[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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| 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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| 25 | include "Globalenvs.ma". |
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| 26 | include "Csyntax.ma". |
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| 27 | include "Maps.ma". |
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| 28 | include "Events.ma". |
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| 29 | include "Smallstep.ma". |
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| 30 | |
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| 31 | (* * * Semantics of type-dependent operations *) |
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| 32 | |
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| 33 | (* * Interpretation of values as truth values. |
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| 34 | Non-zero integers, non-zero floats and non-null pointers are |
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| 35 | considered as true. The integer zero (which also represents |
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| 36 | the null pointer) and the float 0.0 are false. *) |
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| 37 | |
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| 38 | ninductive is_false: val → type → Prop ≝ |
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| 39 | | is_false_int: ∀sz,sg. |
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| 40 | is_false (Vint zero) (Tint sz sg) |
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[124] | 41 | | is_false_pointer: ∀s,t. |
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| 42 | is_false (Vint zero) (Tpointer s t) |
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[3] | 43 | | is_false_float: ∀sz. |
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| 44 | is_false (Vfloat Fzero) (Tfloat sz). |
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| 45 | |
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| 46 | ninductive is_true: val → type → Prop ≝ |
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| 47 | | is_true_int_int: ∀n,sz,sg. |
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| 48 | n ≠ zero → |
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| 49 | is_true (Vint n) (Tint sz sg) |
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[124] | 50 | | is_true_pointer_int: ∀pcl,b,ofs,sz,sg. |
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| 51 | is_true (Vptr pcl b ofs) (Tint sz sg) |
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| 52 | | is_true_int_pointer: ∀n,s,t. |
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[3] | 53 | n ≠ zero → |
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[124] | 54 | is_true (Vint n) (Tpointer s t) |
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| 55 | | is_true_pointer_pointer: ∀pcl,b,ofs,s,t. |
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| 56 | is_true (Vptr pcl b ofs) (Tpointer s t) |
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[3] | 57 | | is_true_float: ∀f,sz. |
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| 58 | f ≠ Fzero → |
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| 59 | is_true (Vfloat f) (Tfloat sz). |
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| 60 | |
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| 61 | ninductive bool_of_val : val → type → val → Prop ≝ |
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| 62 | | bool_of_val_true: ∀v,ty. |
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| 63 | is_true v ty → |
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| 64 | bool_of_val v ty Vtrue |
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| 65 | | bool_of_val_false: ∀v,ty. |
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| 66 | is_false v ty → |
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| 67 | bool_of_val v ty Vfalse. |
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| 68 | |
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| 69 | (* * The following [sem_] functions compute the result of an operator |
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| 70 | application. Since operators are overloaded, the result depends |
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| 71 | both on the static types of the arguments and on their run-time values. |
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| 72 | Unlike in C, automatic conversions between integers and floats |
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| 73 | are not performed. For instance, [e1 + e2] is undefined if [e1] |
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| 74 | is a float and [e2] an integer. The Clight producer must have explicitly |
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| 75 | promoted [e2] to a float. *) |
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| 76 | |
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| 77 | nlet rec sem_neg (v: val) (ty: type) : option val ≝ |
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| 78 | match ty with |
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| 79 | [ Tint _ _ ⇒ |
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| 80 | match v with |
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| 81 | [ Vint n ⇒ Some ? (Vint (neg n)) |
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| 82 | | _ => None ? |
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| 83 | ] |
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| 84 | | Tfloat _ ⇒ |
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| 85 | match v with |
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| 86 | [ Vfloat f ⇒ Some ? (Vfloat (Fneg f)) |
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| 87 | | _ ⇒ None ? |
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| 88 | ] |
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| 89 | | _ ⇒ None ? |
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| 90 | ]. |
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| 91 | |
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| 92 | nlet rec sem_notint (v: val) : option val ≝ |
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| 93 | match v with |
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| 94 | [ Vint n ⇒ Some ? (Vint (xor n mone)) |
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| 95 | | _ ⇒ None ? |
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| 96 | ]. |
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| 97 | |
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| 98 | nlet rec sem_notbool (v: val) (ty: type) : option val ≝ |
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| 99 | match ty with |
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| 100 | [ Tint _ _ ⇒ |
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| 101 | match v with |
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| 102 | [ Vint n ⇒ Some ? (of_bool (eq n zero)) |
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[124] | 103 | | Vptr _ _ _ ⇒ Some ? Vfalse |
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[3] | 104 | | _ ⇒ None ? |
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| 105 | ] |
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[124] | 106 | | Tpointer _ _ ⇒ |
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[3] | 107 | match v with |
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| 108 | [ Vint n ⇒ Some ? (of_bool (eq n zero)) |
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[124] | 109 | | Vptr _ _ _ ⇒ Some ? Vfalse |
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[3] | 110 | | _ ⇒ None ? |
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| 111 | ] |
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| 112 | | Tfloat _ ⇒ |
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| 113 | match v with |
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| 114 | [ Vfloat f ⇒ Some ? (of_bool (Fcmp Ceq f Fzero)) |
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| 115 | | _ ⇒ None ? |
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| 116 | ] |
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| 117 | | _ ⇒ None ? |
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| 118 | ]. |
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| 119 | |
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| 120 | nlet rec sem_add (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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| 121 | match classify_add t1 t2 with |
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| 122 | [ add_case_ii ⇒ (**r integer addition *) |
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| 123 | match v1 with |
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| 124 | [ Vint n1 ⇒ match v2 with |
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| 125 | [ Vint n2 ⇒ Some ? (Vint (add n1 n2)) |
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| 126 | | _ ⇒ None ? ] |
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| 127 | | _ ⇒ None ? ] |
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| 128 | | add_case_ff ⇒ (**r float addition *) |
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| 129 | match v1 with |
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| 130 | [ Vfloat n1 ⇒ match v2 with |
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| 131 | [ Vfloat n2 ⇒ Some ? (Vfloat (Fadd n1 n2)) |
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| 132 | | _ ⇒ None ? ] |
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| 133 | | _ ⇒ None ? ] |
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| 134 | | add_case_pi ty ⇒ (**r pointer plus integer *) |
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| 135 | match v1 with |
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[124] | 136 | [ Vptr pcl1 b1 ofs1 ⇒ match v2 with |
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| 137 | [ Vint n2 ⇒ Some ? (Vptr pcl1 b1 (add ofs1 (mul (repr (sizeof ty)) n2))) |
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[3] | 138 | | _ ⇒ None ? ] |
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| 139 | | _ ⇒ None ? ] |
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| 140 | | add_case_ip ty ⇒ (**r integer plus pointer *) |
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| 141 | match v1 with |
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| 142 | [ Vint n1 ⇒ match v2 with |
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[124] | 143 | [ Vptr pcl2 b2 ofs2 ⇒ Some ? (Vptr pcl2 b2 (add ofs2 (mul (repr (sizeof ty)) n1))) |
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[3] | 144 | | _ ⇒ None ? ] |
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| 145 | | _ ⇒ None ? ] |
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| 146 | | add_default ⇒ None ? |
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| 147 | ]. |
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| 148 | |
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| 149 | nlet rec sem_sub (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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| 150 | match classify_sub t1 t2 with |
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| 151 | [ sub_case_ii ⇒ (**r integer subtraction *) |
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| 152 | match v1 with |
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| 153 | [ Vint n1 ⇒ match v2 with |
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| 154 | [ Vint n2 ⇒ Some ? (Vint (sub n1 n2)) |
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| 155 | | _ ⇒ None ? ] |
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| 156 | | _ ⇒ None ? ] |
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| 157 | | sub_case_ff ⇒ (**r float subtraction *) |
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| 158 | match v1 with |
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| 159 | [ Vfloat f1 ⇒ match v2 with |
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| 160 | [ Vfloat f2 ⇒ Some ? (Vfloat (Fsub f1 f2)) |
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| 161 | | _ ⇒ None ? ] |
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| 162 | | _ ⇒ None ? ] |
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| 163 | | sub_case_pi ty ⇒ (**r pointer minus integer *) |
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| 164 | match v1 with |
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[124] | 165 | [ Vptr pcl1 b1 ofs1 ⇒ match v2 with |
