[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 | (* * This module defines the type of values that is used in the dynamic |
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| 17 | semantics of all our intermediate languages. *) |
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| 18 | |
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[700] | 19 | include "utilities/Coqlib.ma". |
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| 20 | include "common/Floats.ma". |
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| 21 | include "common/Errors.ma". |
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[1213] | 22 | include "common/Pointers.ma". |
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[487] | 23 | include "basics/logic.ma". |
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[3] | 24 | |
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| 25 | (* * A value is either: |
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| 26 | - a machine integer; |
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| 27 | - a floating-point number; |
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[482] | 28 | - a pointer: a triple giving the representation of the pointer (in terms of the |
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| 29 | memory regions such a pointer could address), a memory address and |
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| 30 | an integer offset with respect to this address; |
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[484] | 31 | - a null pointer: the region denotes the representation (i.e., pointer size) |
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[3] | 32 | - the [Vundef] value denoting an arbitrary bit pattern, such as the |
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| 33 | value of an uninitialized variable. |
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| 34 | *) |
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| 35 | |
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[487] | 36 | inductive val: Type[0] ≝ |
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[3] | 37 | | Vundef: val |
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[961] | 38 | | Vint: ∀sz:intsize. bvint sz → val |
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[482] | 39 | | Vfloat: float → val |
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[484] | 40 | | Vnull: region → val |
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[1545] | 41 | | Vptr: pointer → val. |
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[3] | 42 | |
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[961] | 43 | definition Vzero : intsize → val ≝ λsz. Vint sz (zero ?). |
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| 44 | definition Vone: intsize → val ≝ λsz. Vint sz (repr sz 1). |
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| 45 | definition mone ≝ λsz. bitvector_of_Z (bitsize_of_intsize sz) (neg one). |
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| 46 | definition Vmone: intsize → val ≝ λsz. Vint sz (mone ?). |
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[3] | 47 | |
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[961] | 48 | (* XXX 32bit booleans are Clight specific. *) |
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| 49 | definition Vtrue: val ≝ Vone I32. |
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| 50 | definition Vfalse: val ≝ Vzero I32. |
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[3] | 51 | |
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[1369] | 52 | inductive val_typ : val → typ → Prop ≝ |
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| 53 | | VTint: ∀sz,sg,i. val_typ (Vint sz i) (ASTint sz sg) |
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| 54 | | VTfloat: ∀sz,f. val_typ (Vfloat f) (ASTfloat sz) |
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| 55 | | VTnull: ∀r. val_typ (Vnull r) (ASTptr r) |
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[1545] | 56 | | VTptr: ∀r,b,c,o. val_typ (Vptr (mk_pointer r b c o)) (ASTptr r). |
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[1369] | 57 | |
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[797] | 58 | (* |
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[3] | 59 | (** The module [Val] defines a number of arithmetic and logical operations |
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| 60 | over type [val]. Most of these operations are straightforward extensions |
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| 61 | of the corresponding integer or floating-point operations. *) |
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| 62 | |
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| 63 | Module Val. |
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| 64 | *) |
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[487] | 65 | definition of_bool : bool → val ≝ λb. if b then Vtrue else Vfalse. |
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[3] | 66 | |
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| 67 | (* * Truth values. Pointers and non-zero integers are treated as [True]. |
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| 68 | The integer 0 (also used to represent the null pointer) is [False]. |
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| 69 | [Vundef] and floats are neither true nor false. *) |
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| 70 | |
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[487] | 71 | definition is_true : val → Prop ≝ λv. |
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[3] | 72 | match v with |
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[961] | 73 | [ Vint _ n ⇒ n ≠ (zero ?) |
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[1545] | 74 | | Vptr _ ⇒ True |
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[3] | 75 | | _ ⇒ False |
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| 76 | ]. |
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| 77 | |
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[487] | 78 | definition is_false : val → Prop ≝ λv. |
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[3] | 79 | match v with |
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[961] | 80 | [ Vint _ n ⇒ n = (zero ?) |
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[484] | 81 | | Vnull _ ⇒ True |
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[3] | 82 | | _ ⇒ False |
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| 83 | ]. |
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| 84 | |
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[487] | 85 | inductive bool_of_val: val → bool → Prop ≝ |
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[3] | 86 | | bool_of_val_int_true: |
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[961] | 87 | ∀sz,n. n ≠ zero ? → bool_of_val (Vint sz n) true |
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[3] | 88 | | bool_of_val_int_false: |
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[961] | 89 | ∀sz. bool_of_val (Vzero sz) false |
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[3] | 90 | | bool_of_val_ptr: |
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[1545] | 91 | ∀p. bool_of_val (Vptr p) true |
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[484] | 92 | | bool_of_val_null: |
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| 93 | ∀r. bool_of_val (Vnull r) true. |
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[3] | 94 | |
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[797] | 95 | axiom ValueNotABoolean : String. |
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| 96 | |
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[751] | 97 | definition eval_bool_of_val : val → res bool ≝ |
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| 98 | λv. match v with |
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[961] | 99 | [ Vint _ i ⇒ OK ? (notb (eq_bv ? i (zero ?))) |
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[751] | 100 | | Vnull _ ⇒ OK ? false |
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[1545] | 101 | | Vptr _ ⇒ OK ? true |
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[797] | 102 | | _ ⇒ Error ? (msg ValueNotABoolean) |
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[751] | 103 | ]. |
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| 104 | |
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[487] | 105 | definition neg : val → val ≝ λv. |
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[3] | 106 | match v with |
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[961] | 107 | [ Vint sz n ⇒ Vint sz (two_complement_negation ? n) |
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[3] | 108 | | _ ⇒ Vundef |
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| 109 | ]. |
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| 110 | |
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[487] | 111 | definition negf : val → val ≝ λv. |
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[3] | 112 | match v with |
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| 113 | [ Vfloat f ⇒ Vfloat (Fneg f) |
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| 114 | | _ => Vundef |
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| 115 | ]. |
