mirror of https://github.com/nmvdw/HITs-Examples
65 lines
1.8 KiB
Coq
65 lines
1.8 KiB
Coq
(* Properties of the operations on [FSetC A] *)
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Require Import HoTT HitTactics.
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Require Import representations.cons_repr.
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From fsets Require Import operations_cons_repr.
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Section properties.
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Context {A : Type}.
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Definition append_nl : forall (x: FSetC A), ∅ ∪ x = x
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:= fun _ => 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 set_path2).
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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 {H: Funext}:
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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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hinduction
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; try (intros ; apply path_forall ; intro
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; apply path_forall ; intro ; apply set_path2).
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- reflexivity.
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- intros a x HR y z.
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specialize (HR y z).
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apply (ap (fun y => a;;y) HR).
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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 set_path2).
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- reflexivity.
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- intros b x HR. refine (_ @ _).
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+ apply comm.
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+ apply (ap (fun y => b ;; y) HR).
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Defined.
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Lemma append_comm {H: Funext}:
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forall (x1 x2: FSetC A), x1 ∪ x2 = x2 ∪ x1.
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Proof.
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hinduction ; try (intros ; apply path_forall ; intro ; apply set_path2).
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- intros. symmetry. apply append_nr.
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- intros a x1 HR x2.
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etransitivity.
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apply (ap (fun y => a;;y) (HR x2)).
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transitivity ((x2 ∪ x1) ∪ (a;;∅)).
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+ apply append_singleton.
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+ etransitivity.
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* symmetry. apply append_assoc.
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* simple 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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unfold singleton.
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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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