use have instead of assert
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2 changed files with 30 additions and 18 deletions
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@ -105,12 +105,14 @@ namespace group
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definition is_set_homomorphism [instance] (G₁ G₂ : Group) : is_set (G₁ →g G₂) :=
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definition is_set_homomorphism [instance] (G₁ G₂ : Group) : is_set (G₁ →g G₂) :=
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begin
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begin
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assert H : G₁ →g G₂ ≃ Σ(f : G₁ → G₂), Π(g₁ g₂ : G₁), f (g₁ * g₂) = f g₁ * f g₂,
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have H : G₁ →g G₂ ≃ Σ(f : G₁ → G₂), Π(g₁ g₂ : G₁), f (g₁ * g₂) = f g₁ * f g₂,
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{ fapply equiv.MK,
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begin
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fapply equiv.MK,
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{ intro φ, induction φ, constructor, assumption},
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{ intro φ, induction φ, constructor, assumption},
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{ intro v, induction v, constructor, assumption},
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{ intro v, induction v, constructor, assumption},
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{ intro v, induction v, reflexivity},
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{ intro v, induction v, reflexivity},
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{ intro φ, induction φ, reflexivity}},
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{ intro φ, induction φ, reflexivity}
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end,
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apply is_trunc_equiv_closed_rev, exact H
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apply is_trunc_equiv_closed_rev, exact H
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end
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end
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@ -206,10 +206,12 @@ namespace chain_complex
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: is_exact_at_t (transfer_type_chain_complex2 X f c @g @e @p) m :=
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: is_exact_at_t (transfer_type_chain_complex2 X f c @g @e @p) m :=
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begin
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begin
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intro y q, esimp at *,
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intro y q, esimp at *,
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assert H2 : tcc_to_fn X (f m) ((equiv_of_eq (ap (λx, X x) (c m)))⁻¹ᵉ (e⁻¹ y)) = pt,
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have H2 : tcc_to_fn X (f m) ((equiv_of_eq (ap (λx, X x) (c m)))⁻¹ᵉ (e⁻¹ y)) = pt,
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{ refine _ ⬝ ap e⁻¹ᵉ* q ⬝ (respect_pt (e⁻¹ᵉ*)), apply eq_inv_of_eq, clear q, revert y,
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begin
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refine _ ⬝ ap e⁻¹ᵉ* q ⬝ (respect_pt (e⁻¹ᵉ*)), apply eq_inv_of_eq, clear q, revert y,
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refine inv_homotopy_of_homotopy (pequiv.to_equiv e) _,
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refine inv_homotopy_of_homotopy (pequiv.to_equiv e) _,
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apply inv_homotopy_of_homotopy, apply p},
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apply inv_homotopy_of_homotopy, apply p
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end,
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induction (H _ H2) with x r,
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induction (H _ H2) with x r,
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refine fiber.mk (e (cast (ap (λx, X x) (c (S m))) (cast (ap (λx, X (S x)) (c m)) x))) _,
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refine fiber.mk (e (cast (ap (λx, X x) (c (S m))) (cast (ap (λx, X (S x)) (c m)) x))) _,
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refine (p _)⁻¹ ⬝ _,
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refine (p _)⁻¹ ⬝ _,
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@ -293,10 +295,12 @@ namespace chain_complex
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{n : N} (H : is_exact_at X n) : is_exact_at (transfer_chain_complex X @g @e @p) n :=
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{n : N} (H : is_exact_at X n) : is_exact_at (transfer_chain_complex X @g @e @p) n :=
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begin
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begin
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intro y q, esimp at *,
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intro y q, esimp at *,
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assert H2 : cc_to_fn X n (e⁻¹ᵉ* y) = pt,
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have H2 : cc_to_fn X n (e⁻¹ᵉ* y) = pt,
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{ refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
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begin
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refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
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refine ap _ q ⬝ _,
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refine ap _ q ⬝ _,
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exact respect_pt e⁻¹ᵉ*},
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exact respect_pt e⁻¹ᵉ*
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end,
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induction (H _ H2) with x,
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induction (H _ H2) with x,
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induction x with x r,
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induction x with x r,
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refine image.mk (e x) _,
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refine image.mk (e x) _,
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@ -336,12 +340,14 @@ namespace chain_complex