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| 116 | |
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[487] | 117 | definition absf : val → val ≝ λv. |
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[3] | 118 | match v with |
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| 119 | [ Vfloat f ⇒ Vfloat (Fabs f) |
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| 120 | | _ ⇒ Vundef |
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| 121 | ]. |
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| 122 | |
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[961] | 123 | definition intoffloat : intsize → val → val ≝ λsz,v. |
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[3] | 124 | match v with |
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[961] | 125 | [ Vfloat f ⇒ Vint sz (intoffloat ? f) |
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[3] | 126 | | _ ⇒ Vundef |
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| 127 | ]. |
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| 128 | |
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[961] | 129 | definition intuoffloat : intsize → val → val ≝ λsz,v. |
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[3] | 130 | match v with |
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[961] | 131 | [ Vfloat f ⇒ Vint sz (intuoffloat ? f) |
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[3] | 132 | | _ ⇒ Vundef |
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| 133 | ]. |
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| 134 | |
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[487] | 135 | definition floatofint : val → val ≝ λv. |
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[3] | 136 | match v with |
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[961] | 137 | [ Vint sz n ⇒ Vfloat (floatofint ? n) |
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[3] | 138 | | _ ⇒ Vundef |
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| 139 | ]. |
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| 140 | |
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[487] | 141 | definition floatofintu : val → val ≝ λv. |
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[3] | 142 | match v with |
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[961] | 143 | [ Vint sz n ⇒ Vfloat (floatofintu ? n) |
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[3] | 144 | | _ ⇒ Vundef |
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| 145 | ]. |
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| 146 | |
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[487] | 147 | definition notint : val → val ≝ λv. |
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[3] | 148 | match v with |
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[961] | 149 | [ Vint sz n ⇒ Vint sz (exclusive_disjunction_bv ? n (mone ?)) |
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[3] | 150 | | _ ⇒ Vundef |
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| 151 | ]. |
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| 152 | |
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[487] | 153 | definition notbool : val → val ≝ λv. |
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[3] | 154 | match v with |
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[961] | 155 | [ Vint sz n ⇒ of_bool (eq_bv ? n (zero ?)) |
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[1545] | 156 | | Vptr _ ⇒ Vfalse |
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[484] | 157 | | Vnull _ ⇒ Vtrue |
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[3] | 158 | | _ ⇒ Vundef |
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| 159 | ]. |
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| 160 | |
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[961] | 161 | definition zero_ext ≝ λrsz: intsize. λv: val. |
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[3] | 162 | match v with |
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[961] | 163 | [ Vint sz n ⇒ Vint rsz (zero_ext … n) |
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[3] | 164 | | _ ⇒ Vundef |
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| 165 | ]. |
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| 166 | |
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[961] | 167 | definition sign_ext ≝ λrsz:intsize. λv:val. |
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[3] | 168 | match v with |
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[961] | 169 | [ Vint sz i ⇒ Vint rsz (sign_ext … i) |
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[3] | 170 | | _ ⇒ Vundef |
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| 171 | ]. |
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| 172 | |
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[487] | 173 | definition singleoffloat : val → val ≝ λv. |
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[3] | 174 | match v with |
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| 175 | [ Vfloat f ⇒ Vfloat (singleoffloat f) |
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| 176 | | _ ⇒ Vundef |
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| 177 | ]. |
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| 178 | |
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[484] | 179 | (* TODO: add zero to null? *) |
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[487] | 180 | definition add ≝ λv1,v2: val. |
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[3] | 181 | match v1 with |
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[961] | 182 | [ Vint sz1 n1 ⇒ match v2 with |
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| 183 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 184 | (λn1. Vint sz2 (addition_n ? n1 n2)) |
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| 185 | Vundef |
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[1545] | 186 | | Vptr ptr ⇒ Vptr (shift_pointer ? ptr n1) |
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[3] | 187 | | _ ⇒ Vundef ] |
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[1545] | 188 | | Vptr ptr ⇒ match v2 with |
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| 189 | [ Vint _ n2 ⇒ Vptr (shift_pointer ? ptr n2) |
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[3] | 190 | | _ ⇒ Vundef ] |
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| 191 | | _ ⇒ Vundef ]. |
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| 192 | |
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[961] | 193 | (* XXX Is I32 the best answer for ptr subtraction? *) |
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| 194 | |
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[487] | 195 | definition sub ≝ λv1,v2: val. |
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[3] | 196 | match v1 with |
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[961] | 197 | [ Vint sz1 n1 ⇒ match v2 with |
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| 198 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 199 | (λn1. Vint sz2 (subtraction ? n1 n2)) |
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| 200 | Vundef |
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[3] | 201 | | _ ⇒ Vundef ] |
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[1545] | 202 | | Vptr ptr1 ⇒ match v2 with |
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| 203 | [ Vint sz2 n2 ⇒ Vptr (neg_shift_pointer ? ptr1 n2) |
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| 204 | | Vptr ptr2 ⇒ |
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| 205 | if eq_block (pblock ptr1) (pblock ptr2) |
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| 206 | then Vint I32 (sub_offset ? (poff ptr1) (poff ptr2)) |
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| 207 | else Vundef |
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[3] | 208 | | _ ⇒ Vundef ] |
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[961] | 209 | | Vnull r ⇒ match v2 with [ Vnull r' ⇒ Vzero I32 | _ ⇒ Vundef ] |
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[3] | 210 | | _ ⇒ Vundef ]. |
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| 211 | |
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[487] | 212 | definition mul ≝ λv1, v2: val. |
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[3] | 213 | match v1 with |
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[961] | 214 | [ Vint sz1 n1 ⇒ match v2 with |
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| 215 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 216 | (λn1. Vint sz2 (\snd (split … (multiplication ? n1 n2)))) |
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| 217 | Vundef |
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[3] | 218 | | _ ⇒ Vundef ] |
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| 219 | | _ ⇒ Vundef ]. |
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| 220 | (* |
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[487] | 221 | definition divs ≝ λv1, v2: val. |
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[3] | 222 | match v1 with |
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| 223 | [ Vint n1 ⇒ match v2 with |
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| 224 | [ Vint n2 ⇒ Vint (divs n1 n2) |