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| 166 | [ Vint n2 ⇒ Some ? (Vptr pcl1 b1 (sub ofs1 (mul (repr (sizeof ty)) n2))) |
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[3] | 167 | | _ ⇒ None ? ] |
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| 168 | | _ ⇒ None ? ] |
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| 169 | | sub_case_pp ty ⇒ (**r pointer minus pointer *) |
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| 170 | match v1 with |
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[124] | 171 | [ Vptr pcl1 b1 ofs1 ⇒ match v2 with |
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| 172 | [ Vptr pcl2 b2 ofs2 ⇒ |
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[3] | 173 | if eqZb b1 b2 then |
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| 174 | if eq (repr (sizeof ty)) zero then None ? |
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| 175 | else Some ? (Vint (divu (sub ofs1 ofs2) (repr (sizeof ty)))) |
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| 176 | else None ? |
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| 177 | | _ ⇒ None ? ] |
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| 178 | | _ ⇒ None ? ] |
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| 179 | | sub_default ⇒ None ? |
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| 180 | ]. |
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[124] | 181 | |
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[3] | 182 | nlet rec sem_mul (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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| 183 | match classify_mul t1 t2 with |
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| 184 | [ mul_case_ii ⇒ |
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| 185 | match v1 with |
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| 186 | [ Vint n1 ⇒ match v2 with |
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| 187 | [ Vint n2 ⇒ Some ? (Vint (mul n1 n2)) |
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| 188 | | _ ⇒ None ? ] |
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| 189 | | _ ⇒ None ? ] |
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| 190 | | mul_case_ff ⇒ |
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| 191 | match v1 with |
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| 192 | [ Vfloat f1 ⇒ match v2 with |
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| 193 | [ Vfloat f2 ⇒ Some ? (Vfloat (Fmul f1 f2)) |
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| 194 | | _ ⇒ None ? ] |
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| 195 | | _ ⇒ None ? ] |
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| 196 | | mul_default ⇒ |
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| 197 | None ? |
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| 198 | ]. |
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| 199 | |
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| 200 | nlet rec sem_div (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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| 201 | match classify_div t1 t2 with |
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| 202 | [ div_case_I32unsi ⇒ |
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| 203 | match v1 with |
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| 204 | [ Vint n1 ⇒ match v2 with |
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| 205 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (divu n1 n2)) |
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| 206 | | _ ⇒ None ? ] |
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| 207 | | _ ⇒ None ? ] |
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| 208 | | div_case_ii ⇒ |
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| 209 | match v1 with |
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| 210 | [ Vint n1 ⇒ match v2 with |
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| 211 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint(divs n1 n2)) |
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| 212 | | _ ⇒ None ? ] |
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| 213 | | _ ⇒ None ? ] |
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| 214 | | div_case_ff ⇒ |
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| 215 | match v1 with |
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| 216 | [ Vfloat f1 ⇒ match v2 with |
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| 217 | [ Vfloat f2 ⇒ Some ? (Vfloat(Fdiv f1 f2)) |
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| 218 | | _ ⇒ None ? ] |
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| 219 | | _ ⇒ None ? ] |
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| 220 | | div_default ⇒ |
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| 221 | None ? |
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| 222 | ]. |
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| 223 | |
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| 224 | nlet rec sem_mod (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝ |
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| 225 | match classify_mod t1 t2 with |
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| 226 | [ mod_case_I32unsi ⇒ |
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| 227 | match v1 with |
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| 228 | [ Vint n1 ⇒ match v2 with |
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| 229 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (modu n1 n2)) |
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| 230 | | _ ⇒ None ? ] |
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| 231 | | _ ⇒ None ? ] |
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| 232 | | mod_case_ii ⇒ |
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| 233 | match v1 with |
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| 234 | [ Vint n1 ⇒ match v2 with |
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| 235 | [ Vint n2 ⇒ if eq n2 zero then None ? else Some ? (Vint (mods n1 n2)) |
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| 236 | | _ ⇒ None ? ] |
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| 237 | | _ ⇒ None ? ] |
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| 238 | | mod_default ⇒ |
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| 239 | None ? |
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| 240 | ]. |
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| 241 | |
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| 242 | nlet rec sem_and (v1,v2: val) : option val ≝ |
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| 243 | match v1 with |
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| 244 | [ Vint n1 ⇒ match v2 with |
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| 245 | [ Vint n2 ⇒ Some ? (Vint(i_and n1 n2)) |
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| 246 | | _ ⇒ None ? ] |
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| 247 | | _ ⇒ None ? |
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| 248 | ]. |
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| 249 | |
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| 250 | nlet rec sem_or (v1,v2: val) : option val ≝ |
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| 251 | match v1 with |
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| 252 | [ Vint n1 ⇒ match v2 with |
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| 253 | [ Vint n2 ⇒ Some ? (Vint(or n1 n2)) |
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| 254 | | _ ⇒ None ? ] |
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| 255 | | _ ⇒ None ? |
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| 256 | ]. |
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| 257 | |
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| 258 | nlet rec sem_xor (v1,v2: val) : option val ≝ |
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| 259 | match v1 with |
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| 260 | [ Vint n1 ⇒ match v2 with |
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| 261 | [ Vint n2 ⇒ Some ? (Vint(xor n1 n2)) |
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| 262 | | _ ⇒ None ? ] |
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| 263 | | _ ⇒ None ? |
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| 264 | ]. |
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| 265 | |
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| 266 | nlet rec sem_shl (v1,v2: val): option val ≝ |
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| 267 | match v1 with |
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| 268 | [ Vint n1 ⇒ match v2 with |
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| 269 | [ Vint n2 ⇒ |
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| 270 | if ltu n2 iwordsize then Some ? (Vint(shl n1 n2)) else None ? |
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| 271 | | _ ⇒ None ? ] |
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| 272 | | _ ⇒ None ? ]. |
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| 273 | |
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| 274 | nlet rec sem_shr (v1: val) (t1: type) (v2: val) (t2: type): option val ≝ |
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| 275 | match classify_shr t1 t2 with |
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| 276 | [ shr_case_I32unsi ⇒ |
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| 277 | match v1 with |
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| 278 | [ Vint n1 ⇒ match v2 with |
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| 279 | [ Vint n2 ⇒ |
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| 280 | if ltu n2 iwordsize then Some ? (Vint (shru n1 n2)) else None ? |
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| 281 | | _ ⇒ None ? ] |
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| 282 | | _ ⇒ None ? ] |
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| 283 | | shr_case_ii => |
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| 284 | match v1 with |
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| 285 | [ Vint n1 ⇒ match v2 with |
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| 286 | [ Vint n2 ⇒ |
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| 287 | if ltu n2 iwordsize then Some ? (Vint (shr n1 n2)) else None ? |
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| 288 | | _ ⇒ None ? ] |
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| 289 | | _ ⇒ None ? ] |
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| 290 | | shr_default ⇒ |
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| 291 | None ? |
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| 292 | ]. |
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| 293 | |
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| 294 | nlet rec sem_cmp_mismatch (c: comparison): option val ≝ |
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| 295 | match c with |
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| 296 | [ Ceq => Some ? Vfalse |
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| 297 | | Cne => Some ? Vtrue |
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| 298 | | _ => None ? |
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| 299 | ]. |
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| 300 | |
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| 301 | nlet rec sem_cmp (c:comparison) |
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| 302 | (v1: val) (t1: type) (v2: val) (t2: type) |
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| 303 | (m: mem): option val ≝ |
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| 304 | match classify_cmp t1 t2 with |
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| 305 | [ cmp_case_I32unsi ⇒ |
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| 306 | match v1 with |
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| 307 | [ Vint n1 ⇒ match v2 with |
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| 308 | [ Vint n2 ⇒ Some ? (of_bool (cmpu c n1 n2)) |
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| 309 | | _ ⇒ None ? ] |
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| 310 | | _ ⇒ None ? ] |
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| 311 | | cmp_case_ipip ⇒ |
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| 312 | match v1 with |
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| 313 | [ Vint n1 ⇒ match v2 with |
