/usr/bin/diffoscope --timeout 7200 --html /srv/reproducible-results/rbuild-debian/r-b-build.kqCFkQWL/coq_8.12.0-3.diffoscope.html --profile=- /srv/reproducible-results/rbuild-debian/r-b-build.kqCFkQWL/b1/coq_8.12.0-3_i386.changes /srv/reproducible-results/rbuild-debian/r-b-build.kqCFkQWL/b2/coq_8.12.0-3

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file list ¶

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Offset 1, 10 lines modified		Offset 1, 10 lines modified

1	·4cd951d34360f2fd17ecb10cad4819a0·8198668·debug·optional·coq-dbgsym_8.12.0-3_i386.deb	1	·4cd951d34360f2fd17ecb10cad4819a0·8198668·debug·optional·coq-dbgsym_8.12.0-3_i386.deb
2	·e7c346247b1485c2a1da3b230cb9c631·948652·debug·optional·coq-theories-dbgsym_8.12.0-3_i386.deb	2	·e7c346247b1485c2a1da3b230cb9c631·948652·debug·optional·coq-theories-dbgsym_8.12.0-3_i386.deb
3	·501c~~7cf9c9~~9~~e5d998833f~~5~~8c79d~~f8027·35241~~228~~·math·optional·coq-theories_8.12.0-3_i386.deb	3	·3350c3bc231908138656d252726ef303·35241784·math·optional·coq-theories_8.12.0-3_i386.deb
4	·af4c54c0ae0d5e545d913ebcbf734199·87883220·math·optional·coq_8.12.0-3_i386.deb	4	·af4c54c0ae0d5e545d913ebcbf734199·87883220·math·optional·coq_8.12.0-3_i386.deb
5	·a765ed403b7b61da111e8b6a353ab5cd·539572·debug·optional·coqide-dbgsym_8.12.0-3_i386.deb	5	·a765ed403b7b61da111e8b6a353ab5cd·539572·debug·optional·coqide-dbgsym_8.12.0-3_i386.deb
6	·81d25ebcf1e30668304dc44d92c8fcf8·1721436·math·optional·coqide_8.12.0-3_i386.deb	6	·81d25ebcf1e30668304dc44d92c8fcf8·1721436·math·optional·coqide_8.12.0-3_i386.deb
7	·aaabfac46c067580e833f3f160d159f3·653304·debug·optional·libcoq-ocaml-dbgsym_8.12.0-3_i386.deb	7	·aaabfac46c067580e833f3f160d159f3·653304·debug·optional·libcoq-ocaml-dbgsym_8.12.0-3_i386.deb
8	·6170e973092fa319f94bb097229c6479·18669088·ocaml·optional·libcoq-ocaml-dev_8.12.0-3_i386.deb	8	·6170e973092fa319f94bb097229c6479·18669088·ocaml·optional·libcoq-ocaml-dev_8.12.0-3_i386.deb
9	·a9c3ebb0930f48a643638b12b3b26fb5·7569016·ocaml·optional·libcoq-ocaml_8.12.0-3_i386.deb	9	·a9c3ebb0930f48a643638b12b3b26fb5·7569016·ocaml·optional·libcoq-ocaml_8.12.0-3_i386.deb


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1	Package:·coq-theories	1	Package:·coq-theories
2	Source:·coq	2	Source:·coq
3	Version:·8.12.0-3	3	Version:·8.12.0-3
4	Architecture:·i386	4	Architecture:·i386
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:·259190	6	Installed-Size:·259189
7	Depends:·coq-8.12.0+4.11.1	7	Depends:·coq-8.12.0+4.11.1
8	Recommends:·coq·(>=·8.0)	8	Recommends:·coq·(>=·8.0)
9	Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1)	9	Breaks:·coq-doc·(<=·8.0pl1.0-2),·coq-libs·(<<·8.2.pl1)
10	Replaces:·coq-libs·(<<·8.2.pl1)	10	Replaces:·coq-libs·(<<·8.2.pl1)
11	Section:·math	11	Section:·math
12	Priority:·optional	12	Priority:·optional
13	Homepage:·http://coq.inria.fr/	13	Homepage:·http://coq.inria.fr/


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22	<br/>	22	<br/>
23	</div>	23	</div>

24	<div·class="doc">	24	<div·class="doc">
25	Equality·on·natural·numbers·	25	Equality·on·natural·numbers·
26	<div·class="paragraph">·</div>	26	<div·class="paragraph">·</div>

27	<a·id="lab~~277~~"></a><h1·class="section">Propositional·equality</h1>	27	<a·id="lab641"></a><h1·class="section">Propositional·equality</h1>

28	</div>	28	</div>
29	<div·class="code">	29	<div·class="code">

30	<br/>	30	<br/>
31	<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/>	31	<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/>
32	<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/>	32	<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/>


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4702	-rw-r--r--···0·root·········(0)·root·········(0)····44957·2020-09-09·19:16:07.000000·./usr/share/doc/coq-theories/html/Coq.Arith.Plus.html	4702	-rw-r--r--···0·root·········(0)·root·········(0)····44957·2020-09-09·19:16:07.000000·./usr/share/doc/coq-theories/html/Coq.Arith.Plus.html
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39	<br/>	39	<br/>
40	<span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·<span·class="id"·title="var">n</span>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/>	40	<span·class="id"·title="keyword">Implicit</span>·<span·class="id"·title="keyword">Types</span>·<span·class="id"·title="var">m</span>·<span·class="id"·title="var">n</span>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>.<br/>

41	<br/>	41	<br/>
42	</div>	42	</div>

43	<div·class="doc">	43	<div·class="doc">
44	<a·id="lab~~238~~"></a><h1·class="section">Inductive·definition·of·<span·class="inlinecode"><span·class="id"·title="var">even</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">odd</span></span></h1>	44	<a·id="lab667"></a><h1·class="section">Inductive·definition·of·<span·class="inlinecode"><span·class="id"·title="var">even</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">odd</span></span></h1>

