docs/proof/Vericertlib.html


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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) 2019-2020 Yann Herklotz &lt;yann@yannherklotz.com&gt;</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 &lt;https://www.gnu.org/licenses/&gt;.</span>
<span class="c"> *)</span>

</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">From</span> Coq <span class="kn">Require Export</span>
     String
     ZArith
     Znumtheory
     List
     Bool.</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">Require Import</span> Lia.</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">From</span> vericert <span class="kn">Require Import</span> <span class="kn">Show</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
<span class="c">(* Depend on CompCert for the basic library, as they declare and prove some</span>
<span class="c">   useful theorems. *)</span>
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">From</span> compcert.lib <span class="kn">Require Export</span> Coqlib.</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> Integers.</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">Local Open Scope</span> Z_scope.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
<span class="c">(* This tactic due to Clement Pit-Claudel with some minor additions by JDP to</span>
<span class="c">   allow the result to be named: https://pit-claudel.fr/clement/MSc/#org96a1b5f *)</span>
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Inductive</span> <span class="nf">Learnt</span> {<span class="nv">A</span>: <span class="kt">Type</span>} (<span class="nv">a</span>: A) :=
  | AlreadyKnown : Learnt a.</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">learn_tac</span> fact name :=
  <span class="kr">lazymatch goal with</span>
  | [ H: Learnt fact |- _ ] =&gt;
    <span class="kp">fail</span> <span class="mi">0</span> <span class="s2">&quot;fact&quot;</span> fact <span class="s2">&quot;has already been learnt&quot;</span>
  | _ =&gt; <span class="kr">let</span> <span class="nv">type</span> := <span class="kp">type of</span> fact <span class="kr">in</span>
        <span class="kr">lazymatch goal with</span>
        | [ H: @Learnt type _ |- _ ] =&gt;
          <span class="kp">fail</span> <span class="mi">0</span> <span class="s2">&quot;fact&quot;</span> fact <span class="s2">&quot;of type&quot;</span> type <span class="s2">&quot;was already learnt through&quot;</span> H
        | _ =&gt; <span class="kr">let</span> <span class="nv">learnt</span> := <span class="kp">fresh</span> <span class="s2">&quot;Learn&quot;</span> <span class="kr">in</span>
              <span class="nb">pose proof</span> (AlreadyKnown fact) <span class="kr">as</span> learnt; <span class="nb">pose proof</span> fact <span class="kr">as</span> name
        <span class="kr">end</span>
  <span class="kr">end</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">Tactic Notation</span> <span class="s2">&quot;learn&quot;</span> <span class="kp">constr</span>(fact) := <span class="kr">let</span> <span class="nv">name</span> := <span class="kp">fresh</span> <span class="s2">&quot;H&quot;</span> <span class="kr">in</span> learn_tac fact name.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Tactic Notation</span> <span class="s2">&quot;learn&quot;</span> <span class="kp">constr</span>(fact) <span class="s2">&quot;as&quot;</span> simple_intropattern(name) := learn_tac fact name.</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">unfold_rec</span> c := <span class="nb">unfold</span> c; <span class="nb">fold</span> c.</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">solve_by_inverts</span> n :=
  <span class="kr">match goal with</span> | H : <span class="nl">?T</span> |- _ =&gt;
    <span class="kr">match</span> <span class="kp">type of</span> T <span class="kr">with</span> <span class="kt">Prop</span> =&gt;
      <span class="nb">inversion</span> H;
      <span class="kr">match</span> n <span class="kr">with</span> S (S (<span class="nl">?n&#39;</span>)) =&gt; <span class="nb">subst</span>; <span class="kp">try</span> <span class="nb">constructor</span>; solve_by_inverts (S n&#39;) <span class="kr">end</span>
    <span class="kr">end</span>
  <span class="kr">end</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">solve_by_invert</span> := solve_by_inverts <span class="mi">1</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">invert</span> x := <span class="nb">inversion</span> x; <span class="nb">subst</span>; <span class="nb">clear</span> x.</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">destruct_match</span> :=
  <span class="kr">match goal with</span> | [ |- <span class="kp">context</span>[<span class="kr">match</span> <span class="nl">?x</span> <span class="kr">with</span> | _ =&gt; _ <span class="kr">end</span> ] ] =&gt; <span class="nb">destruct</span> x <span class="kr">end</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">nicify_hypotheses</span> :=
  <span class="kp">repeat</span> <span class="kr">match goal with</span>
         | [ H : ex _ |- _ ] =&gt; invert H
         | [ H : Some _ = Some _ |- _ ] =&gt; invert H
         | [ H : <span class="nl">?x</span> = <span class="nl">?x</span> |- _ ] =&gt; <span class="nb">clear</span> H
         | [ H : _ /\ _ |- _ ] =&gt; invert H
         <span class="kr">end</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">nicify_goals</span> :=
  <span class="kp">repeat</span> <span class="kr">match goal with</span>
         | [ |- _ /\ _ ] =&gt; <span class="nb">split</span>
         | [ |- Some _ = Some _ ] =&gt; <span class="nb">f_equal</span>
         | [ |- S _ = S _ ] =&gt; <span class="nb">f_equal</span>
         <span class="kr">end</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">kill_bools</span> :=
  <span class="kp">repeat</span> <span class="kr">match goal with</span>
         | [ H : _ &amp;&amp; _ = true |- _ ] =&gt; <span class="nb">apply</span> andb_prop <span class="kr">in</span> H
         | [ H : _ || _ = false |- _ ] =&gt; <span class="nb">apply</span> orb_false_elim <span class="kr">in</span> H

