feat(homotopy): add results about infty-connectedness and loops of EM-spaces
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
3213b1b3b0
commit
2cc8914874
6 changed files with 158 additions and 13 deletions
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@ -199,16 +199,23 @@ namespace EM
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rexact add_mul_le_mul_add n 1 1
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end
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section
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local attribute EMadd1 [reducible]
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definition is_conn_EMadd1 (G : CommGroup) (n : ℕ) : is_conn n (EMadd1 G n) := _
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definition is_conn_EMadd1 [instance] (G : CommGroup) (n : ℕ) : is_conn n (EMadd1 G n) := _
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definition is_trunc_EMadd1 (G : CommGroup) (n : ℕ) : is_trunc (n+1) (EMadd1 G n) := _
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definition is_trunc_EMadd1 [instance] (G : CommGroup) (n : ℕ) : is_trunc (n+1) (EMadd1 G n) := _
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end
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/- K(G, n) -/
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definition EM (G : CommGroup) : ℕ → Type*
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| 0 := pType_of_Group G
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| (k+1) := EMadd1 G k
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namespace ops
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abbreviation K := EM
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end ops
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open ops
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definition phomotopy_group_EM (G : CommGroup) (n : ℕ) : π*[n] (EM G n) ≃* pType_of_Group G :=
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begin
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cases n with n,
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@ -277,11 +284,55 @@ namespace EM
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[is_conn 0 X] [is_trunc 1 X] : pEM1 G ≃* X :=
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begin
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apply pEM1_pequiv' (!trunc_equiv⁻¹ᵉ ⬝e equiv_of_isomorphism e),
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intro p q, esimp, exact respect_mul e (tr p) (tr q)
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intro p q, esimp, exact to_respect_mul e (tr p) (tr q)
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end
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definition pEM1_pequiv_type {X : Type*} [is_conn 0 X] [is_trunc 1 X] : pEM1 (π₁ X) ≃* X :=
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pEM1_pequiv !isomorphism.refl
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definition EM_pequiv_1.{u} {G : CommGroup.{u}} {X : pType.{u}} (e : π₁ X ≃g G)
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[is_conn 0 X] [is_trunc 1 X] : EM G 1 ≃* X :=
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begin
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refine _ ⬝e* pEM1_pequiv e,
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apply ptrunc_pequiv,
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apply is_trunc_pEM1
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end
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definition EMadd1_pequiv_pEM1 (G : CommGroup) : EMadd1 G 0 ≃* pEM1 G :=
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begin apply ptrunc_pequiv, apply is_trunc_pEM1 end
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definition EM1add1_pequiv_0.{u} {G : CommGroup.{u}} {X : pType.{u}} (e : π₁ X ≃g G)
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[is_conn 0 X] [is_trunc 1 X] : EMadd1 G 0 ≃* X :=
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EMadd1_pequiv_pEM1 G ⬝e* pEM1_pequiv e
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definition KG1_pequiv.{u} {X Y : pType.{u}} (e : π₁ X ≃g π₁ Y)
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[is_conn 0 X] [is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y] : X ≃* Y :=
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(pEM1_pequiv e)⁻¹ᵉ* ⬝e* pEM1_pequiv !isomorphism.refl
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open circle int
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definition EM_pequiv_circle : K agℤ 1 ≃* S¹. :=
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!EMadd1_pequiv_pEM1 ⬝e* pEM1_pequiv fundamental_group_of_circle
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/- loops of EM-spaces -/
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definition loop_EMadd1 {G : CommGroup} (n : ℕ) : Ω (EMadd1 G (succ n)) ≃* EMadd1 G n :=
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begin
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cases n with n,
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{ symmetry, apply EM1add1_pequiv_0, rexact homotopy_group_EMadd1 G 1,
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-- apply is_conn_loop, apply is_conn_EMadd1,
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apply is_trunc_loop, apply is_trunc_EMadd1},
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{ refine loop_ptrunc_pequiv _ _ ⬝e* _,
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rewrite [add_one, succ_sub_two],
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have succ n + 1 ≤ 2 * succ n, from add_mul_le_mul_add n 1 1,
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symmetry, refine freudenthal_pequiv _ this, }
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end
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definition loop_EM (G : CommGroup) (n : ℕ) : Ω (K G (succ n)) ≃* K G n :=
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begin
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cases n with n,
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{ refine _ ⬝e* pequiv_of_isomorphism (fundamental_group_pEM1 G),
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refine loop_pequiv_loop (EMadd1_pequiv_pEM1 G) ⬝e* _,
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symmetry, apply ptrunc_pequiv, exact _},
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{ apply loop_EMadd1}
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end
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end EM
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@ -10,7 +10,7 @@ import .sphere
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import types.int.hott
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import algebra.homotopy_group .connectedness
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open eq susp bool sphere_index is_equiv equiv is_trunc is_conn pi algebra
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open eq susp bool sphere_index is_equiv equiv is_trunc is_conn pi algebra pointed
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definition circle : Type₀ := sphere 1
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@ -337,6 +337,9 @@ namespace circle
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proposition is_conn_circle [instance] : is_conn 0 S¹ :=
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sphere.is_conn_sphere -1.+2
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definition is_conn_pcircle [instance] : is_conn 0 S¹. := !is_conn_circle
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definition is_trunc_pcircle [instance] : is_trunc 1 S¹. := !is_trunc_circle
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definition circle_mul [reducible] (x y : S¹) : S¹ :=
