akjadhav/leandojo-lean4-formal-informal-strings-split

formal stringlengths 41 427k	informal stringclasses 1 value
Int.le.dest a b : Int h : a ≤ b n : Nat h₁ : b - a = ↑n ⊢ a + ↑n = b rw [← h₁, Int.add_comm] a b : Int h : a ≤ b n : Nat h₁ : b - a = ↑n ⊢ b - a + a = b simp [Int.sub_eq_add_neg, Int.add_assoc] ** Qed
Int.ofNat_le m n : Nat h : m ≤ n k : Nat hk : m + k = n ⊢ ↑m + ↑k = ↑n rw [← hk] m n : Nat h : m ≤ n k : Nat hk : m + k = n ⊢ ↑m + ↑k = ↑(m + k) rfl ** Qed
Int.eq_ofNat_of_zero_le a : Int h : 0 ≤ a ⊢ ∃ n, a = ↑n have t := le.dest_sub h a : Int h : 0 ≤ a t : ∃ n, a - 0 = ↑n ⊢ ∃ n, a = ↑n rwa [Int.sub_zero] at t ** Qed
Int.eq_succ_of_zero_lt a : Int h✝ : 0 < a n : Nat h : ↑(1 + n) = a ⊢ a = ↑(succ n) rw [Nat.add_comm] at h a : Int h✝ : 0 < a n : Nat h : ↑(n + 1) = a ⊢ a = ↑(succ n) exact h.symm ** Qed
Int.lt_add_succ a : Int n : Nat ⊢ a + 1 + ↑n = a + ↑(succ n) rw [Int.add_comm, Int.add_left_comm] a : Int n : Nat ⊢ a + (↑n + 1) = a + ↑(succ n) rfl ** Qed
Int.lt.dest a b : Int h✝ : a < b n : Nat h : a + 1 + ↑n = b ⊢ a + ↑(succ n) = b rwa [Int.add_comm, Int.add_left_comm] at h ** Qed
Int.ofNat_lt n m : Nat ⊢ ↑n < ↑m ↔ n < m rw [lt_iff_add_one_le, ← ofNat_succ, ofNat_le] n m : Nat ⊢ succ n ≤ m ↔ n < m rfl ** Qed
Int.le_trans a b c : Int h₁ : a ≤ b h₂ : b ≤ c n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = c ⊢ a + ↑(n + m) = c rw [← hm, ← hn, Int.add_assoc, ofNat_add] ** Qed
Int.le_antisymm a b : Int h₁ : a ≤ b h₂ : b ≤ a ⊢ a = b let ⟨n, hn⟩ := le.dest h₁ a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b ⊢ a = b let ⟨m, hm⟩ := le.dest h₂ a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a ⊢ a = b have := hn a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a this : a + ↑n = b ⊢ a = b rw [← hm, Int.add_assoc, ← ofNat_add] at this a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a this : b + ↑(m + n) = b ⊢ a = b have := Int.ofNat.inj <\| Int.add_left_cancel <\| this.trans (Int.add_zero _).symm a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a this✝ : b + ↑(m + n) = b this : m + n = 0 ⊢ a = b rw [← hn, Nat.eq_zero_of_add_eq_zero_left this, ofNat_zero, Int.add_zero a] ** Qed
Int.lt_irrefl a : Int H : a < a n : Nat hn : a + ↑(succ n) = a ⊢ a + ↑(succ n) = a + 0 rw [hn, Int.add_zero] ** Qed
Int.mul_nonneg ** a b : Int ha : 0 ≤ a hb : 0 ≤ b ⊢ 0 ≤ a * b let ⟨n, hn⟩ := eq_ofNat_of_zero_le ha a b : Int ha : 0 ≤ a hb : 0 ≤ b n : Nat hn : a = ↑n ⊢ 0 ≤ a * b let ⟨m, hm⟩ := eq_ofNat_of_zero_le hb a b : Int ha : 0 ≤ a hb : 0 ≤ b n : Nat hn : a = ↑n m : Nat hm : b = ↑m ⊢ 0 ≤ a * b rw [hn, hm, ← ofNat_mul] a b : Int ha : 0 ≤ a hb : 0 ≤ b n : Nat hn : a = ↑n m : Nat hm : b = ↑m ⊢ 0 ≤ ↑(n * m) apply ofNat_nonneg Qed
Int.lt_iff_le_not_le a b : Int ⊢ a < b ↔ a ≤ b ∧ ¬b ≤ a rw [Int.lt_iff_le_and_ne] a b : Int ⊢ a ≤ b ∧ a ≠ b ↔ a ≤ b ∧ ¬b ≤ a constructor <;> refine fun ⟨h, h'⟩ => ⟨h, h'.imp fun h' => ?_⟩ case mp a b : Int x✝ : a ≤ b ∧ a ≠ b h : a ≤ b h'✝ : a ≠ b h' : b ≤ a ⊢ a = b exact Int.le_antisymm h h' case mpr a b : Int x✝ : a ≤ b ∧ ¬b ≤ a h : a ≤ b h'✝ : ¬b ≤ a h' : a = b ⊢ b ≤ a subst h' case mpr a : Int x✝ : a ≤ a ∧ ¬a ≤ a h : a ≤ a h' : ¬a ≤ a ⊢ a ≤ a apply Int.le_refl ** Qed
Int.not_lt a b : Int ⊢ ¬a < b ↔ b ≤ a rw [← Int.not_le, Decidable.not_not] ** Qed
Int.min_comm a b : Int ⊢ min a b = min b a simp [Int.min_def] a b : Int ⊢ (if a ≤ b then a else b) = if b ≤ a then b else a by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂] case pos a b : Int h₁ : a ≤ b h₂ : b ≤ a ⊢ a = b exact Int.le_antisymm h₁ h₂ case neg a b : Int h₁ : ¬a ≤ b h₂ : ¬b ≤ a ⊢ b = a cases not_or_intro h₁ h₂ <\| Int.le_total .. ** Qed
Int.min_le_right a b : Int ⊢ min a b ≤ b rw [Int.min_def] a b : Int ⊢ (if a ≤ b then a else b) ≤ b split <;> simp [] * Qed
Int.min_eq_left a b : Int h : a ≤ b ⊢ min a b = a simp [Int.min_def, h] ** Qed
Int.max_comm a b : Int ⊢ max a b = max b a simp only [Int.max_def] a b : Int ⊢ (if a ≤ b then b else a) = if b ≤ a then a else b by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂] case pos a b : Int h₁ : a ≤ b h₂ : b ≤ a ⊢ b = a exact Int.le_antisymm h₂ h₁ case neg a b : Int h₁ : ¬a ≤ b h₂ : ¬b ≤ a ⊢ a = b cases not_or_intro h₁ h₂ <\| Int.le_total .. ** Qed
Int.le_max_left a b : Int ⊢ a ≤ max a b rw [Int.max_def] a b : Int ⊢ a ≤ if a ≤ b then b else a split <;> simp [] * Qed
Int.max_eq_right a b : Int h : a ≤ b ⊢ max a b = b simp [Int.max_def, h, Int.not_lt.2 h] ** Qed
Int.max_eq_left a b : Int h : b ≤ a ⊢ max a b = a rw [← Int.max_comm b a] a b : Int h : b ≤ a ⊢ max b a = a exact Int.max_eq_right h ** Qed
Int.eq_natAbs_of_zero_le a : Int h : 0 ≤ a ⊢ a = ↑(natAbs a) let ⟨n, e⟩ := eq_ofNat_of_zero_le h a : Int h : 0 ≤ a n : Nat e : a = ↑n ⊢ a = ↑(natAbs a) rw [e] a : Int h : 0 ≤ a n : Nat e : a = ↑n ⊢ ↑n = ↑(natAbs ↑n) rfl ** Qed
Int.le_natAbs a : Int h : 0 ≤ a ⊢ a ≤ ↑(natAbs a) rw [eq_natAbs_of_zero_le h] a : Int h : 0 ≤ a ⊢ ↑(natAbs a) ≤ ↑(natAbs ↑(natAbs a)) apply Int.le_refl ** Qed
Int.negSucc_not_nonneg n : Nat ⊢ 0 ≤ -[n+1] ↔ False simp only [Int.not_le, iff_false] n : Nat ⊢ -[n+1] < 0 exact Int.negSucc_lt_zero n ** Qed
Int.negSucc_not_pos n : Nat ⊢ 0 < -[n+1] ↔ False simp only [Int.not_lt, iff_false] n : Nat ⊢ -[n+1] ≤ 0 constructor ** Qed
Int.add_le_add_left a b : Int h : a ≤ b c : Int n : Nat hn : a + ↑n = b ⊢ c + a + ↑n = c + b rw [Int.add_assoc, hn] ** Qed
Int.add_lt_add_left a b : Int h : a < b c : Int heq : c + a = c + b ⊢ b < b rwa [Int.add_left_cancel heq] at h ** Qed
