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Distributions.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Distributions</title>
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<body>
<div id="page">
<div id="header">
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<div id="main">
<h1 class="libtitle">Library Distributions</h1>
<div class="code">
</div>
<div class="doc">
<a name="lab4"></a><h1 class="section">Reasoning About Distributions</h1>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="Intervals.html#"><span class="id" type="library">Intervals</span></a>.<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <a name="dist"><span class="id" type="inductive">dist</span></a> <span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="Unit"><span class="id" type="constructor">Unit</span></a> : <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <span class="id" type="var">T</span><br/>
| <a name="Combine"><span class="id" type="constructor">Combine</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> (<span class="id" type="var">p</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" type="axiom">R</span></a>), 0 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<'_x_'<'_x"><span class="id" type="notation"><</span></a> <a class="idref" href="Distributions.html#p"><span class="id" type="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<'_x_'<'_x"><span class="id" type="notation"><</span></a> 1 <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <span class="id" type="var">T</span>.<br/>
<br/>
<span class="id" type="var">Arguments</span> <a class="idref" href="Distributions.html#Unit"><span class="id" type="constructor">Unit</span></a> {<span class="id" type="var">T</span>} <span class="id" type="var">_</span>.<br/>
<span class="id" type="var">Arguments</span> <a class="idref" href="Distributions.html#Combine"><span class="id" type="constructor">Combine</span></a> {<span class="id" type="var">T</span>} <span class="id" type="var">p</span> <span class="id" type="var">v</span> <span class="id" type="var">d1</span> <span class="id" type="var">d2</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> <a name="::'~'_x"><span class="id" type="notation">"</span></a>~ X" := (<span class="id" type="keyword">fun</span> <span class="id" type="var">t</span> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" type="definition">negb</span></a> (<span class="id" type="var">X</span> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a>)).<br/>
<span class="id" type="keyword">Notation</span> <a name="::x_'&&'_x"><span class="id" type="notation">"</span></a>X1 && X2 " := (<span class="id" type="keyword">fun</span> <span class="id" type="var">t</span> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#andb"><span class="id" type="definition">andb</span></a> (<span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a>) (<span class="id" type="var">X2</span> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a>)).<br/>
<span class="id" type="keyword">Notation</span> <a name="::x_'||'_x"><span class="id" type="notation">"</span></a>X1 || X2 " := (<span class="id" type="keyword">fun</span> <span class="id" type="var">t</span> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#orb"><span class="id" type="definition">orb</span></a> (<span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a>) (<span class="id" type="var">X2</span> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a>)).<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="probability"><span class="id" type="definition">probability</span></a> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span>:<a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) (<span class="id" type="var">X</span>: <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" type="inductive">bool</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" type="axiom">R</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Distributions.html#Unit"><span class="id" type="constructor">Unit</span></a> <span class="id" type="var">t</span> => <span class="id" type="keyword">if</span> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="var">t</span> <span class="id" type="keyword">then</span> 1 <span class="id" type="keyword">else</span> 0<br/>
| <a class="idref" href="Distributions.html#Combine"><span class="id" type="constructor">Combine</span></a> <span class="id" type="var">p</span> <span class="id" type="var">v</span> <span class="id" type="var">d1</span> <span class="id" type="var">d2</span> =><br/>
<span class="id" type="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#probability"><span class="id" type="definition">probability</span></a> <span class="id" type="var">d1</span> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a><span class="id" type="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#probability"><span class="id" type="definition">probability</span></a> <span class="id" type="var">d2</span> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> <a name="::'Pr'_x_'in'_x"><span class="id" type="notation">"</span></a>'Pr' X 'in' d" := (<a class="idref" href="Distributions.html#probability"><span class="id" type="definition">probability</span></a> <span class="id" type="var">d</span> <span class="id" type="var">X</span>) (<span class="id" type="tactic">at</span> <span class="id" type="keyword">level</span> 40).<br/>
<br/>
</div>
<div class="doc">
<a name="lab5"></a><h2 class="section">Basic Theorems of Probability Theory</h2>
<div class="paragraph"> </div>
Lemma 2.4: Normality
</div>
<div class="code">
<span class="id" type="keyword">Theorem</span> <a name="pr_normality"><span class="id" type="lemma">pr_normality</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>,<br/>
0 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<='_x_'<='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<='_x_'<='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t</span> |].<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>); <span class="id" type="var">lra</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Intervals.html#in_0_1_closed"><span class="id" type="lemma">in_0_1_closed</span></a>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHd1</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHd2</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.5: Tautology
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_tautology"><span class="id" type="lemma">pr_tautology</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>,<br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t</span> |].<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>, <span class="id" type="var">IHd2</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.6: Contradiction