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circle.elim y (circle_turn y) x
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@ -2,16 +2,24 @@
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Copyright (c) 2015 Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Ulrik Buchholtz, Floris van Doorn
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Connectedness of types and functions
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-/
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import types.trunc types.arrow_2 types.lift
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open eq is_trunc is_equiv nat equiv trunc function fiber funext pi
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namespace is_conn
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open eq is_trunc is_equiv nat equiv trunc function fiber funext pi pointed
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definition is_conn [reducible] (n : ℕ₋₂) (A : Type) : Type :=
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is_contr (trunc n A)
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definition is_conn_fun [reducible] (n : ℕ₋₂) {A B : Type} (f : A → B) : Type :=
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Πb : B, is_conn n (fiber f b)
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definition is_conn_inf [reducible] (A : Type) : Type := Πn, is_conn n A
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definition is_conn_fun_inf [reducible] {A B : Type} (f : A → B) : Type := Πn, is_conn_fun n f
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namespace is_conn
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definition is_conn_equiv_closed (n : ℕ₋₂) {A B : Type}
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: A ≃ B → is_conn n A → is_conn n B :=
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begin
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@ -21,9 +29,6 @@ namespace is_conn
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assumption
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end
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definition is_conn_fun [reducible] (n : ℕ₋₂) {A B : Type} (f : A → B) : Type :=
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Πb : B, is_conn n (fiber f b)
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theorem is_conn_of_le (A : Type) {n k : ℕ₋₂} (H : n ≤ k) [is_conn k A] : is_conn n A :=
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begin
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apply is_contr_equiv_closed,
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@ -34,6 +39,9 @@ namespace is_conn
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[is_conn_fun k f] : is_conn_fun n f :=
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λb, is_conn_of_le _ H
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definition is_conn_of_is_conn_succ (n : ℕ₋₂) (A : Type) [is_conn (n.+1) A] : is_conn n A :=
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is_trunc_trunc_of_le A -2 (trunc_index.self_le_succ n)
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namespace is_conn_fun
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section
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parameters (n : ℕ₋₂) {A B : Type} {h : A → B}
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@ -267,6 +275,15 @@ namespace is_conn
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apply is_trunc_equiv_closed, apply trunc_trunc_equiv_trunc_trunc
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end
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definition is_conn_eq [instance] (n : ℕ₋₂) {A : Type} (a a' : A) [is_conn (n.+1) A] :
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is_conn n (a = a') :=
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begin
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apply is_trunc_equiv_closed, apply tr_eq_tr_equiv,
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end
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definition is_conn_loop [instance] (n : ℕ₋₂) (A : Type*) [is_conn (n.+1) A] : is_conn n (Ω A) :=
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!is_conn_eq
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open pointed
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definition is_conn_ptrunc [instance] (A : Type*) (n k : ℕ₋₂) [H : is_conn n A]
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: is_conn n (ptrunc k A) :=
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@ -330,4 +347,22 @@ namespace is_conn
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{ apply trunc_equiv_trunc, apply fiber_lift_functor}
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end
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open trunc_index
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definition is_conn_fun_inf.mk_nat {A B : Type} {f : A → B} (H : Π(n : ℕ), is_conn_fun n f)
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: is_conn_fun_inf f :=
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begin
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intro n,
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cases n with n, { exact _},
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cases n with n, { have -1 ≤ of_nat 0, from dec_star, apply is_conn_fun_of_le f this},
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rewrite -of_nat_add_two, exact _
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end
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definition is_conn_inf.mk_nat {A : Type} (H : Π(n : ℕ), is_conn n A) : is_conn_inf A :=
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begin
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intro n,
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cases n with n, { exact _},
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cases n with n, { have -1 ≤ of_nat 0, from dec_star, apply is_conn_of_le A this},
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rewrite -of_nat_add_two, exact _
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end
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end is_conn
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@ -244,3 +244,25 @@ namespace is_trunc
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end
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end is_trunc
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open is_trunc function
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/- applications to infty-connected types and maps -/
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namespace is_conn
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definition is_conn_fun_inf_of_equiv_on_homotopy_groups.{u} {A B : Type.{u}} (f : A → B)
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[is_equiv (trunc_functor 0 f)]
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(H1 : Πa k, is_equiv (homotopy_group_functor k (pmap_of_map f a))) : is_conn_fun_inf f :=
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begin
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apply is_conn_fun_inf.mk_nat, intro n, apply is_conn_fun_of_equiv_on_homotopy_groups,
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{ intro a k H, exact H1 a k},
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{ intro a, apply is_surjective_of_is_equiv}
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end
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definition is_equiv_trunc_functor_of_is_conn_fun_inf.{u} (n : ℕ₋₂) {A B : Type.{u}} (f : A → B)
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[is_conn_fun_inf f] : is_equiv (trunc_functor n f) :=
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_