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| 225 | | _ ⇒ Vundef ] |
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| 226 | | _ ⇒ Vundef ]. |
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| 227 | |
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| 228 | Definition mods (v1 v2: val): val := |
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| 229 | match v1, v2 with |
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| 230 | | Vint n1, Vint n2 => |
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| 231 | if Int.eq n2 Int.zero then Vundef else Vint(Int.mods n1 n2) |
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| 232 | | _, _ => Vundef |
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| 233 | end. |
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| 234 | |
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| 235 | Definition divu (v1 v2: val): val := |
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| 236 | match v1, v2 with |
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| 237 | | Vint n1, Vint n2 => |
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| 238 | if Int.eq n2 Int.zero then Vundef else Vint(Int.divu n1 n2) |
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| 239 | | _, _ => Vundef |
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| 240 | end. |
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| 241 | |
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| 242 | Definition modu (v1 v2: val): val := |
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| 243 | match v1, v2 with |
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| 244 | | Vint n1, Vint n2 => |
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| 245 | if Int.eq n2 Int.zero then Vundef else Vint(Int.modu n1 n2) |
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| 246 | | _, _ => Vundef |
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| 247 | end. |
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| 248 | *) |
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[487] | 249 | definition v_and ≝ λv1, v2: val. |
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[3] | 250 | match v1 with |
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[961] | 251 | [ Vint sz1 n1 ⇒ match v2 with |
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| 252 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 253 | (λn1. Vint ? (conjunction_bv ? n1 n2)) |
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| 254 | Vundef |
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[3] | 255 | | _ ⇒ Vundef ] |
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| 256 | | _ ⇒ Vundef ]. |
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| 257 | |
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[487] | 258 | definition or ≝ λv1, v2: val. |
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[3] | 259 | match v1 with |
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[961] | 260 | [ Vint sz1 n1 ⇒ match v2 with |
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| 261 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 262 | (λn1. Vint ? (inclusive_disjunction_bv ? n1 n2)) |
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| 263 | Vundef |
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[3] | 264 | | _ ⇒ Vundef ] |
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| 265 | | _ ⇒ Vundef ]. |
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| 266 | |
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[487] | 267 | definition xor ≝ λv1, v2: val. |
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[3] | 268 | match v1 with |
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[961] | 269 | [ Vint sz1 n1 ⇒ match v2 with |
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| 270 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 271 | (λn1. Vint ? (exclusive_disjunction_bv ? n1 n2)) |
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| 272 | Vundef |
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[3] | 273 | | _ ⇒ Vundef ] |
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| 274 | | _ ⇒ Vundef ]. |
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| 275 | (* |
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| 276 | Definition shl (v1 v2: val): val := |
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| 277 | match v1, v2 with |
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| 278 | | Vint n1, Vint n2 => |
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| 279 | if Int.ltu n2 Int.iwordsize |
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| 280 | then Vint(Int.shl n1 n2) |
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| 281 | else Vundef |
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| 282 | | _, _ => Vundef |
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| 283 | end. |
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| 284 | |
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| 285 | Definition shr (v1 v2: val): val := |
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| 286 | match v1, v2 with |
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| 287 | | Vint n1, Vint n2 => |
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| 288 | if Int.ltu n2 Int.iwordsize |
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| 289 | then Vint(Int.shr n1 n2) |
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| 290 | else Vundef |
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| 291 | | _, _ => Vundef |
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| 292 | end. |
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| 293 | |
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| 294 | Definition shr_carry (v1 v2: val): val := |
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| 295 | match v1, v2 with |
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| 296 | | Vint n1, Vint n2 => |
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| 297 | if Int.ltu n2 Int.iwordsize |
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| 298 | then Vint(Int.shr_carry n1 n2) |
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| 299 | else Vundef |
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| 300 | | _, _ => Vundef |
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| 301 | end. |
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| 302 | |
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| 303 | Definition shrx (v1 v2: val): val := |
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| 304 | match v1, v2 with |
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| 305 | | Vint n1, Vint n2 => |
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| 306 | if Int.ltu n2 Int.iwordsize |
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| 307 | then Vint(Int.shrx n1 n2) |
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| 308 | else Vundef |
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| 309 | | _, _ => Vundef |
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| 310 | end. |
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| 311 | |
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| 312 | Definition shru (v1 v2: val): val := |
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| 313 | match v1, v2 with |
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| 314 | | Vint n1, Vint n2 => |
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| 315 | if Int.ltu n2 Int.iwordsize |
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| 316 | then Vint(Int.shru n1 n2) |
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| 317 | else Vundef |
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| 318 | | _, _ => Vundef |
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| 319 | end. |
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| 320 | |
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| 321 | Definition rolm (v: val) (amount mask: int): val := |
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| 322 | match v with |
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| 323 | | Vint n => Vint(Int.rolm n amount mask) |
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| 324 | | _ => Vundef |
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| 325 | end. |
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| 326 | |
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| 327 | Definition ror (v1 v2: val): val := |
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| 328 | match v1, v2 with |
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| 329 | | Vint n1, Vint n2 => |
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| 330 | if Int.ltu n2 Int.iwordsize |
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| 331 | then Vint(Int.ror n1 n2) |
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| 332 | else Vundef |
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| 333 | | _, _ => Vundef |
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| 334 | end. |
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| 335 | *) |
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[487] | 336 | definition addf ≝ λv1,v2: val. |
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[3] | 337 | match v1 with |
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| 338 | [ Vfloat f1 ⇒ match v2 with |
