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</pre>
<p>In a CCC free elgot algebras are automatically stable.</p>
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<a id="1232" class="Keyword">private</a>
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<a id="1971" href="Algebra.Elgot.Stable.html#1783" class="Field">IsStableFreeElgotAlgebraˡ.♯ˡ-law</a> <a id="2004" class="Symbol">(</a><a id="2005" href="Algebra.Elgot.Properties.html#1709" class="Function">free-isStableˡ</a> <a id="2020" class="Symbol">{</a><a id="2021" href="Algebra.Elgot.Properties.html#2021" class="Bound">Y</a><a id="2022" class="Symbol">}</a> <a id="2024" href="Algebra.Elgot.Properties.html#2024" class="Bound">fe</a><a id="2026" class="Symbol">)</a> <a id="2028" class="Symbol">{</a><a id="2029" href="Algebra.Elgot.Properties.html#2029" class="Bound">X</a><a id="2030" class="Symbol">}</a> <a id="2032" class="Symbol">{</a><a id="2033" href="Algebra.Elgot.Properties.html#2033" class="Bound">A</a><a id="2034" class="Symbol">}</a> <a id="2036" href="Algebra.Elgot.Properties.html#2036" class="Bound">f</a> <a id="2038" class="Symbol">=</a> <a id="2040" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="2044" class="Symbol">(</a><a id="2045" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="3807" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="3810" class="Symbol">(</a><a id="3811" href="Algebra.Elgot.Properties.html#3666" class="Bound">g</a> <a id="3813" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3815" class="Symbol">(</a><a id="3816" href="Algebra.Elgot.Properties.html#3757" class="Bound">h</a> <a id="3818" href="Algebra.Elgot.Properties.html#4683" class="Function Operator">#ʸ</a> <a id="3821" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3823" href="Categories.Category.Core.html#630" class="Function">id</a><a id="3825" class="Symbol">))</a> <a id="3828" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3831" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="3838" class="Symbol">(</a><a id="3839" href="Algebra.Elgot.Properties.html#3672" class="Bound">eq₂</a> <a id="3843" href="Algebra.Elgot.Properties.html#3757" class="Bound">h</a><a id="3844" class="Symbol">)</a> <a id="3846" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3854" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="3857" class="Symbol">(((</a><a id="3860" href="Algebra.Elgot.Properties.html#3666" class="Bound">g</a> <a id="3862" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3865" href="Categories.Category.Core.html#630" class="Function">id</a><a id="3867" class="Symbol">)</a> <a id="3869" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3871" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="3885" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3887" class="Symbol">(</a><a id="3888" href="Algebra.Elgot.Properties.html#3757" class="Bound">h</a> <a id="3890" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3892" href="Categories.Category.Core.html#630" class="Function">id</a><a id="3894" class="Symbol">))</a> <a id="3897" href="Algebra.Elgot.Properties.html#4602" class="Function Operator">#ᵇ</a><a id="3899" class="Symbol">)</a> <a id="3901" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3905" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="3912" class="Symbol">(</a><a id="3913" href="Algebra.Elgot.Properties.html#4619" class="Function">#ᵇ-resp-≈</a> <a id="3923" class="Symbol">(</a><a id="3924" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3930" class="Symbol">(</a><a id="3931" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="3937" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="3939" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="3948" href="Categories.Category.CartesianClosed.html#4217" class="Function">β′</a> <a id="3951" class="Symbol">(</a><a id="3952" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">elimʳ</a> <a id="3958" class="Symbol">(</a><a id="3959" href="Algebra.Elgot.Properties.html#1188" class="Function">⟨⟩-unique</a> <a id="3969" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a> <a id="3977" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a><a id="3984" class="Symbol">)))))</a> <a id="3990" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="3998" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4001" class="Symbol">(((</a><a id="4004" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="4010" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4013" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4015" class="Symbol">)</a> <a id="4017" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4019" class="Symbol">(</a><a id="4020" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4023" href="Algebra.Elgot.Properties.html#3666" class="Bound">g</a> <a id="4025" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4027" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4030" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4033" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4036" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4038" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4040" class="Symbol">)</a> <a id="4042" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4044" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4058" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4060" class="Symbol">(</a><a