80 lines
2.2 KiB
Text
80 lines
2.2 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.coeq
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Authors: Floris van Doorn
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Declaration of the coequalizer
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-/
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import .colimit
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open colimit bool eq
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namespace coeq
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context
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universe u
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parameters {A B : Type.{u}} (f g : A → B)
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definition coeq_diag [reducible] : diagram :=
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diagram.mk bool
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bool
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(λi, bool.rec_on i A B)
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(λj, ff)
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(λj, tt)
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(λj, bool.rec_on j f g)
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local notation `D` := coeq_diag
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definition coeq := colimit coeq_diag -- TODO: define this in root namespace
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local attribute coeq_diag [instance]
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definition coeq_i (x : B) : coeq :=
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@ι _ _ x
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/- cp is the name Coq uses. I don't know what it abbreviates, but at least it's short :-) -/
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definition cp (x : A) : coeq_i (f x) = coeq_i (g x) :=
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@cglue _ _ x ⬝ (@cglue _ _ x)⁻¹
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protected definition rec {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x))
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(Pcp : Π(x : A), cp x ▹ P_i (f x) = P_i (g x)) (y : coeq) : P y :=
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begin
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fapply (@colimit.rec_on _ _ y),
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{ intros [i, x], cases i,
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exact (@cglue _ _ x ▹ P_i (f x)),
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apply P_i},
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{ intros [j, x], cases j,
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exact idp,
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esimp [coeq_diag],
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apply tr_eq_of_eq_inv_tr,
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apply concat, rotate 1, apply tr_con,
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rewrite -Pcp}
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end
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protected definition rec_on [reducible] {P : coeq → Type} (y : coeq)
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(P_i : Π(x : B), P (coeq_i x)) (Pcp : Π(x : A), cp x ▹ P_i (f x) = P_i (g x)) : P y :=
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rec P_i Pcp y
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protected definition elim {P : Type} (P_i : B → P)
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(Pcp : Π(x : A), P_i (f x) = P_i (g x)) (y : coeq) : P :=
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rec P_i (λx, !tr_constant ⬝ Pcp x) y
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protected definition elim_on [reducible] {P : Type} (y : coeq) (P_i : B → P)
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(Pcp : Π(x : A), P_i (f x) = P_i (g x)) : P :=
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elim P_i Pcp y
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definition rec_cp {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x))
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(Pcp : Π(x : A), cp x ▹ P_i (f x) = P_i (g x))
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(x : A) : apD (rec P_i Pcp) (cp x) = sorry ⬝ Pcp x ⬝ sorry :=
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sorry
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definition elim_cp {P : Type} (P_i : B → P) (Pcp : Π(x : A), P_i (f x) = P_i (g x))
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(x : A) : ap (elim P_i Pcp) (cp x) = sorry ⬝ Pcp x ⬝ sorry :=
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sorry
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end
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end coeq
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