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<html><head><meta charset="utf-8"><title>Algebra.Consequences.Propositional</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Relations between properties of functions, such as associativity and</a>
<a id="178" class="Comment">-- commutativity (specialised to propositional equality)</a>
<a id="235" class="Comment">------------------------------------------------------------------------</a>
<a id="309" class="Symbol">{-#</a> <a id="313" class="Keyword">OPTIONS</a> <a id="321" class="Pragma">--cubical-compatible</a> <a id="342" class="Pragma">--safe</a> <a id="349" class="Symbol">#-}</a>
<a id="354" class="Keyword">module</a> <a id="361" href="Algebra.Consequences.Propositional.html" class="Module">Algebra.Consequences.Propositional</a>
<a id="398" class="Symbol">{</a><a id="399" href="Algebra.Consequences.Propositional.html#399" class="Bound">a</a><a id="400" class="Symbol">}</a> <a id="402" class="Symbol">{</a><a id="403" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a> <a id="405" class="Symbol">:</a> <a id="407" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="411" href="Algebra.Consequences.Propositional.html#399" class="Bound">a</a><a id="412" class="Symbol">}</a> <a id="414" class="Keyword">where</a>
<a id="421" class="Keyword">open</a> <a id="426" class="Keyword">import</a> <a id="433" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="447" class="Keyword">using</a> <a id="453" class="Symbol">(</a><a id="454" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a><a id="458" class="Symbol">;</a> <a id="460" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="464" class="Symbol">)</a>
<a id="466" class="Keyword">open</a> <a id="471" class="Keyword">import</a> <a id="478" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="499" class="Keyword">using</a> <a id="505" class="Symbol">(</a><a id="506" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="509" class="Symbol">)</a>
<a id="511" class="Keyword">open</a> <a id="516" class="Keyword">import</a> <a id="523" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a> <a id="547" class="Keyword">using</a> <a id="553" class="Symbol">(</a><a id="554" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="560" class="Symbol">)</a>
<a id="562" class="Keyword">open</a> <a id="567" class="Keyword">import</a> <a id="574" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a> <a id="602" class="Keyword">using</a> <a id="608" class="Symbol">(</a><a id="609" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a><a id="618" class="Symbol">;</a> <a id="620" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a><a id="625" class="Symbol">)</a>
<a id="627" class="Keyword">open</a> <a id="632" class="Keyword">import</a> <a id="639" href="Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a>
<a id="677" class="Keyword">open</a> <a id="682" class="Keyword">import</a> <a id="689" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="704" class="Keyword">using</a> <a id="710" class="Symbol">(</a><a id="711" href="Relation.Unary.html#1232" class="Function">Pred</a><a id="715" class="Symbol">)</a>
<a id="718" class="Keyword">open</a> <a id="723" class="Keyword">import</a> <a id="730" href="Algebra.Core.html" class="Module">Algebra.Core</a>
<a id="743" class="Keyword">open</a> <a id="748" class="Keyword">import</a> <a id="755" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="775" class="Symbol">{</a><a id="776" class="Argument">A</a> <a id="778" class="Symbol">=</a> <a id="780" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="781" class="Symbol">}</a> <a id="783" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="787" class="Keyword">import</a> <a id="794" href="Algebra.Consequences.Setoid.html" class="Module">Algebra.Consequences.Setoid</a> <a id="822" class="Symbol">(</a><a id="823" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="830" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="831" class="Symbol">)</a> as <a id="Base"></a><a id="836" href="Algebra.Consequences.Propositional.html#836" class="Module">Base</a>
<a id="842" class="Comment">------------------------------------------------------------------------</a>
<a id="915" class="Comment">-- Re-export all proofs that don&#39;t require congruence or substitutivity</a>
<a id="988" class="Keyword">open</a> <a id="993" href="Algebra.Consequences.Propositional.html#836" class="Module">Base</a> <a id="998" class="Keyword">public</a>
<a id="1007" class="Keyword">hiding</a>
<a id="1016" class="Symbol">(</a> <a id="1018" href="Algebra.Consequences.Setoid.html#1335" class="Function">comm∧assoc⇒middleFour</a>
<a id="1042" class="Symbol">;</a> <a id="1044" href="Algebra.Consequences.Setoid.html#1790" class="Function">identity∧middleFour⇒assoc</a>
<a id="1072" class="Symbol">;</a> <a id="1074" href="Algebra.Consequences.Setoid.html#2212" class="Function">identity∧middleFour⇒comm</a>
<a id="1101" class="Symbol">;</a> <a id="1103" href="Algebra.Consequences.Setoid.html#10215" class="Function">assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ</a>
<a id="1133" class="Symbol">;</a> <a id="1135" href="Algebra.Consequences.Setoid.html#10957" class="Function">assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ</a>
<a id="1165" class="Symbol">;</a> <a id="1167" href="Algebra.Consequences.Setoid.html#6752" class="Function">assoc∧id∧invʳ⇒invˡ-unique</a>
<a id="1195" class="Symbol">;</a> <a id="1197" href="Algebra.Consequences.Setoid.html#7239" class="Function">assoc∧id∧invˡ⇒invʳ-unique</a>
