/usr/bin/diffoscope --timeout 7200 --html /srv/reproducible-results/rbuild-debian/r-b-build.B0dHHlZU/coq_8.20.1+dfsg-1.diffoscope.html --text /srv/reproducible-results/rbuild-debian/r-b-build.B0dHHlZU/coq_8.20.1+dfsg-1.diffoscope.txt --json /srv/reproducible-results/rbuild-debian/r-b-build.B0dHHlZU/coq_8.20.1+dfsg-1.diffoscope.json --profile=- /srv/reproducible-results/rbuild-debian/r-b-build.B0dHHlZU/b1/coq_8.20.1+dfsg-1_amd64.changes /srv/reproducible-results/rbuild-debian/r-b-build.B0dHHlZU/b2/coq_8.20.1+dfsg-1

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/srv/reproducible-results/rbuild-debian/r-b-build.B0dHHlZU/b2/coq_8.20.1+dfsg-1_amd64.changes ¶

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libcoq-core-ocaml-dev_8.20.1+dfsg-1_amd64.deb ¶

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/cSet.cmti ¶

Format-specific differences are supported for OCaml interface files but no file-specific differences were detected; falling back to a binary diff. file(1) reports: OCaml interface file (.cmi) (Version 035)

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/diff2.cmti ¶

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./usr/lib/x86_64-linux-gnu/ocaml/5.3.0/coq-core/clib/unicodetable.cmti ¶

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libcoq-stdlib_8.20.1+dfsg-1_amd64.deb ¶

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Offset 1, 9 lines modified		Offset 1, 9 lines modified

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00000600:·~~4e32~~·~~3b8b~~·8~~e28~~·~~010~~0·5~~09f~~·~~0022~~·1d20·3~~ab6~~··~~N2;.~~.(..P..".·:.		00000600:·93cc·4678·85ac·a7c0·5103·46e5·cdb6·3baf··..Fx....Q.F...;.
00000610:·~~39ad~~·~~d246~~·~~925a~~·c96d·~~be47~~·~~f46f~~·8~~d9a~~·9~~05c~~··9..~~F.Z.m~~.~~G.o~~...\		00000610:·06c6·0600·412d·3930·ef1a·62d4·84e4·92de··....A-90..b.....
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Offset 63, 352 lines modified		Offset 63, 351 lines modified
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000003f0:·37cc·bdfb·6821·c2b7·a0a0·3843·616d·6c69··7...h!....8Camli		000003f0:·37cc·bdfb·6821·c2b7·a0a0·3843·616d·6c69··7...h!....8Camli
00000400:·6e74·6572·6e61·6c46·6f72·6d61·7442·6173··nternalFormatBas		00000400:·6e74·6572·6e61·6c46·6f72·6d61·7442·6173··nternalFormatBas
00000410:·6963·7390·30aa·550b·dab5·21d6·0e76·9a9a··ics.0.U...!..v..		00000410:·6963·7390·30aa·550b·dab5·21d6·0e76·9a9a··ics.0.U...!..v..
00000420:·d43a·677e·6540·8495·a6be·0000·0004·0000··.:g~e@..........		00000420:·d43a·677e·6540·8495·a6be·0000·0004·0000··.:g~e@..........
00000430:·0002·0000·0005·0000·0005·a090·4040·4361··............@@Ca		00000430:·0002·0000·0005·0000·0005·a090·4040·4361··............@@Ca
00000440:·6d6c·3139·3939·5430·3335·8495·a6bd·0faa··ml1999T035......		00000440:·6d6c·3139·3939·5430·3335·8495·a6bd·0faa··ml1999T035......
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		00000460:·ab00·5a0f·8939·5620·485a·35e9·96ec·ee8e··..Z..9V·HZ5.....
		00000470:·1151·3313·f819·acb4·0a96·c8a6·1aaa·ce04··.Q3.............
		00000480:·1b76·20cd·0c4e·26b3·17b3·8bd2·7ac9·fc1d··.v·..N&.....z...
		00000490:·9ad8·1fbe·91c5·4143·5913·b2db·967c·a411··......ACY....\|..
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		000004d0:·b0b2·50a3·1edb·4aa3·5ea8·3e93·f52b·c0a8··..P...J.^.>..+..
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		000004f0:·aba0·c52c·03b4·492a·8c0a·d126·305f·6893··...,..I*...&0_h.
		00000500:·9f3e·96d1·8d8e·f571·e81f·316b·fbb3·b262··.>.....q..1k...b
		00000510:·1ae2·85be·0d87·7e0a·78fa·b84d·1f28·474d··......~.x..M.(GM
		00000520:·b08f·c5fa·76fa·75fa·1987·beaf·07ae·2e56··....v.u........V
		00000530:·4c73·fa72·fa79·76cf·f44b·8b64·f33d·6106··Ls.r.yv..K.d.=a.
		00000540:·2d2c·2b26·de69·c514·bc15·70b1·6282·61c5··-,+&.i....p.b.a.
		00000550:·320a·7dcc·4077·0e4d·5f88·bd17·8775·403f··2.}.@w.M_....u@?
		00000560:·df7b·dbfe·ee0d·a11f·bc17·878d·1262·4d48··.{...........bMH
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Offset 1, 13 lines modified		Offset 1, 13 lines modified
1	Package:·libcoq-stdlib	1	Package:·libcoq-stdlib
2	Source:·coq	2	Source:·coq
3	Version:·8.20.1+dfsg-1	3	Version:·8.20.1+dfsg-1
4	Architecture:·amd64	4	Architecture:·amd64
5	Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org>	5	Maintainer:·Debian·OCaml·Maintainers·<debian-ocaml-maint@lists.debian.org>
6	Installed-Size:·146164	6	Installed-Size:·146163
7	Recommends:·coq	7	Recommends:·coq
8	Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1),·coq-theories	8	Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1),·coq-theories
9	Replaces:·coq-libs·(<<·8.2.pl1),·coq-theories	9	Replaces:·coq-libs·(<<·8.2.pl1),·coq-theories
10	Provides:·coq-theories·(=·8.20.1+dfsg-1),·libcoq-stdlib-8ndp2	10	Provides:·coq-theories·(=·8.20.1+dfsg-1),·libcoq-stdlib-8ndp2
11	Section:·math	11	Section:·math
12	Priority:·optional	12	Priority:·optional
13	Homepage:·http://coq.inria.fr/	13	Homepage:·http://coq.inria.fr/


Offset 54, 15 lines modified		Offset 54, 15 lines modified
54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#"><span·class="id"·title="library">EqNat</span></a>.<br/>	54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#"><span·class="id"·title="library">EqNat</span></a>.<br/>
55	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Wf_nat.html#"><span·class="id"·title="library">Wf_nat</span></a>.<br/>	55	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Arith.Wf_nat.html#"><span·class="id"·title="library">Wf_nat</span></a>.<br/>

56	<br/>	56	<br/>
57	</div>	57	</div>

58	<div·class="doc">	58	<div·class="doc">
59	<a·id="lab294"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">arith</span></span>·hint·database</h1>	59	<a·id="lab291"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">arith</span></span>·hint·database</h1>

60	</div>	60	</div>
61	<div·class="code">	61	<div·class="code">

62	<br/>	62	<br/>
63	#[<span·class="id"·title="var">global</span>]<br/>	63	#[<span·class="id"·title="var">global</span>]<br/>
64	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">eq_nat_refl</span>:·<span·class="id"·title="var">arith</span>.<br/>	64	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">eq_nat_refl</span>:·<span·class="id"·title="var">arith</span>.<br/>
Offset 76, 15 lines modified		Offset 76, 15 lines modified
76	#[<span·class="id"·title="var">global</span>]<br/>	76	#[<span·class="id"·title="var">global</span>]<br/>
77	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">in_int_Sp_q</span>·<span·class="id"·title="var">exists_le_S</span>·<span·class="id"·title="var">exists_S_le</span>:·<span·class="id"·title="var">arith</span>.<br/>	77	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">in_int_Sp_q</span>·<span·class="id"·title="var">exists_le_S</span>·<span·class="id"·title="var">exists_S_le</span>:·<span·class="id"·title="var">arith</span>.<br/>

78	<br/>	78	<br/>
79	</div>	79	</div>

80	<div·class="doc">	80	<div·class="doc">
81	<a·id="lab295"></a><h2·class="section"><span·class="inlinecode"><span·class="id"·title="var">lt</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h2>	81	<a·id="lab292"></a><h2·class="section"><span·class="inlinecode"><span·class="id"·title="var">lt</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h2>

82	</div>	82	</div>
83	<div·class="code">	83	<div·class="code">
84	#[<span·class="id"·title="var">global</span>]<br/>	84	#[<span·class="id"·title="var">global</span>]<br/>
85	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_trans</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>	85	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_trans</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>
86	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">Nat.le_antisymm</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>	86	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">Nat.le_antisymm</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>
87	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_0_l</span>·<span·class="id"·title="var">Nat.nle_succ_0</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>	87	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Resolve</span>·<span·class="id"·title="var">Nat.le_0_l</span>·<span·class="id"·title="var">Nat.nle_succ_0</span>:·<span·class="id"·title="var">arith</span>.·#[<span·class="id"·title="var">global</span>]<br/>


Offset 52, 15 lines modified		Offset 52, 15 lines modified
52	<br/>	52	<br/>
53	</div>	53	</div>

54	<div·class="doc">	54	<div·class="doc">
55	Equality·on·natural·numbers·	55	Equality·on·natural·numbers·
56	<div·class="paragraph">·</div>	56	<div·class="paragraph">·</div>

57	<a·id="lab278"></a><h1·class="section">Propositional·equality</h1>	57	<a·id="lab293"></a><h1·class="section">Propositional·equality</h1>

58	</div>	58	</div>
59	<div·class="code">	59	<div·class="code">

60	<br/>	60	<br/>
61	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="eq_nat"·class="idref"·href="#eq_nat"><span·class="id"·title="definition">eq_nat</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<span·class="id"·title="keyword">Prop</span>·:=<br/>	61	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="eq_nat"·class="idref"·href="#eq_nat"><span·class="id"·title="definition">eq_nat</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·<a·id="m:2"·class="idref"·href="#m:2"><span·class="id"·title="binder">m</span></a>·:·<span·class="id"·title="keyword">Prop</span>·:=<br/>
62	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#n:1"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Arith.EqNat.html#m:2"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	62	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Arith.EqNat.html#n:1"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Arith.EqNat.html#m:2"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>


Offset 62, 15 lines modified		Offset 62, 15 lines modified
62	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations"><span·class="id"·title="module">BoolNotations</span></a>.<br/>	62	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations"><span·class="id"·title="module">BoolNotations</span></a>.<br/>
63	<span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">typeclass_instances</span>·<span·class="id"·title="var">relations</span>.<br/>	63	<span·class="id"·title="keyword">Local·Ltac</span>·<span·class="id"·title="var">Tauto.intuition_solver</span>·::=·<span·class="id"·title="tactic">auto</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">typeclass_instances</span>·<span·class="id"·title="var">relations</span>.<br/>

64	<br/>	64	<br/>
65	</div>	65	</div>

66	<div·class="doc">	66	<div·class="doc">
67	<a·id="lab~~243~~"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1>	67	<a·id="lab726"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1>

68	</div>	68	</div>
69	<div·class="code">	69	<div·class="code">

70	<br/>	70	<br/>
71	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="lemma">le_refl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:1"·class="idref"·href="#b:1"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>.<br/>	71	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="lemma">le_refl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:1"·class="idref"·href="#b:1"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:1"><span·class="id"·title="variable">b</span></a>.<br/>
72	·	72	·
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88	#[<span·class="id"·title="var">global</span>]<br/>	88	#[<span·class="id"·title="var">global</span>]<br/>
89	<span·class="id"·title="keyword">Instance</span>·<a·id="le_compat"·class="idref"·href="#le_compat"><span·class="id"·title="instance">le_compat</span></a>·:·<a·class="idref"·href="Coq.Classes.Morphisms.html#Proper"><span·class="id"·title="class">Proper</span></a>·(<a·class="idref"·href="Coq.Init.Logic.html#eq"><span·class="id"·title="inductive">eq</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#eq"><span·class="id"·title="inductive">eq</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#iff"><span·class="id"·title="definition">iff</span></a>)·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>	89	<span·class="id"·title="keyword">Instance</span>·<a·id="le_compat"·class="idref"·href="#le_compat"><span·class="id"·title="instance">le_compat</span></a>·:·<a·class="idref"·href="Coq.Classes.Morphisms.html#Proper"><span·class="id"·title="class">Proper</span></a>·(<a·class="idref"·href="Coq.Init.Logic.html#eq"><span·class="id"·title="inductive">eq</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#eq"><span·class="id"·title="inductive">eq</span></a>·<a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a>·<a·class="idref"·href="Coq.Init.Logic.html#iff"><span·class="id"·title="definition">iff</span></a>)·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>
90	·	90	·
91	<br/>	91	<br/>
92	</div>	92	</div>

93	<div·class="doc">	93	<div·class="doc">
94	<a·id="lab~~244~~"></a><h1·class="section">Strict·order·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1>	94	<a·id="lab727"></a><h1·class="section">Strict·order·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1>

95	</div>	95	</div>
96	<div·class="code">	96	<div·class="code">

97	<br/>	97	<br/>
98	<span·class="id"·title="keyword">Lemma</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="lemma">lt_irrefl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:7"·class="idref"·href="#b:7"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>.<br/>	98	<span·class="id"·title="keyword">Lemma</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="lemma">lt_irrefl</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b:7"·class="idref"·href="#b:7"><span·class="id"·title="binder">b</span></a>,·<a·class="idref"·href="Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span·class="id"·title="notation">~</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b:7"><span·class="id"·title="variable">b</span></a>.<br/>
99	·	99	·
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123	<br/>	123	<br/>
124	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_lteq"·class="idref"·href="#le_lteq"><span·class="id"·title="lemma">le_lteq</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:19"·class="idref"·href="#b1:19"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:20"·class="idref"·href="#b2:20"><span·class="id"·title="binder">b2</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>.<br/>	124	<span·class="id"·title="keyword">Lemma</span>·<a·id="le_lteq"·class="idref"·href="#le_lteq"><span·class="id"·title="lemma">le_lteq</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:19"·class="idref"·href="#b1:19"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:20"·class="idref"·href="#b2:20"><span·class="id"·title="binder">b2</span></a>,·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#b6c5208a24d231af77d2816ac22122c4"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#BoolNotations.::bool_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span·class="id"·title="notation">\/</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b1:19"><span·class="id"·title="variable">b1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Bool.BoolOrder.html#b2:20"><span·class="id"·title="variable">b2</span></a>.<br/>
125	·	125	·
126	<br/>	126	<br/>
127	</div>	127	</div>

128	<div·class="doc">	128	<div·class="doc">
129	<a·id="lab~~245~~"></a><h1·class="section">Order·structures</h1>	129	<a·id="lab728"></a><h1·class="section">Order·structures</h1>

130	</div>	130	</div>
131	<div·class="code">	131	<div·class="code">

132	<br/>	132	<br/>
133	#[<span·class="id"·title="var">global</span>]<br/>	133	#[<span·class="id"·title="var">global</span>]<br/>
134	<span·class="id"·title="keyword">Instance</span>·<a·id="le_preorder"·class="idref"·href="#le_preorder"><span·class="id"·title="instance">le_preorder</span></a>·:·<a·class="idref"·href="Coq.Classes.RelationClasses.html#PreOrder"><span·class="id"·title="class">PreOrder</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>	134	<span·class="id"·title="keyword">Instance</span>·<a·id="le_preorder"·class="idref"·href="#le_preorder"><span·class="id"·title="instance">le_preorder</span></a>·:·<a·class="idref"·href="Coq.Classes.RelationClasses.html#PreOrder"><span·class="id"·title="class">PreOrder</span></a>·<a·class="idref"·href="Coq.Bool.Bool.html#le"><span·class="id"·title="definition">Bool.le</span></a>.<br/>


Offset 45, 15 lines modified		Offset 45, 15 lines modified

45	<div·class="code">	45	<div·class="code">

46	<br/>	46	<br/>
47	</div>	47	</div>

48	<div·class="doc">	48	<div·class="doc">
49	<a·id="lab21"></a><h1·class="section">Typeclass-based·setoids.·Definitions·on·<span·class="inlinecode"><span·class="id"·title="var">Equivalence</span></span>.</h1>	49	<a·id="lab1141"></a><h1·class="section">Typeclass-based·setoids.·Definitions·on·<span·class="inlinecode"><span·class="id"·title="var">Equivalence</span></span>.</h1>


50	<div·class="paragraph">·</div>	50	<div·class="paragraph">·</div>

51	···Author:·Matthieu·Sozeau	51	···Author:·Matthieu·Sozeau
52	···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud	52	···Institution:·LRI,·CNRS·UMR·8623·-·University·Paris·Sud


Offset 47, 15 lines modified		Offset 47, 15 lines modified

47	<br/>	47	<br/>

48	<br/>	48	<br/>
49	</div>	49	</div>

50	<div·class="doc">	50	<div·class="doc">
51	<a·id="lab~~580~~"></a><h1·class="section">FMapAVL</h1>	51	<a·id="lab1064"></a><h1·class="section">FMapAVL</h1>

52	<div·class="paragraph">·</div>	52	<div·class="paragraph">·</div>

53	·This·module·implements·maps·using·AVL·trees.	53	·This·module·implements·maps·using·AVL·trees.
54	····It·follows·the·implementation·from·Ocaml's·standard·library.	54	····It·follows·the·implementation·from·Ocaml's·standard·library.