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| 314 | [ Vint n2 ⇒ Some ? (of_bool (cmp c n1 n2)) |
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[124] | 315 | | Vptr pcl b ofs ⇒ if eq n1 zero then sem_cmp_mismatch c else None ? |
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[3] | 316 | | _ ⇒ None ? |
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| 317 | ] |
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[124] | 318 | | Vptr pcl1 b1 ofs1 ⇒ |
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[3] | 319 | match v2 with |
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[124] | 320 | [ Vptr pcl2 b2 ofs2 ⇒ |
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| 321 | if valid_pointer m pcl1 b1 (signed ofs1) |
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| 322 | ∧ valid_pointer m pcl2 b2 (signed ofs2) then |
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[3] | 323 | if eqZb b1 b2 |
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| 324 | then Some ? (of_bool (cmp c ofs1 ofs2)) |
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| 325 | else sem_cmp_mismatch c |
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| 326 | else None ? |
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| 327 | | Vint n ⇒ |
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| 328 | if eq n zero then sem_cmp_mismatch c else None ? |
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| 329 | | _ ⇒ None ? ] |
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| 330 | | _ ⇒ None ? ] |
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| 331 | | cmp_case_ff ⇒ |
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| 332 | match v1 with |
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| 333 | [ Vfloat f1 ⇒ |
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| 334 | match v2 with |
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| 335 | [ Vfloat f2 ⇒ Some ? (of_bool (Fcmp c f1 f2)) |
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| 336 | | _ ⇒ None ? ] |
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| 337 | | _ ⇒ None ? ] |
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| 338 | | cmp_default ⇒ None ? |
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| 339 | ]. |
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| 340 | |
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| 341 | ndefinition sem_unary_operation |
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| 342 | : unary_operation → val → type → option val ≝ |
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| 343 | λop,v,ty. |
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| 344 | match op with |
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| 345 | [ Onotbool => sem_notbool v ty |
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| 346 | | Onotint => sem_notint v |
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| 347 | | Oneg => sem_neg v ty |
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| 348 | ]. |
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| 349 | |
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| 350 | nlet rec sem_binary_operation |
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| 351 | (op: binary_operation) |
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| 352 | (v1: val) (t1: type) (v2: val) (t2:type) |
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| 353 | (m: mem): option val ≝ |
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| 354 | match op with |
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| 355 | [ Oadd ⇒ sem_add v1 t1 v2 t2 |
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| 356 | | Osub ⇒ sem_sub v1 t1 v2 t2 |
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| 357 | | Omul ⇒ sem_mul v1 t1 v2 t2 |
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| 358 | | Omod ⇒ sem_mod v1 t1 v2 t2 |
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| 359 | | Odiv ⇒ sem_div v1 t1 v2 t2 |
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| 360 | | Oand ⇒ sem_and v1 v2 |
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| 361 | | Oor ⇒ sem_or v1 v2 |
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| 362 | | Oxor ⇒ sem_xor v1 v2 |
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| 363 | | Oshl ⇒ sem_shl v1 v2 |
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| 364 | | Oshr ⇒ sem_shr v1 t1 v2 t2 |
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| 365 | | Oeq ⇒ sem_cmp Ceq v1 t1 v2 t2 m |
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| 366 | | One ⇒ sem_cmp Cne v1 t1 v2 t2 m |
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| 367 | | Olt ⇒ sem_cmp Clt v1 t1 v2 t2 m |
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| 368 | | Ogt ⇒ sem_cmp Cgt v1 t1 v2 t2 m |
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| 369 | | Ole ⇒ sem_cmp Cle v1 t1 v2 t2 m |
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| 370 | | Oge ⇒ sem_cmp Cge v1 t1 v2 t2 m |
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| 371 | ]. |
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| 372 | |
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| 373 | (* * Semantic of casts. [cast v1 t1 t2 v2] holds if value [v1], |
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| 374 | viewed with static type [t1], can be cast to type [t2], |
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| 375 | resulting in value [v2]. *) |
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| 376 | |
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| 377 | nlet rec cast_int_int (sz: intsize) (sg: signedness) (i: int) : int ≝ |
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| 378 | match sz with |
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| 379 | [ I8 ⇒ match sg with [ Signed ⇒ sign_ext 8 i | Unsigned ⇒ zero_ext 8 i ] |
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| 380 | | I16 ⇒ match sg with [ Signed => sign_ext 16 i | Unsigned ⇒ zero_ext 16 i ] |
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| 381 | | I32 ⇒ i |
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| 382 | ]. |
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| 383 | |
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| 384 | nlet rec cast_int_float (si : signedness) (i: int) : float ≝ |
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| 385 | match si with |
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| 386 | [ Signed ⇒ floatofint i |
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| 387 | | Unsigned ⇒ floatofintu i |
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| 388 | ]. |
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| 389 | |
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| 390 | nlet rec cast_float_int (si : signedness) (f: float) : int ≝ |
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| 391 | match si with |
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| 392 | [ Signed ⇒ intoffloat f |
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| 393 | | Unsigned ⇒ intuoffloat f |
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| 394 | ]. |
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| 395 | |
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| 396 | nlet rec cast_float_float (sz: floatsize) (f: float) : float ≝ |
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| 397 | match sz with |
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| 398 | [ F32 ⇒ singleoffloat f |
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| 399 | | F64 ⇒ f |
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| 400 | ]. |
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| 401 | |
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[124] | 402 | (* XXX should go somewhere else? *) |
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| 403 | ndefinition ptr_cls_spc : ptr_class → mem_space ≝ |
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| 404 | λp.match p with |
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| 405 | [ Universal ⇒ Generic |
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| 406 | | Data ⇒ Data |
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| 407 | | IData ⇒ IData |
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| 408 | | XData ⇒ XData |
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| 409 | | Code ⇒ Code |
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| 410 | ]. |
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| 411 | ndefinition ptr_spc_cls : mem_space → ptr_class ≝ |
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| 412 | λp.match p with |
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| 413 | [ Generic ⇒ Universal |
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| 414 | | Data ⇒ Data |
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| 415 | | IData ⇒ IData |
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| 416 | | XData ⇒ XData |
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| 417 | | Code ⇒ Code |
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| 418 | ]. |
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[3] | 419 | |
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[124] | 420 | ndefinition blk_ptr_cls : block_class → ptr_class ≝ |
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| 421 | λb.match b with |
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| 422 | [ UnspecifiedB ⇒ Universal |
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| 423 | | DataB ⇒ Data |
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| 424 | | IDataB ⇒ IData |
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| 425 | | XDataB ⇒ XData |
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| 426 | | CodeB ⇒ Code |
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| 427 | ]. |
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| 428 | |
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| 429 | ninductive type_pointable : type → Prop ≝ |
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| 430 | | type_ptr_int : type_pointable (Tint I32 Unsigned) |
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| 431 | | type_ptr_pointer : ∀s,t. type_pointable (Tpointer s t) |
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| 432 | | type_ptr_array : ∀s,t,n. type_pointable (Tarray s t n) |
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| 433 | | type_ptr_function : ∀tys,ty. type_pointable (Tfunction tys ty). |
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| 434 | |
---|
| 435 | ninductive type_space : type → mem_space → Prop ≝ |
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| 436 | | type_spc_int : type_space (Tint I32 Unsigned) Generic |
---|
| 437 | | type_spc_pointer : ∀s,t. type_space (Tpointer s t) s |
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| 438 | | type_spc_array : ∀s,t,n. type_space (Tarray s t n) s |
---|
| 439 | | type_spc_code : ∀tys,ty. type_space (Tfunction tys ty) Code. |
---|
| 440 | |
---|
| 441 | ninductive cast : mem → val → type → type → val → Prop ≝ |
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| 442 | | cast_ii: ∀m,i,sz2,sz1,si1,si2. (**r int to int *) |
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| 443 | cast m (Vint i) (Tint sz1 si1) (Tint sz2 si2) |
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[3] | 444 | (Vint (cast_int_int sz2 si2 i)) |
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[124] | 445 | | cast_fi: ∀m,f,sz1,sz2,si2. (**r float to int *) |
---|
| 446 | cast m (Vfloat f) (Tfloat sz1) (Tint sz2 si2) |
---|
[3] | 447 | (Vint (cast_int_int sz2 si2 (cast_float_int si2 f))) |
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[124] | 448 | | cast_if: ∀m,i,sz1,sz2,si1. (**r int to float *) |
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| 449 | cast m (Vint i) (Tint sz1 si1) (Tfloat sz2) |
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[3] | 450 | (Vfloat (cast_float_float sz2 (cast_int_float si1 i))) |
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[124] | 451 | | cast_ff: ∀m,f,sz1,sz2. (**r float to float *) |
---|
| 452 | cast m (Vfloat f) (Tfloat sz1) (Tfloat sz2) |
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[3] | 453 | (Vfloat (cast_float_float sz2 f)) |
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[124] | 454 | | cast_pp: ∀m,pcl,psp,ty,ty',b,ofs. |
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| 455 | type_pointable ty → |
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| 456 | type_space ty' psp → |
---|
| 457 | class_compat (blockclass m b) (ptr_spc_cls psp) → |
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| 458 | cast m (Vptr pcl b ofs) ty ty' (Vptr (ptr_spc_cls psp) b ofs) |
