feat(homotopy/long_exact_sequence): change to chain complexes as basic structure
Also prove that the equivalences are pointed equivalences
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Floris van Doorn
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1 changed files with 102 additions and 62 deletions
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import types.pointed types.int types.fiber
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open algebra nat int pointed unit sigma fiber sigma.ops eq equiv prod is_trunc
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open algebra nat int pointed unit sigma fiber sigma.ops eq equiv prod is_trunc equiv.ops
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namespace tactic
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definition replace (old : expr) (new : with_expr) (loc : location) : tactic := builtin
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end tactic
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namespace chain_complex
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namespace LES
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structure LES : Type :=
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structure chain_complex : Type :=
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(car : ℤ → Type*)
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(fn : Π{n : ℤ}, car (n + 1) →* car n)
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(chain_complex : Π{n : ℤ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
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(is_exact : Π{n : ℤ} (x : car (n + 1)), fn x = pt → Σ(y : car ((n + 1) + 1)), fn y = x)
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(is_chain_complex : Π{n : ℤ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
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structure LLES : Type := -- "left" long exact sequence
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structure left_chain_complex : Type := -- chain complex on the naturals with maps going down
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(car : ℕ → Type*)
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(fn : Π{n : ℕ}, car (n + 1) →* car n)
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(chain_complex : Π{n : ℕ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
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(is_exact : Π{n : ℕ} (x : car (n + 1)), fn x = pt → Σ(y : car ((n + 1) + 1)), fn y = x)
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(is_chain_complex : Π{n : ℕ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
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structure RLES : Type := -- "right" long exact sequence
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structure right_chain_complex : Type := -- chain complex on the naturals with maps going up
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(car : ℕ → Type*)
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(fn : Π{n : ℕ}, car n →* car (n + 1))
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(chain_complex : Π{n : ℕ} (x : car n), fn (fn x) = pt)
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(is_exact : Π{n : ℕ} (x : car (n + 1)), fn x = pt → Σ(y : car n), fn y = x)
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(is_chain_complex : Π{n : ℕ} (x : car n), fn (fn x) = pt)
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open LES LLES RLES
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definition cc_to_car [coercion] := @chain_complex.car
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definition cc_to_fn := @chain_complex.fn
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definition cc_is_chain_complex := @chain_complex.is_chain_complex
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definition lcc_to_car [coercion] := @left_chain_complex.car
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definition lcc_to_fn := @left_chain_complex.fn
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definition lcc_is_chain_complex := @left_chain_complex.is_chain_complex
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definition rcc_to_car [coercion] := @right_chain_complex.car
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definition rcc_to_fn := @right_chain_complex.fn
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definition rcc_is_chain_complex := @right_chain_complex.is_chain_complex
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/-
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this construction is currently wrong. We need to add one element between the sequence
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-/
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/- definition LES_of_LLES (X : LLES) : LES :=
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LES.mk (int.rec (car X) (λn, Unit))
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-- note: these notions are shifted by one!
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definition is_exact_at [reducible] (X : chain_complex) (n : ℤ) : Type :=
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Π(x : X (n + 1)), cc_to_fn X x = pt → Σ(y : X ((n + 1) + 1)), cc_to_fn X y = x
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definition is_exact_at_l [reducible] (X : left_chain_complex) (n : ℕ) : Type :=
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Π(x : X (n + 1)), lcc_to_fn X x = pt → Σ(y : X ((n + 1) + 1)), lcc_to_fn X y = x
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definition is_exact_at_r [reducible] (X : right_chain_complex) (n : ℕ) : Type :=
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Π(x : X (n + 1)), rcc_to_fn X x = pt → Σ(y : X n), rcc_to_fn X y = x
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definition is_exact [reducible] (X : chain_complex) : Type := Π(n : ℤ), is_exact_at X n
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definition is_exact_l [reducible] (X : left_chain_complex) : Type := Π(n : ℕ), is_exact_at_l X n
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definition is_exact_r [reducible] (X : right_chain_complex) : Type := Π(n : ℕ), is_exact_at_r X n
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definition chain_complex_from_left (X : left_chain_complex) : chain_complex :=
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chain_complex.mk (int.rec X (λn, Unit))
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begin
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intro n, fconstructor,
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{ induction n with n n,
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{ exact @fn X n},
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{ exact @lcc_to_fn X n},
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{ esimp, intro x, exact star}},
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{ induction n with n n,
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{ apply respect_pt},
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@ -44,99 +54,129 @@ namespace LES
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end
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begin