Int.le_of_add_le_add_left a b c : Int h : a + b ≤ a + c ⊢ b ≤ c have : -a + (a + b) ≤ -a + (a + c) := Int.add_le_add_left h _ a b c : Int h : a + b ≤ a + c this : -a + (a + b) ≤ -a + (a + c) ⊢ b ≤ c simp [Int.neg_add_cancel_left] at this a b c : Int h : a + b ≤ a + c this : b ≤ c ⊢ b ≤ c assumption ** Qed
Int.lt_of_add_lt_add_left a b c : Int h : a + b < a + c ⊢ b < c have : -a + (a + b) < -a + (a + c) := Int.add_lt_add_left h _ a b c : Int h : a + b < a + c this : -a + (a + b) < -a + (a + c) ⊢ b < c simp [Int.neg_add_cancel_left] at this a b c : Int h : a + b < a + c this : b < c ⊢ b < c assumption ** Qed
Int.le_of_add_le_add_right a b c : Int h : a + b ≤ c + b ⊢ b + a ≤ b + c rwa [Int.add_comm b a, Int.add_comm b c] ** Qed
Int.lt_of_add_lt_add_right a b c : Int h : a + b < c + b ⊢ b + a < b + c rwa [Int.add_comm b a, Int.add_comm b c] ** Qed
Int.le_add_of_nonneg_right a b : Int h : 0 ≤ b ⊢ a ≤ a + b have : a + b ≥ a + 0 := Int.add_le_add_left h a a b : Int h : 0 ≤ b this : a + b ≥ a + 0 ⊢ a ≤ a + b rwa [Int.add_zero] at this ** Qed
Int.le_add_of_nonneg_left a b : Int h : 0 ≤ b ⊢ a ≤ b + a have : 0 + a ≤ b + a := Int.add_le_add_right h a a b : Int h : 0 ≤ b this : 0 + a ≤ b + a ⊢ a ≤ b + a rwa [Int.zero_add] at this ** Qed
Int.lt_add_of_pos_right a b : Int h : 0 < b ⊢ a < a + b have : a + 0 < a + b := Int.add_lt_add_left h a a b : Int h : 0 < b this : a + 0 < a + b ⊢ a < a + b rwa [Int.add_zero] at this ** Qed
Int.lt_add_of_pos_left a b : Int h : 0 < b ⊢ a < b + a have : 0 + a < b + a := Int.add_lt_add_right h a a b : Int h : 0 < b this : 0 + a < b + a ⊢ a < b + a rwa [Int.zero_add] at this ** Qed
Int.neg_le_neg a b : Int h : a ≤ b ⊢ -b ≤ -a have : 0 ≤ -a + b := Int.add_left_neg a ▸ Int.add_le_add_left h (-a) a b : Int h : a ≤ b this : 0 ≤ -a + b ⊢ -b ≤ -a have : 0 + -b ≤ -a + b + -b := Int.add_le_add_right this (-b) a b : Int h : a ≤ b this✝ : 0 ≤ -a + b this : 0 + -b ≤ -a + b + -b ⊢ -b ≤ -a rwa [Int.add_neg_cancel_right, Int.zero_add] at this ** Qed
Int.le_of_neg_le_neg a b : Int h : -b ≤ -a this : - -a ≤ - -b ⊢ a ≤ b simp [Int.neg_neg] at this a b : Int h : -b ≤ -a this : a ≤ b ⊢ a ≤ b assumption ** Qed
Int.nonneg_of_neg_nonpos a : Int h : -a ≤ 0 ⊢ -a ≤ -0 rwa [Int.neg_zero] ** Qed
Int.neg_nonpos_of_nonneg a : Int h : 0 ≤ a ⊢ -a ≤ 0 have : -a ≤ -0 := Int.neg_le_neg h a : Int h : 0 ≤ a this : -a ≤ -0 ⊢ -a ≤ 0 rwa [Int.neg_zero] at this ** Qed
Int.nonpos_of_neg_nonneg a : Int h : 0 ≤ -a ⊢ -0 ≤ -a rwa [Int.neg_zero] ** Qed
Int.neg_nonneg_of_nonpos a : Int h : a ≤ 0 ⊢ 0 ≤ -a have : -0 ≤ -a := Int.neg_le_neg h a : Int h : a ≤ 0 this : -0 ≤ -a ⊢ 0 ≤ -a rwa [Int.neg_zero] at this ** Qed
Int.neg_lt_neg a b : Int h : a < b ⊢ -b < -a have : 0 < -a + b := Int.add_left_neg a ▸ Int.add_lt_add_left h (-a) a b : Int h : a < b this : 0 < -a + b ⊢ -b < -a have : 0 + -b < -a + b + -b := Int.add_lt_add_right this (-b) a b : Int h : a < b this✝ : 0 < -a + b this : 0 + -b < -a + b + -b ⊢ -b < -a rwa [Int.add_neg_cancel_right, Int.zero_add] at this ** Qed
Int.pos_of_neg_neg a : Int h : -a < 0 ⊢ -a < -0 rwa [Int.neg_zero] ** Qed
Int.neg_neg_of_pos a : Int h : 0 < a ⊢ -a < 0 have : -a < -0 := Int.neg_lt_neg h a : Int h : 0 < a this : -a < -0 ⊢ -a < 0 rwa [Int.neg_zero] at this ** Qed
Int.neg_of_neg_pos a : Int h : 0 < -a ⊢ -0 < -a rwa [Int.neg_zero] ** Qed
Int.neg_pos_of_neg a : Int h : a < 0 ⊢ 0 < -a have : -0 < -a := Int.neg_lt_neg h a : Int h : a < 0 this : -0 < -a ⊢ 0 < -a rwa [Int.neg_zero] at this ** Qed
Int.le_neg_of_le_neg a b : Int h : a ≤ -b ⊢ b ≤ -a have h := Int.neg_le_neg h a b : Int h✝ : a ≤ -b h : - -b ≤ -a ⊢ b ≤ -a rwa [Int.neg_neg] at h ** Qed
Int.neg_le_of_neg_le a b : Int h : -a ≤ b ⊢ -b ≤ a have h := Int.neg_le_neg h a b : Int h✝ : -a ≤ b h : -b ≤ - -a ⊢ -b ≤ a rwa [Int.neg_neg] at h ** Qed
Int.lt_neg_of_lt_neg a b : Int h : a < -b ⊢ b < -a have h := Int.neg_lt_neg h a b : Int h✝ : a < -b h : - -b < -a ⊢ b < -a rwa [Int.neg_neg] at h ** Qed
Int.neg_lt_of_neg_lt a b : Int h : -a < b ⊢ -b < a have h := Int.neg_lt_neg h a b : Int h✝ : -a < b h : -b < - -a ⊢ -b < a rwa [Int.neg_neg] at h ** Qed
Int.sub_nonneg_of_le a b : Int h : b ≤ a ⊢ 0 ≤ a - b have h := Int.add_le_add_right h (-b) a b : Int h✝ : b ≤ a h : b + -b ≤ a + -b ⊢ 0 ≤ a - b rwa [Int.add_right_neg] at h ** Qed
Int.le_of_sub_nonneg a b : Int h : 0 ≤ a - b ⊢ b ≤ a have h := Int.add_le_add_right h b a b : Int h✝ : 0 ≤ a - b h : 0 + b ≤ a - b + b ⊢ b ≤ a rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed
Int.sub_nonpos_of_le a b : Int h : a ≤ b ⊢ a - b ≤ 0 have h := Int.add_le_add_right h (-b) a b : Int h✝ : a ≤ b h : a + -b ≤ b + -b ⊢ a - b ≤ 0 rwa [Int.add_right_neg] at h ** Qed
Int.le_of_sub_nonpos a b : Int h : a - b ≤ 0 ⊢ a ≤ b have h := Int.add_le_add_right h b a b : Int h✝ : a - b ≤ 0 h : a - b + b ≤ 0 + b ⊢ a ≤ b rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed
Int.sub_pos_of_lt a b : Int h : b < a ⊢ 0 < a - b have h := Int.add_lt_add_right h (-b) a b : Int h✝ : b < a h : b + -b < a + -b ⊢ 0 < a - b rwa [Int.add_right_neg] at h ** Qed
Int.lt_of_sub_pos a b : Int h : 0 < a - b ⊢ b < a have h := Int.add_lt_add_right h b a b : Int h✝ : 0 < a - b h : 0 + b < a - b + b ⊢ b < a rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed
Int.sub_neg_of_lt a b : Int h : a < b ⊢ a - b < 0 have h := Int.add_lt_add_right h (-b) a b : Int h✝ : a < b h : a + -b < b + -b ⊢ a - b < 0 rwa [Int.add_right_neg] at h ** Qed