<a id="1225" class="Symbol">;</a> <a id="1227" href="Algebra.Consequences.Setoid.html#7938" class="Function">comm∧distrˡ⇒distrʳ</a>
<a id="1248" class="Symbol">;</a> <a id="1250" href="Algebra.Consequences.Setoid.html#8225" class="Function">comm∧distrʳ⇒distrˡ</a>
<a id="1271" class="Symbol">;</a> <a id="1273" href="Algebra.Consequences.Setoid.html#8789" class="Function">comm⇒sym[distribˡ]</a>
<a id="1294" class="Symbol">;</a> <a id="1296" href="Algebra.Consequences.Setoid.html#11932" class="Function">subst∧comm⇒sym</a>
<a id="1313" class="Symbol">;</a> <a id="1315" href="Algebra.Consequences.Setoid.html#12022" class="Function">wlog</a>
<a id="1322" class="Symbol">;</a> <a id="1324" href="Algebra.Consequences.Base.html#653" class="Function">sel⇒idem</a>
<a id="1333" class="Comment">-- plus all the deprecated versions of the above</a>
<a id="1384" class="Symbol">;</a> <a id="1386" href="Algebra.Consequences.Setoid.html#12455" class="Function">comm+assoc⇒middleFour</a>
<a id="1410" class="Symbol">;</a> <a id="1412" href="Algebra.Consequences.Setoid.html#12647" class="Function">identity+middleFour⇒assoc</a>
<a id="1440" class="Symbol">;</a> <a id="1442" href="Algebra.Consequences.Setoid.html#12859" class="Function">identity+middleFour⇒comm</a>
<a id="1469" class="Symbol">;</a> <a id="1471" href="Algebra.Consequences.Setoid.html#15494" class="Function">assoc+distribʳ+idʳ+invʳ⇒zeˡ</a>
<a id="1501" class="Symbol">;</a> <a id="1503" href="Algebra.Consequences.Setoid.html#15716" class="Function">assoc+distribˡ+idʳ+invʳ⇒zeʳ</a>
<a id="1533" class="Symbol">;</a> <a id="1535" href="Algebra.Consequences.Setoid.html#15938" class="Function">assoc+id+invʳ⇒invˡ-unique</a>
<a id="1563" class="Symbol">;</a> <a id="1565" href="Algebra.Consequences.Setoid.html#16150" class="Function">assoc+id+invˡ⇒invʳ-unique</a>
<a id="1593" class="Symbol">;</a> <a id="1595" href="Algebra.Consequences.Setoid.html#15140" class="Function">comm+distrˡ⇒distrʳ</a>
<a id="1616" class="Symbol">;</a> <a id="1618" href="Algebra.Consequences.Setoid.html#15317" class="Function">comm+distrʳ⇒distrˡ</a>
<a id="1639" class="Symbol">;</a> <a id="1641" href="Algebra.Consequences.Setoid.html#16362" class="Function">subst+comm⇒sym</a>
<a id="1658" class="Symbol">)</a>
<a id="1661" class="Comment">------------------------------------------------------------------------</a>
<a id="1734" class="Comment">-- Group-like structures</a>
<a id="1760" class="Keyword">module</a> <a id="1767" href="Algebra.Consequences.Propositional.html#1767" class="Module">_</a> <a id="1769" class="Symbol">{</a><a id="1770" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="1774" href="Algebra.Consequences.Propositional.html#1774" class="Bound Operator">_⁻¹</a> <a id="1778" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a><a id="1779" class="Symbol">}</a> <a id="1781" class="Keyword">where</a>
<a id="1790" href="Algebra.Consequences.Propositional.html#1790" class="Function">assoc∧id∧invʳ⇒invˡ-unique</a> <a id="1816" class="Symbol">:</a> <a id="1818" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="1830" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="1834" class="Symbol">→</a> <a id="1836" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="1845" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a> <a id="1847" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="1851" class="Symbol">→</a>
<a id="1883" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="1896" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a> <a id="1898" href="Algebra.Consequences.Propositional.html#1774" class="Bound Operator">_⁻¹</a> <a id="1902" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="1906" class="Symbol">→</a>
<a id="1938" class="Symbol">∀</a> <a id="1940" href="Algebra.Consequences.Propositional.html#1940" class="Bound">x</a> <a id="1942" href="Algebra.Consequences.Propositional.html#1942" class="Bound">y</a> <a id="1944" class="Symbol">→</a> <a id="1946" class="Symbol">(</a><a id="1947" href="Algebra.Consequences.Propositional.html#1940" class="Bound">x</a> <a id="1949" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">∙</a> <a id="1951" href="Algebra.Consequences.Propositional.html#1942" class="Bound">y</a><a id="1952" class="Symbol">)</a> <a id="1954" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1956" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a> <a id="1958" class="Symbol">→</a> <a id="1960" href="Algebra.Consequences.Propositional.html#1940" class="Bound">x</a> <a id="1962" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1964" class="Symbol">(</a><a id="1965" href="Algebra.Consequences.Propositional.html#1942" class="Bound">y</a> <a id="1967" href="Algebra.Consequences.Propositional.html#1774" class="Bound Operator">⁻¹</a><a id="1969" class="Symbol">)</a>
<a id="1973" href="Algebra.Consequences.Propositional.html#1790" class="Function">assoc∧id∧invʳ⇒invˡ-unique</a> <a id="1999" class="Symbol">=</a> <a id="2001" href="Algebra.Consequences.Setoid.html#6752" class="Function">Base.assoc∧id∧invʳ⇒invˡ-unique</a> <a id="2032" class="Symbol">(</a><a id="2033" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="2039" class="Symbol">_)</a>
<a id="2045" href="Algebra.Consequences.Propositional.html#2045" class="Function">assoc∧id∧invˡ⇒invʳ-unique</a> <a id="2071" class="Symbol">:</a> <a id="2073" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="2085" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="2089" class="Symbol">→</a> <a id="2091" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="2100" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a> <a id="2102" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="2106" class="Symbol">→</a>
