Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
x y z : EInt
f : x ⟶ y
g : y ⟶ z
f' : x ≤ y
g' : y ≤ z
⊢ WithBotTop.rec (motive := fun {x} ↦
      (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
      (fun a f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (WithBotTop.coe a ⟶ y) →
            (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0) (fun b f ↦ 0)
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
      x (f ≫ g) =
    WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) y f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) y f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) y f)
        x f ≫
      WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
        y g

Tactic:
induction x using WithBotTop.rec

Goal state after:
case bot
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
y z : EInt
g : y ⟶ z
g' : y ≤ z
f : ⊥ ⟶ y
f' : ⊥ ≤ y
⊢ WithBotTop.rec (motive := fun {x} ↦
      (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
      (fun a f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (WithBotTop.coe a ⟶ y) →
            (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0) (fun b f ↦ 0)
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
      ⊥ (f ≫ g) =
    WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) y f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) y f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) y f)
        ⊥ f ≫
      WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
        y g

case coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
y z : EInt
g : y ⟶ z
g' : y ≤ z
a✝ : ℤ
f : WithBotTop.coe a✝ ⟶ y
f' : WithBotTop.coe a✝ ≤ y
⊢ WithBotTop.rec (motive := fun {x} ↦
      (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
      (fun a f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (WithBotTop.coe a ⟶ y) →
            (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0) (fun b f ↦ 0)
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
      (WithBotTop.coe a✝) (f ≫ g) =
    WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) y f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) y f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) y f)
        (WithBotTop.coe a✝) f ≫
      WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
        y g

case top
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
y z : EInt
g : y ⟶ z
g' : y ≤ z
f : ⊤ ⟶ y
f' : ⊤ ≤ y
⊢ WithBotTop.rec (motive := fun {x} ↦
      (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
      (fun a f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (WithBotTop.coe a ⟶ y) →
            (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0) (fun b f ↦ 0)
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
      ⊤ (f ≫ g) =
    WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) y f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) y f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) y f)
        ⊤ f ≫
      WithBotTop.rec (motive := fun {x} ↦
        (x ⟶ z) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) z))
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) z f)
        (fun a f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (WithBotTop.coe a ⟶ y) →
              (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
            (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) z f)
        (fun f ↦
          WithBotTop.rec (motive := fun {y} ↦
            (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0)
            (fun b f ↦ 0) (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) z f)
        y g