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laoziSolution2.html
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laoziSolution2.html
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<p><a href="index.html">top</a> | <a href="theindex.html">index</a> | <a href="sitemap.html">sitemap</a> | <a href="https://github.com/1337777/OOO1337777">edit</a></p>
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<div id="content">
<h1 class="title">laoziSolution.v</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgbc37ed8">1. Grammatical description of polymorph grammatical-coparametrism functor</a>
<ul>
<li><a href="#org0016568">1.1. Importing the functional-monoidal logic</a></li>
<li><a href="#orgcbb684d">1.2. Base generating graph</a></li>
<li><a href="#org6559f38">1.3. Grammatical generation of the morhisms</a></li>
<li><a href="#org525b7f9">1.4. Decoding into the common sense : grammatical is indeed free</a></li>
<li><a href="#orgb266784">1.5. Some notations</a></li>
<li><a href="#org967844b">1.6. More functional-monoidal logic</a></li>
<li><a href="#org6cd96c2">1.7. The generated conversion relations over the morhisms</a></li>
</ul>
</li>
<li><a href="#orgef5e84f">2. Iterated constructors</a>
<ul>
<li><a href="#orgb3fff68">2.1. Indexed list, lemmas</a></li>
<li><a href="#org8177098">2.2. Chained lists, lemmas</a></li>
<li><a href="#org9a5722b">2.3. Iterated constructors</a></li>
</ul>
</li>
<li><a href="#org8391fb0">3. Grade</a></li>
<li><a href="#orgdd57311">4. Reduction</a>
<ul>
<li><a href="#org72c51cd">4.1. Grammatical generation of the reduction relation</a></li>
<li><a href="#orgd6b578b">4.2. Degradation lemmas</a></li>
</ul>
</li>
<li><a href="#orgd94353e">5. Solution</a>
<ul>
<li><a href="#org8317458">5.1. Grammatical generation of the solution morphisms</a></li>
<li><a href="#org49c7fff">5.2. Containment of the solution morphisms into all the morphisms</a></li>
<li><a href="#org0232603">5.3. Destruction of morphisms with inner-instantiation of object-indices</a></li>
<li><a href="#orgf36d000">5.4. Iterated <code>DeClassifying</code> prefix</a></li>
</ul>
</li>
<li><a href="#org197d6a2">6. Resolution</a></li>
</ul>
</div>
</div>
<p>
Proph
</p>
<p>
<a href="https://github.com/1337777/laozi/blob/master/laoziSolution2.v">https://github.com/1337777/laozi/blob/master/laoziSolution2.v</a>
</p>
<p>
solves some question of 老子 <sup><a id="fnr.1" class="footref" href="#fn.1">1</a></sup> which is how to program polymorph coparametrism
functors ( "comonad" ) … Whatever is discovered, its format, its communication is
simultaneously some predictable logical discovery and some random dia-para-logical
discovery.
</p>
<p>
In particular this text programs the grammatical / inductive (therefore free)
description of polymorph coparametrism functor and the conversion relations over the
generated morphisms. This text is based on earlier texts which describe
functional-monoidal logic and the decidable coherence of this logic.
</p>
<p>
Next this text programs the iterated comultiplication (DeClassifying) and the
corresponding deduced conversion relations over the morphisms. Then the reduction
relation and degradation lemmas are programmed and deduced.
</p>
<p>
Finally the solution morphisms are programmed with their (dependent-)destruction
lemmas such to inner-instantiate the object-indices. And the (non-congruent)
resolution by cut elimination / desintegration technique is programmed and deduced :
this deduction is mostly-automated.
</p>
<p>
For instant first impression, the common saying that the counit inner-cancels the
comultiplication is written as :
</p>
<pre class="example">
| IterCancelInner :
forall {trf : obV log -> obV log}
(Vb Vb' : obV log) (vb : V(0 Vb' |- Vb )0)
(W W_dft : obV log) (V0Vb' : obV log) (Vs : list (obV log))
(vs : (hlist (toArrowV (trf:=trf)) (chain Vs)))
(va : V(0 trf(last W_dft Vs) |- W )0)
(v0b : V(0 V0Vb' |- (0 trf(head W_dft Vs) & Vb' )0 )0)
(B : obMod) (A : obMod) (b : 'D(0 Vb |- [0 B ~> A ]0 )0)
(A' : obMod)(a : 'D(0 W |- [0 A ~> A' ]0 )0),
( v0b o>| ( vb o>' b) o>D (iterDeClassifying (V_dft := W_dft) vs ( va o>' a)) )
<~~ ( v0b o>| (vb o>| b o>Mod 'declfy )
o>D (iterDeClassifying (V_dft := W_dft) vs (va o>| 'clfy o>Mod a) )
: 'D(0 V0Vb' |- [0 B ~> 'D0| (iterDeClass0 (length Vs).-1 A') ]0)0 )
</pre>
<div id="outline-container-orgbc37ed8" class="outline-2">
<h2 id="orgbc37ed8"><span class="section-number-2">1</span> Grammatical description of polymorph grammatical-coparametrism functor</h2>
<div class="outline-text-2" id="text-1">
</div><div id="outline-container-org0016568" class="outline-3">
<h3 id="org0016568"><span class="section-number-3">1.1</span> Importing the functional-monoidal logic</h3>
<div class="outline-text-3" id="text-1-1">
<p>
Importing the functional-monoidal logic, the ssreflect grammar and deduction tactics
and math definitions, the congruent rewriting tactics, and the linear arithmetic
decidability-solver. (Memo that <code>COQ</code> –version > 8.5)
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Require Import</span> borceuxSolution_half_old.
<span style="font-weight: bold;">Require Import</span> mathcomp.ssreflect.ssreflect.
From mathcomp <span style="font-weight: bold;">Require Import</span> ssrbool ssrfun eqtype ssrnat seq.
<span style="font-weight: bold;">Require Import</span> Setoid.
<span style="font-weight: bold;">Require</span> Omega.
<span style="font-weight: bold;">Set Implicit Arguments</span>.
Unset Strict Implicits.
<span style="font-weight: bold;">Unset Printing Implicit</span> Defensive.
</pre>
</div>
</div>
</div>
<div id="outline-container-orgcbb684d" class="outline-3">
<h3 id="orgcbb684d"><span class="section-number-3">1.2</span> Base generating graph</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Module</span> <span style="font-weight: bold;">COPARAM</span>.
<span style="font-weight: bold;">Import</span> LOGIC.
<span style="font-weight: bold;">Import</span> LOGIC.Ex_Notations3.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">obMod_gen</span> : <span style="font-weight: bold; text-decoration: underline;">Set</span>.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">Mod_gen</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, obV log -> obMod_gen -> obMod_gen -> <span style="font-weight: bold; text-decoration: underline;">Set</span>.