45	</div>	45	</div>
46	<div·class="code">	46	<div·class="code">

47	<br/>	47	<br/>
48	<span·class="id"·title="keyword">Inductive</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="inductive">even</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</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/>	48	<span·class="id"·title="keyword">Inductive</span>·<a·id="even"·class="idref"·href="#even"><span·class="id"·title="inductive">even</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</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/>
49	\|·<a·id="even_O"·class="idref"·href="#even_O"><span·class="id"·title="constructor">even_O</span></a>·:·<a·class="idref"·href="Coq.Arith.Even.html#even:1"><span·class="id"·title="inductive">even</span></a>·0<br/>	49	\|·<a·id="even_O"·class="idref"·href="#even_O"><span·class="id"·title="constructor">even_O</span></a>·:·<a·class="idref"·href="Coq.Arith.Even.html#even:1"><span·class="id"·title="inductive">even</span></a>·0<br/>


Offset 40, 15 lines modified		Offset 40, 15 lines modified
40	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>·<a·class="idref"·href="Coq.Arith.Le.html#"><span·class="id"·title="library">Le</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#"><span·class="id"·title="library">Lt</span></a>·<a·class="idref"·href="Coq.Arith.Plus.html#"><span·class="id"·title="library">Plus</span></a>.<br/>	40	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#"><span·class="id"·title="library">PeanoNat</span></a>·<a·class="idref"·href="Coq.Arith.Le.html#"><span·class="id"·title="library">Le</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#"><span·class="id"·title="library">Lt</span></a>·<a·class="idref"·href="Coq.Arith.Plus.html#"><span·class="id"·title="library">Plus</span></a>.<br/>
41	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	41	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

42	<br/>	42	<br/>
43	</div>	43	</div>

44	<div·class="doc">	44	<div·class="doc">
45	<a·id="lab242"></a><h1·class="section">Order·and·successor</h1>	45	<a·id="lab643"></a><h1·class="section">Order·and·successor</h1>

46	</div>	46	</div>
47	<div·class="code">	47	<div·class="code">

48	<br/>	48	<br/>
49	<span·class="id"·title="keyword">Theorem</span>·<a·id="gt_Sn_O"·class="idref"·href="#gt_Sn_O"><span·class="id"·title="lemma">gt_Sn_O</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.Gt.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·0.<br/>	49	<span·class="id"·title="keyword">Theorem</span>·<a·id="gt_Sn_O"·class="idref"·href="#gt_Sn_O"><span·class="id"·title="lemma">gt_Sn_O</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.Gt.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'>'_x"><span·class="id"·title="notation">></span></a>·0.<br/>


Offset 45, 15 lines modified		Offset 45, 15 lines modified
45	<br/>	45	<br/>
46	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	46	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

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

49	<div·class="doc">	49	<div·class="doc">
50	<a·id="lab~~264~~"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·an·order·on·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span></h1>	50	<a·id="lab671"></a><h1·class="section"><span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·an·order·on·<span·class="inlinecode"><span·class="id"·title="var">nat</span></span></h1>

51	</div>	51	</div>
52	<div·class="code">	52	<div·class="code">

53	<br/>	53	<br/>
54	<span·class="id"·title="keyword">Notation</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="abbreviation">le_refl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.le_refl"><span·class="id"·title="lemma">Nat.le_refl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>	54	<span·class="id"·title="keyword">Notation</span>·<a·id="le_refl"·class="idref"·href="#le_refl"><span·class="id"·title="abbreviation">le_refl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.le_refl"><span·class="id"·title="lemma">Nat.le_refl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>
55	<span·class="id"·title="keyword">Notation</span>·<a·id="le_trans"·class="idref"·href="#le_trans"><span·class="id"·title="abbreviation">le_trans</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.le_trans"><span·class="id"·title="lemma">Nat.le_trans</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>	55	<span·class="id"·title="keyword">Notation</span>·<a·id="le_trans"·class="idref"·href="#le_trans"><span·class="id"·title="abbreviation">le_trans</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.le_trans"><span·class="id"·title="lemma">Nat.le_trans</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>


Offset 42, 29 lines modified		Offset 42, 29 lines modified
42	<br/>	42	<br/>
43	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	43	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

44	<br/>	44	<br/>
45	</div>	45	</div>

46	<div·class="doc">	46	<div·class="doc">
47	<a·id="lab~~298~~"></a><h1·class="section">Irreflexivity</h1>	47	<a·id="lab651"></a><h1·class="section">Irreflexivity</h1>

48	</div>	48	</div>
49	<div·class="code">	49	<div·class="code">

50	<br/>	50	<br/>
51	<span·class="id"·title="keyword">Notation</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="abbreviation">lt_irrefl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.lt_irrefl"><span·class="id"·title="lemma">Nat.lt_irrefl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).·	51	<span·class="id"·title="keyword">Notation</span>·<a·id="lt_irrefl"·class="idref"·href="#lt_irrefl"><span·class="id"·title="abbreviation">lt_irrefl</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.lt_irrefl"><span·class="id"·title="lemma">Nat.lt_irrefl</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).·
52	<br/>	52	<br/>
53	<span·class="id"·title="keyword">Hint·Resolve</span>·<span·class="id"·title="var">lt_irrefl</span>:·<span·class="id"·title="var">arith</span>.<br/>	53	<span·class="id"·title="keyword">Hint·Resolve</span>·<span·class="id"·title="var">lt_irrefl</span>:·<span·class="id"·title="var">arith</span>.<br/>

54	<br/>	54	<br/>
55	</div>	55	</div>

56	<div·class="doc">	56	<div·class="doc">
57	<a·id="lab~~299~~"></a><h1·class="section">Relationship·between·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1>	57	<a·id="lab652"></a><h1·class="section">Relationship·between·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·and·<span·class="inlinecode"><span·class="id"·title="var">lt</span></span></h1>

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

60	<br/>	60	<br/>
61	<span·class="id"·title="keyword">Theorem</span>·<a·id="lt_le_S"·class="idref"·href="#lt_le_S"><span·class="id"·title="lemma">lt_le_S</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>·:·<a·class="idref"·href="Coq.Arith.Lt.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#m:2"><span·class="id"·title="variable">m</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#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#m:2"><span·class="id"·title="variable">m</span></a>.<br/>	61	<span·class="id"·title="keyword">Theorem</span>·<a·id="lt_le_S"·class="idref"·href="#lt_le_S"><span·class="id"·title="lemma">lt_le_S</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>·:·<a·class="idref"·href="Coq.Arith.Lt.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'<'_x"><span·class="id"·title="notation"><</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#m:2"><span·class="id"·title="variable">m</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#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Arith.Lt.html#m:2"><span·class="id"·title="variable">m</span></a>.<br/>