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| 339 | [ Vfloat f2 ⇒ Vfloat (Fadd f1 f2) |
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| 340 | | _ ⇒ Vundef ] |
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| 341 | | _ ⇒ Vundef ]. |
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| 342 | |
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[487] | 343 | definition subf ≝ λv1,v2: val. |
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[3] | 344 | match v1 with |
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| 345 | [ Vfloat f1 ⇒ match v2 with |
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| 346 | [ Vfloat f2 ⇒ Vfloat (Fsub f1 f2) |
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| 347 | | _ ⇒ Vundef ] |
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| 348 | | _ ⇒ Vundef ]. |
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| 349 | |
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[487] | 350 | definition mulf ≝ λv1,v2: val. |
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[3] | 351 | match v1 with |
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| 352 | [ Vfloat f1 ⇒ match v2 with |
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| 353 | [ Vfloat f2 ⇒ Vfloat (Fmul f1 f2) |
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| 354 | | _ ⇒ Vundef ] |
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| 355 | | _ ⇒ Vundef ]. |
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| 356 | |
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[487] | 357 | definition divf ≝ λv1,v2: val. |
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[3] | 358 | match v1 with |
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| 359 | [ Vfloat f1 ⇒ match v2 with |
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| 360 | [ Vfloat f2 ⇒ Vfloat (Fdiv f1 f2) |
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| 361 | | _ ⇒ Vundef ] |
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| 362 | | _ ⇒ Vundef ]. |
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| 363 | |
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[487] | 364 | definition cmp_match : comparison → val ≝ λc. |
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[484] | 365 | match c with |
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| 366 | [ Ceq ⇒ Vtrue |
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| 367 | | Cne ⇒ Vfalse |
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| 368 | | _ ⇒ Vundef |
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| 369 | ]. |
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| 370 | |
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[487] | 371 | definition cmp_mismatch : comparison → val ≝ λc. |
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[3] | 372 | match c with |
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| 373 | [ Ceq ⇒ Vfalse |
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| 374 | | Cne ⇒ Vtrue |
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| 375 | | _ ⇒ Vundef |
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| 376 | ]. |
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| 377 | |
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[583] | 378 | definition cmp_offset ≝ |
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| 379 | λc: comparison. λx,y:offset. |
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| 380 | match c with |
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| 381 | [ Ceq ⇒ eq_offset x y |
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| 382 | | Cne ⇒ ¬eq_offset x y |
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| 383 | | Clt ⇒ lt_offset x y |
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| 384 | | Cle ⇒ ¬lt_offset y x |
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| 385 | | Cgt ⇒ lt_offset y x |
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| 386 | | Cge ⇒ ¬lt_offset x y |
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| 387 | ]. |
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| 388 | |
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[961] | 389 | definition cmp_int : ∀n. comparison → BitVector n → BitVector n → bool ≝ |
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| 390 | λn,c,x,y. |
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| 391 | match c with |
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| 392 | [ Ceq ⇒ eq_bv ? x y |
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| 393 | | Cne ⇒ notb (eq_bv ? x y) |
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| 394 | | Clt ⇒ lt_s ? x y |
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| 395 | | Cle ⇒ notb (lt_s ? y x) |
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| 396 | | Cgt ⇒ lt_s ? y x |
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| 397 | | Cge ⇒ notb (lt_s ? x y) |
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| 398 | ]. |
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| 399 | |
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| 400 | definition cmpu_int : ∀n. comparison → BitVector n → BitVector n → bool ≝ |
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| 401 | λn,c,x,y. |
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| 402 | match c with |
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| 403 | [ Ceq ⇒ eq_bv ? x y |
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| 404 | | Cne ⇒ notb (eq_bv ? x y) |
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| 405 | | Clt ⇒ lt_u ? x y |
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| 406 | | Cle ⇒ notb (lt_u ? y x) |
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| 407 | | Cgt ⇒ lt_u ? y x |
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| 408 | | Cge ⇒ notb (lt_u ? x y) |
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| 409 | ]. |
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| 410 | |
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[487] | 411 | definition cmp ≝ λc: comparison. λv1,v2: val. |
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[3] | 412 | match v1 with |
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[961] | 413 | [ Vint sz1 n1 ⇒ match v2 with |
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| 414 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 415 | (λn1. of_bool (cmp_int ? c n1 n2)) |
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| 416 | Vundef |
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[3] | 417 | | _ ⇒ Vundef ] |
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[1545] | 418 | | Vptr ptr1 ⇒ match v2 with |
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| 419 | [ Vptr ptr2 ⇒ |
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| 420 | if eq_block (pblock ptr1) (pblock ptr2) |
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| 421 | then of_bool (cmp_offset c (poff ptr1) (poff ptr2)) |
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[3] | 422 | else cmp_mismatch c |
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[484] | 423 | | Vnull r2 ⇒ cmp_mismatch c |
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[3] | 424 | | _ ⇒ Vundef ] |
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[484] | 425 | | Vnull r1 ⇒ match v2 with |
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[1545] | 426 | [ Vptr _ ⇒ cmp_mismatch c |
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[484] | 427 | | Vnull r2 ⇒ cmp_match c |
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| 428 | | _ ⇒ Vundef |
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| 429 | ] |
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[3] | 430 | | _ ⇒ Vundef ]. |
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| 431 | |
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[487] | 432 | definition cmpu ≝ λc: comparison. λv1,v2: val. |
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[3] | 433 | match v1 with |
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[961] | 434 | [ Vint sz1 n1 ⇒ match v2 with |
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| 435 | [ Vint sz2 n2 ⇒ intsize_eq_elim ? sz1 sz2 ? n1 |
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| 436 | (λn1. of_bool (cmpu_int ? c n1 n2)) |
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| 437 | Vundef |
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[3] | 438 | | _ ⇒ Vundef ] |
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[1545] | 439 | | Vptr ptr1 ⇒ match v2 with |
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| 440 | [ Vptr ptr2 ⇒ |
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| 441 | if eq_block (pblock ptr1) (pblock ptr2) |
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| 442 | then of_bool (cmp_offset c (poff ptr1) (poff ptr2)) |
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[3] | 443 | else cmp_mismatch c |
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[484] | 444 | | Vnull r2 ⇒ cmp_mismatch c |
---|
[3] | 445 | | _ ⇒ Vundef ] |
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[484] | 446 | | Vnull r1 ⇒ match v2 with |
---|
[1545] | 447 | [ Vptr _ ⇒ cmp_mismatch c |
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[484] | 448 | | Vnull r2 ⇒ cmp_match c |
---|
| 449 | | _ ⇒ Vundef |
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| 450 | ] |