Int.lt_of_sub_neg a b : Int h : a - b < 0 ⊢ a < b have h := Int.add_lt_add_right h b a b : Int h✝ : a - b < 0 h : a - b + b < 0 + b ⊢ a < b rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed
Int.add_le_of_le_neg_add a b c : Int h : b ≤ -a + c ⊢ a + b ≤ c have h := Int.add_le_add_left h a a b c : Int h✝ : b ≤ -a + c h : a + b ≤ a + (-a + c) ⊢ a + b ≤ c rwa [Int.add_neg_cancel_left] at h ** Qed
Int.le_neg_add_of_add_le a b c : Int h : a + b ≤ c ⊢ b ≤ -a + c have h := Int.add_le_add_left h (-a) a b c : Int h✝ : a + b ≤ c h : -a + (a + b) ≤ -a + c ⊢ b ≤ -a + c rwa [Int.neg_add_cancel_left] at h ** Qed
Int.add_le_of_le_sub_left a b c : Int h : b ≤ c - a ⊢ a + b ≤ c have h := Int.add_le_add_left h a a b c : Int h✝ : b ≤ c - a h : a + b ≤ a + (c - a) ⊢ a + b ≤ c rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h ** Qed
Int.le_sub_left_of_add_le a b c : Int h : a + b ≤ c ⊢ b ≤ c - a have h := Int.add_le_add_right h (-a) a b c : Int h✝ : a + b ≤ c h : a + b + -a ≤ c + -a ⊢ b ≤ c - a rwa [Int.add_comm a b, Int.add_neg_cancel_right] at h ** Qed
Int.add_le_of_le_sub_right a b c : Int h : a ≤ c - b ⊢ a + b ≤ c have h := Int.add_le_add_right h b a b c : Int h✝ : a ≤ c - b h : a + b ≤ c - b + b ⊢ a + b ≤ c rwa [Int.sub_add_cancel] at h ** Qed
Int.le_sub_right_of_add_le a b c : Int h : a + b ≤ c ⊢ a ≤ c - b have h := Int.add_le_add_right h (-b) a b c : Int h✝ : a + b ≤ c h : a + b + -b ≤ c + -b ⊢ a ≤ c - b rwa [Int.add_neg_cancel_right] at h ** Qed
Int.le_add_of_neg_add_le a b c : Int h : -b + a ≤ c ⊢ a ≤ b + c have h := Int.add_le_add_left h b a b c : Int h✝ : -b + a ≤ c h : b + (-b + a) ≤ b + c ⊢ a ≤ b + c rwa [Int.add_neg_cancel_left] at h ** Qed
Int.neg_add_le_of_le_add a b c : Int h : a ≤ b + c ⊢ -b + a ≤ c have h := Int.add_le_add_left h (-b) a b c : Int h✝ : a ≤ b + c h : -b + a ≤ -b + (b + c) ⊢ -b + a ≤ c rwa [Int.neg_add_cancel_left] at h ** Qed
Int.le_add_of_sub_left_le a b c : Int h : a - b ≤ c ⊢ a ≤ b + c have h := Int.add_le_add_right h b a b c : Int h✝ : a - b ≤ c h : a - b + b ≤ c + b ⊢ a ≤ b + c rwa [Int.sub_add_cancel, Int.add_comm] at h ** Qed
Int.sub_left_le_of_le_add a b c : Int h : a ≤ b + c ⊢ a - b ≤ c have h := Int.add_le_add_right h (-b) a b c : Int h✝ : a ≤ b + c h : a + -b ≤ b + c + -b ⊢ a - b ≤ c rwa [Int.add_comm b c, Int.add_neg_cancel_right] at h ** Qed
Int.le_add_of_sub_right_le a b c : Int h : a - c ≤ b ⊢ a ≤ b + c have h := Int.add_le_add_right h c a b c : Int h✝ : a - c ≤ b h : a - c + c ≤ b + c ⊢ a ≤ b + c rwa [Int.sub_add_cancel] at h ** Qed
Int.sub_right_le_of_le_add a b c : Int h : a ≤ b + c ⊢ a - c ≤ b have h := Int.add_le_add_right h (-c) a b c : Int h✝ : a ≤ b + c h : a + -c ≤ b + c + -c ⊢ a - c ≤ b rwa [Int.add_neg_cancel_right] at h ** Qed
Int.neg_add_le_left_of_le_add a b c : Int h : a ≤ b + c ⊢ -b + a ≤ c rw [Int.add_comm] a b c : Int h : a ≤ b + c ⊢ a + -b ≤ c exact Int.sub_left_le_of_le_add h ** Qed
Int.le_add_of_neg_add_le_right a b c : Int h : -c + a ≤ b ⊢ a ≤ b + c rw [Int.add_comm] at h a b c : Int h : a + -c ≤ b ⊢ a ≤ b + c exact Int.le_add_of_sub_right_le h ** Qed
Int.add_lt_of_lt_neg_add a b c : Int h : b < -a + c ⊢ a + b < c have h := Int.add_lt_add_left h a a b c : Int h✝ : b < -a + c h : a + b < a + (-a + c) ⊢ a + b < c rwa [Int.add_neg_cancel_left] at h ** Qed
Int.lt_neg_add_of_add_lt a b c : Int h : a + b < c ⊢ b < -a + c have h := Int.add_lt_add_left h (-a) a b c : Int h✝ : a + b < c h : -a + (a + b) < -a + c ⊢ b < -a + c rwa [Int.neg_add_cancel_left] at h ** Qed
Int.add_lt_of_lt_sub_left a b c : Int h : b < c - a ⊢ a + b < c have h := Int.add_lt_add_left h a a b c : Int h✝ : b < c - a h : a + b < a + (c - a) ⊢ a + b < c rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h ** Qed
Int.lt_sub_left_of_add_lt a b c : Int h : a + b < c ⊢ b < c - a have h := Int.add_lt_add_right h (-a) a b c : Int h✝ : a + b < c h : a + b + -a < c + -a ⊢ b < c - a rwa [Int.add_comm a b, Int.add_neg_cancel_right] at h ** Qed
Int.add_lt_of_lt_sub_right a b c : Int h : a < c - b ⊢ a + b < c have h := Int.add_lt_add_right h b a b c : Int h✝ : a < c - b h : a + b < c - b + b ⊢ a + b < c rwa [Int.sub_add_cancel] at h ** Qed
Int.lt_sub_right_of_add_lt a b c : Int h : a + b < c ⊢ a < c - b have h := Int.add_lt_add_right h (-b) a b c : Int h✝ : a + b < c h : a + b + -b < c + -b ⊢ a < c - b rwa [Int.add_neg_cancel_right] at h ** Qed
Int.lt_add_of_neg_add_lt a b c : Int h : -b + a < c ⊢ a < b + c have h := Int.add_lt_add_left h b a b c : Int h✝ : -b + a < c h : b + (-b + a) < b + c ⊢ a < b + c rwa [Int.add_neg_cancel_left] at h ** Qed
Int.neg_add_lt_of_lt_add a b c : Int h : a < b + c ⊢ -b + a < c have h := Int.add_lt_add_left h (-b) a b c : Int h✝ : a < b + c h : -b + a < -b + (b + c) ⊢ -b + a < c rwa [Int.neg_add_cancel_left] at h ** Qed
Int.lt_add_of_sub_left_lt a b c : Int h : a - b < c ⊢ a < b + c have h := Int.add_lt_add_right h b a b c : Int h✝ : a - b < c h : a - b + b < c + b ⊢ a < b + c rwa [Int.sub_add_cancel, Int.add_comm] at h ** Qed
Int.sub_left_lt_of_lt_add a b c : Int h : a < b + c ⊢ a - b < c have h := Int.add_lt_add_right h (-b) a b c : Int h✝ : a < b + c h : a + -b < b + c + -b ⊢ a - b < c rwa [Int.add_comm b c, Int.add_neg_cancel_right] at h ** Qed
Int.lt_add_of_sub_right_lt a b c : Int h : a - c < b ⊢ a < b + c have h := Int.add_lt_add_right h c a b c : Int h✝ : a - c < b h : a - c + c < b + c ⊢ a < b + c rwa [Int.sub_add_cancel] at h ** Qed