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46	<br/>	46	<br/>
47	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	47	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

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

50	<div·class="doc">	50	<div·class="doc">
51	<a·id="lab~~279~~"></a><h1·class="section">0·is·right·neutral</h1>	51	<a·id="lab660"></a><h1·class="section">0·is·right·neutral</h1>

52	</div>	52	</div>
53	<div·class="code">	53	<div·class="code">

54	<br/>	54	<br/>
55	<span·class="id"·title="keyword">Lemma</span>·<a·id="minus_n_O"·class="idref"·href="#minus_n_O"><span·class="id"·title="lemma">minus_n_O</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Arith.Minus.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·0.<br/>	55	<span·class="id"·title="keyword">Lemma</span>·<a·id="minus_n_O"·class="idref"·href="#minus_n_O"><span·class="id"·title="lemma">minus_n_O</span></a>·<a·id="n:1"·class="idref"·href="#n:1"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Arith.Minus.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#n:1"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·0.<br/>

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

58	<div·class="doc">	58	<div·class="doc">
59	<a·id="lab~~280~~"></a><h1·class="section">Permutation·with·successor</h1>	59	<a·id="lab661"></a><h1·class="section">Permutation·with·successor</h1>

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

62	<br/>	62	<br/>
63	<span·class="id"·title="keyword">Lemma</span>·<a·id="minus_Sn_m"·class="idref"·href="#minus_Sn_m"><span·class="id"·title="lemma">minus_Sn_m</span></a>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>·<a·id="m:3"·class="idref"·href="#m:3"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Arith.Minus.html#m:3"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#n:2"><span·class="id"·title="variable">n</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#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.Arith.Minus.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#m:3"><span·class="id"·title="variable">m</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#m:3"><span·class="id"·title="variable">m</span></a>.<br/>	63	<span·class="id"·title="keyword">Lemma</span>·<a·id="minus_Sn_m"·class="idref"·href="#minus_Sn_m"><span·class="id"·title="lemma">minus_Sn_m</span></a>·<a·id="n:2"·class="idref"·href="#n:2"><span·class="id"·title="binder">n</span></a>·<a·id="m:3"·class="idref"·href="#m:3"><span·class="id"·title="binder">m</span></a>·:·<a·class="idref"·href="Coq.Arith.Minus.html#m:3"><span·class="id"·title="variable">m</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span·class="id"·title="notation"><=</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#n:2"><span·class="id"·title="variable">n</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#S"><span·class="id"·title="constructor">S</span></a>·(<a·class="idref"·href="Coq.Arith.Minus.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#m:3"><span·class="id"·title="variable">m</span></a>)·<a·class="idref"·href="Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span·class="id"·title="notation">=</span></a>·<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#n:2"><span·class="id"·title="variable">n</span></a>·<a·class="idref"·href="Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span·class="id"·title="notation">-</span></a>·<a·class="idref"·href="Coq.Arith.Minus.html#m:3"><span·class="id"·title="variable">m</span></a>.<br/>


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

15	<div·class="code">	15	<div·class="code">

16	<br/>	16	<br/>
17	</div>	17	</div>

18	<div·class="doc">	18	<div·class="doc">
19	<a·id="lab~~286~~"></a><h1·class="section">Properties·of·multiplication.</h1>	19	<a·id="lab690"></a><h1·class="section">Properties·of·multiplication.</h1>


20	<div·class="paragraph">·</div>	20	<div·class="paragraph">·</div>

21	·This·file·is·mostly·OBSOLETE·now,·see·module·<span·class="inlinecode"><span·class="id"·title="var">PeanoNat.Nat</span></span>·instead.	21	·This·file·is·mostly·OBSOLETE·now,·see·module·<span·class="inlinecode"><span·class="id"·title="var">PeanoNat.Nat</span></span>·instead.

22	<div·class="paragraph">·</div>	22	<div·class="paragraph">·</div>


Offset 114, 15 lines modified		Offset 114, 15 lines modified
114	<a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><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.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.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a><a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a><a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>)·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/>	114	<a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><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.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.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a><a·class="idref"·href="Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span·class="id"·title="notation">==></span></a><a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>)·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Init.Logic.html#::type_scope:x_'->'_x"><span·class="id"·title="notation">-></span></a><br/>
115	<span·class="id"·title="keyword">forall</span>·<a·id="n:14"·class="idref"·href="#n:14"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>))·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>)).<br/>	115	<span·class="id"·title="keyword">forall</span>·<a·id="n:14"·class="idref"·href="#n:14"><span·class="id"·title="binder">n</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#nat"><span·class="id"·title="inductive">nat</span></a>,·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Aeq:11"><span·class="id"·title="variable">Aeq</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·(<a·class="idref"·href="Coq.Init.Datatypes.html#S"><span·class="id"·title="constructor">S</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>))·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>·(<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.recursion"><span·class="id"·title="definition">recursion</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#a:12"><span·class="id"·title="variable">a</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#f:13"><span·class="id"·title="variable">f</span></a>·<a·class="idref"·href="Coq.Arith.PeanoNat.html#n:14"><span·class="id"·title="variable">n</span></a>)).<br/>

116	<br/>	116	<br/>
117	</div>	117	</div>

118	<div·class="doc">	118	<div·class="doc">
119	<a·id="lab~~250~~"></a><h2·class="section">Remaining·constants·not·defined·in·Coq.Init.Nat</h2>	119	<a·id="lab676"></a><h2·class="section">Remaining·constants·not·defined·in·Coq.Init.Nat</h2>

120	<div·class="paragraph">·</div>	120	<div·class="paragraph">·</div>

121	·NB:·Aliasing·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·mandatory,·since·only·a·Definition·can·implement	121	·NB:·Aliasing·<span·class="inlinecode"><span·class="id"·title="var">le</span></span>·is·mandatory,·since·only·a·Definition·can·implement
122	····an·interface·Parameter...·	122	····an·interface·Parameter...·
123	</div>	123	</div>
124	<div·class="code">	124	<div·class="code">


