m |
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| 410 | | Ole ⇒ sem_cmp Cle v1 t1 v2 t2 m |
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| 411 | | Oge ⇒ sem_cmp Cge v1 t1 v2 t2 m |
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| 412 | ]. |
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| 413 | |
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| 414 | (* * Semantic of casts. [cast v1 t1 t2 v2] holds if value [v1], |
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| 415 | viewed with static type [t1], can be cast to type [t2], |
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| 416 | resulting in value [v2]. *) |
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| 417 | |
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[964] | 418 | let rec cast_int_int (sz: intsize) (sg: signedness) (dstsz: intsize) (i: BitVector (bitsize_of_intsize sz)) : BitVector (bitsize_of_intsize dstsz) ≝ |
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[961] | 419 | match sg with [ Signed ⇒ sign_ext ?? i | Unsigned ⇒ zero_ext ?? i ]. |
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[3] | 420 | |
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[961] | 421 | let rec cast_int_float (si : signedness) (n:nat) (i: BitVector n) : float ≝ |
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[3] | 422 | match si with |
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[961] | 423 | [ Signed ⇒ floatofint ? i |
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| 424 | | Unsigned ⇒ floatofintu ? i |
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[3] | 425 | ]. |
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| 426 | |
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[961] | 427 | let rec cast_float_int (sz : intsize) (si : signedness) (f: float) : BitVector (bitsize_of_intsize sz) ≝ |
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[3] | 428 | match si with |
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[961] | 429 | [ Signed ⇒ intoffloat ? f |
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| 430 | | Unsigned ⇒ intuoffloat ? f |
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[3] | 431 | ]. |
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| 432 | |
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[487] | 433 | let rec cast_float_float (sz: floatsize) (f: float) : float ≝ |
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[3] | 434 | match sz with |
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| 435 | [ F32 ⇒ singleoffloat f |
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| 436 | | F64 ⇒ f |
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| 437 | ]. |
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| 438 | |
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[487] | 439 | inductive type_region : type → region → Prop ≝ |
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[484] | 440 | | type_rgn_pointer : ∀s,t. type_region (Tpointer s t) s |
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| 441 | | type_rgn_array : ∀s,t,n. type_region (Tarray s t n) s |
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[155] | 442 | (* XXX Is the following necessary? *) |
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[484] | 443 | | type_rgn_code : ∀tys,ty. type_region (Tfunction tys ty) Code. |
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[124] | 444 | |
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[487] | 445 | inductive cast : mem → val → type → type → val → Prop ≝ |
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[961] | 446 | | cast_ii: ∀m,sz2,sz1,si1,si2,i. (**r int to int *) |
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| 447 | cast m (Vint sz1 i) (Tint sz1 si1) (Tint sz2 si2) |
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[964] | 448 | (Vint sz2 (cast_int_int sz1 si1 sz2 i)) |
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[124] | 449 | | cast_fi: ∀m,f,sz1,sz2,si2. (**r float to int *) |
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| 450 | cast m (Vfloat f) (Tfloat sz1) (Tint sz2 si2) |
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[961] | 451 | (Vint sz2 (cast_float_int sz2 si2 f)) |
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| 452 | | cast_if: ∀m,sz1,sz2,si1,i. (**r int to float *) |
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| 453 | cast m (Vint sz1 i) (Tint sz1 si1) (Tfloat sz2) |
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| 454 | (Vfloat (cast_float_float sz2 (cast_int_float si1 ? i))) |
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[124] | 455 | | cast_ff: ∀m,f,sz1,sz2. (**r float to float *) |
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| 456 | cast m (Vfloat f) (Tfloat sz1) (Tfloat sz2) |
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[3] | 457 | (Vfloat (cast_float_float sz2 f)) |
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[1545] | 458 | | cast_pp: ∀m,ty,ty',ptr,r'. |
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| 459 | type_region ty (ptype ptr) → |
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[484] | 460 | type_region ty' r' → |
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[1545] | 461 | ∀pc':pointer_compat (pblock ptr) r'. |
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| 462 | cast m (Vptr ptr) ty ty' (Vptr (mk_pointer r' (pblock ptr) pc' (poff ptr))) |
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[484] | 463 | | cast_ip_z: ∀m,sz,sg,ty',r. |
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| 464 | type_region ty' r → |
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[961] | 465 | cast m (Vint sz (zero ?)) (Tint sz sg) ty' (Vnull r) |
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[484] | 466 | | cast_pp_z: ∀m,ty,ty',r,r'. |
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| 467 | type_region ty r → |
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| 468 | type_region ty' r' → |
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| 469 | cast m (Vnull r) ty ty' (Vnull r'). |
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[127] | 470 | |
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[3] | 471 | (* * * Operational semantics *) |
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| 472 | |
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| 473 | (* * The semantics uses two environments. The global environment |
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| 474 | maps names of functions and global variables to memory block references, |
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| 475 | and function pointers to their definitions. (See module [Globalenvs].) *) |
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| 476 | |
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[725] | 477 | definition genv ≝ (genv_t Genv) clight_fundef. |
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[3] | 478 | |
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| 479 | (* * The local environment maps local variables to block references. |
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| 480 | The current value of the variable is stored in the associated memory |
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| 481 | block. *) |
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| 482 | |