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_contradiction"><span class="id" type="lemma">pr_contradiction</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>,<br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t</span> |].<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>, <span class="id" type="var">IHd2</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.7 Equivalence
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_equivalence"><span class="id" type="lemma">pr_equivalence</span></a>: <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">X'</span>,<br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>, <span class="id" type="var">IHd2</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_totality"><span class="id" type="lemma">pr_totality</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>, <br/>
<a class="idref" href="Distributions.html#probability"><span class="id" type="definition">probability</span></a> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <a class="idref" href="Distributions.html#probability"><span class="id" type="definition">probability</span></a> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> (<a class="idref" href="Distributions.html#::'~'_x"><span class="id" type="notation">~</span></a><a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">simpl</span>; <span class="id" type="var">lra</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rplus_assoc"><span class="id" type="axiom">Rplus_assoc</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rplus_assoc"><span class="id" type="axiom">Rplus_assoc</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">r1</span> := <span class="id" type="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <span class="id" type="var">X</span> <span class="id" type="keyword">in</span> <span class="id" type="var">d1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a>) <br/>
(<span class="id" type="var">r2</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a> <span class="id" type="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <span class="id" type="var">X</span> <span class="id" type="keyword">in</span> <span class="id" type="var">d2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a>) <br/>
(<span class="id" type="var">r3</span> := <span class="id" type="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'~'_x"><span class="id" type="notation">~</span></a><span class="id" type="var">X</span> <span class="id" type="keyword">in</span> <span class="id" type="var">d1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a>).<br/>
<span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rplus_comm"><span class="id" type="axiom">Rplus_comm</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">r1</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a> <span class="id" type="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <span class="id" type="var">X</span> <span class="id" type="keyword">in</span> <span class="id" type="var">d2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a>) <br/>
(<span class="id" type="var">r2</span> := <span class="id" type="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'~'_x"><span class="id" type="notation">~</span></a><span class="id" type="var">X</span> <span class="id" type="keyword">in</span> <span class="id" type="var">d1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'*'_x"><span class="id" type="notation">)</span></a>).<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rplus_assoc"><span class="id" type="axiom">Rplus_assoc</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rplus_assoc"><span class="id" type="axiom">Rplus_assoc</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rmult_plus_distr_l"><span class="id" type="axiom">Rmult_plus_distr_l</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Raxioms.html#Rmult_plus_distr_l"><span class="id" type="axiom">Rmult_plus_distr_l</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd2</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.8: Complement
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_complement"><span class="id" type="lemma">pr_complement</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'~'_x"><span class="id" type="notation">~</span></a><a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_totality"><span class="id" type="lemma">pr_totality</span></a> <span class="id" type="var">d</span> <span class="id" type="var">X</span>). <span class="id" type="tactic">intros</span> <span class="id" type="var">Eq</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.9: Disjunction
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_disjunction"><span class="id" type="lemma">pr_disjunction</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>; <span class="id" type="tactic">intros</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">simpl</span>; <span class="id" type="var">lra</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd2</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_addition"><span class="id" type="lemma">pr_addition</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, (<a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a>) <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.RIneq.html#Rminus_0_r"><span class="id" type="lemma">Rminus_0_r</span></a>.<br/>
<span class="id" type="tactic">replace</span> 0 <span class="id" type="keyword">with</span> (<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <span class="id" type="var">X2</span><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">d</span>).<br/>
<span class="id" type="var">Focus</span> 2. <span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_contradiction"><span class="id" type="lemma">pr_contradiction</span></a>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_disjunction"><span class="id" type="lemma">pr_disjunction</span></a>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab6"></a><h2 class="section">Rules for Bounding Probabilities</h2>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_union_bound"><span class="id" type="lemma">pr_union_bound</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_disjunction"><span class="id" type="lemma">pr_disjunction</span></a> <span class="id" type="var">d</span> <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>); <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a> <span class="id" type="var">d</span> (<span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <span class="id" type="var">X2</span>)); <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_weaken"><span class="id" type="lemma">pr_weaken</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">X'</span>, <br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">specialize</span> (<span class="id" type="var">H</span> <span class="id" type="var">t</span>).<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X'</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">try</span> <span class="id" type="var">lra</span>.<br/>