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">obMod</span> : <span style="font-weight: bold; text-decoration: underline;">Set</span> :=
| GenObMod : <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">A</span> : obMod_gen, obMod
| DeClass0 : <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">A</span> : obMod, obMod.
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"#0| A"</span> := (GenObMod A) (<span style="font-weight: bold; text-decoration: underline;">at</span> level 4, <span style="text-decoration: underline;">right</span> associativity).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"'D0| A"</span> := (DeClass0 A) (<span style="font-weight: bold; text-decoration: underline;">at</span> level 4, <span style="text-decoration: underline;">right</span> associativity).
</pre>
</div>
</div>
</div>
<div id="outline-container-org6559f38" class="outline-3">
<h3 id="org6559f38"><span class="section-number-3">1.3</span> Grammatical generation of the morhisms</h3>
<div class="outline-text-3" id="text-1-3">
<p>
The grammatical description of polymorph coparametrism functor, primo the morphisms :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Reserved Notation</span> <span style="font-style: italic;">"''Mod' (0 V |- [0 A1 ~> A2 ]0 )0"</span>
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, format <span style="font-style: italic;">"''Mod' (0 V |- [0 A1 ~> A2 ]0 )0"</span>).
<span style="font-weight: bold;">Reserved Notation</span> <span style="font-style: italic;">"''D' (0 V |- [0 A1 ~> A2 ]0 )0"</span>
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, format <span style="font-style: italic;">"''D' (0 V |- [0 A1 ~> A2 ]0 )0"</span>).
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">Mod00</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} : obV log -> obMod -> obMod -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| PolyV_Mod : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log),
V(0 V' |- V )0 -> <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod), 'Mod(0 V |- [0 A1 ~> A2 ]0 )0 ->
'Mod(0 V' |- [0 A1 ~> A2 ]0 )0
| GenArrowsMod : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log), <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod_gen
, Mod_gen V A1 A2 -> V(0 V' |- V )0 ->
'Mod(0 V' |- [0 (#0| A1) ~> (#0| A2) ]0 )0
| UnitMod : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log), <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">A</span> : obMod}
, V(0 V |- log.-I )0 -> 'Mod(0 V |- [0 A ~> A ]0 )0
| PolyMod : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A1</span> : obMod)
, 'Mod(0 V |- [0 A2 ~> A1 ]0 )0 -> <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">A1'</span> : obMod, <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W WV</span> : obV log),
V(0 WV |- (0 W & V )0 )0 ->
'Mod(0 W |- [0 A1 ~> A1' ]0 )0 -> 'Mod(0 WV |- [0 A2 ~> A1' ]0 )0
| UnitDeClass : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log)
, V(0 W' |- (0 W & log.-I)0 )0 ->
'Mod(0 W |- [0 A ~> A' ]0 )0 -> 'D(0 W' |- [0 'D0| A ~> A' ]0 )0
| PolyDeClass : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod),
'D(0 V |- [0 B ~> A ]0 )0 -> <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">A'</span> : obMod, <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W WV</span> : obV log),
V(0 WV |- (0 W & V )0 )0 ->
'Mod(0 W |- [0 A ~> A' ]0 )0 -> 'D(0 WV |- [0 B ~> A' ]0 )0
<span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">common CoUnit, errata: Unit </span><span style="font-weight: bold; font-style: italic;">*)</span>
| Classifying : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log), <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod),
V(0 V' |- V )0 ->
'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> 'Mod(0 V' |- [0 ('D0| A1) ~> A2 ]0 )0
| DeClassifying : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log), <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod),
V(0 V' |- V )0 ->
'D(0 V |- [0 A1 ~> A2 ]0 )0 -> 'D(0 V' |- [0 A1 ~> ('D0| A2) ]0 )0
<span style="font-weight: bold; text-decoration: underline;">where</span>
<span style="font-style: italic;">"''Mod' (0 V |- [0 A1 ~> A2 ]0 )0"</span>
:= (@Mod00 _ V A1 A2) and <span style="font-style: italic;">"''D' (0 V |- [0 A1 ~> A2 ]0 )0"</span>
:= (@Mod00 _ V A1 ('D0| A2)).
</pre>
</div>
</div>
</div>
<div id="outline-container-org525b7f9" class="outline-3">
<h3 id="org525b7f9"><span class="section-number-3">1.4</span> Decoding into the common sense : grammatical is indeed free</h3>
<div class="outline-text-3" id="text-1-4">
<p>
How to decode from the grammatical description to the non-grammatical description such
to deduce that it is indeed some instance of polymorph coparametrism functor, in the
common sense :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">Mod'00</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, obMod -> obMod -> obV log.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">decode</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod},
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">V</span> : obV log}, 'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> V(0 V |- Mod'00 A1 A2 )0.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">encode</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod},
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">V</span> : obV log}, V(0 V |- Mod'00 A1 A2 )0 -> 'Mod(0 V |- [0 A1 ~> A2 ]0 )0.
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">decodeK</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">V</span> : obV log),
cancel (@decode _ A1 A2 V) (@encode _ A1 A2 V).
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">encodeK</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">V</span> : obV log),
cancel (@encode _ A1 A2 V) (@decode _ A1 A2 V).
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">decode_metaPoly</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic}, <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">f</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( v o> (decode f) )
~~ ( decode (PolyV_Mod v f)
: log.-V(0 V' |- Mod'00 A1 A2 )0 ) .
</pre>
</div>
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</div>
<div id="outline-container-orgb266784" class="outline-3">
<h3 id="orgb266784"><span class="section-number-3">1.5</span> Some notations</h3>
<div class="outline-text-3" id="text-1-5">
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">PolyV_Mod_rewrite</span> log V V' A1 A2 v a :=
(@PolyV_Mod log V V' v A1 A2 a).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>' a"</span> := (@PolyV_Mod_rewrite _ _ _ _ _ v a)
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, <span style="text-decoration: underline;">right</span> associativity, format <span style="font-style: italic;">"v o>' a"</span>).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| #1| a"</span> :=
(@GenArrowsMod _ _ _ _ _ a v) (<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, <span style="text-decoration: underline;">right</span> associativity).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| 'uMod'"</span> := (@UnitMod _ _ _ v)(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| @ 'uMod' A"</span> :=
(@UnitMod _ _ A v) (<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, only parsing).
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">PolyMod_rewrite</span> log V A2 A1 A1' W WV wv a_ a' :=
(@PolyMod log V A2 A1 a_ A1' W WV wv a').
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| a_ o>Mod a'"</span> :=
(@PolyMod_rewrite _ _ _ _ _ _ _ v a_ a')
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, <span style="text-decoration: underline;">right</span> associativity, a_ <span style="font-weight: bold; text-decoration: underline;">at</span> next level,
format <span style="font-style: italic;">"v o>| a_ o>Mod a'"</span>).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| ''D1|' a"</span> := (@UnitDeClass _ _ _ _ _ v a)
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, <span style="text-decoration: underline;">right</span> associativity).