55	<div·class="paragraph">·</div>	55	<div·class="paragraph">·</div>
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84	<span·class="id"·title="keyword">Notation</span>·<a·id="0ecba978737ceb119698c5cbbee795eb"·class="idref"·href="#0ecba978737ceb119698c5cbbee795eb"><span·class="id"·title="notation">"</span></a>s·#1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#fst"><span·class="id"·title="definition">fst</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#1'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>	84	<span·class="id"·title="keyword">Notation</span>·<a·id="0ecba978737ceb119698c5cbbee795eb"·class="idref"·href="#0ecba978737ceb119698c5cbbee795eb"><span·class="id"·title="notation">"</span></a>s·#1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#fst"><span·class="id"·title="definition">fst</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#1'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>
85	<span·class="id"·title="keyword">Notation</span>·<a·id="3434f18bfdd992fed799ae34b564c3f9"·class="idref"·href="#3434f18bfdd992fed799ae34b564c3f9"><span·class="id"·title="notation">"</span></a>s·#2"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#snd"><span·class="id"·title="definition">snd</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#2'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>	85	<span·class="id"·title="keyword">Notation</span>·<a·id="3434f18bfdd992fed799ae34b564c3f9"·class="idref"·href="#3434f18bfdd992fed799ae34b564c3f9"><span·class="id"·title="notation">"</span></a>s·#2"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#snd"><span·class="id"·title="definition">snd</span></a>·<span·class="id"·title="var">s</span>)·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·9,·<span·class="id"·title="var">format</span>·"s·'#2'")·:·<span·class="id"·title="var">pair_scope</span>.<br/>

86	<br/>	86	<br/>
87	</div>	87	</div>

88	<div·class="doc">	88	<div·class="doc">
89	<a·id="lab~~581~~"></a><h1·class="section">The·Raw·functor</h1>	89	<a·id="lab1065"></a><h1·class="section">The·Raw·functor</h1>


90	<div·class="paragraph">·</div>	90	<div·class="paragraph">·</div>

91	···Functor·of·pure·functions·+·separate·proofs·of·invariant	91	···Functor·of·pure·functions·+·separate·proofs·of·invariant
92	···preservation·	92	···preservation·
93	</div>	93	</div>
Offset 110, 19 lines modified		Offset 110, 19 lines modified
110	#[<span·class="id"·title="var">global</span>]<br/>	110	#[<span·class="id"·title="var">global</span>]<br/>
111	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Transparent</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.key"><span·class="id"·title="definition">key</span></a>·:·<span·class="id"·title="var">core</span>.<br/>	111	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Transparent</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.key"><span·class="id"·title="definition">key</span></a>·:·<span·class="id"·title="var">core</span>.<br/>

112	<br/>	112	<br/>
113	</div>	113	</div>

114	<div·class="doc">	114	<div·class="doc">
115	<a·id="lab~~582~~"></a><h1·class="section">Trees</h1>	115	<a·id="lab1066"></a><h1·class="section">Trees</h1>

116	<div·class="paragraph">·</div>	116	<div·class="paragraph">·</div>

117	<a·id="lab~~583~~"></a><h1·class="section">Trees</h1>	117	<a·id="lab1067"></a><h1·class="section">Trees</h1>


118	<div·class="paragraph">·</div>	118	<div·class="paragraph">·</div>

119	···The·fifth·field·of·<span·class="inlinecode"><span·class="id"·title="var">Node</span></span>·is·the·height·of·the·tree·	119	···The·fifth·field·of·<span·class="inlinecode"><span·class="id"·title="var">Node</span></span>·is·the·height·of·the·tree·
120	</div>	120	</div>
121	<div·class="code">	121	<div·class="code">
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146	<br/>	146	<br/>
147	<span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>.<br/>	147	<span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>.<br/>

148	<br/>	148	<br/>
149	</div>	149	</div>

150	<div·class="doc">	150	<div·class="doc">
151	<a·id="lab~~584~~"></a><h1·class="section">Basic·functions·on·trees:·height·and·cardinal</h1>	151	<a·id="lab1068"></a><h1·class="section">Basic·functions·on·trees:·height·and·cardinal</h1>

152	</div>	152	</div>
153	<div·class="code">	153	<div·class="code">

154	<br/>	154	<br/>
155	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.height"·class="idref"·href="#Raw.height"><span·class="id"·title="definition">height</span></a>·(<a·id="m:6"·class="idref"·href="#m:6"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>)·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.int"><span·class="id"·title="abbreviation">int</span></a>·:=<br/>	155	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.height"·class="idref"·href="#Raw.height"><span·class="id"·title="definition">height</span></a>·(<a·id="m:6"·class="idref"·href="#m:6"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>)·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.int"><span·class="id"·title="abbreviation">int</span></a>·:=<br/>
156	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:6"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	156	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:6"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 169, 39 lines modified		Offset 169, 39 lines modified
169	\|·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Node"><span·class="id"·title="constructor">Node</span></a>·<span·class="id"·title="var">l</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">r</span>·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">l</span>·<a·class="idref"·href="Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">r</span>)<br/>	169	\|·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Node"><span·class="id"·title="constructor">Node</span></a>·<span·class="id"·title="var">l</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">_</span>·<span·class="id"·title="var">r</span>·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">l</span>·<a·class="idref"·href="Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#cardinal:9"><span·class="id"·title="definition">cardinal</span></a>·<span·class="id"·title="var">r</span>)<br/>
170	<span·class="id"·title="keyword">end</span>.<br/>	170	<span·class="id"·title="keyword">end</span>.<br/>

171	<br/>	171	<br/>
172	</div>	172	</div>

173	<div·class="doc">	173	<div·class="doc">
174	<a·id="lab~~585~~"></a><h1·class="section">Empty·Map</h1>	174	<a·id="lab1069"></a><h1·class="section">Empty·Map</h1>

175	</div>	175	</div>
176	<div·class="code">	176	<div·class="code">

177	<br/>	177	<br/>
178	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.empty"·class="idref"·href="#Raw.empty"><span·class="id"·title="definition">empty</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>·:=·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>.<br/>	178	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.empty"·class="idref"·href="#Raw.empty"><span·class="id"·title="definition">empty</span></a>·:·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.t"><span·class="id"·title="abbreviation">t</span></a>·:=·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>.<br/>

179	<br/>	179	<br/>
180	</div>	180	</div>

181	<div·class="doc">	181	<div·class="doc">
182	<a·id="lab~~586~~"></a><h1·class="section">Emptyness·test</h1>	182	<a·id="lab1070"></a><h1·class="section">Emptyness·test</h1>

183	</div>	183	</div>
184	<div·class="code">	184	<div·class="code">

185	<br/>	185	<br/>
186	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.is_empty"·class="idref"·href="#Raw.is_empty"><span·class="id"·title="definition">is_empty</span></a>·<a·id="m:11"·class="idref"·href="#m:11"><span·class="id"·title="binder">m</span></a>·:=·<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:11"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</span></a>·\|·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·<span·class="id"·title="keyword">end</span>.<br/>	186	<span·class="id"·title="keyword">Definition</span>·<a·id="Raw.is_empty"·class="idref"·href="#Raw.is_empty"><span·class="id"·title="definition">is_empty</span></a>·<a·id="m:11"·class="idref"·href="#m:11"><span·class="id"·title="binder">m</span></a>·:=·<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#m:11"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.Leaf"><span·class="id"·title="constructor">Leaf</span></a>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</span></a>·\|·<span·class="id"·title="var">_</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·<span·class="id"·title="keyword">end</span>.<br/>

187	<br/>	187	<br/>
188	</div>	188	</div>

189	<div·class="doc">	189	<div·class="doc">
190	<a·id="lab587"></a><h1·class="section">Membership</h1>	190	<a·id="lab1071"></a><h1·class="section">Membership</h1>

191	<div·class="paragraph">·</div>	191	<div·class="paragraph">·</div>

192	·The·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·function·is·deciding·membership.·It·exploits·the·<span·class="inlinecode"><span·class="id"·title="var">bst</span></span>·property	192	·The·<span·class="inlinecode"><span·class="id"·title="var">mem</span></span>·function·is·deciding·membership.·It·exploits·the·<span·class="inlinecode"><span·class="id"·title="var">bst</span></span>·property
193	····to·achieve·logarithmic·complexity.·	193	····to·achieve·logarithmic·complexity.·
194	</div>	194	</div>
195	<div·class="code">	195	<div·class="code">
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228	<span·class="id"·title="keyword">end</span><br/>	228	<span·class="id"·title="keyword">end</span><br/>
229	<span·class="id"·title="keyword">end</span>.<br/>	229	<span·class="id"·title="keyword">end</span>.<br/>

230	<br/>	230	<br/>
231	</div>	231	</div>

232	<div·class="doc">	232	<div·class="doc">
233	<a·id="lab~~588~~"></a><h1·class="section">Helper·functions</h1>	233	<a·id="lab1072"></a><h1·class="section">Helper·functions</h1>

234	<div·class="paragraph">·</div>	234	<div·class="paragraph">·</div>

235	·<span·class="inlinecode"><span·class="id"·title="var">create</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·creates·a·node,·assuming·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>	235	·<span·class="inlinecode"><span·class="id"·title="var">create</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode"><span·class="id"·title="var">x</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>·creates·a·node,·assuming·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">r</span></span>
236	····to·be·balanced·and·<span·class="inlinecode">\|<span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode">-</span>·<span·class="inlinecode"><span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span>\|</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode">2</span>.·	236	····to·be·balanced·and·<span·class="inlinecode">\|<span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">l</span></span>·<span·class="inlinecode">-</span>·<span·class="inlinecode"><span·class="id"·title="var">height</span></span>·<span·class="inlinecode"><span·class="id"·title="var">r</span>\|</span>·<span·class="inlinecode"><=</span>·<span·class="inlinecode">2</span>.·
237	</div>	237	</div>
238	<div·class="code">	238	<div·class="code">
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291	<span·class="id"·title="keyword">else</span><br/>	291	<span·class="id"·title="keyword">else</span><br/>
292	<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.create"><span·class="id"·title="definition">create</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#l:25"><span·class="id"·title="variable">l</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#x:26"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#d:27"><span·class="id"·title="variable">d</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#r:28"><span·class="id"·title="variable">r</span></a>.<br/>	292	<a·class="idref"·href="Coq.FSets.FMapAVL.html#Raw.create"><span·class="id"·title="definition">create</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#l:25"><span·class="id"·title="variable">l</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#x:26"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#d:27"><span·class="id"·title="variable">d</span></a>·<a·class="idref"·href="Coq.FSets.FMapAVL.html#r:28"><span·class="id"·title="variable">r</span></a>.<br/>

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Offset 71, 15 lines modified		Offset 71, 15 lines modified
71	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun"·class="idref"·href="#WDecide_fun"><span·class="id"·title="module">WDecide_fun</span></a>·(<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>)(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">M</span>·:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>).<br/>	71	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun"·class="idref"·href="#WDecide_fun"><span·class="id"·title="module">WDecide_fun</span></a>·(<span·class="id"·title="var">E</span>·:·<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType</span></a>)(<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">M</span>·:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>).<br/>
72	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.F"·class="idref"·href="#WDecide_fun.F"><span·class="id"·title="module">F</span></a>·:=·<a·class="idref"·href="Coq.FSets.FSetFacts.html#WFacts_fun"><span·class="id"·title="module">FSetFacts.WFacts_fun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#M"><span·class="id"·title="module">M</span></a>.<br/>	72	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.F"·class="idref"·href="#WDecide_fun.F"><span·class="id"·title="module">F</span></a>·:=·<a·class="idref"·href="Coq.FSets.FSetFacts.html#WFacts_fun"><span·class="id"·title="module">FSetFacts.WFacts_fun</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#E"><span·class="id"·title="module">E</span></a>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#M"><span·class="id"·title="module">M</span></a>.<br/>

73	<br/>	73	<br/>
74	</div>	74	</div>

75	<div·class="doc">	75	<div·class="doc">
76	<a·id="lab~~625~~"></a><h1·class="section">Overview</h1>	76	<a·id="lab1009"></a><h1·class="section">Overview</h1>

77	····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will	77	····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will
78	····solve·any·valid·goal·of·the·form	78	····solve·any·valid·goal·of·the·form
79	<pre>	79	<pre>
80	····forall·s1·...·sn,	80	····forall·s1·...·sn,
81	····forall·x1·...·xm,	81	····forall·x1·...·xm,
82	····P1·->·...·->·Pk·->·P	82	····P1·->·...·->·Pk·->·P
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177	</pre>	177	</pre>

178	</li>	178	</li>
179	</ul>	179	</ul>

180	<div·class="paragraph">·</div>	180	<div·class="paragraph">·</div>

181	<a·id="lab~~626~~"></a><h1·class="section">Facts·and·Tactics·for·Propositional·Logic</h1>	181	<a·id="lab1010"></a><h1·class="section">Facts·and·Tactics·for·Propositional·Logic</h1>

182	······These·lemmas·and·tactics·are·in·a·module·so·that·they·do	182	······These·lemmas·and·tactics·are·in·a·module·so·that·they·do
183	······not·affect·the·namespace·if·you·import·the·enclosing	183	······not·affect·the·namespace·if·you·import·the·enclosing
184	······module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.·	184	······module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.·
185	</div>	185	</div>
186	<div·class="code">	186	<div·class="code">
187	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetLogicalFacts"·class="idref"·href="#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>	187	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetLogicalFacts"·class="idref"·href="#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>
188	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Logic.Decidable.html#"><span·class="id"·title="module">Decidable</span></a>.<br/>	188	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Logic.Decidable.html#"><span·class="id"·title="module">Decidable</span></a>.<br/>
189	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="module">Setoid</span></a>.<br/>	189	<span·class="id"·title="keyword">Export</span>·<a·class="idref"·href="Coq.Setoids.Setoid.html#"><span·class="id"·title="module">Setoid</span></a>.<br/>

190	<br/>	190	<br/>
191	</div>	191	</div>

192	<div·class="doc">	192	<div·class="doc">
193	<a·id="lab~~627~~"></a><h2·class="section">Lemmas·and·Tactics·About·Decidable·Propositions</h2>	193	<a·id="lab1011"></a><h2·class="section">Lemmas·and·Tactics·About·Decidable·Propositions</h2>

194	<div·class="paragraph">·</div>	194	<div·class="paragraph">·</div>

195	<a·id="lab~~628~~"></a><h2·class="section">Propositional·Equivalences·Involving·Negation</h2>	195	<a·id="lab1012"></a><h2·class="section">Propositional·Equivalences·Involving·Negation</h2>

196	········These·are·all·written·with·the·unfolded·form·of	196	········These·are·all·written·with·the·unfolded·form·of
197	········negation,·since·I·am·not·sure·if·setoid·rewriting·will	197	········negation,·since·I·am·not·sure·if·setoid·rewriting·will
198	········always·perform·conversion.·	198	········always·perform·conversion.·
199	<div·class="paragraph">·</div>	199	<div·class="paragraph">·</div>

200	<a·id="lab~~629~~"></a><h2·class="section">Tactics·for·Negations</h2>	200	<a·id="lab1013"></a><h2·class="section">Tactics·for·Negations</h2>

201	</div>	201	</div>
202	<div·class="code">	202	<div·class="code">

203	<br/>	203	<br/>
204	<span·class="id"·title="keyword">Tactic·Notation</span>·"fold"·"any"·"not"·:=<br/>	204	<span·class="id"·title="keyword">Tactic·Notation</span>·"fold"·"any"·"not"·:=<br/>
205	<span·class="id"·title="tactic">repeat</span>·(<br/>	205	<span·class="id"·title="tactic">repeat</span>·(<br/>
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454	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>	454	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>
455	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>	455	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetLogicalFacts"><span·class="id"·title="module">FSetLogicalFacts</span></a>.<br/>

456	<br/>	456	<br/>
457	</div>	457	</div>

458	<div·class="doc">	458	<div·class="doc">
459	<a·id="lab~~630~~"></a><h1·class="section">Auxiliary·Tactics</h1>	459	<a·id="lab1014"></a><h1·class="section">Auxiliary·Tactics</h1>

460	······Again,·these·lemmas·and·tactics·are·in·a·module·so·that	460	······Again,·these·lemmas·and·tactics·are·in·a·module·so·that
461	······they·do·not·affect·the·namespace·if·you·import·the	461	······they·do·not·affect·the·namespace·if·you·import·the
462	······enclosing·module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.··	462	······enclosing·module·<span·class="inlinecode"><span·class="id"·title="var">Decide</span></span>.··
463	</div>	463	</div>
464	<div·class="code">	464	<div·class="code">
465	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetDecideAuxiliary"·class="idref"·href="#WDecide_fun.FSetDecideAuxiliary"><span·class="id"·title="module">FSetDecideAuxiliary</span></a>.<br/>	465	<span·class="id"·title="keyword">Module</span>·<a·id="WDecide_fun.FSetDecideAuxiliary"·class="idref"·href="#WDecide_fun.FSetDecideAuxiliary"><span·class="id"·title="module">FSetDecideAuxiliary</span></a>.<br/>