<span class="id" type="tactic">assert</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span class="id" type="keyword">as</span> <span class="id" type="var">contra</span> <span class="id" type="tactic">by</span> (<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">reflexivity</span>).<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">contra</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.RIneq.html#Rmult_le_compat_l"><span class="id" type="lemma">Rmult_le_compat_l</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">r</span>:=<span class="id" type="var">p</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">IHd1</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">lra</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.RIneq.html#Rmult_le_compat_l"><span class="id" type="lemma">Rmult_le_compat_l</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">r</span>:=(1<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a><span class="id" type="var">p</span>)) <span class="id" type="keyword">in</span> <span class="id" type="var">IHd2</span>; <span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_strengthen"><span class="id" type="lemma">pr_strengthen</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">X'</span>, <br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'>='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">specialize</span> (<span class="id" type="var">H</span> <span class="id" type="var">t</span>).<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X'</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">try</span> <span class="id" type="var">lra</span>.<br/>
<span class="id" type="tactic">assert</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span class="id" type="keyword">as</span> <span class="id" type="var">contra</span> <span class="id" type="tactic">by</span> (<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">reflexivity</span>).<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">contra</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.RIneq.html#Rmult_ge_compat_l"><span class="id" type="lemma">Rmult_ge_compat_l</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">r</span>:=<span class="id" type="var">p</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">IHd1</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">lra</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.RIneq.html#Rmult_ge_compat_l"><span class="id" type="lemma">Rmult_ge_compat_l</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">r</span>:=(1<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'-'_x"><span class="id" type="notation">-</span></a><span class="id" type="var">p</span>)) <span class="id" type="keyword">in</span> <span class="id" type="var">IHd2</span>; <span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_disj_ge_l"><span class="id" type="lemma">pr_disj_ge_l</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'>='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_strengthen"><span class="id" type="lemma">pr_strengthen</span></a>.<br/>
<span class="id" type="tactic">simpl</span>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_disj_ge_r"><span class="id" type="lemma">pr_disj_ge_r</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'>='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_strengthen"><span class="id" type="lemma">pr_strengthen</span></a>.<br/>
<span class="id" type="tactic">simpl</span>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_le_l"><span class="id" type="lemma">pr_conj_le_l</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_weaken"><span class="id" type="lemma">pr_weaken</span></a>.<br/>
<span class="id" type="tactic">simpl</span>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_le_r"><span class="id" type="lemma">pr_conj_le_r</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'<='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_weaken"><span class="id" type="lemma">pr_weaken</span></a>.<br/>
<span class="id" type="tactic">simpl</span>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_ge_bound"><span class="id" type="lemma">pr_ge_bound</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span> <span class="id" type="var">p1</span> <span class="id" type="var">p2</span>,<br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, (<a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a>) <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'>='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#p1"><span class="id" type="variable">p1</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'>='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <a class="idref" href="Distributions.html#p2"><span class="id" type="variable">p2</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#p1"><span class="id" type="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <a class="idref" href="Distributions.html#p2"><span class="id" type="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#p1"><span class="id" type="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation"><span style="font-family: arial;">∧</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#p2"><span class="id" type="variable">p2</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_addition"><span class="id" type="lemma">pr_addition</span></a> <span class="id" type="var">d</span> <span class="id" type="var">X1</span> <span class="id" type="var">X2</span> <span class="id" type="var">H</span>); <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">assert</span> (<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <span class="id" type="var">X2</span><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">d</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'>='_x"><span class="id" type="notation"><span style="font-family: serif;">≥</span></span></a> <span class="id" type="var">p1</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Reals.Rdefinitions.html#:R_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <span class="id" type="var">p2</span>) <span