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">PolyDeClass_rewrite</span> log V B A A' W WV wv b a :=
(@PolyDeClass log V B A b A' W WV wv a).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| b o>D a"</span> :=
(@PolyDeClass_rewrite _ _ _ _ _ _ _ v b a)
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, <span style="text-decoration: underline;">right</span> associativity, b <span style="font-weight: bold; text-decoration: underline;">at</span> next level, format <span style="font-style: italic;">"v o>| b o>D a"</span>).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| 'clfy o>Mod a'"</span> :=
(@Classifying _ _ _ _ _ v a') (<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, <span style="text-decoration: underline;">right</span> associativity).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"v o>| a_ o>Mod 'declfy"</span> :=
(@DeClassifying _ _ _ _ _ v a_) (<span style="font-weight: bold; text-decoration: underline;">at</span> level 25, a_ <span style="font-weight: bold; text-decoration: underline;">at</span> next level,
<span style="text-decoration: underline;">right</span> associativity).
</pre>
</div>
</div>
</div>
<div id="outline-container-org967844b" class="outline-3">
<h3 id="org967844b"><span class="section-number-3">1.6</span> More functional-monoidal logic</h3>
<div class="outline-text-3" id="text-1-6">
<p>
More description of the functional-monoidal logic which is assumed. These equations
are decidable by coherence lemmas …
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">PolyV_unitPre</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log} (<span style="font-weight: bold; font-style: italic;">v</span> : log.-V(0 V |- V')0), log.-1 o> v ~~ v.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">PolyV_unitPost</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log} (<span style="font-weight: bold; font-style: italic;">v</span> : log.-V(0 V |- V')0), v o> log.-1 ~~ v.
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">desIdenObLK</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V</span> : obV log}, log.-V(0 log.-(0 log.-I & V )0 |- V )0
:= <span style="font-weight: bold; text-decoration: underline;">fun</span> <span style="font-weight: bold; font-style: italic;">log V</span> => Des (log.-uV) .
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">desIdenObLKV</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V</span> : obV log}, log.-V(0 V |- log.-(0 log.-I & V )0 )0 .
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">desIdenObLK_K</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V</span> : obV log},
log.-1 ~~ (@desIdenObLK log V) o> (@desIdenObLKV log V).
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">desIdenObLKV_K</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V</span> : obV log},
log.-1 ~~ (@desIdenObLKV log V) o> (@desIdenObLK log V).
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">desIdenObLKV_Assoc_Rev_desIdenObLK</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V W</span> : obV log),
log.-1 ~~ ( ( log.-(1 desIdenObLKV & V )0 o> Assoc_Rev ) o> desIdenObLK
: log.-V(0 log.-(0 W & V )0 |- log.-(0 W & V )0 )0 ).
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">desIdenObRK</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V</span> : obV log}, log.-V(0 log.-(0 V & log.-I )0 |- V )0.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">desIdenObRKV</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V</span> : obV log}, log.-V(0 V |- log.-(0 V & log.-I )0 )0.
<span style="font-weight: bold;">Parameter</span> <span style="font-weight: bold; font-style: italic;">desV01</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">V2 V2' V1</span> : obV log},
log.-V(0 V2 |- V2' )0 -> log.-V(0 log.-(0 V1 & V2 )0 |- log.-(0 V1 & V2' )0 )0.
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"dat .-(0 V1 & v )1"</span> := (@desV01 dat _ _ V1 v)
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 30, format <span style="font-style: italic;">"dat .-(0 V1 & v )1"</span>).
<span style="font-weight: bold;">Notation</span> <span style="font-style: italic;">"(0 V1 & v )1"</span> := (_ .-(0 V1 & v )1)
(<span style="font-weight: bold; text-decoration: underline;">at</span> level 30, format <span style="font-style: italic;">"(0 V1 & v )1"</span>).
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">desV01_consV10</span> :
<span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V2 V2' V1</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : log.-V(0 V2' |- V2 )0) (<span style="font-weight: bold; font-style: italic;">W</span> : obV log)
(<span style="font-weight: bold; font-style: italic;">w</span> : log.-V(0 log.-(0 V1 & V2 )0 |- W )0),
Des( [1 v ~> W ]0 <o (Cons w) ) ~~ w <o log.-(0 V1 & v )1 .
<span style="font-weight: bold;">Axiom</span> <span style="font-weight: bold; font-style: italic;">desIdenObLKV_IdenOb_Assoc_Rev_desIdenObLK</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V</span> : obV log),
log.-1 ~~ ( ( (log.-(1 desIdenObLKV & V )0)
o> Assoc_Rev ) o> (log.-(0 log.-I & desIdenObLK )1)
: log.-V(0 log.-(0 log.-I & V )0 |- log.-(0 log.-I & V )0 )0 ).
</pre>
</div>
</div>
</div>
<div id="outline-container-org6cd96c2" class="outline-3">
<h3 id="org6cd96c2"><span class="section-number-3">1.7</span> The generated conversion relations over the morhisms</h3>
<div class="outline-text-3" id="text-1-7">
<p>
The grammatical description of polymorph coparametrism functor, secondo the conversion
relations over the morphisms :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Reserved Notation</span> <span style="font-style: italic;">"f2 ~~~ f1"</span> (<span style="font-weight: bold; text-decoration: underline;">at</span> level 70).
