73 lines
3 KiB
Text
73 lines
3 KiB
Text
import homotopy.susp types.pointed2 ..move_to_lib
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open susp eq pointed function is_equiv lift equiv is_trunc nat
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namespace susp
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variables {X X' Y Y' Z : Type*}
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definition iterate_susp_iterate_susp_rev (n m : ℕ) (A : Type*) :
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iterate_susp n (iterate_susp m A) ≃* iterate_susp (m + n) A :=
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begin
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induction n with n e,
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{ reflexivity },
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{ exact susp_pequiv e }
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end
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definition iterate_susp_pequiv [constructor] (n : ℕ) {X Y : Type*} (f : X ≃* Y) :
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iterate_susp n X ≃* iterate_susp n Y :=
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begin
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induction n with n e,
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{ exact f },
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{ exact susp_pequiv e }
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end
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open algebra nat
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definition iterate_susp_iterate_susp (n m : ℕ) (A : Type*) :
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iterate_susp n (iterate_susp m A) ≃* iterate_susp (n + m) A :=
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iterate_susp_iterate_susp_rev n m A ⬝e* pequiv_of_eq (ap (λk, iterate_susp k A) (add.comm m n))
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definition plift_susp.{u v} : Π(A : Type*), plift.{u v} (susp A) ≃* susp (plift.{u v} A) :=
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begin
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intro A,
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calc
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plift.{u v} (susp A) ≃* susp A : by exact (pequiv_plift (susp A))⁻¹ᵉ*
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... ≃* susp (plift.{u v} A) : by exact susp_pequiv (pequiv_plift.{u v} A)
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end
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definition is_contr_susp [instance] (A : Type) [H : is_contr A] : is_contr (susp A) :=
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begin
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apply is_contr.mk north,
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intro x, induction x,
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reflexivity,
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exact merid !center,
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apply eq_pathover_constant_left_id_right, apply square_of_eq,
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exact whisker_left idp (ap merid !eq_of_is_contr)
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end
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definition loop_susp_pintro_phomotopy {X Y : Type*} {f g : ⅀ X →* Y} (p : f ~* g) :
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loop_susp_pintro X Y f ~* loop_susp_pintro X Y g :=
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pwhisker_right (loop_susp_unit X) (Ω⇒ p)
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variables {A₀₀ A₂₀ A₀₂ A₂₂ : Type*}
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{f₁₀ : A₀₀ →* A₂₀} {f₁₂ : A₀₂ →* A₂₂}
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{f₀₁ : A₀₀ →* A₀₂} {f₂₁ : A₂₀ →* A₂₂}
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-- rename: susp_functor_psquare
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definition suspend_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : psquare (⅀→ f₁₀) (⅀→ f₁₂) (⅀→ f₀₁) (⅀→ f₂₁) :=
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sorry
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definition susp_to_loop_psquare (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂) (f₀₁ : susp A₀₀ →* A₀₂) (f₂₁ : susp A₂₀ →* A₂₂) : (psquare (⅀→ f₁₀) f₁₂ f₀₁ f₂₁) → (psquare f₁₀ (Ω→ f₁₂) ((loop_susp_pintro A₀₀ A₀₂) f₀₁) ((loop_susp_pintro A₂₀ A₂₂) f₂₁)) :=
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begin
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intro p,
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refine pvconcat _ (ap1_psquare p),
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exact loop_susp_unit_natural f₁₀
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end
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definition loop_to_susp_square (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂) (f₀₁ : A₀₀ →* Ω A₀₂) (f₂₁ : A₂₀ →* Ω A₂₂) : (psquare f₁₀ (Ω→ f₁₂) f₀₁ f₂₁) → (psquare (⅀→ f₁₀) f₁₂ ((susp_pelim A₀₀ A₀₂) f₀₁) ((susp_pelim A₂₀ A₂₂) f₂₁)) :=
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begin
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intro p,
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refine pvconcat (suspend_psquare p) _,
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exact psquare_transpose (loop_susp_counit_natural f₁₂)
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end
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end susp
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