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intro n, induction n with n n,
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{ exact chain_complex X},
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{ exact lcc_is_chain_complex X},
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{ esimp, intro x, reflexivity}
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end
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begin
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intro n, induction n with n n,
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{ exact is_exact X},
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{ esimp, intro x p, exact sorry}
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end-/
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definition pfiber {X Y : Type*} (f : X →* Y) : Type* :=
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definition is_exact_from_left {X : left_chain_complex} {n : ℕ} (H : is_exact_at_l X n)
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: is_exact_at (chain_complex_from_left X) n :=
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H
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-- move to pointed
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definition pfiber [constructor] {X Y : Type*} (f : X →* Y) : Type* :=
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pointed.MK (fiber f pt) (fiber.mk pt !respect_pt)
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definition fiber_sequence_helper (v : Σ(X Y : Type*), X →* Y) : Σ(Z X : Type*), Z →* X :=
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definition pequiv_of_equiv [constructor] {A B : Type*} (f : A ≃ B) (H : f pt = pt) : A ≃* B :=
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pequiv.mk' (pmap.mk f H)
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definition fiber_sequence_helper [constructor] (v : Σ(X Y : Type*), X →* Y)
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: Σ(Z X : Type*), Z →* X :=
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⟨pfiber v.2.2, v.1, pmap.mk point rfl⟩
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definition fiber_sequence_carrier {X Y : Type*} (f : X →* Y) (n : ℕ) : Type* :=
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nat.cases_on n Y (λk, (iterate fiber_sequence_helper k ⟨X, Y, f⟩).1)
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definition fiber_sequence_arrow {X Y : Type*} (f : X →* Y) (n : ℕ)
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definition fiber_sequence_fun {X Y : Type*} (f : X →* Y) (n : ℕ)
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: fiber_sequence_carrier f (n + 1) →* fiber_sequence_carrier f n :=
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nat.cases_on n f proof (λk, pmap.mk point rfl) qed
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/- Definition 8.4.3 -/
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definition fiber_sequence.{u} {X Y : Pointed.{u}} (f : X →* Y) : LLES.{u} :=
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definition fiber_sequence.{u} {X Y : Pointed.{u}} (f : X →* Y) : left_chain_complex.{u} :=
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begin
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fconstructor,
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{ exact fiber_sequence_carrier f},
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{ exact fiber_sequence_arrow f},
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{ exact fiber_sequence_fun f},
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{ intro n x, cases n with n,
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{ exact point_eq x},
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{ exact point_eq x}},
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{ intro n x p, cases n with n,
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{ exact ⟨fiber.mk x p, rfl⟩},
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{ exact ⟨fiber.mk x p, rfl⟩}}
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{ exact point_eq x}}
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end
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definition is_exact_fiber_sequence {X Y : Type*} (f : X →* Y) : is_exact_l (fiber_sequence f) :=
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begin
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intro n x p, cases n with n,
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{ exact ⟨fiber.mk x p, rfl⟩},
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{ exact ⟨fiber.mk x p, rfl⟩}
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end
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-- move to types.sigma
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definition sigma_assoc_comm_equiv {A : Type} (B C : A → Type)
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definition sigma_assoc_comm_equiv [constructor] {A : Type} (B C : A → Type)
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: (Σ(v : Σa, B a), C v.1) ≃ (Σ(u : Σa, C a), B u.1) :=
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calc (Σ(v : Σa, B a), C v.1)
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≃ (Σa (b : B a), C a) : sigma_assoc_equiv
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≃ (Σa (b : B a), C a) : !sigma_assoc_equiv⁻¹
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... ≃ (Σa, B a × C a) : sigma_equiv_sigma_id (λa, !equiv_prod)
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... ≃ (Σa, C a × B a) : sigma_equiv_sigma_id (λa, !prod_comm_equiv)
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... ≃ (Σa (c : C a), B a) : sigma_equiv_sigma_id (λa, !equiv_prod)
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... ≃ (Σ(u : Σa, C a), B u.1) : sigma_assoc_equiv
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attribute is_equiv_sigma_functor is_equiv.is_equiv_id pequiv.mk' [constructor]
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attribute sigma.eta [unfold 3]
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-- set_option pp.notation false
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/- Lemma 8.4.4(i) -/
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definition fiber_sequence_carrier_equiv0.{u} {X Y : Pointed.{u}} (f : X →* Y)
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: fiber_sequence_carrier f 3 ≃ Ω Y :=
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calc
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fiber_sequence_carrier f 3 ≃ fiber (fiber_sequence_arrow f 1) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f 2), fiber_sequence_arrow f 1 x = pt : fiber.sigma_char
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... ≃ Σ(v : fiber f pt), fiber_sequence_arrow f 1 v = pt : erfl
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... ≃ Σ(v : Σ(x : X), f x = pt), fiber_sequence_arrow f 1 (fiber.mk v.1 v.2) = pt
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: fiber_sequence_carrier f 3 ≃* Ω Y :=
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pequiv_of_equiv
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(calc
