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| <title>Bonak.νType.LeInductive</title> | ||
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| <h1 class="libtitle">Bonak.νType.LeInductive</h1> | ||
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| <div class="code"> | ||
| <span class="id" title="keyword">From</span> <span class="id" title="var">Bonak</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Bonak.νType.Notation.html#"><span class="id" title="library">Notation</span></a>.<br/> | ||
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| An inductive definition of le | ||
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| <span class="id" title="keyword">Reserved Infix</span> "<~" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<br/> | ||
| <span class="id" title="keyword">Inductive</span> <a id="leI" class="idref" href="#leI"><span class="id" title="definition, inductive"><span id="leI_rect" class="id"><span id="leI_ind" class="id"><span id="leI_rec" class="id"><span id="leI_sind" class="id">leI</span></span></span></span></span></a>: <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> | ||
| | <a id="leI_refl" class="idref" href="#leI_refl"><span class="id" title="constructor">leI_refl</span></a> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a>: <a class="idref" href="Bonak.νType.LeInductive.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:3"><span class="id" title="variable">n</span></a><br/> | ||
| | <a id="leI_down" class="idref" href="#leI_down"><span class="id" title="constructor">leI_down</span></a> {<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> <a id="p:5" class="idref" href="#p:5"><span class="id" title="binder">p</span></a>}: <a class="idref" href="Bonak.νType.LeInductive.html#p:5"><span class="id" title="variable">p</span></a><a class="idref" href="Bonak.νType.Notation.html#9c103920f8e0be789ba768cf55671c49"><span class="id" title="notation">.+1</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:5"><span class="id" title="variable">p</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:4"><span class="id" title="variable">n</span></a><br/> | ||
| <span class="id" title="keyword">where</span> <a id="688dc3f7fbd5d2f2aa067be3d6c27f36" class="idref" href="#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation">"</span></a>n <~ m" := (<a class="idref" href="Bonak.νType.LeInductive.html#leI:2"><span class="id" title="inductive">leI</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span>): <span class="id" title="var">nat_scope</span>.<br/> | ||
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| <span class="id" title="keyword">Lemma</span> <a id="leI_up" class="idref" href="#leI_up"><span class="id" title="lemma">leI_up</span></a> {<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a> <a id="p:7" class="idref" href="#p:7"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}: <a class="idref" href="Bonak.νType.LeInductive.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:7"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:7"><span class="id" title="variable">p</span></a><a class="idref" href="Bonak.νType.Notation.html#9c103920f8e0be789ba768cf55671c49"><span class="id" title="notation">.+1</span></a>.<br/> | ||
| <span class="id" title="tactic">induction</span> 1. <span class="id" title="tactic">constructor</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>.<br/> | ||
| <span class="id" title="keyword">Defined</span>.<br/> | ||
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| <br/> | ||
| <span class="id" title="keyword">Lemma</span> <a id="leI_0" class="idref" href="#leI_0"><span class="id" title="lemma">leI_0</span></a> {<a id="p:8" class="idref" href="#p:8"><span class="id" title="binder">p</span></a>}: <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:8"><span class="id" title="variable">p</span></a>.<br/> | ||
| <span class="id" title="tactic">induction</span> <span class="id" title="var">p</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="Bonak.νType.LeInductive.html#leI_up"><span class="id" title="lemma">leI_up</span></a>.<br/> | ||
| <span class="id" title="keyword">Defined</span>.<br/> | ||
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| <br/> | ||
| <span class="id" title="keyword">Lemma</span> <a id="leI_lower_both" class="idref" href="#leI_lower_both"><span class="id" title="lemma">leI_lower_both</span></a> {<a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> <a id="p:10" class="idref" href="#p:10"><span class="id" title="binder">p</span></a>}: <a class="idref" href="Bonak.νType.LeInductive.html#p:10"><span class="id" title="variable">p</span></a><a class="idref" href="Bonak.νType.Notation.html#9c103920f8e0be789ba768cf55671c49"><span class="id" title="notation">.+1</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:9"><span class="id" title="variable">n</span></a><a class="idref" href="Bonak.νType.Notation.html#9c103920f8e0be789ba768cf55671c49"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:10"><span class="id" title="variable">p</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:9"><span class="id" title="variable">n</span></a>.<br/> | ||
| <span class="id" title="tactic">induction</span> 1 <span class="id" title="keyword">using</span> <a class="idref" href="Bonak.νType.LeInductive.html#leI_rec"><span class="id" title="definition">leI_rec</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">P</span> := <span class="id" title="keyword">fun</span> <a id="p:11" class="idref" href="#p:11"><span class="id" title="binder"><span id="p:13" class="id">p</span></span></a> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder"><span id="n:14" class="id">n</span></span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:11"><span class="id" title="variable">p</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:12"><span class="id" title="variable">n</span></a>).<br/> | ||
| <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">p0</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="Bonak.νType.LeInductive.html#leI_0"><span class="id" title="lemma">leI_0</span></a>. <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>.<br/> | ||
| <span class="id" title="keyword">Defined</span>.<br/> | ||
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| <br/> | ||
| <span class="id" title="keyword">Lemma</span> <a id="leI_raise_both" class="idref" href="#leI_raise_both"><span class="id" title="lemma">leI_raise_both</span></a> {<a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> <a id="p:16" class="idref" href="#p:16"><span class="id" title="binder">p</span></a>}: <a class="idref" href="Bonak.νType.LeInductive.html#p:16"><span class="id" title="variable">p</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:15"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/doc/V8.20.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#p:16"><span class="id" title="variable">p</span></a><a class="idref" href="Bonak.νType.Notation.html#9c103920f8e0be789ba768cf55671c49"><span class="id" title="notation">.+1</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#688dc3f7fbd5d2f2aa067be3d6c27f36"><span class="id" title="notation"><~</span></a> <a class="idref" href="Bonak.νType.LeInductive.html#n:15"><span class="id" title="variable">n</span></a><a class="idref" href="Bonak.νType.Notation.html#9c103920f8e0be789ba768cf55671c49"><span class="id" title="notation">.+1</span></a>.<br/> | ||
| <span class="id" title="tactic">induction</span> 1. <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>.<br/> | ||
| <span class="id" title="keyword">Defined</span>.<br/> | ||
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