mirror of https://github.com/nmvdw/HITs-Examples
40 lines
1.2 KiB
Coq
40 lines
1.2 KiB
Coq
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(* The length function for finite sets *)
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Require Import HoTT HitTactics.
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From representations Require Import cons_repr definition.
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From fsets Require Import operations_decidable isomorphism properties_decidable.
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Section Length.
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Context {A : Type}.
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Context {A_deceq : DecidablePaths A}.
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Context `{Univalence}.
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Opaque isIn_b.
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Definition length (x: FSetC A) : nat.
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Proof.
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simple refine (FSetC_ind A _ _ _ _ _ _ x ); simpl.
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- exact 0.
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- intros a y n.
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pose (y' := FSetC_to_FSet y).
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exact (if isIn_b a y' then n else (S n)).
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- intros. rewrite transport_const. cbn.
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simplify_isIn. simpl. reflexivity.
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- intros. rewrite transport_const. cbn.
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simplify_isIn.
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destruct (dec (a = b)) as [Hab | Hab].
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+ rewrite Hab. simplify_isIn. simpl. reflexivity.
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+ rewrite ?L_isIn_b_false; auto. simpl.
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destruct (isIn_b a (FSetC_to_FSet x0)), (isIn_b b (FSetC_to_FSet x0)) ; reflexivity.
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intro p. contradiction (Hab p^).
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Defined.
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Definition length_FSet (x: FSet A) := length (FSet_to_FSetC x).
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Lemma length_singleton: forall (a: A), length_FSet (L a) = 1.
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Proof.
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intro a.
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cbn. reflexivity.
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Defined.
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End Length.
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