---|
| 459 | | cast_pp_z: ∀m,ty,ty'. |
---|
| 460 | type_pointable ty → (* Don't care which space it is for the source type *) |
---|
| 461 | type_pointable ty' → |
---|
| 462 | cast m (Vint zero) ty ty' (Vint zero). |
---|
| 463 | (* XXX: other integers? |
---|
| 464 | | cast_nn_i: ∀m,n,t1,t2. (**r no change in data representation *) |
---|
[3] | 465 | neutral_for_cast t1 → neutral_for_cast t2 → |
---|
[124] | 466 | cast m (Vint n) t1 t2 (Vint n). |
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| 467 | *) |
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[3] | 468 | (* * * Operational semantics *) |
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| 469 | |
---|
| 470 | (* * The semantics uses two environments. The global environment |
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| 471 | maps names of functions and global variables to memory block references, |
---|
| 472 | and function pointers to their definitions. (See module [Globalenvs].) *) |
---|
| 473 | |
---|
| 474 | ndefinition genv ≝ (genv_t Genv) fundef. |
---|
| 475 | |
---|
| 476 | (* * The local environment maps local variables to block references. |
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| 477 | The current value of the variable is stored in the associated memory |
---|
| 478 | block. *) |
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| 479 | |
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[124] | 480 | ndefinition env ≝ (tree_t ? PTree) (block_class × block). (* map variable -> location *) |
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[3] | 481 | |
---|
[13] | 482 | ndefinition empty_env: env ≝ (empty …). |
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[3] | 483 | |
---|
| 484 | (* * [load_value_of_type ty m b ofs] computes the value of a datum |
---|
| 485 | of type [ty] residing in memory [m] at block [b], offset [ofs]. |
---|
| 486 | If the type [ty] indicates an access by value, the corresponding |
---|
| 487 | memory load is performed. If the type [ty] indicates an access by |
---|
| 488 | reference, the pointer [Vptr b ofs] is returned. *) |
---|
| 489 | |
---|
[124] | 490 | nlet rec load_value_of_type (ty: type) (m: mem) (pcl:ptr_class) (b: block) (ofs: int) : option val ≝ |
---|
[3] | 491 | match access_mode ty with |
---|
[124] | 492 | [ By_value chunk ⇒ loadv chunk m (Vptr pcl b ofs) |
---|
| 493 | | By_reference ⇒ Some ? (Vptr pcl b ofs) |
---|
[3] | 494 | | By_nothing ⇒ None ? |
---|
| 495 | ]. |
---|
| 496 | |
---|
| 497 | (* * Symmetrically, [store_value_of_type ty m b ofs v] returns the |
---|
| 498 | memory state after storing the value [v] in the datum |
---|
| 499 | of type [ty] residing in memory [m] at block [b], offset [ofs]. |
---|
| 500 | This is allowed only if [ty] indicates an access by value. *) |
---|
| 501 | |
---|
[124] | 502 | nlet rec store_value_of_type (ty_dest: type) (m: mem) (pcl:ptr_class) (loc: block) (ofs: int) (v: val) : option mem ≝ |
---|
[3] | 503 | match access_mode ty_dest with |
---|
[124] | 504 | [ By_value chunk ⇒ storev chunk m (Vptr pcl loc ofs) v |
---|
[3] | 505 | | By_reference ⇒ None ? |
---|
| 506 | | By_nothing ⇒ None ? |
---|
| 507 | ]. |
---|
| 508 | |
---|
| 509 | (* * Allocation of function-local variables. |
---|
| 510 | [alloc_variables e1 m1 vars e2 m2] allocates one memory block |
---|
| 511 | for each variable declared in [vars], and associates the variable |
---|
| 512 | name with this block. [e1] and [m1] are the initial local environment |
---|
| 513 | and memory state. [e2] and [m2] are the final local environment |
---|
| 514 | and memory state. *) |
---|
| 515 | |
---|
| 516 | ninductive alloc_variables: env → mem → |
---|
| 517 | list (ident × type) → |
---|
| 518 | env → mem → Prop ≝ |
---|
| 519 | | alloc_variables_nil: |
---|
| 520 | ∀e,m. |
---|
| 521 | alloc_variables e m (nil ?) e m |
---|
| 522 | | alloc_variables_cons: |
---|
| 523 | ∀e,m,id,ty,vars,m1,b1,m2,e2. |
---|
[124] | 524 | alloc m 0 (sizeof ty) UnspecifiedB = 〈m1, b1〉 → (* XXX *) |
---|
| 525 | alloc_variables (set … id 〈UnspecifiedB, b1〉 e) m1 vars e2 m2 → |
---|
[3] | 526 | alloc_variables e m (〈id, ty〉 :: vars) e2 m2. |
---|
| 527 | |
---|
| 528 | (* * Initialization of local variables that are parameters to a function. |
---|
| 529 | [bind_parameters e m1 params args m2] stores the values [args] |
---|
| 530 | in the memory blocks corresponding to the variables [params]. |
---|
| 531 | [m1] is the initial memory state and [m2] the final memory state. *) |
---|
| 532 | |
---|
| 533 | ninductive bind_parameters: env → |
---|
| 534 | mem → list (ident × type) → list val → |
---|
| 535 | mem → Prop ≝ |
---|
| 536 | | bind_parameters_nil: |
---|
| 537 | ∀e,m. |
---|
| 538 | bind_parameters e m (nil ?) (nil ?) m |
---|
| 539 | | bind_parameters_cons: |
---|
[124] | 540 | ∀e,m,id,ty,params,v1,vl,bcls,b,m1,m2. |
---|
| 541 | get ??? id e = Some ? 〈bcls,b〉 → |
---|
| 542 | store_value_of_type ty m (blk_ptr_cls bcls) b zero v1 = Some ? m1 → |
---|
[3] | 543 | bind_parameters e m1 params vl m2 → |
---|
| 544 | bind_parameters e m (〈id, ty〉 :: params) (v1 :: vl) m2. |
---|
| 545 | |
---|
| 546 | (* * Return the list of blocks in the codomain of [e]. *) |
---|
| 547 | |
---|
| 548 | ndefinition blocks_of_env : env → list block ≝ λe. |
---|
[124] | 549 | map ?? (λx. snd ?? (snd ?? x)) (elements ??? e). |
---|
[3] | 550 | |
---|
| 551 | (* * Selection of the appropriate case of a [switch], given the value [n] |
---|
| 552 | of the selector expression. *) |
---|
| 553 | |
---|
| 554 | nlet rec select_switch (n: int) (sl: labeled_statements) |
---|
| 555 | on sl : labeled_statements ≝ |
---|
| 556 | match sl with |
---|
| 557 | [ LSdefault _ ⇒ sl |
---|
| 558 | | LScase c s sl' ⇒ if eq c n then sl else select_switch n sl' |
---|
| 559 | ]. |
---|
| 560 | |
---|
| 561 | (* * Turn a labeled statement into a sequence *) |
---|
| 562 | |
---|
| 563 | nlet rec seq_of_labeled_statement (sl: labeled_statements) : statement ≝ |
---|
| 564 | match sl with |
---|
| 565 | [ LSdefault s ⇒ s |
---|
| 566 | | LScase c s sl' ⇒ Ssequence s (seq_of_labeled_statement sl') |
---|
| 567 | ]. |
---|
| 568 | |
---|
| 569 | (* |
---|
| 570 | Section SEMANTICS. |
---|
| 571 | |
---|
| 572 | Variable ge: genv. |
---|
| 573 | |
---|
| 574 | (** ** Evaluation of expressions *) |
---|
| 575 | |
---|
| 576 | Section EXPR. |
---|
| 577 | |
---|
| 578 | Variable e: env. |
---|
| 579 | Variable m: mem. |
---|
| 580 | *) |
---|
| 581 | (* * [eval_expr ge e m a v] defines the evaluation of expression [a] |
---|
| 582 | in r-value position. [v] is the value of the expression. |
---|
| 583 | [e] is the current environment and [m] is the current memory state. *) |
---|
| 584 | |
---|
| 585 | ninductive eval_expr (ge:genv) (e:env) (m:mem) : expr → val → Prop ≝ |
---|
| 586 | | eval_Econst_int: ∀i,ty. |
---|
| 587 | eval_expr ge e m (Expr (Econst_int i) ty) (Vint i) |
---|
| 588 | | eval_Econst_float: ∀f,ty. |
---|
| 589 | eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f) |
---|
[124] | 590 | | eval_Elvalue: ∀a,ty,pcl,loc,ofs,v. |
---|
| 591 | eval_lvalue ge e m (Expr a ty) pcl loc ofs -> |
---|
| 592 | load_value_of_type ty m pcl loc ofs = Some ? v -> |
---|
[3] | 593 | eval_expr ge e m (Expr a ty) v |
---|
[124] | 594 | | eval_Eaddrof: ∀a,ty,pcl,loc,ofs. |
---|
| 595 | eval_lvalue ge e m a pcl loc ofs -> |
---|
| 596 | eval_expr ge e m (Expr (Eaddrof a) ty) (Vptr pcl loc ofs) |
---|
[3] | 597 | | eval_Esizeof: ∀ty',ty. |
---|
| 598 | eval_expr ge e m (Expr (Esizeof ty') ty) (Vint (repr (sizeof ty'))) |
---|
| 599 | | eval_Eunop: ∀op,a,ty,v1,v. |
---|
| 600 | eval_expr ge e m a v1 -> |
---|
| 601 | sem_unary_operation op v1 (typeof a) = Some ? v -> |
---|
| 602 | eval_expr ge e m (Expr (Eunop op a) ty) v |
---|
| 603 | | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v. |
---|
| 604 | eval_expr ge e m a1 v1 -> |
---|
| 605 | eval_expr ge e m a2 v2 -> |
---|
| 606 | sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some ? v -> |
---|
| 607 | eval_expr ge e m (Expr (Ebinop op a1 a2) ty) v |
---|
| 608 | | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2. |
---|
| 609 | eval_expr ge e m a1 v1 -> |
---|
| 610 | is_true v1 (typeof a1) -> |
---|
| 611 | eval_expr ge e m a2 v2 -> |
---|
| 612 | eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2 |
---|
| 613 | | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3. |
---|
| 614 | eval_expr ge e m a1 v1 -> |
---|
| 615 | is_false v1 (typeof a1) -> |
---|
| 616 | eval_expr ge e m a3 v3 -> |
---|
| 617 | eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3 |
---|
| 618 | | eval_Eorbool_1: ∀a1,a2,ty,v1. |
---|
| 619 | eval_expr ge e m a1 v1 -> |
---|
| 620 | is_true v1 (typeof a1) -> |
---|
| 621 | eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue |
---|
| 622 | | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v. |
---|
| 623 | eval_expr ge e m a1 v1 -> |
---|
| 624 | is_false v1 (typeof a1) -> |
---|
| 625 | eval_expr ge e m a2 v2 -> |
---|
| 626 | bool_of_val v2 (typeof a2) v -> |
---|
| 627 | eval_expr ge e m (Expr (Eorbool a1 a2) ty) v |
---|
| 628 | | eval_Eandbool_1: ∀a1,a2,ty,v1. |
---|
| 629 | eval_expr ge e m a1 v1 -> |
---|
| 630 | is_false v1 (typeof a1) -> |
---|
| 631 | eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse |
---|
| 632 | | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v. |
---|
| 633 | eval_expr ge e m a1 v1 -> |
---|
| 634 | is_true v1 (typeof a1) -> |
---|
| 635 | eval_expr ge e m a2 v2 -> |
---|
| 636 | bool_of_val v2 (typeof a2) v -> |
---|
| 637 | eval_expr ge e m (Expr (Eandbool a1 a2) ty) v |
---|
| 638 | | eval_Ecast: ∀a,ty,ty',v1,v. |
---|
| 639 | eval_expr ge e m a v1 -> |
---|
[124] | 640 | cast m v1 (typeof a) ty v -> |
---|
[3] | 641 | eval_expr ge e m (Expr (Ecast ty a) ty') v |
---|
| 642 | |
---|
| 643 | (* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a] |
---|
| 644 | in l-value position. The result is the memory location [b, ofs] |
---|
| 645 | that contains the value of the expression [a]. *) |
---|
| 646 | |
---|
[124] | 647 | with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → ptr_class → block -> int -> Prop ≝ |
---|
| 648 | | eval_Evar_local: ∀id,bcl,l,ty. |
---|
| 649 | (* XXX notation? e!id*) get ??? id e = Some ? 〈bcl,l〉 -> |
---|
| 650 | eval_lvalue ge e m (Expr (Evar id) ty) (blk_ptr_cls bcl) l zero |
---|
[3] | 651 | | eval_Evar_global: ∀id,l,ty. |
---|
[13] | 652 | (* XXX e!id *) get ??? id e = None ? -> |
---|
| 653 | find_symbol ?? ge id = Some ? l -> |
---|
[124] | 654 | eval_lvalue ge e m (Expr (Evar id) ty) Universal l zero (* XXX *) |
---|
| 655 | | eval_Ederef: ∀a,ty,pcl,l,ofs. |
---|
| 656 | eval_expr ge e m a (Vptr pcl l ofs) -> |
---|
| 657 | eval_lvalue ge e m (Expr (Ederef a) ty) pcl l ofs |
---|
| 658 | | eval_Efield_struct: ∀a,i,ty,pcl,l,ofs,id,fList,delta. |
---|
| 659 | eval_lvalue ge e m a pcl l ofs -> |
---|
[3] | 660 | typeof a = Tstruct id fList -> |
---|
| 661 | field_offset i fList = OK ? delta -> |
---|
[124] | 662 | eval_lvalue ge e m (Expr (Efield a i) ty) pcl l (add ofs (repr delta)) |
---|
| 663 | | eval_Efield_union: ∀a,i,ty,pcl,l,ofs,id,fList. |
---|
| 664 | eval_lvalue ge e m a pcl l ofs -> |
---|
[3] | 665 | typeof a = Tunion id fList -> |
---|
[124] | 666 | eval_lvalue ge e m (Expr (Efield a i) ty) pcl l ofs. |
---|
[3] | 667 | |
---|
| 668 | (* |
---|
| 669 | Scheme eval_expr_ind2 := Minimality for eval_expr Sort Prop |
---|
| 670 | with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop. |
---|
| 671 | *) |
---|
| 672 | |
---|
| 673 | (* * [eval_exprlist ge e m al vl] evaluates a list of r-value |
---|
| 674 | expressions [al] to their values [vl]. *) |
---|
| 675 | |
---|
| 676 | ninductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr -> list val -> Prop := |
---|
| 677 | | eval_Enil: |
---|
| 678 | eval_exprlist ge e m (nil ?) (nil ?) |
---|
| 679 | | eval_Econs: ∀a,bl,v,vl. |
---|
| 680 | eval_expr ge e m a v -> |
---|
| 681 | eval_exprlist ge e m bl vl -> |
---|
| 682 | eval_exprlist ge e m (a :: bl) (v :: vl). |
---|
| 683 | |
---|
| 684 | (*End EXPR.*) |
---|
| 685 | |
---|
| 686 | (* * ** Transition semantics for statements and functions *) |
---|
| 687 | |
---|
| 688 | (* * Continuations *) |
---|
| 689 | |
---|
| 690 | ninductive cont: Type := |
---|
| 691 | | Kstop: cont |
---|
| 692 | | Kseq: statement -> cont -> cont |
---|
| 693 | (**r [Kseq s2 k] = after [s1] in [s1;s2] *) |
---|
| 694 | | Kwhile: expr -> statement -> cont -> cont |
---|
| 695 | (**r [Kwhile e s k] = after [s] in [while (e) s] *) |
---|
| 696 | | Kdowhile: expr -> statement -> cont -> cont |
---|
| 697 | (**r [Kdowhile e s k] = after [s] in [do s while (e)] *) |
---|