Int.sub_right_lt_of_lt_add a b c : Int h : a < b + c ⊢ a - c < b have h := Int.add_lt_add_right h (-c) a b c : Int h✝ : a < b + c h : a + -c < b + c + -c ⊢ a - c < b rwa [Int.add_neg_cancel_right] at h ** Qed
Int.lt_add_of_neg_add_lt_left a b c : Int h : -b + a < c ⊢ a < b + c rw [Int.add_comm] at h a b c : Int h : a + -b < c ⊢ a < b + c exact Int.lt_add_of_sub_left_lt h ** Qed
Int.neg_add_lt_left_of_lt_add a b c : Int h : a < b + c ⊢ -b + a < c rw [Int.add_comm] a b c : Int h : a < b + c ⊢ a + -b < c exact Int.sub_left_lt_of_lt_add h ** Qed
Int.lt_add_of_neg_add_lt_right a b c : Int h : -c + a < b ⊢ a < b + c rw [Int.add_comm] at h a b c : Int h : a + -c < b ⊢ a < b + c exact Int.lt_add_of_sub_right_lt h ** Qed
Int.sub_le_self a b : Int h : 0 ≤ b ⊢ a + 0 = a rw [Int.add_zero] ** Qed
Int.mul_lt_mul_of_pos_left ** a b c : Int h₁ : a < b h₂ : 0 < c ⊢ c * a < c * b ** have : 0 < c * (b - a) := Int.mul_pos h₂ (Int.sub_pos_of_lt h₁) ** a b c : Int h₁ : a < b h₂ : 0 < c this : 0 < c * (b - a) ⊢ c * a < c * b rw [Int.mul_sub] at this a b c : Int h₁ : a < b h₂ : 0 < c this : 0 < c * b - c * a ⊢ c * a < c * b exact Int.lt_of_sub_pos this Qed
Int.mul_lt_mul_of_pos_right ** a b c : Int h₁ : a < b h₂ : 0 < c ⊢ a * c < b * c have : 0 < b - a := Int.sub_pos_of_lt h₁ a b c : Int h₁ : a < b h₂ : 0 < c this : 0 < b - a ⊢ a * c < b * c ** have : 0 < (b - a) * c := Int.mul_pos this h₂ ** a b c : Int h₁ : a < b h₂ : 0 < c this✝ : 0 < b - a this : 0 < (b - a) * c ⊢ a * c < b * c rw [Int.sub_mul] at this a b c : Int h₁ : a < b h₂ : 0 < c this✝ : 0 < b - a this : 0 < b * c - a * c ⊢ a * c < b * c exact Int.lt_of_sub_pos this Qed
Int.mul_le_mul_of_nonneg_left ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c * a ≤ c * b if hba : b ≤ a then rw [Int.le_antisymm hba h₁]; apply Int.le_refl else if hc0 : c ≤ 0 then simp [Int.le_antisymm hc0 h₂, Int.zero_mul] else exact Int.le_of_lt <\| Int.mul_lt_mul_of_pos_left (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩) a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : b ≤ a ⊢ c * a ≤ c * b rw [Int.le_antisymm hba h₁] a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : b ≤ a ⊢ c * a ≤ c * a apply Int.le_refl a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : ¬b ≤ a ⊢ c * a ≤ c * b if hc0 : c ≤ 0 then simp [Int.le_antisymm hc0 h₂, Int.zero_mul] else exact Int.le_of_lt <\| Int.mul_lt_mul_of_pos_left (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩) a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : ¬b ≤ a hc0 : c ≤ 0 ⊢ c * a ≤ c * b simp [Int.le_antisymm hc0 h₂, Int.zero_mul] a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : ¬b ≤ a hc0 : ¬c ≤ 0 ⊢ c * a ≤ c * b exact Int.le_of_lt <\| Int.mul_lt_mul_of_pos_left (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩) Qed
Int.mul_le_mul_of_nonneg_right ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c ⊢ a * c ≤ b * c rw [Int.mul_comm, Int.mul_comm b] a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c * a ≤ c * b exact Int.mul_le_mul_of_nonneg_left h₁ h₂ Qed
Int.mul_nonpos_of_nonneg_of_nonpos ** a b : Int ha : 0 ≤ a hb : b ≤ 0 ⊢ a * b ≤ 0 ** have h : a * b ≤ a * 0 := Int.mul_le_mul_of_nonneg_left hb ha ** a b : Int ha : 0 ≤ a hb : b ≤ 0 h : a * b ≤ a * 0 ⊢ a * b ≤ 0 rwa [Int.mul_zero] at h Qed
Int.mul_nonpos_of_nonpos_of_nonneg ** a b : Int ha : a ≤ 0 hb : 0 ≤ b ⊢ a * b ≤ 0 ** have h : a * b ≤ 0 * b := Int.mul_le_mul_of_nonneg_right ha hb ** a b : Int ha : a ≤ 0 hb : 0 ≤ b h : a * b ≤ 0 * b ⊢ a * b ≤ 0 rwa [Int.zero_mul] at h Qed
Int.mul_neg_of_pos_of_neg ** a b : Int ha : 0 < a hb : b < 0 ⊢ a * b < 0 ** have h : a * b < a * 0 := Int.mul_lt_mul_of_pos_left hb ha ** a b : Int ha : 0 < a hb : b < 0 h : a * b < a * 0 ⊢ a * b < 0 rwa [Int.mul_zero] at h Qed
Int.mul_neg_of_neg_of_pos ** a b : Int ha : a < 0 hb : 0 < b ⊢ a * b < 0 ** have h : a * b < 0 * b := Int.mul_lt_mul_of_pos_right ha hb ** a b : Int ha : a < 0 hb : 0 < b h : a * b < 0 * b ⊢ a * b < 0 rwa [Int.zero_mul] at h Qed
Int.mul_le_mul_of_nonpos_right ** a b c : Int h : b ≤ a hc : c ≤ 0 this✝ : -c ≥ 0 this : b * -c ≤ a * -c ⊢ -(b * c) ≤ -(a * c) rwa [← Int.neg_mul_eq_mul_neg, ← Int.neg_mul_eq_mul_neg] at this Qed
Int.mul_lt_mul_of_neg_left ** a b c : Int h : b < a hc : c < 0 this✝ : -c > 0 this : -c * b < -c * a ⊢ -(c * b) < -(c * a) rwa [← Int.neg_mul_eq_neg_mul, ← Int.neg_mul_eq_neg_mul] at this Qed
Int.mul_lt_mul_of_neg_right ** a b c : Int h : b < a hc : c < 0 this✝ : -c > 0 this : b * -c < a * -c ⊢ -(b * c) < -(a * c) rwa [← Int.neg_mul_eq_mul_neg, ← Int.neg_mul_eq_mul_neg] at this Qed
Int.ofNat_natAbs_of_nonpos a : Int H : a ≤ 0 ⊢ ↑(natAbs a) = -a rw [← natAbs_neg, natAbs_of_nonneg (Int.neg_nonneg_of_nonpos H)] ** Qed
Int.mul_eq_zero ** a b : Int h : a * b = 0 ⊢ a = 0 ∨ b = 0 exact match Int.lt_trichotomy a 0, Int.lt_trichotomy b 0 with \| .inr (.inl heq₁), _ => .inl heq₁ \| _, .inr (.inl heq₂) => .inr heq₂ \| .inl hlt₁, .inl hlt₂ => absurd h <\| Int.ne_of_gt <\| Int.mul_pos_of_neg_of_neg hlt₁ hlt₂ \| .inl hlt₁, .inr (.inr hgt₂) => absurd h <\| Int.ne_of_lt <\| Int.mul_neg_of_neg_of_pos hlt₁ hgt₂ \| .inr (.inr hgt₁), .inl hlt₂ => absurd h <\| Int.ne_of_lt <\| Int.mul_neg_of_pos_of_neg hgt₁ hlt₂ \| .inr (.inr hgt₁), .inr (.inr hgt₂) => absurd h <\| Int.ne_of_gt <\| Int.mul_pos hgt₁ hgt₂ Qed