466	<br/>	466	<br/>
467	</div>	467	</div>

468	<div·class="doc">	468	<div·class="doc">
469	<a·id="lab631"></a><h2·class="section">Generic·Tactics</h2>	469	<a·id="lab1015"></a><h2·class="section">Generic·Tactics</h2>

470	········We·begin·by·defining·a·few·generic,·useful·tactics.·	470	········We·begin·by·defining·a·few·generic,·useful·tactics.·
471	<div·class="paragraph">·</div>	471	<div·class="paragraph">·</div>

472	·remove·logical·hypothesis·inter-dependencies·(fix·2136).·	472	·remove·logical·hypothesis·inter-dependencies·(fix·2136).·
473	</div>	473	</div>
474	<div·class="code">	474	<div·class="code">
Offset 584, 15 lines modified		Offset 584, 15 lines modified
584	\|·<span·class="id"·title="var">H</span>:·<span·class="id"·title="var">_</span>·\|-·<span·class="id"·title="var">_</span>·=>·<span·class="id"·title="tactic">progress</span>·(<span·class="id"·title="var">decompose</span>·<span·class="id"·title="var">record</span>·<span·class="id"·title="var">H</span>);·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>	584	\|·<span·class="id"·title="var">H</span>:·<span·class="id"·title="var">_</span>·\|-·<span·class="id"·title="var">_</span>·=>·<span·class="id"·title="tactic">progress</span>·(<span·class="id"·title="var">decompose</span>·<span·class="id"·title="var">record</span>·<span·class="id"·title="var">H</span>);·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>
585	<span·class="id"·title="keyword">end</span>).<br/>	585	<span·class="id"·title="keyword">end</span>).<br/>

586	<br/>	586	<br/>
587	</div>	587	</div>

588	<div·class="doc">	588	<div·class="doc">
589	<a·id="lab~~632~~"></a><h2·class="section">Discarding·Irrelevant·Hypotheses</h2>	589	<a·id="lab1016"></a><h2·class="section">Discarding·Irrelevant·Hypotheses</h2>

590	········We·will·want·to·clear·the·context·of·any	590	········We·will·want·to·clear·the·context·of·any
591	········non-FSet-related·hypotheses·in·order·to·increase·the	591	········non-FSet-related·hypotheses·in·order·to·increase·the
592	········speed·of·the·tactic.··To·do·this,·we·will·need·to·be	592	········speed·of·the·tactic.··To·do·this,·we·will·need·to·be
593	········able·to·decide·which·are·relevant.··We·do·this·by·making	593	········able·to·decide·which·are·relevant.··We·do·this·by·making
594	········a·simple·inductive·definition·classifying·the	594	········a·simple·inductive·definition·classifying·the
595	········propositions·of·interest.·	595	········propositions·of·interest.·
Offset 664, 15 lines modified		Offset 664, 15 lines modified
664	<span·class="id"·title="keyword">else</span>·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>	664	<span·class="id"·title="keyword">else</span>·<span·class="id"·title="tactic">clear</span>·<span·class="id"·title="var">H</span><br/>
665	<span·class="id"·title="keyword">end</span>).<br/>	665	<span·class="id"·title="keyword">end</span>).<br/>

666	<br/>	666	<br/>
667	</div>	667	</div>

668	<div·class="doc">	668	<div·class="doc">
669	<a·id="lab~~633~~"></a><h2·class="section">Turning·Set·Operators·into·Propositional·Connectives</h2>	669	<a·id="lab1017"></a><h2·class="section">Turning·Set·Operators·into·Propositional·Connectives</h2>

670	········The·lemmas·from·<span·class="inlinecode"><span·class="id"·title="var">FSetFacts</span></span>·will·be·used·to·break·down	670	········The·lemmas·from·<span·class="inlinecode"><span·class="id"·title="var">FSetFacts</span></span>·will·be·used·to·break·down
671	········set·operations·into·propositional·formulas·built·over	671	········set·operations·into·propositional·formulas·built·over
672	········the·predicates·<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">E.eq</span></span>·applied·only·to	672	········the·predicates·<span·class="inlinecode"><span·class="id"·title="var">In</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">E.eq</span></span>·applied·only·to
673	········variables.··We·are·going·to·use·them·with·<span·class="inlinecode"><span·class="id"·title="tactic">autorewrite</span></span>.	673	········variables.··We·are·going·to·use·them·with·<span·class="inlinecode"><span·class="id"·title="tactic">autorewrite</span></span>.
674	········	674	········
675	</div>	675	</div>
Offset 690, 15 lines modified		Offset 690, 15 lines modified
690	<br/>	690	<br/>
691	<span·class="id"·title="keyword">Global·Hint·Rewrite</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetDecideAuxiliary.eq_refl_iff"><span·class="id"·title="lemma">eq_refl_iff</span></a>·:·<span·class="id"·title="var">set_eq_simpl</span>.<br/>	691	<span·class="id"·title="keyword">Global·Hint·Rewrite</span>·<a·class="idref"·href="Coq.FSets.FSetDecide.html#WDecide_fun.FSetDecideAuxiliary.eq_refl_iff"><span·class="id"·title="lemma">eq_refl_iff</span></a>·:·<span·class="id"·title="var">set_eq_simpl</span>.<br/>

692	<br/>	692	<br/>
693	</div>	693	</div>

694	<div·class="doc">	694	<div·class="doc">
695	<a·id="lab~~634~~"></a><h2·class="section">Decidability·of·FSet·Propositions</h2>	695	<a·id="lab1018"></a><h2·class="section">Decidability·of·FSet·Propositions</h2>
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Offset 45, 28 lines modified		Offset 45, 28 lines modified

45	<div·class="code">	45	<div·class="code">

46	<br/>	46	<br/>
47	</div>	47	</div>

48	<div·class="doc">	48	<div·class="doc">
49	<a·id="lab~~578~~"></a><h1·class="section">Finite·sets·library·:·conversion·to·old·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span></h1>	49	<a·id="lab1114"></a><h1·class="section">Finite·sets·library·:·conversion·to·old·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span></h1>

50	</div>	50	</div>
51	<div·class="code">	51	<div·class="code">

52	<br/>	52	<br/>
53	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Sets.Ensembles.html#"><span·class="id"·title="library">Ensembles</span></a>·<a·class="idref"·href="Coq.Sets.Finite_sets.html#"><span·class="id"·title="library">Finite_sets</span></a>.<br/>	53	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Sets.Ensembles.html#"><span·class="id"·title="library">Ensembles</span></a>·<a·class="idref"·href="Coq.Sets.Finite_sets.html#"><span·class="id"·title="library">Finite_sets</span></a>.<br/>
54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetInterface.html#"><span·class="id"·title="library">FSetInterface</span></a>·<a·class="idref"·href="Coq.FSets.FSetProperties.html#"><span·class="id"·title="library">FSetProperties</span></a>·<a·class="idref"·href="Coq.Structures.OrderedTypeEx.html#"><span·class="id"·title="library">OrderedTypeEx</span></a>·<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#"><span·class="id"·title="library">DecidableTypeEx</span></a>.<br/>	54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.FSets.FSetInterface.html#"><span·class="id"·title="library">FSetInterface</span></a>·<a·class="idref"·href="Coq.FSets.FSetProperties.html#"><span·class="id"·title="library">FSetProperties</span></a>·<a·class="idref"·href="Coq.Structures.OrderedTypeEx.html#"><span·class="id"·title="library">OrderedTypeEx</span></a>·<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#"><span·class="id"·title="library">DecidableTypeEx</span></a>.<br/>

55	<br/>	55	<br/>
56	</div>	56	</div>

57	<div·class="doc">	57	<div·class="doc">
58	<a·id="lab~~579~~"></a><h1·class="section">Going·from·<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·with·usual·Leibniz·equality</h1>	58	<a·id="lab1115"></a><h1·class="section">Going·from·<span·class="inlinecode"><span·class="id"·title="var">FSets</span></span>·with·usual·Leibniz·equality</h1>

59	····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.·	59	····to·the·good·old·<span·class="inlinecode"><span·class="id"·title="var">Ensembles</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">Finite_sets</span></span>·theory.·
60	</div>	60	</div>
61	<div·class="code">	61	<div·class="code">

62	<br/>	62	<br/>
63	<span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/>	63	<span·class="id"·title="keyword">Module</span>·<a·id="WS_to_Finite_set"·class="idref"·href="#WS_to_Finite_set"><span·class="id"·title="module">WS_to_Finite_set</span></a>·(<span·class="id"·title="var">U</span>:<a·class="idref"·href="Coq.Structures.DecidableTypeEx.html#UsualDecidableType"><span·class="id"·title="module">UsualDecidableType</span></a>)(<span·class="id"·title="var">M</span>:·<a·class="idref"·href="Coq.FSets.FSetInterface.html#WSfun"><span·class="id"·title="module">WSfun</span></a>·<a·class="idref"·href="Coq.FSets.FSetToFiniteSet.html#U"><span·class="id"·title="module">U</span></a>).<br/>


Offset 48, 15 lines modified		Offset 48, 15 lines modified
48	<br/>	48	<br/>
49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>.<br/>	49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>.<br/>

50	<br/>	50	<br/>
51	</div>	51	</div>

52	<div·class="doc">	52	<div·class="doc">
53	<a·id="lab~~406~~"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1>	53	<a·id="lab957"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1>

54	</div>	54	</div>
55	<div·class="code">	55	<div·class="code">

56	<br/>	56	<br/>
57	<span·class="id"·title="keyword">Notation</span>·<a·id="valid_binary"·class="idref"·href="#valid_binary"><span·class="id"·title="abbreviation">valid_binary</span></a>·:=·(<a·class="idref"·href="Coq.Floats.SpecFloat.html#valid_binary"><span·class="id"·title="definition">valid_binary</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>).<br/>	57	<span·class="id"·title="keyword">Notation</span>·<a·id="valid_binary"·class="idref"·href="#valid_binary"><span·class="id"·title="abbreviation">valid_binary</span></a>·:=·(<a·class="idref"·href="Coq.Floats.SpecFloat.html#valid_binary"><span·class="id"·title="definition">valid_binary</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>).<br/>


Offset 49, 15 lines modified		Offset 49, 15 lines modified
49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>·<a·class="idref"·href="Coq.Floats.FloatAxioms.html#"><span·class="id"·title="library">FloatAxioms</span></a>.<br/>	49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.ZArith.ZArith.html#"><span·class="id"·title="library">ZArith</span></a>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Uint63.html#"><span·class="id"·title="library">Uint63</span></a>·<a·class="idref"·href="Coq.Floats.SpecFloat.html#"><span·class="id"·title="library">SpecFloat</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#"><span·class="id"·title="library">PrimFloat</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#"><span·class="id"·title="library">FloatOps</span></a>·<a·class="idref"·href="Coq.Floats.FloatAxioms.html#"><span·class="id"·title="library">FloatAxioms</span></a>.<br/>
50	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Psatz.html#"><span·class="id"·title="library">Psatz</span></a>.<br/>	50	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.micromega.Psatz.html#"><span·class="id"·title="library">Psatz</span></a>.<br/>

51	<br/>	51	<br/>
52	</div>	52	</div>

53	<div·class="doc">	53	<div·class="doc">
54	<a·id="lab~~407~~"></a><h1·class="section">Support·results·involving·frexp·and·ldexp</h1>	54	<a·id="lab959"></a><h1·class="section">Support·results·involving·frexp·and·ldexp</h1>

55	</div>	55	</div>
56	<div·class="code">	56	<div·class="code">

57	<br/>	57	<br/>
58	<span·class="id"·title="keyword">Lemma</span>·<a·id="shift_value"·class="idref"·href="#shift_value"><span·class="id"·title="lemma">shift_value</span></a>·:·<a·class="idref"·href="Coq.Floats.FloatOps.html#shift"><span·class="id"·title="definition">shift</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·(2<a·class="idref"·href="Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>)%<span·class="id"·title="var">Z</span>.<br/>	58	<span·class="id"·title="keyword">Lemma</span>·<a·id="shift_value"·class="idref"·href="#shift_value"><span·class="id"·title="lemma">shift_value</span></a>·:·<a·class="idref"·href="Coq.Floats.FloatOps.html#shift"><span·class="id"·title="definition">shift</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·(2<a·class="idref"·href="Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span·class="id"·title="notation">*</span></a><a·class="idref"·href="Coq.Floats.FloatOps.html#emax"><span·class="id"·title="definition">emax</span></a>·<a·class="idref"·href="Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Floats.FloatOps.html#prec"><span·class="id"·title="definition">prec</span></a>)%<span·class="id"·title="var">Z</span>.<br/>


Offset 48, 24 lines modified		Offset 48, 24 lines modified
48	<br/>	48	<br/>
49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.PrimInt63.html#"><span·class="id"·title="library">PrimInt63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>	49	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.PrimInt63.html#"><span·class="id"·title="library">PrimInt63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>

50	<br/>	50	<br/>
51	</div>	51	</div>

52	<div·class="doc">	52	<div·class="doc">
53	<a·id="lab~~409~~"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1>	53	<a·id="lab946"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1>


54	<div·class="paragraph">·</div>	54	<div·class="paragraph">·</div>

55	This·interface·provides·processor·operators·for·the·Binary64·format·of·the	55	This·interface·provides·processor·operators·for·the·Binary64·format·of·the
56	IEEE·754-2008·standard.·	56	IEEE·754-2008·standard.·
57	<div·class="paragraph">·</div>	57	<div·class="paragraph">·</div>

58	<a·id="lab~~410~~"></a><h2·class="section">Type·definition·for·the·co-domain·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h2>	58	<a·id="lab947"></a><h2·class="section">Type·definition·for·the·co-domain·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h2>

59	</div>	59	</div>
60	<div·class="code">	60	<div·class="code">
61	<span·class="id"·title="keyword">Variant</span>·<a·id="float_comparison"·class="idref"·href="#float_comparison"><span·class="id"·title="inductive">float_comparison</span></a>·:·<span·class="id"·title="keyword">Set</span>·:=·<a·id="FEq"·class="idref"·href="#FEq"><span·class="id"·title="constructor">FEq</span></a>·\|·<a·id="FLt"·class="idref"·href="#FLt"><span·class="id"·title="constructor">FLt</span></a>·\|·<a·id="FGt"·class="idref"·href="#FGt"><span·class="id"·title="constructor">FGt</span></a>·\|·<a·id="FNotComparable"·class="idref"·href="#FNotComparable"><span·class="id"·title="constructor">FNotComparable</span></a>.<br/>	61	<span·class="id"·title="keyword">Variant</span>·<a·id="float_comparison"·class="idref"·href="#float_comparison"><span·class="id"·title="inductive">float_comparison</span></a>·:·<span·class="id"·title="keyword">Set</span>·:=·<a·id="FEq"·class="idref"·href="#FEq"><span·class="id"·title="constructor">FEq</span></a>·\|·<a·id="FLt"·class="idref"·href="#FLt"><span·class="id"·title="constructor">FLt</span></a>·\|·<a·id="FGt"·class="idref"·href="#FGt"><span·class="id"·title="constructor">FGt</span></a>·\|·<a·id="FNotComparable"·class="idref"·href="#FNotComparable"><span·class="id"·title="constructor">FNotComparable</span></a>.<br/>

62	<br/>	62	<br/>
63	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_comparison</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_cmp</span>.<br/>	63	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_comparison</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_cmp</span>.<br/>
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73	<br/>	73	<br/>
74	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_class</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_class</span>.<br/>	74	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float_class</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">kernel.ind_f_class</span>.<br/>

75	<br/>	75	<br/>
76	</div>	76	</div>

77	<div·class="doc">	77	<div·class="doc">
78	<a·id="lab~~411~~"></a><h2·class="section">The·main·type</h2>	78	<a·id="lab948"></a><h2·class="section">The·main·type</h2>
79	·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>:·primitive·type·for·Binary64·floating-point·numbers.·	79	·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>:·primitive·type·for·Binary64·floating-point·numbers.·
80	</div>	80	</div>
81	<div·class="code">	81	<div·class="code">
82	<span·class="id"·title="keyword">Primitive</span>·<a·id="float"·class="idref"·href="#float"><span·class="id"·title="axiom">float</span></a>·:=·#<span·class="id"·title="var">float64_type</span>.<br/>	82	<span·class="id"·title="keyword">Primitive</span>·<a·id="float"·class="idref"·href="#float"><span·class="id"·title="axiom">float</span></a>·:=·#<span·class="id"·title="var">float64_type</span>.<br/>
83	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.float.type</span>.<br/>	83	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">float</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.float.type</span>.<br/>

