mirror of https://github.com/nmvdw/HITs-Examples
63 lines
1.4 KiB
Coq
63 lines
1.4 KiB
Coq
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Require Import HoTT.
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Require Export HoTT.
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Require Import definition.
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(*Set Implicit Arguments.*)
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Arguments E {_}.
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Arguments U {_} _ _.
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Arguments L {_} _.
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Arguments assoc {_} _ _ _.
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Arguments comm {_} _ _.
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Arguments nl {_} _.
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Arguments nr {_} _.
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Arguments idem {_} _.
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Section operations.
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Variable A : Type.
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Parameter A_eqdec : forall (x y : A), Decidable (x = y).
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Definition deceq (x y : A) :=
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if dec (x = y) then true else false.
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Definition isIn : A -> FSet A -> Bool.
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Proof.
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intros a.
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simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- exact false.
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- intro a'. apply (deceq a a').
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- apply orb.
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- intros x y z. destruct x; reflexivity.
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- intros x y. destruct x, y; reflexivity.
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- intros x. reflexivity.
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- intros x. destruct x; reflexivity.
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- intros a'. destruct (deceq a a'); reflexivity.
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Defined.
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Infix "∈" := isIn (at level 9, right associativity).
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Definition comprehension :
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(A -> Bool) -> FSet A -> FSet A.
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Proof.
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intros P.
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simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- apply E.
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- intro a.
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refine (if (P a) then L a else E).
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- apply U.
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- intros. cbv. apply assoc.
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- intros. cbv. apply comm.
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- intros. cbv. apply nl.
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- intros. cbv. apply nr.
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- intros. cbv.
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destruct (P x); simpl.
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+ apply idem.
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+ apply nl.
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Defined.
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Definition intersection :
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FSet A -> FSet A -> FSet A.
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Proof.
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intros X Y.
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apply (comprehension (fun (a : A) => isIn a X) Y).
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Defined.
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End operations.
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