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chain_complex.mk Y @g
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chain_complex.mk Y @g
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begin
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begin
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refine equiv_rect f _ _, intro n,
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refine equiv_rect f _ _, intro n,
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assert H : Π (x : Y (f (S (S n)))), g (c n ▸ g (c (S n) ▸ x)) = pt,
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have H : Π (x : Y (f (S (S n)))), g (c n ▸ g (c (S n) ▸ x)) = pt,
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{ apply equiv_rect (equiv_of_pequiv e), intro x,
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begin
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apply equiv_rect (equiv_of_pequiv e), intro x,
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refine ap (λx, g (c n ▸ x)) (@p (S n) x)⁻¹ᵖ ⬝ _,
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refine ap (λx, g (c n ▸ x)) (@p (S n) x)⁻¹ᵖ ⬝ _,
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refine (p _)⁻¹ ⬝ _,
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refine (p _)⁻¹ ⬝ _,
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refine ap e (cc_is_chain_complex X n _) ⬝ _,
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refine ap e (cc_is_chain_complex X n _) ⬝ _,
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apply respect_pt},
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apply respect_pt
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end,
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refine pi.pi_functor _ _ H,
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refine pi.pi_functor _ _ H,
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{ intro x, exact (c (S n))⁻¹ ▸ (c n)⁻¹ ▸ x}, -- with implicit arguments, this is:
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{ intro x, exact (c (S n))⁻¹ ▸ (c n)⁻¹ ▸ x}, -- with implicit arguments, this is:
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-- transport (λx, Y x) (c (S n))⁻¹ (transport (λx, Y (S x)) (c n)⁻¹ x)
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-- transport (λx, Y x) (c (S n))⁻¹ (transport (λx, Y (S x)) (c n)⁻¹ x)
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@ -355,10 +361,12 @@ namespace chain_complex
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{n : N} (H : is_exact_at X n) : is_exact_at (transfer_chain_complex2' X f c @g @e @p) (f n) :=
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{n : N} (H : is_exact_at X n) : is_exact_at (transfer_chain_complex2' X f c @g @e @p) (f n) :=
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begin
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begin
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intro y q, esimp at *,
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intro y q, esimp at *,
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assert H2 : cc_to_fn X n (e⁻¹ᵉ* ((c n)⁻¹ ▸ y)) = pt,
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have H2 : cc_to_fn X n (e⁻¹ᵉ* ((c n)⁻¹ ▸ y)) = pt,
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{ refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
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begin
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refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
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rewrite [tr_inv_tr, q],
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rewrite [tr_inv_tr, q],
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exact respect_pt e⁻¹ᵉ*},
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exact respect_pt e⁻¹ᵉ*
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end,
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induction (H _ H2) with x,
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induction (H _ H2) with x,
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induction x with x r,
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induction x with x r,
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refine image.mk (c n ▸ c (S n) ▸ e x) _,
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refine image.mk (c n ▸ c (S n) ▸ e x) _,
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@ -448,10 +456,12 @@ namespace chain_complex
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-- (H : is_exact_at_lg X n) : is_exact_at_lg (transfer_left_group_chain_complex X @g @e @p) n :=
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-- (H : is_exact_at_lg X n) : is_exact_at_lg (transfer_left_group_chain_complex X @g @e @p) n :=
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-- begin
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-- begin
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-- intro y q, esimp at *,
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-- intro y q, esimp at *,
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-- assert H2 : lgcc_to_fn X n (e⁻¹ᵉ* y) = pt,
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-- have H2 : lgcc_to_fn X n (e⁻¹ᵉ* y) = pt,
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-- { refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
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-- begin
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-- refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
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-- refine ap _ q ⬝ _,
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-- refine ap _ q ⬝ _,
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-- exact respect_pt e⁻¹ᵉ*},
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-- exact respect_pt e⁻¹ᵉ*
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-- end,
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-- cases (H _ H2) with x r,
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-- cases (H _ H2) with x r,
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-- refine image.mk (e x) _,
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-- refine image.mk (e x) _,
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-- refine (p x)⁻¹ ⬝ _,
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-- refine (p x)⁻¹ ⬝ _,
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