59 lines
1.9 KiB
Text
59 lines
1.9 KiB
Text
/-
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Copyright (c) 2015 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, Floris van Doorn
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Proof of the theorem that (is_trunc n A) is a mere proposition
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We prove this here to avoid circular dependency of files
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We want to use this in .equiv; .equiv is imported by .function and .function is imported by .trunc
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-/
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import types.pi
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open equiv sigma sigma.ops eq function pi
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namespace is_trunc
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definition is_contr.sigma_char (A : Type) :
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(Σ (center : A), Π (a : A), center = a) ≃ (is_contr A) :=
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begin
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fapply equiv.MK,
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{ intro S, exact (is_contr.mk S.1 S.2)},
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{ intro H, cases H with H', cases H' with ce co, exact ⟨ce, co⟩},
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{ intro H, cases H with H', cases H' with ce co, exact idp},
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{ intro S, cases S, apply idp}
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end
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definition is_trunc.pi_char (n : trunc_index) (A : Type) :
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(Π (x y : A), is_trunc n (x = y)) ≃ (is_trunc (n .+1) A) :=
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begin
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fapply equiv.MK,
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{ intro H, apply is_trunc_succ_intro},
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{ intro H x y, apply is_trunc_eq},
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{ intro H, cases H, apply idp},
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{ intro P, apply eq_of_homotopy, intro a, apply eq_of_homotopy, intro b,
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change is_trunc.mk (to_internal n (a = b)) = P a b,
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induction (P a b), apply idp},
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end
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definition is_prop_is_trunc (n : trunc_index) :
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Π (A : Type), is_prop (is_trunc n A) :=
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begin
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induction n,
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{ intro A,
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apply is_trunc_is_equiv_closed,
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{ apply equiv.to_is_equiv, apply is_contr.sigma_char},
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apply is_prop.mk, intros,
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fapply sigma_eq, apply x.2,
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apply is_prop.elimo},
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{ intro A,
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apply is_trunc_is_equiv_closed,
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apply equiv.to_is_equiv,
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apply is_trunc.pi_char},
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end
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local attribute is_prop_is_trunc [instance]
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definition is_trunc_succ_is_trunc [instance] (n m : ℕ₋₂) (A : Type) :
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is_trunc (n.+1) (is_trunc m A) :=
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!is_trunc_succ_of_is_prop
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end is_trunc
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