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">convMod</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod),
'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> 'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> <span style="font-weight: bold; text-decoration: underline;">Prop</span> :=
| Mod_ReflV : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
a ~~~ a
| Mod_TransV : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">uTrans a</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
uTrans ~~~ a -> <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">a0</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
a0 ~~~ uTrans -> a0 ~~~ a
| Mod_SymV : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a a0</span> : 'Mod(0 V |- [0 A1 ~> A2]0 )0),
a ~~~ a0 -> a0 ~~~ a
| PolyV_Mod_cong : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 V' |- V )0)
(<span style="font-weight: bold; font-style: italic;">a a0</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
v0 ~~ v -> a0 ~~~ a -> ( v0 o>' a0 ) ~~~ ( v o>' a )
| GenArrowsMod_cong : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log), <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod_gen)
(<span style="font-weight: bold; font-style: italic;">aGen</span> : Mod_gen V A1 A2)
, <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 V' |- V )0), v0 ~~ v -> v0 o>| #1| aGen ~~~ v o>| #1| aGen
| UnitMod_cong : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log), <span style="font-weight: bold; text-decoration: underline;">forall</span> {<span style="font-weight: bold; font-style: italic;">A</span> : obMod} (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 V |- log.-I )0),
v0 ~~ v -> v0 o>| @uMod A ~~~ v o>| @uMod A
| Mod_cong :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a_ a_0</span> : 'Mod(0 V |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A''</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a' a'0</span> : 'Mod(0 W |- [0 A' ~> A'' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 WV |- (0 W & V )0 )0),
v0 ~~ v -> a_0 ~~~ a_ -> a'0 ~~~ a' ->
( v0 o>| a_0 o>Mod a'0 ) ~~~ ( v o>| a_ o>Mod a' )
| UnitDeClass_cong :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log)
(<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 W' |- (0 W & log.-I )0 )0) (<span style="font-weight: bold; font-style: italic;">a a0</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
v0 ~~ v -> a0 ~~~ a -> ( v0 o>| 'D1| a0 ) ~~~ ( v o>| 'D1| a )
| PolyDeClass_cong :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b b0</span> : 'D(0 V |- [0 B ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a a0</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 WV |- (0 W & V )0 )0),
v0 ~~ v -> b0 ~~~ b -> a0 ~~~ a -> ( v0 o>| b0 o>D a0 ) ~~~ ( v o>| b o>D a )
| Classifying_cong :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a a0</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
v0 ~~ v -> a0 ~~~ a -> (v0 o>| 'clfy o>Mod a0 ) ~~~ (v o>| 'clfy o>Mod a )
| DeClassifying_cong :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v v0</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a a0</span> : 'D(0 V |- [0 A1 ~> A2 ]0 )0),
v0 ~~ v -> a0 ~~~ a -> (v0 o>| a0 o>Mod 'declfy ) ~~~ (v o>| a o>Mod 'declfy )
| GenArrowsMod_arrowLog : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V' V''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod_gen)
(<span style="font-weight: bold; font-style: italic;">aGen</span> : Mod_gen V A1 A2)
(<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">v'</span> : V(0 V'' |- V' )0),
( ( v' o> v) o>| #1| aGen )
~~~ (v' o>' (v o>| #1| aGen)
: 'Mod(0 V'' |- [0 #0| A1 ~> #0| A2 ]0)0 )
| UnitMod_arrowLog : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V |- log.-I )0)
(<span style="font-weight: bold; font-style: italic;">v'</span> : V(0 V' |- V )0),
( ( v' o> v ) o>| @uMod A )
~~~ (v' o>' (v o>| @uMod A)
: 'Mod(0 V' |- [0 A ~> A ]0)0 )
| Mod_arrowLog :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A0</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 WV |- (0 W & V )0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV0</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV0 |- WV )0),
( ( v0 o> v ) o>| a_ o>Mod a' )
~~~ ( v0 o>' ( v o>| a_ o>Mod a' )
: 'Mod(0 WV0 |- [0 A0 ~> A' ]0)0 )
| Mod_arrowPre :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">A0</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV' |- (0 W & V' )0 )0),
( ( v0 o> log.-(0 _ & v )1 ) o>| a_ o>Mod a' )
~~~ ( v0 o>| ( v o>' a_ ) o>Mod a'
: 'Mod(0 WV' |- [0 A0 ~> A' ]0)0 )
| Mod_arrowPost :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A0</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w0</span> : V(0 W'V |- (0 W' & V )0 )0),
( ( w0 o> log.-(1 w & _ )0 ) o>| a_ o>Mod a' )
~~~ ( w0 o>| a_ o>Mod ( w o>' a' )
: 'Mod(0 W'V |- [0 A0 ~> A' ]0)0 )
| UnitDeClass_arrowLog :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W W' W''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- (0 W & log.-I )0 )0)
(<span style="font-weight: bold; font-style: italic;">w'</span> : V(0 W'' |- W' )0)
(<span style="font-weight: bold; font-style: italic;">A A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
( ( w' o> w ) o>| 'D1| a )
~~~ ( w' o>' ( w o>| 'D1| a )
: 'D(0 W'' |- [0 'D0| A ~> A' ]0)0 )
| UnitDeClass_arrow :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W W' W''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0)
(<span style="font-weight: bold; font-style: italic;">w'</span> : V(0 W'' |- (0 W' & log.-I )0 )0)
(<span style="font-weight: bold; font-style: italic;">A A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
( ( w' o> log.-(1 w & log.-I )0 ) o>| 'D1| a )
~~~ ( w' o>| 'D1| ( w o>' a)
: 'D(0 W'' |- [0 'D0| A ~> A' ]0)0 )
| DeClass_arrowLog :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 WV |- (0 W & V )0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV0</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV0 |- WV )0),
( ( v0 o> v ) o>| b o>D a )
~~~ ( v0 o>' ( v o>| b o>D a )
: 'D(0 WV0 |- [0 B ~> A' ]0)0 )
| DeClass_arrowPre :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV' |- (0 W & V' )0 )0),
( ( v0 o> log.-(0 _ & v )1 ) o>| b o>D a )
~~~ ( v0 o>| ( v o>' b ) o>D a
: 'D(0 WV' |- [0 B ~> A' ]0)0 )
| DeClass_arrowPost :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w0</span> : V(0 W'V |- (0 W' & V )0 )0),
( ( w0 o> log.-(1 w & _ )0 ) o>| b o>D a )
~~~ ( w0 o>| b o>D ( w o>' a )
: 'D(0 W'V |- [0 B ~> A' ]0)0 )
| Classifying_arrowLog : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V' V''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0)
(<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 V'' |- V' )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| 'clfy o>Mod a )
~~~ ( v0 o>' ( v o>| 'clfy o>Mod a )
: 'Mod(0 V'' |- [0 'D0| A1 ~> A2 ]0)0 )
| Classifying_arrow : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V' V''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0)
(<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 V'' |- V' )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| 'clfy o>Mod a )
~~~ ( v0 o>| 'clfy o>Mod ( v o>' a )
: 'Mod(0 V'' |- [0 'D0| A1 ~> A2 ]0)0 )
| DeClassifying_arrowLog :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V' V''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 V'' |- V' )0)
(<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'D(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| a o>Mod 'declfy )
~~~ ( v0 o>' ( v o>| a o>Mod 'declfy )
: 'D(0 V'' |- [0 A1 ~> 'D0| A2 ]0)0 )
| DeClassifying_arrow :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V' V''</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 V'' |- V' )0)
(<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'D(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| a o>Mod 'declfy )
~~~ ( v0 o>| ( v o>' a ) o>Mod 'declfy
: 'D(0 V'' |- [0 A1 ~> 'D0| A2 ]0)0 )
| PolyV_Mod_arrowLog :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V'' V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v'</span> : V(0 V'' |- V' )0) (<span style="font-weight: bold; font-style: italic;">V</span> : obV log)
(<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v' o> v ) o>' a )
~~~ ( v' o>' ( v o>' a )
: 'Mod(0 V'' |- [0 A1 ~> A2 ]0)0 )
<span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">non for reduction </span><span style="font-weight: bold; font-style: italic;">*)</span>