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[1058] | 483 | definition env ≝ identifier_map SymbolTag block. (* map variable -> location *) |
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[3] | 484 | |
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[1058] | 485 | definition empty_env: env ≝ (empty_map …). |
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[3] | 486 | |
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| 487 | (* * [load_value_of_type ty m b ofs] computes the value of a datum |
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| 488 | of type [ty] residing in memory [m] at block [b], offset [ofs]. |
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| 489 | If the type [ty] indicates an access by value, the corresponding |
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| 490 | memory load is performed. If the type [ty] indicates an access by |
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| 491 | reference, the pointer [Vptr b ofs] is returned. *) |
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| 492 | |
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[583] | 493 | let rec load_value_of_type (ty: type) (m: mem) (b: block) (ofs: offset) : option val ≝ |
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[3] | 494 | match access_mode ty with |
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[1545] | 495 | [ By_value chunk ⇒ loadv chunk m (Vptr (mk_pointer Any b ? ofs)) |
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[500] | 496 | | By_reference r ⇒ |
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| 497 | match pointer_compat_dec b r with |
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[1545] | 498 | [ inl p ⇒ Some ? (Vptr (mk_pointer r b p ofs)) |
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[500] | 499 | | inr _ ⇒ None ? |
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| 500 | ] |
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[3] | 501 | | By_nothing ⇒ None ? |
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| 502 | ]. |
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[500] | 503 | cases b // |
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| 504 | qed. |
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[3] | 505 | |
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| 506 | (* * Symmetrically, [store_value_of_type ty m b ofs v] returns the |
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| 507 | memory state after storing the value [v] in the datum |
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| 508 | of type [ty] residing in memory [m] at block [b], offset [ofs]. |
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| 509 | This is allowed only if [ty] indicates an access by value. *) |
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| 510 | |
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[583] | 511 | let rec store_value_of_type (ty_dest: type) (m: mem) (loc: block) (ofs: offset) (v: val) : option mem ≝ |
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[3] | 512 | match access_mode ty_dest with |
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[1545] | 513 | [ By_value chunk ⇒ storev chunk m (Vptr (mk_pointer Any loc ? ofs)) v |
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[498] | 514 | | By_reference _ ⇒ None ? |
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[3] | 515 | | By_nothing ⇒ None ? |
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| 516 | ]. |
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[500] | 517 | cases loc // |
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| 518 | qed. |
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[3] | 519 | |
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| 520 | (* * Allocation of function-local variables. |
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| 521 | [alloc_variables e1 m1 vars e2 m2] allocates one memory block |
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| 522 | for each variable declared in [vars], and associates the variable |
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| 523 | name with this block. [e1] and [m1] are the initial local environment |
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| 524 | and memory state. [e2] and [m2] are the final local environment |
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| 525 | and memory state. *) |
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| 526 | |
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[487] | 527 | inductive alloc_variables: env → mem → |
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[3] | 528 | list (ident × type) → |
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| 529 | env → mem → Prop ≝ |
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| 530 | | alloc_variables_nil: |
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| 531 | ∀e,m. |
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| 532 | alloc_variables e m (nil ?) e m |
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| 533 | | alloc_variables_cons: |
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| 534 | ∀e,m,id,ty,vars,m1,b1,m2,e2. |
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[125] | 535 | alloc m 0 (sizeof ty) Any = 〈m1, b1〉 → |
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[1058] | 536 | alloc_variables (add … e id b1) m1 vars e2 m2 → |
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[3] | 537 | alloc_variables e m (〈id, ty〉 :: vars) e2 m2. |
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| 538 | |
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| 539 | (* * Initialization of local variables that are parameters to a function. |
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| 540 | [bind_parameters e m1 params args m2] stores the values [args] |
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| 541 | in the memory blocks corresponding to the variables [params]. |
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| 542 | [m1] is the initial memory state and [m2] the final memory state. *) |
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| 543 | |
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[487] | 544 | inductive bind_parameters: env → |
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[3] | 545 | mem → list (ident × type) → list val → |
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| 546 | mem → Prop ≝ |
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| 547 | | bind_parameters_nil: |
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| 548 | ∀e,m. |
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| 549 | bind_parameters e m (nil ?) (nil ?) m |
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| 550 | | bind_parameters_cons: |
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[125] | 551 | ∀e,m,id,ty,params,v1,vl,b,m1,m2. |
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[1058] | 552 | lookup ?? e id = Some ? b → |
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[583] | 553 | store_value_of_type ty m b zero_offset v1 = Some ? m1 → |
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[3] | 554 | bind_parameters e m1 params vl m2 → |
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| 555 | bind_parameters e m (〈id, ty〉 :: params) (v1 :: vl) m2. |
