mirror of https://github.com/nmvdw/HITs-Examples
68 lines
1.5 KiB
Coq
68 lines
1.5 KiB
Coq
Require Import HoTT HitTactics.
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Require Import definition.
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Section operations.
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Context {A : Type}.
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Context {A_deceq : DecidablePaths A}.
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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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hrecursion.
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- exact false.
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- intro a'. destruct (dec (a = a')); [exact true | exact false].
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- apply orb.
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- intros x y z. compute. destruct x; reflexivity.
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- intros x y. compute. destruct x, y; reflexivity.
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- intros x. compute. reflexivity.
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- intros x. compute. destruct x; reflexivity.
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- intros a'. compute. destruct (A_deceq a a'); reflexivity.
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Defined.
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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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hrecursion.
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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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Definition subset :
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FSet A -> FSet A -> Bool.
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Proof.
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intros X Y.
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hrecursion X.
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- exact true.
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- exact (fun a => (isIn a Y)).
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- exact andb.
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- intros. compute. destruct x; reflexivity.
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- intros x y; compute; destruct x, y; reflexivity.
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- intros x; compute; destruct x; reflexivity.
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- intros x; compute; destruct x; reflexivity.
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- intros x; cbn; destruct (isIn x Y); reflexivity.
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Defined.
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End operations.
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Infix "∈" := isIn (at level 9, right associativity).
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Infix "⊆" := subset (at level 10, right associativity). |