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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 Import</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></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">value</span> : <span class="kt">Type</span> := int.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
<span class="sd">(** ** Value conversions</span>
<span class="sd">Various conversions to different number types such as [N], [Z], [positive] and</span>
<span class="sd">[int], where the last one is a theory of integers of powers of 2 in CompCert. *)</span>
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToNat</span> (<span class="nv">v</span> : value) : nat :=
Z.to_nat (Int.unsigned v).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">natToValue</span> (<span class="nv">n</span> : nat) : value :=
Int.repr (Z.of_nat n).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToN</span> (<span class="nv">v</span> : value) : N :=
Z.to_N (Int.unsigned v).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">NToValue</span> (<span class="nv">n</span> : N) : value :=
Int.repr (Z.of_N n).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">ZToValue</span> (<span class="nv">z</span> : Z) : value :=
Int.repr z.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToZ</span> (<span class="nv">v</span> : value) : Z :=
Int.signed v.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">uvalueToZ</span> (<span class="nv">v</span> : value) : Z :=
Int.unsigned v.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">posToValue</span> (<span class="nv">p</span> : positive) : value :=
Int.repr (Z.pos p).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToPos</span> (<span class="nv">v</span> : value) : positive :=
Z.to_pos (Int.unsigned v).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">intToValue</span> (<span class="nv">i</span> : Integers.int) : value := i.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToInt</span> (<span class="nv">i</span> : value) : Integers.int := i.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">ptrToValue</span> (<span class="nv">i</span> : ptrofs) : value := Ptrofs.to_int i.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToPtr</span> (<span class="nv">i</span> : value) : Integers.ptrofs :=
Ptrofs.of_int i.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valToValue</span> (<span class="nv">v</span> : Values.val) : option value :=
<span class="kr">match</span> v <span class="kr">with</span>
| Values.Vint i => Some (intToValue i)
| Values.Vptr b off => Some (ptrToValue off)
| Values.Vundef => Some (ZToValue <span class="mi">0</span>%Z)
| _ => None
<span class="kr">end</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
<span class="sd">(** Convert a [value] to a [bool], so that choices can be made based on the</span>
<span class="sd">result. This is also because comparison operators will give back [value] instead</span>
<span class="sd">of [bool], so if they are in a condition, they will have to be converted before</span>
<span class="sd">they can be used. *)</span>
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">valueToBool</span> (<span class="nv">v</span> : value) : bool :=
<span class="kr">if</span> Z.eqb (uvalueToZ v) <span class="mi">0</span> <span class="kr">then</span> false <span class="kr">else</span> true.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Definition</span> <span class="nf">boolToValue</span> (<span class="nv">b</span> : bool) : value :=
natToValue (<span class="kr">if</span> b <span class="kr">then</span> <span class="mi">1</span> <span class="kr">else</span> <span class="mi">0</span>).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
<span class="sd">(** ** Arithmetic operations *)</span>
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Inductive</span> <span class="nf">val_value_lessdef</span>: val -> value -> <span class="kt">Prop</span> :=
| val_value_lessdef_int:
<span class="kr">forall</span> <span class="nv">i</span> <span class="nv">v'</span>,
i = valueToInt v' ->
val_value_lessdef (Vint i) v'
| val_value_lessdef_ptr:
<span class="kr">forall</span> <span class="nv">b</span> <span class="nv">off</span> <span class="nv">v'</span>,
off = valueToPtr v' ->
val_value_lessdef (Vptr b off) v'
| lessdef_undef: <span class="kr">forall</span> <span class="nv">v</span>, val_value_lessdef Vundef v.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Inductive</span> <span class="nf">opt_val_value_lessdef</span>: option val -> value -> <span class="kt">Prop</span> :=
| opt_lessdef_some:
<span class="kr">forall</span> <span class="nv">v</span> <span class="nv">v'</span>, val_value_lessdef v v' -> opt_val_value_lessdef (Some v) v'
| opt_lessdef_none: <span class="kr">forall</span> <span class="nv">v</span>, opt_val_value_lessdef None v.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk0" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk0"><span class="highlight"><span class="kn">Lemma</span> <span class="nf">valueToZ_ZToValue</span> :
<span class="kr">forall</span> <span class="nv">z</span>,
(Int.min_signed <= z <= Int.max_signed)%Z ->
valueToZ (ZToValue z) = z.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">z</span> : Z,
Int.min_signed <= z <= Int.max_signed ->
valueToZ (ZToValue z) = z</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk1" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk1"><span class="highlight"><span class="kn">Proof</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">z</span> : Z,
Int.min_signed <= z <= Int.max_signed ->
valueToZ (ZToValue z) = z</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="nb">auto using</span> Int.signed_repr.</span></span><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Qed</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk2" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk2"><span class="highlight"><span class="kn">Lemma</span> <span class="nf">uvalueToZ_ZToValue</span> :
<span class="kr">forall</span> <span class="nv">z</span>,
(<span class="mi">0</span> <= z <= Int.max_unsigned)%Z ->
uvalueToZ (ZToValue z) = z.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">z</span> : Z,
<span class="mi">0</span> <= z <= Int.max_unsigned ->
uvalueToZ (ZToValue z) = z</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk3" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk3"><span class="highlight"><span class="kn">Proof</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">z</span> : Z,
<span class="mi">0</span> <= z <= Int.max_unsigned ->
uvalueToZ (ZToValue z) = z</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="nb">auto using</span> Int.unsigned_repr.</span></span><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Qed</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk4" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk4"><span class="highlight"><span class="kn">Lemma</span> <span class="nf">valueToPos_posToValue</span> :