id="4061" href="Algebra.Elgot.Properties.html#3757" class="Bound">h</a> <a id="4063" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4065" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4067" class="Symbol">))</a> <a id="4070" href="Algebra.Elgot.Properties.html#4602" class="Function Operator">#ᵇ</a><a id="4072" class="Symbol">)</a> <a id="4074" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="4078" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="4085" class="Symbol">(</a><a id="4086" href="Algebra.Elgot.Properties.html#4619" class="Function">#ᵇ-resp-≈</a> <a id="4096" class="Symbol">(</a><a id="4097" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4105" class="Symbol">(</a><a id="4106" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="4112" class="Symbol">(</a><a id="4113" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4117" class="Symbol">(</a><a id="4118" href="Categories.Category.Distributive.Properties.html#4502" class="Function">distributeʳ⁻¹-natural</a> <a id="4140" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4143" class="Symbol">(</a><a id="4144" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4147" href="Algebra.Elgot.Properties.html#3666" class="Bound">g</a><a id="4148" class="Symbol">)</a> <a id="4150" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4152" class="Symbol">))</a> <a id="4155" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="4157" href="Categories.Category.Core.html#715" class="Function">assoc</a><a id="4162" class="Symbol">)))</a> <a id="4166" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="4174" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4177" class="Symbol">(((</a><a id="4180" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="4186" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4189" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4191" class="Symbol">)</a> <a id="4193" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4195" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4209" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4211" class="Symbol">((</a><a id="4213" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4216" href="Algebra.Elgot.Properties.html#3666" class="Bound">g</a> <a id="4218" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4221" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4223" class="Symbol">)</a> <a id="4225" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4227" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4229" class="Symbol">)</a> <a id="4231" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4233" class="Symbol">(</a><a id="4234" href="Algebra.Elgot.Properties.html#3757" class="Bound">h</a> <a id="4236" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4238" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4240" class="Symbol">))</a> <a id="4243" href="Algebra.Elgot.Properties.html#4602" class="Function Operator">#ᵇ</a><a id="4245" class="Symbol">)</a> <a id="4247" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4250" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="4257" class="Symbol">(</a><a id="4258" href="Algebra.Elgot.Properties.html#4619" class="Function">#ᵇ-resp-≈</a> <a id="4268" class="Symbol">(</a><a id="4269" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4277" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4285" class="Symbol">(</a><a id="4286" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a> <a id="4290" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="4292" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="4300" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="4305" href="Categories.Category.Core.html#1339" class="Function">identity²</a><a id="4314" class="Symbol">)))</a> <a id="4318" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4326" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4329" class="Symbol">(((</a><a id="4332" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="4338" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4341" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4343" class="Symbol">)</a> <a id="4345" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4347" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4361" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4363" class="Symbol">((</a><a id="4365" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4368" href="Algebra.Elgot.Properties.html#3666" class="Bound">g</a> <a id="4370" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4373" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4375" class="Symbol">)</a> <a id="4377" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4379" href="Algebra.Elgot.Properties.html#3757" class="Bound">h</a> <a id="4381" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4383" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4385" class="Symbol">))</a> <a id="4388" href="Algebra.Elgot.Properties.html#4602" class="Function Operator">#ᵇ</a><a id="4390" class="Symbol">)</a> <a id="4392" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a> <a id="4394" class="Symbol">})</a> <a id="4397" class="Symbol">(</a><a id="4398" href="Categories.Category.CartesianClosed.html#2454" class="Function">subst</a> <a