84	<br/>	84	<br/>
Offset 91, 30 lines modified		Offset 91, 30 lines modified
91	<span·class="id"·title="keyword">Definition</span>·<a·id="printer"·class="idref"·href="#printer"><span·class="id"·title="definition">printer</span></a>·(<a·id="x:5"·class="idref"·href="#x:5"><span·class="id"·title="binder">x</span></a>·:·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrapper"><span·class="id"·title="record">float_wrapper</span></a>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrap"><span·class="id"·title="projection">float_wrap</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:5"><span·class="id"·title="variable">x</span></a>.<br/>	91	<span·class="id"·title="keyword">Definition</span>·<a·id="printer"·class="idref"·href="#printer"><span·class="id"·title="definition">printer</span></a>·(<a·id="x:5"·class="idref"·href="#x:5"><span·class="id"·title="binder">x</span></a>·:·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrapper"><span·class="id"·title="record">float_wrapper</span></a>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#float_wrap"><span·class="id"·title="projection">float_wrap</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:5"><span·class="id"·title="variable">x</span></a>.<br/>
92	<span·class="id"·title="keyword">Definition</span>·<a·id="parser"·class="idref"·href="#parser"><span·class="id"·title="definition">parser</span></a>·(<a·id="x:6"·class="idref"·href="#x:6"><span·class="id"·title="binder">x</span></a>·:·<span·class="id"·title="var">float</span>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:6"><span·class="id"·title="variable">x</span></a>.<br/>	92	<span·class="id"·title="keyword">Definition</span>·<a·id="parser"·class="idref"·href="#parser"><span·class="id"·title="definition">parser</span></a>·(<a·id="x:6"·class="idref"·href="#x:6"><span·class="id"·title="binder">x</span></a>·:·<span·class="id"·title="var">float</span>)·:·<span·class="id"·title="var">float</span>·:=·<a·class="idref"·href="Coq.Floats.PrimFloat.html#x:6"><span·class="id"·title="variable">x</span></a>.<br/>

93	<br/>	93	<br/>
94	</div>	94	</div>

95	<div·class="doc">	95	<div·class="doc">
96	<a·id="lab~~412~~"></a><h2·class="section">Syntax·support</h2>	96	<a·id="lab949"></a><h2·class="section">Syntax·support</h2>

97	</div>	97	</div>
98	<div·class="code">	98	<div·class="code">
99	<span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Import</span>·<a·id="PrimFloatNotationsInternalA"·class="idref"·href="#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>	99	<span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Import</span>·<a·id="PrimFloatNotationsInternalA"·class="idref"·href="#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>
100	<span·class="id"·title="keyword">Declare·Scope</span>·<span·class="id"·title="var">float_scope</span>.<br/>	100	<span·class="id"·title="keyword">Declare·Scope</span>·<span·class="id"·title="var">float_scope</span>.<br/>
101	<span·class="id"·title="keyword">Delimit</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>	101	<span·class="id"·title="keyword">Delimit</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>
102	<span·class="id"·title="keyword">Bind·Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>	102	<span·class="id"·title="keyword">Bind·Scope</span>·<span·class="id"·title="var">float_scope</span>·<span·class="id"·title="keyword">with</span>·<span·class="id"·title="var">float</span>.<br/>
103	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>	103	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalA"><span·class="id"·title="module">PrimFloatNotationsInternalA</span></a>.<br/>
104	<span·class="id"·title="keyword">Number·Notation</span>·<span·class="id"·title="var">float</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#parser"><span·class="id"·title="definition">parser</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#printer"><span·class="id"·title="definition">printer</span></a>·:·<span·class="id"·title="var">float_scope</span>.<br/>	104	<span·class="id"·title="keyword">Number·Notation</span>·<span·class="id"·title="var">float</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#parser"><span·class="id"·title="definition">parser</span></a>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#printer"><span·class="id"·title="definition">printer</span></a>·:·<span·class="id"·title="var">float_scope</span>.<br/>

105	<br/>	105	<br/>
106	</div>	106	</div>

107	<div·class="doc">	107	<div·class="doc">
108	<a·id="lab~~413~~"></a><h2·class="section">Floating-point·operators</h2>	108	<a·id="lab950"></a><h2·class="section">Floating-point·operators</h2>

109	</div>	109	</div>
110	<div·class="code">	110	<div·class="code">
111	<span·class="id"·title="keyword">Primitive</span>·<a·id="classify"·class="idref"·href="#classify"><span·class="id"·title="axiom">classify</span></a>·:=·#<span·class="id"·title="var">float64_classify</span>.<br/>	111	<span·class="id"·title="keyword">Primitive</span>·<a·id="classify"·class="idref"·href="#classify"><span·class="id"·title="axiom">classify</span></a>·:=·#<span·class="id"·title="var">float64_classify</span>.<br/>

112	<br/>	112	<br/>
113	<span·class="id"·title="keyword">Primitive</span>·<a·id="abs"·class="idref"·href="#abs"><span·class="id"·title="axiom">abs</span></a>·:=·#<span·class="id"·title="var">float64_abs</span>.<br/>	113	<span·class="id"·title="keyword">Primitive</span>·<a·id="abs"·class="idref"·href="#abs"><span·class="id"·title="axiom">abs</span></a>·:=·#<span·class="id"·title="var">float64_abs</span>.<br/>
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181	<span·class="id"·title="keyword">Notation</span>·<a·id="6e32edb097de448d43e1d8a509d61efe"·class="idref"·href="#6e32edb097de448d43e1d8a509d61efe"><span·class="id"·title="notation">"</span></a>x·/·y"·:=·(<span·class="id"·title="var">div</span>·<span·class="id"·title="var">x</span>·<span·class="id"·title="var">y</span>)·:·<span·class="id"·title="var">float_scope</span>.<br/>	181	<span·class="id"·title="keyword">Notation</span>·<a·id="6e32edb097de448d43e1d8a509d61efe"·class="idref"·href="#6e32edb097de448d43e1d8a509d61efe"><span·class="id"·title="notation">"</span></a>x·/·y"·:=·(<span·class="id"·title="var">div</span>·<span·class="id"·title="var">x</span>·<span·class="id"·title="var">y</span>)·:·<span·class="id"·title="var">float_scope</span>.<br/>
182	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalB"><span·class="id"·title="module">PrimFloatNotationsInternalB</span></a>.<br/>	182	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Floats.PrimFloat.html#PrimFloatNotationsInternalB"><span·class="id"·title="module">PrimFloatNotationsInternalB</span></a>.<br/>

183	<br/>	183	<br/>
184	</div>	184	</div>

185	<div·class="doc">	185	<div·class="doc">
186	<a·id="lab~~414~~"></a><h2·class="section">Conversions</h2>	186	<a·id="lab951"></a><h2·class="section">Conversions</h2>

187	<div·class="paragraph">·</div>	187	<div·class="paragraph">·</div>

188	·<span·class="inlinecode"><span·class="id"·title="var">of_uint63</span></span>:·convert·a·primitive·unsigned·integer·into·a·float·value.	188	·<span·class="inlinecode"><span·class="id"·title="var">of_uint63</span></span>:·convert·a·primitive·unsigned·integer·into·a·float·value.
189	····The·value·is·rounded·if·need·be.·	189	····The·value·is·rounded·if·need·be.·
190	</div>	190	</div>
191	<div·class="code">	191	<div·class="code">
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220	<div·class="code">	220	<div·class="code">
221	<span·class="id"·title="keyword">Primitive</span>·<a·id="normfr_mantissa"·class="idref"·href="#normfr_mantissa"><span·class="id"·title="axiom">normfr_mantissa</span></a>·:=·#<span·class="id"·title="var">float64_normfr_mantissa</span>.<br/>	221	<span·class="id"·title="keyword">Primitive</span>·<a·id="normfr_mantissa"·class="idref"·href="#normfr_mantissa"><span·class="id"·title="axiom">normfr_mantissa</span></a>·:=·#<span·class="id"·title="var">float64_normfr_mantissa</span>.<br/>

222	<br/>	222	<br/>
223	</div>	223	</div>

224	<div·class="doc">	224	<div·class="doc">
225	<a·id="lab~~415~~"></a><h2·class="section">Exponent·manipulation·functions</h2>	225	<a·id="lab952"></a><h2·class="section">Exponent·manipulation·functions</h2>
226	·<span·class="inlinecode"><span·class="id"·title="var">frshiftexp</span></span>:·convert·a·float·to·fractional·part·in·[0.5,·1.)	226	·<span·class="inlinecode"><span·class="id"·title="var">frshiftexp</span></span>:·convert·a·float·to·fractional·part·in·[0.5,·1.)
227	and·integer·part.·	227	and·integer·part.·
228	</div>	228	</div>
229	<div·class="code">	229	<div·class="code">
230	<span·class="id"·title="keyword">Primitive</span>·<a·id="frshiftexp"·class="idref"·href="#frshiftexp"><span·class="id"·title="axiom">frshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_frshiftexp</span>.<br/>	230	<span·class="id"·title="keyword">Primitive</span>·<a·id="frshiftexp"·class="idref"·href="#frshiftexp"><span·class="id"·title="axiom">frshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_frshiftexp</span>.<br/>

231	<br/>	231	<br/>
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240	<div·class="code">	240	<div·class="code">
241	<span·class="id"·title="keyword">Primitive</span>·<a·id="ldshiftexp"·class="idref"·href="#ldshiftexp"><span·class="id"·title="axiom">ldshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_ldshiftexp</span>.<br/>	241	<span·class="id"·title="keyword">Primitive</span>·<a·id="ldshiftexp"·class="idref"·href="#ldshiftexp"><span·class="id"·title="axiom">ldshiftexp</span></a>·:=·#<span·class="id"·title="var">float64_ldshiftexp</span>.<br/>

242	<br/>	242	<br/>
243	</div>	243	</div>

244	<div·class="doc">	244	<div·class="doc">
245	<a·id="lab~~416~~"></a><h2·class="section">Predecesor/Successor·functions</h2>	245	<a·id="lab953"></a><h2·class="section">Predecesor/Successor·functions</h2>

246	<div·class="paragraph">·</div>	246	<div·class="paragraph">·</div>

247	·<span·class="inlinecode"><span·class="id"·title="var">next_up</span></span>:·return·the·next·float·towards·positive·infinity.·	247	·<span·class="inlinecode"><span·class="id"·title="var">next_up</span></span>:·return·the·next·float·towards·positive·infinity.·
248	</div>	248	</div>
249	<div·class="code">	249	<div·class="code">
250	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_up"·class="idref"·href="#next_up"><span·class="id"·title="axiom">next_up</span></a>·:=·#<span·class="id"·title="var">float64_next_up</span>.<br/>	250	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_up"·class="idref"·href="#next_up"><span·class="id"·title="axiom">next_up</span></a>·:=·#<span·class="id"·title="var">float64_next_up</span>.<br/>
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262	<div·class="code">	262	<div·class="code">
263	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_down"·class="idref"·href="#next_down"><span·class="id"·title="axiom">next_down</span></a>·:=·#<span·class="id"·title="var">float64_next_down</span>.<br/>	263	<span·class="id"·title="keyword">Primitive</span>·<a·id="next_down"·class="idref"·href="#next_down"><span·class="id"·title="axiom">next_down</span></a>·:=·#<span·class="id"·title="var">float64_next_down</span>.<br/>

264	<br/>	264	<br/>
265	</div>	265	</div>

266	<div·class="doc">	266	<div·class="doc">
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53	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Ltac</span></a>.<br/>	53	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Ltac.html#"><span·class="id"·title="library">Ltac</span></a>.<br/>
54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/>	54	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/>

55	<br/>	55	<br/>
56	</div>	56	</div>

57	<div·class="doc">	57	<div·class="doc">
58	<a·id="lab246"></a><h1·class="section">Datatypes·with·zero·and·one·element</h1>	58	<a·id="lab269"></a><h1·class="section">Datatypes·with·zero·and·one·element</h1>

59	<div·class="paragraph">·</div>	59	<div·class="paragraph">·</div>

60	·<span·class="inlinecode"><span·class="id"·title="var">Empty_set</span></span>·is·a·datatype·with·no·inhabitant·	60	·<span·class="inlinecode"><span·class="id"·title="var">Empty_set</span></span>·is·a·datatype·with·no·inhabitant·
61	</div>	61	</div>
62	<div·class="code">	62	<div·class="code">

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87	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">unit</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.type</span>.<br/>	87	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">unit</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.type</span>.<br/>
88	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">tt</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.tt</span>.<br/>	88	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">tt</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.unit.tt</span>.<br/>

89	<br/>	89	<br/>
90	</div>	90	</div>

91	<div·class="doc">	91	<div·class="doc">
92	<a·id="lab247"></a><h1·class="section">The·boolean·datatype</h1>	92	<a·id="lab270"></a><h1·class="section">The·boolean·datatype</h1>

93	<div·class="paragraph">·</div>	93	<div·class="paragraph">·</div>

94	·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·is·the·datatype·of·the·boolean·values·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>·	94	·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·is·the·datatype·of·the·boolean·values·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>·
95	</div>	95	</div>
96	<div·class="code">	96	<div·class="code">

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258	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecT</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecT</span>.<br/>	258	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecT</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecT</span>.<br/>
259	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecF</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecF</span>.<br/>	259	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">BoolSpecF</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">core.BoolSpec.BoolSpecF</span>.<br/>

260	<br/>	260	<br/>
261	</div>	261	</div>

262	<div·class="doc">	262	<div·class="doc">
263	<a·id="lab248"></a><h1·class="section">Peano·natural·numbers</h1>	263	<a·id="lab271"></a><h1·class="section">Peano·natural·numbers</h1>

264	<div·class="paragraph">·</div>	264	<div·class="paragraph">·</div>

265	·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·is·the·datatype·of·natural·numbers·built·from·<span·class="inlinecode"><span·class="id"·title="var">O</span></span>·and·successor·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>;	265	·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·is·the·datatype·of·natural·numbers·built·from·<span·class="inlinecode"><span·class="id"·title="var">O</span></span>·and·successor·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>;
266	····note·that·the·constructor·name·is·the·letter·O.	266	····note·that·the·constructor·name·is·the·letter·O.
267	····Numbers·in·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·can·be·denoted·using·a·decimal·notation;	267	····Numbers·in·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span>·can·be·denoted·using·a·decimal·notation;
268	····e.g.·<span·class="inlinecode">3%<span·class="id"·title="var">nat</span></span>·abbreviates·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode"><span·class="id"·title="var">O</span>))</span>·	268	····e.g.·<span·class="inlinecode">3%<span·class="id"·title="var">nat</span></span>·abbreviates·<span·class="inlinecode"><span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode">(<span·class="id"·title="var">S</span></span>·<span·class="inlinecode"><span·class="id"·title="var">O</span>))</span>·
Offset 293, 15 lines modified		Offset 293, 15 lines modified
293	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">O</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.O</span>.<br/>	293	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">O</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.O</span>.<br/>
294	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">S</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.S</span>.<br/>	294	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">S</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.S</span>.<br/>

295	<br/>	295	<br/>
296	</div>	296	</div>

297	<div·class="doc">	297	<div·class="doc">
298	<a·id="lab249"></a><h1·class="section">Container·datatypes</h1>	298	<a·id="lab272"></a><h1·class="section">Container·datatypes</h1>

299	</div>	299	</div>
300	<div·class="code">	300	<div·class="code">

301	<br/>	301	<br/>

302	<br/>	302	<br/>
Offset 498, 15 lines modified		Offset 498, 15 lines modified

498	<br/>	498	<br/>

499	<br/>	499	<br/>
500	</div>	500	</div>

501	<div·class="doc">	501	<div·class="doc">
502	<a·id="lab250"></a><h1·class="section">The·comparison·datatype</h1>	502	<a·id="lab273"></a><h1·class="section">The·comparison·datatype</h1>

503	</div>	503	</div>
504	<div·class="code">	504	<div·class="code">

505	<br/>	505	<br/>
506	<span·class="id"·title="keyword">Inductive</span>·<a·id="comparison"·class="idref"·href="#comparison"><span·class="id"·title="definition,·inductive"><span·id="comparison_rect"·class="id"><span·id="comparison_ind"·class="id"><span·id="comparison_rec"·class="id"><span·id="comparison_sind"·class="id">comparison</span></span></span></span></span></a>·:·<span·class="id"·title="keyword">Set</span>·:=<br/>	506	<span·class="id"·title="keyword">Inductive</span>·<a·id="comparison"·class="idref"·href="#comparison"><span·class="id"·title="definition,·inductive"><span·id="comparison_rect"·class="id"><span·id="comparison_ind"·class="id"><span·id="comparison_rec"·class="id"><span·id="comparison_sind"·class="id">comparison</span></span></span></span></span></a>·:·<span·class="id"·title="keyword">Set</span>·:=<br/>
507	\|·<a·id="Eq"·class="idref"·href="#Eq"><span·class="id"·title="constructor">Eq</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#comparison:103"><span·class="id"·title="inductive">comparison</span></a><br/>	507	\|·<a·id="Eq"·class="idref"·href="#Eq"><span·class="id"·title="constructor">Eq</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#comparison:103"><span·class="id"·title="inductive">comparison</span></a><br/>
Offset 616, 15 lines modified		Offset 616, 15 lines modified
616	<span·class="id"·title="keyword">Lemma</span>·<a·id="CompSpec2Type"·class="idref"·href="#CompSpec2Type"><span·class="id"·title="lemma">CompSpec2Type</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="A:138"·class="idref"·href="#A:138"><span·class="id"·title="binder">A</span></a>·(<a·id="eq:139"·class="idref"·href="#eq:139"><span·class="id"·title="binder">eq</span></a>·<a·id="lt:140"·class="idref"·href="#lt:140"><span·class="id"·title="binder">lt</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#A:138"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Init.Datatypes.html#A:138"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>)·<a·id="x:141"·class="idref"·href="#x:141"><span·class="id"·title="binder">x</span></a>·<a·id="y:142"·class="idref"·href="#y:142"><span·class="id"·title="binder">y</span></a>·<a·id="c:143"·class="idref"·href="#c:143"><span·class="id"·title="binder">c</span></a>,<br/>	616	<span·class="id"·title="keyword">Lemma</span>·<a·id="CompSpec2Type"·class="idref"·href="#CompSpec2Type"><span·class="id"·title="lemma">CompSpec2Type</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="A:138"·class="idref"·href="#A:138"><span·class="id"·title="binder">A</span></a>·(<a·id="eq:139"·class="idref"·href="#eq:139"><span·class="id"·title="binder">eq</span></a>·<a·id="lt:140"·class="idref"·href="#lt:140"><span·class="id"·title="binder">lt</span></a>:<a·class="idref"·href="Coq.Init.Datatypes.html#A:138"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Init.Datatypes.html#A:138"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>)·<a·id="x:141"·class="idref"·href="#x:141"><span·class="id"·title="binder">x</span></a>·<a·id="y:142"·class="idref"·href="#y:142"><span·class="id"·title="binder">y</span></a>·<a·id="c:143"·class="idref"·href="#c:143"><span·class="id"·title="binder">c</span></a>,<br/>
617	<a·class="idref"·href="Coq.Init.Datatypes.html#CompSpec"><span·class="id"·title="definition">CompSpec</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#CompSpecT"><span·class="id"·title="definition">CompSpecT</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</span></a>.<br/>	617	<a·class="idref"·href="Coq.Init.Datatypes.html#CompSpec"><span·class="id"·title="definition">CompSpec</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#CompSpecT"><span·class="id"·title="definition">CompSpecT</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#eq:139"><span·class="id"·title="variable">eq</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#lt:140"><span·class="id"·title="variable">lt</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#x:141"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#y:142"><span·class="id"·title="variable">y</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#c:143"><span·class="id"·title="variable">c</span></a>.<br/>
618	·	618	·
619	<br/>	619	<br/>
620	</div>	620	</div>