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[3] | 451 | | _ ⇒ Vundef ]. |
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| 452 | |
---|
[961] | 453 | definition cmpf ≝ λc: comparison. λsz:intsize. λv1,v2: val. |
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[3] | 454 | match v1 with |
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| 455 | [ Vfloat f1 ⇒ match v2 with |
---|
| 456 | [ Vfloat f2 ⇒ of_bool (Fcmp c f1 f2) |
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| 457 | | _ ⇒ Vundef ] |
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| 458 | | _ ⇒ Vundef ]. |
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| 459 | |
---|
| 460 | (* * [load_result] is used in the memory model (library [Mem]) |
---|
| 461 | to post-process the results of a memory read. For instance, |
---|
| 462 | consider storing the integer value [0xFFF] on 1 byte at a |
---|
| 463 | given address, and reading it back. If it is read back with |
---|
| 464 | chunk [Mint8unsigned], zero-extension must be performed, resulting |
---|
| 465 | in [0xFF]. If it is read back as a [Mint8signed], sign-extension |
---|
| 466 | is performed and [0xFFFFFFFF] is returned. Type mismatches |
---|
| 467 | (e.g. reading back a float as a [Mint32]) read back as [Vundef]. *) |
---|
[961] | 468 | (* XXX update comment *) |
---|
| 469 | (* XXX is this even necessary now? |
---|
| 470 | should we be able to extract bytes? *) |
---|
[3] | 471 | |
---|
[487] | 472 | let rec load_result (chunk: memory_chunk) (v: val) ≝ |
---|
[3] | 473 | match v with |
---|
[961] | 474 | [ Vint sz n ⇒ |
---|
[3] | 475 | match chunk with |
---|
[961] | 476 | [ Mint8signed ⇒ match sz with [ I8 ⇒ v | _ ⇒ Vundef ] |
---|
| 477 | | Mint8unsigned ⇒ match sz with [ I8 ⇒ v | _ ⇒ Vundef ] |
---|
| 478 | | Mint16signed ⇒ match sz with [ I16 ⇒ v | _ ⇒ Vundef ] |
---|
| 479 | | Mint16unsigned ⇒ match sz with [ I16 ⇒ v | _ ⇒ Vundef ] |
---|
| 480 | | Mint32 ⇒ match sz with [ I32 ⇒ v | _ ⇒ Vundef ] |
---|
[3] | 481 | | _ ⇒ Vundef |
---|
| 482 | ] |
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[1545] | 483 | | Vptr ptr ⇒ |
---|
[483] | 484 | match chunk with |
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[1545] | 485 | [ Mpointer r ⇒ if eq_region (ptype ptr) r then Vptr ptr else Vundef |
---|
[483] | 486 | | _ ⇒ Vundef |
---|
[3] | 487 | ] |
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[484] | 488 | | Vnull r ⇒ |
---|
| 489 | match chunk with |
---|
| 490 | [ Mpointer r' ⇒ if eq_region r r' then Vnull r else Vundef |
---|
| 491 | | _ ⇒ Vundef |
---|
| 492 | ] |
---|
[3] | 493 | | Vfloat f ⇒ |
---|
| 494 | match chunk with |
---|
| 495 | [ Mfloat32 ⇒ Vfloat(singleoffloat f) |
---|
| 496 | | Mfloat64 ⇒ Vfloat f |
---|
| 497 | | _ ⇒ Vundef |
---|
| 498 | ] |
---|
| 499 | | _ ⇒ Vundef |
---|
| 500 | ]. |
---|
| 501 | |
---|
| 502 | (* |
---|
| 503 | (** Theorems on arithmetic operations. *) |
---|
| 504 | |
---|
| 505 | Theorem cast8unsigned_and: |
---|
| 506 | forall x, zero_ext 8 x = and x (Vint(Int.repr 255)). |
---|
| 507 | Proof. |
---|
| 508 | destruct x; simpl; auto. decEq. |
---|
| 509 | change 255 with (two_p 8 - 1). apply Int.zero_ext_and. vm_compute; auto. |
---|
| 510 | Qed. |
---|
| 511 | |
---|
| 512 | Theorem cast16unsigned_and: |
---|
| 513 | forall x, zero_ext 16 x = and x (Vint(Int.repr 65535)). |
---|
| 514 | Proof. |
---|
| 515 | destruct x; simpl; auto. decEq. |
---|
| 516 | change 65535 with (two_p 16 - 1). apply Int.zero_ext_and. vm_compute; auto. |
---|
| 517 | Qed. |
---|
| 518 | |
---|
| 519 | Theorem istrue_not_isfalse: |
---|
| 520 | forall v, is_false v -> is_true (notbool v). |
---|
| 521 | Proof. |
---|
| 522 | destruct v; simpl; try contradiction. |
---|
| 523 | intros. subst i. simpl. discriminate. |
---|
| 524 | Qed. |
---|
| 525 | |
---|
| 526 | Theorem isfalse_not_istrue: |
---|
| 527 | forall v, is_true v -> is_false (notbool v). |
---|
| 528 | Proof. |
---|
| 529 | destruct v; simpl; try contradiction. |
---|
| 530 | intros. generalize (Int.eq_spec i Int.zero). |
---|
| 531 | case (Int.eq i Int.zero); intro. |
---|
| 532 | contradiction. simpl. auto. |
---|
| 533 | auto. |
---|
| 534 | Qed. |
---|
| 535 | |
---|
| 536 | Theorem bool_of_true_val: |
---|
| 537 | forall v, is_true v -> bool_of_val v true. |
---|
| 538 | Proof. |
---|
| 539 | intro. destruct v; simpl; intros; try contradiction. |
---|
| 540 | constructor; auto. constructor. |
---|
| 541 | Qed. |
---|
| 542 | |
---|
| 543 | Theorem bool_of_true_val2: |
---|
| 544 | forall v, bool_of_val v true -> is_true v. |
---|
| 545 | Proof. |
---|
| 546 | intros. inversion H; simpl; auto. |
---|
| 547 | Qed. |
---|
| 548 | |
---|
| 549 | Theorem bool_of_true_val_inv: |
---|
| 550 | forall v b, is_true v -> bool_of_val v b -> b = true. |
---|
| 551 | Proof. |
---|
| 552 | intros. inversion H0; subst v b; simpl in H; auto. |
---|
| 553 | Qed. |
---|
| 554 | |
---|
| 555 | Theorem bool_of_false_val: |
---|
| 556 | forall v, is_false v -> bool_of_val v false. |
---|
| 557 | Proof. |
---|
| 558 | intro. destruct v; simpl; intros; try contradiction. |
---|
| 559 | subst i; constructor. |
---|
| 560 | Qed. |
---|
| 561 | |
---|
| 562 | Theorem bool_of_false_val2: |
---|
| 563 | forall v, bool_of_val v false -> is_false v. |
---|
| 564 | Proof. |
---|
| 565 | intros. inversion H; simpl; auto. |
---|
| 566 | Qed. |
---|
| 567 | |
---|
| 568 | Theorem bool_of_false_val_inv: |
---|
| 569 | forall v b, is_false v -> bool_of_val v b -> b = false. |
---|
| 570 | Proof. |
---|
| 571 | intros. inversion H0; subst v b; simpl in H. |
---|
| 572 | congruence. auto. contradiction. |
---|
| 573 | Qed. |
---|
| 574 | |
---|
| 575 | Theorem notbool_negb_1: |
---|
| 576 | forall b, of_bool (negb b) = notbool (of_bool b). |
---|
| 577 | Proof. |
---|
| 578 | destruct b; reflexivity. |
---|
| 579 | Qed. |
---|
| 580 | |
---|
| 581 | Theorem notbool_negb_2: |
---|
| 582 | forall b, of_bool b = notbool (of_bool (negb b)). |
---|
| 583 | Proof. |
---|
| 584 | destruct b; reflexivity. |
---|
| 585 | Qed. |
---|
| 586 | |
---|
| 587 | Theorem notbool_idem2: |
---|
| 588 | forall b, notbool(notbool(of_bool b)) = of_bool b. |
---|
| 589 | Proof. |
---|
| 590 | destruct b; reflexivity. |
---|
| 591 | Qed. |
---|
| 592 | |
---|
| 593 | Theorem notbool_idem3: |
---|
| 594 | forall x, notbool(notbool(notbool x)) = notbool x. |
---|
| 595 | Proof. |
---|
| 596 | destruct x; simpl; auto. |
---|
| 597 | case (Int.eq i Int.zero); reflexivity. |
---|
| 598 | Qed. |
---|
| 599 | |
---|
| 600 | Theorem add_commut: forall x y, add x y = add y x. |
---|
| 601 | Proof. |
---|
| 602 | destruct x; destruct y; simpl; auto. |
---|
| 603 | decEq. apply Int.add_commut. |
---|
| 604 | Qed. |
---|
| 605 | |
---|
| 606 | Theorem add_assoc: forall x y z, add (add x y) z = add x (add y z). |
---|
| 607 | Proof. |
---|
| 608 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 609 | rewrite Int.add_assoc; auto. |
---|
| 610 | rewrite Int.add_assoc; auto. |
---|
| 611 | decEq. decEq. apply Int.add_commut. |
---|
| 612 | decEq. rewrite Int.add_commut. rewrite <- Int.add_assoc. |
---|
| 613 | decEq. apply Int.add_commut. |
---|
| 614 | decEq. rewrite Int.add_assoc. auto. |
---|
| 615 | Qed. |
---|
| 616 | |
---|
| 617 | Theorem add_permut: forall x y z, add x (add y z) = add y (add x z). |
---|
| 618 | Proof. |
---|
| 619 | intros. rewrite (add_commut y z). rewrite <- add_assoc. apply add_commut. |
---|
| 620 | Qed. |
---|
| 621 | |
---|
| 622 | Theorem add_permut_4: |
---|
| 623 | forall x y z t, add (add x y) (add z t) = add (add x z) (add y t). |
---|
| 624 | Proof. |
---|
| 625 | intros. rewrite add_permut. rewrite add_assoc. |
---|
| 626 | rewrite add_permut. symmetry. apply add_assoc. |
---|
| 627 | Qed. |
---|
| 628 | |
---|
| 629 | Theorem neg_zero: neg Vzero = Vzero. |
---|
| 630 | Proof. |
---|
| 631 | reflexivity. |
---|
| 632 | Qed. |
---|
| 633 | |
---|
| 634 | Theorem neg_add_distr: forall x y, neg(add x y) = add (neg x) (neg y). |
---|
| 635 | Proof. |
---|
| 636 | destruct x; destruct y; simpl; auto. decEq. apply Int.neg_add_distr. |
---|
| 637 | Qed. |
---|
| 638 | |
---|
| 639 | Theorem sub_zero_r: forall x, sub Vzero x = neg x. |
---|
| 640 | Proof. |
---|
| 641 | destruct x; simpl; auto. |
---|
| 642 | Qed. |
---|
| 643 | |
---|
| 644 | Theorem sub_add_opp: forall x y, sub x (Vint y) = add x (Vint (Int.neg y)). |
---|
| 645 | Proof. |
---|
| 646 | destruct x; intro y; simpl; auto; rewrite Int.sub_add_opp; auto. |
---|
| 647 | Qed. |
---|
| 648 | |
---|
| 649 | Theorem sub_opp_add: forall x y, sub x (Vint (Int.neg y)) = add x (Vint y). |
---|
| 650 | Proof. |
---|
| 651 | intros. unfold sub, add. |
---|
| 652 | destruct x; auto; rewrite Int.sub_add_opp; rewrite Int.neg_involutive; auto. |
---|
| 653 | Qed. |
---|
| 654 | |
---|
| 655 | Theorem sub_add_l: |
---|
| 656 | forall v1 v2 i, sub (add v1 (Vint i)) v2 = add (sub v1 v2) (Vint i). |
---|
| 657 | Proof. |
---|
| 658 | destruct v1; destruct v2; intros; simpl; auto. |
---|
| 659 | rewrite Int.sub_add_l. auto. |
---|
| 660 | rewrite Int.sub_add_l. auto. |
---|
| 661 | case (zeq b b0); intro. rewrite Int.sub_add_l. auto. reflexivity. |
---|
| 662 | Qed. |
---|
| 663 | |
---|
| 664 | Theorem sub_add_r: |
---|
| 665 | forall v1 v2 i, sub v1 (add v2 (Vint i)) = add (sub v1 v2) (Vint (Int.neg i)). |
---|
| 666 | Proof. |
---|
| 667 | destruct v1; destruct v2; intros; simpl; auto. |