Int.le.dest ** a b : Int h : a ≤ b n : Nat h₁ : b - a = ↑n ⊢ a + ↑n = b ** rw [← h₁, Int.add_comm] ** a b : Int h : a ≤ b n : Nat h₁ : b - a = ↑n ⊢ b - a + a = b ** simp [Int.sub_eq_add_neg, Int.add_assoc] ** Qed

Int.ofNat_le ** m n : Nat h : m ≤ n k : Nat hk : m + k = n ⊢ ↑m + ↑k = ↑n ** rw [← hk] ** m n : Nat h : m ≤ n k : Nat hk : m + k = n ⊢ ↑m + ↑k = ↑(m + k) ** rfl ** Qed

Int.eq_ofNat_of_zero_le ** a : Int h : 0 ≤ a ⊢ ∃ n, a = ↑n ** have t := le.dest_sub h ** a : Int h : 0 ≤ a t : ∃ n, a - 0 = ↑n ⊢ ∃ n, a = ↑n ** rwa [Int.sub_zero] at t ** Qed

Int.eq_succ_of_zero_lt ** a : Int h✝ : 0 < a n : Nat h : ↑(1 + n) = a ⊢ a = ↑(succ n) ** rw [Nat.add_comm] at h ** a : Int h✝ : 0 < a n : Nat h : ↑(n + 1) = a ⊢ a = ↑(succ n) ** exact h.symm ** Qed

Int.lt_add_succ ** a : Int n : Nat ⊢ a + 1 + ↑n = a + ↑(succ n) ** rw [Int.add_comm, Int.add_left_comm] ** a : Int n : Nat ⊢ a + (↑n + 1) = a + ↑(succ n) ** rfl ** Qed

Int.lt.dest ** a b : Int h✝ : a < b n : Nat h : a + 1 + ↑n = b ⊢ a + ↑(succ n) = b ** rwa [Int.add_comm, Int.add_left_comm] at h ** Qed

Int.ofNat_lt ** n m : Nat ⊢ ↑n < ↑m ↔ n < m ** rw [lt_iff_add_one_le, ← ofNat_succ, ofNat_le] ** n m : Nat ⊢ succ n ≤ m ↔ n < m ** rfl ** Qed

Int.le_trans ** a b c : Int h₁ : a ≤ b h₂ : b ≤ c n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = c ⊢ a + ↑(n + m) = c ** rw [← hm, ← hn, Int.add_assoc, ofNat_add] ** Qed

Int.le_antisymm ** a b : Int h₁ : a ≤ b h₂ : b ≤ a ⊢ a = b ** let ⟨n, hn⟩ := le.dest h₁ ** a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b ⊢ a = b ** let ⟨m, hm⟩ := le.dest h₂ ** a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a ⊢ a = b ** have := hn ** a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a this : a + ↑n = b ⊢ a = b ** rw [← hm, Int.add_assoc, ← ofNat_add] at this ** a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a this : b + ↑(m + n) = b ⊢ a = b ** have := Int.ofNat.inj <| Int.add_left_cancel <| this.trans (Int.add_zero _).symm ** a b : Int h₁ : a ≤ b h₂ : b ≤ a n : Nat hn : a + ↑n = b m : Nat hm : b + ↑m = a this✝ : b + ↑(m + n) = b this : m + n = 0 ⊢ a = b ** rw [← hn, Nat.eq_zero_of_add_eq_zero_left this, ofNat_zero, Int.add_zero a] ** Qed

Int.lt_irrefl ** a : Int H : a < a n : Nat hn : a + ↑(succ n) = a ⊢ a + ↑(succ n) = a + 0 ** rw [hn, Int.add_zero] ** Qed

Int.mul_nonneg ** a b : Int ha : 0 ≤ a hb : 0 ≤ b ⊢ 0 ≤ a * b ** let ⟨n, hn⟩ := eq_ofNat_of_zero_le ha ** a b : Int ha : 0 ≤ a hb : 0 ≤ b n : Nat hn : a = ↑n ⊢ 0 ≤ a * b ** let ⟨m, hm⟩ := eq_ofNat_of_zero_le hb ** a b : Int ha : 0 ≤ a hb : 0 ≤ b n : Nat hn : a = ↑n m : Nat hm : b = ↑m ⊢ 0 ≤ a * b ** rw [hn, hm, ← ofNat_mul] ** a b : Int ha : 0 ≤ a hb : 0 ≤ b n : Nat hn : a = ↑n m : Nat hm : b = ↑m ⊢ 0 ≤ ↑(n * m) ** apply ofNat_nonneg ** Qed

Int.lt_iff_le_not_le ** a b : Int ⊢ a < b ↔ a ≤ b ∧ ¬b ≤ a ** rw [Int.lt_iff_le_and_ne] ** a b : Int ⊢ a ≤ b ∧ a ≠ b ↔ a ≤ b ∧ ¬b ≤ a ** constructor <;> refine fun ⟨h, h'⟩ => ⟨h, h'.imp fun h' => ?_⟩ ** case mp a b : Int x✝ : a ≤ b ∧ a ≠ b h : a ≤ b h'✝ : a ≠ b h' : b ≤ a ⊢ a = b ** exact Int.le_antisymm h h' ** case mpr a b : Int x✝ : a ≤ b ∧ ¬b ≤ a h : a ≤ b h'✝ : ¬b ≤ a h' : a = b ⊢ b ≤ a ** subst h' ** case mpr a : Int x✝ : a ≤ a ∧ ¬a ≤ a h : a ≤ a h' : ¬a ≤ a ⊢ a ≤ a ** apply Int.le_refl ** Qed

Int.not_lt ** a b : Int ⊢ ¬a < b ↔ b ≤ a ** rw [← Int.not_le, Decidable.not_not] ** Qed

Int.min_comm ** a b : Int ⊢ min a b = min b a ** simp [Int.min_def] ** a b : Int ⊢ (if a ≤ b then a else b) = if b ≤ a then b else a ** by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂] ** case pos a b : Int h₁ : a ≤ b h₂ : b ≤ a ⊢ a = b ** exact Int.le_antisymm h₁ h₂ ** case neg a b : Int h₁ : ¬a ≤ b h₂ : ¬b ≤ a ⊢ b = a ** cases not_or_intro h₁ h₂ <| Int.le_total .. ** Qed

Int.min_le_right ** a b : Int ⊢ min a b ≤ b ** rw [Int.min_def] ** a b : Int ⊢ (if a ≤ b then a else b) ≤ b ** split <;> simp [*] ** Qed

Int.min_eq_left ** a b : Int h : a ≤ b ⊢ min a b = a ** simp [Int.min_def, h] ** Qed

Int.max_comm ** a b : Int ⊢ max a b = max b a ** simp only [Int.max_def] ** a b : Int ⊢ (if a ≤ b then b else a) = if b ≤ a then a else b ** by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂] ** case pos a b : Int h₁ : a ≤ b h₂ : b ≤ a ⊢ b = a ** exact Int.le_antisymm h₂ h₁ ** case neg a b : Int h₁ : ¬a ≤ b h₂ : ¬b ≤ a ⊢ a = b ** cases not_or_intro h₁ h₂ <| Int.le_total .. ** Qed

Int.le_max_left ** a b : Int ⊢ a ≤ max a b ** rw [Int.max_def] ** a b : Int ⊢ a ≤ if a ≤ b then b else a ** split <;> simp [*] ** Qed

Int.max_eq_right ** a b : Int h : a ≤ b ⊢ max a b = b ** simp [Int.max_def, h, Int.not_lt.2 h] ** Qed

Int.max_eq_left ** a b : Int h : b ≤ a ⊢ max a b = a ** rw [← Int.max_comm b a] ** a b : Int h : b ≤ a ⊢ max b a = a ** exact Int.max_eq_right h ** Qed

Int.eq_natAbs_of_zero_le ** a : Int h : 0 ≤ a ⊢ a = ↑(natAbs a) ** let ⟨n, e⟩ := eq_ofNat_of_zero_le h ** a : Int h : 0 ≤ a n : Nat e : a = ↑n ⊢ a = ↑(natAbs a) ** rw [e] ** a : Int h : 0 ≤ a n : Nat e : a = ↑n ⊢ ↑n = ↑(natAbs ↑n) ** rfl ** Qed

Int.le_natAbs ** a : Int h : 0 ≤ a ⊢ a ≤ ↑(natAbs a) ** rw [eq_natAbs_of_zero_le h] ** a : Int h : 0 ≤ a ⊢ ↑(natAbs a) ≤ ↑(natAbs ↑(natAbs a)) ** apply Int.le_refl ** Qed

Int.negSucc_not_nonneg ** n : Nat ⊢ 0 ≤ -[n+1] ↔ False ** simp only [Int.not_le, iff_false] ** n : Nat ⊢ -[n+1] < 0 ** exact Int.negSucc_lt_zero n ** Qed