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| 556 | |
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| 557 | (* * Return the list of blocks in the codomain of [e]. *) |
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| 558 | |
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[487] | 559 | definition blocks_of_env : env → list block ≝ λe. |
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[1058] | 560 | map ?? (λx. snd ?? x) (elements ?? e). |
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[3] | 561 | |
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| 562 | (* * Selection of the appropriate case of a [switch], given the value [n] |
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| 563 | of the selector expression. *) |
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[961] | 564 | (* FIXME: now that we have several sizes of integer, it isn't clear whether we |
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| 565 | should allow case labels to be of a different size to the switch expression. *) |
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| 566 | let rec select_switch (sz:intsize) (n: BitVector (bitsize_of_intsize sz)) (sl: labeled_statements) |
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[3] | 567 | on sl : labeled_statements ≝ |
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| 568 | match sl with |
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| 569 | [ LSdefault _ ⇒ sl |
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[961] | 570 | | LScase sz' c s sl' ⇒ intsize_eq_elim ? sz sz' ? n |
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| 571 | (λn. if eq_bv ? c n then sl else select_switch sz' n sl') (select_switch sz n sl') |
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[3] | 572 | ]. |
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| 573 | |
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| 574 | (* * Turn a labeled statement into a sequence *) |
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| 575 | |
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[487] | 576 | let rec seq_of_labeled_statement (sl: labeled_statements) : statement ≝ |
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[3] | 577 | match sl with |
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| 578 | [ LSdefault s ⇒ s |
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[961] | 579 | | LScase _ c s sl' ⇒ Ssequence s (seq_of_labeled_statement sl') |
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[3] | 580 | ]. |
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| 581 | |
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| 582 | (* |
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| 583 | Section SEMANTICS. |
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| 584 | |
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| 585 | Variable ge: genv. |
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| 586 | |
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| 587 | (** ** Evaluation of expressions *) |
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| 588 | |
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| 589 | Section EXPR. |
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| 590 | |
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| 591 | Variable e: env. |
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| 592 | Variable m: mem. |
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| 593 | *) |
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| 594 | (* * [eval_expr ge e m a v] defines the evaluation of expression [a] |
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| 595 | in r-value position. [v] is the value of the expression. |
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| 596 | [e] is the current environment and [m] is the current memory state. *) |
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| 597 | |
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[487] | 598 | inductive eval_expr (ge:genv) (e:env) (m:mem) : expr → val → trace → Prop ≝ |
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[961] | 599 | | eval_Econst_int: ∀sz,sg,i. |
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| 600 | eval_expr ge e m (Expr (Econst_int sz i) (Tint sz sg)) (Vint sz i) E0 |
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[3] | 601 | | eval_Econst_float: ∀f,ty. |
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[175] | 602 | eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f) E0 |
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[498] | 603 | | eval_Elvalue: ∀a,ty,loc,ofs,v,tr. |
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| 604 | eval_lvalue ge e m (Expr a ty) loc ofs tr → |
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| 605 | load_value_of_type ty m loc ofs = Some ? v → |
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[175] | 606 | eval_expr ge e m (Expr a ty) v tr |
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[496] | 607 | | eval_Eaddrof: ∀a,ty,r,loc,ofs,tr. |
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[498] | 608 | eval_lvalue ge e m a loc ofs tr → |
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[500] | 609 | ∀pc:pointer_compat loc r. |
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[1545] | 610 | eval_expr ge e m (Expr (Eaddrof a) (Tpointer r ty)) (Vptr (mk_pointer r loc pc ofs)) tr |
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[961] | 611 | | eval_Esizeof: ∀ty',sz,sg. |
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| 612 | eval_expr ge e m (Expr (Esizeof ty') (Tint sz sg)) (Vint sz (repr ? (sizeof ty'))) E0 |
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[175] | 613 | | eval_Eunop: ∀op,a,ty,v1,v,tr. |
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| 614 | eval_expr ge e m a v1 tr → |
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| 615 | sem_unary_operation op v1 (typeof a) = Some ? v → |
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| 616 | eval_expr ge e m (Expr (Eunop op a) ty) v tr |
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| 617 | | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v,tr1,tr2. |
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| 618 | eval_expr ge e m a1 v1 tr1 → |
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| 619 | eval_expr ge e m a2 v2 tr2 → |
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| 620 | sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some ? v → |
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| 621 | eval_expr ge e m (Expr (Ebinop op a1 a2) ty) v (tr1⧺tr2) |
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| 622 | | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2,tr1,tr2. |
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| 623 | eval_expr ge e m a1 v1 tr1 → |
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| 624 | is_true v1 (typeof a1) → |
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| 625 | eval_expr ge e m a2 v2 tr2 → |
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| 626 | eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2 (tr1⧺tr2) |
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| 627 | | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3,tr1,tr2. |
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| 628 | eval_expr ge e m a1 v1 tr1 → |