621	<div·class="doc">	621	<div·class="doc">
622	<a·id="lab251"></a><h1·class="section">Misc·Other·Datatypes</h1>	622	<a·id="lab274"></a><h1·class="section">Misc·Other·Datatypes</h1>

623	<div·class="paragraph">·</div>	623	<div·class="paragraph">·</div>

624	·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·is·the·family·of·datatypes·on·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·whose·sole·non-empty	624	·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·is·the·family·of·datatypes·on·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·whose·sole·non-empty
625	····member·is·the·singleton·datatype·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·whose	625	····member·is·the·singleton·datatype·<span·class="inlinecode"><span·class="id"·title="var">identity</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·whose
626	····sole·inhabitant·is·denoted·<span·class="inlinecode"><span·class="id"·title="var">identity_refl</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·	626	····sole·inhabitant·is·denoted·<span·class="inlinecode"><span·class="id"·title="var">identity_refl</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">a</span></span>·
627	</div>	627	</div>


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000009e0:·7222·3e42·3c2f·7370·616e·3e29·203a·203c··r">B</span>)·:·<		000009e0:·7222·3e42·3c2f·7370·616e·3e29·203a·203c··r">B</span>)·:·<
000009f0:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"·		000009f0:·7370·616e·2063·6c61·7373·3d22·6964·2220··span·class="id"·
00000a00:·7469·746c·653d·2276·6172·223e·7479·7065··title="var">type		00000a00:·7469·746c·653d·2276·6172·223e·7479·7065··title="var">type
00000a10:·5f73·636f·7065·3c2f·7370·616e·3e2e·3c62··_scope</span>.<b		00000a10:·5f73·636f·7065·3c2f·7370·616e·3e2e·3c62··_scope</span>.<b
00000a20:·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64·6976··r/>..<br/>.</div		00000a20:·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64·6976··r/>..<br/>.</div
00000a30:·3e0a·0a3c·6469·7620·636c·6173·733d·2264··>..<div·class="d		00000a30:·3e0a·0a3c·6469·7620·636c·6173·733d·2264··>..<div·class="d
00000a40:·6f63·223e·0a3c·6120·6964·3d22·6c61·6232··oc">.<a·id="lab2		00000a40:·6f63·223e·0a3c·6120·6964·3d22·6c61·6232··oc">.<a·id="lab2
00000a50:·3734·223e·3c2f·613e·3c68·3120·636c·6173··74"></a><h1·clas		00000a50:·3631·223e·3c2f·613e·3c68·3120·636c·6173··61"></a><h1·clas
00000a60:·733d·2273·6563·7469·6f6e·223e·5072·6f70··s="section">Prop		00000a60:·733d·2273·6563·7469·6f6e·223e·5072·6f70··s="section">Prop
00000a70:·6f73·6974·696f·6e61·6c20·636f·6e6e·6563··ositional·connec		00000a70:·6f73·6974·696f·6e61·6c20·636f·6e6e·6563··ositional·connec
00000a80:·7469·7665·733c·2f68·313e·0a0a·3c64·6976··tives</h1>..<div		00000a80:·7469·7665·733c·2f68·313e·0a0a·3c64·6976··tives</h1>..<div
00000a90:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap		00000a90:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap
00000aa0:·6822·3e20·3c2f·6469·763e·0a0a·203c·7370··h">·</div>..·<sp		00000aa0:·6822·3e20·3c2f·6469·763e·0a0a·203c·7370··h">·</div>..·<sp
00000ab0:·616e·2063·6c61·7373·3d22·696e·6c69·6e65··an·class="inline		00000ab0:·616e·2063·6c61·7373·3d22·696e·6c69·6e65··an·class="inline
00000ac0:·636f·6465·223e·3c73·7061·6e20·636c·6173··code"><span·clas		00000ac0:·636f·6465·223e·3c73·7061·6e20·636c·6173··code"><span·clas
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00011590:·3436·3735·6130·6163·6633·6561·3932·6161··4675a0acf3ea92aa		00011590:·3436·3735·6130·6163·6633·6561·3932·6161··4675a0acf3ea92aa
000115a0:·6261·6464·223e·3c73·7061·6e20·636c·6173··badd"><span·clas		000115a0:·6261·6464·223e·3c73·7061·6e20·636c·6173··badd"><span·clas
000115b0:·733d·2269·6422·2074·6974·6c65·3d22·6e6f··s="id"·title="no		000115b0:·733d·2269·6422·2074·6974·6c65·3d22·6e6f··s="id"·title="no
000115c0:·7461·7469·6f6e·223e·293c·2f73·7061·6e3e··tation">)</span>		000115c0:·7461·7469·6f6e·223e·293c·2f73·7061·6e3e··tation">)</span>
000115d0:·3c2f·613e·2e3c·6272·2f3e·0a0a·3c62·722f··</a>.<br/>..<br/		000115d0:·3c2f·613e·2e3c·6272·2f3e·0a0a·3c62·722f··</a>.<br/>..<br/
000115e0:·3e0a·3c2f·6469·763e·0a0a·3c64·6976·2063··>.</div>..<div·c		000115e0:·3e0a·3c2f·6469·763e·0a0a·3c64·6976·2063··>.</div>..<div·c
000115f0:·6c61·7373·3d22·646f·6322·3e0a·3c61·2069··lass="doc">.<a·i		000115f0:·6c61·7373·3d22·646f·6322·3e0a·3c61·2069··lass="doc">.<a·i
00011600:·643d·226c·6162·3237·3522·3e3c·2f61·3e3c··d="lab275"></a><		00011600:·643d·226c·6162·3236·3222·3e3c·2f61·3e3c··d="lab262"></a><
00011610:·6831·2063·6c61·7373·3d22·7365·6374·696f··h1·class="sectio		00011610:·6831·2063·6c61·7373·3d22·7365·6374·696f··h1·class="sectio
00011620:·6e22·3e46·6972·7374·2d6f·7264·6572·2071··n">First-order·q		00011620:·6e22·3e46·6972·7374·2d6f·7264·6572·2071··n">First-order·q
00011630:·7561·6e74·6966·6965·7273·3c2f·6831·3e0a··uantifiers</h1>.		00011630:·7561·6e74·6966·6965·7273·3c2f·6831·3e0a··uantifiers</h1>.
00011640:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par		00011640:·0a3c·6469·7620·636c·6173·733d·2270·6172··.<div·class="par
00011650:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>.		00011650:·6167·7261·7068·223e·203c·2f64·6976·3e0a··agraph">·</div>.
00011660:·0a20·3c73·7061·6e20·636c·6173·733d·2269··.·<span·class="i		00011660:·0a20·3c73·7061·6e20·636c·6173·733d·2269··.·<span·class="i
00011670:·6e6c·696e·6563·6f64·6522·3e3c·7370·616e··nlinecode"><span		00011670:·6e6c·696e·6563·6f64·6522·3e3c·7370·616e··nlinecode"><span
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00019270:·2063·6c61·7373·3d22·6964·2220·7469·746c···class="id"·titl		00019270:·2063·6c61·7373·3d22·6964·2220·7469·746c···class="id"·titl
00019280:·653d·2273·6563·7469·6f6e·223e·756e·6976··e="section">univ		00019280:·653d·2273·6563·7469·6f6e·223e·756e·6976··e="section">univ
00019290:·6572·7361·6c5f·7175·616e·7469·6669·6361··ersal_quantifica		00019290:·6572·7361·6c5f·7175·616e·7469·6669·6361··ersal_quantifica
000192a0:·7469·6f6e·3c2f·7370·616e·3e3c·2f61·3e2e··tion</span></a>.		000192a0:·7469·6f6e·3c2f·7370·616e·3e3c·2f61·3e2e··tion</span></a>.
000192b0:·3c62·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64··<br/>..<br/>.</d		000192b0:·3c62·722f·3e0a·0a3c·6272·2f3e·0a3c·2f64··<br/>..<br/>.</d
000192c0:·6976·3e0a·0a3c·6469·7620·636c·6173·733d··iv>..<div·class=		000192c0:·6976·3e0a·0a3c·6469·7620·636c·6173·733d··iv>..<div·class=
000192d0:·2264·6f63·223e·0a3c·6120·6964·3d22·6c61··"doc">.<a·id="la		000192d0:·2264·6f63·223e·0a3c·6120·6964·3d22·6c61··"doc">.<a·id="la
000192e0:·6232·3736·223e·3c2f·613e·3c68·3120·636c··b276"></a><h1·cl		000192e0:·6232·3633·223e·3c2f·613e·3c68·3120·636c··b263"></a><h1·cl
000192f0:·6173·733d·2273·6563·7469·6f6e·223e·4571··ass="section">Eq		000192f0:·6173·733d·2273·6563·7469·6f6e·223e·4571··ass="section">Eq
00019300:·7561·6c69·7479·3c2f·6831·3e0a·0a3c·6469··uality</h1>..<di		00019300:·7561·6c69·7479·3c2f·6831·3e0a·0a3c·6469··uality</h1>..<di
00019310:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra		00019310:·7620·636c·6173·733d·2270·6172·6167·7261··v·class="paragra
00019320:·7068·223e·203c·2f64·6976·3e0a·0a20·3c73··ph">·</div>..·<s		00019320:·7068·223e·203c·2f64·6976·3e0a·0a20·3c73··ph">·</div>..·<s
00019330:·7061·6e20·636c·6173·733d·2269·6e6c·696e··pan·class="inlin		00019330:·7061·6e20·636c·6173·733d·2269·6e6c·696e··pan·class="inlin
00019340:·6563·6f64·6522·3e3c·7370·616e·2063·6c61··ecode"><span·cla		00019340:·6563·6f64·6522·3e3c·7370·616e·2063·6c61··ecode"><span·cla
00019350:·7373·3d22·6964·2220·7469·746c·653d·2276··ss="id"·title="v		00019350:·7373·3d22·6964·2220·7469·746c·653d·2276··ss="id"·title="v
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00041780:·3a74·7970·655f·7363·6f70·653a·785f·272d··:type_scope:x_'-		00041780:·3a74·7970·655f·7363·6f70·653a·785f·272d··:type_scope:x_'-
00041790:·2667·743b·275f·7822·3e3c·7370·616e·2063··>'_x"><span·c		00041790:·2667·743b·275f·7822·3e3c·7370·616e·2063··>'_x"><span·c
000417a0:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title=		000417a0:·6c61·7373·3d22·6964·2220·7469·746c·653d··lass="id"·title=
000417b0:·226e·6f74·6174·696f·6e22·3e29·3c2f·7370··"notation">)</sp		000417b0:·226e·6f74·6174·696f·6e22·3e29·3c2f·7370··"notation">)</sp
000417c0:·616e·3e3c·2f61·3e2e·3c62·722f·3e0a·0a3c··an></a>.<br/>..<		000417c0:·616e·3e3c·2f61·3e2e·3c62·722f·3e0a·0a3c··an></a>.<br/>..<
000417d0:·6272·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469··br/>.</div>..<di		000417d0:·6272·2f3e·0a3c·2f64·6976·3e0a·0a3c·6469··br/>.</div>..<di
000417e0:·7620·636c·6173·733d·2264·6f63·223e·0a3c··v·class="doc">.<		000417e0:·7620·636c·6173·733d·2264·6f63·223e·0a3c··v·class="doc">.<
000417f0:·6120·6964·3d22·6c61·6232·3737·223e·3c2f··a·id="lab277"></		000417f0:·6120·6964·3d22·6c61·6232·3634·223e·3c2f··a·id="lab264"></
00041800:·613e·3c68·3120·636c·6173·733d·2273·6563··a><h1·class="sec		00041800:·613e·3c68·3120·636c·6173·733d·2273·6563··a><h1·class="sec
00041810:·7469·6f6e·223e·4265·696e·6720·696e·6861··tion">Being·inha		00041810:·7469·6f6e·223e·4265·696e·6720·696e·6861··tion">Being·inha
00041820:·6269·7465·643c·2f68·313e·0a0a·3c64·6976··bited</h1>..<div		00041820:·6269·7465·643c·2f68·313e·0a0a·3c64·6976··bited</h1>..<div
00041830:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap		00041830:·2063·6c61·7373·3d22·7061·7261·6772·6170···class="paragrap
00041840:·6822·3e20·3c2f·6469·763e·0a0a·2054·6865··h">·</div>..·The		00041840:·6822·3e20·3c2f·6469·763e·0a0a·2054·6865··h">·</div>..·The
00041850:·2070·7265·6469·6361·7465·203c·7370·616e···predicate·<span		00041850:·2070·7265·6469·6361·7465·203c·7370·616e···predicate·<span
00041860:·2063·6c61·7373·3d22·696e·6c69·6e65·636f···class="inlineco		00041860:·2063·6c61·7373·3d22·696e·6c69·6e65·636f···class="inlineco


Offset 50, 15 lines modified		Offset 50, 15 lines modified
50	<span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Init.Number.html#"><span·class="id"·title="library">Number</span></a>.<br/>	50	<span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Init.Decimal.html#"><span·class="id"·title="library">Decimal</span></a>·<a·class="idref"·href="Coq.Init.Hexadecimal.html#"><span·class="id"·title="library">Hexadecimal</span></a>·<a·class="idref"·href="Coq.Init.Number.html#"><span·class="id"·title="library">Number</span></a>.<br/>
51	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	51	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

52	<br/>	52	<br/>
53	</div>	53	</div>

54	<div·class="doc">	54	<div·class="doc">
55	<a·id="lab252"></a><h1·class="section">Peano·natural·numbers,·definitions·of·operations</h1>	55	<a·id="lab247"></a><h1·class="section">Peano·natural·numbers,·definitions·of·operations</h1>

56	<div·class="paragraph">·</div>	56	<div·class="paragraph">·</div>

57	·This·file·is·meant·to·be·used·as·a·whole·module,	57	·This·file·is·meant·to·be·used·as·a·whole·module,
58	····without·importing·it,·leading·to·qualified·definitions	58	····without·importing·it,·leading·to·qualified·definitions
59	····(e.g.·Nat.pred)·	59	····(e.g.·Nat.pred)·
60	</div>	60	</div>
Offset 67, 15 lines modified		Offset 67, 15 lines modified
67	<br/>	67	<br/>
68	<span·class="id"·title="keyword">Definition</span>·<a·id="t"·class="idref"·href="#t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/>	68	<span·class="id"·title="keyword">Definition</span>·<a·id="t"·class="idref"·href="#t"><span·class="id"·title="definition">t</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/>

69	<br/>	69	<br/>
70	</div>	70	</div>

71	<div·class="doc">	71	<div·class="doc">
72	<a·id="lab253"></a><h2·class="section">Constants</h2>	72	<a·id="lab248"></a><h2·class="section">Constants</h2>