Offset 46, 15 lines modified		Offset 46, 15 lines modified
46	<br/>	46	<br/>
47	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	47	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

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

50	<div·class="doc">	50	<div·class="doc">
51	<a·id="lab~~269~~"></a><h1·class="section">Neutrality·of·0,·commutativity,·associativity</h1>	51	<a·id="lab702"></a><h1·class="section">Neutrality·of·0,·commutativity,·associativity</h1>

52	</div>	52	</div>
53	<div·class="code">	53	<div·class="code">

54	<br/>	54	<br/>
55	<span·class="id"·title="keyword">Notation</span>·<a·id="plus_0_l"·class="idref"·href="#plus_0_l"><span·class="id"·title="abbreviation">plus_0_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.add_0_l"><span·class="id"·title="lemma">Nat.add_0_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>	55	<span·class="id"·title="keyword">Notation</span>·<a·id="plus_0_l"·class="idref"·href="#plus_0_l"><span·class="id"·title="abbreviation">plus_0_l</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.add_0_l"><span·class="id"·title="lemma">Nat.add_0_l</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>
56	<span·class="id"·title="keyword">Notation</span>·<a·id="plus_0_r"·class="idref"·href="#plus_0_r"><span·class="id"·title="abbreviation">plus_0_r</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.add_0_r"><span·class="id"·title="lemma">Nat.add_0_r</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>	56	<span·class="id"·title="keyword">Notation</span>·<a·id="plus_0_r"·class="idref"·href="#plus_0_r"><span·class="id"·title="abbreviation">plus_0_r</span></a>·:=·<a·class="idref"·href="Coq.Arith.PeanoNat.html#Nat.add_0_r"><span·class="id"·title="lemma">Nat.add_0_r</span></a>·(<span·class="id"·title="var">only</span>·<span·class="id"·title="var">parsing</span>).<br/>


Offset 50, 27 lines modified		Offset 50, 27 lines modified
50	\|·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·=>·<a·class="idref"·href="Coq.Init.Logic.html#False"><span·class="id"·title="inductive">False</span></a><br/>	50	\|·<a·class="idref"·href="Coq.Init.Datatypes.html#false"><span·class="id"·title="constructor">false</span></a>·=>·<a·class="idref"·href="Coq.Init.Logic.html#False"><span·class="id"·title="inductive">False</span></a><br/>
51	<span·class="id"·title="keyword">end</span>.<br/>	51	<span·class="id"·title="keyword">end</span>.<br/>

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

54	<div·class="doc">	54	<div·class="doc">
55	<a·id="lab~~769~~"></a><h1·class="section">Decidability</h1>	55	<a·id="lab1007"></a><h1·class="section">Decidability</h1>

56	</div>	56	</div>
57	<div·class="code">	57	<div·class="code">

58	<br/>	58	<br/>
59	<span·class="id"·title="keyword">Lemma</span>·<a·id="bool_dec"·class="idref"·href="#bool_dec"><span·class="id"·title="lemma">bool_dec</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:3"·class="idref"·href="#b1:3"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:4"·class="idref"·href="#b2:4"><span·class="id"·title="binder">b2</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>,·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><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.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><span·class="id"·title="variable">b1</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.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/>	59	<span·class="id"·title="keyword">Lemma</span>·<a·id="bool_dec"·class="idref"·href="#bool_dec"><span·class="id"·title="lemma">bool_dec</span></a>·:·<span·class="id"·title="keyword">forall</span>·<a·id="b1:3"·class="idref"·href="#b1:3"><span·class="id"·title="binder">b1</span></a>·<a·id="b2:4"·class="idref"·href="#b2:4"><span·class="id"·title="binder">b2</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#bool"><span·class="id"·title="inductive">bool</span></a>,·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><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.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">+</span></a>·<a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">{</span></a><a·class="idref"·href="Coq.Bool.Bool.html#b1:3"><span·class="id"·title="variable">b1</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.Bool.html#b2:4"><span·class="id"·title="variable">b2</span></a><a·class="idref"·href="Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span·class="id"·title="notation">}</span></a>.<br/>

60	<br/>	60	<br/>
61	</div>	61	</div>

62	<div·class="doc">	62	<div·class="doc">
63	<a·id="lab770"></a><h1·class="section">Discrimination</h1>	63	<a·id="lab1008"></a><h1·class="section">Discrimination</h1>

64	</div>	64	</div>
65	<div·class="code">	65	<div·class="code">

66	<br/>	66	<br/>
67	<span·class="id"·title="keyword">Lemma</span>·<a·id="diff_true_false"·class="idref"·href="#diff_true_false"><span·class="id"·title="lemma">diff_true_false</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</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#false"><span·class="id"·title="constructor">false</span></a>.<br/>	67	<span·class="id"·title="keyword">Lemma</span>·<a·id="diff_true_false"·class="idref"·href="#diff_true_false"><span·class="id"·title="lemma">diff_true_false</span></a>·:·<a·class="idref"·href="Coq.Init.Datatypes.html#true"><span·class="id"·title="constructor">true</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#false"><span·class="id"·title="constructor">false</span></a>.<br/>
68	<span·class="id"·title="keyword">Hint·Resolve</span>·<span·class="id"·title="var">diff_true_false</span>·:·<span·class="id"·title="var">bool</span>.<br/>	68	<span·class="id"·title="keyword">Hint·Resolve</span>·<span·class="id"·title="var">diff_true_false</span>·:·<span·class="id"·title="var">bool</span>.<br/>


Offset 31, 15 lines modified		Offset 31, 15 lines modified
31	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>.<br/>	31	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Structures.Orders.html#"><span·class="id"·title="library">Orders</span></a>.<br/>
32	<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">BoolNotations</span>.<br/>	32	<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">BoolNotations</span>.<br/>

33	<br/>	33	<br/>
34	</div>	34	</div>

35	<div·class="doc">	35	<div·class="doc">
36	<a·id="lab784"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1>	36	<a·id="lab1004"></a><h1·class="section">Order·<span·class="inlinecode"><span·class="id"·title="var">le</span></span></h1>