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| 629 | is_false v1 (typeof a1) → |
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| 630 | eval_expr ge e m a3 v3 tr2 → |
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| 631 | eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3 (tr1⧺tr2) |
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| 632 | | eval_Eorbool_1: ∀a1,a2,ty,v1,tr. |
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| 633 | eval_expr ge e m a1 v1 tr → |
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| 634 | is_true v1 (typeof a1) → |
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| 635 | eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue tr |
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| 636 | | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2. |
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| 637 | eval_expr ge e m a1 v1 tr1 → |
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| 638 | is_false v1 (typeof a1) → |
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| 639 | eval_expr ge e m a2 v2 tr2 → |
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| 640 | bool_of_val v2 (typeof a2) v → |
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| 641 | eval_expr ge e m (Expr (Eorbool a1 a2) ty) v (tr1⧺tr2) |
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| 642 | | eval_Eandbool_1: ∀a1,a2,ty,v1,tr. |
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| 643 | eval_expr ge e m a1 v1 tr → |
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| 644 | is_false v1 (typeof a1) → |
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| 645 | eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse tr |
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| 646 | | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2. |
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| 647 | eval_expr ge e m a1 v1 tr1 → |
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| 648 | is_true v1 (typeof a1) → |
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| 649 | eval_expr ge e m a2 v2 tr2 → |
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| 650 | bool_of_val v2 (typeof a2) v → |
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| 651 | eval_expr ge e m (Expr (Eandbool a1 a2) ty) v (tr1⧺tr2) |
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| 652 | | eval_Ecast: ∀a,ty,ty',v1,v,tr. |
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| 653 | eval_expr ge e m a v1 tr → |
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| 654 | cast m v1 (typeof a) ty v → |
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| 655 | eval_expr ge e m (Expr (Ecast ty a) ty') v tr |
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| 656 | | eval_Ecost: ∀a,ty,v,l,tr. |
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| 657 | eval_expr ge e m a v tr → |
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| 658 | eval_expr ge e m (Expr (Ecost l a) ty) v (tr⧺Echarge l) |
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[3] | 659 | |
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[496] | 660 | (* * [eval_lvalue ge e m a r b ofs] defines the evaluation of expression [a] |
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[3] | 661 | in l-value position. The result is the memory location [b, ofs] |
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[496] | 662 | that contains the value of the expression [a]. The memory location should |
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| 663 | be representable in a pointer of region r. *) |
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[3] | 664 | |
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[583] | 665 | with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → block → offset → trace → Prop ≝ |
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[125] | 666 | | eval_Evar_local: ∀id,l,ty. |
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[1058] | 667 | (* XXX notation? e!id*) lookup ?? e id = Some ? l → |
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[583] | 668 | eval_lvalue ge e m (Expr (Evar id) ty) l zero_offset E0 |
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[498] | 669 | | eval_Evar_global: ∀id,l,ty. |
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[1058] | 670 | (* XXX e!id *) lookup ?? e id = None ? → |
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[498] | 671 | find_symbol ?? ge id = Some ? l → |
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[583] | 672 | eval_lvalue ge e m (Expr (Evar id) ty) l zero_offset E0 |
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[500] | 673 | | eval_Ederef: ∀a,ty,r,l,p,ofs,tr. |
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[1545] | 674 | eval_expr ge e m a (Vptr (mk_pointer r l p ofs)) tr → |
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[498] | 675 | eval_lvalue ge e m (Expr (Ederef a) ty) l ofs tr |
---|
| 676 | (* Aside: note that each block of memory is entirely contained within one |
---|
| 677 | memory region; hence adding a field offset will not produce a location |
---|
| 678 | outside of the original location's region. *) |
---|
| 679 | | eval_Efield_struct: ∀a,i,ty,l,ofs,id,fList,delta,tr. |
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| 680 | eval_lvalue ge e m a l ofs tr → |
---|
[175] | 681 | typeof a = Tstruct id fList → |
---|
| 682 | field_offset i fList = OK ? delta → |
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[961] | 683 | eval_lvalue ge e m (Expr (Efield a i) ty) l (shift_offset ? ofs (repr I32 delta)) tr |
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[498] | 684 | | eval_Efield_union: ∀a,i,ty,l,ofs,id,fList,tr. |
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| 685 | eval_lvalue ge e m a l ofs tr → |
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[175] | 686 | typeof a = Tunion id fList → |
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[498] | 687 | eval_lvalue ge e m (Expr (Efield a i) ty) l ofs tr. |
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[3] | 688 | |
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[487] | 689 | let rec eval_expr_ind (ge:genv) (e:env) (m:mem) |
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[226] | 690 | (P:∀a,v,tr. eval_expr ge e m a v tr → Prop) |
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[961] | 691 | (eci:∀sz,sg,i. P ??? (eval_Econst_int ge e m sz sg i)) |
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[226] | 692 | (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty)) |
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[498] | 693 | (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] | 694 | (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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[961] | 695 | (esz:∀ty',sz,sg. P ??? (eval_Esizeof ge e m ty' sz sg)) |
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[226] | 696 | (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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| 697 | (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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| 698 | (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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| 699 | (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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| 700 | (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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| 701 | (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)) |