---|
| 668 | rewrite Int.sub_add_r. auto. |
---|
| 669 | repeat rewrite Int.sub_add_opp. decEq. |
---|
| 670 | repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. |
---|
| 671 | decEq. repeat rewrite Int.sub_add_opp. |
---|
| 672 | rewrite Int.add_assoc. decEq. apply Int.neg_add_distr. |
---|
| 673 | case (zeq b b0); intro. simpl. decEq. |
---|
| 674 | repeat rewrite Int.sub_add_opp. rewrite Int.add_assoc. decEq. |
---|
| 675 | apply Int.neg_add_distr. |
---|
| 676 | reflexivity. |
---|
| 677 | Qed. |
---|
| 678 | |
---|
| 679 | Theorem mul_commut: forall x y, mul x y = mul y x. |
---|
| 680 | Proof. |
---|
| 681 | destruct x; destruct y; simpl; auto. decEq. apply Int.mul_commut. |
---|
| 682 | Qed. |
---|
| 683 | |
---|
| 684 | Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z). |
---|
| 685 | Proof. |
---|
| 686 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 687 | decEq. apply Int.mul_assoc. |
---|
| 688 | Qed. |
---|
| 689 | |
---|
| 690 | Theorem mul_add_distr_l: |
---|
| 691 | forall x y z, mul (add x y) z = add (mul x z) (mul y z). |
---|
| 692 | Proof. |
---|
| 693 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 694 | decEq. apply Int.mul_add_distr_l. |
---|
| 695 | Qed. |
---|
| 696 | |
---|
| 697 | |
---|
| 698 | Theorem mul_add_distr_r: |
---|
| 699 | forall x y z, mul x (add y z) = add (mul x y) (mul x z). |
---|
| 700 | Proof. |
---|
| 701 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 702 | decEq. apply Int.mul_add_distr_r. |
---|
| 703 | Qed. |
---|
| 704 | |
---|
| 705 | Theorem mul_pow2: |
---|
| 706 | forall x n logn, |
---|
| 707 | Int.is_power2 n = Some logn -> |
---|
| 708 | mul x (Vint n) = shl x (Vint logn). |
---|
| 709 | Proof. |
---|
| 710 | intros; destruct x; simpl; auto. |
---|
| 711 | change 32 with (Z_of_nat Int.wordsize). |
---|
| 712 | rewrite (Int.is_power2_range _ _ H). decEq. apply Int.mul_pow2. auto. |
---|
| 713 | Qed. |
---|
| 714 | |
---|
| 715 | Theorem mods_divs: |
---|
| 716 | forall x y, mods x y = sub x (mul (divs x y) y). |
---|
| 717 | Proof. |
---|
| 718 | destruct x; destruct y; simpl; auto. |
---|
| 719 | case (Int.eq i0 Int.zero); simpl. auto. decEq. apply Int.mods_divs. |
---|
| 720 | Qed. |
---|
| 721 | |
---|
| 722 | Theorem modu_divu: |
---|
| 723 | forall x y, modu x y = sub x (mul (divu x y) y). |
---|
| 724 | Proof. |
---|
| 725 | destruct x; destruct y; simpl; auto. |
---|
| 726 | generalize (Int.eq_spec i0 Int.zero); |
---|
| 727 | case (Int.eq i0 Int.zero); simpl. auto. |
---|
| 728 | intro. decEq. apply Int.modu_divu. auto. |
---|
| 729 | Qed. |
---|
| 730 | |
---|
| 731 | Theorem divs_pow2: |
---|
| 732 | forall x n logn, |
---|
| 733 | Int.is_power2 n = Some logn -> |
---|
| 734 | divs x (Vint n) = shrx x (Vint logn). |
---|
| 735 | Proof. |
---|
| 736 | intros; destruct x; simpl; auto. |
---|
| 737 | change 32 with (Z_of_nat Int.wordsize). |
---|
| 738 | rewrite (Int.is_power2_range _ _ H). |
---|
| 739 | generalize (Int.eq_spec n Int.zero); |
---|
| 740 | case (Int.eq n Int.zero); intro. |
---|
| 741 | subst n. compute in H. discriminate. |
---|
| 742 | decEq. apply Int.divs_pow2. auto. |
---|
| 743 | Qed. |
---|
| 744 | |
---|
| 745 | Theorem divu_pow2: |
---|
| 746 | forall x n logn, |
---|
| 747 | Int.is_power2 n = Some logn -> |
---|
| 748 | divu x (Vint n) = shru x (Vint logn). |
---|
| 749 | Proof. |
---|
| 750 | intros; destruct x; simpl; auto. |
---|
| 751 | change 32 with (Z_of_nat Int.wordsize). |
---|
| 752 | rewrite (Int.is_power2_range _ _ H). |
---|
| 753 | generalize (Int.eq_spec n Int.zero); |
---|
| 754 | case (Int.eq n Int.zero); intro. |
---|
| 755 | subst n. compute in H. discriminate. |
---|
| 756 | decEq. apply Int.divu_pow2. auto. |
---|
| 757 | Qed. |
---|
| 758 | |
---|
| 759 | Theorem modu_pow2: |
---|
| 760 | forall x n logn, |
---|
| 761 | Int.is_power2 n = Some logn -> |
---|
| 762 | modu x (Vint n) = and x (Vint (Int.sub n Int.one)). |
---|
| 763 | Proof. |
---|
| 764 | intros; destruct x; simpl; auto. |
---|
| 765 | generalize (Int.eq_spec n Int.zero); |
---|
| 766 | case (Int.eq n Int.zero); intro. |
---|
| 767 | subst n. compute in H. discriminate. |
---|
| 768 | decEq. eapply Int.modu_and; eauto. |
---|
| 769 | Qed. |
---|
| 770 | |
---|
| 771 | Theorem and_commut: forall x y, and x y = and y x. |
---|
| 772 | Proof. |
---|
| 773 | destruct x; destruct y; simpl; auto. decEq. apply Int.and_commut. |
---|
| 774 | Qed. |
---|
| 775 | |
---|
| 776 | Theorem and_assoc: forall x y z, and (and x y) z = and x (and y z). |
---|
| 777 | Proof. |
---|
| 778 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 779 | decEq. apply Int.and_assoc. |
---|
| 780 | Qed. |
---|
| 781 | |
---|
| 782 | Theorem or_commut: forall x y, or x y = or y x. |
---|
| 783 | Proof. |
---|
| 784 | destruct x; destruct y; simpl; auto. decEq. apply Int.or_commut. |
---|
| 785 | Qed. |
---|
| 786 | |
---|
| 787 | Theorem or_assoc: forall x y z, or (or x y) z = or x (or y z). |
---|
| 788 | Proof. |
---|
| 789 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 790 | decEq. apply Int.or_assoc. |
---|
| 791 | Qed. |
---|
| 792 | |
---|
| 793 | Theorem xor_commut: forall x y, xor x y = xor y x. |
---|
| 794 | Proof. |
---|
| 795 | destruct x; destruct y; simpl; auto. decEq. apply Int.xor_commut. |
---|
| 796 | Qed. |
---|
| 797 | |
---|
| 798 | Theorem xor_assoc: forall x y z, xor (xor x y) z = xor x (xor y z). |
---|
| 799 | Proof. |
---|
| 800 | destruct x; destruct y; destruct z; simpl; auto. |
---|
| 801 | decEq. apply Int.xor_assoc. |
---|
| 802 | Qed. |
---|
| 803 | |
---|
| 804 | Theorem shl_mul: forall x y, Val.mul x (Val.shl Vone y) = Val.shl x y. |
---|
| 805 | Proof. |
---|
| 806 | destruct x; destruct y; simpl; auto. |
---|
| 807 | case (Int.ltu i0 Int.iwordsize); auto. |
---|
| 808 | decEq. symmetry. apply Int.shl_mul. |
---|
| 809 | Qed. |
---|
| 810 | |
---|
| 811 | Theorem shl_rolm: |
---|
| 812 | forall x n, |
---|
| 813 | Int.ltu n Int.iwordsize = true -> |
---|
| 814 | shl x (Vint n) = rolm x n (Int.shl Int.mone n). |
---|
| 815 | Proof. |
---|
| 816 | intros; destruct x; simpl; auto. |
---|
| 817 | rewrite H. decEq. apply Int.shl_rolm. exact H. |
---|
| 818 | Qed. |
---|
| 819 | |
---|
| 820 | Theorem shru_rolm: |
---|
| 821 | forall x n, |
---|
| 822 | Int.ltu n Int.iwordsize = true -> |
---|
| 823 | shru x (Vint n) = rolm x (Int.sub Int.iwordsize n) (Int.shru Int.mone n). |
---|
| 824 | Proof. |
---|
| 825 | intros; destruct x; simpl; auto. |
---|
| 826 | rewrite H. decEq. apply Int.shru_rolm. exact H. |
---|
| 827 | Qed. |
---|
| 828 | |
---|
| 829 | Theorem shrx_carry: |
---|
| 830 | forall x y, |
---|
| 831 | add (shr x y) (shr_carry x y) = shrx x y. |
---|
| 832 | Proof. |
---|
| 833 | destruct x; destruct y; simpl; auto. |
---|
| 834 | case (Int.ltu i0 Int.iwordsize); auto. |
---|
| 835 | simpl. decEq. apply Int.shrx_carry. |
---|
| 836 | Qed. |
---|
| 837 | |
---|
| 838 | Theorem or_rolm: |
---|
| 839 | forall x n m1 m2, |
---|
| 840 | or (rolm x n m1) (rolm x n m2) = rolm x n (Int.or m1 m2). |
---|
| 841 | Proof. |
---|
| 842 | intros; destruct x; simpl; auto. |
---|
| 843 | decEq. apply Int.or_rolm. |
---|
| 844 | Qed. |
---|
| 845 | |
---|
| 846 | Theorem rolm_rolm: |
---|
| 847 | forall x n1 m1 n2 m2, |
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| 848 | rolm (rolm x n1 m1) n2 m2 = |
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| 849 | rolm x (Int.modu (Int.add n1 n2) Int.iwordsize) |
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| 850 | (Int.and (Int.rol m1 n2) m2). |
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| 851 | Proof. |
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| 852 | intros; destruct x; simpl; auto. |
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| 853 | decEq. |
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| 854 | apply Int.rolm_rolm. apply int_wordsize_divides_modulus. |
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| 855 | Qed. |
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| 856 | |
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| 857 | Theorem rolm_zero: |
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| 858 | forall x m, |
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| 859 | rolm x Int.zero m = and x (Vint m). |
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| 860 | Proof. |
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| 861 | intros; destruct x; simpl; auto. decEq. apply Int.rolm_zero. |
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| 862 | Qed. |
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| 863 | |
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| 864 | Theorem addf_commut: forall x y, addf x y = addf y x. |
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| 865 | Proof. |
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| 866 | destruct x; destruct y; simpl; auto. decEq. apply Float.addf_commut. |
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| 867 | Qed. |
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| 868 | |
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| 869 | Lemma negate_cmp_mismatch: |
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| 870 | forall c, |
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| 871 | cmp_mismatch (negate_comparison c) = notbool(cmp_mismatch c). |
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| 872 | Proof. |