Int.negSucc_not_pos ** n : Nat ⊢ 0 < -[n+1] ↔ False ** simp only [Int.not_lt, iff_false] ** n : Nat ⊢ -[n+1] ≤ 0 ** constructor ** Qed

Int.add_le_add_left ** a b : Int h : a ≤ b c : Int n : Nat hn : a + ↑n = b ⊢ c + a + ↑n = c + b ** rw [Int.add_assoc, hn] ** Qed

Int.add_lt_add_left ** a b : Int h : a < b c : Int heq : c + a = c + b ⊢ b < b ** rwa [Int.add_left_cancel heq] at h ** Qed

Int.le_of_add_le_add_left ** a b c : Int h : a + b ≤ a + c ⊢ b ≤ c ** have : -a + (a + b) ≤ -a + (a + c) := Int.add_le_add_left h _ ** a b c : Int h : a + b ≤ a + c this : -a + (a + b) ≤ -a + (a + c) ⊢ b ≤ c ** simp [Int.neg_add_cancel_left] at this ** a b c : Int h : a + b ≤ a + c this : b ≤ c ⊢ b ≤ c ** assumption ** Qed

Int.lt_of_add_lt_add_left ** a b c : Int h : a + b < a + c ⊢ b < c ** have : -a + (a + b) < -a + (a + c) := Int.add_lt_add_left h _ ** a b c : Int h : a + b < a + c this : -a + (a + b) < -a + (a + c) ⊢ b < c ** simp [Int.neg_add_cancel_left] at this ** a b c : Int h : a + b < a + c this : b < c ⊢ b < c ** assumption ** Qed

Int.le_of_add_le_add_right ** a b c : Int h : a + b ≤ c + b ⊢ b + a ≤ b + c ** rwa [Int.add_comm b a, Int.add_comm b c] ** Qed

Int.lt_of_add_lt_add_right ** a b c : Int h : a + b < c + b ⊢ b + a < b + c ** rwa [Int.add_comm b a, Int.add_comm b c] ** Qed

Int.le_add_of_nonneg_right ** a b : Int h : 0 ≤ b ⊢ a ≤ a + b ** have : a + b ≥ a + 0 := Int.add_le_add_left h a ** a b : Int h : 0 ≤ b this : a + b ≥ a + 0 ⊢ a ≤ a + b ** rwa [Int.add_zero] at this ** Qed

Int.le_add_of_nonneg_left ** a b : Int h : 0 ≤ b ⊢ a ≤ b + a ** have : 0 + a ≤ b + a := Int.add_le_add_right h a ** a b : Int h : 0 ≤ b this : 0 + a ≤ b + a ⊢ a ≤ b + a ** rwa [Int.zero_add] at this ** Qed

Int.lt_add_of_pos_right ** a b : Int h : 0 < b ⊢ a < a + b ** have : a + 0 < a + b := Int.add_lt_add_left h a ** a b : Int h : 0 < b this : a + 0 < a + b ⊢ a < a + b ** rwa [Int.add_zero] at this ** Qed

Int.lt_add_of_pos_left ** a b : Int h : 0 < b ⊢ a < b + a ** have : 0 + a < b + a := Int.add_lt_add_right h a ** a b : Int h : 0 < b this : 0 + a < b + a ⊢ a < b + a ** rwa [Int.zero_add] at this ** Qed

Int.neg_le_neg ** a b : Int h : a ≤ b ⊢ -b ≤ -a ** have : 0 ≤ -a + b := Int.add_left_neg a ▸ Int.add_le_add_left h (-a) ** a b : Int h : a ≤ b this : 0 ≤ -a + b ⊢ -b ≤ -a ** have : 0 + -b ≤ -a + b + -b := Int.add_le_add_right this (-b) ** a b : Int h : a ≤ b this✝ : 0 ≤ -a + b this : 0 + -b ≤ -a + b + -b ⊢ -b ≤ -a ** rwa [Int.add_neg_cancel_right, Int.zero_add] at this ** Qed

Int.le_of_neg_le_neg ** a b : Int h : -b ≤ -a this : - -a ≤ - -b ⊢ a ≤ b ** simp [Int.neg_neg] at this ** a b : Int h : -b ≤ -a this : a ≤ b ⊢ a ≤ b ** assumption ** Qed

Int.nonneg_of_neg_nonpos ** a : Int h : -a ≤ 0 ⊢ -a ≤ -0 ** rwa [Int.neg_zero] ** Qed

Int.neg_nonpos_of_nonneg ** a : Int h : 0 ≤ a ⊢ -a ≤ 0 ** have : -a ≤ -0 := Int.neg_le_neg h ** a : Int h : 0 ≤ a this : -a ≤ -0 ⊢ -a ≤ 0 ** rwa [Int.neg_zero] at this ** Qed

Int.nonpos_of_neg_nonneg ** a : Int h : 0 ≤ -a ⊢ -0 ≤ -a ** rwa [Int.neg_zero] ** Qed

Int.neg_nonneg_of_nonpos ** a : Int h : a ≤ 0 ⊢ 0 ≤ -a ** have : -0 ≤ -a := Int.neg_le_neg h ** a : Int h : a ≤ 0 this : -0 ≤ -a ⊢ 0 ≤ -a ** rwa [Int.neg_zero] at this ** Qed

Int.neg_lt_neg ** a b : Int h : a < b ⊢ -b < -a ** have : 0 < -a + b := Int.add_left_neg a ▸ Int.add_lt_add_left h (-a) ** a b : Int h : a < b this : 0 < -a + b ⊢ -b < -a ** have : 0 + -b < -a + b + -b := Int.add_lt_add_right this (-b) ** a b : Int h : a < b this✝ : 0 < -a + b this : 0 + -b < -a + b + -b ⊢ -b < -a ** rwa [Int.add_neg_cancel_right, Int.zero_add] at this ** Qed

Int.pos_of_neg_neg ** a : Int h : -a < 0 ⊢ -a < -0 ** rwa [Int.neg_zero] ** Qed

Int.neg_neg_of_pos ** a : Int h : 0 < a ⊢ -a < 0 ** have : -a < -0 := Int.neg_lt_neg h ** a : Int h : 0 < a this : -a < -0 ⊢ -a < 0 ** rwa [Int.neg_zero] at this ** Qed

Int.neg_of_neg_pos ** a : Int h : 0 < -a ⊢ -0 < -a ** rwa [Int.neg_zero] ** Qed

Int.neg_pos_of_neg ** a : Int h : a < 0 ⊢ 0 < -a ** have : -0 < -a := Int.neg_lt_neg h ** a : Int h : a < 0 this : -0 < -a ⊢ 0 < -a ** rwa [Int.neg_zero] at this ** Qed

Int.le_neg_of_le_neg ** a b : Int h : a ≤ -b ⊢ b ≤ -a ** have h := Int.neg_le_neg h ** a b : Int h✝ : a ≤ -b h : - -b ≤ -a ⊢ b ≤ -a ** rwa [Int.neg_neg] at h ** Qed

Int.neg_le_of_neg_le ** a b : Int h : -a ≤ b ⊢ -b ≤ a ** have h := Int.neg_le_neg h ** a b : Int h✝ : -a ≤ b h : -b ≤ - -a ⊢ -b ≤ a ** rwa [Int.neg_neg] at h ** Qed

Int.lt_neg_of_lt_neg ** a b : Int h : a < -b ⊢ b < -a ** have h := Int.neg_lt_neg h ** a b : Int h✝ : a < -b h : - -b < -a ⊢ b < -a ** rwa [Int.neg_neg] at h ** Qed

Int.neg_lt_of_neg_lt ** a b : Int h : -a < b ⊢ -b < a ** have h := Int.neg_lt_neg h ** a b : Int h✝ : -a < b h : -b < - -a ⊢ -b < a ** rwa [Int.neg_neg] at h ** Qed

Int.sub_nonneg_of_le ** a b : Int h : b ≤ a ⊢ 0 ≤ a - b ** have h := Int.add_le_add_right h (-b) ** a b : Int h✝ : b ≤ a h : b + -b ≤ a + -b ⊢ 0 ≤ a - b ** rwa [Int.add_right_neg] at h ** Qed

Int.le_of_sub_nonneg ** a b : Int h : 0 ≤ a - b ⊢ b ≤ a ** have h := Int.add_le_add_right h b ** a b : Int h✝ : 0 ≤ a - b h : 0 + b ≤ a - b + b ⊢ b ≤ a ** rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed

Int.sub_nonpos_of_le ** a b : Int h : a ≤ b ⊢ a - b ≤ 0 ** have h := Int.add_le_add_right h (-b) ** a b : Int h✝ : a ≤ b h : a + -b ≤ b + -b ⊢ a - b ≤ 0 ** rwa [Int.add_right_neg] at h ** Qed

