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Added disjunction.

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Niels
2017-07-31 14:52:41 +02:00
parent f4d89f810c
commit 8ff9089d3d
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1
FiniteSets/_CoqProject
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@@ -1,6 +1,7 @@
-R . "" COQC = hoqc COQDEP = hoqdep -R . "" COQC = hoqc COQDEP = hoqdep
-R ../prelude "" -R ../prelude ""
definition.v definition.v
disjunction.v
operations.v operations.v
Ext.v Ext.v
properties.v properties.v

69
FiniteSets/disjunction.v Normal file
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@@ -0,0 +1,69 @@
Require Import HoTT.
Definition lor (X Y : hProp) : hProp := BuildhProp (Trunc (-1) (sum X Y)).
Infix "\/" := lor.
Section lor_props.
Variable X Y Z : hProp.
Context `{Univalence}.
Theorem lor_assoc : (X \/ (Y \/ Z)) = ((X \/ Y) \/ Z).
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
intros [x | yz] ; cbn.
+ apply (tr (inl (tr (inl x)))).
+ simple refine (Trunc_ind _ _ yz).
intros [y | z].
++ apply (tr (inl (tr (inr y)))).
++ apply (tr (inr z)).
* simple refine (Trunc_ind _ _).
intros [xy | z] ; cbn.
+ simple refine (Trunc_ind _ _ xy).
intros [x | y].
++ apply (tr (inl x)).
++ apply (tr (inr (tr (inl y)))).
+ apply (tr (inr (tr (inr z)))).
Defined.
Theorem lor_comm : (X \/ Y) = (Y \/ X).
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
intros [x | y].
+ apply (tr (inr x)).
+ apply (tr (inl y)).
* simple refine (Trunc_ind _ _).
intros [y | x].
+ apply (tr (inr y)).
+ apply (tr (inl x)).
Defined.
Theorem lor_nl : (False_hp \/ X) = X.
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
intros [ | x].
+ apply Empty_rec.
+ apply x.
* apply (fun x => tr (inr x)).
Defined.
Theorem lor_nr : (X \/ False_hp) = X.
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
intros [x | ].
+ apply x.
+ apply Empty_rec.
* apply (fun x => tr (inl x)).
Defined.
Theorem lor_idem : (X \/ X) = X.
Proof.
apply path_iff_hprop ; cbn.
- simple refine (Trunc_ind _ _).
intros [x | x] ; apply x.
- apply (fun x => tr (inl x)).
Defined.