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| 873 | destruct c; reflexivity. |
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| 874 | Qed. |
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| 875 | |
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| 876 | Theorem negate_cmp: |
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| 877 | forall c x y, |
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| 878 | cmp (negate_comparison c) x y = notbool (cmp c x y). |
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| 879 | Proof. |
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| 880 | destruct x; destruct y; simpl; auto. |
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| 881 | rewrite Int.negate_cmp. apply notbool_negb_1. |
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| 882 | case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity. |
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| 883 | case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity. |
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| 884 | case (zeq b b0); intro. |
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| 885 | rewrite Int.negate_cmp. apply notbool_negb_1. |
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| 886 | apply negate_cmp_mismatch. |
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| 887 | Qed. |
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| 888 | |
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| 889 | Theorem negate_cmpu: |
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| 890 | forall c x y, |
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| 891 | cmpu (negate_comparison c) x y = notbool (cmpu c x y). |
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| 892 | Proof. |
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| 893 | destruct x; destruct y; simpl; auto. |
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| 894 | rewrite Int.negate_cmpu. apply notbool_negb_1. |
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| 895 | case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity. |
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| 896 | case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity. |
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| 897 | case (zeq b b0); intro. |
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| 898 | rewrite Int.negate_cmpu. apply notbool_negb_1. |
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| 899 | apply negate_cmp_mismatch. |
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| 900 | Qed. |
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| 901 | |
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| 902 | Lemma swap_cmp_mismatch: |
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| 903 | forall c, cmp_mismatch (swap_comparison c) = cmp_mismatch c. |
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| 904 | Proof. |
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| 905 | destruct c; reflexivity. |
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| 906 | Qed. |
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| 907 | |
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| 908 | Theorem swap_cmp: |
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| 909 | forall c x y, |
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| 910 | cmp (swap_comparison c) x y = cmp c y x. |
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| 911 | Proof. |
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| 912 | destruct x; destruct y; simpl; auto. |
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| 913 | rewrite Int.swap_cmp. auto. |
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| 914 | case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto. |
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| 915 | case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto. |
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| 916 | case (zeq b b0); intro. |
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| 917 | subst b0. rewrite zeq_true. rewrite Int.swap_cmp. auto. |
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| 918 | rewrite zeq_false. apply swap_cmp_mismatch. auto. |
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| 919 | Qed. |
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| 920 | |
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| 921 | Theorem swap_cmpu: |
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| 922 | forall c x y, |
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| 923 | cmpu (swap_comparison c) x y = cmpu c y x. |
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| 924 | Proof. |
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| 925 | destruct x; destruct y; simpl; auto. |
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| 926 | rewrite Int.swap_cmpu. auto. |
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| 927 | case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto. |
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| 928 | case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto. |
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| 929 | case (zeq b b0); intro. |
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| 930 | subst b0. rewrite zeq_true. rewrite Int.swap_cmpu. auto. |
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| 931 | rewrite zeq_false. apply swap_cmp_mismatch. auto. |
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| 932 | Qed. |
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| 933 | |
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| 934 | Theorem negate_cmpf_eq: |
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| 935 | forall v1 v2, notbool (cmpf Cne v1 v2) = cmpf Ceq v1 v2. |
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| 936 | Proof. |
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| 937 | destruct v1; destruct v2; simpl; auto. |
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| 938 | rewrite Float.cmp_ne_eq. rewrite notbool_negb_1. |
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| 939 | apply notbool_idem2. |
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| 940 | Qed. |
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| 941 | |
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| 942 | Theorem negate_cmpf_ne: |
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| 943 | forall v1 v2, notbool (cmpf Ceq v1 v2) = cmpf Cne v1 v2. |
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| 944 | Proof. |
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| 945 | destruct v1; destruct v2; simpl; auto. |
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| 946 | rewrite Float.cmp_ne_eq. rewrite notbool_negb_1. auto. |
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| 947 | Qed. |
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| 948 | |
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| 949 | Lemma or_of_bool: |
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| 950 | forall b1 b2, or (of_bool b1) (of_bool b2) = of_bool (b1 || b2). |
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| 951 | Proof. |
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| 952 | destruct b1; destruct b2; reflexivity. |
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| 953 | Qed. |
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| 954 | |
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| 955 | Theorem cmpf_le: |
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| 956 | forall v1 v2, cmpf Cle v1 v2 = or (cmpf Clt v1 v2) (cmpf Ceq v1 v2). |
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| 957 | Proof. |
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| 958 | destruct v1; destruct v2; simpl; auto. |
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| 959 | rewrite or_of_bool. decEq. apply Float.cmp_le_lt_eq. |
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| 960 | Qed. |
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| 961 | |
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| 962 | Theorem cmpf_ge: |
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| 963 | forall v1 v2, cmpf Cge v1 v2 = or (cmpf Cgt v1 v2) (cmpf Ceq v1 v2). |
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| 964 | Proof. |
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| 965 | destruct v1; destruct v2; simpl; auto. |
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| 966 | rewrite or_of_bool. decEq. apply Float.cmp_ge_gt_eq. |
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| 967 | Qed. |
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| 968 | |
---|
| 969 | Definition is_bool (v: val) := |
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| 970 | v = Vundef \/ v = Vtrue \/ v = Vfalse. |
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| 971 | |
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| 972 | Lemma of_bool_is_bool: |
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| 973 | forall b, is_bool (of_bool b). |
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| 974 | Proof. |
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| 975 | destruct b; unfold is_bool; simpl; tauto. |
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| 976 | Qed. |