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definition is_equiv_homotopy_group_functor_of_is_conn_fun_inf.{u} {A B : pType.{u}} (f : A →* B)
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[is_conn_fun_inf f] (a : A) (k : ℕ) : is_equiv (homotopy_group_functor k f) :=
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is_equiv_π_of_is_connected f (le.refl k)
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end is_conn
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@ -18,6 +18,10 @@ namespace int
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Group.mk ℤ (group_of_add_group _)
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notation `gℤ` := group_integers
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definition CommGroup_int [constructor] : CommGroup :=
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CommGroup.mk ℤ ⦃comm_group, group_of_add_group ℤ, mul_comm := add.comm⦄
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notation `agℤ` := CommGroup_int
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end
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definition is_equiv_succ [instance] : is_equiv succ :=
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@ -182,6 +182,9 @@ namespace trunc_index
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definition sub_two_succ_succ (n : ℕ) : n.-2.+1.+1 = n := rfl
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definition succ_sub_two_succ (n : ℕ) : (nat.succ n).-2.+1 = n := rfl
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definition of_nat_add_two (n : ℕ₋₂) : of_nat (add_two n) = n.+2 :=
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begin induction n with n IH, reflexivity, exact ap succ IH end
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definition of_nat_le_of_nat {n m : ℕ} (H : n ≤ m) : (of_nat n ≤ of_nat m) :=
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begin
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induction H with m H IH,
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@ -224,6 +227,17 @@ namespace trunc_index
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{ exact succ_le_succ H2'}}
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end
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definition trunc_index.decidable_le [instance] : Π(n m : ℕ₋₂), decidable (n ≤ m) :=
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begin
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intro n, induction n with n IH: intro m,
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{ left, apply minus_two_le},
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cases m with m,
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{ right, apply not_succ_le_minus_two},
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cases IH m with H H,
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{ left, apply succ_le_succ H},
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right, intro H2, apply H, exact le_of_succ_le_succ H2
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end
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end trunc_index open trunc_index
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namespace is_trunc
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@ -401,6 +415,7 @@ namespace is_trunc
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is_trunc (n.+1) A ↔ Π(a : A), is_trunc n (a = a) :=
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iff.intro _ (is_trunc_succ_of_is_trunc_loop Hn)
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-- use loopn in name
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theorem is_trunc_iff_is_contr_loop_succ (n : ℕ) (A : Type)
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: is_trunc n A ↔ Π(a : A), is_contr (Ω[succ n](pointed.Mk a)) :=
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begin
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@ -417,6 +432,7 @@ namespace is_trunc
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apply imp_iff, reflexivity}
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end
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-- use loopn in name
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theorem is_trunc_iff_is_contr_loop (n : ℕ) (A : Type)
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: is_trunc (n.-2.+1) A ↔ (Π(a : A), is_contr (Ω[n](pointed.Mk a))) :=
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begin
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@ -427,6 +443,7 @@ namespace is_trunc
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{ apply is_trunc_iff_is_contr_loop_succ},
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end
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-- rename to is_contr_loopn_of_is_trunc
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theorem is_contr_loop_of_is_trunc (n : ℕ) (A : Type*) [H : is_trunc (n.-2.+1) A] :
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is_contr (Ω[n] A) :=
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begin
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@ -434,6 +451,7 @@ namespace is_trunc
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apply iff.mp !is_trunc_iff_is_contr_loop H
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end
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-- rename to is_trunc_loopn_of_is_trunc
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theorem is_trunc_loop_of_is_trunc (n : ℕ₋₂) (k : ℕ) (A : Type*) [H : is_trunc n A] :
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is_trunc n (Ω[k] A) :=
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begin
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@ -442,6 +460,17 @@ namespace is_trunc
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{ apply is_trunc_eq}
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end
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definition is_trunc_loopn (k : ℕ₋₂) (n : ℕ) (A : Type*) [H : is_trunc (k+n) A]
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: is_trunc k (Ω[n] A) :=
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begin
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revert k H, induction n with n IH: intro k H, exact _,
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apply is_trunc_eq, apply IH, rewrite [succ_add_nat, add_nat_succ at H], exact H
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end
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definition is_set_loopn (n : ℕ) (A : Type*) [is_trunc n A] : is_set (Ω[n] A) :=
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have is_trunc (0+[ℕ₋₂]n) A, by rewrite [trunc_index.zero_add]; exact _,
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is_trunc_loopn 0 n A
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end is_trunc open is_trunc
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namespace trunc
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@ -496,7 +525,8 @@ namespace trunc
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intro p, induction p, induction aa with a, esimp, exact (tr idp)
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end
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protected definition decode {n : ℕ₋₂} (aa aa' : trunc n.+1 A) : trunc.code n aa aa' → aa = aa' :=
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protected definition decode [unfold 3 4 5] {n : ℕ₋₂} (aa aa' : trunc n.+1 A) :
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trunc.code n aa aa' → aa = aa' :=
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begin
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induction aa' with a', induction aa with a,
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esimp [trunc.code, trunc.rec_on], intro x,
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