73	</div>	73	</div>
74	<div·class="code">	74	<div·class="code">

75	<br/>	75	<br/>
76	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'0'"·class="idref"·href="#:::'0'"><span·class="id"·title="notation">"</span></a>0"·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>.<br/>	76	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'0'"·class="idref"·href="#:::'0'"><span·class="id"·title="notation">"</span></a>0"·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>.<br/>
77	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>).<br/>	77	<span·class="id"·title="keyword">Local·Notation</span>·<a·id=":::'1'"·class="idref"·href="#:::'1'"><span·class="id"·title="notation">"</span></a>1"·:=·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#O"><span·class="id"·title="constructor">O</span></a>).<br/>
Offset 86, 15 lines modified		Offset 86, 15 lines modified
86	<span·class="id"·title="keyword">Definition</span>·<a·id="one"·class="idref"·href="#one"><span·class="id"·title="definition">one</span></a>·:=·1.<br/>	86	<span·class="id"·title="keyword">Definition</span>·<a·id="one"·class="idref"·href="#one"><span·class="id"·title="definition">one</span></a>·:=·1.<br/>
87	<span·class="id"·title="keyword">Definition</span>·<a·id="two"·class="idref"·href="#two"><span·class="id"·title="definition">two</span></a>·:=·2.<br/>	87	<span·class="id"·title="keyword">Definition</span>·<a·id="two"·class="idref"·href="#two"><span·class="id"·title="definition">two</span></a>·:=·2.<br/>

88	<br/>	88	<br/>
89	</div>	89	</div>

90	<div·class="doc">	90	<div·class="doc">
91	<a·id="lab254"></a><h2·class="section">Basic·operations</h2>	91	<a·id="lab249"></a><h2·class="section">Basic·operations</h2>

92	</div>	92	</div>
93	<div·class="code">	93	<div·class="code">

94	<br/>	94	<br/>
95	<span·class="id"·title="keyword">Definition</span>·<a·id="succ"·class="idref"·href="#succ"><span·class="id"·title="definition">succ</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>.<br/>	95	<span·class="id"·title="keyword">Definition</span>·<a·id="succ"·class="idref"·href="#succ"><span·class="id"·title="definition">succ</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>.<br/>

Offset 155, 15 lines modified		Offset 155, 15 lines modified
155	<br/>	155	<br/>
156	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">sub</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.sub</span>.<br/>	156	<span·class="id"·title="keyword">Register</span>·<span·class="id"·title="var">sub</span>·<span·class="id"·title="keyword">as</span>·<span·class="id"·title="var">num.nat.sub</span>.<br/>

157	<br/>	157	<br/>
158	</div>	158	</div>

159	<div·class="doc">	159	<div·class="doc">
160	<a·id="lab255"></a><h2·class="section">Comparisons</h2>	160	<a·id="lab250"></a><h2·class="section">Comparisons</h2>

161	</div>	161	</div>
162	<div·class="code">	162	<div·class="code">

163	<br/>	163	<br/>
164	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="eqb"·class="idref"·href="#eqb"><span·class="id"·title="definition">eqb</span></a>·<a·id="n:17"·class="idref"·href="#n:17"><span·class="id"·title="binder">n</span></a>·<a·id="m:18"·class="idref"·href="#m:18"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>	164	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="eqb"·class="idref"·href="#eqb"><span·class="id"·title="definition">eqb</span></a>·<a·id="n:17"·class="idref"·href="#n:17"><span·class="id"·title="binder">n</span></a>·<a·id="m:18"·class="idref"·href="#m:18"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>
165	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:17"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:18"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	165	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:17"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:18"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 201, 15 lines modified		Offset 201, 15 lines modified
201	<br/>	201	<br/>
202	<span·class="id"·title="keyword">Infix</span>·<a·id="4a243c6f4b9487508e7110341eeedb0b"·class="idref"·href="#4a243c6f4b9487508e7110341eeedb0b"><span·class="id"·title="notation">"</span></a>?="·:=·<a·class="idref"·href="Coq.Init.Nat.html#compare"><span·class="id"·title="definition">compare</span></a>·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·70)·:·<span·class="id"·title="var">nat_scope</span>.<br/>	202	<span·class="id"·title="keyword">Infix</span>·<a·id="4a243c6f4b9487508e7110341eeedb0b"·class="idref"·href="#4a243c6f4b9487508e7110341eeedb0b"><span·class="id"·title="notation">"</span></a>?="·:=·<a·class="idref"·href="Coq.Init.Nat.html#compare"><span·class="id"·title="definition">compare</span></a>·(<span·class="id"·title="tactic">at</span>·<span·class="id"·title="keyword">level</span>·70)·:·<span·class="id"·title="var">nat_scope</span>.<br/>

203	<br/>	203	<br/>
204	</div>	204	</div>

205	<div·class="doc">	205	<div·class="doc">
206	<a·id="lab256"></a><h2·class="section">Minimum,·maximum</h2>	206	<a·id="lab251"></a><h2·class="section">Minimum,·maximum</h2>

207	</div>	207	</div>
208	<div·class="code">	208	<div·class="code">

209	<br/>	209	<br/>
210	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="max"·class="idref"·href="#max"><span·class="id"·title="definition">max</span></a>·<a·id="n:34"·class="idref"·href="#n:34"><span·class="id"·title="binder">n</span></a>·<a·id="m:35"·class="idref"·href="#m:35"><span·class="id"·title="binder">m</span></a>·:=<br/>	210	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="max"·class="idref"·href="#max"><span·class="id"·title="definition">max</span></a>·<a·id="n:34"·class="idref"·href="#n:34"><span·class="id"·title="binder">n</span></a>·<a·id="m:35"·class="idref"·href="#m:35"><span·class="id"·title="binder">m</span></a>·:=<br/>
211	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:34"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:35"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	211	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:34"><span·class="id"·title="variable">n</span></a>,·<a·class="idref"·href="Coq.Init.Nat.html#m:35"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 226, 15 lines modified		Offset 226, 15 lines modified
226	\|·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">n'</span>,·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">m'</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#min:41"><span·class="id"·title="definition">min</span></a>·<span·class="id"·title="var">n'</span>·<span·class="id"·title="var">m'</span>)<br/>	226	\|·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">n'</span>,·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<span·class="id"·title="var">m'</span>·=>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#min:41"><span·class="id"·title="definition">min</span></a>·<span·class="id"·title="var">n'</span>·<span·class="id"·title="var">m'</span>)<br/>
227	<span·class="id"·title="keyword">end</span>.<br/>	227	<span·class="id"·title="keyword">end</span>.<br/>

228	<br/>	228	<br/>
229	</div>	229	</div>

230	<div·class="doc">	230	<div·class="doc">
231	<a·id="lab257"></a><h2·class="section">Parity·tests</h2>	231	<a·id="lab252"></a><h2·class="section">Parity·tests</h2>

232	</div>	232	</div>
233	<div·class="code">	233	<div·class="code">

234	<br/>	234	<br/>
235	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="definition">even</span></a>·<a·id="n:44"·class="idref"·href="#n:44"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>	235	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="definition">even</span></a>·<a·id="n:44"·class="idref"·href="#n:44"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>·:=<br/>
236	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:44"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>	236	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:44"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 246, 15 lines modified		Offset 246, 15 lines modified
246	<br/>	246	<br/>
247	<span·class="id"·title="keyword">Definition</span>·<a·id="odd"·class="idref"·href="#odd"><span·class="id"·title="definition">odd</span></a>·<a·id="n:47"·class="idref"·href="#n:47"><span·class="id"·title="binder">n</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#negb"><span·class="id"·title="definition">negb</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#even"><span·class="id"·title="definition">even</span></a>·<a·class="idref"·href="Coq.Init.Nat.html#n:47"><span·class="id"·title="variable">n</span></a>).<br/>	247	<span·class="id"·title="keyword">Definition</span>·<a·id="odd"·class="idref"·href="#odd"><span·class="id"·title="definition">odd</span></a>·<a·id="n:47"·class="idref"·href="#n:47"><span·class="id"·title="binder">n</span></a>·:=·<a·class="idref"·href="Coq.Init.Datatypes.html#negb"><span·class="id"·title="definition">negb</span></a>·(<a·class="idref"·href="Coq.Init.Nat.html#even"><span·class="id"·title="definition">even</span></a>·<a·class="idref"·href="Coq.Init.Nat.html#n:47"><span·class="id"·title="variable">n</span></a>).<br/>

248	<br/>	248	<br/>
249	</div>	249	</div>

250	<div·class="doc">	250	<div·class="doc">
251	<a·id="lab258"></a><h2·class="section">Power</h2>	251	<a·id="lab253"></a><h2·class="section">Power</h2>

252	</div>	252	</div>
253	<div·class="code">	253	<div·class="code">

254	<br/>	254	<br/>
255	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="pow"·class="idref"·href="#pow"><span·class="id"·title="definition">pow</span></a>·<a·id="n:48"·class="idref"·href="#n:48"><span·class="id"·title="binder">n</span></a>·<a·id="m:49"·class="idref"·href="#m:49"><span·class="id"·title="binder">m</span></a>·:=<br/>	255	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="pow"·class="idref"·href="#pow"><span·class="id"·title="definition">pow</span></a>·<a·id="n:48"·class="idref"·href="#n:48"><span·class="id"·title="binder">n</span></a>·<a·id="m:49"·class="idref"·href="#m:49"><span·class="id"·title="binder">m</span></a>·:=<br/>
256	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#m:49"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>	256	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#m:49"><span·class="id"·title="variable">m</span></a>·<span·class="id"·title="keyword">with</span><br/>
Offset 264, 15 lines modified		Offset 264, 15 lines modified
264	<br/>	264	<br/>
265	<span·class="id"·title="keyword">where</span>·<a·id="81fd94e251a61ee523cdd7855774ae7c"·class="idref"·href="#81fd94e251a61ee523cdd7855774ae7c"><span·class="id"·title="notation">"</span></a>n·^·m"·:=·(<a·class="idref"·href="Coq.Init.Nat.html#pow:50"><span·class="id"·title="definition">pow</span></a>·<span·class="id"·title="var">n</span>·<span·class="id"·title="var">m</span>)·:·<span·class="id"·title="var">nat_scope</span>.<br/>	265	<span·class="id"·title="keyword">where</span>·<a·id="81fd94e251a61ee523cdd7855774ae7c"·class="idref"·href="#81fd94e251a61ee523cdd7855774ae7c"><span·class="id"·title="notation">"</span></a>n·^·m"·:=·(<a·class="idref"·href="Coq.Init.Nat.html#pow:50"><span·class="id"·title="definition">pow</span></a>·<span·class="id"·title="var">n</span>·<span·class="id"·title="var">m</span>)·:·<span·class="id"·title="var">nat_scope</span>.<br/>

266	<br/>	266	<br/>
267	</div>	267	</div>

268	<div·class="doc">	268	<div·class="doc">
269	<a·id="lab259"></a><h2·class="section">Tail-recursive·versions·of·<span·class="inlinecode"><span·class="id"·title="var">add</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">mul</span></span></h2>	269	<a·id="lab254"></a><h2·class="section">Tail-recursive·versions·of·<span·class="inlinecode"><span·class="id"·title="var">add</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">mul</span></span></h2>

270	</div>	270	</div>
271	<div·class="code">	271	<div·class="code">

272	<br/>	272	<br/>
273	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="tail_add"·class="idref"·href="#tail_add"><span·class="id"·title="definition">tail_add</span></a>·<a·id="n:52"·class="idref"·href="#n:52"><span·class="id"·title="binder">n</span></a>·<a·id="m:53"·class="idref"·href="#m:53"><span·class="id"·title="binder">m</span></a>·:=<br/>	273	<span·class="id"·title="keyword">Fixpoint</span>·<a·id="tail_add"·class="idref"·href="#tail_add"><span·class="id"·title="definition">tail_add</span></a>·<a·id="n:52"·class="idref"·href="#n:52"><span·class="id"·title="binder">n</span></a>·<a·id="m:53"·class="idref"·href="#m:53"><span·class="id"·title="binder">m</span></a>·:=<br/>
274	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:52"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>	274	<span·class="id"·title="keyword">match</span>·<a·class="idref"·href="Coq.Init.Nat.html#n:52"><span·class="id"·title="variable">n</span></a>·<span·class="id"·title="keyword">with</span><br/>
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Offset 51, 15 lines modified		Offset 51, 15 lines modified
51	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/>	51	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#"><span·class="id"·title="library">Logic</span></a>.<br/>
52	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Specif.html#"><span·class="id"·title="library">Specif</span></a>.<br/>	52	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Specif.html#"><span·class="id"·title="library">Specif</span></a>.<br/>

53	<br/>	53	<br/>
54	</div>	54	</div>

55	<div·class="doc">	55	<div·class="doc">
56	<a·id="lab268"></a><h1·class="section">Useful·tactics</h1>	56	<a·id="lab266"></a><h1·class="section">Useful·tactics</h1>

57	<div·class="paragraph">·</div>	57	<div·class="paragraph">·</div>

58	·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal.	58	·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal.
59	····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.·	59	····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.·
60	</div>	60	</div>
61	<div·class="code">	61	<div·class="code">
Offset 372, 15 lines modified		Offset 372, 15 lines modified
372	<span·class="id"·title="keyword">Tactic·Notation</span>·"dependent"·"destruction"·<span·class="id"·title="var">ident</span>(<span·class="id"·title="var">H</span>)·:=<br/>	372	<span·class="id"·title="keyword">Tactic·Notation</span>·"dependent"·"destruction"·<span·class="id"·title="var">ident</span>(<span·class="id"·title="var">H</span>)·:=<br/>
373	<span·class="id"·title="tactic">fail</span>·"To·use·dependent·destruction,·first·[Require·Import·Coq.Program.Equality.]".<br/>	373	<span·class="id"·title="tactic">fail</span>·"To·use·dependent·destruction,·first·[Require·Import·Coq.Program.Equality.]".<br/>

374	<br/>	374	<br/>
375	</div>	375	</div>

376	<div·class="doc">	376	<div·class="doc">
377	<a·id="lab269"></a><h3·class="section"><span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span></h3>	377	<a·id="lab267"></a><h3·class="section"><span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span></h3>
378	·The·built-in·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·will·frequently·leave·equalities·of	378	·The·built-in·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·will·frequently·leave·equalities·of
379	····dependent·pairs.··When·the·first·type·in·the·pair·is·an·hProp·or	379	····dependent·pairs.··When·the·first·type·in·the·pair·is·an·hProp·or
380	····otherwise·simplifies,·<span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span>·is·useful;·it·will·replace	380	····otherwise·simplifies,·<span·class="inlinecode"><span·class="id"·title="var">inversion_sigma</span></span>·is·useful;·it·will·replace
381	····the·equality·of·pairs·with·a·pair·of·equalities,·one·involving·a	381	····the·equality·of·pairs·with·a·pair·of·equalities,·one·involving·a
382	····term·casted·along·the·other.··This·might·also·prove·useful·for	382	····term·casted·along·the·other.··This·might·also·prove·useful·for
383	····writing·a·version·of·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·/·<span·class="inlinecode"><span·class="id"·title="tactic">dependent</span></span>·<span·class="inlinecode"><span·class="id"·title="tactic">destruction</span></span>·which	383	····writing·a·version·of·<span·class="inlinecode"><span·class="id"·title="tactic">inversion</span></span>·/·<span·class="inlinecode"><span·class="id"·title="tactic">dependent</span></span>·<span·class="inlinecode"><span·class="id"·title="tactic">destruction</span></span>·which
384	····does·not·lose·information,·i.e.,·does·not·turn·a·goal·which·is	384	····does·not·lose·information,·i.e.,·does·not·turn·a·goal·which·is


Offset 46, 15 lines modified		Offset 46, 15 lines modified
46	<div·class="code">	46	<div·class="code">

47	<br/>	47	<br/>
48	</div>	48	</div>

49	<div·class="doc">	49	<div·class="doc">
50	Some·facts·and·definitions·concerning·choice·and·description·in	50	Some·facts·and·definitions·concerning·choice·and·description·in
51	·······intuitionistic·logic.·<a·id="lab962"></a><h1·class="section">References:</h1>	51	·······intuitionistic·logic.·<a·id="lab568"></a><h1·class="section">References:</h1>

52	<div·class="paragraph">·</div>	52	<div·class="paragraph">·</div>

53	<span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory,	53	<span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory,
54	unpublished.	54	unpublished.