| Mod_morphism :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b</span> : 'Mod(0 V |- [0 B ~> A ]0 )0)
(<span style="font-weight: bold; font-style: italic;">W_</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 W_ |- [0 A ~> A' ]0 )0)
(<span style="font-weight: bold; font-style: italic;">W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A''</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W' |- [0 A' ~> A'' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W_V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 W_V |- (0 W_ & V )0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'W_V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 W'W_V |- (0 W' & W_V )0 )0),
( ( v0 o> (0 W' & v )1 o> Assoc ) o>| b o>Mod ( log.-1 o>| a_ o>Mod a' ) )
~~~ ( v0 o>| ( v o>| b o>Mod a_ ) o>Mod a'
: 'Mod(0 W'W_V |- [0 B ~> A'' ]0)0 )
| DeClass_morphismPost :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">W_ W_'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 W_' |- (0 W_ & log.-I )0 )0) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 W_ |- [0 A ~> A' ]0 )0)
(<span style="font-weight: bold; font-style: italic;">W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A''</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W' |- [0 A' ~> A'' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'W_'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 W'W_' |- (0 W' & W_' )0 )0),
( ( v0 o> desIdenObRKV ) o>|
'D1| ( (log.-1) o>| ( ( v o> desIdenObRK ) o>' a_ ) o>Mod a' ) )
~~~ ( v0 o>| ( v o>| 'D1| a_ ) o>D a'
: 'D(0 W'W_' |- [0 'D0| A ~> A'' ]0)0 )
| DeClass_morphismPre :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b'</span> : 'D(0 V' |- [0 B' ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 W' |- (0 W & log.-I )0 )0)
(<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 W'V' |- (0 W' & V' )0 )0),
( v0 o>| b' o>D ( ( v o> desIdenObRK ) o>' a ) )
~~~ ( v0 o>| b' o>Mod ( v o>| 'D1| a )
: 'D(0 W'V' |- [0 B' ~> A' ]0)0 )
| PolyV_Mod_unit :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( a ) ~~~ ( log.-1 o>' a
: 'Mod(0 V |- [0 A1 ~> A2 ]0)0 )
| Mod_unit :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V |- log.-I )0)
(<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV |- (0 W & V )0 )0),
( ( v0 o> log.-(0 W & v )1 o> desIdenObRK ) o>' a )
~~~ ( v0 o>| ( v o>| uMod ) o>Mod a
: 'Mod(0 WV |- [0 A ~> A' ]0)0 )
| Mod_inputUnitMod :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b</span> : 'Mod(0 V |- [0 B ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W |- log.-I )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w0</span> : V(0 WV |- (0 W & V )0 )0),
( ( w0 o> log.-(1 w & V )0 o> desIdenObLK ) o>' b )
~~~ ( w0 o>| b o>Mod ( w o>| uMod )
: 'Mod(0 WV |- [0 B ~> A ]0)0 )
| DeClass_unit :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V |- log.-I )0) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV |- (0 W & V )0 )0),
( ( v0 o> log.-(0 W & v )1 ) o>| 'D1| a )
~~~ ( v0 o>| ( v o>| uMod ) o>D a
: 'D(0 WV |- [0 'D0| A ~> A' ]0 )0 )
| DeClass_inputUnitMod :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W |- log.-I )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w0</span> : V(0 WV |- (0 W & V )0 )0),
( ( w0 o> ( log.-(1 w & _ )0 ) o> desIdenObLK ) o>' b )
~~~ ( w0 o>| b o>D ( w o>| uMod )
: 'D(0 WV |- [0 B ~> A ]0)0 )
| Classifying_morphismPre :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0 ) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0)
(<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A3</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W |- [0 A2 ~> A3 ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV' |- (0 W & V' )0 )0),
( ( log.-1 ) o>| 'clfy o>Mod ( ( v0 o> (0 _ & v )1 ) o>| a_ o>Mod a' ) )
~~~ ( v0 o>| (v o>| 'clfy o>Mod a_ ) o>Mod a'
: 'Mod(0 WV' |- [0 'D0| A1 ~> A3 ]0)0 )
<span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">non-necessary, deductible </span><span style="font-weight: bold; font-style: italic;">*)</span>
| Classifying_morphismPre_DeClass :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0 ) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 A1 ~> A2 ]0 )0)
(<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A3</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W |- [0 A2 ~> A3 ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">WV'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 WV' |- (0 W & V' )0 )0),
( ( log.-1 ) o>| 'clfy o>Mod ( ( v0 o> (0 _ & v )1 ) o>| b o>D a' ) )
~~~ ( v0 o>| (v o>| 'clfy o>Mod b ) o>D a'
: 'D(0 WV' |- [0 'D0| A1 ~> A3 ]0)0 )
| Classifying_morphismPost :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- (0 V & log.-I )0 )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">a_</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0) (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log)
(<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">A3</span> : obMod) (<span style="font-weight: bold; font-style: italic;">a'</span> : 'Mod(0 W |- [0 A2 ~> A3 ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v0</span> : V(0 W'V' |- (0 W' & V' )0 )0),
( ( log.-1 )
o>| 'clfy o>Mod ( v0 o>| ( ( v o> desIdenObRK ) o>' a_ ) o>Mod (w o>' a') ) )
~~~ ( v0 o>| ( v o>| 'D1| a_ ) o>Mod ( w o>| 'clfy o>Mod a' )
: 'Mod(0 W'V' |- [0 'D0| A1 ~> A3 ]0)0 )
| DeClassifying_morphismPost :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b_</span> : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0)
(<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">A3</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b'</span> : 'D(0 W |- [0 A2 ~> A3 ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w0</span> : V(0 W'V |- (0 W' & V )0 )0),
( log.-1 o>| ( ( w0 o> (1 w & _ )0 ) o>| b_ o>Mod b' ) o>Mod 'declfy )
~~~ ( w0 o>| b_ o>Mod ( w o>| b' o>Mod 'declfy )
: 'D(0 W'V |- [0 A1 ~> 'D0| A3 ]0)0 )
| DeClassifying_morphismPre :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b_</span> : 'D(0 V |- [0 A1 ~> A2 ]0 )0) (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log)
(<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- (0 W & log.-I )0 )0) (<span style="font-weight: bold; font-style: italic;">A4</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b'</span> : 'Mod(0 W |- [0 A2 ~> A4 ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">wv</span> : V(0 W'V' |- (0 W' & V' )0 )0),
( log.-1 o>|
( wv o>| ( v o>' b_ ) o>D ( ( w o> desIdenObRK ) o>' b') ) o>Mod 'declfy )
~~~ ( wv o>| ( v o>| b_ o>Mod 'declfy ) o>D ( w o>| 'D1| b' )
: 'D(0 W'V' |- [0 A1 ~> 'D0| A4 ]0)0 )
| CancelOuter : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 'D0| A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">wv</span> : V(0 W'V' |- (0 W' & V' )0 )0),
( wv o>| ( v o>' b ) o>Mod ( w o>' a ) )
~~~ ( wv o>| ( v o>| b o>Mod 'declfy ) o>Mod ( w o>| 'clfy o>Mod a )
: 'Mod(0 W'V' |- [0 B ~> A' ]0)0 )
| CancelInner : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0) (<span style="font-weight: bold; font-style: italic;">A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">a</span> : 'Mod(0 W |- [0 A ~> A' ]0 )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">wv</span> : V(0 W'V' |- (0 W' & V' )0 )0),
( wv o>| (v o>' b) o>D (w o>' a) )
~~~ ( wv o>| (v o>| b o>Mod 'declfy ) o>D (w o>| 'clfy o>Mod a )
: 'D(0 W'V' |- [0 B ~> A' ]0)0 )
| PermOuterInner : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">v</span> : V(0 V' |- V )0) (<span style="font-weight: bold; font-style: italic;">B</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">A</span> : obMod) (<span style="font-weight: bold; font-style: italic;">b</span> : 'D(0 V |- [0 B ~> A ]0 )0) (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log)
(<span style="font-weight: bold; font-style: italic;">w</span> : V(0 W' |- W )0) (<span style="font-weight: bold; font-style: italic;">u</span> : V(0 W |- log.-I )0),
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">W'V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">wv</span> : V(0 W'V' |- (0 W' & V' )0 )0),
( wv o>| ( ( log.-(1 w & _ )0 o> log.-(1 u & _ )0 o> desIdenObLK ) o>|
( v o>' b ) o>Mod 'declfy ) o>Mod 'declfy )
~~~ ( wv o>| ( v o>| b o>Mod 'declfy ) o>D ( w o>| (u o>| uMod) o>Mod 'declfy )
: 'D(0 W'V' |- [0 B ~> 'D0| 'D0| A ]0)0 )
<span style="font-weight: bold; text-decoration: underline;">where</span> <span style="font-style: italic;">"f2 ~~~ f1"</span> := (@convMod _ _ _ _ f2 f1).