class="id" type="tactic">by</span> <span class="id" type="var">lra</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a> <span class="id" type="var">d</span> (<span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#::x_'||'_x"><span class="id" type="notation"><span style="font-family: arial;">⇓</span></span></a> <span class="id" type="var">X2</span>)). <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab7"></a><h3 class="section">The base states are deterministic</h3>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="pr_unit"><span class="id" type="lemma">pr_unit</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">t</span>:<a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#Unit"><span class="id" type="constructor">Unit</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'\/'_x"><span class="id" type="notation"><span style="font-family: arial;">∨</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#Unit"><span class="id" type="constructor">Unit</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>); <span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab8"></a><h3 class="section">Combining and Splitting Distributions</h3>
<div class="paragraph"> </div>
Lemma 2.10: Combine
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_combine"><span class="id" type="lemma">pr_combine</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d1</span> <span class="id" type="var">d2</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">p</span> <span class="id" type="var">v</span> <span class="id" type="var">r</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d1"><span class="id" type="variable">d1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#r"><span class="id" type="variable">r</span></a> <span style="font-family: arial;">→</span> <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d2"><span class="id" type="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#r"><span class="id" type="variable">r</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#Combine"><span class="id" type="constructor">Combine</span></a> <a class="idref" href="Distributions.html#p"><span class="id" type="variable">p</span></a> <a class="idref" href="Distributions.html#v"><span class="id" type="variable">v</span></a> <a class="idref" href="Distributions.html#d1"><span class="id" type="variable">d1</span></a> <a class="idref" href="Distributions.html#d2"><span class="id" type="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#r"><span class="id" type="variable">r</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span>, <span class="id" type="var">H0</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.11: Split (0)
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_split_0"><span class="id" type="lemma">pr_split_0</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d1</span> <span class="id" type="var">d2</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">p</span> <span class="id" type="var">v</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#Combine"><span class="id" type="constructor">Combine</span></a> <a class="idref" href="Distributions.html#p"><span class="id" type="variable">p</span></a> <a class="idref" href="Distributions.html#v"><span class="id" type="variable">v</span></a> <a class="idref" href="Distributions.html#d1"><span class="id" type="variable">d1</span></a> <a class="idref" href="Distributions.html#d2"><span class="id" type="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d1"><span class="id" type="variable">d1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation"><span style="font-family: arial;">∧</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d2"><span class="id" type="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Intervals.html#sum_to_0"><span class="id" type="lemma">sum_to_0</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">p</span>:=<span class="id" type="var">p</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Lemma 2.11: Split (1)
</div>
<div class="code">
<span class="id" type="keyword">Lemma</span> <a name="pr_split_1"><span class="id" type="lemma">pr_split_1</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d1</span> <span class="id" type="var">d2</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">p</span> <span class="id" type="var">v</span>,<br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#Combine"><span class="id" type="constructor">Combine</span></a> <a class="idref" href="Distributions.html#p"><span class="id" type="variable">p</span></a> <a class="idref" href="Distributions.html#v"><span class="id" type="variable">v</span></a> <a class="idref" href="Distributions.html#d1"><span class="id" type="variable">d1</span></a> <a class="idref" href="Distributions.html#d2"><span class="id" type="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d1"><span class="id" type="variable">d1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation"><span style="font-family: arial;">∧</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d2"><span class="id" type="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Intervals.html#sum_to_1"><span class="id" type="lemma">sum_to_1</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">p</span>:=<span class="id" type="var">p</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab9"></a><h3 class="section">Useful Rules for Zero and One</h3>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_complement_1"><span class="id" type="lemma">pr_complement_1</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'<->'_x"><span class="id" type="notation"><span style="font-family: arial;">↔</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'~'_x"><span class="id" type="notation">~</span></a><a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span>. <span class="id" type="tactic">rewrite</span> <a class="idref" href="Distributions.html#pr_complement"><span class="id" type="lemma">pr_complement</span></a>. <span class="id" type="var">lra</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_complement_0"><span class="id" type="lemma">pr_complement_0</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'<->'_x"><span class="id" type="notation"><span style="font-family: arial;">↔</span></span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#::'~'_x"><span class="id" type="notation">~</span></a><a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span>. <span class="id" type="tactic">rewrite</span> <a class="idref" href="Distributions.html#pr_complement"><span class="id" type="lemma">pr_complement</span></a>. <span class="id" type="var">lra</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_weaken_1"><span class="id" type="lemma">pr_weaken_1</span></a>: <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">X'</span>,<br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eqx</span>, (<span class="id" type="var">X'</span> <span class="id" type="var">t</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eqx'</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Eqx</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Eqx</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Eqx'</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Eqx'</span>.<br/>