| 698 | | Kfor2: expr -> statement -> statement -> cont -> cont |
---|
| 699 | (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *) |
---|
| 700 | | Kfor3: expr -> statement -> statement -> cont -> cont |
---|
| 701 | (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *) |
---|
| 702 | | Kswitch: cont -> cont |
---|
| 703 | (**r catches [break] statements arising out of [switch] *) |
---|
[124] | 704 | | Kcall: option (ptr_class × block × int × type) -> (**r where to store result *) |
---|
[3] | 705 | function -> (**r calling function *) |
---|
| 706 | env -> (**r local env of calling function *) |
---|
| 707 | cont -> cont. |
---|
| 708 | |
---|
| 709 | (* * Pop continuation until a call or stop *) |
---|
| 710 | |
---|
| 711 | nlet rec call_cont (k: cont) : cont := |
---|
| 712 | match k with |
---|
| 713 | [ Kseq s k => call_cont k |
---|
| 714 | | Kwhile e s k => call_cont k |
---|
| 715 | | Kdowhile e s k => call_cont k |
---|
| 716 | | Kfor2 e2 e3 s k => call_cont k |
---|
| 717 | | Kfor3 e2 e3 s k => call_cont k |
---|
| 718 | | Kswitch k => call_cont k |
---|
| 719 | | _ => k |
---|
| 720 | ]. |
---|
| 721 | |
---|
| 722 | ndefinition is_call_cont : cont → Prop ≝ λk. |
---|
| 723 | match k with |
---|
| 724 | [ Kstop => True |
---|
| 725 | | Kcall _ _ _ _ => True |
---|
| 726 | | _ => False |
---|
| 727 | ]. |
---|
| 728 | |
---|
| 729 | (* * States *) |
---|
| 730 | |
---|
| 731 | ninductive state: Type := |
---|
| 732 | | State: |
---|
| 733 | ∀f: function. |
---|
| 734 | ∀s: statement. |
---|
| 735 | ∀k: cont. |
---|
| 736 | ∀e: env. |
---|
| 737 | ∀m: mem. state |
---|
| 738 | | Callstate: |
---|
| 739 | ∀fd: fundef. |
---|
| 740 | ∀args: list val. |
---|
| 741 | ∀k: cont. |
---|
| 742 | ∀m: mem. state |
---|
| 743 | | Returnstate: |
---|
| 744 | ∀res: val. |
---|
| 745 | ∀k: cont. |
---|
| 746 | ∀m: mem. state. |
---|
| 747 | |
---|
| 748 | (* * Find the statement and manufacture the continuation |
---|
| 749 | corresponding to a label *) |
---|
| 750 | |
---|
| 751 | nlet rec find_label (lbl: label) (s: statement) (k: cont) |
---|
| 752 | on s: option (statement × cont) := |
---|
| 753 | match s with |
---|
| 754 | [ Ssequence s1 s2 => |
---|
| 755 | match find_label lbl s1 (Kseq s2 k) with |
---|
| 756 | [ Some sk => Some ? sk |
---|
| 757 | | None => find_label lbl s2 k |
---|
| 758 | ] |
---|
| 759 | | Sifthenelse a s1 s2 => |
---|
| 760 | match find_label lbl s1 k with |
---|
| 761 | [ Some sk => Some ? sk |
---|
| 762 | | None => find_label lbl s2 k |
---|
| 763 | ] |
---|
| 764 | | Swhile a s1 => |
---|
| 765 | find_label lbl s1 (Kwhile a s1 k) |
---|
| 766 | | Sdowhile a s1 => |
---|
| 767 | find_label lbl s1 (Kdowhile a s1 k) |
---|
| 768 | | Sfor a1 a2 a3 s1 => |
---|
| 769 | match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with |
---|
| 770 | [ Some sk => Some ? sk |
---|
| 771 | | None => |
---|
| 772 | match find_label lbl s1 (Kfor2 a2 a3 s1 k) with |
---|
| 773 | [ Some sk => Some ? sk |
---|
| 774 | | None => find_label lbl a3 (Kfor3 a2 a3 s1 k) |
---|
| 775 | ] |
---|
| 776 | ] |
---|
| 777 | | Sswitch e sl => |
---|
| 778 | find_label_ls lbl sl (Kswitch k) |
---|
| 779 | | Slabel lbl' s' => |
---|
| 780 | match ident_eq lbl lbl' with |
---|
| 781 | [ inl _ ⇒ Some ? 〈s', k〉 |
---|
| 782 | | inr _ ⇒ find_label lbl s' k |
---|
| 783 | ] |
---|
| 784 | | _ => None ? |
---|
| 785 | ] |
---|
| 786 | |
---|
| 787 | and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont) |
---|
| 788 | on sl: option (statement × cont) := |
---|
| 789 | match sl with |
---|
| 790 | [ LSdefault s => find_label lbl s k |
---|
| 791 | | LScase _ s sl' => |
---|
| 792 | match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with |
---|
| 793 | [ Some sk => Some ? sk |
---|
| 794 | | None => find_label_ls lbl sl' k |
---|
| 795 | ] |
---|
| 796 | ]. |
---|
| 797 | |
---|
| 798 | (* * Transition relation *) |
---|
| 799 | |
---|
| 800 | ninductive step (ge:genv) : state -> trace -> state -> Prop := |
---|
| 801 | |
---|
[124] | 802 | | step_assign: ∀f,a1,a2,k,e,m,pcl,loc,ofs,v2,m'. |
---|
| 803 | eval_lvalue ge e m a1 pcl loc ofs -> |
---|
[3] | 804 | eval_expr ge e m a2 v2 -> |
---|
[124] | 805 | store_value_of_type (typeof a1) m pcl loc ofs v2 = Some ? m' -> |
---|
[3] | 806 | step ge (State f (Sassign a1 a2) k e m) |
---|
| 807 | E0 (State f Sskip k e m') |
---|
| 808 | |
---|
| 809 | | step_call_none: ∀f,a,al,k,e,m,vf,vargs,fd. |
---|
| 810 | eval_expr ge e m a vf -> |
---|
| 811 | eval_exprlist ge e m al vargs -> |
---|
[13] | 812 | find_funct ?? ge vf = Some ? fd -> |
---|
[3] | 813 | type_of_fundef fd = typeof a -> |
---|
| 814 | step ge (State f (Scall (None ?) a al) k e m) |
---|
| 815 | E0 (Callstate fd vargs (Kcall (None ?) f e k) m) |
---|
| 816 | |
---|
[124] | 817 | | step_call_some: ∀f,lhs,a,al,k,e,m,pcl,loc,ofs,vf,vargs,fd. |
---|
| 818 | eval_lvalue ge e m lhs pcl loc ofs -> |
---|
[3] | 819 | eval_expr ge e m a vf -> |
---|
| 820 | eval_exprlist ge e m al vargs -> |
---|
[13] | 821 | find_funct ?? ge vf = Some ? fd -> |
---|
[3] | 822 | type_of_fundef fd = typeof a -> |
---|
| 823 | step ge (State f (Scall (Some ? lhs) a al) k e m) |
---|
[124] | 824 | E0 (Callstate fd vargs (Kcall (Some ? 〈〈〈pcl, loc〉, ofs〉, typeof lhs〉) f e k) m) |
---|
[3] | 825 | |
---|
| 826 | | step_seq: ∀f,s1,s2,k,e,m. |
---|
| 827 | step ge (State f (Ssequence s1 s2) k e m) |
---|
| 828 | E0 (State f s1 (Kseq s2 k) e m) |
---|
| 829 | | step_skip_seq: ∀f,s,k,e,m. |
---|
| 830 | step ge (State f Sskip (Kseq s k) e m) |
---|
| 831 | E0 (State f s k e m) |
---|
| 832 | | step_continue_seq: ∀f,s,k,e,m. |
---|
| 833 | step ge (State f Scontinue (Kseq s k) e m) |
---|
| 834 | E0 (State f Scontinue k e m) |
---|
| 835 | | step_break_seq: ∀f,s,k,e,m. |
---|
| 836 | step ge (State f Sbreak (Kseq s k) e m) |
---|
| 837 | E0 (State f Sbreak k e m) |
---|
| 838 | |
---|
| 839 | | step_ifthenelse_true: ∀f,a,s1,s2,k,e,m,v1. |
---|
| 840 | eval_expr ge e m a v1 -> |
---|
| 841 | is_true v1 (typeof a) -> |
---|
| 842 | step ge (State f (Sifthenelse a s1 s2) k e m) |
---|
| 843 | E0 (State f s1 k e m) |
---|
| 844 | | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1. |
---|
| 845 | eval_expr ge e m a v1 -> |
---|
| 846 | is_false v1 (typeof a) -> |
---|
| 847 | step ge (State f (Sifthenelse a s1 s2) k e m) |
---|
| 848 | E0 (State f s2 k e m) |
---|
| 849 | |
---|
| 850 | | step_while_false: ∀f,a,s,k,e,m,v. |
---|
| 851 | eval_expr ge e m a v -> |
---|
| 852 | is_false v (typeof a) -> |
---|
| 853 | step ge (State f (Swhile a s) k e m) |
---|
| 854 | E0 (State f Sskip k e m) |
---|
| 855 | | step_while_true: ∀f,a,s,k,e,m,v. |
---|
| 856 | eval_expr ge e m a v -> |
---|
| 857 | is_true v (typeof a) -> |
---|
| 858 | step ge (State f (Swhile a s) k e m) |
---|
| 859 | E0 (State f s (Kwhile a s k) e m) |
---|
| 860 | | step_skip_or_continue_while: ∀f,x,a,s,k,e,m. |
---|
| 861 | x = Sskip ∨ x = Scontinue -> |
---|
| 862 | step ge (State f x (Kwhile a s k) e m) |
---|
| 863 | E0 (State f (Swhile a s) k e m) |
---|
| 864 | | step_break_while: ∀f,a,s,k,e,m. |
---|
| 865 | step ge (State f Sbreak (Kwhile a s k) e m) |
---|
| 866 | E0 (State f Sskip k e m) |
---|
| 867 | |
---|
| 868 | | step_dowhile: ∀f,a,s,k,e,m. |
---|
| 869 | step ge (State f (Sdowhile a s) k e m) |
---|
| 870 | E0 (State f s (Kdowhile a s k) e m) |
---|
| 871 | | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v. |
---|
| 872 | x = Sskip ∨ x = Scontinue -> |
---|
| 873 | eval_expr ge e m a v -> |
---|
| 874 | is_false v (typeof a) -> |
---|
| 875 | step ge (State f x (Kdowhile a s k) e m) |
---|
| 876 | E0 (State f Sskip k e m) |
---|
| 877 | | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v. |
---|
| 878 | x = Sskip ∨ x = Scontinue -> |
---|
| 879 | eval_expr ge e m a v -> |
---|
| 880 | is_true v (typeof a) -> |
---|
| 881 | step ge (State f x (Kdowhile a s k) e m) |
---|
| 882 | E0 (State f (Sdowhile a s) k e m) |
---|
| 883 | | step_break_dowhile: ∀f,a,s,k,e,m. |
---|
| 884 | step ge (State f Sbreak (Kdowhile a s k) e m) |
---|
| 885 | E0 (State f Sskip k e m) |
---|
| 886 | |
---|
| 887 | | step_for_start: ∀f,a1,a2,a3,s,k,e,m. |
---|
| 888 | a1 ≠ Sskip -> |
---|
| 889 | step ge (State f (Sfor a1 a2 a3 s) k e m) |
---|
| 890 | E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m) |
---|
| 891 | | step_for_false: ∀f,a2,a3,s,k,e,m,v. |
---|
| 892 | eval_expr ge e m a2 v -> |
---|
| 893 | is_false v (typeof a2) -> |
---|
| 894 | step ge (State f (Sfor Sskip a2 a3 s) k e m) |
---|
| 895 | E0 (State f Sskip k e m) |
---|
| 896 | | step_for_true: ∀f,a2,a3,s,k,e,m,v. |
---|
| 897 | eval_expr ge e m a2 v -> |
---|
| 898 | is_true v (typeof a2) -> |
---|
| 899 | step ge (State f (Sfor Sskip a2 a3 s) k e m) |
---|
| 900 | E0 (State f s (Kfor2 a2 a3 s k) e m) |
---|
| 901 | | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m. |
---|
| 902 | x = Sskip ∨ x = Scontinue -> |
---|
| 903 | step ge (State f x (Kfor2 a2 a3 s k) e m) |
---|
| 904 | E0 (State f a3 (Kfor3 a2 a3 s k) e m) |
---|
| 905 | | step_break_for2: ∀f,a2,a3,s,k,e,m. |
---|
| 906 | step ge (State f Sbreak (Kfor2 a2 a3 s k) e m) |
---|
| 907 | E0 (State f Sskip k e m) |
---|
| 908 | | step_skip_for3: ∀f,a2,a3,s,k,e,m. |
---|
| 909 | step ge (State f Sskip (Kfor3 a2 a3 s k) e m) |
---|
| 910 | E0 (State f (Sfor Sskip a2 a3 s) k e m) |
---|
| 911 | |
---|
| 912 | | step_return_0: ∀f,k,e,m. |
---|
| 913 | fn_return f = Tvoid -> |
---|
| 914 | step ge (State f (Sreturn (None ?)) k e m) |
---|
| 915 | E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e))) |
---|
| 916 | | step_return_1: ∀f,a,k,e,m,v. |
---|
| 917 | fn_return f ≠ Tvoid -> |
---|
| 918 | eval_expr ge e m a v -> |
---|
| 919 | step ge (State f (Sreturn (Some ? a)) k e m) |
---|
| 920 | E0 (Returnstate v (call_cont k) (free_list m (blocks_of_env e))) |
---|
| 921 | | step_skip_call: ∀f,k,e,m. |
---|
| 922 | is_call_cont k -> |
---|
| 923 | fn_return f = Tvoid -> |
---|
| 924 | step ge (State f Sskip k e m) |
---|
| 925 | E0 (Returnstate Vundef k (free_list m (blocks_of_env e))) |
---|
| 926 | |
---|
| 927 | | step_switch: ∀f,a,sl,k,e,m,n. |
---|
| 928 | eval_expr ge e m a (Vint n) -> |
---|
| 929 | step ge (State f (Sswitch a sl) k e m) |
---|
| 930 | E0 (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m) |
---|
| 931 | | step_skip_break_switch: ∀f,x,k,e,m. |
---|
| 932 | x = Sskip ∨ x = Sbreak -> |
---|
| 933 | step ge (State f x (Kswitch k) e m) |
---|
| 934 | E0 (State f Sskip k e m) |
---|
| 935 | | step_continue_switch: ∀f,k,e,m. |
---|
| 936 | step ge (State f Scontinue (Kswitch k) e m) |
---|
| 937 | E0 (State f Scontinue k e m) |
---|
| 938 | |
---|
| 939 | | step_label: ∀f,lbl,s,k,e,m. |
---|
| 940 | step ge (State f (Slabel lbl s) k e m) |
---|
| 941 | E0 (State f s k e m) |
---|
| 942 | |
---|
| 943 | | step_goto: ∀f,lbl,k,e,m,s',k'. |
---|
| 944 | find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 -> |
---|
| 945 | step ge (State f (Sgoto lbl) k e m) |
---|
| 946 | E0 (State f s' k' e m) |
---|
| 947 | |
---|
| 948 | | step_internal_function: ∀f,vargs,k,m,e,m1,m2. |
---|
| 949 | alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 -> |
---|
| 950 | bind_parameters e m1 (fn_params f) vargs m2 -> |
---|
| 951 | step ge (Callstate (Internal f) vargs k m) |
---|
| 952 | E0 (State f (fn_body f) k e m2) |
---|
| 953 | |
---|
| 954 | | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t. |
---|
| 955 | event_match (external_function id targs tres) vargs t vres -> |
---|
| 956 | step ge (Callstate (External id targs tres) vargs k m) |
---|
| 957 | t (Returnstate vres k m) |
---|
| 958 | |
---|
| 959 | | step_returnstate_0: ∀v,f,e,k,m. |
---|
| 960 | step ge (Returnstate v (Kcall (None ?) f e k) m) |
---|
| 961 | E0 (State f Sskip k e m) |
---|
| 962 | |
---|
[124] | 963 | | step_returnstate_1: ∀v,f,e,k,m,m',pcl,loc,ofs,ty. |
---|
| 964 | store_value_of_type ty m pcl loc ofs v = Some ? m' -> |
---|
| 965 | step ge (Returnstate v (Kcall (Some ? 〈〈〈pcl,loc〉, ofs〉, ty〉) f e k) m) |
---|
[3] | 966 | E0 (State f Sskip k e m'). |
---|
| 967 | (* |
---|
| 968 | (** * Alternate big-step semantics *) |
---|
| 969 | |
---|
| 970 | (** ** Big-step semantics for terminating statements and functions *) |
---|
| 971 | |
---|
| 972 | (** The execution of a statement produces an ``outcome'', indicating |
---|
| 973 | how the execution terminated: either normally or prematurely |
---|
| 974 | through the execution of a [break], [continue] or [return] statement. *) |
---|
| 975 | |
---|
| 976 | ninductive outcome: Type := |
---|
| 977 | | Out_break: outcome (**r terminated by [break] *) |
---|
| 978 | | Out_continue: outcome (**r terminated by [continue] *) |
---|
| 979 | | Out_normal: outcome (**r terminated normally *) |