37	</div>	37	</div>
38	<div·class="code">	38	<div·class="code">

39	<br/>	39	<br/>
40	<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/>	40	<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/>
41	·	41	·


Offset 17, 15 lines modified		Offset 17, 15 lines modified

17	<br/>	17	<br/>

18	<br/>	18	<br/>
19	</div>	19	</div>

20	<div·class="doc">	20	<div·class="doc">
21	<a·id="lab~~109~~"></a><h1·class="section">FMapAVL</h1>	21	<a·id="lab242"></a><h1·class="section">FMapAVL</h1>

22	<div·class="paragraph">·</div>	22	<div·class="paragraph">·</div>

23	·This·module·implements·maps·using·AVL·trees.	23	·This·module·implements·maps·using·AVL·trees.
24	····It·follows·the·implementation·from·Ocaml's·standard·library.	24	····It·follows·the·implementation·from·Ocaml's·standard·library.

25	<div·class="paragraph">·</div>	25	<div·class="paragraph">·</div>


Offset 15, 28 lines modified		Offset 15, 28 lines modified

15	<div·class="code">	15	<div·class="code">

16	<br/>	16	<br/>
17	</div>	17	</div>

18	<div·class="doc">	18	<div·class="doc">
19	<a·id="lab89"></a><h1·class="section">Compatibility·functors·between·FSetInterface·and·MSetInterface.</h1>	19	<a·id="lab396"></a><h1·class="section">Compatibility·functors·between·FSetInterface·and·MSetInterface.</h1>

20	</div>	20	</div>
21	<div·class="code">	21	<div·class="code">

22	<br/>	22	<br/>
23	<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.FSetFacts.html#"><span·class="id"·title="library">FSetFacts</span></a>·<a·class="idref"·href="Coq.MSets.MSetInterface.html#"><span·class="id"·title="library">MSetInterface</span></a>·<a·class="idref"·href="Coq.MSets.MSetFacts.html#"><span·class="id"·title="library">MSetFacts</span></a>.<br/>	23	<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.FSetFacts.html#"><span·class="id"·title="library">FSetFacts</span></a>·<a·class="idref"·href="Coq.MSets.MSetInterface.html#"><span·class="id"·title="library">MSetInterface</span></a>·<a·class="idref"·href="Coq.MSets.MSetFacts.html#"><span·class="id"·title="library">MSetFacts</span></a>.<br/>
24	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>	24	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>

25	<br/>	25	<br/>
26	</div>	26	</div>

27	<div·class="doc">	27	<div·class="doc">
28	<a·id="lab90"></a><h1·class="section">From·new·Weak·Sets·to·old·ones</h1>	28	<a·id="lab397"></a><h1·class="section">From·new·Weak·Sets·to·old·ones</h1>

29	</div>	29	</div>
30	<div·class="code">	30	<div·class="code">

31	<br/>	31	<br/>
32	<span·class="id"·title="keyword">Module</span>·<a·id="Backport_WSets"·class="idref"·href="#Backport_WSets"><span·class="id"·title="module">Backport_WSets</span></a><br/>	32	<span·class="id"·title="keyword">Module</span>·<a·id="Backport_WSets"·class="idref"·href="#Backport_WSets"><span·class="id"·title="module">Backport_WSets</span></a><br/>
33	(<span·class="id"·title="var">E</span>:<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType.DecidableType</span></a>)<br/>	33	(<span·class="id"·title="var">E</span>:<a·class="idref"·href="Coq.Structures.DecidableType.html#DecidableType"><span·class="id"·title="module">DecidableType.DecidableType</span></a>)<br/>


Offset 41, 15 lines modified		Offset 41, 15 lines modified
41	<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/>	41	<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/>
42	<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/>	42	<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/>

43	<br/>	43	<br/>
44	</div>	44	</div>

45	<div·class="doc">	45	<div·class="doc">
46	<a·id="lab~~158~~"></a><h1·class="section">Overview</h1>	46	<a·id="lab380"></a><h1·class="section">Overview</h1>

47	····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will	47	····This·functor·defines·the·tactic·<span·class="inlinecode"><span·class="id"·title="var">fsetdec</span></span>,·which·will
48	····solve·any·valid·goal·of·the·form	48	····solve·any·valid·goal·of·the·form
49	<pre>	49	<pre>
50	····forall·s1·...·sn,	50	····forall·s1·...·sn,
51	····forall·x1·...·xm,	51	····forall·x1·...·xm,
52	····P1·->·...·->·Pk·->·P	52	····P1·->·...·->·Pk·->·P


Offset 18, 15 lines modified		Offset 18, 15 lines modified
18	<br/>	18	<br/>
19	<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.Int63.html#"><span·class="id"·title="library">Int63</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/>	19	<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.Int63.html#"><span·class="id"·title="library">Int63</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/>

20	<br/>	20	<br/>
21	</div>	21	</div>

22	<div·class="doc">	22	<div·class="doc">
23	<a·id="lab~~799~~"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1>	23	<a·id="lab165"></a><h1·class="section">Properties·of·the·primitive·operators·for·the·Binary64·format</h1>

24	</div>	24	</div>
25	<div·class="code">	25	<div·class="code">

26	<br/>	26	<br/>
27	<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/>	27	<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 19, 15 lines modified		Offset 19, 15 lines modified
19	<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.Int63.html#"><span·class="id"·title="library">Int63</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/>	19	<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.Int63.html#"><span·class="id"·title="library">Int63</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/>
20	<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/>	20	<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/>

21	<br/>	21	<br/>
22	</div>	22	</div>

23	<div·class="doc">	23	<div·class="doc">
24	<a·id="lab815"></a><h1·class="section">Support·results·involving·frexp·and·ldexp</h1>	24	<a·id="lab153"></a><h1·class="section">Support·results·involving·frexp·and·ldexp</h1>