         | [ H : _ &lt;=? _ = true |- _ ] =&gt; <span class="nb">apply</span> Z.leb_le <span class="kr">in</span> H
         | [ H : _ &lt;=? _ = false |- _ ] =&gt; <span class="nb">apply</span> Z.leb_gt <span class="kr">in</span> H
         | [ H : _ &lt;? _ = true |- _ ] =&gt; <span class="nb">apply</span> Z.ltb_lt <span class="kr">in</span> H
         | [ H : _ &lt;? _ = false |- _ ] =&gt; <span class="nb">apply</span> Z.ltb_ge <span class="kr">in</span> H
         | [ H : _ &gt;=? _ = _ |- _ ] =&gt; <span class="nb">rewrite</span> Z.geb_leb <span class="kr">in</span> H
         | [ H : _ &gt;? _ = _ |- _ ] =&gt; <span class="nb">rewrite</span> Z.gtb_ltb <span class="kr">in</span> H

         | [ H : _ =? _ = true |- _ ] =&gt; <span class="nb">apply</span> Z.eqb_eq <span class="kr">in</span> H
         | [ H : _ =? _ = false |- _ ] =&gt; <span class="nb">apply</span> Z.eqb_neq <span class="kr">in</span> H
         <span class="kr">end</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">unfold_constants</span> :=
  <span class="kp">repeat</span> <span class="kr">match goal with</span>
         | [ |- <span class="kp">context</span>[Integers.Ptrofs.modulus] ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.modulus <span class="kr">with</span> <span class="mi">4294967296</span> <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Ptrofs.modulus] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.modulus <span class="kr">with</span> <span class="mi">4294967296</span> <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Ptrofs.min_signed] ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.min_signed <span class="kr">with</span> (-<span class="mi">2147483648</span>) <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Ptrofs.min_signed] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.min_signed <span class="kr">with</span> (-<span class="mi">2147483648</span>) <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Ptrofs.max_signed] ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.max_signed <span class="kr">with</span> <span class="mi">2147483647</span> <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Ptrofs.max_signed] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.max_signed <span class="kr">with</span> <span class="mi">2147483647</span> <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Ptrofs.max_unsigned] ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.max_unsigned <span class="kr">with</span> <span class="mi">4294967295</span> <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Ptrofs.max_unsigned] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Ptrofs.max_unsigned <span class="kr">with</span> <span class="mi">4294967295</span> <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Int.modulus] ] =&gt;
           <span class="nb">replace</span> Integers.Int.modulus <span class="kr">with</span> <span class="mi">4294967296</span> <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Int.modulus] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Int.modulus <span class="kr">with</span> <span class="mi">4294967296</span> <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Int.min_signed] ] =&gt;
           <span class="nb">replace</span> Integers.Int.min_signed <span class="kr">with</span> (-<span class="mi">2147483648</span>) <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Int.min_signed] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Int.min_signed <span class="kr">with</span> (-<span class="mi">2147483648</span>) <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Int.max_signed] ] =&gt;
           <span class="nb">replace</span> Integers.Int.max_signed <span class="kr">with</span> <span class="mi">2147483647</span> <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Int.max_signed] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Int.max_signed <span class="kr">with</span> <span class="mi">2147483647</span> <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Int.max_unsigned] ] =&gt;
           <span class="nb">replace</span> Integers.Int.max_unsigned <span class="kr">with</span> <span class="mi">4294967295</span> <span class="bp">by</span> <span class="bp">reflexivity</span>
         | [ H : <span class="kp">context</span>[Integers.Int.max_unsigned] |- _ ] =&gt;
           <span class="nb">replace</span> Integers.Int.max_unsigned <span class="kr">with</span> <span class="mi">4294967295</span> <span class="kr">in</span> H <span class="bp">by</span> <span class="bp">reflexivity</span>