+ <span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_split_1"><span class="id" type="lemma">pr_split_1</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> (<span class="id" type="var">H1</span> & <span class="id" type="var">H2</span>).<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>, <span class="id" type="var">IHd2</span>; <span class="id" type="tactic">trivial</span>. <span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_strengthen_0"><span class="id" type="lemma">pr_strengthen_0</span></a>: <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X</span> <span class="id" type="var">X'</span>,<br/>
(<span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">t</span>, <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <a class="idref" href="Distributions.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X"><span class="id" type="variable">X</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X'"><span class="id" type="variable">X'</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X</span> <span class="id" type="var">t</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eqx</span>, (<span class="id" type="var">X'</span> <span class="id" type="var">t</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eqx'</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Eqx'</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Eqx</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Eqx'</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Eqx'</span>.<br/>
+ <span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_split_0"><span class="id" type="lemma">pr_split_0</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> (<span class="id" type="var">H1</span> & <span class="id" type="var">H2</span>).<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>, <span class="id" type="var">IHd2</span>; <span class="id" type="tactic">trivial</span>. <span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_0_l"><span class="id" type="lemma">pr_conj_0_l</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_conj_le_l"><span class="id" type="lemma">pr_conj_le_l</span></a> <span class="id" type="var">d</span> <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>).<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a> <span class="id" type="var">d</span> (<span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <span class="id" type="var">X2</span>)).<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_0_r"><span class="id" type="lemma">pr_conj_0_r</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_conj_le_r"><span class="id" type="lemma">pr_conj_le_r</span></a> <span class="id" type="var">d</span> <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>).<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a> <span class="id" type="var">d</span> (<span class="id" type="var">X1</span> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <span class="id" type="var">X2</span>)).<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="var">lra</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_1_l"><span class="id" type="lemma">pr_conj_1_l</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">d</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">trivial</span>; <span class="id" type="var">lra</span>.<br/>
+ <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Intervals.html#sum_to_1"><span class="id" type="lemma">sum_to_1</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">trivial</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_normality"><span class="id" type="lemma">pr_normality</span></a>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHd1</span>, <span class="id" type="var">IHd2</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_1_r"><span class="id" type="lemma">pr_conj_1_r</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">rewrite</span> <a class="idref" href="Distributions.html#pr_equivalence"><span class="id" type="lemma">pr_equivalence</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">X'</span> := (<span class="id" type="var">X2</span> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <span class="id" type="var">X1</span>)).<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_conj_1_l"><span class="id" type="lemma">pr_conj_1_l</span></a>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">X1</span> <span class="id" type="var">t</span>), (<span class="id" type="var">X2</span> <span class="id" type="var">t</span>); <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="pr_conj_1"><span class="id" type="lemma">pr_conj_1</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> {<span class="id" type="var">T</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">d</span> : <a class="idref" href="Distributions.html#dist"><span class="id" type="inductive">dist</span></a> <a class="idref" href="Distributions.html#T"><span class="id" type="variable">T</span></a>) <span class="id" type="var">X1</span> <span class="id" type="var">X2</span>, <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span> <br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1 <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">Pr</span></a> <a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">(</span></a><a class="idref" href="Distributions.html#X1"><span class="id" type="variable">X1</span></a> <a class="idref" href="Distributions.html#::x_'&&'_x"><span class="id" type="notation">&&</span></a> <a class="idref" href="Distributions.html#X2"><span class="id" type="variable">X2</span></a><a class="idref" href="Distributions.html#::'Pr'_x_'in'_x"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">in</span> <a class="idref" href="Distributions.html#d"><span class="id" type="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Distributions.html#pr_conj_1_r"><span class="id" type="lemma">pr_conj_1_r</span></a>.<br/>
<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
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