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<pre class="alectryon-io"><!-- Generator: Alectryon v1.0 --><span class="coq-wsp"><span class="highlight"><span class="c">(*</span>
<span class="c"> * Vericert: Verified high-level synthesis.</span>
<span class="c"> * Copyright (C) 2020 Yann Herklotz <yann@yannherklotz.com></span>
<span class="c"> *</span>
<span class="c"> * This program is free software: you can redistribute it and/or modify</span>
<span class="c"> * it under the terms of the GNU General Public License as published by</span>
<span class="c"> * the Free Software Foundation, either version 3 of the License, or</span>
<span class="c"> * (at your option) any later version.</span>
<span class="c"> *</span>
<span class="c"> * This program is distributed in the hope that it will be useful,</span>
<span class="c"> * but WITHOUT ANY WARRANTY; without even the implied warranty of</span>
<span class="c"> * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the</span>
<span class="c"> * GNU General Public License for more details.</span>
<span class="c"> *</span>
<span class="c"> * You should have received a copy of the GNU General Public License</span>
<span class="c"> * along with this program. If not, see <https://www.gnu.org/licenses/>.</span>
<span class="c"> *)</span>
<span class="c">(* begin hide *)</span>
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">From</span> Coq <span class="kn">Require Import</span> ZArith.ZArith FSets.FMapPositive Lia.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">From</span> compcert <span class="kn">Require Export</span> lib.Integers common.Values.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">From</span> vericert <span class="kn">Require Import</span> Vericertlib.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight"><span class="c">(* end hide *)</span>
<span class="sd">(** * Value</span>
<span class="sd">A [value] is a bitvector with a specific size. We are using the implementation</span>
<span class="sd">of the bitvector by mit-plv/bbv, because it has many theorems that we can reuse.</span>
<span class="sd">However, we need to wrap it with an [Inductive] so that we can specify and match</span>
<span class="sd">on the size of the [value]. This is necessary so that we can easily store</span>
<span class="sd">[value]s of different sizes in a list or in a map.</span>
<span class="sd">Using the default [word], this would not be possible, as the size is part of the type. *)</span>
<span class="c">(* Definition value : Type := val.</span>
<span class="c">(** ** Value conversions</span>
<span class="c">Various conversions to different number types such as [N], [Z], [positive] and</span>
<span class="c">[int], where the last one is a theory of integers of powers of 2 in CompCert. *)</span>
<span class="c">Definition valueToNat (v : value) : nat :=</span>
<span class="c"> match v with</span>
<span class="c"> | value_bool b => Nat.b2n b</span>
<span class="c"> | value_int i => Z.to_nat (Int.unsigned i)</span>
<span class="c"> | value_int64 i => Z.to_nat (Int64.unsigned i)</span>
<span class="c"> end.</span>
<span class="c">Definition natToValue (n : nat) : value :=</span>
<span class="c"> value_int (Int.repr (Z.of_nat n)).</span>
<span class="c">Definition natToValue64 (n : nat) : value :=</span>
<span class="c"> value_int64 (Int64.repr (Z.of_nat n)).</span>
<span class="c">Definition valueToN (v : value) : N :=</span>
<span class="c"> match v with</span>
<span class="c"> | value_bool b => N.b2n b</span>
<span class="c"> | value_int i => Z.to_N (Int.unsigned i)</span>
<span class="c"> | value_int64 i => Z.to_N (Int64.unsigned i)</span>
<span class="c"> end.</span>
<span class="c">Definition NToValue (n : N) : value :=</span>
<span class="c"> value_int (Int.repr (Z.of_N n)).</span>
<span class="c">Definition NToValue64 (n : N) : value :=</span>
<span class="c"> value_int64 (Int64.repr (Z.of_N n)).</span>
<span class="c">Definition ZToValue (z : Z) : value :=</span>
<span class="c"> value_int (Int.repr z).</span>
<span class="c">Definition ZToValue64 (z : Z) : value :=</span>
<span class="c"> value_int64 (Int64.repr z).</span>
<span class="c">Definition valueToZ (v : value) : Z :=</span>
<span class="c"> match v with</span>
<span class="c"> | value_bool b => Z.b2z b</span>
<span class="c"> | value_int i => Int.signed i</span>
<span class="c"> | value_int64 i => Int64.signed i</span>
<span class="c"> end.</span>
<span class="c">Definition uvalueToZ (v : value) : Z :=</span>
<span class="c"> match v with</span>
<span class="c"> | value_bool b => Z.b2z b</span>
<span class="c"> | value_int i => Int.unsigned i</span>
<span class="c"> | value_int64 i => Int64.unsigned i</span>
<span class="c"> end.</span>
<span class="c">Definition posToValue (p : positive) : value :=</span>
<span class="c"> value_int (Int.repr (Z.pos p)).</span>
<span class="c">Definition posToValue64 (p : positive) : value :=</span>
<span class="c"> value_int64 (Int64.repr (Z.pos p)).</span>
<span class="c">Definition valueToPos (v : value) : positive :=</span>
<span class="c"> match v with</span>
<span class="c"> | value_bool b => 1%positive</span>
<span class="c"> | value_int i => Z.to_pos (Int.unsigned i)</span>