Int.le_of_sub_nonpos ** a b : Int h : a - b ≤ 0 ⊢ a ≤ b ** have h := Int.add_le_add_right h b ** a b : Int h✝ : a - b ≤ 0 h : a - b + b ≤ 0 + b ⊢ a ≤ b ** rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed

Int.sub_pos_of_lt ** a b : Int h : b < a ⊢ 0 < a - b ** have h := Int.add_lt_add_right h (-b) ** a b : Int h✝ : b < a h : b + -b < a + -b ⊢ 0 < a - b ** rwa [Int.add_right_neg] at h ** Qed

Int.lt_of_sub_pos ** a b : Int h : 0 < a - b ⊢ b < a ** have h := Int.add_lt_add_right h b ** a b : Int h✝ : 0 < a - b h : 0 + b < a - b + b ⊢ b < a ** rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed

Int.sub_neg_of_lt ** a b : Int h : a < b ⊢ a - b < 0 ** have h := Int.add_lt_add_right h (-b) ** a b : Int h✝ : a < b h : a + -b < b + -b ⊢ a - b < 0 ** rwa [Int.add_right_neg] at h ** Qed

Int.lt_of_sub_neg ** a b : Int h : a - b < 0 ⊢ a < b ** have h := Int.add_lt_add_right h b ** a b : Int h✝ : a - b < 0 h : a - b + b < 0 + b ⊢ a < b ** rwa [Int.sub_add_cancel, Int.zero_add] at h ** Qed

Int.add_le_of_le_neg_add ** a b c : Int h : b ≤ -a + c ⊢ a + b ≤ c ** have h := Int.add_le_add_left h a ** a b c : Int h✝ : b ≤ -a + c h : a + b ≤ a + (-a + c) ⊢ a + b ≤ c ** rwa [Int.add_neg_cancel_left] at h ** Qed

Int.le_neg_add_of_add_le ** a b c : Int h : a + b ≤ c ⊢ b ≤ -a + c ** have h := Int.add_le_add_left h (-a) ** a b c : Int h✝ : a + b ≤ c h : -a + (a + b) ≤ -a + c ⊢ b ≤ -a + c ** rwa [Int.neg_add_cancel_left] at h ** Qed

Int.add_le_of_le_sub_left ** a b c : Int h : b ≤ c - a ⊢ a + b ≤ c ** have h := Int.add_le_add_left h a ** a b c : Int h✝ : b ≤ c - a h : a + b ≤ a + (c - a) ⊢ a + b ≤ c ** rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h ** Qed

Int.le_sub_left_of_add_le ** a b c : Int h : a + b ≤ c ⊢ b ≤ c - a ** have h := Int.add_le_add_right h (-a) ** a b c : Int h✝ : a + b ≤ c h : a + b + -a ≤ c + -a ⊢ b ≤ c - a ** rwa [Int.add_comm a b, Int.add_neg_cancel_right] at h ** Qed

Int.add_le_of_le_sub_right ** a b c : Int h : a ≤ c - b ⊢ a + b ≤ c ** have h := Int.add_le_add_right h b ** a b c : Int h✝ : a ≤ c - b h : a + b ≤ c - b + b ⊢ a + b ≤ c ** rwa [Int.sub_add_cancel] at h ** Qed

Int.le_sub_right_of_add_le ** a b c : Int h : a + b ≤ c ⊢ a ≤ c - b ** have h := Int.add_le_add_right h (-b) ** a b c : Int h✝ : a + b ≤ c h : a + b + -b ≤ c + -b ⊢ a ≤ c - b ** rwa [Int.add_neg_cancel_right] at h ** Qed

Int.le_add_of_neg_add_le ** a b c : Int h : -b + a ≤ c ⊢ a ≤ b + c ** have h := Int.add_le_add_left h b ** a b c : Int h✝ : -b + a ≤ c h : b + (-b + a) ≤ b + c ⊢ a ≤ b + c ** rwa [Int.add_neg_cancel_left] at h ** Qed

Int.neg_add_le_of_le_add ** a b c : Int h : a ≤ b + c ⊢ -b + a ≤ c ** have h := Int.add_le_add_left h (-b) ** a b c : Int h✝ : a ≤ b + c h : -b + a ≤ -b + (b + c) ⊢ -b + a ≤ c ** rwa [Int.neg_add_cancel_left] at h ** Qed

Int.le_add_of_sub_left_le ** a b c : Int h : a - b ≤ c ⊢ a ≤ b + c ** have h := Int.add_le_add_right h b ** a b c : Int h✝ : a - b ≤ c h : a - b + b ≤ c + b ⊢ a ≤ b + c ** rwa [Int.sub_add_cancel, Int.add_comm] at h ** Qed

Int.sub_left_le_of_le_add ** a b c : Int h : a ≤ b + c ⊢ a - b ≤ c ** have h := Int.add_le_add_right h (-b) ** a b c : Int h✝ : a ≤ b + c h : a + -b ≤ b + c + -b ⊢ a - b ≤ c ** rwa [Int.add_comm b c, Int.add_neg_cancel_right] at h ** Qed

Int.le_add_of_sub_right_le ** a b c : Int h : a - c ≤ b ⊢ a ≤ b + c ** have h := Int.add_le_add_right h c ** a b c : Int h✝ : a - c ≤ b h : a - c + c ≤ b + c ⊢ a ≤ b + c ** rwa [Int.sub_add_cancel] at h ** Qed

Int.sub_right_le_of_le_add ** a b c : Int h : a ≤ b + c ⊢ a - c ≤ b ** have h := Int.add_le_add_right h (-c) ** a b c : Int h✝ : a ≤ b + c h : a + -c ≤ b + c + -c ⊢ a - c ≤ b ** rwa [Int.add_neg_cancel_right] at h ** Qed

Int.neg_add_le_left_of_le_add ** a b c : Int h : a ≤ b + c ⊢ -b + a ≤ c ** rw [Int.add_comm] ** a b c : Int h : a ≤ b + c ⊢ a + -b ≤ c ** exact Int.sub_left_le_of_le_add h ** Qed

Int.le_add_of_neg_add_le_right ** a b c : Int h : -c + a ≤ b ⊢ a ≤ b + c ** rw [Int.add_comm] at h ** a b c : Int h : a + -c ≤ b ⊢ a ≤ b + c ** exact Int.le_add_of_sub_right_le h ** Qed

Int.add_lt_of_lt_neg_add ** a b c : Int h : b < -a + c ⊢ a + b < c ** have h := Int.add_lt_add_left h a ** a b c : Int h✝ : b < -a + c h : a + b < a + (-a + c) ⊢ a + b < c ** rwa [Int.add_neg_cancel_left] at h ** Qed

Int.lt_neg_add_of_add_lt ** a b c : Int h : a + b < c ⊢ b < -a + c ** have h := Int.add_lt_add_left h (-a) ** a b c : Int h✝ : a + b < c h : -a + (a + b) < -a + c ⊢ b < -a + c ** rwa [Int.neg_add_cancel_left] at h ** Qed

Int.add_lt_of_lt_sub_left ** a b c : Int h : b < c - a ⊢ a + b < c ** have h := Int.add_lt_add_left h a ** a b c : Int h✝ : b < c - a h : a + b < a + (c - a) ⊢ a + b < c ** rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h ** Qed

Int.lt_sub_left_of_add_lt ** a b c : Int h : a + b < c ⊢ b < c - a ** have h := Int.add_lt_add_right h (-a) ** a b c : Int h✝ : a + b < c h : a + b + -a < c + -a ⊢ b < c - a ** rwa [Int.add_comm a b, Int.add_neg_cancel_right] at h ** Qed

Int.add_lt_of_lt_sub_right ** a b c : Int h : a < c - b ⊢ a + b < c ** have h := Int.add_lt_add_right h b ** a b c : Int h✝ : a < c - b h : a + b < c - b + b ⊢ a + b < c ** rwa [Int.sub_add_cancel] at h ** Qed

Int.lt_sub_right_of_add_lt ** a b c : Int h : a + b < c ⊢ a < c - b ** have h := Int.add_lt_add_right h (-b) ** a b c : Int h✝ : a + b < c h : a + b + -b < c + -b ⊢ a < c - b ** rwa [Int.add_neg_cancel_right] at h ** Qed

