awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
inr₂₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₆ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
inr₂₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → x ≤ b
inr₂₆ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₇ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
inr₂₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → x ≤ b
inr₂₆ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₇ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
inr₂₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → x ≤ b
inr₂₆ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₇ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₂₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → x ≤ b
Tactic: tauto
-- Lean 4 proof step from Mathlib.Order.Interval.Set.UnorderedInterval (goal)
Goal:
α : Type u_1
inst : LinearOrder α
a b c x : α
h2 : b ≤ x
inr : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₂ : a ≤ x → x ≤ c → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₃ : c ≤ x → x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₄ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₅ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₆ inr₇ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₈ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₉ : a ≤ x → x ≤ c → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₀ inr₁₁ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
inr₁₂ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → c ≤ x
inr₁₃ : c ≤ x → x ≤ a → ¬(a ≤ x ∧ x ≤ b) → ¬(b ≤ x ∧ x ≤ a) → ¬(b ≤ x ∧ x ≤ c) → x ≤ b
inr₁₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₁₆ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₁₇ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₁₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → c ≤ x
inr₁₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → c ≤ x
inr₂₀ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → c ≤ x
inr₂₁ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → c ≤ x
inr₂₂ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₃ : a ≤ x → x ≤ c → ¬a ≤ x → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₄ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬b ≤ x → x ≤ b
inr₂₅ : a ≤ x → x ≤ c → ¬a ≤ x → ¬x ≤ a → ¬x ≤ c → x ≤ b
inr₂₆ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬b ≤ x → x ≤ b
inr₂₇ : a ≤ x → x ≤ c → ¬x ≤ b → ¬b ≤ x → ¬x ≤ c → x ≤ b
inr₂₈ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬b ≤ x → x ≤ b
inr₂₉ : a ≤ x → x ≤ c → ¬x ≤ b → ¬x ≤ a → ¬x ≤ c → x ≤ b
⊢ a ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ a → (a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a) ∨ b ≤ x ∧ x ≤ c ∨ c ≤ x ∧ x ≤ b
have inr₃₀ : c ≤ x → x ≤ a → ¬a ≤ x → ¬b ≤ x → ¬b ≤ x → x ≤ b
Tactic: tauto