<span style="font-weight: bold;">Hint Constructors</span> convMod.
<span style="font-weight: bold;">Hint Extern</span> 4 (_ ~~?lo` _) => <span style="text-decoration: underline;">eapply</span> (@ReflV lo _).
<span style="font-weight: bold;">Ltac</span> <span style="font-weight: bold;">rewriterMod</span> := repeat <span style="font-weight: bold; text-decoration: underline;">match</span> goal <span style="font-weight: bold; text-decoration: underline;">with</span>
| [ HH : @eq (Mod00 _ _ _) _ _ |- _ ] => try <span style="text-decoration: underline;">rewrite</span> -> HH <span style="font-weight: bold; text-decoration: underline;">in</span> *; <span style="text-decoration: underline;">clear</span> HH <span style="font-weight: bold; text-decoration: underline;">end</span>.
</pre>
</div>
<p>
Descriptions for the congruent rewriting tactics :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Add Parametric Relation</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) :
('Mod(0 V |- [0 A1 ~> A2 ]0 )0) (@convMod log V A1 A2)
reflexivity proved by (@Mod_ReflV log V A1 A2)
<span style="text-decoration: underline;">symmetry</span> proved by (@Mod_SymV log V A1 A2)
<span style="text-decoration: underline;">transitivity</span> proved by
(<span style="font-weight: bold; text-decoration: underline;">fun</span> <span style="font-weight: bold; font-style: italic;">x y z r1 r2</span> => ((@Mod_TransV log V A1 A2) y z r2 x r1))
<span style="font-weight: bold; text-decoration: underline;">as</span> convMod_rewrite.
<span style="font-weight: bold;">Add Parametric Relation</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V1 V2</span> : obV log) :
(log.-V(0 V1 |- V2 )0) (@convV log V1 V2)
reflexivity proved by (@ReflV log log V1 V2)
<span style="text-decoration: underline;">symmetry</span> proved by (<span style="font-weight: bold; text-decoration: underline;">fun</span> <span style="font-weight: bold; font-style: italic;">x</span> => (@SymV log log V1 V2)^~ x)
<span style="text-decoration: underline;">transitivity</span> proved by
(<span style="font-weight: bold; text-decoration: underline;">fun</span> <span style="font-weight: bold; font-style: italic;">x y z r1 r2</span> => ((@TransV log log V1 V2) y z r2 x r1))
<span style="font-weight: bold; text-decoration: underline;">as</span> convV_rewrite.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) :
(@PolyV_Mod_rewrite log V V' A1 A2) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log V' V)
==> (@convMod log V A1 A2)
==> (@convMod log V' A1 A2))
<span style="font-weight: bold; text-decoration: underline;">as</span> PolyV_Mod_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: PolyV_Mod_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod_gen)
(<span style="font-weight: bold; font-style: italic;">aGen</span> : Mod_gen V A1 A2) :
(@GenArrowsMod log V V' A1 A2 aGen) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log V' V)
==> (@convMod log V' (#0| A1) (#0| A2)))
<span style="font-weight: bold; text-decoration: underline;">as</span> GenArrowsMod_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: GenArrowsMod_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A</span> : obMod)
: (@UnitMod log V A) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log V log.-I)
==> (@convMod log V A A))
<span style="font-weight: bold; text-decoration: underline;">as</span> UnitMod_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: UnitMod_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">W</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A''</span> : obMod) (<span style="font-weight: bold; font-style: italic;">WV</span> : obV log) :
(@PolyMod_rewrite log V A A' A'' W WV ) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log WV ((0 W & V)0) )
==>(@convMod log V A A')
==> (@convMod log W A' A'')
==> (@convMod log WV A A''))
<span style="font-weight: bold; text-decoration: underline;">as</span> Mod_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: Mod_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">A A'</span> : obMod) (<span style="font-weight: bold; font-style: italic;">W W'</span> : obV log) :
(@UnitDeClass log A A' W W') <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ( (@convV log W' ((0 W & log.-I )0) )
==> (@convMod log W A A')
==> (@convMod log W' ('D0| A) ('D0| A')))
<span style="font-weight: bold; text-decoration: underline;">as</span> UnitDeClass_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: UnitDeClass_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V</span> : obV log) (<span style="font-weight: bold; font-style: italic;">B A A'</span> : obMod)
(<span style="font-weight: bold; font-style: italic;">W WV</span> : obV log) :
(@PolyDeClass_rewrite log V B A A' W WV) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log WV ((0 W & V )0) )
==> (@convMod log V B ('D0| A))
==> (@convMod log W A A')
==> (@convMod log WV B ('D0| A')))
<span style="font-weight: bold; text-decoration: underline;">as</span> PolyDeClass_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: PolyDeClass_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) :
(@Classifying log V V' A1 A2) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log V' V)
==> (@convMod log V A1 A2)
==> (@convMod log V' ('D0| A1) A2))
<span style="font-weight: bold; text-decoration: underline;">as</span> Classifying_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: Classifying_cong. <span style="font-weight: bold;">Qed</span>.