<span class="kr">forall</span> <span class="nv">v</span>,
<span class="mi">0</span> <= Z.pos v <= Int.max_unsigned ->
valueToPos (posToValue v) = v.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">v</span> : positive,
<span class="mi">0</span> <= Z.pos v <= Int.max_unsigned ->
valueToPos (posToValue v) = v</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk5" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk5"><span class="highlight"><span class="kn">Proof</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">v</span> : positive,
<span class="mi">0</span> <= Z.pos v <= Int.max_unsigned ->
valueToPos (posToValue v) = v</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk6" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk6"><span class="highlight"><span class="nb">unfold</span> valueToPos, posToValue.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">v</span> : positive,
<span class="mi">0</span> <= Z.pos v <= Int.max_unsigned ->
Z.to_pos (Int.unsigned (Int.repr (Z.pos v))) = v</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk7" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk7"><span class="highlight"><span class="nb">intros</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">v</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">positive</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight"><span class="mi">0</span> <= Z.pos v <= Int.max_unsigned</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">Z.to_pos (Int.unsigned (Int.repr (Z.pos v))) = v</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk8" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk8"><span class="highlight"><span class="nb">rewrite</span> Int.unsigned_repr.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">v</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">positive</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight"><span class="mi">0</span> <= Z.pos v <= Int.max_unsigned</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">Z.to_pos (Z.pos v) = v</span></div></blockquote><div class="coq-extra-goals"><input class="coq-extra-goal-toggle" id="valueint-v-chk9" style="display: none" type="checkbox"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">v</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">positive</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight"><span class="mi">0</span> <= Z.pos v <= Int.max_unsigned</span></span></span></span></div></div><label class="goal-separator coq-extra-goal-label" for="valueint-v-chk9"><hr></label><div class="goal-conclusion"><span class="highlight"><span class="mi">0</span> <= Z.pos v <= Int.max_unsigned</span></div></blockquote></div></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chka" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chka"><span class="highlight"><span class="nb">apply</span> Pos2Z.id.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">v</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">positive</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight"><span class="mi">0</span> <= Z.pos v <= Int.max_unsigned</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">0</span> <= Z.pos v <= Int.max_unsigned</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="bp">assumption</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Qed</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chkb" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chkb"><span class="highlight"><span class="kn">Lemma</span> <span class="nf">valueToInt_intToValue</span> :
<span class="kr">forall</span> <span class="nv">v</span>,
valueToInt (intToValue v) = v.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">v</span> : int, valueToInt (intToValue v) = v</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chkc" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chkc"><span class="highlight"><span class="kn">Proof</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> <span class="nv">v</span> : int, valueToInt (intToValue v) = v</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="nb">auto</span>.</span></span><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Qed</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chkd" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chkd"><span class="highlight"><span class="kn">Lemma</span> <span class="nf">valToValue_lessdef</span> :
<span class="kr">forall</span> <span class="nv">v</span> <span class="nv">v'</span>,
valToValue v = Some v' ->
val_value_lessdef v v'.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> (<span class="nv">v</span> : val) (<span class="nv">v'</span> : value),
valToValue v = Some v' -> val_value_lessdef v v'</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chke" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chke"><span class="highlight"><span class="kn">Proof</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">forall</span> (<span class="nv">v</span> : val) (<span class="nv">v'</span> : value),
valToValue v = Some v' -> val_value_lessdef v v'</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chkf" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chkf"><span class="highlight"><span class="nb">intros</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">v</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">val</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue v = Some v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">val_value_lessdef v v'</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk10" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk10"><span class="highlight"><span class="nb">destruct</span> v; <span class="kp">try</span> <span class="bp">discriminate</span>; <span class="nb">constructor</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">int</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vint i) = Some v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = valueToInt v'</span></div></blockquote><div class="coq-extra-goals"><input class="coq-extra-goal-toggle" id="valueint-v-chk11" style="display: none" type="checkbox"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vptr b i) = Some v'</span></span></span></span></div></div><label class="goal-separator coq-extra-goal-label" for="valueint-v-chk11"><hr></label><div class="goal-conclusion"><span class="highlight">i = valueToPtr v'</span></div></blockquote></div></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk12" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk12"><span class="highlight"><span class="nb">unfold</span> valToValue <span class="kr">in</span> H.