---|
| 702 | (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)) |
---|
| 703 | (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)) |
---|
| 704 | (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)) |
---|
| 705 | (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H)) |
---|
| 706 | (a:expr) (v:val) (tr:trace) (ev:eval_expr ge e m a v tr) on ev : P a v tr ev ≝ |
---|
| 707 | match ev with |
---|
[961] | 708 | [ eval_Econst_int sz sg i ⇒ eci sz sg i |
---|
[226] | 709 | | eval_Econst_float f ty ⇒ ecF f ty |
---|
[498] | 710 | | eval_Elvalue a ty loc ofs v tr H1 H2 ⇒ elv a ty loc ofs v tr H1 H2 |
---|
[500] | 711 | | eval_Eaddrof a ty r loc pc ofs tr H ⇒ ead a ty r loc pc ofs tr H |
---|
[961] | 712 | | eval_Esizeof ty' sz sg ⇒ esz ty' sz sg |
---|
[226] | 713 | | 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) |
---|
| 714 | | 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) |
---|
| 715 | | 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) |
---|
| 716 | | 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) |
---|
| 717 | | 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) |
---|
| 718 | | 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) |
---|
| 719 | | 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) |
---|
| 720 | | 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) |
---|
| 721 | | 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) |
---|
| 722 | | 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) |
---|
| 723 | ]. |
---|
[1672] | 724 | (* |
---|
[487] | 725 | inverter eval_expr_inv_ind for eval_expr : Prop. |
---|
[1672] | 726 | *) |
---|
[487] | 727 | let rec eval_lvalue_ind (ge:genv) (e:env) (m:mem) |
---|
[498] | 728 | (P:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop) |
---|
| 729 | (lvl:∀id,l,ty,H. P ???? (eval_Evar_local ge e m id l ty H)) |
---|
| 730 | (lvg:∀id,l,ty,H1,H2. P ???? (eval_Evar_global ge e m id l ty H1 H2)) |
---|
[500] | 731 | (lde:∀a,ty,r,l,pc,ofs,tr,H. P ???? (eval_Ederef ge e m a ty r l pc ofs tr H)) |
---|
[498] | 732 | (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)) |
---|
| 733 | (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)) |
---|
[583] | 734 | (a:expr) (loc:block) (ofs:offset) (tr:trace) (ev:eval_lvalue ge e m a loc ofs tr) on ev : P a loc ofs tr ev ≝ |
---|
[226] | 735 | match ev with |
---|
| 736 | [ eval_Evar_local id l ty H ⇒ lvl id l ty H |
---|
[498] | 737 | | eval_Evar_global id l ty H1 H2 ⇒ lvg id l ty H1 H2 |
---|
[500] | 738 | | eval_Ederef a ty r l pc ofs tr H ⇒ lde a ty r l pc ofs tr H |
---|
[498] | 739 | | 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) |
---|
| 740 | | 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] | 741 | ]. |
---|
| 742 | |
---|
[3] | 743 | (* |
---|
[226] | 744 | ninverter eval_lvalue_inv_ind for eval_lvalue : Prop. |
---|
| 745 | *) |
---|
[1672] | 746 | (* |
---|
[487] | 747 | definition eval_lvalue_inv_ind : |
---|
[226] | 748 | ∀x1: genv. |
---|
| 749 | ∀x2: env. |
---|
| 750 | ∀x3: mem. |
---|
| 751 | ∀x4: expr. |
---|
| 752 | ∀x6: block. |
---|
[583] | 753 | ∀x7: offset. |
---|
[226] | 754 | ∀x8: trace. |
---|
| 755 | ∀P: |
---|
| 756 | ∀_z1430: expr. |
---|
[583] | 757 | ∀_z1428: block. ∀_z1427: offset. ∀_z1426: trace. Prop. |
---|
[226] | 758 | ∀_H1: ?. |
---|
| 759 | ∀_H2: ?. |
---|
| 760 | ∀_H3: ?. |
---|
| 761 | ∀_H4: ?. |
---|
| 762 | ∀_H5: ?. |
---|
[498] | 763 | ∀_Hterm: eval_lvalue x1 x2 x3 x4 x6 x7 x8. |
---|
| 764 | P x4 x6 x7 x8 |
---|
[226] | 765 | := |
---|
| 766 | (λx1:genv. |
---|
| 767 | (λx2:env. |
---|
| 768 | (λx3:mem. |
---|
| 769 | (λx4:expr. |
---|
| 770 | (λx6:block. |
---|
[583] | 771 | (λx7:offset. |
---|
[226] | 772 | (λx8:trace. |
---|
| 773 | (λP:∀_z1430: expr. |
---|
| 774 | ∀_z1428: block. |
---|
[583] | 775 | ∀_z1427: offset. ∀_z1426: trace. Prop. |
---|
[226] | 776 | (λH1:?. |
---|
| 777 | (λH2:?. |
---|
| 778 | (λH3:?. |
---|
| 779 | (λH4:?. |
---|
| 780 | (λH5:?. |
---|
[498] | 781 | (λHterm:eval_lvalue x1 x2 x3 x4 x6 x7 x8. |
---|
[226] | 782 | ((λHcut:∀z1435: eq expr x4 x4. |
---|
| 783 | ∀z1433: eq block x6 x6. |
---|
[583] | 784 | ∀z1432: eq offset x7 x7. |
---|
[226] | 785 | ∀z1431: eq trace x8 x8. |
---|
[498] | 786 | P x4 x6 x7 x8. |
---|
[226] | 787 | (Hcut (refl expr x4) |
---|
[498] | 788 | (refl block x6) |
---|
[583] | 789 | (refl offset x7) (refl trace x8))) |
---|
[498] | 790 | ?))))))))))))))). |
---|
| 791 | [ @(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] | 792 | [ @H1 | @H2 | @H3 | @H4 | @H5 ] |
---|
| 793 | | *: skip |
---|
| 794 | ] qed. |
---|
[1672] | 795 | *) |
---|
[487] | 796 | let rec eval_expr_ind2 (ge:genv) (e:env) (m:mem) |
---|
[226] | 797 | (P:∀a,v,tr. eval_expr ge e m a v tr → Prop) |
---|
[498] | 798 | (Q:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop) |
---|
[961] | 799 | (eci:∀sz,sg,i. P ??? (eval_Econst_int ge e m sz sg i)) |
---|
[226] | 800 | (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty)) |
---|
[498] | 801 | (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] | 802 | (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)) |
---|
[961] | 803 | (esz:∀ty',sz,sg. P ??? (eval_Esizeof ge e m ty' sz sg)) |
---|
[226] | 804 | (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)) |
---|
| 805 | (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)) |
---|
| 806 | (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)) |
---|
| 807 | (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)) |
---|
| 808 | (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)) |
---|
| 809 | (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)) |
---|
| 810 | (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)) |
---|
| 811 | (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)) |
---|
| 812 | (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)) |
---|
| 813 | (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] | 814 | (lvl:∀id,l,ty,H. Q ???? (eval_Evar_local ge e m id l ty H)) |
---|
| 815 | (lvg:∀id,l,ty,H1,H2. Q ???? (eval_Evar_global ge e m id l ty H1 H2)) |
---|
[1545] | 816 | (lde:∀a,ty,r,l,pc,ofs,tr,H. P a (Vptr (mk_pointer r l pc ofs)) tr H → Q ???? (eval_Ederef ge e m a ty r l pc ofs tr H)) |
---|
[498] | 817 | (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)) |
---|
| 818 | (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] | 819 | |
---|
| 820 | (a:expr) (v:val) (tr:trace) (ev:eval_expr ge e m a v tr) on ev : P a v tr ev ≝ |
---|
| 821 | match ev with |
---|
[961] | 822 | [ eval_Econst_int sz sg i ⇒ eci sz sg i |
---|
[226] | 823 | | eval_Econst_float f ty ⇒ ecF f ty |
---|
[498] | 824 | | 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] | 825 | | 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) |
---|
[961] | 826 | | eval_Esizeof ty' sz sg ⇒ esz ty' sz sg |
---|
[226] | 827 | | 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) |
---|
| 828 | | 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) |
---|
| 829 | | 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) |
---|
| 830 | | 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) |
---|
