awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddLeftMono α
z : WithBot α
a_1 : α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddLeftMono α
z : WithBot α
a_1 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ ⊥ →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
    ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddLeftMono α
z : WithBot α
a_1 : α
coe :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddLeftMono α
z : WithBot α
a_1 a_2 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_2 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) ⊥ →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_2 : WithBot α))
    ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddLeftMono α
z : WithBot α
a_1 a_2 : α
coe :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_2 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_2 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_2 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddLeftMono α
a_1 a_2 a_3 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) (↑a_3 : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_2 : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a_3 : WithBot α))
have coe : a_2 ≤ a_3 → a_1 + a_2 ≤ a_1 + a_3
Tactic: simp [← coe_add]
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x_1 y_1 z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
x y z : WithBot α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) x y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) x z)
have bot : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
    (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x_1 y_1 z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
x y z : WithBot α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) x y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) x z)
have coe : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a : WithBot α) y)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a : WithBot α) z)
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
y z : WithBot α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
    (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
y z : WithBot α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : WithBot α))
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
y z : WithBot α
a_1 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) y)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
    (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
y z : WithBot α
a_1 : α
coe :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) y z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) y)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a : WithBot α))
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ ⊥ →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
    (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
a_1 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) ⊥ →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
    (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
a_1 : α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z)
have bot₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_1 : WithBot α) (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a_1 : WithBot α))
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
a_1 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ ⊥ →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
    (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
a_1 : α
coe :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      (swap (φ := fun (a a_2 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) ⊥ (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
a_1 a_2 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a_2 : WithBot α))
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe : (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) ⊥ →
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
    (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
      (↑a_2 : WithBot α))
    (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z_1 : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
z : WithBot α
a_1 a_2 : α
coe :
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) ⊥ →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a_2 : WithBot α))
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) z →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a_2 : WithBot α))
      (swap (φ := fun (a a_3 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z)
have coe₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) (↑a : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_4 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a_2 : WithBot α))
      (swap (φ := fun (a a_4 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a : WithBot α))
Tactic: cases z
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y z : WithBot α
a b : α
inst_1 : LE α
inst_2 : AddRightMono α
a_1 a_2 a_3 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2) (↑a_2 : WithBot α) (↑a_3 : WithBot α) →
    (fun (x1 x2 : WithBot α) ↦ x1 ≤ x2)
      (swap (φ := fun (a a_4 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a_2 : WithBot α))
      (swap (φ := fun (a a_4 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α)
        (↑a_3 : WithBot α))
have coe : a_2 ≤ a_3 → a_2 + a_1 ≤ a_3 + a_1
Tactic: simp [← coe_add, swap]
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x_1 y_1 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
x y z : WithBot α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) x y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) x z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have bot : (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
    ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x_1 y_1 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
x y z : WithBot α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
⊢ (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) x y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) x z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have coe : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a : WithBot α) y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a : WithBot α) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
y z : WithBot α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have bot : (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
    ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ z
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
y z : WithBot α
bot :
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ z
⊢ (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have bot₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a : WithBot α) z
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
β : Type v
inst : Add α
w x y_1 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
y z : WithBot α
a_1 : α
⊢ (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) y)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have coe : (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) ⊥)
    ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) z) →
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ z
Tactic: cases y
Cleared hypotheses: y