<span style="font-weight: bold;">Add Parametric Morphism</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} (<span style="font-weight: bold; font-style: italic;">V V'</span> : obV log) (<span style="font-weight: bold; font-style: italic;">A1 A2</span> : obMod) :
(@DeClassifying log V V' A1 A2) <span style="font-weight: bold; text-decoration: underline;">with</span>
<span style="font-weight: bold; text-decoration: underline;"> signature</span> ((@convV log V' V)
==> (@convMod log V A1 ('D0| A2))
==> (@convMod log V' A1 ('D0| ('D0| A2))))
<span style="font-weight: bold; text-decoration: underline;">as</span> DeClassifying_cong_rewrite.
by <span style="text-decoration: underline;">move</span> => *; <span style="text-decoration: underline;">apply</span>: DeClassifying_cong. <span style="font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgef5e84f" class="outline-2">
<h2 id="orgef5e84f"><span class="section-number-2">2</span> Iterated constructors</h2>
<div class="outline-text-2" id="text-2">
</div><div id="outline-container-orgb3fff68" class="outline-3">
<h3 id="orgb3fff68"><span class="section-number-3">2.1</span> Indexed list, lemmas</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Some definitions :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Module</span> <span style="font-weight: bold;">Import</span> Destruct_hlist.
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">hlist</span> (<span style="font-weight: bold; font-style: italic;">A</span> : <span style="font-weight: bold; text-decoration: underline;">Type</span>) (<span style="font-weight: bold; font-style: italic;">B</span> : A -> <span style="font-weight: bold; text-decoration: underline;">Type</span>)
: list A -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| HNil : hlist B nil
| HCons : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">x</span> : A) (<span style="font-weight: bold; font-style: italic;">ls</span> : list A), B x -> hlist B ls -> hlist B (x :: ls).
<span style="font-weight: bold;">Implicit Arguments</span> HNil [A B].
<span style="font-weight: bold;">Implicit Arguments</span> HCons [A B x ls].
<span style="font-weight: bold;">Infix</span> <span style="font-style: italic;">":::"</span> := HCons (<span style="text-decoration: underline;">right</span> associativity, <span style="font-weight: bold; text-decoration: underline;">at</span> level 60).
<span style="font-weight: bold;">Section</span> <span style="font-weight: bold;">Section1</span>.
<span style="font-weight: bold;">Variable</span> (<span style="font-weight: bold; font-style: italic;">A</span> : <span style="font-weight: bold; text-decoration: underline;">Type</span>) (<span style="font-weight: bold; font-style: italic;">B1 B2</span> : A -> <span style="font-weight: bold; text-decoration: underline;">Type</span>).
<span style="font-weight: bold;">Variable</span> <span style="font-weight: bold; font-style: italic;">f</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">x</span>, B1 x -> B2 x.
<span style="font-weight: bold;">Fixpoint</span> <span style="font-weight: bold;">hmap</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hl</span> : hlist B1 ls) : hlist B2 ls :=
<span style="font-weight: bold; text-decoration: underline;">match</span> hl <span style="font-weight: bold; text-decoration: underline;">with</span>
| HNil => HNil
| HCons _ _ x hl' => f x ::: hmap hl'
<span style="font-weight: bold; text-decoration: underline;">end</span>.
<span style="font-weight: bold;">End</span> <span style="font-weight: bold;">Section1</span>.
<span style="font-weight: bold;">Implicit Arguments</span> hmap [A B1 B2 ls].
</pre>
</div>
<p>
Lemmas for (dependent-)destruction by inner instantiations of indices :
</p>
<div class="org-src-container">
<pre class="src src-coq"> <span style="font-weight: bold;">Section</span> <span style="font-weight: bold;">Section2</span>.
<span style="font-weight: bold;">Variables</span> (<span style="font-weight: bold; font-style: italic;">A</span> : <span style="font-weight: bold; text-decoration: underline;">Type</span>) (<span style="font-weight: bold; font-style: italic;">B</span> : A -> <span style="font-weight: bold; text-decoration: underline;">Type</span>).
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">hlist_nil</span> : hlist B ( [::]) -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| Hlist_nil : hlist_nil (<span style="font-weight: bold; font-style: italic;">HNil</span> : hlist B [::]).
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">hlist_cons</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">a</span> : A) (<span style="font-weight: bold; font-style: italic;">ls'</span> : list A), hlist B (a :: ls') -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| Hlist_cons : <span style="font-weight: bold; text-decoration: underline;">forall</span> a ls' (<span style="font-weight: bold; font-style: italic;">ba</span> : B a) (<span style="font-weight: bold; font-style: italic;">hls'</span> : hlist B ls'),
@hlist_cons a ls' (ba ::: hls' : hlist B ([:: a & ls'])).
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">hlist_destructP_type</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hls</span> : hlist B ls), <span style="font-weight: bold; text-decoration: underline;">Type</span>.
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">move</span> => ls. <span style="text-decoration: underline;">case</span> : ls => [ | a ls'] hls.
- <span style="text-decoration: underline;">refine</span> (@hlist_nil hls).
- <span style="text-decoration: underline;">refine</span> (@hlist_cons a ls' hls).
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">hlist_destructP</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hls</span> : hlist B ls),
@hlist_destructP_type ls hls.
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">move</span> => ls hls. <span style="text-decoration: underline;">case</span>: ls / hls.
<span style="text-decoration: underline;">rewrite</span> /hlist_destructP_type /=.
- <span style="text-decoration: underline;">constructor</span>.
- <span style="text-decoration: underline;">constructor</span>.
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">type indeed computes: ;;;</span>
<span style="font-weight: bold; font-style: italic;"> Definition hlist_cons_destructP (a : A) (ls' : list A)</span>
<span style="font-weight: bold; font-style: italic;"> (hls : hlist B (a :: ls')) :</span>
<span style="font-weight: bold; font-style: italic;"> (@hlist_cons a ls' hls).(**)</span>
<span style="font-weight: bold; font-style: italic;"> Proof.(**)</span>
<span style="font-weight: bold; font-style: italic;"> apply: (hlist_destructP hls ).(* *)</span>
<span style="font-weight: bold; font-style: italic;"> Defined.(**) </span><span style="font-weight: bold; font-style: italic;">*)</span>
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">hlist_eta_type</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hls</span> : hlist B ( ls)), <span style="font-weight: bold; text-decoration: underline;">Type</span>.
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">move</span> => ls hls. <span style="text-decoration: underline;">refine</span> (hls = _ ). <span style="text-decoration: underline;">move</span>: hls. <span style="text-decoration: underline;">case</span>: ls.
- <span style="text-decoration: underline;">move</span> => _ . exact: HNil.
- <span style="text-decoration: underline;">move</span> => a ls' hls'. <span style="text-decoration: underline;">refine</span> (_ ::: _).