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">int</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">Some (intToValue i) = Some v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = valueToInt v'</span></div></blockquote><div class="coq-extra-goals"><input class="coq-extra-goal-toggle" id="valueint-v-chk13" style="display: none" type="checkbox"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vptr b i) = Some v'</span></span></span></span></div></div><label class="goal-separator coq-extra-goal-label" for="valueint-v-chk13"><hr></label><div class="goal-conclusion"><span class="highlight">i = valueToPtr v'</span></div></blockquote></div></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk14" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk14"><span class="highlight"><span class="nb">inversion</span> H.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">int</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">Some (intToValue i) = Some v'</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H1</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">intToValue i = v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = valueToInt v'</span></div></blockquote><div class="coq-extra-goals"><input class="coq-extra-goal-toggle" id="valueint-v-chk15" style="display: none" type="checkbox"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vptr b i) = Some v'</span></span></span></span></div></div><label class="goal-separator coq-extra-goal-label" for="valueint-v-chk15"><hr></label><div class="goal-conclusion"><span class="highlight">i = valueToPtr v'</span></div></blockquote></div></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk16" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk16"><span class="highlight"><span class="nb">unfold</span> valueToInt.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">int</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">Some (intToValue i) = Some v'</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H1</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">intToValue i = v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = v'</span></div></blockquote><div class="coq-extra-goals"><input class="coq-extra-goal-toggle" id="valueint-v-chk17" style="display: none" type="checkbox"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vptr b i) = Some v'</span></span></span></span></div></div><label class="goal-separator coq-extra-goal-label" for="valueint-v-chk17"><hr></label><div class="goal-conclusion"><span class="highlight">i = valueToPtr v'</span></div></blockquote></div></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk18" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk18"><span class="highlight"><span class="nb">unfold</span> intToValue <span class="kr">in</span> H1.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">int</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">Some (intToValue i) = Some v'</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H1</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">i = v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = v'</span></div></blockquote><div class="coq-extra-goals"><input class="coq-extra-goal-toggle" id="valueint-v-chk19" style="display: none" type="checkbox"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vptr b i) = Some v'</span></span></span></span></div></div><label class="goal-separator coq-extra-goal-label" for="valueint-v-chk19"><hr></label><div class="goal-conclusion"><span class="highlight">i = valueToPtr v'</span></div></blockquote></div></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk1a" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk1a"><span class="highlight"><span class="nb">auto</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">v'</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">value</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">H</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">valToValue (Vptr b i) = Some v'</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = valueToPtr v'</span></div></blockquote></div></div></small><span class="coq-wsp">
</span></span><span class="coq-wsp"> </span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk1b" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk1b"><span class="highlight">inv H.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">i = valueToPtr (ptrToValue i)</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk1c" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk1c"><span class="highlight"><span class="nb">symmetry</span>.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">valueToPtr (ptrToValue i) = i</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk1d" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk1d"><span class="highlight"><span class="nb">unfold</span> valueToPtr, ptrToValue.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">Ptrofs.of_int (Ptrofs.to_int i) = i</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><input class="coq-toggle" id="valueint-v-chk1e" style="display: none" type="checkbox"><label class="coq-input" for="valueint-v-chk1e"><span class="highlight"><span class="nb">apply</span> Ptrofs.of_int_to_int.</span></label><small class="coq-output"><div class="coq-output-sticky-wrapper"><div class="coq-goals"><blockquote class="coq-goal"><div class="goal-hyps"><div class="goal-hyp"><span class="hyp-names">b</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">block</span></span></span></span></div><div class="goal-hyp"><span class="hyp-names">i</span><span><span class="hyp-type-block"><span class="hyp-punct">:</span><span class="hyp-type"><span class="highlight">ptrofs</span></span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight">Archi.ptr64 = false</span></div></blockquote></div></div></small><span class="coq-wsp"> </span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="nb">trivial</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Qed</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Ltac</span> <span class="nf">simplify_val</span> := <span class="kp">repeat</span> (simplify; <span class="nb">unfold</span> uvalueToZ, valueToPtr, Ptrofs.of_int, valueToInt, intToValue,
ptrToValue <span class="kr">in</span> *).</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Ltac</span> <span class="nf">crush_val</span> := simplify_val; <span class="kp">try</span> <span class="bp">discriminate</span>; <span class="kp">try</span> <span class="bp">congruence</span>; <span class="kp">try</span> <span class="bp">lia</span>; liapp; <span class="kp">try</span> <span class="bp">assumption</span>.</span></span></span></pre>
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