id="4404" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="4406" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="4413" class="Symbol">(</a><a id="4414" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4418" href="Algebra.Elgot.Properties.html#3668" class="Bound">eq₁</a><a id="4421" class="Symbol">))</a> <a id="4424" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4430" href="Algebra.Elgot.Properties.html#1701" class="Field Operator">&lt;</a> <a id="4432" class="Symbol">(</a><a id="4433" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4436" href="Algebra.Elgot.Properties.html#3664" class="Bound">f</a><a id="4437" class="Symbol">)</a><a id="4438" href="Categories.FreeObjects.Free.html#880" class="Function Operator">*</a> <a id="4440" href="Algebra.Elgot.Properties.html#1701" class="Field Operator">&gt;</a> <a id="4442" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="4446" href="Categories.Category.CartesianClosed.html#4406" class="Function">η-exp′</a> <a id="4453" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="4459" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4462" class="Symbol">(</a><a id="4463" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="4469" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4471" class="Symbol">(</a><a id="4472" href="Algebra.Elgot.Properties.html#1701" class="Field Operator">&lt;</a> <a id="4474" class="Symbol">(</a><a id="4475" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="4478" href="Algebra.Elgot.Properties.html#3664" class="Bound">f</a><a id="4479" class="Symbol">)</a><a id="4480" href="Categories.FreeObjects.Free.html#880" class="Function Operator">*</a> <a id="4482" href="Algebra.Elgot.Properties.html#1701" class="Field Operator">&gt;</a> <a id="4484" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4486" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4488" class="Symbol">))</a> <a id="4491" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="4492" class="Symbol">)</a>
<a id="4498" class="Keyword">where</a>
<a id="4511" class="Keyword">open</a> <a id="4516" href="Categories.FreeObjects.Free.html#485" class="Module">FreeObject</a> <a id="4527" href="Algebra.Elgot.Properties.html#3652" class="Bound">fe</a> <a id="4530" class="Keyword">renaming</a> <a id="4539" class="Symbol">(</a><a id="4540" href="Categories.FreeObjects.Free.html#838" class="Field">FX</a> <a id="4543" class="Symbol">to</a> <a id="4546" class="Field">KY</a><a id="4548" class="Symbol">)</a>
<a id="4556" class="Keyword">open</a> <a id="4561" href="Algebra.Elgot.html#15964" class="Module">Elgot-Algebra</a> <a id="4575" href="Algebra.Elgot.Properties.html#3661" class="Bound">B</a> <a id="4577" class="Keyword">using</a> <a id="4583" class="Symbol">()</a> <a id="4586" class="Keyword">renaming</a> <a id="4595" class="Symbol">(</a><a id="4596" href="Algebra.Elgot.html#1938" class="Function Operator">_#</a> <a id="4599" class="Symbol">to</a> <a id="4602" class="Function Operator">_#ᵇ</a><a id="4605" class="Symbol">;</a> <a id="4607" href="Algebra.Elgot.html#2331" class="Function">#-resp-≈</a> <a id="4616" class="Symbol">to</a> <a id="4619" class="Function">#ᵇ-resp-≈</a><a id="4628" class="Symbol">)</a>
<a id="4636" class="Keyword">open</a> <a id="4641" href="Algebra.Elgot.html#15964" class="Module">Elgot-Algebra</a> <a id="4655" href="Algebra.Elgot.Properties.html#4546" class="Function">KY</a> <a id="4658" class="Keyword">using</a> <a id="4664" class="Symbol">()</a> <a id="4667" class="Keyword">renaming</a> <a id="4676" class="Symbol">(</a><a id="4677" href="Algebra.Elgot.html#1938" class="Function Operator">_#</a> <a id="4680" class="Symbol">to</a> <a id="4683" class="Function Operator">_#ʸ</a><a id="4686" class="Symbol">)</a>
<a id="free-isStable"></a><a id="4691" href="Algebra.Elgot.Properties.html#4691" class="Function">free-isStable</a> <a id="4705" class="Symbol">:</a> <a id="4707" class="Symbol">∀</a> <a id="4709" class="Symbol">{</a><a id="4710" href="Algebra.Elgot.Properties.html#4710" class="Bound">Y</a><a id="4711" class="Symbol">}</a> <a id="4713" class="Symbol">(</a><a id="4714" href="Algebra.Elgot.Properties.html#4714" class="Bound">fe</a> <a id="4717" class="Symbol">:</a> <a id="4719" href="Algebra.Elgot.Free.html#1441" class="Function">FreeElgotAlgebra</a> <a id="4736" href="Algebra.Elgot.Properties.html#4710" class="Bound">Y</a><a id="4737" class="Symbol">)</a> <a id="4739" class="Symbol">→</a> <a id="4741" href="Algebra.Elgot.Stable.html#2437" class="Record">IsStableFreeElgotAlgebra</a> <a id="4766" href="Algebra.Elgot.Properties.html#4714" class="Bound">fe</a>
<a id="4771" href="Algebra.Elgot.Properties.html#4691" class="Function">free-isStable</a> <a id="4785" class="Symbol">{</a><a id="4786" href="Algebra.Elgot.Properties.html#4786" class="Bound">Y</a><a id="4787" class="Symbol">}</a> <a id="4789" href="Algebra.Elgot.Properties.html#4789" class="Bound">fe</a> <a id="4792" class="Symbol">=</a> <a id="4794" href="Algebra.Elgot.Stable.html#7211" class="Function">isStableˡ⇒isStable</a> <a id="4813" href="Algebra.Elgot.Properties.html#4789" class="Bound">fe</a> <a id="4816" class="Symbol">(</a><a id="4817" href="Algebra.Elgot.Properties.html#1709" class="Function">free-isStableˡ</a> <a id="4832" href="Algebra.Elgot.Properties.html#4789" class="Bound">fe</a><a id="4834" class="Symbol">)</a>
</pre>
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