mirror of https://github.com/nmvdw/HITs-Examples
69 lines
1.7 KiB
Coq
69 lines
1.7 KiB
Coq
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Require Import HoTT.
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Definition lor (X Y : hProp) : hProp := BuildhProp (Trunc (-1) (sum X Y)).
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Infix "\/" := lor.
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Section lor_props.
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Variable X Y Z : hProp.
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Context `{Univalence}.
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Theorem lor_assoc : (X \/ (Y \/ Z)) = ((X \/ Y) \/ Z).
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [x | yz] ; cbn.
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+ apply (tr (inl (tr (inl x)))).
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+ simple refine (Trunc_ind _ _ yz).
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intros [y | z].
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++ apply (tr (inl (tr (inr y)))).
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++ apply (tr (inr z)).
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* simple refine (Trunc_ind _ _).
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intros [xy | z] ; cbn.
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+ simple refine (Trunc_ind _ _ xy).
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intros [x | y].
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++ apply (tr (inl x)).
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++ apply (tr (inr (tr (inl y)))).
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+ apply (tr (inr (tr (inr z)))).
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Defined.
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Theorem lor_comm : (X \/ Y) = (Y \/ X).
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [x | y].
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+ apply (tr (inr x)).
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+ apply (tr (inl y)).
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* simple refine (Trunc_ind _ _).
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intros [y | x].
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+ apply (tr (inr y)).
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+ apply (tr (inl x)).
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Defined.
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Theorem lor_nl : (False_hp \/ X) = X.
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [ | x].
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+ apply Empty_rec.
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+ apply x.
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* apply (fun x => tr (inr x)).
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Defined.
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Theorem lor_nr : (X \/ False_hp) = X.
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [x | ].
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+ apply x.
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+ apply Empty_rec.
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* apply (fun x => tr (inl x)).
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Defined.
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Theorem lor_idem : (X \/ X) = X.
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Proof.
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apply path_iff_hprop ; cbn.
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- simple refine (Trunc_ind _ _).
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intros [x | x] ; apply x.
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- apply (fun x => tr (inl x)).
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Defined.
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