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Some cleanup

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This commit is contained in:
Niels
2017-08-01 15:12:59 +02:00
parent 37e3017cfc
commit fed9546d11
16 changed files with 659 additions and 1733 deletions
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20
FiniteSets/disjunction.v
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@@ -1,14 +1,18 @@
(* Logical disjunction in HoTT (see ch. 3 of the book) *)
Require Import HoTT.
Definition lor (X Y : hProp) : hProp := BuildhProp (Trunc (-1) (sum X Y)).
Infix "\/" := lor.
Delimit Scope logic_scope with L.
Notation "A ∨ B" := (lor A B) (at level 20, right associativity) : logic_scope.
Arguments lor _%L _%L.
Open Scope logic_scope.
Section lor_props.
Section lor_props.
Variable X Y Z : hProp.
Context `{Univalence}.
Theorem lor_assoc : (X \/ (Y \/ Z)) = ((X \/ Y) \/ Z).
Theorem lor_assoc : X ∨ Y ∨ Z = (X ∨ Y) ∨ Z.
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
@@ -27,7 +31,7 @@ Section lor_props.
+ apply (tr (inr (tr (inr z)))).
Defined.
Theorem lor_comm : (X \/ Y) = (Y \/ X).
Theorem lor_comm : X ∨ Y = Y ∨ X.
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
@@ -40,7 +44,7 @@ Section lor_props.
+ apply (tr (inl x)).
Defined.
Theorem lor_nl : (False_hp \/ X) = X.
Theorem lor_nl : False_hp ∨ X = X.
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
@@ -50,7 +54,7 @@ Section lor_props.
* apply (fun x => tr (inr x)).
Defined.
Theorem lor_nr : (X \/ False_hp) = X.
Theorem lor_nr : X ∨ False_hp = X.
Proof.
apply path_iff_hprop ; cbn.
* simple refine (Trunc_ind _ _).
@@ -60,7 +64,7 @@ Section lor_props.
* apply (fun x => tr (inl x)).
Defined.
Theorem lor_idem : (X \/ X) = X.
Theorem lor_idem : X ∨ X = X.
Proof.
apply path_iff_hprop ; cbn.
- simple refine (Trunc_ind _ _).