         | [ |- <span class="kp">context</span>[Integers.Ptrofs.unsigned (Integers.Ptrofs.repr <span class="nl">?x</span>) ] ] =&gt;
           <span class="kr">match</span> (<span class="kp">eval</span> <span class="nb">compute</span> <span class="kr">in</span> (<span class="mi">0</span> &lt;=? x)) <span class="kr">with</span>
           | true =&gt; <span class="nb">replace</span> (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr x))
                    <span class="kr">with</span> x <span class="bp">by</span> <span class="bp">reflexivity</span>
           | false =&gt; <span class="kp">idtac</span>
           <span class="kr">end</span>
         <span class="kr">end</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">substpp</span> :=
  <span class="kp">repeat</span> <span class="kr">match goal with</span>
         | [ H1 : <span class="nl">?x</span> = Some _, H2 : <span class="nl">?x</span> = Some _ |- _ ] =&gt;
           <span class="kr">let</span> <span class="nv">EQ</span> := <span class="kp">fresh</span> <span class="s2">&quot;EQ&quot;</span> <span class="kr">in</span>
           learn H1 <span class="kr">as</span> EQ; <span class="nb">rewrite</span> H2 <span class="kr">in</span> EQ; invert EQ
         | _ =&gt; <span class="kp">idtac</span>
         <span class="kr">end</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</span> := <span class="nb">intros</span>; unfold_constants; <span class="nb">simpl</span> <span class="kr">in</span> *;
                 <span class="kp">repeat</span> (nicify_hypotheses; nicify_goals; kill_bools; substpp);
                 <span class="nb">simpl</span> <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">Infix</span> <span class="s2">&quot;==nat&quot;</span> := eq_nat_dec (<span class="kn">no associativity</span>, <span class="kn">at level</span> <span class="mi">50</span>).</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Infix</span> <span class="s2">&quot;==Z&quot;</span> := Z.eq_dec (<span class="kn">no associativity</span>, <span class="kn">at level</span> <span class="mi">50</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">liapp</span> :=
  <span class="kp">repeat</span> <span class="kr">match goal with</span>
         | [ |- (<span class="nl">?x</span> | <span class="nl">?y</span>) ] =&gt;
           <span class="kr">match</span> (<span class="kp">eval</span> <span class="nb">compute</span> <span class="kr">in</span> (Z.rem y x ==Z <span class="mi">0</span>)) <span class="kr">with</span>
           | <span class="nb">left</span> _ =&gt;
             <span class="kr">let</span> <span class="nv">q</span> := (<span class="kp">eval</span> <span class="nb">compute</span> <span class="kr">in</span> (Z.div y x))
             <span class="kr">in</span> <span class="kr">exists</span> <span class="nv">q</span>; <span class="bp">reflexivity</span>
           | _ =&gt; <span class="kp">idtac</span>
          <span class="kr">end</span>
         | _ =&gt; <span class="kp">idtac</span>
         <span class="kr">end</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</span> := simplify; <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 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">Global Opaque</span> Nat.div.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Global Opaque</span> Z.mul.</span></span><span class="coq-wsp">
</span></span><span class="coq-wsp"><span class="highlight">
<span class="c">(* Definition const (A B : Type) (a : A) (b : B) : A := a.</span>