<a id="2138" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="2150" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a> <a id="2152" href="Algebra.Consequences.Propositional.html#1774" class="Bound Operator">_⁻¹</a> <a id="2156" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">_∙_</a> <a id="2160" class="Symbol">→</a>
<a id="2192" class="Symbol">∀</a> <a id="2194" href="Algebra.Consequences.Propositional.html#2194" class="Bound">x</a> <a id="2196" href="Algebra.Consequences.Propositional.html#2196" class="Bound">y</a> <a id="2198" class="Symbol">→</a> <a id="2200" class="Symbol">(</a><a id="2201" href="Algebra.Consequences.Propositional.html#2194" class="Bound">x</a> <a id="2203" href="Algebra.Consequences.Propositional.html#1770" class="Bound Operator">∙</a> <a id="2205" href="Algebra.Consequences.Propositional.html#2196" class="Bound">y</a><a id="2206" class="Symbol">)</a> <a id="2208" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2210" href="Algebra.Consequences.Propositional.html#1778" class="Bound">ε</a> <a id="2212" class="Symbol">→</a> <a id="2214" href="Algebra.Consequences.Propositional.html#2196" class="Bound">y</a> <a id="2216" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2218" class="Symbol">(</a><a id="2219" href="Algebra.Consequences.Propositional.html#2194" class="Bound">x</a> <a id="2221" href="Algebra.Consequences.Propositional.html#1774" class="Bound Operator">⁻¹</a><a id="2223" class="Symbol">)</a>
<a id="2227" href="Algebra.Consequences.Propositional.html#2045" class="Function">assoc∧id∧invˡ⇒invʳ-unique</a> <a id="2253" class="Symbol">=</a> <a id="2255" href="Algebra.Consequences.Setoid.html#7239" class="Function">Base.assoc∧id∧invˡ⇒invʳ-unique</a> <a id="2286" class="Symbol">(</a><a id="2287" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="2293" class="Symbol">_)</a>
<a id="2297" class="Comment">------------------------------------------------------------------------</a>
<a id="2370" class="Comment">-- Ring-like structures</a>
<a id="2395" class="Keyword">module</a> <a id="2402" href="Algebra.Consequences.Propositional.html#2402" class="Module">_</a> <a id="2404" class="Symbol">{</a><a id="2405" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2409" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a> <a id="2413" href="Algebra.Consequences.Propositional.html#2413" class="Bound Operator">-_</a> <a id="2416" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a><a id="2418" class="Symbol">}</a> <a id="2420" class="Keyword">where</a>
<a id="2429" href="Algebra.Consequences.Propositional.html#2429" class="Function">assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ</a> <a id="2457" class="Symbol">:</a> <a id="2459" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="2471" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2475" class="Symbol">→</a> <a id="2477" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a> <a id="2481" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="2498" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2502" class="Symbol">→</a>
<a id="2536" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="2550" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a> <a id="2553" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2557" class="Symbol">→</a> <a id="2559" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="2572" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a> <a id="2575" href="Algebra.Consequences.Propositional.html#2413" class="Bound Operator">-_</a> <a id="2578" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2582" class="Symbol">→</a>
<a id="2616" href="Algebra.Definitions.html#1942" class="Function">LeftZero</a> <a id="2625" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a> <a id="2628" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a>
<a id="2634" href="Algebra.Consequences.Propositional.html#2429" class="Function">assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ</a> <a id="2662" class="Symbol">=</a>
<a id="2668" href="Algebra.Consequences.Setoid.html#10215" class="Function">Base.assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ</a> <a id="2701" class="Symbol">(</a><a id="2702" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="2708" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a><a id="2711" class="Symbol">)</a> <a id="2713" class="Symbol">(</a><a id="2714" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="2720" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a><a id="2723" class="Symbol">)</a>
<a id="2728" href="Algebra.Consequences.Propositional.html#2728" class="Function">assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ</a> <a id="2756" class="Symbol">:</a> <a id="2758" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="2770" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2774" class="Symbol">→</a> <a id="2776" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a> <a id="2780" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="2797" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2801" class="Symbol">→</a>
<a id="2835" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="2849" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a> <a id="2852" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2856" class="Symbol">→</a> <a id="2858" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="2871" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a> <a id="2874" href="Algebra.Consequences.Propositional.html#2413" class="Bound Operator">-_</a> <a id="2877" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a> <a id="2881" class="Symbol">→</a>