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fiber_sequence_carrier f 3 ≃ fiber (fiber_sequence_fun f 1) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f 2), fiber_sequence_fun f 1 x = pt : fiber.sigma_char
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... ≃ Σ(v : fiber f pt), fiber_sequence_fun f 1 v = pt : erfl
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... ≃ Σ(v : Σ(x : X), f x = pt), fiber_sequence_fun f 1 (fiber.mk v.1 v.2) = pt
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: sigma_equiv_sigma_left !fiber.sigma_char
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... ≃ Σ(v : Σ(x : X), f x = pt), v.1 = pt : erfl
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... ≃ Σ(v : Σ(x : X), x = pt), f v.1 = pt : sigma_assoc_comm_equiv
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... ≃ f !center.1 = pt : sigma_equiv_of_is_contr_left _
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... ≃ f pt = pt : erfl
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... ≃ pt = pt : by exact !equiv_eq_closed_left !respect_pt
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... ≃ Ω Y : erfl
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... ≃ Ω Y : erfl)
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begin
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change (respect_pt f)⁻¹ ⬝
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((center_eq ⟨Pointed.Point X, refl (Pointed.Point X)⟩)⁻¹ ▸ respect_pt f) = idp,
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rewrite tr_constant,
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apply con.left_inv
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end
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/- (generalization of) Lemma 8.4.4(ii) -/
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definition fiber_sequence_carrier_equiv1.{u} {X Y : Pointed.{u}} (f : X →* Y) (n : ℕ)
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: fiber_sequence_carrier f (n+4) ≃ Ω(fiber_sequence_carrier f (n+1)) :=
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calc
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fiber_sequence_carrier f (n+4) ≃ fiber (fiber_sequence_arrow f (n+2)) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f _), fiber_sequence_arrow f (n+2) x = pt
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: fiber_sequence_carrier f (n+4) ≃* Ω(fiber_sequence_carrier f (n+1)) :=
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pequiv_of_equiv
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(calc
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fiber_sequence_carrier f (n+4) ≃ fiber (fiber_sequence_fun f (n+2)) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f (n+2) x = pt
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: fiber.sigma_char
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... ≃ Σ(x : fiber (fiber_sequence_arrow f (n+1)) pt), fiber_sequence_arrow f _ x = pt
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... ≃ Σ(x : fiber (fiber_sequence_fun f (n+1)) pt), fiber_sequence_fun f _ x = pt
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: erfl
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_arrow f _ x = pt),
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fiber_sequence_arrow f _ (fiber.mk v.1 v.2) = pt
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
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fiber_sequence_fun f _ (fiber.mk v.1 v.2) = pt
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: by exact sigma_equiv_sigma !fiber.sigma_char (λa, erfl)
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_arrow f _ x = pt),
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
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v.1 = pt
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: erfl
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), x = pt),
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fiber_sequence_arrow f _ v.1 = pt
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fiber_sequence_fun f _ v.1 = pt
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: sigma_assoc_comm_equiv
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... ≃ fiber_sequence_arrow f _ !center.1 = pt
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... ≃ fiber_sequence_fun f _ !center.1 = pt
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: @(sigma_equiv_of_is_contr_left _) !is_contr_sigma_eq'
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... ≃ fiber_sequence_arrow f _ pt = pt
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... ≃ fiber_sequence_fun f _ pt = pt
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: erfl
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... ≃ pt = pt
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: by exact !equiv_eq_closed_left !respect_pt
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... ≃ Ω(fiber_sequence_carrier f (n+1)) : erfl
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... ≃ Ω(fiber_sequence_carrier f (n+1)) : erfl)
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begin reflexivity end
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/- Lemma 8.4.4 (i)(ii), combined -/
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attribute bit0 bit1 add nat.add [reducible]
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definition fiber_sequence_carrier_equiv {X Y : Type*} (f : X →* Y) (n : ℕ)
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: fiber_sequence_carrier f (n+3) ≃ Ω(fiber_sequence_carrier f n) :=
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: fiber_sequence_carrier f (n+3) ≃* Ω(fiber_sequence_carrier f n) :=
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nat.cases_on n (fiber_sequence_carrier_equiv0 f) (fiber_sequence_carrier_equiv1 f)
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exit
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/- Lemma 8.4.4(iii) -/
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definition fiber_sequence_function0 {X Y : Type*} (f : X →* Y)
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: Π(x : fiber_sequence_carrier f 4), ap1 f (fiber_sequence_carrier_equiv f 1 x)⁻¹ᵖ =
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fiber_sequence_carrier_equiv f 0 (fiber_sequence_fun f 3 x) :=
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take (x : fiber (fiber_sequence_fun f 2) pt),
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obtain (v : fiber (fiber_sequence_fun f 1) pt) (q : _), from x,
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begin
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unfold [fiber_sequence_carrier_equiv,fiber_sequence_carrier_equiv0,fiber_sequence_carrier_equiv1,equiv.trans, equiv.symm, pequiv._trans_of_to_pmap],
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esimp [sigma_assoc_equiv, equiv.symm, equiv.trans], unfold [fiber_sequence_fun, fiber_sequence_carrier]
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end
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end LES
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end chain_complex
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