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| 977 | |
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| 978 | Lemma undef_is_bool: is_bool Vundef. |
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| 979 | Proof. |
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| 980 | unfold is_bool; tauto. |
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| 981 | Qed. |
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| 982 | |
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| 983 | Lemma cmp_mismatch_is_bool: |
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| 984 | forall c, is_bool (cmp_mismatch c). |
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| 985 | Proof. |
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| 986 | destruct c; simpl; unfold is_bool; tauto. |
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| 987 | Qed. |
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| 988 | |
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| 989 | Lemma cmp_is_bool: |
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| 990 | forall c v1 v2, is_bool (cmp c v1 v2). |
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| 991 | Proof. |
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| 992 | destruct v1; destruct v2; simpl; try apply undef_is_bool. |
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| 993 | apply of_bool_is_bool. |
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| 994 | case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. |
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| 995 | case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. |
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| 996 | case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool. |
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| 997 | Qed. |
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| 998 | |
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| 999 | Lemma cmpu_is_bool: |
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| 1000 | forall c v1 v2, is_bool (cmpu c v1 v2). |
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| 1001 | Proof. |
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| 1002 | destruct v1; destruct v2; simpl; try apply undef_is_bool. |
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| 1003 | apply of_bool_is_bool. |
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| 1004 | case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. |
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| 1005 | case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. |
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| 1006 | case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool. |
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| 1007 | Qed. |
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| 1008 | |
---|
| 1009 | Lemma cmpf_is_bool: |
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| 1010 | forall c v1 v2, is_bool (cmpf c v1 v2). |
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| 1011 | Proof. |
---|
| 1012 | destruct v1; destruct v2; simpl; |
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| 1013 | apply undef_is_bool || apply of_bool_is_bool. |
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| 1014 | Qed. |
---|
| 1015 | |
---|
| 1016 | Lemma notbool_is_bool: |
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| 1017 | forall v, is_bool (notbool v). |
---|
| 1018 | Proof. |
---|
| 1019 | destruct v; simpl. |
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| 1020 | apply undef_is_bool. apply of_bool_is_bool. |
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| 1021 | apply undef_is_bool. unfold is_bool; tauto. |
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| 1022 | Qed. |
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| 1023 | |
---|
| 1024 | Lemma notbool_xor: |
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| 1025 | forall v, is_bool v -> v = xor (notbool v) Vone. |
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| 1026 | Proof. |
---|
| 1027 | intros. elim H; intro. |
---|
| 1028 | subst v. reflexivity. |
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| 1029 | elim H0; intro; subst v; reflexivity. |
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| 1030 | Qed. |
---|
| 1031 | |
---|
| 1032 | Lemma rolm_lt_zero: |
---|
| 1033 | forall v, rolm v Int.one Int.one = cmp Clt v (Vint Int.zero). |
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| 1034 | Proof. |
---|
| 1035 | intros. destruct v; simpl; auto. |
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| 1036 | transitivity (Vint (Int.shru i (Int.repr (Z_of_nat Int.wordsize - 1)))). |
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| 1037 | decEq. symmetry. rewrite Int.shru_rolm. auto. auto. |
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| 1038 | rewrite Int.shru_lt_zero. destruct (Int.lt i Int.zero); auto. |
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| 1039 | Qed. |
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| 1040 | |
---|
| 1041 | Lemma rolm_ge_zero: |
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| 1042 | forall v, |
---|
| 1043 | xor (rolm v Int.one Int.one) (Vint Int.one) = cmp Cge v (Vint Int.zero). |
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| 1044 | Proof. |
---|
| 1045 | intros. rewrite rolm_lt_zero. destruct v; simpl; auto. |
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| 1046 | destruct (Int.lt i Int.zero); auto. |
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| 1047 | Qed. |
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| 1048 | *) |
---|
| 1049 | (* * The ``is less defined'' relation between values. |
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| 1050 | A value is less defined than itself, and [Vundef] is |
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| 1051 | less defined than any value. *) |
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| 1052 | |
---|
[487] | 1053 | inductive Val_lessdef: val → val → Prop ≝ |
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[3] | 1054 | | lessdef_refl: ∀v. Val_lessdef v v |
---|
| 1055 | | lessdef_undef: ∀v. Val_lessdef Vundef v. |
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| 1056 | |
---|
[487] | 1057 | inductive lessdef_list: list val → list val → Prop ≝ |
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[3] | 1058 | | lessdef_list_nil: |
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| 1059 | lessdef_list (nil ?) (nil ?) |
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| 1060 | | lessdef_list_cons: |
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| 1061 | ∀v1,v2,vl1,vl2. |
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| 1062 | Val_lessdef v1 v2 → lessdef_list vl1 vl2 → |
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| 1063 | lessdef_list (v1 :: vl1) (v2 :: vl2). |
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| 1064 | |
---|
| 1065 | (*Hint Resolve lessdef_refl lessdef_undef lessdef_list_nil lessdef_list_cons.*) |
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[487] | 1066 | (* |
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| 1067 | lemma lessdef_list_inv: |
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[3] | 1068 | ∀vl1,vl2. lessdef_list vl1 vl2 → vl1 = vl2 ∨ in_list ? Vundef vl1. |
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[487] | 1069 | #vl1 elim vl1; |
---|
| 1070 | [ #vl2 #H inversion H; /2/; #h1 #h2 #t1 #t2 #H1 #H2 #H3 #Hbad destruct |
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| 1071 | | #h #t #IH #vl2 #H |
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| 1072 | inversion H; |
---|
| 1073 | [ #H' destruct |
---|
| 1074 | | #h1 #h2 #t1 #t2 #H1 #H2 #H3 #e1 #e2 destruct; |
---|
| 1075 | elim H1; |
---|
| 1076 | [ elim (IH t2 H2); |
---|
| 1077 | [ #e destruct; /2/; |
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| 1078 | | /3/ ] |
---|
| 1079 | | /3/ ] |
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| 1080 | ] |
---|
| 1081 | ] qed. |
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| 1082 | *) |
---|
| 1083 | lemma load_result_lessdef: |
---|
[3] | 1084 | ∀chunk,v1,v2. |
---|
| 1085 | Val_lessdef v1 v2 → Val_lessdef (load_result chunk v1) (load_result chunk v2). |
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[1510] | 1086 | #chunk #v1 #v2 #H inversion H; //; #v #e1 #e2 #e3 cases chunk |
---|
| 1087 | [ 8: #r ] whd in ⊢ (?%?); //; |
---|
[487] | 1088 | qed. |
---|
[3] | 1089 | |
---|
[744] | 1090 | lemma zero_ext_lessdef: |
---|
| 1091 | ∀n,v1,v2. Val_lessdef v1 v2 → Val_lessdef (zero_ext n v1) (zero_ext n v2). |
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[1339] | 1092 | #n #v1 #v2 #H inversion H // |
---|
[744] | 1093 | qed. |
---|
| 1094 | |
---|
[487] | 1095 | lemma sign_ext_lessdef: |
---|
[3] | 1096 | ∀n,v1,v2. Val_lessdef v1 v2 → Val_lessdef (sign_ext n v1) (sign_ext n v2). |
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[1339] | 1097 | #n #v1 #v2 #H inversion H // |
---|
[487] | 1098 | qed. |
---|
[3] | 1099 | (* |
---|
| 1100 | Lemma singleoffloat_lessdef: |
---|
| 1101 | forall v1 v2, lessdef v1 v2 -> lessdef (singleoffloat v1) (singleoffloat v2). |
---|
| 1102 | Proof. |
---|
| 1103 | intros; inv H; simpl; auto. |
---|
| 1104 | Qed. |
---|
| 1105 | |
---|
| 1106 | End Val. |
---|
| 1107 | *) |
---|