<a id="2915" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="2925" href="Algebra.Consequences.Propositional.html#2416" class="Bound">0#</a> <a id="2928" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a>
<a id="2934" href="Algebra.Consequences.Propositional.html#2728" class="Function">assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ</a> <a id="2962" class="Symbol">=</a>
<a id="2968" href="Algebra.Consequences.Setoid.html#10957" class="Function">Base.assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ</a> <a id="3001" class="Symbol">(</a><a id="3002" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3008" href="Algebra.Consequences.Propositional.html#2405" class="Bound Operator">_+_</a><a id="3011" class="Symbol">)</a> <a id="3013" class="Symbol">(</a><a id="3014" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3020" href="Algebra.Consequences.Propositional.html#2409" class="Bound Operator">_*_</a><a id="3023" class="Symbol">)</a>
<a id="3026" class="Comment">------------------------------------------------------------------------</a>
<a id="3099" class="Comment">-- Bisemigroup-like structures</a>
<a id="3131" class="Keyword">module</a> <a id="3138" href="Algebra.Consequences.Propositional.html#3138" class="Module">_</a> <a id="3140" class="Symbol">{</a><a id="3141" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">_∙_</a> <a id="3145" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">_◦_</a> <a id="3149" class="Symbol">:</a> <a id="3151" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3155" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="3156" class="Symbol">}</a> <a id="3158" class="Symbol">(</a><a id="3159" href="Algebra.Consequences.Propositional.html#3159" class="Bound">∙-comm</a> <a id="3166" class="Symbol">:</a> <a id="3168" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="3180" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">_∙_</a><a id="3183" class="Symbol">)</a> <a id="3185" class="Keyword">where</a>
<a id="3194" href="Algebra.Consequences.Propositional.html#3194" class="Function">comm∧distrˡ⇒distrʳ</a> <a id="3213" class="Symbol">:</a> <a id="3215" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">_∙_</a> <a id="3219" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="3236" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">_◦_</a> <a id="3240" class="Symbol">→</a> <a id="3242" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">_∙_</a> <a id="3246" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="3263" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">_◦_</a>
<a id="3269" href="Algebra.Consequences.Propositional.html#3194" class="Function">comm∧distrˡ⇒distrʳ</a> <a id="3288" class="Symbol">=</a> <a id="3290" href="Algebra.Consequences.Setoid.html#15140" class="Function">Base.comm+distrˡ⇒distrʳ</a> <a id="3314" class="Symbol">(</a><a id="3315" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3321" class="Symbol">_)</a> <a id="3324" href="Algebra.Consequences.Propositional.html#3159" class="Bound">∙-comm</a>
<a id="3334" href="Algebra.Consequences.Propositional.html#3334" class="Function">comm∧distrʳ⇒distrˡ</a> <a id="3353" class="Symbol">:</a> <a id="3355" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">_∙_</a> <a id="3359" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="3376" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">_◦_</a> <a id="3380" class="Symbol">→</a> <a id="3382" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">_∙_</a> <a id="3386" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="3403" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">_◦_</a>
<a id="3409" href="Algebra.Consequences.Propositional.html#3334" class="Function">comm∧distrʳ⇒distrˡ</a> <a id="3428" class="Symbol">=</a> <a id="3430" href="Algebra.Consequences.Setoid.html#8225" class="Function">Base.comm∧distrʳ⇒distrˡ</a> <a id="3454" class="Symbol">(</a><a id="3455" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3461" class="Symbol">_)</a> <a id="3464" href="Algebra.Consequences.Propositional.html#3159" class="Bound">∙-comm</a>
<a id="3474" href="Algebra.Consequences.Propositional.html#3474" class="Function">comm⇒sym[distribˡ]</a> <a id="3493" class="Symbol">:</a> <a id="3495" class="Symbol">∀</a> <a id="3497" href="Algebra.Consequences.Propositional.html#3497" class="Bound">x</a> <a id="3499" class="Symbol">→</a> <a id="3501" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="3511" class="Symbol">(λ</a> <a id="3514" href="Algebra.Consequences.Propositional.html#3514" class="Bound">y</a> <a id="3516" href="Algebra.Consequences.Propositional.html#3516" class="Bound">z</a> <a id="3518" class="Symbol">→</a> <a id="3520" class="Symbol">(</a><a id="3521" href="Algebra.Consequences.Propositional.html#3497" class="Bound">x</a> <a id="3523" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">◦</a> <a id="3525" class="Symbol">(</a><a id="3526" href="Algebra.Consequences.Propositional.html#3514" class="Bound">y</a> <a id="3528" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">∙</a> <a id="3530" href="Algebra.Consequences.Propositional.html#3516" class="Bound">z</a><a id="3531" class="Symbol">))</a> <a id="3534" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3536" class="Symbol">((</a><a id="3538" href="Algebra.Consequences.Propositional.html#3497" class="Bound">x</a> <a id="3540" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">◦</a> <a id="3542" href="Algebra.Consequences.Propositional.html#3514" class="Bound">y</a><a id="3543" class="Symbol">)</a> <a id="3545" href="Algebra.Consequences.Propositional.html#3141" class="Bound Operator">∙</a> <a id="3547" class="Symbol">(</a><a id="3548" href="Algebra.Consequences.Propositional.html#3497" class="Bound">x</a> <a id="3550" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">◦</a> <a id="3552" href="Algebra.Consequences.Propositional.html#3516" class="Bound">z</a><a id="3553" class="Symbol">)))</a>