<span class="c"> | value_int64 i => Z.to_pos (Int64.unsigned i)</span>
<span class="c"> end.</span>
<span class="c">Definition intToValue (i : Integers.int) : value := value_int i.</span>
<span class="c">Definition int64ToValue (i : Integers.int64) : value := value_int64 i.</span>
<span class="c">Definition valueToInt (v : value) : Integers.int :=</span>
<span class="c"> match v with</span>
<span class="c"> | value_bool b => Int.repr (if b then 1 else 0)</span>
<span class="c"> | value_int i => i</span>
<span class="c"> | value_int64 i => Int.repr (Int64.unsigned i)</span>
<span class="c"> end.</span>
<span class="c">(*Definition ptrToValue (i : ptrofs) : value :=</span>
<span class="c"> value_int (Ptrofs.to_int i).</span>
<span class="c">Definition valueToPtr (i : value) : Integers.ptrofs :=</span>
<span class="c"> Ptrofs.of_int i.</span>
<span class="c">Definition valToValue (v : Values.val) : option value :=</span>
<span class="c"> match v with</span>
<span class="c"> | Values.Vint i => Some (intToValue i)</span>
<span class="c"> | Values.Vint64 i => Some (intToValue i)</span>
<span class="c"> | Values.Vptr b off => Some (ptrToValue off)</span>
<span class="c"> | Values.Vundef => Some (ZToValue 0%Z)</span>
<span class="c"> | _ => None</span>
<span class="c"> end.</span>
<span class="c">(** Convert a [value] to a [bool], so that choices can be made based on the</span>
<span class="c">result. This is also because comparison operators will give back [value] instead</span>
<span class="c">of [bool], so if they are in a condition, they will have to be converted before</span>
<span class="c">they can be used. *)</span>
<span class="c">Definition valueToBool (v : value) : bool :=</span>
<span class="c"> if Z.eqb (uvalueToZ v) 0 then false else true.</span>
<span class="c">Definition boolToValue (b : bool) : value :=</span>
<span class="c"> natToValue (if b then 1 else 0).</span>
<span class="c">(** ** Arithmetic operations *)</span>
<span class="c">Definition unify_word (sz1 sz2 : nat) (w1 : word sz2): sz1 = sz2 -> word sz1.</span>
<span class="c">intros; subst; assumption. Defined.</span>
<span class="c">Lemma unify_word_unfold :</span>
<span class="c"> forall sz w,</span>
<span class="c"> unify_word sz sz w eq_refl = w.</span>
<span class="c">Proof. auto. Qed.</span>
<span class="c">Inductive val_value_lessdef: val -> value -> Prop :=</span>
<span class="c">| val_value_lessdef_int:</span>
<span class="c"> forall i v',</span>
<span class="c"> i = valueToInt v' -></span>
<span class="c"> val_value_lessdef (Vint i) v'</span>
<span class="c">| val_value_lessdef_ptr:</span>
<span class="c"> forall b off v',</span>
<span class="c"> off = valueToPtr v' -></span>
<span class="c"> val_value_lessdef (Vptr b off) v'</span>
<span class="c">| lessdef_undef: forall v, val_value_lessdef Vundef v.</span>
<span class="c">Inductive opt_val_value_lessdef: option val -> value -> Prop :=</span>
<span class="c">| opt_lessdef_some:</span>
<span class="c"> forall v v', val_value_lessdef v v' -> opt_val_value_lessdef (Some v) v'</span>
<span class="c">| opt_lessdef_none: forall v, opt_val_value_lessdef None v.</span>
<span class="c">Lemma valueToZ_ZToValue :</span>
<span class="c"> forall z,</span>
<span class="c"> (Int.min_signed <= z <= Int.max_signed)%Z -></span>
<span class="c"> valueToZ (ZToValue z) = z.</span>
<span class="c">Proof. auto using Int.signed_repr. Qed.</span>
<span class="c">Lemma uvalueToZ_ZToValue :</span>
<span class="c"> forall z,</span>
<span class="c"> (0 <= z <= Int.max_unsigned)%Z -></span>
<span class="c"> uvalueToZ (ZToValue z) = z.</span>
<span class="c">Proof. auto using Int.unsigned_repr. Qed.</span>
<span class="c">Lemma valueToPos_posToValue :</span>
<span class="c"> forall v,</span>
<span class="c"> 0 <= Z.pos v <= Int.max_unsigned -></span>
<span class="c"> valueToPos (posToValue v) = v.</span>
<span class="c">Proof.</span>
<span class="c"> unfold valueToPos, posToValue.</span>
<span class="c"> intros. rewrite Int.unsigned_repr.</span>
<span class="c"> apply Pos2Z.id. assumption.</span>
<span class="c">Qed.</span>
<span class="c">Lemma valueToInt_intToValue :</span>
<span class="c"> forall v,</span>
<span class="c"> valueToInt (intToValue v) = v.</span>
<span class="c">Proof. auto. Qed.</span>
<span class="c">Lemma valToValue_lessdef :</span>
<span class="c"> forall v v',</span>
<span class="c"> valToValue v = Some v' -></span>
<span class="c"> val_value_lessdef v v'.</span>
<span class="c">Proof.</span>
<span class="c"> intros.</span>
<span class="c"> destruct v; try discriminate; constructor.</span>
<span class="c"> unfold valToValue in H. inversion H.</span>
<span class="c"> unfold valueToInt. unfold intToValue in H1. auto.</span>
<span class="c"> inv H. symmetry. unfold valueToPtr, ptrToValue. apply Ptrofs.of_int_to_int. trivial.</span>
<span class="c">Qed.</span>
<span class="c">Ltac simplify_val := repeat (simplify; unfold uvalueToZ, valueToPtr, Ptrofs.of_int, valueToInt, intToValue,</span>
<span class="c"> ptrToValue in *)</span>
<span class="c">(*Ltac crush_val := simplify_val; try discriminate; try congruence; try lia; liapp; try assumption.*)</span>
<span class="c">*)</span></span></span></pre>
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