mirror of https://github.com/nmvdw/HITs-Examples
61 lines
1.5 KiB
Coq
61 lines
1.5 KiB
Coq
(** Properties of the operations on [FSetC A]. These are needed to prove that the
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representations are isomorphic. *)
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Require Import HoTT HitTactics.
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Require Import list_representation list_representation.operations.
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Section properties.
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Context {A : Type}.
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Definition append_nl (x: FSetC A) : ∅ ∪ x = x
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:= idpath.
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Lemma append_nr : forall (x: FSetC A), x ∪ ∅ = x.
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Proof.
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hinduction; try (intros; apply path_ishprop).
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- reflexivity.
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- intros. apply (ap (fun y => a;;y) X).
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Defined.
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Lemma append_assoc :
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forall (x y z: FSetC A), x ∪ (y ∪ z) = (x ∪ y) ∪ z.
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Proof.
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intros x y z.
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hinduction x ; try (intros ; apply path_ishprop).
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- reflexivity.
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- intros.
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cbn.
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f_ap.
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Defined.
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Lemma append_singleton: forall (a: A) (x: FSetC A),
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a ;; x = x ∪ (a ;; ∅).
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Proof.
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intro a. hinduction; try (intros; apply path_ishprop).
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- reflexivity.
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- intros b x HR.
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refine (comm_s _ _ _ @ ap (fun y => b ;; y) HR).
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Defined.
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Lemma append_comm :
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forall (x1 x2: FSetC A), x1 ∪ x2 = x2 ∪ x1.
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Proof.
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intros x1 x2.
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hinduction x1 ; try (intros ; apply path_ishprop).
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- intros.
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apply (append_nr _)^.
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- intros a x HR.
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refine (ap (fun y => a;;y) HR @ _).
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refine (append_singleton _ _ @ _).
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refine ((append_assoc _ _ _)^ @ _).
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refine (ap (x2 ∪) (append_singleton _ _)^).
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Defined.
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Lemma singleton_idem: forall (a: A),
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{|a|} ∪ {|a|} = {|a|}.
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Proof.
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intro.
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apply dupl.
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Defined.
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End properties.
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