<a id="3559" href="Algebra.Consequences.Propositional.html#3474" class="Function">comm⇒sym[distribˡ]</a> <a id="3578" class="Symbol">=</a> <a id="3580" href="Algebra.Consequences.Setoid.html#8789" class="Function">Base.comm⇒sym[distribˡ]</a> <a id="3604" class="Symbol">(</a><a id="3605" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3611" href="Algebra.Consequences.Propositional.html#3145" class="Bound Operator">_◦_</a><a id="3614" class="Symbol">)</a> <a id="3616" href="Algebra.Consequences.Propositional.html#3159" class="Bound">∙-comm</a>
<a id="3624" class="Comment">------------------------------------------------------------------------</a>
<a id="3697" class="Comment">-- Selectivity</a>
<a id="3713" class="Keyword">module</a> <a id="3720" href="Algebra.Consequences.Propositional.html#3720" class="Module">_</a> <a id="3722" class="Symbol">{</a><a id="3723" href="Algebra.Consequences.Propositional.html#3723" class="Bound Operator">_∙_</a> <a id="3727" class="Symbol">:</a> <a id="3729" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3733" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="3734" class="Symbol">}</a> <a id="3736" class="Keyword">where</a>
<a id="3745" href="Algebra.Consequences.Propositional.html#3745" class="Function">sel⇒idem</a> <a id="3754" class="Symbol">:</a> <a id="3756" href="Algebra.Definitions.html#3841" class="Function">Selective</a> <a id="3766" href="Algebra.Consequences.Propositional.html#3723" class="Bound Operator">_∙_</a> <a id="3770" class="Symbol">→</a> <a id="3772" href="Algebra.Definitions.html#3706" class="Function">Idempotent</a> <a id="3783" href="Algebra.Consequences.Propositional.html#3723" class="Bound Operator">_∙_</a>
<a id="3789" href="Algebra.Consequences.Propositional.html#3745" class="Function">sel⇒idem</a> <a id="3798" class="Symbol">=</a> <a id="3800" href="Algebra.Consequences.Base.html#653" class="Function">Base.sel⇒idem</a> <a id="3814" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="3819" class="Comment">------------------------------------------------------------------------</a>
<a id="3892" class="Comment">-- Middle-Four Exchange</a>
<a id="3917" class="Keyword">module</a> <a id="3924" href="Algebra.Consequences.Propositional.html#3924" class="Module">_</a> <a id="3926" class="Symbol">{</a><a id="3927" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="3931" class="Symbol">:</a> <a id="3933" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3937" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="3938" class="Symbol">}</a> <a id="3940" class="Keyword">where</a>
<a id="3949" href="Algebra.Consequences.Propositional.html#3949" class="Function">comm∧assoc⇒middleFour</a> <a id="3971" class="Symbol">:</a> <a id="3973" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="3985" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="3989" class="Symbol">→</a> <a id="3991" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="4003" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="4007" class="Symbol">→</a>
<a id="4035" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="4039" href="Algebra.Definitions.html#3506" class="Function Operator">MiddleFourExchange</a> <a id="4058" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a>
<a id="4064" href="Algebra.Consequences.Propositional.html#3949" class="Function">comm∧assoc⇒middleFour</a> <a id="4086" class="Symbol">=</a> <a id="4088" href="Algebra.Consequences.Setoid.html#1335" class="Function">Base.comm∧assoc⇒middleFour</a> <a id="4115" class="Symbol">(</a><a id="4116" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="4122" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a><a id="4125" class="Symbol">)</a>
<a id="4130" href="Algebra.Consequences.Propositional.html#4130" class="Function">identity∧middleFour⇒assoc</a> <a id="4156" class="Symbol">:</a> <a id="4158" class="Symbol">{</a><a id="4159" href="Algebra.Consequences.Propositional.html#4159" class="Bound">e</a> <a id="4161" class="Symbol">:</a> <a id="4163" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="4164" class="Symbol">}</a> <a id="4166" class="Symbol">→</a> <a id="4168" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="4177" href="Algebra.Consequences.Propositional.html#4159" class="Bound">e</a> <a id="4179" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="4183" class="Symbol">→</a>
<a id="4215" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="4219" href="Algebra.Definitions.html#3506" class="Function Operator">MiddleFourExchange</a> <a id="4238" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="4242" class="Symbol">→</a>
<a id="4274" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="4286" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a>
<a id="4292" href="Algebra.Consequences.Propositional.html#4130" class="Function">identity∧middleFour⇒assoc</a> <a id="4318" class="Symbol">=</a> <a id="4320" href="Algebra.Consequences.Setoid.html#1790" class="Function">Base.identity∧middleFour⇒assoc</a> <a id="4351" class="Symbol">(</a><a id="4352" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="4358" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a><a id="4361" class="Symbol">)</a>
