89 lines
3 KiB
Text
89 lines
3 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: hit.colimit
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Authors: Floris van Doorn
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Colimits.
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-/
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/- The hit colimit is primitive, declared in init.hit. -/
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open eq colimit colimit.diagram function
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namespace colimit
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protected definition elim [D : diagram] {P : Type} (Pincl : Π⦃i : Iob⦄ (x : ob i), P)
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(Pglue : Π(j : Ihom) (x : ob (dom j)), Pincl (hom j x) = Pincl x) : colimit D → P :=
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rec Pincl (λj x, !tr_constant ⬝ Pglue j x)
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protected definition elim_on [reducible] [D : diagram] {P : Type} (y : colimit D)
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(Pincl : Π⦃i : Iob⦄ (x : ob i), P)
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(Pglue : Π(j : Ihom) (x : ob (dom j)), Pincl (hom j x) = Pincl x) : P :=
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elim Pincl Pglue y
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definition elim_cglue [D : diagram] {P : Type} (Pincl : Π⦃i : Iob⦄ (x : ob i), P)
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(Pglue : Π(j : Ihom) (x : ob (dom j)), Pincl (hom j x) = Pincl x) {j : Ihom} (x : ob (dom j)) :
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ap (elim Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
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sorry
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end colimit
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/- definition of a sequential colimit -/
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open nat
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namespace seq_colimit
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context
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parameters {A : ℕ → Type} (f : Π⦃n⦄, A n → A (succ n))
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variables {n : ℕ} (a : A n)
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definition seq_diagram : diagram :=
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diagram.mk ℕ ℕ A id succ f
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local attribute seq_diagram [instance]
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-- TODO: define this in root namespace
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definition seq_colim {A : ℕ → Type} (f : Π⦃n⦄, A n → A (succ n)) : Type :=
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colimit seq_diagram
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definition inclusion : seq_colim f :=
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@colimit.inclusion _ _ a
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abbreviation sι := @inclusion
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definition glue : sι (f a) = sι a :=
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@cglue _ _ a
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protected definition rec [reducible] {P : seq_colim f → Type}
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(Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
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(Pglue : Π(n : ℕ) (a : A n), glue a ▹ Pincl (f a) = Pincl a) : Πaa, P aa :=
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@colimit.rec _ _ Pincl Pglue
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protected definition rec_on [reducible] {P : seq_colim f → Type} (aa : seq_colim f)
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(Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
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(Pglue : Π⦃n : ℕ⦄ (a : A n), glue a ▹ Pincl (f a) = Pincl a)
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: P aa :=
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rec Pincl Pglue aa
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protected definition elim {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
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(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) = Pincl a) : seq_colim f → P :=
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@colimit.elim _ _ Pincl Pglue
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protected definition elim_on [reducible] {P : Type} (aa : seq_colim f)
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(Pincl : Π⦃n : ℕ⦄ (a : A n), P)
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(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) = Pincl a) : P :=
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elim Pincl Pglue aa
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definition rec_glue {P : seq_colim f → Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
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(Pglue : Π⦃n : ℕ⦄ (a : A n), glue a ▹ Pincl (f a) = Pincl a) {n : ℕ} (a : A n)
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: apD (rec Pincl Pglue) (glue a) = sorry ⬝ Pglue a ⬝ sorry :=
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sorry
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definition elim_glue {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
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(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) = Pincl a) {n : ℕ} (a : A n)
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: ap (elim Pincl Pglue) (glue a) = sorry ⬝ Pglue a ⬝ sorry :=
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sorry
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end
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end seq_colimit
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