---|
| 980 | | Out_return: option val -> outcome. (**r terminated by [return] *) |
---|
| 981 | |
---|
| 982 | ninductive out_normal_or_continue : outcome -> Prop := |
---|
| 983 | | Out_normal_or_continue_N: out_normal_or_continue Out_normal |
---|
| 984 | | Out_normal_or_continue_C: out_normal_or_continue Out_continue. |
---|
| 985 | |
---|
| 986 | ninductive out_break_or_return : outcome -> outcome -> Prop := |
---|
| 987 | | Out_break_or_return_B: out_break_or_return Out_break Out_normal |
---|
| 988 | | Out_break_or_return_R: ∀ov. |
---|
| 989 | out_break_or_return (Out_return ov) (Out_return ov). |
---|
| 990 | |
---|
| 991 | Definition outcome_switch (out: outcome) : outcome := |
---|
| 992 | match out with |
---|
| 993 | | Out_break => Out_normal |
---|
| 994 | | o => o |
---|
| 995 | end. |
---|
| 996 | |
---|
| 997 | Definition outcome_result_value (out: outcome) (t: type) (v: val) : Prop := |
---|
| 998 | match out, t with |
---|
| 999 | | Out_normal, Tvoid => v = Vundef |
---|
| 1000 | | Out_return None, Tvoid => v = Vundef |
---|
| 1001 | | Out_return (Some v'), ty => ty <> Tvoid /\ v'=v |
---|
| 1002 | | _, _ => False |
---|
| 1003 | end. |
---|
| 1004 | |
---|
| 1005 | (** [exec_stmt ge e m1 s t m2 out] describes the execution of |
---|
| 1006 | the statement [s]. [out] is the outcome for this execution. |
---|
| 1007 | [m1] is the initial memory state, [m2] the final memory state. |
---|
| 1008 | [t] is the trace of input/output events performed during this |
---|
| 1009 | evaluation. *) |
---|
| 1010 | |
---|
| 1011 | ninductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop := |
---|
| 1012 | | exec_Sskip: ∀e,m. |
---|
| 1013 | exec_stmt e m Sskip |
---|
| 1014 | E0 m Out_normal |
---|
| 1015 | | exec_Sassign: ∀e,m,a1,a2,loc,ofs,v2,m'. |
---|
| 1016 | eval_lvalue e m a1 loc ofs -> |
---|
| 1017 | eval_expr e m a2 v2 -> |
---|
| 1018 | store_value_of_type (typeof a1) m loc ofs v2 = Some m' -> |
---|
| 1019 | exec_stmt e m (Sassign a1 a2) |
---|
| 1020 | E0 m' Out_normal |
---|
| 1021 | | exec_Scall_none: ∀e,m,a,al,vf,vargs,f,t,m',vres. |
---|
| 1022 | eval_expr e m a vf -> |
---|
| 1023 | eval_exprlist e m al vargs -> |
---|
| 1024 | Genv.find_funct ge vf = Some f -> |
---|
| 1025 | type_of_fundef f = typeof a -> |
---|
| 1026 | eval_funcall m f vargs t m' vres -> |
---|
| 1027 | exec_stmt e m (Scall None a al) |
---|
| 1028 | t m' Out_normal |
---|
| 1029 | | exec_Scall_some: ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t,m',vres,m''. |
---|
| 1030 | eval_lvalue e m lhs loc ofs -> |
---|
| 1031 | eval_expr e m a vf -> |
---|
| 1032 | eval_exprlist e m al vargs -> |
---|
| 1033 | Genv.find_funct ge vf = Some f -> |
---|
| 1034 | type_of_fundef f = typeof a -> |
---|
| 1035 | eval_funcall m f vargs t m' vres -> |
---|
| 1036 | store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' -> |
---|
| 1037 | exec_stmt e m (Scall (Some lhs) a al) |
---|
| 1038 | t m'' Out_normal |
---|
| 1039 | | exec_Sseq_1: ∀e,m,s1,s2,t1,m1,t2,m2,out. |
---|
| 1040 | exec_stmt e m s1 t1 m1 Out_normal -> |
---|
| 1041 | exec_stmt e m1 s2 t2 m2 out -> |
---|
| 1042 | exec_stmt e m (Ssequence s1 s2) |
---|
| 1043 | (t1 ** t2) m2 out |
---|
| 1044 | | exec_Sseq_2: ∀e,m,s1,s2,t1,m1,out. |
---|
| 1045 | exec_stmt e m s1 t1 m1 out -> |
---|
| 1046 | out <> Out_normal -> |
---|
| 1047 | exec_stmt e m (Ssequence s1 s2) |
---|
| 1048 | t1 m1 out |
---|
| 1049 | | exec_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t,m',out. |
---|
| 1050 | eval_expr e m a v1 -> |
---|
| 1051 | is_true v1 (typeof a) -> |
---|
| 1052 | exec_stmt e m s1 t m' out -> |
---|
| 1053 | exec_stmt e m (Sifthenelse a s1 s2) |
---|
| 1054 | t m' out |
---|
| 1055 | | exec_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t,m',out. |
---|
| 1056 | eval_expr e m a v1 -> |
---|
| 1057 | is_false v1 (typeof a) -> |
---|
| 1058 | exec_stmt e m s2 t m' out -> |
---|
| 1059 | exec_stmt e m (Sifthenelse a s1 s2) |
---|
| 1060 | t m' out |
---|
| 1061 | | exec_Sreturn_none: ∀e,m. |
---|
| 1062 | exec_stmt e m (Sreturn None) |
---|
| 1063 | E0 m (Out_return None) |
---|
| 1064 | | exec_Sreturn_some: ∀e,m,a,v. |
---|
| 1065 | eval_expr e m a v -> |
---|
| 1066 | exec_stmt e m (Sreturn (Some a)) |
---|
| 1067 | E0 m (Out_return (Some v)) |
---|
| 1068 | | exec_Sbreak: ∀e,m. |
---|
| 1069 | exec_stmt e m Sbreak |
---|
| 1070 | E0 m Out_break |
---|
| 1071 | | exec_Scontinue: ∀e,m. |
---|
| 1072 | exec_stmt e m Scontinue |
---|
| 1073 | E0 m Out_continue |
---|
| 1074 | | exec_Swhile_false: ∀e,m,a,s,v. |
---|
| 1075 | eval_expr e m a v -> |
---|
| 1076 | is_false v (typeof a) -> |
---|
| 1077 | exec_stmt e m (Swhile a s) |
---|
| 1078 | E0 m Out_normal |
---|
| 1079 | | exec_Swhile_stop: ∀e,m,a,v,s,t,m',out',out. |
---|
| 1080 | eval_expr e m a v -> |
---|
| 1081 | is_true v (typeof a) -> |
---|
| 1082 | exec_stmt e m s t m' out' -> |
---|
| 1083 | out_break_or_return out' out -> |
---|
| 1084 | exec_stmt e m (Swhile a s) |
---|
| 1085 | t m' out |
---|
| 1086 | | exec_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2,m2,out. |
---|
| 1087 | eval_expr e m a v -> |
---|
| 1088 | is_true v (typeof a) -> |
---|
| 1089 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1090 | out_normal_or_continue out1 -> |
---|
| 1091 | exec_stmt e m1 (Swhile a s) t2 m2 out -> |
---|
| 1092 | exec_stmt e m (Swhile a s) |
---|
| 1093 | (t1 ** t2) m2 out |
---|
| 1094 | | exec_Sdowhile_false: ∀e,m,s,a,t,m1,out1,v. |
---|
| 1095 | exec_stmt e m s t m1 out1 -> |
---|
| 1096 | out_normal_or_continue out1 -> |
---|
| 1097 | eval_expr e m1 a v -> |
---|
| 1098 | is_false v (typeof a) -> |
---|
| 1099 | exec_stmt e m (Sdowhile a s) |
---|
| 1100 | t m1 Out_normal |
---|
| 1101 | | exec_Sdowhile_stop: ∀e,m,s,a,t,m1,out1,out. |
---|
| 1102 | exec_stmt e m s t m1 out1 -> |
---|
| 1103 | out_break_or_return out1 out -> |
---|
| 1104 | exec_stmt e m (Sdowhile a s) |
---|
| 1105 | t m1 out |
---|
| 1106 | | exec_Sdowhile_loop: ∀e,m,s,a,m1,m2,t1,t2,out,out1,v. |
---|
| 1107 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1108 | out_normal_or_continue out1 -> |
---|
| 1109 | eval_expr e m1 a v -> |
---|
| 1110 | is_true v (typeof a) -> |
---|
| 1111 | exec_stmt e m1 (Sdowhile a s) t2 m2 out -> |
---|
| 1112 | exec_stmt e m (Sdowhile a s) |
---|
| 1113 | (t1 ** t2) m2 out |
---|
| 1114 | | exec_Sfor_start: ∀e,m,s,a1,a2,a3,out,m1,m2,t1,t2. |
---|
| 1115 | a1 <> Sskip -> |
---|
| 1116 | exec_stmt e m a1 t1 m1 Out_normal -> |
---|
| 1117 | exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out -> |
---|
| 1118 | exec_stmt e m (Sfor a1 a2 a3 s) |
---|
| 1119 | (t1 ** t2) m2 out |
---|
| 1120 | | exec_Sfor_false: ∀e,m,s,a2,a3,v. |
---|
| 1121 | eval_expr e m a2 v -> |
---|
| 1122 | is_false v (typeof a2) -> |
---|
| 1123 | exec_stmt e m (Sfor Sskip a2 a3 s) |
---|
| 1124 | E0 m Out_normal |
---|
| 1125 | | exec_Sfor_stop: ∀e,m,s,a2,a3,v,m1,t,out1,out. |
---|
| 1126 | eval_expr e m a2 v -> |
---|
| 1127 | is_true v (typeof a2) -> |
---|
| 1128 | exec_stmt e m s t m1 out1 -> |
---|
| 1129 | out_break_or_return out1 out -> |
---|
| 1130 | exec_stmt e m (Sfor Sskip a2 a3 s) |
---|
| 1131 | t m1 out |
---|
| 1132 | | exec_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,m3,t1,t2,t3,out1,out. |
---|
| 1133 | eval_expr e m a2 v -> |
---|
| 1134 | is_true v (typeof a2) -> |
---|
| 1135 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1136 | out_normal_or_continue out1 -> |
---|
| 1137 | exec_stmt e m1 a3 t2 m2 Out_normal -> |
---|
| 1138 | exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out -> |
---|
| 1139 | exec_stmt e m (Sfor Sskip a2 a3 s) |
---|
| 1140 | (t1 ** t2 ** t3) m3 out |
---|
| 1141 | | exec_Sswitch: ∀e,m,a,t,n,sl,m1,out. |
---|
| 1142 | eval_expr e m a (Vint n) -> |
---|
| 1143 | exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out -> |
---|
| 1144 | exec_stmt e m (Sswitch a sl) |
---|
| 1145 | t m1 (outcome_switch out) |
---|
| 1146 | |
---|
| 1147 | (** [eval_funcall m1 fd args t m2 res] describes the invocation of |
---|
| 1148 | function [fd] with arguments [args]. [res] is the value returned |
---|
| 1149 | by the call. *) |
---|
| 1150 | |
---|
| 1151 | with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop := |
---|
| 1152 | | eval_funcall_internal: ∀m,f,vargs,t,e,m1,m2,m3,out,vres. |
---|
| 1153 | alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 -> |
---|
| 1154 | bind_parameters e m1 f.(fn_params) vargs m2 -> |
---|
| 1155 | exec_stmt e m2 f.(fn_body) t m3 out -> |
---|
| 1156 | outcome_result_value out f.(fn_return) vres -> |
---|
| 1157 | eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres |
---|
| 1158 | | eval_funcall_external: ∀m,id,targs,tres,vargs,t,vres. |
---|
| 1159 | event_match (external_function id targs tres) vargs t vres -> |
---|
| 1160 | eval_funcall m (External id targs tres) vargs t m vres. |
---|
| 1161 | |
---|
| 1162 | Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop |
---|
| 1163 | with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop. |
---|
| 1164 | |
---|
| 1165 | (** ** Big-step semantics for diverging statements and functions *) |
---|
| 1166 | |
---|
| 1167 | (** Coinductive semantics for divergence. |
---|
| 1168 | [execinf_stmt ge e m s t] holds if the execution of statement [s] |
---|
| 1169 | diverges, i.e. loops infinitely. [t] is the possibly infinite |
---|
| 1170 | trace of observable events performed during the execution. *) |
---|
| 1171 | |
---|
| 1172 | Coninductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop := |
---|
| 1173 | | execinf_Scall_none: ∀e,m,a,al,vf,vargs,f,t. |
---|
| 1174 | eval_expr e m a vf -> |
---|
| 1175 | eval_exprlist e m al vargs -> |
---|
| 1176 | Genv.find_funct ge vf = Some f -> |
---|
| 1177 | type_of_fundef f = typeof a -> |
---|
| 1178 | evalinf_funcall m f vargs t -> |
---|
| 1179 | execinf_stmt e m (Scall None a al) t |
---|
| 1180 | | execinf_Scall_some: ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t. |
---|
| 1181 | eval_lvalue e m lhs loc ofs -> |
---|
| 1182 | eval_expr e m a vf -> |
---|
| 1183 | eval_exprlist e m al vargs -> |
---|
| 1184 | Genv.find_funct ge vf = Some f -> |
---|
| 1185 | type_of_fundef f = typeof a -> |
---|
| 1186 | evalinf_funcall m f vargs t -> |
---|
| 1187 | execinf_stmt e m (Scall (Some lhs) a al) t |
---|
| 1188 | | execinf_Sseq_1: ∀e,m,s1,s2,t. |
---|
| 1189 | execinf_stmt e m s1 t -> |
---|
| 1190 | execinf_stmt e m (Ssequence s1 s2) t |
---|
| 1191 | | execinf_Sseq_2: ∀e,m,s1,s2,t1,m1,t2. |
---|
| 1192 | exec_stmt e m s1 t1 m1 Out_normal -> |
---|
| 1193 | execinf_stmt e m1 s2 t2 -> |
---|
| 1194 | execinf_stmt e m (Ssequence s1 s2) (t1 *** t2) |
---|
| 1195 | | execinf_Sifthenelse_true: ∀e,m,a,s1,s2,v1,t. |
---|
| 1196 | eval_expr e m a v1 -> |
---|
| 1197 | is_true v1 (typeof a) -> |
---|
| 1198 | execinf_stmt e m s1 t -> |
---|
| 1199 | execinf_stmt e m (Sifthenelse a s1 s2) t |
---|
| 1200 | | execinf_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t. |
---|
| 1201 | eval_expr e m a v1 -> |
---|
| 1202 | is_false v1 (typeof a) -> |
---|
| 1203 | execinf_stmt e m s2 t -> |
---|
| 1204 | execinf_stmt e m (Sifthenelse a s1 s2) t |
---|
| 1205 | | execinf_Swhile_body: ∀e,m,a,v,s,t. |
---|
| 1206 | eval_expr e m a v -> |
---|
| 1207 | is_true v (typeof a) -> |
---|
| 1208 | execinf_stmt e m s t -> |
---|
| 1209 | execinf_stmt e m (Swhile a s) t |
---|
| 1210 | | execinf_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2. |
---|
| 1211 | eval_expr e m a v -> |
---|
| 1212 | is_true v (typeof a) -> |
---|
| 1213 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1214 | out_normal_or_continue out1 -> |
---|
| 1215 | execinf_stmt e m1 (Swhile a s) t2 -> |
---|
| 1216 | execinf_stmt e m (Swhile a s) (t1 *** t2) |
---|
| 1217 | | execinf_Sdowhile_body: ∀e,m,s,a,t. |
---|
| 1218 | execinf_stmt e m s t -> |
---|
| 1219 | execinf_stmt e m (Sdowhile a s) t |
---|
| 1220 | | execinf_Sdowhile_loop: ∀e,m,s,a,m1,t1,t2,out1,v. |
---|
| 1221 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1222 | out_normal_or_continue out1 -> |
---|
| 1223 | eval_expr e m1 a v -> |
---|
| 1224 | is_true v (typeof a) -> |
---|
| 1225 | execinf_stmt e m1 (Sdowhile a s) t2 -> |
---|
| 1226 | execinf_stmt e m (Sdowhile a s) (t1 *** t2) |
---|
| 1227 | | execinf_Sfor_start_1: ∀e,m,s,a1,a2,a3,t. |
---|
| 1228 | execinf_stmt e m a1 t -> |
---|
| 1229 | execinf_stmt e m (Sfor a1 a2 a3 s) t |
---|
| 1230 | | execinf_Sfor_start_2: ∀e,m,s,a1,a2,a3,m1,t1,t2. |
---|
| 1231 | a1 <> Sskip -> |
---|
| 1232 | exec_stmt e m a1 t1 m1 Out_normal -> |
---|
| 1233 | execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 -> |
---|
| 1234 | execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2) |
---|