<a id="4366" href="Algebra.Consequences.Propositional.html#4366" class="Function">identity∧middleFour⇒comm</a> <a id="4391" class="Symbol">:</a> <a id="4393" class="Symbol">{</a><a id="4394" href="Algebra.Consequences.Propositional.html#4394" class="Bound Operator">_+_</a> <a id="4398" class="Symbol">:</a> <a id="4400" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="4404" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="4405" class="Symbol">}</a> <a id="4407" class="Symbol">{</a><a id="4408" href="Algebra.Consequences.Propositional.html#4408" class="Bound">e</a> <a id="4410" class="Symbol">:</a> <a id="4412" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a><a id="4413" class="Symbol">}</a> <a id="4415" class="Symbol">→</a> <a id="4417" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="4426" href="Algebra.Consequences.Propositional.html#4408" class="Bound">e</a> <a id="4428" href="Algebra.Consequences.Propositional.html#4394" class="Bound Operator">_+_</a> <a id="4432" class="Symbol">→</a>
<a id="4463" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a> <a id="4467" href="Algebra.Definitions.html#3506" class="Function Operator">MiddleFourExchange</a> <a id="4486" href="Algebra.Consequences.Propositional.html#4394" class="Bound Operator">_+_</a> <a id="4490" class="Symbol">→</a>
<a id="4521" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="4533" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a>
<a id="4539" href="Algebra.Consequences.Propositional.html#4366" class="Function">identity∧middleFour⇒comm</a> <a id="4564" class="Symbol">=</a> <a id="4566" href="Algebra.Consequences.Setoid.html#2212" class="Function">Base.identity∧middleFour⇒comm</a> <a id="4596" class="Symbol">(</a><a id="4597" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="4603" href="Algebra.Consequences.Propositional.html#3927" class="Bound Operator">_∙_</a><a id="4606" class="Symbol">)</a>
<a id="4609" class="Comment">------------------------------------------------------------------------</a>
<a id="4682" class="Comment">-- Without Loss of Generality</a>
<a id="4713" class="Keyword">module</a> <a id="4720" href="Algebra.Consequences.Propositional.html#4720" class="Module">_</a> <a id="4722" class="Symbol">{</a><a id="4723" href="Algebra.Consequences.Propositional.html#4723" class="Bound">p</a><a id="4724" class="Symbol">}</a> <a id="4726" class="Symbol">{</a><a id="4727" href="Algebra.Consequences.Propositional.html#4727" class="Bound">P</a> <a id="4729" class="Symbol">:</a> <a id="4731" href="Relation.Unary.html#1232" class="Function">Pred</a> <a id="4736" href="Algebra.Consequences.Propositional.html#403" class="Bound">A</a> <a id="4738" href="Algebra.Consequences.Propositional.html#4723" class="Bound">p</a><a id="4739" class="Symbol">}</a> <a id="4741" class="Keyword">where</a>
<a id="4750" href="Algebra.Consequences.Propositional.html#4750" class="Function">subst∧comm⇒sym</a> <a id="4765" class="Symbol">:</a> <a id="4767" class="Symbol">∀</a> <a id="4769" class="Symbol">{</a><a id="4770" href="Algebra.Consequences.Propositional.html#4770" class="Bound">f</a><a id="4771" class="Symbol">}</a> <a id="4773" class="Symbol">(</a><a id="4774" href="Algebra.Consequences.Propositional.html#4774" class="Bound">f-comm</a> <a id="4781" class="Symbol">:</a> <a id="4783" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="4795" href="Algebra.Consequences.Propositional.html#4770" class="Bound">f</a><a id="4796" class="Symbol">)</a> <a id="4798" class="Symbol">→</a>
<a id="4819" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="4829" class="Symbol">(λ</a> <a id="4832" href="Algebra.Consequences.Propositional.html#4832" class="Bound">a</a> <a id="4834" href="Algebra.Consequences.Propositional.html#4834" class="Bound">b</a> <a id="4836" class="Symbol">→</a> <a id="4838" href="Algebra.Consequences.Propositional.html#4727" class="Bound">P</a> <a id="4840" class="Symbol">(</a><a id="4841" href="Algebra.Consequences.Propositional.html#4770" class="Bound">f</a> <a id="4843" href="Algebra.Consequences.Propositional.html#4832" class="Bound">a</a> <a id="4845" href="Algebra.Consequences.Propositional.html#4834" class="Bound">b</a><a id="4846" class="Symbol">))</a>
<a id="4851" href="Algebra.Consequences.Propositional.html#4750" class="Function">subst∧comm⇒sym</a> <a id="4866" class="Symbol">=</a> <a id="4868" href="Algebra.Consequences.Setoid.html#11932" class="Function">Base.subst∧comm⇒sym</a> <a id="4888" class="Symbol">{</a><a id="4889" class="Argument">P</a> <a id="4891" class="Symbol">=</a> <a id="4893" href="Algebra.Consequences.Propositional.html#4727" class="Bound">P</a><a id="4894" class="Symbol">}</a> <a id="4896" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a>
<a id="4905" href="Algebra.Consequences.Propositional.html#4905" class="Function">wlog</a> <a id="4910" class="Symbol">:</a> <a id="4912" class="Symbol">∀</a> <a id="4914" class="Symbol">{</a><a id="4915" href="Algebra.Consequences.Propositional.html#4915" class="Bound">f</a><a id="4916" class="Symbol">}</a> <a id="4918" class="Symbol">(</a><a id="4919" href="Algebra.Consequences.Propositional.html#4919" class="Bound">f-comm</a> <a id="4926" class="Symbol">:</a> <a id="4928" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="4940" href="Algebra.Consequences.Propositional.html#4915" class="Bound">f</a><a id="4941" class="Symbol">)</a> <a id="4943" class="Symbol">→</a>
