lean2/hott/hprop_trunc.hlean

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/-
Copyright (c) 2015 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Floris van Doorn
Proof of the theorem that (is_trunc n A) is a mere proposition
We prove this here to avoid circular dependency of files
We want to use this in .equiv; .equiv is imported by .function and .function is imported by .trunc
-/
import types.pi
open equiv sigma sigma.ops eq function pi
namespace is_trunc
definition is_contr.sigma_char (A : Type) :
(Σ (center : A), Π (a : A), center = a) ≃ (is_contr A) :=
begin
fapply equiv.MK,
{ intro S, exact (is_contr.mk S.1 S.2)},
{ intro H, cases H with H', cases H' with ce co, exact ⟨ce, co⟩},
{ intro H, cases H with H', cases H' with ce co, exact idp},
{ intro S, cases S, apply idp}
end
definition is_trunc.pi_char (n : trunc_index) (A : Type) :
(Π (x y : A), is_trunc n (x = y)) ≃ (is_trunc (n .+1) A) :=
begin
fapply equiv.MK,
{ intro H, apply is_trunc_succ_intro},
{ intro H x y, apply is_trunc_eq},
{ intro H, cases H, apply idp},
{ intro P, apply eq_of_homotopy, intro a, apply eq_of_homotopy, intro b,
change is_trunc.mk (to_internal n (a = b)) = P a b,
induction (P a b), apply idp},
end
definition is_hprop_is_trunc [instance] (n : trunc_index) :
Π (A : Type), is_hprop (is_trunc n A) :=
begin
induction n,
{ intro A,
apply is_trunc_is_equiv_closed,
{ apply equiv.to_is_equiv, apply is_contr.sigma_char},
apply is_hprop.mk, intros,
fapply sigma_eq, apply x.2,
apply is_hprop.elimo},
{ intro A,
apply is_trunc_is_equiv_closed,
apply equiv.to_is_equiv,
apply is_trunc.pi_char},
end
end is_trunc