55	<div·class="paragraph">·</div>	55	<div·class="paragraph">·</div>
Offset 85, 15 lines modified		Offset 85, 15 lines modified
85	<br/>	85	<br/>
86	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>	86	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>

87	<br/>	87	<br/>
88	</div>	88	</div>

89	<div·class="doc">	89	<div·class="doc">
90	<a·id="lab963"></a><h1·class="section">Definitions</h1>	90	<a·id="lab569"></a><h1·class="section">Definitions</h1>

91	<div·class="paragraph">·</div>	91	<div·class="paragraph">·</div>

92	·Choice,·reification·and·description·schemes·	92	·Choice,·reification·and·description·schemes·
93	<div·class="paragraph">·</div>	93	<div·class="paragraph">·</div>

94	·We·make·them·all·polymorphic.·Most·of·them·have·existentials·as·conclusion	94	·We·make·them·all·polymorphic.·Most·of·them·have·existentials·as·conclusion
Offset 112, 15 lines modified		Offset 112, 15 lines modified
112	<br/>	112	<br/>
113	<span·class="id"·title="keyword">Variable</span>·<a·id="ChoiceSchemes.P"·class="idref"·href="#ChoiceSchemes.P"><span·class="id"·title="variable">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>.<br/>	113	<span·class="id"·title="keyword">Variable</span>·<a·id="ChoiceSchemes.P"·class="idref"·href="#ChoiceSchemes.P"><span·class="id"·title="variable">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>.<br/>

114	<br/>	114	<br/>
115	</div>	115	</div>

116	<div·class="doc">	116	<div·class="doc">
117	<a·id="lab~~964~~"></a><h2·class="section">Constructive·choice·and·description</h2>	117	<a·id="lab570"></a><h2·class="section">Constructive·choice·and·description</h2>

118	<div·class="paragraph">·</div>	118	<div·class="paragraph">·</div>

119	·AC_rel·=·relational·form·of·the·(non·extensional)·axiom·of·choice	119	·AC_rel·=·relational·form·of·the·(non·extensional)·axiom·of·choice
120	·············(a·"set-theoretic"·axiom·of·choice)·	120	·············(a·"set-theoretic"·axiom·of·choice)·
121	</div>	121	</div>
122	<div·class="code">	122	<div·class="code">
Offset 342, 15 lines modified		Offset 342, 15 lines modified
342	<span·class="id"·title="keyword">forall</span>·<a·id="P:100"·class="idref"·href="#P:100"><span·class="id"·title="binder">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>,<br/>	342	<span·class="id"·title="keyword">forall</span>·<a·id="P:100"·class="idref"·href="#P:100"><span·class="id"·title="binder">P</span></a>:<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>,<br/>
343	<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">exists</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">!</span></a>·<a·id="x:101"·class="idref"·href="#x:101"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">,</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:101"><span·class="id"·title="variable">x</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">{</span></a>·<a·id="x:102"·class="idref"·href="#x:102"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">:</span></a><a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">\|</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:102"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">}</span></a>.<br/>	343	<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">(</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">exists</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">!</span></a>·<a·id="x:101"·class="idref"·href="#x:101"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span·class="id"·title="notation">,</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:101"><span·class="id"·title="variable">x</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">)</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">{</span></a>·<a·id="x:102"·class="idref"·href="#x:102"><span·class="id"·title="binder">x</span></a><a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">:</span></a><a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ChoiceSchemes.A"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">\|</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#P:100"><span·class="id"·title="variable">P</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#x:102"><span·class="id"·title="variable">x</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span·class="id"·title="notation">}</span></a>.<br/>

344	<br/>	344	<br/>
345	</div>	345	</div>

346	<div·class="doc">	346	<div·class="doc">
347	<a·id="lab~~965~~"></a><h2·class="section">Weakly·classical·choice·and·description</h2>	347	<a·id="lab571"></a><h2·class="section">Weakly·classical·choice·and·description</h2>

348	<div·class="paragraph">·</div>	348	<div·class="paragraph">·</div>

349	·GAC_rel·=·guarded·relational·form·of·the·(non·extensional)·axiom·of·choice·	349	·GAC_rel·=·guarded·relational·form·of·the·(non·extensional)·axiom·of·choice·
350	</div>	350	</div>
351	<div·class="code">	351	<div·class="code">

Offset 621, 15 lines modified		Offset 621, 15 lines modified

621	<div·class="paragraph">·</div>	621	<div·class="paragraph">·</div>

622	with·no·prerequisite·on·the·non-emptiness·of·domains	622	with·no·prerequisite·on·the·non-emptiness·of·domains

623	<div·class="paragraph">·</div>	623	<div·class="paragraph">·</div>

624	<a·id="lab~~966~~"></a><h1·class="section">Table·of·contents</h1>	624	<a·id="lab572"></a><h1·class="section">Table·of·contents</h1>

625	</div>	625	</div>
626	<div·class="code">	626	<div·class="code">

627	<br/>	627	<br/>
628	</div>	628	</div>

Offset 678, 15 lines modified		Offset 678, 15 lines modified

678	<div·class="paragraph">·</div>	678	<div·class="paragraph">·</div>

679	9.2.·AC_fun_setoid·=·AC_fun·+·Ext_pred_repr·+·PI	679	9.2.·AC_fun_setoid·=·AC_fun·+·Ext_pred_repr·+·PI
680	·	680	·
681	<div·class="paragraph">·</div>	681	<div·class="paragraph">·</div>

682	<a·id="lab~~967~~"></a><h1·class="section">AC_rel·+·AC!·=·AC_fun</h1>	682	<a·id="lab573"></a><h1·class="section">AC_rel·+·AC!·=·AC_fun</h1>


683	<div·class="paragraph">·</div>	683	<div·class="paragraph">·</div>

684	···We·show·that·the·functional·formulation·of·the·axiom·of·Choice	684	···We·show·that·the·functional·formulation·of·the·axiom·of·Choice
685	···(usual·formulation·in·type·theory)·is·equivalent·to·its·relational	685	···(usual·formulation·in·type·theory)·is·equivalent·to·its·relational
686	···formulation·(only·formulation·of·set·theory)·+·functional·relation	686	···formulation·(only·formulation·of·set·theory)·+·functional·relation
Offset 718, 26 lines modified		Offset 718, 26 lines modified
718	<span·class="id"·title="keyword">forall</span>·<a·id="A:214"·class="idref"·href="#A:214"><span·class="id"·title="binder">A</span></a>·<a·id="B:215"·class="idref"·href="#B:215"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>,·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalChoice_on"><span·class="id"·title="definition">FunctionalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a><br/>	718	<span·class="id"·title="keyword">forall</span>·<a·id="A:214"·class="idref"·href="#A:214"><span·class="id"·title="binder">A</span></a>·<a·id="B:215"·class="idref"·href="#B:215"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>,·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalChoice_on"><span·class="id"·title="definition">FunctionalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a><br/>
719	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice_on"><span·class="id"·title="definition">RelationalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalRelReification_on"><span·class="id"·title="definition">FunctionalRelReification_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>.<br/>	719	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice_on"><span·class="id"·title="definition">RelationalChoice_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span·class="id"·title="notation">/\</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#FunctionalRelReification_on"><span·class="id"·title="definition">FunctionalRelReification_on</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#A:214"><span·class="id"·title="variable">A</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#B:215"><span·class="id"·title="variable">B</span></a>.<br/>

720	<br/>	720	<br/>
721	</div>	721	</div>

722	<div·class="doc">	722	<div·class="doc">
723	<a·id="lab~~968~~"></a><h1·class="section">Connection·between·the·guarded,·non·guarded·and·omniscient·choices</h1>	723	<a·id="lab574"></a><h1·class="section">Connection·between·the·guarded,·non·guarded·and·omniscient·choices</h1>

724	<div·class="paragraph">·</div>	724	<div·class="paragraph">·</div>

725	·We·show·that·the·guarded·formulations·of·the·axiom·of·choice	725	·We·show·that·the·guarded·formulations·of·the·axiom·of·choice
726	···are·equivalent·to·their·"omniscient"·variant·and·comes·from·the·non·guarded	726	···are·equivalent·to·their·"omniscient"·variant·and·comes·from·the·non·guarded
727	···formulation·in·presence·either·of·the·independence·of·general·premises	727	···formulation·in·presence·either·of·the·independence·of·general·premises
728	···or·subset·types·(themselves·derivable·from·subtypes·thanks·to·proof-	728	···or·subset·types·(themselves·derivable·from·subtypes·thanks·to·proof-
729	···irrelevance)·	729	···irrelevance)·
730	<div·class="paragraph">·</div>	730	<div·class="paragraph">·</div>

731	<a·id="lab~~969~~"></a><h2·class="section">AC_rel·+·PI·->·GAC_rel·and·AC_rel·+·IGP·->·GAC_rel·and·GAC_rel·=·OAC_rel</h2>	731	<a·id="lab575"></a><h2·class="section">AC_rel·+·PI·->·GAC_rel·and·AC_rel·+·IGP·->·GAC_rel·and·GAC_rel·=·OAC_rel</h2>

732	</div>	732	</div>
733	<div·class="code">	733	<div·class="code">

734	<br/>	734	<br/>
735	<span·class="id"·title="keyword">Lemma</span>·<a·id="rel_choice_and_proof_irrel_imp_guarded_rel_choice"·class="idref"·href="#rel_choice_and_proof_irrel_imp_guarded_rel_choice"><span·class="id"·title="lemma">rel_choice_and_proof_irrel_imp_guarded_rel_choice</span></a>·:<br/>	735	<span·class="id"·title="keyword">Lemma</span>·<a·id="rel_choice_and_proof_irrel_imp_guarded_rel_choice"·class="idref"·href="#rel_choice_and_proof_irrel_imp_guarded_rel_choice"><span·class="id"·title="lemma">rel_choice_and_proof_irrel_imp_guarded_rel_choice</span></a>·:<br/>
736	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice"><span·class="id"·title="abbreviation">RelationalChoice</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ProofIrrelevance"><span·class="id"·title="definition">ProofIrrelevance</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</span></a>.<br/>	736	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#RelationalChoice"><span·class="id"·title="abbreviation">RelationalChoice</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#ProofIrrelevance"><span·class="id"·title="definition">ProofIrrelevance</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</span></a>.<br/>
Offset 767, 15 lines modified		Offset 767, 15 lines modified
767	<span·class="id"·title="keyword">Corollary</span>·<a·id="guarded_iff_omniscient_rel_choice"·class="idref"·href="#guarded_iff_omniscient_rel_choice"><span·class="id"·title="lemma">guarded_iff_omniscient_rel_choice</span></a>·:<br/>	767	<span·class="id"·title="keyword">Corollary</span>·<a·id="guarded_iff_omniscient_rel_choice"·class="idref"·href="#guarded_iff_omniscient_rel_choice"><span·class="id"·title="lemma">guarded_iff_omniscient_rel_choice</span></a>·:<br/>
768	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#OmniscientRelationalChoice"><span·class="id"·title="abbreviation">OmniscientRelationalChoice</span></a>.<br/>	768	<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#GuardedRelationalChoice"><span·class="id"·title="abbreviation">GuardedRelationalChoice</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'<->'_x"><span·class="id"·title="notation"><-></span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#OmniscientRelationalChoice"><span·class="id"·title="abbreviation">OmniscientRelationalChoice</span></a>.<br/>

769	<br/>	769	<br/>
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Offset 129, 15 lines modified		Offset 129, 15 lines modified
129	<br/>	129	<br/>
130	<span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/>	130	<span·class="id"·title="keyword">Opaque</span>·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#epsilon"><span·class="id"·title="definition">epsilon</span></a>.<br/>

131	<br/>	131	<br/>
132	</div>	132	</div>

133	<div·class="doc">	133	<div·class="doc">
134	<a·id="lab961"></a><h3·class="section">Weaker·lemmas·(compatibility·lemmas)</h3>	134	<a·id="lab560"></a><h3·class="section">Weaker·lemmas·(compatibility·lemmas)</h3>

135	</div>	135	</div>
136	<div·class="code">	136	<div·class="code">

137	<br/>	137	<br/>
138	<span·class="id"·title="keyword">Theorem</span>·<a·id="choice"·class="idref"·href="#choice"><span·class="id"·title="lemma">choice</span></a>·:<br/>	138	<span·class="id"·title="keyword">Theorem</span>·<a·id="choice"·class="idref"·href="#choice"><span·class="id"·title="lemma">choice</span></a>·:<br/>
139	<span·class="id"·title="keyword">forall</span>·(<a·id="A:38"·class="idref"·href="#A:38"><span·class="id"·title="binder">A</span></a>·<a·id="B:39"·class="idref"·href="#B:39"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:40"·class="idref"·href="#R:40"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:38"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#B:39"><span·class="id"·title="variable">B</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>),<br/>	139	<span·class="id"·title="keyword">forall</span>·(<a·id="A:38"·class="idref"·href="#A:38"><span·class="id"·title="binder">A</span></a>·<a·id="B:39"·class="idref"·href="#B:39"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:40"·class="idref"·href="#R:40"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:38"><span·class="id"·title="variable">A</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#B:39"><span·class="id"·title="variable">B</span></a><a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><span·class="id"·title="keyword">Prop</span>),<br/>


Offset 110, 15 lines modified		Offset 110, 15 lines modified

110	<div·class="paragraph">·</div>	110	<div·class="paragraph">·</div>

111	4.2·Classical·logic·=·choice·of·representatives·in·a·partition·of·bool	111	4.2·Classical·logic·=·choice·of·representatives·in·a·partition·of·bool

112	<div·class="paragraph">·</div>	112	<div·class="paragraph">·</div>

113	<a·id="lab~~904~~"></a><h1·class="section">Prop·degeneracy·=·excluded-middle·+·prop·extensionality</h1>	113	<a·id="lab545"></a><h1·class="section">Prop·degeneracy·=·excluded-middle·+·prop·extensionality</h1>

114	<div·class="paragraph">·</div>	114	<div·class="paragraph">·</div>

115	·i.e.········<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">True</span></span>·<span·class="inlinecode">\/</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">False</span>)</span>	115	·i.e.········<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">True</span></span>·<span·class="inlinecode">\/</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">False</span>)</span>
116	·························<span·class="inlinecode"><-></span>	116	·························<span·class="inlinecode"><-></span>
117	·······<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>\/~<span·class="id"·title="var">A</span>)</span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span>,</span>·<span·class="inlinecode">(<span·class="id"·title="var">A</span><-><span·class="id"·title="var">B</span>)</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">B</span>)</span>	117	·······<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>,</span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>\/~<span·class="id"·title="var">A</span>)</span>·<span·class="inlinecode">/\</span>·<span·class="inlinecode">(<span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span></span>·<span·class="inlinecode"><span·class="id"·title="var">B</span>,</span>·<span·class="inlinecode">(<span·class="id"·title="var">A</span><-><span·class="id"·title="var">B</span>)</span>·<span·class="inlinecode">-></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>=<span·class="id"·title="var">B</span>)</span>

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186	<span·class="id"·title="keyword">Lemma</span>·<a·id="provable_prop_ext"·class="idref"·href="#provable_prop_ext"><span·class="id"·title="lemma">provable_prop_ext</span></a>·:<br/>	186	<span·class="id"·title="keyword">Lemma</span>·<a·id="provable_prop_ext"·class="idref"·href="#provable_prop_ext"><span·class="id"·title="lemma">provable_prop_ext</span></a>·:<br/>
187	<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#prop_extensionality"><span·class="id"·title="definition">prop_extensionality</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#provable_prop_extensionality"><span·class="id"·title="definition">provable_prop_extensionality</span></a>.<br/>	187	<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#prop_extensionality"><span·class="id"·title="definition">prop_extensionality</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#provable_prop_extensionality"><span·class="id"·title="definition">provable_prop_extensionality</span></a>.<br/>

188	<br/>	188	<br/>
189	</div>	189	</div>

190	<div·class="doc">	190	<div·class="doc">
191	<a·id="lab~~905~~"></a><h1·class="section">Classical·logic·and·proof-irrelevance</h1>	191	<a·id="lab546"></a><h1·class="section">Classical·logic·and·proof-irrelevance</h1>

192	<div·class="paragraph">·</div>	192	<div·class="paragraph">·</div>

193	<a·id="lab~~906~~"></a><h2·class="section">CC·\|-·prop·ext·+·A·inhabited·->·(A·=·A->A)·->·A·has·fixpoint</h2>	193	<a·id="lab547"></a><h2·class="section">CC·\|-·prop·ext·+·A·inhabited·->·(A·=·A->A)·->·A·has·fixpoint</h2>

194	<div·class="paragraph">·</div>	194	<div·class="paragraph">·</div>

195	·We·successively·show·that:	195	·We·successively·show·that:

196	<div·class="paragraph">·</div>	196	<div·class="paragraph">·</div>

Offset 243, 15 lines modified		Offset 243, 15 lines modified

243	<div·class="doc">	243	<div·class="doc">
244	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma·<span·class="inlinecode"><span·class="id"·title="var">ext_prop_fixpoint</span></span>	244	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma·<span·class="inlinecode"><span·class="id"·title="var">ext_prop_fixpoint</span></span>
245	····by·the·weakest·property·<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.	245	····by·the·weakest·property·<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.