25	</div>	25	</div>
26	<div·class="code">	26	<div·class="code">

27	<br/>	27	<br/>
28	<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/>	28	<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 18, 59 lines modified		Offset 18, 59 lines modified
18	<br/>	18	<br/>
19	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Int63.html#"><span·class="id"·title="library">Int63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>	19	<span·class="id"·title="keyword">Require</span>·<span·class="id"·title="keyword">Import</span>·<a·class="idref"·href="Coq.Numbers.Cyclic.Int63.Int63.html#"><span·class="id"·title="library">Int63</span></a>·<a·class="idref"·href="Coq.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>

20	<br/>	20	<br/>
21	</div>	21	</div>

22	<div·class="doc">	22	<div·class="doc">
23	<a·id="lab~~800~~"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1>	23	<a·id="lab154"></a><h1·class="section">Definition·of·the·interface·for·primitive·floating-point·arithmetic</h1>


24	<div·class="paragraph">·</div>	24	<div·class="paragraph">·</div>

25	This·interface·provides·processor·operators·for·the·Binary64·format·of·the	25	This·interface·provides·processor·operators·for·the·Binary64·format·of·the
26	IEEE·754-2008·standard.·	26	IEEE·754-2008·standard.·
27	<div·class="paragraph">·</div>	27	<div·class="paragraph">·</div>

28	<a·id="lab~~801~~"></a><h2·class="section">Type·definition·for·the·co-domain·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h2>	28	<a·id="lab155"></a><h2·class="section">Type·definition·for·the·co-domain·of·<span·class="inlinecode"><span·class="id"·title="var">compare</span></span></h2>

29	</div>	29	</div>
30	<div·class="code">	30	<div·class="code">
31	<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/>	31	<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/>

32	<br/>	32	<br/>

33	<br/>	33	<br/>

34	<br/>	34	<br/>
35	</div>	35	</div>

36	<div·class="doc">	36	<div·class="doc">
37	<a·id="lab~~802~~"></a><h2·class="section">The·main·type</h2>	37	<a·id="lab156"></a><h2·class="section">The·main·type</h2>
38	·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>:·primitive·type·for·Binary64·floating-point·numbers.·	38	·<span·class="inlinecode"><span·class="id"·title="var">float</span></span>:·primitive·type·for·Binary64·floating-point·numbers.·
39	</div>	39	</div>
40	<div·class="code">	40	<div·class="code">

41	<br/>	41	<br/>
42	</div>	42	</div>

43	<div·class="doc">	43	<div·class="doc">
44	<a·id="lab~~803~~"></a><h2·class="section">Syntax·support</h2>	44	<a·id="lab157"></a><h2·class="section">Syntax·support</h2>

45	</div>	45	</div>
46	<div·class="code">	46	<div·class="code">
47	<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/>	47	<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/>

48	<br/>	48	<br/>

49	<br/>	49	<br/>
50	</div>	50	</div>

51	<div·class="doc">	51	<div·class="doc">
52	<a·id="lab~~804~~"></a><h2·class="section">Floating-point·operators</h2>	52	<a·id="lab158"></a><h2·class="section">Floating-point·operators</h2>

53	</div>	53	</div>
54	<div·class="code">	54	<div·class="code">

55	<br/>	55	<br/>

56	<br/>	56	<br/>


Offset 18, 24 lines modified		Offset 18, 24 lines modified
18	<br/>	18	<br/>
19	<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.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>	19	<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.Floats.FloatClass.html#"><span·class="id"·title="library">FloatClass</span></a>.<br/>

20	<br/>	20	<br/>
21	</div>	21	</div>

22	<div·class="doc">	22	<div·class="doc">
23	<a·id="lab~~811~~"></a><h1·class="section">Specification·of·floating-point·arithmetic</h1>	23	<a·id="lab148"></a><h1·class="section">Specification·of·floating-point·arithmetic</h1>


24	<div·class="paragraph">·</div>	24	<div·class="paragraph">·</div>

25	This·specification·is·mostly·borrowed·from·the·<span·class="inlinecode"><span·class="id"·title="var">IEEE754.Binary</span></span>·module	25	This·specification·is·mostly·borrowed·from·the·<span·class="inlinecode"><span·class="id"·title="var">IEEE754.Binary</span></span>·module
26	of·the·Flocq·library·(see·<a·href="http://flocq.gforge.inria.fr/">http://flocq.gforge.inria.fr/</a>)·	26	of·the·Flocq·library·(see·<a·href="http://flocq.gforge.inria.fr/">http://flocq.gforge.inria.fr/</a>)·
27	<div·class="paragraph">·</div>	27	<div·class="paragraph">·</div>

28	<a·id="lab~~812~~"></a><h2·class="section">Inductive·specification·of·floating-point·numbers</h2>	28	<a·id="lab149"></a><h2·class="section">Inductive·specification·of·floating-point·numbers</h2>


29	<div·class="paragraph">·</div>	29	<div·class="paragraph">·</div>

30	Similar·to·<span·class="inlinecode"><span·class="id"·title="var">Flocq.IEEE754.Binary.full_float</span></span>,·but·with·no·NaN·payload.·	30	Similar·to·<span·class="inlinecode"><span·class="id"·title="var">Flocq.IEEE754.Binary.full_float</span></span>,·but·with·no·NaN·payload.·
31	</div>	31	</div>
32	<div·class="code">	32	<div·class="code">


Offset 23, 15 lines modified		Offset 23, 15 lines modified
23	<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/>	23	<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/>
24	<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/>	24	<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/>

25	<br/>	25	<br/>
26	</div>	26	</div>

27	<div·class="doc">	27	<div·class="doc">
28	<a·id="lab3"></a><h1·class="section">Datatypes·with·zero·and·one·element</h1>	28	<a·id="lab105"></a><h1·class="section">Datatypes·with·zero·and·one·element</h1>

29	<div·class="paragraph">·</div>	29	<div·class="paragraph">·</div>

30	·<span·class="inlinecode"><span·class="id"·title="var">Empty_set</span></span>·is·a·datatype·with·no·inhabitant·	30	·<span·class="inlinecode"><span·class="id"·title="var">Empty_set</span></span>·is·a·datatype·with·no·inhabitant·
31	</div>	31	</div>
32	<div·class="code">	32	<div·class="code">


