mirror of https://github.com/nmvdw/HITs-Examples
109 lines
3.0 KiB
Coq
109 lines
3.0 KiB
Coq
(** Some general prerequisities in homotopy type theory. *)
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Require Import HoTT.
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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.
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destruct p.
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hott_simpl.
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Defined.
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Global Instance hprop_lem `{Univalence} (T : Type) (Ttrunc : IsHProp T) : IsHProp (T + ~T).
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Proof.
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apply (equiv_hprop_allpath _)^-1.
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intros [x | nx] [y | ny] ; try f_ap ; try (apply Ttrunc) ; try contradiction.
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- apply equiv_hprop_allpath. apply _.
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Defined.
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Global Instance inl_embedding (A B : Type) : IsEmbedding (@inl A B).
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Proof.
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- intros [x1 | x2].
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* apply ishprop_hfiber_inl.
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* intros [z p].
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contradiction (inl_ne_inr _ _ p).
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Defined.
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Global Instance inr_embedding (A B : Type) : IsEmbedding (@inr A B).
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Proof.
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- intros [x1 | x2].
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* intros [z p].
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contradiction (inr_ne_inl _ _ p).
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* apply ishprop_hfiber_inr.
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Defined.
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Class MerelyDecidablePaths A :=
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m_dec_path : forall (a b : A), Decidable(Trunc (-1) (a = b)).
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Global Instance DecidableToMerely A (H : DecidablePaths A) : MerelyDecidablePaths A.
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Proof.
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intros x y.
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destruct (dec (x = y)).
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- apply (inl(tr p)).
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- refine (inr(fun p => _)).
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strip_truncations.
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apply (n p).
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Defined.
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Section merely_decidable_operations.
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Variable (A B : Type).
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Context `{MerelyDecidablePaths A} `{MerelyDecidablePaths B}.
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Global Instance merely_decidable_paths_prod : MerelyDecidablePaths(A * B).
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Proof.
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intros x y.
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destruct (m_dec_path (fst x) (fst y)) as [p1 | n1],
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(m_dec_path (snd x) (snd y)) as [p2 | n2].
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- apply inl.
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strip_truncations.
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apply tr.
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apply path_prod ; assumption.
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- apply inr.
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intros pp.
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strip_truncations.
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apply (n2 (tr (ap snd pp))).
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- apply inr.
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intros pp.
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strip_truncations.
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apply (n1 (tr (ap fst pp))).
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- apply inr.
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intros pp.
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strip_truncations.
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apply (n1 (tr (ap fst pp))).
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Defined.
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Global Instance merely_decidable_sum : MerelyDecidablePaths (A + B).
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Proof.
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intros [x1 | x2] [y1 | y2].
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- destruct (m_dec_path x1 y1) as [t | n].
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* apply inl.
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strip_truncations.
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apply (tr(ap inl t)).
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* refine (inr(fun p => _)).
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strip_truncations.
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refine (n(tr _)).
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refine (path_sum_inl _ p).
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- refine (inr(fun p => _)).
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strip_truncations.
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refine (inl_ne_inr x1 y2 p).
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- refine (inr(fun p => _)).
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strip_truncations.
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refine (inr_ne_inl x2 y1 p).
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- destruct (m_dec_path x2 y2) as [t | n].
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* apply inl.
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strip_truncations.
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apply (tr(ap inr t)).
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* refine (inr(fun p => _)).
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strip_truncations.
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refine (n(tr _)).
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refine (path_sum_inr _ p).
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Defined.
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End merely_decidable_operations. |