246	<div·class="paragraph">·</div>	246	<div·class="paragraph">·</div>

247	<a·id="lab~~907~~"></a><h2·class="section">CC·\|-·prop_ext·/\·dep·elim·on·bool·->·proof-irrelevance</h2>	247	<a·id="lab548"></a><h2·class="section">CC·\|-·prop_ext·/\·dep·elim·on·bool·->·proof-irrelevance</h2>

248	<div·class="paragraph">·</div>	248	<div·class="paragraph">·</div>

249	·<span·class="inlinecode"><span·class="id"·title="var">proof_irrelevance</span></span>·asserts·equality·of·all·proofs·of·a·given·formula·	249	·<span·class="inlinecode"><span·class="id"·title="var">proof_irrelevance</span></span>·asserts·equality·of·all·proofs·of·a·given·formula·
250	</div>	250	</div>
251	<div·class="code">	251	<div·class="code">
252	<span·class="id"·title="keyword">Definition</span>·<a·id="proof_irrelevance"·class="idref"·href="#proof_irrelevance"><span·class="id"·title="definition">proof_irrelevance</span></a>·:=·<span·class="id"·title="keyword">forall</span>·(<a·id="A:42"·class="idref"·href="#A:42"><span·class="id"·title="binder">A</span></a>:<span·class="id"·title="keyword">Prop</span>)·(<a·id="a1:43"·class="idref"·href="#a1:43"><span·class="id"·title="binder">a1</span></a>·<a·id="a2:44"·class="idref"·href="#a2:44"><span·class="id"·title="binder">a2</span></a>:<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#A:42"><span·class="id"·title="variable">A</span></a>),·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a1:43"><span·class="id"·title="variable">a1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a2:44"><span·class="id"·title="variable">a2</span></a>.<br/>	252	<span·class="id"·title="keyword">Definition</span>·<a·id="proof_irrelevance"·class="idref"·href="#proof_irrelevance"><span·class="id"·title="definition">proof_irrelevance</span></a>·:=·<span·class="id"·title="keyword">forall</span>·(<a·id="A:42"·class="idref"·href="#A:42"><span·class="id"·title="binder">A</span></a>:<span·class="id"·title="keyword">Prop</span>)·(<a·id="a1:43"·class="idref"·href="#a1:43"><span·class="id"·title="binder">a1</span></a>·<a·id="a2:44"·class="idref"·href="#a2:44"><span·class="id"·title="binder">a2</span></a>:<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#A:42"><span·class="id"·title="variable">A</span></a>),·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a1:43"><span·class="id"·title="variable">a1</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#a2:44"><span·class="id"·title="variable">a2</span></a>.<br/>
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340	<div·class="doc">	340	<div·class="doc">
341	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma	341	Remark:·<span·class="inlinecode"><span·class="id"·title="var">prop_extensionality</span></span>·can·be·replaced·in·lemma
342	····<span·class="inlinecode"><span·class="id"·title="var">ext_prop_dep_proof_irrel_gen</span></span>·by·the·weakest·property	342	····<span·class="inlinecode"><span·class="id"·title="var">ext_prop_dep_proof_irrel_gen</span></span>·by·the·weakest·property
343	····<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.	343	····<span·class="inlinecode"><span·class="id"·title="var">provable_prop_extensionality</span></span>.

344	<div·class="paragraph">·</div>	344	<div·class="paragraph">·</div>

345	<a·id="lab~~908~~"></a><h2·class="section">CIC·\|-·prop.·ext.·->·proof-irrelevance</h2>	345	<a·id="lab549"></a><h2·class="section">CIC·\|-·prop.·ext.·->·proof-irrelevance</h2>

346	<div·class="paragraph">·</div>	346	<div·class="paragraph">·</div>

347	·In·the·Calculus·of·Inductive·Constructions,·inductively·defined·booleans	347	·In·the·Calculus·of·Inductive·Constructions,·inductively·defined·booleans
348	····enjoy·dependent·case·analysis,·hence·directly·proof-irrelevance·from	348	····enjoy·dependent·case·analysis,·hence·directly·proof-irrelevance·from
349	····propositional·extensionality.	349	····propositional·extensionality.

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394	··<span·class="inlinecode">[<span·class="id"·title="var">Berardi90</span>]</span>·Stefano·Berardi,·"Type·dependence·and·constructive	394	··<span·class="inlinecode">[<span·class="id"·title="var">Berardi90</span>]</span>·Stefano·Berardi,·"Type·dependence·and·constructive
395	··mathematics",·Ph.·D.·thesis,·Dipartimento·Matematica,·Università·di	395	··mathematics",·Ph.·D.·thesis,·Dipartimento·Matematica,·Università·di
396	··Torino,·1990.	396	··Torino,·1990.

397	<div·class="paragraph">·</div>	397	<div·class="paragraph">·</div>

398	<a·id="lab~~909~~"></a><h2·class="section">CC·\|-·excluded-middle·+·dep·elim·on·bool·->·proof-irrelevance</h2>	398	<a·id="lab550"></a><h2·class="section">CC·\|-·excluded-middle·+·dep·elim·on·bool·->·proof-irrelevance</h2>

399	<div·class="paragraph">·</div>	399	<div·class="paragraph">·</div>

400	·This·is·a·proof·in·the·pure·Calculus·of·Construction·that	400	·This·is·a·proof·in·the·pure·Calculus·of·Construction·that
401	····classical·logic·in·<span·class="inlinecode"><span·class="id"·title="keyword">Prop</span></span>·+·dependent·elimination·of·disjunction·entails	401	····classical·logic·in·<span·class="inlinecode"><span·class="id"·title="keyword">Prop</span></span>·+·dependent·elimination·of·disjunction·entails
402	····proof-irrelevance.	402	····proof-irrelevance.

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566	<br/>	566	<br/>
567	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#Proof_irrelevance_WEM_CC"><span·class="id"·title="section">Proof_irrelevance_WEM_CC</span></a>.<br/>	567	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#Proof_irrelevance_WEM_CC"><span·class="id"·title="section">Proof_irrelevance_WEM_CC</span></a>.<br/>

568	<br/>	568	<br/>
569	</div>	569	</div>

570	<div·class="doc">	570	<div·class="doc">
571	<a·id="lab~~910~~"></a><h2·class="section">CIC·\|-·excluded-middle·->·proof-irrelevance</h2>	571	<a·id="lab551"></a><h2·class="section">CIC·\|-·excluded-middle·->·proof-irrelevance</h2>

572	<div·class="paragraph">·</div>	572	<div·class="paragraph">·</div>


573	<div·class="paragraph">·</div>	573	<div·class="paragraph">·</div>

574	····Since,·dependent·elimination·is·derivable·in·the·Calculus·of	574	····Since,·dependent·elimination·is·derivable·in·the·Calculus·of
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631	Remark:·in·the·Set-impredicative·CCI,·Hurkens'·paradox·still·holds·with	631	Remark:·in·the·Set-impredicative·CCI,·Hurkens'·paradox·still·holds·with
632	····<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·in·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>·and·since·<span·class="inlinecode">~<span·class="id"·title="var">true</span>=<span·class="id"·title="var">false</span></span>·for·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>	632	····<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·in·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>·and·since·<span·class="inlinecode">~<span·class="id"·title="var">true</span>=<span·class="id"·title="var">false</span></span>·for·<span·class="inlinecode"><span·class="id"·title="var">true</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">false</span></span>
633	····in·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·from·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>,·we·get·the·inconsistency·of	633	····in·<span·class="inlinecode"><span·class="id"·title="var">bool</span></span>·from·<span·class="inlinecode"><span·class="id"·title="keyword">Set</span></span>,·we·get·the·inconsistency·of
634	····<span·class="inlinecode"><span·class="id"·title="var">em</span></span>·<span·class="inlinecode">:</span>·<span·class="inlinecode"><span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>:<span·class="id"·title="keyword">Prop</span>,</span>·<span·class="inlinecode">{<span·class="id"·title="var">A</span>}+{~<span·class="id"·title="var">A</span>}</span>·in·the·Set-impredicative·CCI.	634	····<span·class="inlinecode"><span·class="id"·title="var">em</span></span>·<span·class="inlinecode">:</span>·<span·class="inlinecode"><span·class="id"·title="keyword">forall</span></span>·<span·class="inlinecode"><span·class="id"·title="var">A</span>:<span·class="id"·title="keyword">Prop</span>,</span>·<span·class="inlinecode">{<span·class="id"·title="var">A</span>}+{~<span·class="id"·title="var">A</span>}</span>·in·the·Set-impredicative·CCI.

635	<div·class="paragraph">·</div>	635	<div·class="paragraph">·</div>

636	<a·id="lab~~911~~"></a><h1·class="section">Weak·classical·axioms</h1>	636	<a·id="lab552"></a><h1·class="section">Weak·classical·axioms</h1>

637	<div·class="paragraph">·</div>	637	<div·class="paragraph">·</div>

638	·We·show·the·following·increasing·in·the·strength·of·axioms:	638	·We·show·the·following·increasing·in·the·strength·of·axioms:
639	<ul·class="doclist">	639	<ul·class="doclist">
640	<li>·weak·excluded-middle·and·classical·De·Morgan's·law	640	<li>·weak·excluded-middle·and·classical·De·Morgan's·law

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653	<li>·excluded-middle	653	<li>·excluded-middle

654	</li>	654	</li>
655	</ul>	655	</ul>

656	<div·class="paragraph">·</div>	656	<div·class="paragraph">·</div>

657	<a·id="lab~~912~~"></a><h2·class="section">Weak·excluded-middle</h2>	657	<a·id="lab553"></a><h2·class="section">Weak·excluded-middle</h2>

658	<div·class="paragraph">·</div>	658	<div·class="paragraph">·</div>

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104	<div·class="code">	104	<div·class="code">
105	<span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#"><span·class="id"·title="library">ClassicalFacts</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#"><span·class="id"·title="library">ChoiceFacts</span></a>.<br/>	105	<span·class="id"·title="keyword">Require</span>·<a·class="idref"·href="Coq.Logic.ClassicalFacts.html#"><span·class="id"·title="library">ClassicalFacts</span></a>·<a·class="idref"·href="Coq.Logic.ChoiceFacts.html#"><span·class="id"·title="library">ChoiceFacts</span></a>.<br/>

106	<br/>	106	<br/>
107	</div>	107	</div>

108	<div·class="doc">	108	<div·class="doc">
109	<a·id="lab~~918~~"></a><h1·class="section">Pred.·Ext.·+·Rel.·Axiom·of·Choice·->·Excluded-Middle</h1>	109	<a·id="lab565"></a><h1·class="section">Pred.·Ext.·+·Rel.·Axiom·of·Choice·->·Excluded-Middle</h1>

110	</div>	110	</div>
111	<div·class="code">	111	<div·class="code">

112	<br/>	112	<br/>
113	<span·class="id"·title="keyword">Section</span>·<a·id="PredExt_RelChoice_imp_EM"·class="idref"·href="#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/>	113	<span·class="id"·title="keyword">Section</span>·<a·id="PredExt_RelChoice_imp_EM"·class="idref"·href="#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/>

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200	<br/>	200	<br/>
201	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/>	201	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#PredExt_RelChoice_imp_EM"><span·class="id"·title="section">PredExt_RelChoice_imp_EM</span></a>.<br/>

202	<br/>	202	<br/>
203	</div>	203	</div>

204	<div·class="doc">	204	<div·class="doc">
205	<a·id="lab~~919~~"></a><h1·class="section">Proof-Irrel.·+·Rel.·Axiom·of·Choice·->·Excl.-Middle·for·Equality</h1>	205	<a·id="lab566"></a><h1·class="section">Proof-Irrel.·+·Rel.·Axiom·of·Choice·->·Excl.-Middle·for·Equality</h1>

206	<div·class="paragraph">·</div>	206	<div·class="paragraph">·</div>

207	·This·is·an·adaptation·of·Diaconescu's·theorem,·exploiting·the	207	·This·is·an·adaptation·of·Diaconescu's·theorem,·exploiting·the
208	····form·of·extensionality·provided·by·proof-irrelevance·	208	····form·of·extensionality·provided·by·proof-irrelevance·
209	</div>	209	</div>
210	<div·class="code">	210	<div·class="code">
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307	<br/>	307	<br/>
308	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#ProofIrrel_RelChoice_imp_EqEM"><span·class="id"·title="section">ProofIrrel_RelChoice_imp_EqEM</span></a>.<br/>	308	<span·class="id"·title="keyword">End</span>·<a·class="idref"·href="Coq.Logic.Diaconescu.html#ProofIrrel_RelChoice_imp_EqEM"><span·class="id"·title="section">ProofIrrel_RelChoice_imp_EqEM</span></a>.<br/>

309	<br/>	309	<br/>
310	</div>	310	</div>

311	<div·class="doc">	311	<div·class="doc">
312	<a·id="lab~~920~~"></a><h1·class="section">Extensional·Hilbert's·epsilon·description·operator·->·Excluded-Middle</h1>	312	<a·id="lab567"></a><h1·class="section">Extensional·Hilbert's·epsilon·description·operator·->·Excluded-Middle</h1>

313	<div·class="paragraph">·</div>	313	<div·class="paragraph">·</div>

314	·Proof·sketch·from·Bell·<span·class="inlinecode">[<span·class="id"·title="var">Bell93</span>]</span>·(with·thanks·to·P.·Castéran)·	314	·Proof·sketch·from·Bell·<span·class="inlinecode">[<span·class="id"·title="var">Bell93</span>]</span>·(with·thanks·to·P.·Castéran)·
315	</div>	315	</div>
316	<div·class="code">	316	<div·class="code">


Offset 112, 15 lines modified		Offset 112, 15 lines modified
112	···equalities·assuming·axiom·eq_rect_eq	112	···equalities·assuming·axiom·eq_rect_eq

113	<div·class="paragraph">·</div>	113	<div·class="paragraph">·</div>


114	<div·class="paragraph">·</div>	114	<div·class="paragraph">·</div>

115	<a·id="lab~~889~~"></a><h1·class="section">Definition·of·dependent·equality·and·equivalence·with·equality·of·dependent·pairs</h1>	115	<a·id="lab542"></a><h1·class="section">Definition·of·dependent·equality·and·equivalence·with·equality·of·dependent·pairs</h1>

116	</div>	116	</div>
117	<div·class="code">	117	<div·class="code">

118	<br/>	118	<br/>
119	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#EqNotations"><span·class="id"·title="module">EqNotations</span></a>.<br/>	119	<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Init.Logic.html#EqNotations"><span·class="id"·title="module">EqNotations</span></a>.<br/>

Offset 280, 15 lines modified		Offset 280, 15 lines modified
280	#[<span·class="id"·title="var">global</span>]<br/>	280	#[<span·class="id"·title="var">global</span>]<br/>
281	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">eq_dep_sym</span>:·<span·class="id"·title="var">core</span>.<br/>	281	<span·class="id"·title="keyword">Hint</span>·<span·class="id"·title="keyword">Immediate</span>·<span·class="id"·title="var">eq_dep_sym</span>:·<span·class="id"·title="var">core</span>.<br/>

282	<br/>	282	<br/>
283	</div>	283	</div>

284	<div·class="doc">	284	<div·class="doc">
285	<a·id="lab~~890~~"></a><h1·class="section">Eq_rect_eq·<->·Eq_dep_eq·<->·UIP·<->·UIP_refl·<->·K</h1>	285	<a·id="lab543"></a><h1·class="section">Eq_rect_eq·<->·Eq_dep_eq·<->·UIP·<->·UIP_refl·<->·K</h1>

286	</div>	286	</div>
287	<div·class="code">	287	<div·class="code">

288	<br/>	288	<br/>
289	<span·class="id"·title="keyword">Section</span>·<a·id="Equivalences"·class="idref"·href="#Equivalences"><span·class="id"·title="section">Equivalences</span></a>.<br/>	289	<span·class="id"·title="keyword">Section</span>·<a·id="Equivalences"·class="idref"·href="#Equivalences"><span·class="id"·title="section">Equivalences</span></a>.<br/>

Offset 506, 15 lines modified		Offset 506, 15 lines modified
506	<span·class="id"·title="keyword">Notation</span>·<a·id="Inj_dep_pairT"·class="idref"·href="#Inj_dep_pairT"><span·class="id"·title="abbreviation">Inj_dep_pairT</span></a>·:=·<a·class="idref"·href="Coq.Logic.EqdepFacts.html#Inj_dep_pair"><span·class="id"·title="definition">Inj_dep_pair</span></a>.<br/>	506	<span·class="id"·title="keyword">Notation</span>·<a·id="Inj_dep_pairT"·class="idref"·href="#Inj_dep_pairT"><span·class="id"·title="abbreviation">Inj_dep_pairT</span></a>·:=·<a·class="idref"·href="Coq.Logic.EqdepFacts.html#Inj_dep_pair"><span·class="id"·title="definition">Inj_dep_pair</span></a>.<br/>
507	<span·class="id"·title="keyword">Notation</span>·<a·id="eq_dep_eq__inj_pairT2"·class="idref"·href="#eq_dep_eq__inj_pairT2"><span·class="id"·title="abbreviation">eq_dep_eq__inj_pairT2</span></a>·:=·<a·class="idref"·href="Coq.Logic.EqdepFacts.html#eq_dep_eq__inj_pair2"><span·class="id"·title="lemma">eq_dep_eq__inj_pair2</span></a>.<br/>	507	<span·class="id"·title="keyword">Notation</span>·<a·id="eq_dep_eq__inj_pairT2"·class="idref"·href="#eq_dep_eq__inj_pairT2"><span·class="id"·title="abbreviation">eq_dep_eq__inj_pairT2</span></a>·:=·<a·class="idref"·href="Coq.Logic.EqdepFacts.html#eq_dep_eq__inj_pair2"><span·class="id"·title="lemma">eq_dep_eq__inj_pair2</span></a>.<br/>

508	<br/>	508	<br/>
509	</div>	509	</div>

510	<div·class="doc">	510	<div·class="doc">
511	<a·id="lab~~891~~"></a><h1·class="section">Definition·of·the·functor·that·builds·properties·of·dependent·equalities·assuming·axiom·eq_rect_eq</h1>	511	<a·id="lab544"></a><h1·class="section">Definition·of·the·functor·that·builds·properties·of·dependent·equalities·assuming·axiom·eq_rect_eq</h1>

512	</div>	512	</div>
513	<div·class="code">	513	<div·class="code">

514	<br/>	514	<br/>
515	<span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="EqdepElimination"·class="idref"·href="#EqdepElimination"><span·class="id"·title="module">EqdepElimination</span></a>.<br/>	515	<span·class="id"·title="keyword">Module</span>·<span·class="id"·title="keyword">Type</span>·<a·id="EqdepElimination"·class="idref"·href="#EqdepElimination"><span·class="id"·title="module">EqdepElimination</span></a>.<br/>