<a id="4954" class="Symbol">∀</a> <a id="4956" class="Symbol">{</a><a id="4957" href="Algebra.Consequences.Propositional.html#4957" class="Bound">r</a><a id="4958" class="Symbol">}</a> <a id="4960" class="Symbol">{</a><a id="4961" href="Algebra.Consequences.Propositional.html#4961" class="Bound Operator">_R_</a> <a id="4965" class="Symbol">:</a> <a id="4967" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="4971" class="Symbol">_</a> <a id="4973" href="Algebra.Consequences.Propositional.html#4957" class="Bound">r</a><a id="4974" class="Symbol">}</a> <a id="4976" class="Symbol">→</a> <a id="4978" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="4984" href="Algebra.Consequences.Propositional.html#4961" class="Bound Operator">_R_</a> <a id="4988" class="Symbol">→</a>
<a id="4999" class="Symbol">(∀</a> <a id="5002" href="Algebra.Consequences.Propositional.html#5002" class="Bound">a</a> <a id="5004" href="Algebra.Consequences.Propositional.html#5004" class="Bound">b</a> <a id="5006" class="Symbol">→</a> <a id="5008" href="Algebra.Consequences.Propositional.html#5002" class="Bound">a</a> <a id="5010" href="Algebra.Consequences.Propositional.html#4961" class="Bound Operator">R</a> <a id="5012" href="Algebra.Consequences.Propositional.html#5004" class="Bound">b</a> <a id="5014" class="Symbol">→</a> <a id="5016" href="Algebra.Consequences.Propositional.html#4727" class="Bound">P</a> <a id="5018" class="Symbol">(</a><a id="5019" href="Algebra.Consequences.Propositional.html#4915" class="Bound">f</a> <a id="5021" href="Algebra.Consequences.Propositional.html#5002" class="Bound">a</a> <a id="5023" href="Algebra.Consequences.Propositional.html#5004" class="Bound">b</a><a id="5024" class="Symbol">))</a> <a id="5027" class="Symbol">→</a>
<a id="5038" class="Symbol">∀</a> <a id="5040" href="Algebra.Consequences.Propositional.html#5040" class="Bound">a</a> <a id="5042" href="Algebra.Consequences.Propositional.html#5042" class="Bound">b</a> <a id="5044" class="Symbol">→</a> <a id="5046" href="Algebra.Consequences.Propositional.html#4727" class="Bound">P</a> <a id="5048" class="Symbol">(</a><a id="5049" href="Algebra.Consequences.Propositional.html#4915" class="Bound">f</a> <a id="5051" href="Algebra.Consequences.Propositional.html#5040" class="Bound">a</a> <a id="5053" href="Algebra.Consequences.Propositional.html#5042" class="Bound">b</a><a id="5054" class="Symbol">)</a>
<a id="5058" href="Algebra.Consequences.Propositional.html#4905" class="Function">wlog</a> <a id="5063" class="Symbol">=</a> <a id="5065" href="Algebra.Consequences.Setoid.html#12022" class="Function">Base.wlog</a> <a id="5075" class="Symbol">{</a><a id="5076" class="Argument">P</a> <a id="5078" class="Symbol">=</a> <a id="5080" href="Algebra.Consequences.Propositional.html#4727" class="Bound">P</a><a id="5081" class="Symbol">}</a> <a id="5083" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a>
<a id="5091" class="Comment">------------------------------------------------------------------------</a>
<a id="5164" class="Comment">-- DEPRECATED NAMES</a>
<a id="5184" class="Comment">------------------------------------------------------------------------</a>
<a id="5257" class="Comment">-- Please use the new names as continuing support for the old names is</a>
<a id="5328" class="Comment">-- not guaranteed.</a>
<a id="5348" class="Comment">-- Version 2.0</a>
<a id="comm+assoc⇒middleFour"></a><a id="5364" href="Algebra.Consequences.Propositional.html#5364" class="Function">comm+assoc⇒middleFour</a> <a id="5386" class="Symbol">=</a> <a id="5388" href="Algebra.Consequences.Propositional.html#3949" class="Function">comm∧assoc⇒middleFour</a>
<a id="5410" class="Symbol">{-#</a> <a id="5414" class="Keyword">WARNING_ON_USAGE</a> <a id="5431" class="Pragma">comm+assoc⇒middleFour</a>
<a id="5453" class="String">&quot;Warning: comm+assoc⇒middleFour was deprecated in v2.0.
Please use comm∧assoc⇒middleFour instead.&quot;</a>
<a id="5552" class="Symbol">#-}</a>
<a id="identity+middleFour⇒assoc"></a><a id="5556" href="Algebra.Consequences.Propositional.html#5556" class="Function">identity+middleFour⇒assoc</a> <a id="5582" class="Symbol">=</a> <a id="5584" href="Algebra.Consequences.Propositional.html#4130" class="Function">identity∧middleFour⇒assoc</a>
<a id="5610" class="Symbol">{-#</a> <a id="5614" class="Keyword">WARNING_ON_USAGE</a> <a id="5631" class="Pragma">identity+middleFour⇒assoc</a>
<a id="5657" class="String">&quot;Warning: identity+middleFour⇒assoc was deprecated in v2.0.
Please use identity∧middleFour⇒assoc instead.&quot;</a>
<a id="5764" class="Symbol">#-}</a>
<a id="identity+middleFour⇒comm"></a><a id="5768" href="Algebra.Consequences.Propositional.html#5768" class="Function">identity+middleFour⇒comm</a> <a id="5793" class="Symbol">=</a> <a id="5795" href="Algebra.Consequences.Propositional.html#4366" class="Function">identity∧middleFour⇒comm</a>
<a id="5820" class="Symbol">{-#</a> <a id="5824" class="Keyword">WARNING_ON_USAGE</a> <a id="5841" class="Pragma">identity+middleFour⇒comm</a>
<a id="5866" class="String">&quot;Warning: identity+middleFour⇒comm was deprecated in v2.0.
Please use identity∧middleFour⇒comm instead.&quot;</a>
<a id="5971" class="Symbol">#-}</a>
<a id="comm+distrˡ⇒distrʳ"></a><a id="5975" href="Algebra.Consequences.Propositional.html#5975" class="Function">comm+distrˡ⇒distrʳ</a> <a id="5994" class="Symbol">=</a> <a id="5996" href="Algebra.Consequences.Propositional.html#3194" class="Function">comm∧distrˡ⇒distrʳ</a>
<a id="6015" class="Symbol">{-#</a> <a id="6019" class="Keyword">WARNING_ON_USAGE</a> <a id="6036" class="Pragma">comm+distrˡ⇒distrʳ</a>
<a id="6055" class="String">&quot;Warning: comm+distrˡ⇒distrʳ was deprecated in v2.0.
