35 lines
1.1 KiB
Text
35 lines
1.1 KiB
Text
/-
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Copyright (c) 2016 Jakob von Raumer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jakob von Raumer
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The Wedge Sum of Two Pointed Types
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-/
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import hit.pointed_pushout
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open eq pushout pointed Pointed
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namespace wedge
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section
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variables (A B : Type*)
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definition Wedge : Type* := Pushout (pconst Unit A) (pconst Unit B)
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-- TODO maybe find a cleaner proof
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protected definition unit : A ≃* Wedge Unit A :=
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begin
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fconstructor,
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{ fapply pmap.mk, intro a, apply pinr a, apply respect_pt },
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{ fapply is_equiv.adjointify, intro x, fapply pushout.elim_on x,
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exact λ x, Point A, exact id, intro u, reflexivity,
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intro x, fapply pushout.rec_on x, intro u, cases u, esimp, apply (glue unit.star)⁻¹,
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intro a, reflexivity,
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intro u, cases u, esimp, apply eq_pathover,
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refine _ ⬝hp !ap_id⁻¹, fapply eq_hconcat, apply ap_compose inr,
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krewrite elim_glue, fapply eq_hconcat, apply ap_idp, apply square_of_eq,
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apply con.left_inv,
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intro a, reflexivity },
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end
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end
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end wedge
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