Offset 25, 15 lines modified		Offset 25, 15 lines modified
25	<br/>	25	<br/>
26	<span·class="id"·title="keyword">Notation</span>·<a·id="::type_scope:x_'->'_x"·class="idref"·href="#::type_scope:x_'->'_x"><span·class="id"·title="notation">"</span></a>A·->·B"·:=·(<span·class="id"·title="keyword">forall</span>·(<span·class="id"·title="var">_</span>·:·<span·class="id"·title="var">A</span>),·<span·class="id"·title="var">B</span>)·:·<span·class="id"·title="var">type_scope</span>.<br/>	26	<span·class="id"·title="keyword">Notation</span>·<a·id="::type_scope:x_'->'_x"·class="idref"·href="#::type_scope:x_'->'_x"><span·class="id"·title="notation">"</span></a>A·->·B"·:=·(<span·class="id"·title="keyword">forall</span>·(<span·class="id"·title="var">_</span>·:·<span·class="id"·title="var">A</span>),·<span·class="id"·title="var">B</span>)·:·<span·class="id"·title="var">type_scope</span>.<br/>

27	<br/>	27	<br/>
28	</div>	28	</div>

29	<div·class="doc">	29	<div·class="doc">
30	<a·id="lab27"></a><h1·class="section">Propositional·connectives</h1>	30	<a·id="lab100"></a><h1·class="section">Propositional·connectives</h1>

31	<div·class="paragraph">·</div>	31	<div·class="paragraph">·</div>

32	·<span·class="inlinecode"><span·class="id"·title="var">True</span></span>·is·the·always·true·proposition·	32	·<span·class="inlinecode"><span·class="id"·title="var">True</span></span>·is·the·always·true·proposition·
33	</div>	33	</div>
34	<div·class="code">	34	<div·class="code">


Offset 20, 15 lines modified		Offset 20, 15 lines modified
20	<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.Numeral.html#"><span·class="id"·title="library">Numeral</span></a>.<br/>	20	<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.Numeral.html#"><span·class="id"·title="library">Numeral</span></a>.<br/>
21	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>	21	<span·class="id"·title="keyword">Local·Open</span>·<span·class="id"·title="keyword">Scope</span>·<span·class="id"·title="var">nat_scope</span>.<br/>

22	<br/>	22	<br/>
23	</div>	23	</div>

24	<div·class="doc">	24	<div·class="doc">
25	<a·id="lab12"></a><h1·class="section">Peano·natural·numbers,·definitions·of·operations</h1>	25	<a·id="lab111"></a><h1·class="section">Peano·natural·numbers,·definitions·of·operations</h1>

26	<div·class="paragraph">·</div>	26	<div·class="paragraph">·</div>

27	·This·file·is·meant·to·be·used·as·a·whole·module,	27	·This·file·is·meant·to·be·used·as·a·whole·module,
28	····without·importing·it,·leading·to·qualified·definitions	28	····without·importing·it,·leading·to·qualified·definitions
29	····(e.g.·Nat.pred)·	29	····(e.g.·Nat.pred)·
30	</div>	30	</div>


Offset 21, 15 lines modified		Offset 21, 15 lines modified
21	<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/>	21	<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/>
22	<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/>	22	<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/>

23	<br/>	23	<br/>
24	</div>	24	</div>

25	<div·class="doc">	25	<div·class="doc">
26	<a·id="lab9"></a><h1·class="section">Useful·tactics</h1>	26	<a·id="lab125"></a><h1·class="section">Useful·tactics</h1>

27	<div·class="paragraph">·</div>	27	<div·class="paragraph">·</div>

28	·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal.	28	·Ex·falso·quodlibet·:·a·tactic·for·proving·False·instead·of·the·current·goal.
29	····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">elimtype</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>	29	····This·is·just·a·nicer·name·for·tactics·such·as·<span·class="inlinecode"><span·class="id"·title="tactic">elimtype</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>
30	····and·other·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.·	30	····and·other·<span·class="inlinecode"><span·class="id"·title="tactic">cut</span></span>·<span·class="inlinecode"><span·class="id"·title="var">False</span></span>.·
31	</div>	31	</div>


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16	<div·class="code">	16	<div·class="code">

17	<br/>	17	<br/>
18	</div>	18	</div>

19	<div·class="doc">	19	<div·class="doc">
20	Some·facts·and·definitions·concerning·choice·and·description·in	20	Some·facts·and·definitions·concerning·choice·and·description·in
21	·······intuitionistic·logic.·<a·id="lab544"></a><h1·class="section">References:</h1>	21	·······intuitionistic·logic.·<a·id="lab784"></a><h1·class="section">References:</h1>

22	<div·class="paragraph">·</div>	22	<div·class="paragraph">·</div>

23	<span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory,	23	<span·class="inlinecode">[<span·class="id"·title="var">Bell</span>]</span>·John·L.·Bell,·Choice·principles·in·intuitionistic·set·theory,
24	unpublished.	24	unpublished.

25	<div·class="paragraph">·</div>	25	<div·class="paragraph">·</div>


Offset 99, 15 lines modified		Offset 99, 15 lines modified
99	<br/>	99	<br/>
100	<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/>	100	<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/>

101	<br/>	101	<br/>
102	</div>	102	</div>

103	<div·class="doc">	103	<div·class="doc">
104	<a·id="lab~~604~~"></a><h3·class="section">Weaker·lemmas·(compatibility·lemmas)</h3>	104	<a·id="lab757"></a><h3·class="section">Weaker·lemmas·(compatibility·lemmas)</h3>

105	</div>	105	</div>
106	<div·class="code">	106	<div·class="code">

107	<br/>	107	<br/>
108	<span·class="id"·title="keyword">Theorem</span>·<a·id="choice"·class="idref"·href="#choice"><span·class="id"·title="lemma">choice</span></a>·:<br/>	108	<span·class="id"·title="keyword">Theorem</span>·<a·id="choice"·class="idref"·href="#choice"><span·class="id"·title="lemma">choice</span></a>·:<br/>
109	<span·class="id"·title="keyword">forall</span>·(<a·id="A:37"·class="idref"·href="#A:37"><span·class="id"·title="binder">A</span></a>·<a·id="B:38"·class="idref"·href="#B:38"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:39"·class="idref"·href="#R:39"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:37"><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:38"><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/>	109	<span·class="id"·title="keyword">forall</span>·(<a·id="A:37"·class="idref"·href="#A:37"><span·class="id"·title="binder">A</span></a>·<a·id="B:38"·class="idref"·href="#B:38"><span·class="id"·title="binder">B</span></a>·:·<span·class="id"·title="keyword">Type</span>)·(<a·id="R:39"·class="idref"·href="#R:39"><span·class="id"·title="binder">R</span></a>·:·<a·class="idref"·href="Coq.Logic.ClassicalEpsilon.html#A:37"><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:38"><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 80, 15 lines modified		Offset 80, 15 lines modified

