51 lines
1.3 KiB
Text
51 lines
1.3 KiB
Text
/-
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Copyright (c) 2014 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: types.fiber
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Author: Floris van Doorn
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Ported from Coq HoTT
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Theorems about fibers
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-/
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import types.sigma types.eq
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structure fiber {A B : Type} (f : A → B) (b : B) :=
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(point : A)
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(point_eq : f point = b)
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open equiv sigma sigma.ops eq
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namespace fiber
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variables {A B : Type} {f : A → B} {b : B}
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definition sigma_char (f : A → B) (b : B) : fiber f b ≃ (Σ(a : A), f a = b) :=
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begin
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fapply equiv.MK,
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{intro x, exact ⟨point x, point_eq x⟩},
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{intro x, exact (fiber.mk x.1 x.2)},
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{intro x, cases x, apply idp},
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{intro x, cases x, apply idp},
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end
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definition equiv_fiber_eq (x y : fiber f b)
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: (x = y) ≃ (Σ(p : point x = point y), point_eq x = ap f p ⬝ point_eq y) :=
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begin
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apply equiv.trans,
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{apply eq_equiv_fn_eq_of_equiv, apply sigma_char},
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apply equiv.trans,
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{apply equiv.symm, apply equiv_sigma_eq},
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apply sigma_equiv_sigma_id,
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intro p,
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apply equiv_of_equiv_of_eq,
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rotate 1,
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apply inv_con_eq_equiv_eq_con,
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{apply (ap (λx, x = _)), rewrite transport_eq_Fl}
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end
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definition eq_mk {x y : fiber f b} (p : point x = point y) (q : point_eq x = ap f p ⬝ point_eq y)
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: x = y :=
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to_inv !equiv_fiber_eq ⟨p, q⟩
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end fiber
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