| 831 | | 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) |
---|
| 832 | | 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) |
---|
| 833 | | 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) |
---|
| 834 | | 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) |
---|
| 835 | | 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) |
---|
| 836 | | 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) |
---|
| 837 | ] |
---|
| 838 | and eval_lvalue_ind2 (ge:genv) (e:env) (m:mem) |
---|
| 839 | (P:∀a,v,tr. eval_expr ge e m a v tr → Prop) |
---|
[498] | 840 | (Q:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop) |
---|
[961] | 841 | (eci:∀sz,sg,i. P ??? (eval_Econst_int ge e m sz sg i)) |
---|
[226] | 842 | (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty)) |
---|
[498] | 843 | (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] | 844 | (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)) |
---|
[961] | 845 | (esz:∀ty',sz,sg. P ??? (eval_Esizeof ge e m ty' sz sg)) |
---|
[226] | 846 | (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)) |
---|
| 847 | (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)) |
---|
| 848 | (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)) |
---|
| 849 | (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)) |
---|
| 850 | (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)) |
---|
| 851 | (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)) |
---|
| 852 | (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)) |
---|
| 853 | (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)) |
---|
| 854 | (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)) |
---|
| 855 | (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] | 856 | (lvl:∀id,l,ty,H. Q ???? (eval_Evar_local ge e m id l ty H)) |
---|
| 857 | (lvg:∀id,l,ty,H1,H2. Q ???? (eval_Evar_global ge e m id l ty H1 H2)) |
---|
[1545] | 858 | (lde:∀a,ty,r,l,pc,ofs,tr,H. P a (Vptr (mk_pointer r l pc ofs)) tr H → Q ???? (eval_Ederef ge e m a ty r l pc ofs tr H)) |
---|
[498] | 859 | (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)) |
---|
| 860 | (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)) |
---|
[583] | 861 | (a:expr) (loc:block) (ofs:offset) (tr:trace) (ev:eval_lvalue ge e m a loc ofs tr) on ev : Q a loc ofs tr ev ≝ |
---|
[226] | 862 | match ev with |
---|
| 863 | [ eval_Evar_local id l ty H ⇒ lvl id l ty H |
---|
[498] | 864 | | eval_Evar_global id l ty H1 H2 ⇒ lvg id l ty H1 H2 |
---|
[1545] | 865 | | 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 (mk_pointer r l pc ofs)) tr H) |
---|
[498] | 866 | | 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) |
---|
| 867 | | 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] | 868 | ]. |
---|
| 869 | |
---|
[487] | 870 | definition combined_expr_lvalue_ind ≝ |
---|
[226] | 871 | 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. |
---|
| 872 | conj ?? |
---|
| 873 | (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) |
---|
| 874 | (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). |
---|
| 875 | |
---|
| 876 | (* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a] |
---|
| 877 | in l-value position. The result is the memory location [b, ofs] |
---|
| 878 | that contains the value of the expression [a]. *) |
---|
| 879 | |
---|
| 880 | (* |
---|
| 881 | Scheme eval_expr_ind22 := Minimality for eval_expr Sort Prop |
---|
[3] | 882 | with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop. |
---|
| 883 | *) |
---|
| 884 | |
---|
| 885 | (* * [eval_exprlist ge e m al vl] evaluates a list of r-value |
---|
| 886 | expressions [al] to their values [vl]. *) |
---|
| 887 | |
---|
[487] | 888 | inductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr → list val → trace → Prop ≝ |
---|
[3] | 889 | | eval_Enil: |
---|
[175] | 890 | eval_exprlist ge e m (nil ?) (nil ?) E0 |
---|
| 891 | | eval_Econs: ∀a,bl,v,vl,tr1,tr2. |
---|
| 892 | eval_expr ge e m a v tr1 → |
---|
| 893 | eval_exprlist ge e m bl vl tr2 → |
---|
| 894 | eval_exprlist ge e m (a :: bl) (v :: vl) (tr1⧺tr2). |
---|
[3] | 895 | |
---|
| 896 | (*End EXPR.*) |
---|
| 897 | |
---|
| 898 | (* * ** Transition semantics for statements and functions *) |
---|
| 899 | |
---|
| 900 | (* * Continuations *) |
---|
| 901 | |
---|
[487] | 902 | inductive cont: Type[0] := |
---|
[3] | 903 | | Kstop: cont |
---|
| 904 | | Kseq: statement -> cont -> cont |
---|
| 905 | (**r [Kseq s2 k] = after [s1] in [s1;s2] *) |
---|
| 906 | | Kwhile: expr -> statement -> cont -> cont |
---|
| 907 | (**r [Kwhile e s k] = after [s] in [while (e) s] *) |
---|
| 908 | | Kdowhile: expr -> statement -> cont -> cont |
---|
| 909 | (**r [Kdowhile e s k] = after [s] in [do s while (e)] *) |
---|
| 910 | | Kfor2: expr -> statement -> statement -> cont -> cont |
---|
| 911 | (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *) |
---|
| 912 | | Kfor3: expr -> statement -> statement -> cont -> cont |
---|
| 913 | (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *) |
---|
| 914 | | Kswitch: cont -> cont |
---|
| 915 | (**r catches [break] statements arising out of [switch] *) |
---|
[583] | 916 | | Kcall: option (block × offset × type) -> (**r where to store result *) |
---|
| 917 | function -> (**r calling function *) |
---|
| 918 | env -> (**r local env of calling function *) |
---|
[3] | 919 | cont -> cont. |
---|
| 920 | |
---|
| 921 | (* * Pop continuation until a call or stop *) |
---|
| 922 | |
---|
[487] | 923 | let rec call_cont (k: cont) : cont := |
---|
[3] | 924 | match k with |
---|
| 925 | [ Kseq s k => call_cont k |
---|
| 926 | | Kwhile e s k => call_cont k |
---|
| 927 | | Kdowhile e s k => call_cont k |
---|
| 928 | | Kfor2 e2 e3 s k => call_cont k |
---|
| 929 | | Kfor3 e2 e3 s k => call_cont k |
---|
| 930 | | Kswitch k => call_cont k |
---|
| 931 | | _ => k |
---|
| 932 | ]. |
---|
| 933 | |
---|
[487] | 934 | definition is_call_cont : cont → Prop ≝ λk. |
---|
[3] | 935 | match k with |
---|
| 936 | [ Kstop => True |
---|
| 937 | | Kcall _ _ _ _ => True |
---|
| 938 | | _ => False |
---|
| 939 | ]. |
---|
| 940 | |
---|
| 941 | (* * States *) |
---|
| 942 | |
---|
[487] | 943 | inductive state: Type[0] := |
---|
[3] | 944 | | State: |
---|
| 945 | ∀f: function. |
---|
| 946 | ∀s: statement. |
---|
| 947 | ∀k: cont. |
---|
| 948 | ∀e: env. |
---|
| 949 | ∀m: mem. state |
---|
| 950 | | Callstate: |
---|
[725] | 951 | ∀fd: clight_fundef. |
---|
[3] | 952 | ∀args: list val. |
---|
| 953 | ∀k: cont. |
---|
| 954 | ∀m: mem. state |
---|
| 955 | | Returnstate: |
---|
| 956 | ∀res: val. |
---|
| 957 | ∀k: cont. |
---|
| 958 | ∀m: mem. state. |
---|
| 959 | |
---|
| 960 | (* * Find the statement and manufacture the continuation |
---|
| 961 | corresponding to a label *) |
---|
| 962 | |
---|
[487] | 963 | let rec find_label (lbl: label) (s: statement) (k: cont) |
---|
[3] | 964 | on s: option (statement × cont) := |
---|
| 965 | match s with |
---|
| 966 | [ Ssequence s1 s2 => |
---|
| 967 | match find_label lbl s1 (Kseq s2 k) with |
---|
| 968 | [ Some sk => Some ? sk |
---|
| 969 | | None => find_label lbl s2 k |
---|
| 970 | ] |
---|
| 971 | | Sifthenelse a s1 s2 => |
---|
| 972 | match find_label lbl s1 k with |
---|
| 973 | [ Some sk => Some ? sk |
---|
| 974 | | None => find_label lbl s2 k |
---|
| 975 | ] |
---|
| 976 | | Swhile a s1 => |
---|