Please use comm∧distrˡ⇒distrʳ instead.&quot;</a>
<a id="6148" class="Symbol">#-}</a>
<a id="comm+distrʳ⇒distrˡ"></a><a id="6152" href="Algebra.Consequences.Propositional.html#6152" class="Function">comm+distrʳ⇒distrˡ</a> <a id="6171" class="Symbol">=</a> <a id="6173" href="Algebra.Consequences.Propositional.html#3334" class="Function">comm∧distrʳ⇒distrˡ</a>
<a id="6192" class="Symbol">{-#</a> <a id="6196" class="Keyword">WARNING_ON_USAGE</a> <a id="6213" class="Pragma">comm+distrʳ⇒distrˡ</a>
<a id="6232" class="String">&quot;Warning: comm+distrʳ⇒distrˡ was deprecated in v2.0.
Please use comm∧distrʳ⇒distrˡ instead.&quot;</a>
<a id="6325" class="Symbol">#-}</a>
<a id="assoc+distribʳ+idʳ+invʳ⇒zeˡ"></a><a id="6329" href="Algebra.Consequences.Propositional.html#6329" class="Function">assoc+distribʳ+idʳ+invʳ⇒zeˡ</a> <a id="6357" class="Symbol">=</a> <a id="6359" href="Algebra.Consequences.Propositional.html#2429" class="Function">assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ</a>
<a id="6387" class="Symbol">{-#</a> <a id="6391" class="Keyword">WARNING_ON_USAGE</a> <a id="6408" class="Pragma">assoc+distribʳ+idʳ+invʳ⇒zeˡ</a>
<a id="6436" class="String">&quot;Warning: assoc+distribʳ+idʳ+invʳ⇒zeˡ was deprecated in v2.0.
Please use assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ instead.&quot;</a>
<a id="6547" class="Symbol">#-}</a>
<a id="assoc+distribˡ+idʳ+invʳ⇒zeʳ"></a><a id="6551" href="Algebra.Consequences.Propositional.html#6551" class="Function">assoc+distribˡ+idʳ+invʳ⇒zeʳ</a> <a id="6579" class="Symbol">=</a> <a id="6581" href="Algebra.Consequences.Propositional.html#2728" class="Function">assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ</a>
<a id="6609" class="Symbol">{-#</a> <a id="6613" class="Keyword">WARNING_ON_USAGE</a> <a id="6630" class="Pragma">assoc+distribˡ+idʳ+invʳ⇒zeʳ</a>
<a id="6658" class="String">&quot;Warning: assoc+distribˡ+idʳ+invʳ⇒zeʳ was deprecated in v2.0.
Please use assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ instead.&quot;</a>
<a id="6769" class="Symbol">#-}</a>
<a id="assoc+id+invʳ⇒invˡ-unique"></a><a id="6773" href="Algebra.Consequences.Propositional.html#6773" class="Function">assoc+id+invʳ⇒invˡ-unique</a> <a id="6799" class="Symbol">=</a> <a id="6801" href="Algebra.Consequences.Propositional.html#1790" class="Function">assoc∧id∧invʳ⇒invˡ-unique</a>
<a id="6827" class="Symbol">{-#</a> <a id="6831" class="Keyword">WARNING_ON_USAGE</a> <a id="6848" class="Pragma">assoc+id+invʳ⇒invˡ-unique</a>
<a id="6874" class="String">&quot;Warning: assoc+id+invʳ⇒invˡ-unique was deprecated in v2.0.
Please use assoc∧id∧invʳ⇒invˡ-unique instead.&quot;</a>
<a id="6981" class="Symbol">#-}</a>
<a id="assoc+id+invˡ⇒invʳ-unique"></a><a id="6985" href="Algebra.Consequences.Propositional.html#6985" class="Function">assoc+id+invˡ⇒invʳ-unique</a> <a id="7011" class="Symbol">=</a> <a id="7013" href="Algebra.Consequences.Propositional.html#2045" class="Function">assoc∧id∧invˡ⇒invʳ-unique</a>
<a id="7039" class="Symbol">{-#</a> <a id="7043" class="Keyword">WARNING_ON_USAGE</a> <a id="7060" class="Pragma">assoc+id+invˡ⇒invʳ-unique</a>
<a id="7086" class="String">&quot;Warning: assoc+id+invˡ⇒invʳ-unique was deprecated in v2.0.
Please use assoc∧id∧invˡ⇒invʳ-unique instead.&quot;</a>
<a id="7193" class="Symbol">#-}</a>
<a id="subst+comm⇒sym"></a><a id="7197" href="Algebra.Consequences.Propositional.html#7197" class="Function">subst+comm⇒sym</a> <a id="7212" class="Symbol">=</a> <a id="7214" href="Algebra.Consequences.Propositional.html#4750" class="Function">subst∧comm⇒sym</a>
<a id="7229" class="Symbol">{-#</a> <a id="7233" class="Keyword">WARNING_ON_USAGE</a> <a id="7250" class="Pragma">subst+comm⇒sym</a>
<a id="7265" class="String">&quot;Warning: subst+comm⇒sym was deprecated in v2.0.
Please use subst∧comm⇒sym instead.&quot;</a>
<a id="7350" class="Symbol">#-}</a>
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