<span class="c">Definition compose (A B C : Type) (f : B -&gt; C) (g : A -&gt; B) (x : A) : C := f (g x). *)</span>

</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Module</span> <span class="nf">Option</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">Definition</span> <span class="nf">default</span> {<span class="nv">T</span> : <span class="kt">Type</span>} (<span class="nv">x</span> : T) (<span class="nv">u</span> : option T) : T :=
  <span class="kr">match</span> u <span class="kr">with</span>
  | Some y =&gt; y
  | _ =&gt; x
  <span class="kr">end</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">Definition</span> <span class="nf">map</span> {<span class="nv">S</span> : <span class="kt">Type</span>} {<span class="nv">T</span> : <span class="kt">Type</span>} (<span class="nv">f</span> : S -&gt; T) (<span class="nv">u</span> : option S) : option T :=
  <span class="kr">match</span> u <span class="kr">with</span>
  | Some y =&gt; Some (f y)
  | _ =&gt; None
  <span class="kr">end</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">Definition</span> <span class="nf">liftA2</span> {<span class="nv">T</span> : <span class="kt">Type</span>} (<span class="nv">f</span> : T -&gt; T -&gt; T) (<span class="nv">a</span> : option T) (<span class="nv">b</span> : option T) : option T :=
  <span class="kr">match</span> a <span class="kr">with</span>
  | Some x =&gt; map (f x) b
  | _ =&gt; None
  <span class="kr">end</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">Definition</span> <span class="nf">bind</span> {<span class="nv">A</span> <span class="nv">B</span> : <span class="kt">Type</span>} (<span class="nv">f</span> : option A) (<span class="nv">g</span> : A -&gt; option B) : option B :=
  <span class="kr">match</span> f <span class="kr">with</span>
  | Some a =&gt; g a
  | _ =&gt; None
  <span class="kr">end</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">Definition</span> <span class="nf">join</span> {<span class="nv">A</span> : <span class="kt">Type</span>} (<span class="nv">a</span> : option (option A)) : option A :=
  <span class="kr">match</span> a <span class="kr">with</span>
  | None =&gt; None
  | Some a&#39; =&gt; a&#39;
  <span class="kr">end</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">Module</span> <span class="nf">Notation</span>.</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">Notation</span> <span class="s2">&quot;&#39;do&#39; X &lt;- A ; B&quot;</span> := (bind A (<span class="kr">fun</span> <span class="nv">X</span> =&gt; B))
   (<span class="kn">at level</span> <span class="mi">200</span>, X <span class="kn">ident</span>, A <span class="kn">at level</span> <span class="mi">100</span>, B <span class="kn">at level</span> <span class="mi">200</span>).</span></span><span class="coq-wsp">
</span></span><span class="coq-sentence"><span class="coq-input"><span class="highlight"><span class="kn">End</span> <span class="nf">Notation</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">End</span> <span class="nf">Option</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">Parameter</span> <span class="nv">debug_print</span> : string -&gt; unit.</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">debug_show</span> {<span class="nv">A</span> <span class="nv">B</span> : <span class="kt">Type</span>} `{<span class="kn">Show</span> A} (a : A) (b : B) : B :=
  <span class="kr">let</span> <span class="nv">unused</span> := debug_print (show a) <span class="kr">in</span> b.</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">debug_show_msg</span> {<span class="nv">A</span> <span class="nv">B</span> : <span class="kt">Type</span>} `{<span class="kn">Show</span> A} (s : string) (a : A) (b : B) : B :=
  <span class="kr">let</span> <span class="nv">unused</span> := debug_print (s ++ show a) <span class="kr">in</span> b.</span></span></span></pre>
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