| 977 | find_label lbl s1 (Kwhile a s1 k) |
---|
| 978 | | Sdowhile a s1 => |
---|
| 979 | find_label lbl s1 (Kdowhile a s1 k) |
---|
| 980 | | Sfor a1 a2 a3 s1 => |
---|
| 981 | match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with |
---|
| 982 | [ Some sk => Some ? sk |
---|
| 983 | | None => |
---|
| 984 | match find_label lbl s1 (Kfor2 a2 a3 s1 k) with |
---|
| 985 | [ Some sk => Some ? sk |
---|
| 986 | | None => find_label lbl a3 (Kfor3 a2 a3 s1 k) |
---|
| 987 | ] |
---|
| 988 | ] |
---|
| 989 | | Sswitch e sl => |
---|
| 990 | find_label_ls lbl sl (Kswitch k) |
---|
| 991 | | Slabel lbl' s' => |
---|
| 992 | match ident_eq lbl lbl' with |
---|
| 993 | [ inl _ ⇒ Some ? 〈s', k〉 |
---|
| 994 | | inr _ ⇒ find_label lbl s' k |
---|
| 995 | ] |
---|
| 996 | | _ => None ? |
---|
| 997 | ] |
---|
| 998 | |
---|
| 999 | and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont) |
---|
| 1000 | on sl: option (statement × cont) := |
---|
| 1001 | match sl with |
---|
| 1002 | [ LSdefault s => find_label lbl s k |
---|
[961] | 1003 | | LScase _ _ s sl' => |
---|
[3] | 1004 | match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with |
---|
| 1005 | [ Some sk => Some ? sk |
---|
| 1006 | | None => find_label_ls lbl sl' k |
---|
| 1007 | ] |
---|
| 1008 | ]. |
---|
| 1009 | |
---|
| 1010 | (* * Transition relation *) |
---|
| 1011 | |
---|
[457] | 1012 | (* Strip off outer pointer for use when comparing function types. *) |
---|
[487] | 1013 | definition fun_typeof ≝ |
---|
[457] | 1014 | λe. match typeof e with |
---|
| 1015 | [ Tvoid ⇒ Tvoid |
---|
| 1016 | | Tint a b ⇒ Tint a b |
---|
| 1017 | | Tfloat a ⇒ Tfloat a |
---|
| 1018 | | Tpointer _ ty ⇒ ty |
---|
| 1019 | | Tarray a b c ⇒ Tarray a b c |
---|
| 1020 | | Tfunction a b ⇒ Tfunction a b |
---|
| 1021 | | Tstruct a b ⇒ Tstruct a b |
---|
| 1022 | | Tunion a b ⇒ Tunion a b |
---|
[481] | 1023 | | Tcomp_ptr a b ⇒ Tcomp_ptr a b |
---|
[457] | 1024 | ]. |
---|
| 1025 | |
---|
[175] | 1026 | (* XXX: note that cost labels in exprs expose a particular eval order. *) |
---|
[3] | 1027 | |
---|
[487] | 1028 | inductive step (ge:genv) : state → trace → state → Prop ≝ |
---|
[175] | 1029 | |
---|
[498] | 1030 | | step_assign: ∀f,a1,a2,k,e,m,loc,ofs,v2,m',tr1,tr2. |
---|
| 1031 | eval_lvalue ge e m a1 loc ofs tr1 → |
---|
[175] | 1032 | eval_expr ge e m a2 v2 tr2 → |
---|
[498] | 1033 | store_value_of_type (typeof a1) m loc ofs v2 = Some ? m' → |
---|
[3] | 1034 | step ge (State f (Sassign a1 a2) k e m) |
---|
[175] | 1035 | (tr1⧺tr2) (State f Sskip k e m') |
---|
[3] | 1036 | |
---|
[175] | 1037 | | step_call_none: ∀f,a,al,k,e,m,vf,vargs,fd,tr1,tr2. |
---|
| 1038 | eval_expr ge e m a vf tr1 → |
---|
| 1039 | eval_exprlist ge e m al vargs tr2 → |
---|
| 1040 | find_funct ?? ge vf = Some ? fd → |
---|
[457] | 1041 | type_of_fundef fd = fun_typeof a → |
---|
[3] | 1042 | step ge (State f (Scall (None ?) a al) k e m) |
---|
[175] | 1043 | (tr1⧺tr2) (Callstate fd vargs (Kcall (None ?) f e k) m) |
---|
[3] | 1044 | |
---|
[498] | 1045 | | step_call_some: ∀f,lhs,a,al,k,e,m,loc,ofs,vf,vargs,fd,tr1,tr2,tr3. |
---|
| 1046 | eval_lvalue ge e m lhs loc ofs tr1 → |
---|
[175] | 1047 | eval_expr ge e m a vf tr2 → |
---|
| 1048 | eval_exprlist ge e m al vargs tr3 → |
---|
| 1049 | find_funct ?? ge vf = Some ? fd → |
---|
[457] | 1050 | type_of_fundef fd = fun_typeof a → |
---|
[3] | 1051 | step ge (State f (Scall (Some ? lhs) a al) k e m) |
---|
[498] | 1052 | (tr1⧺tr2⧺tr3) (Callstate fd vargs (Kcall (Some ? 〈〈loc, ofs〉, typeof lhs〉) f e k) m) |
---|
[3] | 1053 | |
---|
| 1054 | | step_seq: ∀f,s1,s2,k,e,m. |
---|
| 1055 | step ge (State f (Ssequence s1 s2) k e m) |
---|
| 1056 | E0 (State f s1 (Kseq s2 k) e m) |
---|
| 1057 | | step_skip_seq: ∀f,s,k,e,m. |
---|
| 1058 | step ge (State f Sskip (Kseq s k) e m) |
---|
| 1059 | E0 (State f s k e m) |
---|
| 1060 | | step_continue_seq: ∀f,s,k,e,m. |
---|
| 1061 | step ge (State f Scontinue (Kseq s k) e m) |
---|
| 1062 | E0 (State f Scontinue k e m) |
---|
| 1063 | | step_break_seq: ∀f,s,k,e,m. |
---|
| 1064 | step ge (State f Sbreak (Kseq s k) e m) |
---|
| 1065 | E0 (State f Sbreak k e m) |
---|
| 1066 | |
---|
[175] | 1067 | | step_ifthenelse_true: ∀f,a,s1,s2,k,e,m,v1,tr. |
---|
| 1068 | eval_expr ge e m a v1 tr → |
---|
| 1069 | is_true v1 (typeof a) → |
---|
[3] | 1070 | step ge (State f (Sifthenelse a s1 s2) k e m) |
---|
[175] | 1071 | tr (State f s1 k e m) |
---|
| 1072 | | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1,tr. |
---|
| 1073 | eval_expr ge e m a v1 tr → |
---|
| 1074 | is_false v1 (typeof a) → |
---|
[3] | 1075 | step ge (State f (Sifthenelse a s1 s2) k e m) |
---|
[175] | 1076 | tr (State f s2 k e m) |
---|
[3] | 1077 | |
---|
[175] | 1078 | | step_while_false: ∀f,a,s,k,e,m,v,tr. |
---|
| 1079 | eval_expr ge e m a v tr → |
---|
| 1080 | is_false v (typeof a) → |
---|
[3] | 1081 | step ge (State f (Swhile a s) k e m) |
---|
[175] | 1082 | tr (State f Sskip k e m) |
---|
| 1083 | | step_while_true: ∀f,a,s,k,e,m,v,tr. |
---|
| 1084 | eval_expr ge e m a v tr → |
---|
| 1085 | is_true v (typeof a) → |
---|
[3] | 1086 | step ge (State f (Swhile a s) k e m) |
---|
[175] | 1087 | tr (State f s (Kwhile a s k) e m) |
---|
[3] | 1088 | | step_skip_or_continue_while: ∀f,x,a,s,k,e,m. |
---|
[175] | 1089 | x = Sskip ∨ x = Scontinue → |
---|
[3] | 1090 | step ge (State f x (Kwhile a s k) e m) |
---|
| 1091 | E0 (State f (Swhile a s) k e m) |
---|
| 1092 | | step_break_while: ∀f,a,s,k,e,m. |
---|
| 1093 | step ge (State f Sbreak (Kwhile a s k) e m) |
---|
| 1094 | E0 (State f Sskip k e m) |
---|
| 1095 | |
---|
| 1096 | | step_dowhile: ∀f,a,s,k,e,m. |
---|
| 1097 | step ge (State f (Sdowhile a s) k e m) |
---|
| 1098 | E0 (State f s (Kdowhile a s k) e m) |
---|
[175] | 1099 | | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v,tr. |
---|
| 1100 | x = Sskip ∨ x = Scontinue → |
---|
| 1101 | eval_expr ge e m a v tr → |
---|
| 1102 | is_false v (typeof a) → |
---|
[3] | 1103 | step ge (State f x (Kdowhile a s k) e m) |
---|
[175] | 1104 | tr (State f Sskip k e m) |
---|
| 1105 | | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v,tr. |
---|
| 1106 | x = Sskip ∨ x = Scontinue → |
---|
| 1107 | eval_expr ge e m a v tr → |
---|
| 1108 | is_true v (typeof a) → |
---|
[3] | 1109 | step ge (State f x (Kdowhile a s k) e m) |
---|
[175] | 1110 | tr (State f (Sdowhile a s) k e m) |
---|
[3] | 1111 | | step_break_dowhile: ∀f,a,s,k,e,m. |
---|
| 1112 | step ge (State f Sbreak (Kdowhile a s k) e m) |
---|
| 1113 | E0 (State f Sskip k e m) |
---|
| 1114 | |
---|
| 1115 | | step_for_start: ∀f,a1,a2,a3,s,k,e,m. |
---|
[175] | 1116 | a1 ≠ Sskip → |
---|
[3] | 1117 | step ge (State f (Sfor a1 a2 a3 s) k e m) |
---|
| 1118 | E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m) |
---|
[175] | 1119 | | step_for_false: ∀f,a2,a3,s,k,e,m,v,tr. |
---|
| 1120 | eval_expr ge e m a2 v tr → |
---|
| 1121 | is_false v (typeof a2) → |
---|
[3] | 1122 | step ge (State f (Sfor Sskip a2 a3 s) k e m) |
---|
[175] | 1123 | tr (State f Sskip k e m) |
---|
| 1124 | | step_for_true: ∀f,a2,a3,s,k,e,m,v,tr. |
---|
| 1125 | eval_expr ge e m a2 v tr → |
---|
| 1126 | is_true v (typeof a2) → |
---|
[3] | 1127 | step ge (State f (Sfor Sskip a2 a3 s) k e m) |
---|
[175] | 1128 | tr (State f s (Kfor2 a2 a3 s k) e m) |
---|
[3] | 1129 | | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m. |
