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<p>It is not true. When you consider the case where y = true, we
have that f true false = 1 by the definition f true x = 1. f y
false = 2 does not hold definitionally because f true x = 1 has
precedence.</p>
<p>To avoid this kind of definition, you can use the --exact-split
option. See
<a class="moz-txt-link-freetext" href="http://agda.readthedocs.io/en/latest/language/function-definitions.html#case-trees">http://agda.readthedocs.io/en/latest/language/function-definitions.html#case-trees</a>
.<br>
</p>
<p>James<code class="samp docutils literal"><span class="pre"><br>
</span></code></p>
<div class="moz-cite-prefix">On 03/07/17 14:35, Manuel Bärenz wrote:<br>
</div>
<blockquote type="cite"
cite="mid:d31e3d21-4a48-e88c-d197-918ce74b7b47@enigmage.de">
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<p>I didn't try, but how about pattern match on y, for example in
a with-clause? Then the proof should be refl.<br>
</p>
<br>
<div class="moz-cite-prefix">On 2017-07-03 15:34, v0id_NULL wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CAE3YR0Xf2M1hV8St1BcjyswNQrUSckfCnzd1cDu3sNbW26E5xQ@mail.gmail.com">
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<div style="font-size:12.8px">Dear list,</div>
<div style="font-size:12.8px"><br>
</div>
<div style="font-size:12.8px">Help me please. How can I prove
type of p2?</div>
<div style="font-size:12.8px"><br>
</div>
<div>
<div><span style="font-size:12.8px">module M1 where</span></div>
<div><span style="font-size:12.8px"><br>
</span></div>
<div>
<div><span style="font-size:12.8px">open import Data.Nat
using (ℕ)</span></div>
<div><span style="font-size:12.8px">open import Data.Bool
using (Bool; true; false)</span></div>
<div><span style="font-size:12.8px">open import
Relation.Binary.PropositionalEquality using (_≡_;
refl)</span></div>
<div><span style="font-size:12.8px"><br>
</span></div>
<div><span style="font-size:12.8px">f : Bool → Bool → ℕ</span></div>
<div><span style="font-size:12.8px">f true x = 1</span></div>
<div><span style="font-size:12.8px">f y false = 2</span></div>
<div><span style="font-size:12.8px">f false true = 3</span></div>
<div><span style="font-size:12.8px"><br>
</span></div>
<div><span style="font-size:12.8px">p1 : {x : Bool} → f
true x ≡ 1</span></div>
<div><span style="font-size:12.8px">p1 = refl</span></div>
<div><span style="font-size:12.8px"><br>
</span></div>
<div><span style="font-size:12.8px">p2 : {y : Bool} → f y
false ≡ 2</span></div>
<div><span style="font-size:12.8px">p2 = {!!}</span></div>
<div><span style="font-size:12.8px"><br>
</span></div>
<div><span style="font-size:12.8px">p3 : f false true ≡ 3</span></div>
<div><span style="font-size:12.8px">p3 = refl</span></div>
</div>
</div>
<div><br>
</div>
</div>
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