| 1235 | | execinf_Sfor_body: ∀e,m,s,a2,a3,v,t. |
---|
| 1236 | eval_expr e m a2 v -> |
---|
| 1237 | is_true v (typeof a2) -> |
---|
| 1238 | execinf_stmt e m s t -> |
---|
| 1239 | execinf_stmt e m (Sfor Sskip a2 a3 s) t |
---|
| 1240 | | execinf_Sfor_next: ∀e,m,s,a2,a3,v,m1,t1,t2,out1. |
---|
| 1241 | eval_expr e m a2 v -> |
---|
| 1242 | is_true v (typeof a2) -> |
---|
| 1243 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1244 | out_normal_or_continue out1 -> |
---|
| 1245 | execinf_stmt e m1 a3 t2 -> |
---|
| 1246 | execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2) |
---|
| 1247 | | execinf_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,t1,t2,t3,out1. |
---|
| 1248 | eval_expr e m a2 v -> |
---|
| 1249 | is_true v (typeof a2) -> |
---|
| 1250 | exec_stmt e m s t1 m1 out1 -> |
---|
| 1251 | out_normal_or_continue out1 -> |
---|
| 1252 | exec_stmt e m1 a3 t2 m2 Out_normal -> |
---|
| 1253 | execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 -> |
---|
| 1254 | execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3) |
---|
| 1255 | | execinf_Sswitch: ∀e,m,a,t,n,sl. |
---|
| 1256 | eval_expr e m a (Vint n) -> |
---|
| 1257 | execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t -> |
---|
| 1258 | execinf_stmt e m (Sswitch a sl) t |
---|
| 1259 | |
---|
| 1260 | (** [evalinf_funcall ge m fd args t] holds if the invocation of function |
---|
| 1261 | [fd] on arguments [args] diverges, with observable trace [t]. *) |
---|
| 1262 | |
---|
| 1263 | with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop := |
---|
| 1264 | | evalinf_funcall_internal: ∀m,f,vargs,t,e,m1,m2. |
---|
| 1265 | alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 -> |
---|
| 1266 | bind_parameters e m1 f.(fn_params) vargs m2 -> |
---|
| 1267 | execinf_stmt e m2 f.(fn_body) t -> |
---|
| 1268 | evalinf_funcall m (Internal f) vargs t. |
---|
| 1269 | |
---|
| 1270 | End SEMANTICS. |
---|
| 1271 | *) |
---|
| 1272 | (* * * Whole-program semantics *) |
---|
| 1273 | |
---|
| 1274 | (* * Execution of whole programs are described as sequences of transitions |
---|
| 1275 | from an initial state to a final state. An initial state is a [Callstate] |
---|
| 1276 | corresponding to the invocation of the ``main'' function of the program |
---|
| 1277 | without arguments and with an empty continuation. *) |
---|
| 1278 | |
---|
| 1279 | ninductive initial_state (p: program): state -> Prop := |
---|
| 1280 | | initial_state_intro: ∀b,f. |
---|
| 1281 | let ge := globalenv Genv ?? p in |
---|
| 1282 | let m0 := init_mem Genv ?? p in |
---|
[13] | 1283 | find_symbol ?? ge (prog_main ?? p) = Some ? b -> |
---|
| 1284 | find_funct_ptr ?? ge b = Some ? f -> |
---|
[3] | 1285 | initial_state p (Callstate f (nil ?) Kstop m0). |
---|
| 1286 | |
---|
| 1287 | (* * A final state is a [Returnstate] with an empty continuation. *) |
---|
| 1288 | |
---|
| 1289 | ninductive final_state: state -> int -> Prop := |
---|
| 1290 | | final_state_intro: ∀r,m. |
---|
| 1291 | final_state (Returnstate (Vint r) Kstop m) r. |
---|
| 1292 | |
---|
| 1293 | (* * Execution of a whole program: [exec_program p beh] |
---|
| 1294 | holds if the application of [p]'s main function to no arguments |
---|
| 1295 | in the initial memory state for [p] has [beh] as observable |
---|
| 1296 | behavior. *) |
---|
| 1297 | |
---|
| 1298 | ndefinition exec_program : program → program_behavior → Prop ≝ λp,beh. |
---|
[13] | 1299 | program_behaves (mk_transrel ?? step) (initial_state p) final_state (globalenv ??? p) beh. |
---|
[3] | 1300 | (* |
---|
| 1301 | (** Big-step execution of a whole program. *) |
---|
| 1302 | |
---|
| 1303 | ninductive bigstep_program_terminates (p: program): trace -> int -> Prop := |
---|
| 1304 | | bigstep_program_terminates_intro: ∀b,f,m1,t,r. |
---|
| 1305 | let ge := Genv.globalenv p in |
---|
| 1306 | let m0 := Genv.init_mem p in |
---|
| 1307 | Genv.find_symbol ge p.(prog_main) = Some b -> |
---|
| 1308 | Genv.find_funct_ptr ge b = Some f -> |
---|
| 1309 | eval_funcall ge m0 f nil t m1 (Vint r) -> |
---|
| 1310 | bigstep_program_terminates p t r. |
---|
| 1311 | |
---|
| 1312 | ninductive bigstep_program_diverges (p: program): traceinf -> Prop := |
---|
| 1313 | | bigstep_program_diverges_intro: ∀b,f,t. |
---|
| 1314 | let ge := Genv.globalenv p in |
---|
| 1315 | let m0 := Genv.init_mem p in |
---|
| 1316 | Genv.find_symbol ge p.(prog_main) = Some b -> |
---|
| 1317 | Genv.find_funct_ptr ge b = Some f -> |
---|
| 1318 | evalinf_funcall ge m0 f nil t -> |
---|
| 1319 | bigstep_program_diverges p t. |
---|
| 1320 | |
---|
| 1321 | (** * Implication from big-step semantics to transition semantics *) |
---|
| 1322 | |
---|
| 1323 | Section BIGSTEP_TO_TRANSITIONS. |
---|
| 1324 | |
---|
| 1325 | Variable prog: program. |
---|
| 1326 | Let ge : genv := Genv.globalenv prog. |
---|
| 1327 | |
---|
| 1328 | Definition exec_stmt_eval_funcall_ind |
---|
| 1329 | (PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop) |
---|
| 1330 | (PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) := |
---|
| 1331 | fun a b c d e f g h i j k l m n o p q r s t u v w x y => |
---|
| 1332 | conj (exec_stmt_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y) |
---|
| 1333 | (eval_funcall_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y). |
---|
| 1334 | |
---|
| 1335 | ninductive outcome_state_match |
---|
| 1336 | (e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop := |
---|
| 1337 | | osm_normal: |
---|
| 1338 | outcome_state_match e m f k Out_normal (State f Sskip k e m) |
---|
| 1339 | | osm_break: |
---|
| 1340 | outcome_state_match e m f k Out_break (State f Sbreak k e m) |
---|
| 1341 | | osm_continue: |
---|
| 1342 | outcome_state_match e m f k Out_continue (State f Scontinue k e m) |
---|
| 1343 | | osm_return_none: ∀k'. |
---|
| 1344 | call_cont k' = call_cont k -> |
---|
| 1345 | outcome_state_match e m f k |
---|
| 1346 | (Out_return None) (State f (Sreturn None) k' e m) |
---|
| 1347 | | osm_return_some: ∀a,v,k'. |
---|
| 1348 | call_cont k' = call_cont k -> |
---|
| 1349 | eval_expr ge e m a v -> |
---|
| 1350 | outcome_state_match e m f k |
---|
| 1351 | (Out_return (Some v)) (State f (Sreturn (Some a)) k' e m). |
---|
| 1352 | |
---|
| 1353 | Lemma is_call_cont_call_cont: |
---|
| 1354 | ∀k. is_call_cont k -> call_cont k = k. |
---|
| 1355 | Proof. |
---|
| 1356 | destruct k; simpl; intros; contradiction || auto. |
---|
| 1357 | Qed. |
---|
| 1358 | |
---|
| 1359 | Lemma exec_stmt_eval_funcall_steps: |
---|
| 1360 | (∀e,m,s,t,m',out. |
---|
| 1361 | exec_stmt ge e m s t m' out -> |
---|
| 1362 | ∀f,k. exists S, |
---|
| 1363 | star step ge (State f s k e m) t S |
---|
| 1364 | /\ outcome_state_match e m' f k out S) |
---|
| 1365 | /\ |
---|
| 1366 | (∀m,fd,args,t,m',res. |
---|
| 1367 | eval_funcall ge m fd args t m' res -> |
---|
| 1368 | ∀k. |
---|
| 1369 | is_call_cont k -> |
---|
| 1370 | star step ge (Callstate fd args k m) t (Returnstate res k m')). |
---|
| 1371 | Proof. |
---|
| 1372 | apply exec_stmt_eval_funcall_ind; intros. |
---|
| 1373 | |
---|
| 1374 | (* skip *) |
---|
| 1375 | econstructor; split. apply star_refl. constructor. |
---|
| 1376 | |
---|
| 1377 | (* assign *) |
---|
| 1378 | econstructor; split. apply star_one. econstructor; eauto. constructor. |
---|
| 1379 | |
---|
| 1380 | (* call none *) |
---|
| 1381 | econstructor; split. |
---|
| 1382 | eapply star_left. econstructor; eauto. |
---|
| 1383 | eapply star_right. apply H4. simpl; auto. econstructor. reflexivity. traceEq. |
---|
| 1384 | constructor. |
---|
| 1385 | |
---|
| 1386 | (* call some *) |
---|
| 1387 | econstructor; split. |
---|
| 1388 | eapply star_left. econstructor; eauto. |
---|
| 1389 | eapply star_right. apply H5. simpl; auto. econstructor; eauto. reflexivity. traceEq. |
---|
| 1390 | constructor. |
---|
| 1391 | |
---|
| 1392 | (* sequence 2 *) |
---|
| 1393 | destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. inv B1. |
---|
| 1394 | destruct (H2 f k) as [S2 [A2 B2]]. |
---|
| 1395 | econstructor; split. |
---|
| 1396 | eapply star_left. econstructor. |
---|
| 1397 | eapply star_trans. eexact A1. |
---|
| 1398 | eapply star_left. constructor. eexact A2. |
---|
| 1399 | reflexivity. reflexivity. traceEq. |
---|
| 1400 | auto. |
---|
| 1401 | |
---|
| 1402 | (* sequence 1 *) |
---|
| 1403 | destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. |
---|
| 1404 | set (S2 := |
---|
| 1405 | match out with |
---|
| 1406 | | Out_break => State f Sbreak k e m1 |
---|
| 1407 | | Out_continue => State f Scontinue k e m1 |
---|
| 1408 | | _ => S1 |
---|
| 1409 | end). |
---|
| 1410 | exists S2; split. |
---|
| 1411 | eapply star_left. econstructor. |
---|
| 1412 | eapply star_trans. eexact A1. |
---|
| 1413 | unfold S2; inv B1. |
---|
| 1414 | congruence. |
---|
| 1415 | apply star_one. apply step_break_seq. |
---|
| 1416 | apply star_one. apply step_continue_seq. |
---|
| 1417 | apply star_refl. |
---|
| 1418 | apply star_refl. |
---|
| 1419 | reflexivity. traceEq. |
---|
| 1420 | unfold S2; inv B1; congruence || econstructor; eauto. |
---|
| 1421 | |
---|
| 1422 | (* ifthenelse true *) |
---|
| 1423 | destruct (H2 f k) as [S1 [A1 B1]]. |
---|
| 1424 | exists S1; split. |
---|
| 1425 | eapply star_left. eapply step_ifthenelse_true; eauto. eexact A1. traceEq. |
---|
| 1426 | auto. |
---|
| 1427 | |
---|
| 1428 | (* ifthenelse false *) |
---|
| 1429 | destruct (H2 f k) as [S1 [A1 B1]]. |
---|
| 1430 | exists S1; split. |
---|
| 1431 | eapply star_left. eapply step_ifthenelse_false; eauto. eexact A1. traceEq. |
---|
| 1432 | auto. |
---|
| 1433 | |
---|
| 1434 | (* return none *) |
---|
| 1435 | econstructor; split. apply star_refl. constructor. auto. |
---|
| 1436 | |
---|
| 1437 | (* return some *) |
---|
| 1438 | econstructor; split. apply star_refl. econstructor; eauto. |
---|
| 1439 | |
---|
| 1440 | (* break *) |
---|
| 1441 | econstructor; split. apply star_refl. constructor. |
---|
| 1442 | |
---|
| 1443 | (* continue *) |
---|
| 1444 | econstructor; split. apply star_refl. constructor. |
---|
| 1445 | |
---|
| 1446 | (* while false *) |
---|
| 1447 | econstructor; split. |
---|
| 1448 | apply star_one. eapply step_while_false; eauto. |
---|
| 1449 | constructor. |
---|
| 1450 | |
---|
| 1451 | (* while stop *) |
---|
| 1452 | destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]]. |
---|
| 1453 | set (S2 := |
---|
| 1454 | match out' with |
---|
| 1455 | | Out_break => State f Sskip k e m' |
---|
| 1456 | | _ => S1 |
---|
| 1457 | end). |
---|
| 1458 | exists S2; split. |
---|
| 1459 | eapply star_left. eapply step_while_true; eauto. |
---|
| 1460 | eapply star_trans. eexact A1. |
---|
| 1461 | unfold S2. inversion H3; subst. |
---|
| 1462 | inv B1. apply star_one. constructor. |
---|
| 1463 | apply star_refl. |
---|
| 1464 | reflexivity. traceEq. |
---|
| 1465 | unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto. |
---|
| 1466 | |
---|
| 1467 | (* while loop *) |
---|
| 1468 | destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]]. |
---|
| 1469 | destruct (H5 f k) as [S2 [A2 B2]]. |
---|
| 1470 | exists S2; split. |
---|
| 1471 | eapply star_left. eapply step_while_true; eauto. |
---|
| 1472 | eapply star_trans. eexact A1. |
---|
| 1473 | eapply star_left. |
---|
| 1474 | inv H3; inv B1; apply step_skip_or_continue_while; auto. |
---|
| 1475 | eexact A2. |
---|
| 1476 | reflexivity. reflexivity. traceEq. |
---|
| 1477 | auto. |
---|
| 1478 | |
---|
| 1479 | (* dowhile false *) |
---|
| 1480 | destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]]. |
---|
| 1481 | exists (State f Sskip k e m1); split. |
---|
| 1482 | eapply star_left. constructor. |
---|
| 1483 | eapply star_right. eexact A1. |
---|
| 1484 | inv H1; inv B1; eapply step_skip_or_continue_dowhile_false; eauto. |
---|
| 1485 | reflexivity. traceEq. |
---|
| 1486 | constructor. |
---|
| 1487 | |
---|
| 1488 | (* dowhile stop *) |
---|
| 1489 | destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]]. |
---|
| 1490 | set (S2 := |
---|
| 1491 | match out1 with |
---|
| 1492 | | Out_break => State f Sskip k e m1 |
---|
| 1493 | | _ => S1 |
---|
| 1494 | end). |
---|
| 1495 | exists S2; split. |
---|
| 1496 | eapply star_left. apply step_dowhile. |
---|
| 1497 | eapply star_trans. eexact A1. |
---|
| 1498 | unfold S2. inversion H1; subst. |
---|
| 1499 | inv B1. apply star_one. constructor. |
---|
| 1500 | apply star_refl. |
---|
| 1501 | reflexivity. traceEq. |
---|
| 1502 | unfold S2. inversion H1; subst. constructor. inv B1; econstructor; eauto. |
---|
| 1503 | |
---|
| 1504 | (* dowhile loop *) |
---|
| 1505 | destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]]. |
---|
| 1506 | destruct (H5 f k) as [S2 [A2 B2]]. |