---|
[175] | 1130 | x = Sskip ∨ x = Scontinue → |
---|
[3] | 1131 | step ge (State f x (Kfor2 a2 a3 s k) e m) |
---|
| 1132 | E0 (State f a3 (Kfor3 a2 a3 s k) e m) |
---|
| 1133 | | step_break_for2: ∀f,a2,a3,s,k,e,m. |
---|
| 1134 | step ge (State f Sbreak (Kfor2 a2 a3 s k) e m) |
---|
| 1135 | E0 (State f Sskip k e m) |
---|
| 1136 | | step_skip_for3: ∀f,a2,a3,s,k,e,m. |
---|
| 1137 | step ge (State f Sskip (Kfor3 a2 a3 s k) e m) |
---|
| 1138 | E0 (State f (Sfor Sskip a2 a3 s) k e m) |
---|
| 1139 | |
---|
| 1140 | | step_return_0: ∀f,k,e,m. |
---|
[175] | 1141 | fn_return f = Tvoid → |
---|
[3] | 1142 | step ge (State f (Sreturn (None ?)) k e m) |
---|
| 1143 | E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e))) |
---|
[175] | 1144 | | step_return_1: ∀f,a,k,e,m,v,tr. |
---|
| 1145 | fn_return f ≠ Tvoid → |
---|
| 1146 | eval_expr ge e m a v tr → |
---|
[3] | 1147 | step ge (State f (Sreturn (Some ? a)) k e m) |
---|
[175] | 1148 | tr (Returnstate v (call_cont k) (free_list m (blocks_of_env e))) |
---|
[3] | 1149 | | step_skip_call: ∀f,k,e,m. |
---|
[175] | 1150 | is_call_cont k → |
---|
| 1151 | fn_return f = Tvoid → |
---|
[3] | 1152 | step ge (State f Sskip k e m) |
---|
| 1153 | E0 (Returnstate Vundef k (free_list m (blocks_of_env e))) |
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| 1154 | |
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[961] | 1155 | | step_switch: ∀f,a,sl,k,e,m,sz,n,tr. |
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| 1156 | eval_expr ge e m a (Vint sz n) tr → |
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[3] | 1157 | step ge (State f (Sswitch a sl) k e m) |
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[961] | 1158 | tr (State f (seq_of_labeled_statement (select_switch ? n sl)) (Kswitch k) e m) |
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[3] | 1159 | | step_skip_break_switch: ∀f,x,k,e,m. |
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[175] | 1160 | x = Sskip ∨ x = Sbreak → |
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[3] | 1161 | step ge (State f x (Kswitch k) e m) |
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| 1162 | E0 (State f Sskip k e m) |
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| 1163 | | step_continue_switch: ∀f,k,e,m. |
---|
| 1164 | step ge (State f Scontinue (Kswitch k) e m) |
---|
| 1165 | E0 (State f Scontinue k e m) |
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| 1166 | |
---|
| 1167 | | step_label: ∀f,lbl,s,k,e,m. |
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| 1168 | step ge (State f (Slabel lbl s) k e m) |
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| 1169 | E0 (State f s k e m) |
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| 1170 | |
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| 1171 | | step_goto: ∀f,lbl,k,e,m,s',k'. |
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[175] | 1172 | find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 → |
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[3] | 1173 | step ge (State f (Sgoto lbl) k e m) |
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| 1174 | E0 (State f s' k' e m) |
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| 1175 | |
---|
| 1176 | | step_internal_function: ∀f,vargs,k,m,e,m1,m2. |
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[175] | 1177 | alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 → |
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| 1178 | bind_parameters e m1 (fn_params f) vargs m2 → |
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[725] | 1179 | step ge (Callstate (CL_Internal f) vargs k m) |
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[3] | 1180 | E0 (State f (fn_body f) k e m2) |
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| 1181 | |
---|
| 1182 | | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t. |
---|
[175] | 1183 | event_match (external_function id targs tres) vargs t vres → |
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[725] | 1184 | step ge (Callstate (CL_External id targs tres) vargs k m) |
---|
[3] | 1185 | t (Returnstate vres k m) |
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| 1186 | |
---|
| 1187 | | step_returnstate_0: ∀v,f,e,k,m. |
---|
| 1188 | step ge (Returnstate v (Kcall (None ?) f e k) m) |
---|
| 1189 | E0 (State f Sskip k e m) |
---|
| 1190 | |
---|
[498] | 1191 | | step_returnstate_1: ∀v,f,e,k,m,m',loc,ofs,ty. |
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| 1192 | store_value_of_type ty m loc ofs v = Some ? m' → |
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| 1193 | step ge (Returnstate v (Kcall (Some ? 〈〈loc, ofs〉, ty〉) f e k) m) |
---|
[175] | 1194 | E0 (State f Sskip k e m') |
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| 1195 | |
---|
| 1196 | | step_cost: ∀f,lbl,s,k,e,m. |
---|
| 1197 | step ge (State f (Scost lbl s) k e m) |
---|
| 1198 | (Echarge lbl) (State f s k e m). |
---|
[1147] | 1199 | |
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[3] | 1200 | (* |
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| 1201 | End SEMANTICS. |
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| 1202 | *) |
---|
[1147] | 1203 | |
---|
[3] | 1204 | (* * * Whole-program semantics *) |
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| 1205 | |
---|
| 1206 | (* * Execution of whole programs are described as sequences of transitions |
---|
| 1207 | from an initial state to a final state. An initial state is a [Callstate] |
---|
| 1208 | corresponding to the invocation of the ``main'' function of the program |
---|
| 1209 | without arguments and with an empty continuation. *) |
---|
| 1210 | |
---|
[487] | 1211 | inductive initial_state (p: clight_program): state -> Prop := |
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[485] | 1212 | | initial_state_intro: ∀b,f,ge,m0. |
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[1231] | 1213 | globalenv Genv ?? (fst ??) p = ge → |
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[1139] | 1214 | init_mem Genv ?? (fst ??) p = OK ? m0 → |
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[496] | 1215 | find_symbol ?? ge (prog_main ?? p) = Some ? b → |
---|
[485] | 1216 | find_funct_ptr ?? ge b = Some ? f → |
---|
[3] | 1217 | initial_state p (Callstate f (nil ?) Kstop m0). |
---|
| 1218 | |
---|
| 1219 | (* * A final state is a [Returnstate] with an empty continuation. *) |
---|
| 1220 | |
---|
[487] | 1221 | inductive final_state: state -> int -> Prop := |
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[3] | 1222 | | final_state_intro: ∀r,m. |
---|
[961] | 1223 | final_state (Returnstate (Vint I32 r) Kstop m) r. |
---|
[3] | 1224 | |
---|
| 1225 | (* * Execution of a whole program: [exec_program p beh] |
---|
| 1226 | holds if the application of [p]'s main function to no arguments |
---|
| 1227 | in the initial memory state for [p] has [beh] as observable |
---|
| 1228 | behavior. *) |
---|
| 1229 | |
---|
[487] | 1230 | definition exec_program : clight_program → program_behavior → Prop ≝ λp,beh. |
---|
[1231] | 1231 | ∀ge. globalenv ??? (fst ??) p = ge → |
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[485] | 1232 | program_behaves (mk_transrel ?? step) (initial_state p) final_state ge beh. |
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[3] | 1233 | |
---|