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Require Import HoTT.
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Require Export HoTT.
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Module Export FinSet.
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Set Implicit Arguments.
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Parameter A: Type.
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Private Inductive FSet : Type :=
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| empty : FSet
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| L : A -> FSet
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| U : FSet -> FSet -> FSet.
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Infix "∪" := U (at level 8, right associativity).
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Notation "∅" := empty.
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Axiom assoc : forall (x y z : FSet),
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x ∪ (y ∪ z) = (x ∪ y) ∪ z.
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Axiom comm : forall (x y : FSet),
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x ∪ y = y ∪ x.
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Axiom neutl : forall (x : FSet),
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∅ ∪ x = x.
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Axiom neutr : forall (x : FSet),
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x ∪ ∅ = x.
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Axiom idem : forall (x : A),
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(L x) ∪ (L x) = L x.
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Axiom trunc : IsHSet FSet.
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Fixpoint FSet_rec
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet)
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{struct x}
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: P
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
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| empty => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
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(FSet_rec H e l u assocP commP nlP nrP idemP y)
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(FSet_rec H e l u assocP commP nlP nrP idemP z)
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end) assocP commP nlP nrP idemP H.
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Axiom FSet_rec_beta_assoc : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y z : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (assoc x y z)
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=
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(assocP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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(FSet_rec H e l u assocP commP nlP nrP idemP y)
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(FSet_rec H e l u assocP commP nlP nrP idemP z)
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).
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Axiom FSet_rec_beta_comm : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (comm x y)
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=
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(commP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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(FSet_rec H e l u assocP commP nlP nrP idemP y)
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).
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Axiom FSet_rec_beta_nl : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (neutl x)
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=
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(nlP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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).
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Axiom FSet_rec_beta_nr : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (neutr x)
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=
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(nrP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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).
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Axiom FSet_rec_beta_idem : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : A),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (idem x)
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=
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idemP x.
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Fixpoint FSet_ind
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P empty)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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neutl x # uP empty x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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neutr x # uP x empty px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet)
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{struct x}
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: P x
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P x with
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| empty => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => eP
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => lP a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => uP y z
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP z)
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end) H assocP commP nlP nrP idemP.
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Axiom FSet_ind_beta_assoc : forall
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(A : Type)
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(P : FSet A -> Type)
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(eP : P (empty A))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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nl x # uP (empty A) x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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nr x # uP x (empty A) px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x y z : FSet A),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP)
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(assoc x y z)
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=
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(assocP x y z
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP z)
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).
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Axiom FSet_ind_beta_comm : forall
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(A : Type)
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(P : FSet A -> Type)
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(eP : P (empty A))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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nl x # uP (empty A) x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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nr x # uP x (empty A) px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x y : FSet A),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (comm x y)
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=
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(commP x y
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP y)
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).
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Axiom FSet_ind_beta_nl : forall
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(A : Type)
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(P : FSet A -> Type)
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(eP : P (empty A))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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nl x # uP (empty A) x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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nr x # uP x (empty A) px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet A),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (nl x)
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=
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(nlP x (FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
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).
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Axiom FSet_ind_beta_nr : forall
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(A : Type)
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(P : FSet A -> Type)
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(eP : P (empty A))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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nl x # uP (empty A) x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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nr x # uP x (empty A) px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet A),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (nr x)
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=
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(nrP x (FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
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).
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Axiom FSet_ind_beta_idem : forall
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(A : Type)
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(P : FSet A -> Type)
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(eP : P (empty A))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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nl x # uP (empty A) x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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nr x # uP x (empty A) px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : A),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (idem x)
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=
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idemP x.
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Parameter A: Type.
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Theorem idemSet :
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forall x: FSet A, U x x = x.
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Proof.
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simple refine (FSet_ind _ _ _ _ _ _ _ _ _).
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- cbn.
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apply nl.
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- cbn.
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apply idem.
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- cbn.
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intros.
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rewrite assoc.
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rewrite (comm (U x y)).
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rewrite assoc.
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rewrite X.
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rewrite <- assoc.
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rewrite X0.
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reflexivity.
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- cbn.
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(* todo optimisation *)
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Theorem FSetRec (A : Type)
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(P : Type)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet A) : P.
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Proof.
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simple refine (FSet_ind _ _ _ _ _ _ _ _ _ x).
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- apply e.
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- apply l.
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- apply (fun _ => fun _ => u).
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- cbn.
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intros.
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transitivity (u px (u py pz)).
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apply transport_const.
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apply assocP.
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- cbn.
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intros.
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transitivity (u px py).
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apply transport_const.
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apply commP.
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- cbn.
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intros.
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transitivity (u e px).
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apply transport_const.
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apply nlP.
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- cbn.
