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_1 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
y z : WithBot α
a_1 : α
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
⊢ (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 coe₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) (↑a : WithBot α))
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) (↑a_1 : WithBot α) 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 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
⊢ (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
have bot : (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
    ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥) →
  (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 : LT α
inst_2 : AddLeftReflectLT α
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) ⊥ ⊥) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ ⊥
⊢ (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
have bot₁ : ∀ (a : α),
  (fun (x1 x2 : WithBot α) ↦ x1 < x2) ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ ⊥)
      ((fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : 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 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
a_1 : 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) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_1 : WithBot α) z
have bot : (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) ⊥
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 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
a_1 : α
bot :
  (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) ⊥
⊢ (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) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_1 : WithBot α) z
have bot₁ : ∀ (a : 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)) →
    (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 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
a_1 : 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) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ 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) ⊥) →
  (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 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
a_1 : α
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) ⊥) →
    (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) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ z
have coe₁ : ∀ (a : 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)) →
    (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 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
a_1 a_2 : 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) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : WithBot α) z
have coe : (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) ⊥
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 : LT α
inst_2 : AddLeftReflectLT α
z : WithBot α
a_1 a_2 : α
coe :
  (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) ⊥
⊢ (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) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : WithBot α) z
have coe₁ : ∀ (a : 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)) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : 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 : LT α
inst_2 : AddLeftReflectLT α
a_1 a_2 a_3 : 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)) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : WithBot α) (↑a_3 : WithBot α)
have coe : a_1 + a_2 < a_1 + a_3 → a_2 < 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 : LT α
inst_2 : AddRightReflectLT α
x y z : 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) x y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (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)
    (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
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 : AddRightReflectLT α
x y z : WithBot α
bot :
  (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) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have coe : ∀ (a : 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) (↑a : WithBot α) y)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (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 : AddRightReflectLT α
y z : 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) ⊥ 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
have bot : (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) ⊥ 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 : AddRightReflectLT α
y z : WithBot α
bot :
  (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) ⊥ 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
have bot₁ : ∀ (a : 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) ⊥ (↑a : WithBot α))
      (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) (↑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 : AddRightReflectLT α
y z : WithBot α
a_1 : 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) y)
      (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
have coe : (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) ⊥ 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 : AddRightReflectLT α
y z : WithBot α
a_1 : α
coe :
  (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) ⊥ 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) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) y z
have coe₁ : ∀ (a : 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)
        (↑a : 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) (↑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 z_1 : WithBot α
a b : α
inst_1 : LT α
inst_2 : AddRightReflectLT α
z : 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) ⊥ ⊥)
      (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) ⊥ z
have bot : (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) ⊥ ⊥
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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
bot :
  (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) ⊥ ⊥
⊢ (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) ⊥ z
have bot₁ : ∀ (a : 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) ⊥ ⊥)
      (swap (φ := fun (a a_1 : WithBot α) ↦ WithBot α) (fun (x1 x2 : WithBot α) ↦ x1 + x2) ⊥ (↑a : 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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
a_1 : 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) ⊥ z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_1 : WithBot α) z
have bot : (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) ⊥
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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
a_1 : α
bot :
  (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) ⊥
⊢ (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) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_1 : WithBot α) z
have bot₁ : ∀ (a : 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) ⊥ (↑a : 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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
a_1 : 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) (↑a_1 : WithBot α) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) ⊥ z
have coe : (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) ⊥ ⊥
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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
a_1 : α
coe :
  (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) ⊥ ⊥
⊢ (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) ⊥ z
have coe₁ : ∀ (a : 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) (↑a_1 : WithBot α)
        (↑a : 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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
a_1 a_2 : 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) z) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : WithBot α) z
have coe : (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) ⊥
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 : LT α
inst_2 : AddRightReflectLT α
z : WithBot α
a_1 a_2 : α
coe :
  (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) ⊥
⊢ (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) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : WithBot α) z
have coe₁ : ∀ (a : 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)
        (↑a : WithBot α)) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : 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 : LT α
inst_2 : AddRightReflectLT α
a_1 a_2 a_3 : 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)) →
    (fun (x1 x2 : WithBot α) ↦ x1 < x2) (↑a_2 : WithBot α) (↑a_3 : WithBot α)
have coe : a_2 + a_1 < a_3 + a_1 → a_2 < a_3
Tactic: simp [← coe_add, swap]
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
inst : Add α
x y z : WithBot α
inst_1 : LE α
inst_2 : AddLeftReflectLE α
hx : x ≠ ⊥
⊢ x + y ≤ x + z → y ≤ z
have lift : ∀ (x : α), (↑x : WithBot α) + y ≤ (↑x : WithBot α) + z → y ≤ z
Tactic: lift x to α using hx
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
inst : Add α
y z : WithBot α
inst_1 : LE α
inst_2 : AddLeftReflectLE α
x : α
⊢ (↑x : WithBot α) + y ≤ (↑x : WithBot α) + z → y ≤ z
have bot : (↑x : WithBot α) + ⊥ ≤ (↑x : WithBot α) + z → ⊥ ≤ z
Tactic: cases y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.Algebra.Order.Monoid.Unbundled.WithTop (goal)
Goal:
α : Type u
inst : Add α
y z : WithBot α
inst_1 : LE α
inst_2 : AddLeftReflectLE α
x : α
bot : (↑x : WithBot α) + ⊥ ≤ (↑x : WithBot α) + z → ⊥ ≤ z
⊢ (↑x : WithBot α) + y ≤ (↑x : WithBot α) + z → y ≤ z
have coe : ∀ (a : α), (↑x : WithBot α) + (↑a : WithBot α) ≤ (↑x : WithBot α) + z → (↑a : WithBot α) ≤ z
Tactic: cases y
Cleared hypotheses: y