---|
| 1507 | exists S2; split. |
---|
| 1508 | eapply star_left. apply step_dowhile. |
---|
| 1509 | eapply star_trans. eexact A1. |
---|
| 1510 | eapply star_left. |
---|
| 1511 | inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto. |
---|
| 1512 | eexact A2. |
---|
| 1513 | reflexivity. reflexivity. traceEq. |
---|
| 1514 | auto. |
---|
| 1515 | |
---|
| 1516 | (* for start *) |
---|
| 1517 | destruct (H1 f (Kseq (Sfor Sskip a2 a3 s) k)) as [S1 [A1 B1]]. inv B1. |
---|
| 1518 | destruct (H3 f k) as [S2 [A2 B2]]. |
---|
| 1519 | exists S2; split. |
---|
| 1520 | eapply star_left. apply step_for_start; auto. |
---|
| 1521 | eapply star_trans. eexact A1. |
---|
| 1522 | eapply star_left. constructor. eexact A2. |
---|
| 1523 | reflexivity. reflexivity. traceEq. |
---|
| 1524 | auto. |
---|
| 1525 | |
---|
| 1526 | (* for false *) |
---|
| 1527 | econstructor; split. |
---|
| 1528 | eapply star_one. eapply step_for_false; eauto. |
---|
| 1529 | constructor. |
---|
| 1530 | |
---|
| 1531 | (* for stop *) |
---|
| 1532 | destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]]. |
---|
| 1533 | set (S2 := |
---|
| 1534 | match out1 with |
---|
| 1535 | | Out_break => State f Sskip k e m1 |
---|
| 1536 | | _ => S1 |
---|
| 1537 | end). |
---|
| 1538 | exists S2; split. |
---|
| 1539 | eapply star_left. eapply step_for_true; eauto. |
---|
| 1540 | eapply star_trans. eexact A1. |
---|
| 1541 | unfold S2. inversion H3; subst. |
---|
| 1542 | inv B1. apply star_one. constructor. |
---|
| 1543 | apply star_refl. |
---|
| 1544 | reflexivity. traceEq. |
---|
| 1545 | unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto. |
---|
| 1546 | |
---|
| 1547 | (* for loop *) |
---|
| 1548 | destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]]. |
---|
| 1549 | destruct (H5 f (Kfor3 a2 a3 s k)) as [S2 [A2 B2]]. inv B2. |
---|
| 1550 | destruct (H7 f k) as [S3 [A3 B3]]. |
---|
| 1551 | exists S3; split. |
---|
| 1552 | eapply star_left. eapply step_for_true; eauto. |
---|
| 1553 | eapply star_trans. eexact A1. |
---|
| 1554 | eapply star_trans with (s2 := State f a3 (Kfor3 a2 a3 s k) e m1). |
---|
| 1555 | inv H3; inv B1. |
---|
| 1556 | apply star_one. constructor. auto. |
---|
| 1557 | apply star_one. constructor. auto. |
---|
| 1558 | eapply star_trans. eexact A2. |
---|
| 1559 | eapply star_left. constructor. |
---|
| 1560 | eexact A3. |
---|
| 1561 | reflexivity. reflexivity. reflexivity. reflexivity. traceEq. |
---|
| 1562 | auto. |
---|
| 1563 | |
---|
| 1564 | (* switch *) |
---|
| 1565 | destruct (H1 f (Kswitch k)) as [S1 [A1 B1]]. |
---|
| 1566 | set (S2 := |
---|
| 1567 | match out with |
---|
| 1568 | | Out_normal => State f Sskip k e m1 |
---|
| 1569 | | Out_break => State f Sskip k e m1 |
---|
| 1570 | | Out_continue => State f Scontinue k e m1 |
---|
| 1571 | | _ => S1 |
---|
| 1572 | end). |
---|
| 1573 | exists S2; split. |
---|
| 1574 | eapply star_left. eapply step_switch; eauto. |
---|
| 1575 | eapply star_trans. eexact A1. |
---|
| 1576 | unfold S2; inv B1. |
---|
| 1577 | apply star_one. constructor. auto. |
---|
| 1578 | apply star_one. constructor. auto. |
---|
| 1579 | apply star_one. constructor. |
---|
| 1580 | apply star_refl. |
---|
| 1581 | apply star_refl. |
---|
| 1582 | reflexivity. traceEq. |
---|
| 1583 | unfold S2. inv B1; simpl; econstructor; eauto. |
---|
| 1584 | |
---|
| 1585 | (* call internal *) |
---|
| 1586 | destruct (H2 f k) as [S1 [A1 B1]]. |
---|
| 1587 | eapply star_left. eapply step_internal_function; eauto. |
---|
| 1588 | eapply star_right. eexact A1. |
---|
| 1589 | inv B1; simpl in H3; try contradiction. |
---|
| 1590 | (* Out_normal *) |
---|
| 1591 | assert (fn_return f = Tvoid /\ vres = Vundef). |
---|
| 1592 | destruct (fn_return f); auto || contradiction. |
---|
| 1593 | destruct H5. subst vres. apply step_skip_call; auto. |
---|
| 1594 | (* Out_return None *) |
---|
| 1595 | assert (fn_return f = Tvoid /\ vres = Vundef). |
---|
| 1596 | destruct (fn_return f); auto || contradiction. |
---|
| 1597 | destruct H6. subst vres. |
---|
| 1598 | rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5. |
---|
| 1599 | apply step_return_0; auto. |
---|
| 1600 | (* Out_return Some *) |
---|
| 1601 | destruct H3. subst vres. |
---|
| 1602 | rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5. |
---|
| 1603 | eapply step_return_1; eauto. |
---|
| 1604 | reflexivity. traceEq. |
---|
| 1605 | |
---|
| 1606 | (* call external *) |
---|
| 1607 | apply star_one. apply step_external_function; auto. |
---|
| 1608 | Qed. |
---|
| 1609 | |
---|
| 1610 | Lemma exec_stmt_steps: |
---|
| 1611 | ∀e,m,s,t,m',out. |
---|
| 1612 | exec_stmt ge e m s t m' out -> |
---|
| 1613 | ∀f,k. exists S, |
---|
| 1614 | star step ge (State f s k e m) t S |
---|
| 1615 | /\ outcome_state_match e m' f k out S. |
---|
| 1616 | Proof (proj1 exec_stmt_eval_funcall_steps). |
---|
| 1617 | |
---|
| 1618 | Lemma eval_funcall_steps: |
---|
| 1619 | ∀m,fd,args,t,m',res. |
---|
| 1620 | eval_funcall ge m fd args t m' res -> |
---|
| 1621 | ∀k. |
---|
| 1622 | is_call_cont k -> |
---|
| 1623 | star step ge (Callstate fd args k m) t (Returnstate res k m'). |
---|
| 1624 | Proof (proj2 exec_stmt_eval_funcall_steps). |
---|
| 1625 | |
---|
| 1626 | Definition order (x y: unit) := False. |
---|
| 1627 | |
---|
| 1628 | Lemma evalinf_funcall_forever: |
---|
| 1629 | ∀m,fd,args,T,k. |
---|
| 1630 | evalinf_funcall ge m fd args T -> |
---|
| 1631 | forever_N step order ge tt (Callstate fd args k m) T. |
---|
| 1632 | Proof. |
---|
| 1633 | cofix CIH_FUN. |
---|
| 1634 | assert (∀e,m,s,T,f,k. |
---|
| 1635 | execinf_stmt ge e m s T -> |
---|
| 1636 | forever_N step order ge tt (State f s k e m) T). |
---|
| 1637 | cofix CIH_STMT. |
---|
| 1638 | intros. inv H. |
---|
| 1639 | |
---|
| 1640 | (* call none *) |
---|
| 1641 | eapply forever_N_plus. |
---|
| 1642 | apply plus_one. eapply step_call_none; eauto. |
---|
| 1643 | apply CIH_FUN. eauto. traceEq. |
---|
| 1644 | (* call some *) |
---|
| 1645 | eapply forever_N_plus. |
---|
| 1646 | apply plus_one. eapply step_call_some; eauto. |
---|
| 1647 | apply CIH_FUN. eauto. traceEq. |
---|
| 1648 | |
---|
| 1649 | (* seq 1 *) |
---|
| 1650 | eapply forever_N_plus. |
---|
| 1651 | apply plus_one. econstructor. |
---|
| 1652 | apply CIH_STMT; eauto. traceEq. |
---|
| 1653 | (* seq 2 *) |
---|
| 1654 | destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kseq s2 k)) as [S1 [A1 B1]]. |
---|
| 1655 | inv B1. |
---|
| 1656 | eapply forever_N_plus. |
---|
| 1657 | eapply plus_left. constructor. eapply star_trans. eexact A1. |
---|
| 1658 | apply star_one. constructor. reflexivity. reflexivity. |
---|
| 1659 | apply CIH_STMT; eauto. traceEq. |
---|
| 1660 | |
---|
| 1661 | (* ifthenelse true *) |
---|
| 1662 | eapply forever_N_plus. |
---|
| 1663 | apply plus_one. eapply step_ifthenelse_true; eauto. |
---|
| 1664 | apply CIH_STMT; eauto. traceEq. |
---|
| 1665 | (* ifthenelse false *) |
---|
| 1666 | eapply forever_N_plus. |
---|
| 1667 | apply plus_one. eapply step_ifthenelse_false; eauto. |
---|
| 1668 | apply CIH_STMT; eauto. traceEq. |
---|
| 1669 | |
---|
| 1670 | (* while body *) |
---|
| 1671 | eapply forever_N_plus. |
---|
| 1672 | eapply plus_one. eapply step_while_true; eauto. |
---|
| 1673 | apply CIH_STMT; eauto. traceEq. |
---|
| 1674 | (* while loop *) |
---|
| 1675 | destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kwhile a s0 k)) as [S1 [A1 B1]]. |
---|
| 1676 | eapply forever_N_plus with (s2 := State f (Swhile a s0) k e m1). |
---|
| 1677 | eapply plus_left. eapply step_while_true; eauto. |
---|
| 1678 | eapply star_right. eexact A1. |
---|
| 1679 | inv H3; inv B1; apply step_skip_or_continue_while; auto. |
---|
| 1680 | reflexivity. reflexivity. |
---|
| 1681 | apply CIH_STMT; eauto. traceEq. |
---|
| 1682 | |
---|
| 1683 | (* dowhile body *) |
---|
| 1684 | eapply forever_N_plus. |
---|
| 1685 | eapply plus_one. eapply step_dowhile. |
---|
| 1686 | apply CIH_STMT; eauto. |
---|
| 1687 | traceEq. |
---|
| 1688 | |
---|
| 1689 | (* dowhile loop *) |
---|
| 1690 | destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kdowhile a s0 k)) as [S1 [A1 B1]]. |
---|
| 1691 | eapply forever_N_plus with (s2 := State f (Sdowhile a s0) k e m1). |
---|
| 1692 | eapply plus_left. eapply step_dowhile. |
---|
| 1693 | eapply star_right. eexact A1. |
---|
| 1694 | inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto. |
---|
| 1695 | reflexivity. reflexivity. |
---|
| 1696 | apply CIH_STMT. eauto. |
---|
| 1697 | traceEq. |
---|
| 1698 | |
---|
| 1699 | (* for start 1 *) |
---|
| 1700 | assert (a1 <> Sskip). red; intros; subst. inv H0. |
---|
| 1701 | eapply forever_N_plus. |
---|
| 1702 | eapply plus_one. apply step_for_start; auto. |
---|
| 1703 | apply CIH_STMT; eauto. |
---|
| 1704 | traceEq. |
---|
| 1705 | |
---|
| 1706 | (* for start 2 *) |
---|
| 1707 | destruct (exec_stmt_steps _ _ _ _ _ _ H1 f (Kseq (Sfor Sskip a2 a3 s0) k)) as [S1 [A1 B1]]. |
---|
| 1708 | inv B1. |
---|
| 1709 | eapply forever_N_plus. |
---|
| 1710 | eapply plus_left. eapply step_for_start; eauto. |
---|
| 1711 | eapply star_right. eexact A1. |
---|
| 1712 | apply step_skip_seq. |
---|
| 1713 | reflexivity. reflexivity. |
---|
| 1714 | apply CIH_STMT; eauto. |
---|
| 1715 | traceEq. |
---|
| 1716 | |
---|
| 1717 | (* for body *) |
---|
| 1718 | eapply forever_N_plus. |
---|
| 1719 | apply plus_one. eapply step_for_true; eauto. |
---|
| 1720 | apply CIH_STMT; eauto. |
---|
| 1721 | traceEq. |
---|
| 1722 | |
---|
| 1723 | (* for next *) |
---|
| 1724 | destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]]. |
---|
| 1725 | eapply forever_N_plus. |
---|
| 1726 | eapply plus_left. eapply step_for_true; eauto. |
---|
| 1727 | eapply star_trans. eexact A1. |
---|
| 1728 | apply star_one. |
---|
| 1729 | inv H3; inv B1; apply step_skip_or_continue_for2; auto. |
---|
| 1730 | reflexivity. reflexivity. |
---|
| 1731 | apply CIH_STMT; eauto. |
---|
| 1732 | traceEq. |
---|
| 1733 | |
---|
| 1734 | (* for body *) |
---|
| 1735 | destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]]. |
---|
| 1736 | destruct (exec_stmt_steps _ _ _ _ _ _ H4 f (Kfor3 a2 a3 s0 k)) as [S2 [A2 B2]]. |
---|
| 1737 | inv B2. |
---|
| 1738 | eapply forever_N_plus. |
---|
| 1739 | eapply plus_left. eapply step_for_true; eauto. |
---|
| 1740 | eapply star_trans. eexact A1. |
---|
| 1741 | eapply star_left. inv H3; inv B1; apply step_skip_or_continue_for2; auto. |
---|
| 1742 | eapply star_right. eexact A2. |
---|
| 1743 | constructor. |
---|
| 1744 | reflexivity. reflexivity. reflexivity. reflexivity. |
---|
| 1745 | apply CIH_STMT; eauto. |
---|
| 1746 | traceEq. |
---|
| 1747 | |
---|
| 1748 | (* switch *) |
---|
| 1749 | eapply forever_N_plus. |
---|
| 1750 | eapply plus_one. eapply step_switch; eauto. |
---|
| 1751 | apply CIH_STMT; eauto. |
---|
| 1752 | traceEq. |
---|
| 1753 | |
---|
| 1754 | (* call internal *) |
---|
| 1755 | intros. inv H0. |
---|
| 1756 | eapply forever_N_plus. |
---|
| 1757 | eapply plus_one. econstructor; eauto. |
---|
| 1758 | apply H; eauto. |
---|
| 1759 | traceEq. |
---|
| 1760 | Qed. |
---|
| 1761 | |
---|
| 1762 | Theorem bigstep_program_terminates_exec: |
---|
| 1763 | ∀t,r. bigstep_program_terminates prog t r -> exec_program prog (Terminates t r). |
---|
| 1764 | Proof. |
---|
| 1765 | intros. inv H. unfold ge0, m0 in *. |
---|
| 1766 | econstructor. |
---|
| 1767 | econstructor. eauto. eauto. |
---|
| 1768 | apply eval_funcall_steps. eauto. red; auto. |
---|
| 1769 | econstructor. |
---|
| 1770 | Qed. |
---|
| 1771 | |
---|
| 1772 | Theorem bigstep_program_diverges_exec: |
---|
| 1773 | ∀T. bigstep_program_diverges prog T -> |
---|
| 1774 | exec_program prog (Reacts T) \/ |
---|
| 1775 | exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T. |
---|
| 1776 | Proof. |
---|
| 1777 | intros. inv H. |
---|
| 1778 | set (st := Callstate f nil Kstop m0). |
---|
| 1779 | assert (forever step ge0 st T). |
---|
| 1780 | eapply forever_N_forever with (order := order). |
---|
| 1781 | red; intros. constructor; intros. red in H. elim H. |
---|
| 1782 | eapply evalinf_funcall_forever; eauto. |
---|
| 1783 | destruct (forever_silent_or_reactive _ _ _ _ _ _ H) |
---|
| 1784 | as [A | [t [s' [T' [B [C D]]]]]]. |
---|
| 1785 | left. econstructor. econstructor. eauto. eauto. auto. |
---|
| 1786 | right. exists t. split. |
---|
| 1787 | econstructor. econstructor; eauto. eauto. auto. |
---|
| 1788 | subst T. rewrite <- (E0_right t) at 1. apply traceinf_prefix_app. constructor. |
---|
| 1789 | Qed. |
---|
| 1790 | |
---|
| 1791 | End BIGSTEP_TO_TRANSITIONS. |
---|
| 1792 | |
---|
| 1793 | |
---|
| 1794 | |
---|
| 1795 | *) |
---|
| 1796 | |
---|
| 1797 | |
---|