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intros.
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transitivity (u px e).
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apply transport_const.
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apply nrP.
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- cbn.
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intros.
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transitivity (u (l x0) (l x0)).
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apply transport_const.
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apply idemP.
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Defined.
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Theorem FSet_rec_beta_assocT : forall
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(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x y z : FSet A),
|
||||
ap (FSetRec e l u assocP commP nlP nrP idemP) (assoc x y z)
|
||||
=
|
||||
(assocP (FSetRec e l u assocP commP nlP nrP idemP x)
|
||||
(FSetRec e l u assocP commP nlP nrP idemP y)
|
||||
(FSetRec e l u assocP commP nlP nrP idemP z)
|
||||
).
|
||||
Proof.
|
||||
intros.
|
||||
eapply (cancelL (transport_const (assoc x y z) _ ) ).
|
||||
simple refine
|
||||
((apD_const
|
||||
(FSetRec e l u assocP commP nlP nrP idemP)
|
||||
(assoc x y z))^ @ _).
|
||||
apply FSet_ind_beta_assoc.
|
||||
|
||||
|
||||
Theorem FSet_rec_beta_commT : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x y : FSet A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (comm x y)
|
||||
=
|
||||
(commP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
||||
(FSet_rec e l u assocP commP nlP nrP idemP y)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_nl : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (nl x)
|
||||
=
|
||||
(nlP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_nr : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (nr x)
|
||||
=
|
||||
(nrP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_idem : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (idem x)
|
||||
=
|
||||
idemP x.
|
||||
|
||||
|
||||
End FinSet.
|
||||
|
||||
Definition isIn : forall
|
||||
(A : Type)
|
||||
(eq : A -> A -> Bool),
|
||||
A -> FSet A -> Bool.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro a.
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
apply false.
|
||||
|
||||
Focus 6.
|
||||
intro a'.
|
||||
apply (eq a a').
|
||||
|
||||
Focus 6.
|
||||
intro b.
|
||||
intro b'.
|
||||
apply (orb b b').
|
||||
|
||||
Focus 3.
|
||||
intros.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
reflexivity.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
intros.
|
||||
compute.
|
||||
destruct (eq a x).
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Definition comprehension : forall
|
||||
(A : Type)
|
||||
(eq : A -> A -> Bool),
|
||||
(A -> Bool) -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro phi.
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
apply empty.
|
||||
|
||||
Focus 6.
|
||||
intro a.
|
||||
apply (if (phi a) then L A a else (empty A)).
|
||||
|
||||
Focus 6.
|
||||
intro x.
|
||||
intro y.
|
||||
apply (U A x y).
|
||||
|
||||
Focus 3.
|
||||
intros.
|
||||
compute.
|
||||
apply nl.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
apply assoc.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
apply comm.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
apply nr.
|
||||
|
||||
intros.
|
||||
compute.
|
||||
destruct (phi x).
|
||||
apply idem.
|
||||
apply nl.
|
||||
Defined.
|
||||
|
||||
Definition intersection : forall (A : Type) (eq : A -> A -> Bool),
|
||||
FSet A -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro x.
|
||||
intro y.
|
||||
apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
|
||||
Defined.
|
||||
|
||||
Definition subset (A : Type) (eq : A -> A -> Bool) (x : FSet A) (y : FSet A) : Bool.
|
||||
Proof.
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
apply x.
|
||||
|
||||
Focus 6.
|
||||
apply true.
|
||||
|
||||
Focus 6.
|
||||
intro a.
|
||||
apply (isIn A eq a y).
|
||||
|
||||
Focus 6.
|
||||
intro b.
|
||||
intro b'.
|
||||
apply (andb b b').
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x0.
|
||||
destruct y0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x0.
|
||||
destruct y0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
destruct y0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
intros.
|
||||
destruct (isIn A eq x0 y).
|
||||
compute.
|
||||
reflexivity.
|
||||
compute.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FSet A) (y : FSet A) : Bool
|
||||
:= andb (subset A eq x y) (subset A eq y x).
|
||||
|
||||
Fixpoint eq_nat n m : Bool :=
|
||||
match n, m with
|
||||
| O, O => true
|
||||
| O, S _ => false
|
||||
| S _, O => false
|
||||
| S n1, S m1 => eq_nat n1 m1
|
||||
end.
|
||||
|
||||
|
||||
|
||||
Theorem test : forall (A:Type) (eq : A -> A -> Bool)
|
||||
(u: FSet A), ~(u = empty _) -> exists (a: A) (v: FSet A),
|
||||
u = U _ (L _ a) v /\ (isIn _ eq a v) = False.
|
||||
Proof.
|
||||
intros A eq.
|
||||
i
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Loading…
Reference in New Issue