+ <span style="text-decoration: underline;">case</span>: a ls' hls' / (hlist_destructP hls') => a ls' ba _.
exact: ba. <span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">hhd </span><span style="font-weight: bold; font-style: italic;">*)</span>
+ <span style="text-decoration: underline;">case</span>: a ls' hls' / (hlist_destructP hls') => a ls' _ hls'.
exact: hls'. <span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">htl </span><span style="font-weight: bold; font-style: italic;">*)</span>
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold;">Lemma</span> <span style="font-weight: bold;">hlist_eta</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hls</span> : hlist B ( ls)),
@hlist_eta_type ls hls.
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">move</span> => ls hls. <span style="text-decoration: underline;">case</span>: ls / hls.
<span style="text-decoration: underline;">rewrite</span> /hlist_eta_type. reflexivity.
<span style="text-decoration: underline;">rewrite</span> /hlist_eta_type. reflexivity.
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold; font-style: italic;">(* </span><span style="font-weight: bold; font-style: italic;">memo: may ;;; </span><span style="font-weight: bold; font-style: italic;">*)</span>
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">tl_hlist_type</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hls</span> : hlist B ( ls)), <span style="font-weight: bold; text-decoration: underline;">Type</span>.
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">move</span> => ls. <span style="text-decoration: underline;">case</span>: ls.
- <span style="text-decoration: underline;">move</span> => _ . <span style="text-decoration: underline;">refine</span> (hlist B [::]).
- <span style="text-decoration: underline;">move</span> => a ls' hls'. <span style="text-decoration: underline;">refine</span> (hlist B ls').
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">tl_hlist</span> : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list A) (<span style="font-weight: bold; font-style: italic;">hls</span> : hlist B ( ls)), tl_hlist_type hls.
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">move</span> => ls. <span style="text-decoration: underline;">case</span>: ls => /=.
- <span style="text-decoration: underline;">move</span> => hls. exact: hls.
- <span style="text-decoration: underline;">move</span> => a ls' hls. <span style="text-decoration: underline;">case</span>: a ls' hls / (hlist_destructP hls) => a ls' _ hls'.
exact: hls'.
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold;">End</span> <span style="font-weight: bold;">Section2</span>.
<span style="font-weight: bold;">End</span> <span style="font-weight: bold;">Destruct_hlist</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org8177098" class="outline-3">
<h3 id="org8177098"><span class="section-number-3">2.2</span> Chained lists, lemmas</h3>
<div class="outline-text-3" id="text-2-2">
<p>
Some definitions :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Fixpoint</span> <span style="font-weight: bold;">chain</span> (<span style="font-weight: bold; font-style: italic;">T</span> : <span style="font-weight: bold; text-decoration: underline;">Type</span>) (<span style="font-weight: bold; font-style: italic;">ls</span> : list T) {<span style="font-weight: bold; text-decoration: underline;">struct</span> ls} : list (prod T T) :=
<span style="font-weight: bold; text-decoration: underline;">match</span> ls <span style="font-weight: bold; text-decoration: underline;">with</span>
| nil => nil
| cons t0 ls' => <span style="font-weight: bold; text-decoration: underline;">match</span> ls' <span style="font-weight: bold; text-decoration: underline;">with</span>
| nil => nil
| cons t1 ls'' => (t0, t1) :: chain ls'
<span style="font-weight: bold; text-decoration: underline;">end</span>
<span style="font-weight: bold; text-decoration: underline;">end</span>.
<span style="font-weight: bold;">Arguments</span> chain : <span style="text-decoration: underline;">simpl</span> nomatch.
<span style="font-weight: bold;">Eval</span> <span style="text-decoration: underline;">compute</span> <span style="font-weight: bold; text-decoration: underline;">in</span> chain [:: 0; 11; 2; 3].
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">chain_graph</span> (<span style="font-weight: bold; font-style: italic;">T</span> : <span style="font-weight: bold; text-decoration: underline;">Type</span>) : list T -> list (prod T T) -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| Chain_nil : chain_graph [::] (chain [::])
| Chain_cons_nil : <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">t0</span> : T, chain_graph [:: t0] (chain [:: t0])
| Chain_cons_cons : <span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">t0 t1</span> : T) (<span style="font-weight: bold; font-style: italic;">ls''</span> : list T),
chain_graph (t1 :: ls'') (chain ([:: t1 & ls'']))
-> chain_graph (t0 :: t1 :: ls'') ((t0 , t1) :: chain ([:: t1 & ls''])) .
<span style="font-weight: bold;">Lemma</span> <span style="font-weight: bold;">chain_graphP</span> (<span style="font-weight: bold; font-style: italic;">T</span> : <span style="font-weight: bold; text-decoration: underline;">Type</span>) :
<span style="font-weight: bold; text-decoration: underline;">forall</span> (<span style="font-weight: bold; font-style: italic;">ls</span> : list T), chain_graph ls (chain ls).
<span style="font-weight: bold;">Proof</span>.
<span style="text-decoration: underline;">induction</span> ls <span style="font-weight: bold; text-decoration: underline;">as</span> [|t0 ls']. <span style="text-decoration: underline;">constructor</span> 1.
<span style="text-decoration: underline;">destruct</span> ls' <span style="font-weight: bold; text-decoration: underline;">as</span> [|t1 ls'']. <span style="text-decoration: underline;">constructor</span> 2.
<span style="text-decoration: underline;">simpl</span>. <span style="text-decoration: underline;">constructor</span> 3. exact: IHls'.
<span style="font-weight: bold;">Defined</span>.
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">toArrowV</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">trf</span> : obV log -> obV log}
(<span style="font-weight: bold; font-style: italic;">V1V2</span> : prod (obV log) (obV log))
:= V(0 trf V1V2.1 |- trf V1V2.2 )0.
<span style="font-weight: bold;">Definition</span> <span style="font-weight: bold;">arrowList</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">trf</span> : obV log -> obV log} ls
:= (hlist (@toArrowV log trf) (chain ls)).
</pre>
</div>
<p>
Lemmas for (dependent-)destruction by inner instantiations of indices :
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="font-weight: bold;">Module</span> <span style="font-weight: bold;">Import</span> Destruct_arrowList.
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">arrowList_nil</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">trf</span> : obV log -> obV log}
: hlist (toArrowV (trf:=trf)) (chain [::]) -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| ArrowList_nil : arrowList_nil (<span style="font-weight: bold; font-style: italic;">HNil</span> : arrowList [::]).
<span style="font-weight: bold;">Inductive</span> <span style="font-weight: bold;">arrowList_cons_nil</span> {<span style="font-weight: bold; font-style: italic;">log</span> : logic} {<span style="font-weight: bold; font-style: italic;">trf</span> : obV log -> obV log}
: <span style="font-weight: bold; text-decoration: underline;">forall</span> <span style="font-weight: bold; font-style: italic;">V0</span> : (obV log), hlist (toArrowV (trf:=trf)) (chain [:: V0]) -> <span style="font-weight: bold; text-decoration: underline;">Type</span> :=
| ArrowList_cons_nil :