Int.lt_add_of_neg_add_lt ** a b c : Int h : -b + a < c ⊢ a < b + c ** have h := Int.add_lt_add_left h b ** a b c : Int h✝ : -b + a < c h : b + (-b + a) < b + c ⊢ a < b + c ** rwa [Int.add_neg_cancel_left] at h ** Qed

Int.neg_add_lt_of_lt_add ** a b c : Int h : a < b + c ⊢ -b + a < c ** have h := Int.add_lt_add_left h (-b) ** a b c : Int h✝ : a < b + c h : -b + a < -b + (b + c) ⊢ -b + a < c ** rwa [Int.neg_add_cancel_left] at h ** Qed

Int.lt_add_of_sub_left_lt ** a b c : Int h : a - b < c ⊢ a < b + c ** have h := Int.add_lt_add_right h b ** a b c : Int h✝ : a - b < c h : a - b + b < c + b ⊢ a < b + c ** rwa [Int.sub_add_cancel, Int.add_comm] at h ** Qed

Int.sub_left_lt_of_lt_add ** a b c : Int h : a < b + c ⊢ a - b < c ** have h := Int.add_lt_add_right h (-b) ** a b c : Int h✝ : a < b + c h : a + -b < b + c + -b ⊢ a - b < c ** rwa [Int.add_comm b c, Int.add_neg_cancel_right] at h ** Qed

Int.lt_add_of_sub_right_lt ** a b c : Int h : a - c < b ⊢ a < b + c ** have h := Int.add_lt_add_right h c ** a b c : Int h✝ : a - c < b h : a - c + c < b + c ⊢ a < b + c ** rwa [Int.sub_add_cancel] at h ** Qed

Int.sub_right_lt_of_lt_add ** a b c : Int h : a < b + c ⊢ a - c < b ** have h := Int.add_lt_add_right h (-c) ** a b c : Int h✝ : a < b + c h : a + -c < b + c + -c ⊢ a - c < b ** rwa [Int.add_neg_cancel_right] at h ** Qed

Int.lt_add_of_neg_add_lt_left ** a b c : Int h : -b + a < c ⊢ a < b + c ** rw [Int.add_comm] at h ** a b c : Int h : a + -b < c ⊢ a < b + c ** exact Int.lt_add_of_sub_left_lt h ** Qed

Int.neg_add_lt_left_of_lt_add ** a b c : Int h : a < b + c ⊢ -b + a < c ** rw [Int.add_comm] ** a b c : Int h : a < b + c ⊢ a + -b < c ** exact Int.sub_left_lt_of_lt_add h ** Qed

Int.lt_add_of_neg_add_lt_right ** a b c : Int h : -c + a < b ⊢ a < b + c ** rw [Int.add_comm] at h ** a b c : Int h : a + -c < b ⊢ a < b + c ** exact Int.lt_add_of_sub_right_lt h ** Qed

Int.sub_le_self ** a b : Int h : 0 ≤ b ⊢ a + 0 = a ** rw [Int.add_zero] ** Qed

Int.mul_lt_mul_of_pos_left ** a b c : Int h₁ : a < b h₂ : 0 < c ⊢ c * a < c * b ** have : 0 < c * (b - a) := Int.mul_pos h₂ (Int.sub_pos_of_lt h₁) ** a b c : Int h₁ : a < b h₂ : 0 < c this : 0 < c * (b - a) ⊢ c * a < c * b ** rw [Int.mul_sub] at this ** a b c : Int h₁ : a < b h₂ : 0 < c this : 0 < c * b - c * a ⊢ c * a < c * b ** exact Int.lt_of_sub_pos this ** Qed

Int.mul_lt_mul_of_pos_right ** a b c : Int h₁ : a < b h₂ : 0 < c ⊢ a * c < b * c ** have : 0 < b - a := Int.sub_pos_of_lt h₁ ** a b c : Int h₁ : a < b h₂ : 0 < c this : 0 < b - a ⊢ a * c < b * c ** have : 0 < (b - a) * c := Int.mul_pos this h₂ ** a b c : Int h₁ : a < b h₂ : 0 < c this✝ : 0 < b - a this : 0 < (b - a) * c ⊢ a * c < b * c ** rw [Int.sub_mul] at this ** a b c : Int h₁ : a < b h₂ : 0 < c this✝ : 0 < b - a this : 0 < b * c - a * c ⊢ a * c < b * c ** exact Int.lt_of_sub_pos this ** Qed

Int.mul_le_mul_of_nonneg_left ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c * a ≤ c * b ** if hba : b ≤ a then rw [Int.le_antisymm hba h₁]; apply Int.le_refl else if hc0 : c ≤ 0 then simp [Int.le_antisymm hc0 h₂, Int.zero_mul] else exact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩) ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : b ≤ a ⊢ c * a ≤ c * b ** rw [Int.le_antisymm hba h₁] ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : b ≤ a ⊢ c * a ≤ c * a ** apply Int.le_refl ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : ¬b ≤ a ⊢ c * a ≤ c * b ** if hc0 : c ≤ 0 then simp [Int.le_antisymm hc0 h₂, Int.zero_mul] else exact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩) ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : ¬b ≤ a hc0 : c ≤ 0 ⊢ c * a ≤ c * b ** simp [Int.le_antisymm hc0 h₂, Int.zero_mul] ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c hba : ¬b ≤ a hc0 : ¬c ≤ 0 ⊢ c * a ≤ c * b ** exact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩) ** Qed

Int.mul_le_mul_of_nonneg_right ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c ⊢ a * c ≤ b * c ** rw [Int.mul_comm, Int.mul_comm b] ** a b c : Int h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c * a ≤ c * b ** exact Int.mul_le_mul_of_nonneg_left h₁ h₂ ** Qed

Int.mul_nonpos_of_nonneg_of_nonpos ** a b : Int ha : 0 ≤ a hb : b ≤ 0 ⊢ a * b ≤ 0 ** have h : a * b ≤ a * 0 := Int.mul_le_mul_of_nonneg_left hb ha ** a b : Int ha : 0 ≤ a hb : b ≤ 0 h : a * b ≤ a * 0 ⊢ a * b ≤ 0 ** rwa [Int.mul_zero] at h ** Qed

Int.mul_nonpos_of_nonpos_of_nonneg ** a b : Int ha : a ≤ 0 hb : 0 ≤ b ⊢ a * b ≤ 0 ** have h : a * b ≤ 0 * b := Int.mul_le_mul_of_nonneg_right ha hb ** a b : Int ha : a ≤ 0 hb : 0 ≤ b h : a * b ≤ 0 * b ⊢ a * b ≤ 0 ** rwa [Int.zero_mul] at h ** Qed

Int.mul_neg_of_pos_of_neg ** a b : Int ha : 0 < a hb : b < 0 ⊢ a * b < 0 ** have h : a * b < a * 0 := Int.mul_lt_mul_of_pos_left hb ha ** a b : Int ha : 0 < a hb : b < 0 h : a * b < a * 0 ⊢ a * b < 0 ** rwa [Int.mul_zero] at h ** Qed

Int.mul_neg_of_neg_of_pos ** a b : Int ha : a < 0 hb : 0 < b ⊢ a * b < 0 ** have h : a * b < 0 * b := Int.mul_lt_mul_of_pos_right ha hb ** a b : Int ha : a < 0 hb : 0 < b h : a * b < 0 * b ⊢ a * b < 0 ** rwa [Int.zero_mul] at h ** Qed

Int.mul_le_mul_of_nonpos_right ** a b c : Int h : b ≤ a hc : c ≤ 0 this✝ : -c ≥ 0 this : b * -c ≤ a * -c ⊢ -(b * c) ≤ -(a * c) ** rwa [← Int.neg_mul_eq_mul_neg, ← Int.neg_mul_eq_mul_neg] at this ** Qed

Int.mul_lt_mul_of_neg_left ** a b c : Int h : b < a hc : c < 0 this✝ : -c > 0 this : -c * b < -c * a ⊢ -(c * b) < -(c * a) ** rwa [← Int.neg_mul_eq_neg_mul, ← Int.neg_mul_eq_neg_mul] at this ** Qed

Int.mul_lt_mul_of_neg_right ** a b c : Int h : b < a hc : c < 0 this✝ : -c > 0 this : b * -c < a * -c ⊢ -(b * c) < -(a * c) ** rwa [← Int.neg_mul_eq_mul_neg, ← Int.neg_mul_eq_mul_neg] at this ** Qed

Int.ofNat_natAbs_of_nonpos ** a : Int H : a ≤ 0 ⊢ ↑(natAbs a) = -a ** rw [← natAbs_neg, natAbs_of_nonneg (Int.neg_nonneg_of_nonpos H)] ** Qed