80	<div·class="paragraph">·</div>	80	<div·class="paragraph">·</div>

81	4.2·Classical·logic·=·choice·of·representatives·in·a·partition·of·bool	81	4.2·Classical·logic·=·choice·of·representatives·in·a·partition·of·bool

82	<div·class="paragraph">·</div>	82	<div·class="paragraph">·</div>

83	<a·id="lab~~578~~"></a><h1·class="section">Prop·degeneracy·=·excluded-middle·+·prop·extensionality</h1>	83	<a·id="lab765"></a><h1·class="section">Prop·degeneracy·=·excluded-middle·+·prop·extensionality</h1>

84	<div·class="paragraph">·</div>	84	<div·class="paragraph">·</div>

85	·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>	85	·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>
86	·························<span·class="inlinecode"><-></span>	86	·························<span·class="inlinecode"><-></span>
87	·······<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>	87	·······<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>


Offset 74, 15 lines modified		Offset 74, 15 lines modified
74	<div·class="code">	74	<div·class="code">
75	<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/>	75	<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/>

76	<br/>	76	<br/>
77	</div>	77	</div>

78	<div·class="doc">	78	<div·class="doc">
79	<a·id="lab~~597~~"></a><h1·class="section">Pred.·Ext.·+·Rel.·Axiom·of·Choice·->·Excluded-Middle</h1>	79	<a·id="lab758"></a><h1·class="section">Pred.·Ext.·+·Rel.·Axiom·of·Choice·->·Excluded-Middle</h1>

80	</div>	80	</div>
81	<div·class="code">	81	<div·class="code">

82	<br/>	82	<br/>
83	<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/>	83	<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/>


Offset 82, 15 lines modified		Offset 82, 15 lines modified
82	···equalities·assuming·axiom·eq_rect_eq	82	···equalities·assuming·axiom·eq_rect_eq

83	<div·class="paragraph">·</div>	83	<div·class="paragraph">·</div>


84	<div·class="paragraph">·</div>	84	<div·class="paragraph">·</div>

85	<a·id="lab575"></a><h1·class="section">Definition·of·dependent·equality·and·equivalence·with·equality·of·dependent·pairs</h1>	85	<a·id="lab753"></a><h1·class="section">Definition·of·dependent·equality·and·equivalence·with·equality·of·dependent·pairs</h1>

86	</div>	86	</div>
87	<div·class="code">	87	<div·class="code">

88	<br/>	88	<br/>
89	<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">EqNotations</span>.<br/>	89	<span·class="id"·title="keyword">Import</span>·<span·class="id"·title="var">EqNotations</span>.<br/>


Offset 54, 15 lines modified		Offset 54, 15 lines modified
54	·····from·a·proof·of·decidability·of·equality·for·a·set·in·Set	54	·····from·a·proof·of·decidability·of·equality·for·a·set·in·Set

55	<div·class="paragraph">·</div>	55	<div·class="paragraph">·</div>


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

57	<a·id="lab~~541~~"></a><h1·class="section">Streicher's·K·and·injectivity·of·dependent·pair·hold·on·decidable·types</h1>	57	<a·id="lab750"></a><h1·class="section">Streicher's·K·and·injectivity·of·dependent·pair·hold·on·decidable·types</h1>

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

60	<br/>	60	<br/>
61	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>	61	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>


Offset 71, 15 lines modified		Offset 71, 15 lines modified
71	<br/>	71	<br/>
72	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>	72	<span·class="id"·title="keyword">Set·Implicit·Arguments</span>.<br/>

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

75	<div·class="doc">	75	<div·class="doc">
76	<a·id="lab~~600~~"></a><h1·class="section">Definitions</h1>	76	<a·id="lab761"></a><h1·class="section">Definitions</h1>

77	<div·class="paragraph">·</div>	77	<div·class="paragraph">·</div>

78	·Being·an·inverse·	78	·Being·an·inverse·
79	</div>	79	</div>
80	<div·class="code">	80	<div·class="code">


Offset 142, 15 lines modified		Offset 142, 15 lines modified

142	<br/>	142	<br/>

143	<br/>	143	<br/>
144	</div>	144	</div>

145	<div·class="doc">	145	<div·class="doc">
146	<a·id="lab~~503~~"></a><h1·class="section">A·modular·proof·of·Hurkens's·paradox.</h1>	146	<a·id="lab711"></a><h1·class="section">A·modular·proof·of·Hurkens's·paradox.</h1>

147	<div·class="paragraph">·</div>	147	<div·class="paragraph">·</div>

148	·It·relies·on·an·axiomatisation·of·a·shallow·embedding·of·system·U-	148	·It·relies·on·an·axiomatisation·of·a·shallow·embedding·of·system·U-
149	····(i.e.··types·of·U-·are·interpreted·by·types·of·Coq).·The	149	····(i.e.··types·of·U-·are·interpreted·by·types·of·Coq).·The
150	····universes·are·encoded·in·a·style,·due·to·Martin-Löf,·where·they	150	····universes·are·encoded·in·a·style,·due·to·Martin-Löf,·where·they
151	····are·given·by·a·set·of·names·and·a·family·<span·class="inlinecode"><span·class="id"·title="var">El</span>:<span·class="id"·title="var">Name</span>-><span·class="id"·title="keyword">Type</span></span>·which	151	····are·given·by·a·set·of·names·and·a·family·<span·class="inlinecode"><span·class="id"·title="var">El</span>:<span·class="id"·title="var">Name</span>-><span·class="id"·title="keyword">Type</span></span>·which