mirror of https://github.com/nmvdw/HITs-Examples
17 lines
558 B
Coq
17 lines
558 B
Coq
Require Import HoTT.
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(* Lemmas from this file do not belong in this project. *)
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(* Some of them should probably be in the HoTT library? *)
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Lemma ap_inl_path_sum_inl {A B} (x y : A) (p : inl x = inl y) :
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ap inl (path_sum_inl B p) = p.
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Proof.
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transitivity (@path_sum _ B (inl x) (inl y) (path_sum_inl B p));
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[ | apply (eisretr_path_sum _) ].
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destruct (path_sum_inl B p).
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reflexivity.
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Defined.
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Lemma ap_equiv {A B} (f : A <~> B) {x y : A} (p : x = y) :
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ap (f^-1 o f) p = eissect f x @ p @ (eissect f y)^.
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Proof. destruct p. hott_simpl. Defined.
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