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leandojo-lean4-formal-informal-strings-split

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formal stringlengths 41 427k	informal stringclasses 1 value
Nat.gcd_eq_left_iff_dvd m n : Nat h : m ∣ n ⊢ gcd m n = m rw [gcd_rec, mod_eq_zero_of_dvd h, gcd_zero_left] ** Qed
Nat.gcd_eq_right_iff_dvd m n : Nat ⊢ m ∣ n ↔ gcd n m = m rw [gcd_comm] m n : Nat ⊢ m ∣ n ↔ gcd m n = m exact gcd_eq_left_iff_dvd ** Qed
Nat.gcd_mul_left ** m n k : Nat ⊢ gcd (m * n) (m * k) = m * gcd n k induction n, k using gcd.induction with \| H0 k => simp \| H1 n k _ IH => rwa [← mul_mod_mul_left, ← gcd_rec, ← gcd_rec] at IH case H0 m k : Nat ⊢ gcd (m * 0) (m * k) = m * gcd 0 k simp case H1 m n k : Nat a✝ : 0 < n IH : gcd (m * (k % n)) (m * n) = m * gcd (k % n) n ⊢ gcd (m * n) (m * k) = m * gcd n k rwa [← mul_mod_mul_left, ← gcd_rec, ← gcd_rec] at IH Qed
Nat.gcd_mul_right ** m n k : Nat ⊢ gcd (m * n) (k * n) = gcd m k * n rw [Nat.mul_comm m n, Nat.mul_comm k n, Nat.mul_comm (gcd m k) n, gcd_mul_left] Qed
Nat.eq_zero_of_gcd_eq_zero_right m n : Nat H : gcd m n = 0 ⊢ n = 0 rw [gcd_comm] at H m n : Nat H : gcd n m = 0 ⊢ n = 0 exact eq_zero_of_gcd_eq_zero_left H ** Qed
Nat.gcd_eq_right m n : Nat H : n ∣ m ⊢ gcd m n = n rw [gcd_comm, gcd_eq_left H] ** Qed
Nat.gcd_mul_left_right ** m n : Nat ⊢ gcd n (m * n) = n rw [gcd_comm, gcd_mul_left_left] Qed
Nat.gcd_mul_right_left ** m n : Nat ⊢ gcd (n * m) n = n rw [Nat.mul_comm, gcd_mul_left_left] Qed
Nat.gcd_add_mul_self ** m n k : Nat ⊢ gcd m (n + k * m) = gcd m n ** simp [gcd_rec m (n + k * m), gcd_rec m n] ** Qed
Nat.gcd_eq_zero_iff i j : Nat h : i = 0 ∧ j = 0 ⊢ gcd i j = 0 simp [h] ** Qed
Nat.gcd_eq_iff g a b : Nat ⊢ gcd a b = g ↔ g ∣ a ∧ g ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g constructor case mp g a b : Nat ⊢ gcd a b = g → g ∣ a ∧ g ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g rintro rfl case mp a b : Nat ⊢ gcd a b ∣ a ∧ gcd a b ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ gcd a b exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _, fun _ => Nat.dvd_gcd⟩ case mpr g a b : Nat ⊢ (g ∣ a ∧ g ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g) → gcd a b = g rintro ⟨ha, hb, hc⟩ case mpr.intro.intro g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b = g apply Nat.dvd_antisymm case mpr.intro.intro.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b ∣ g apply hc case mpr.intro.intro.a.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b ∣ a exact gcd_dvd_left a b case mpr.intro.intro.a.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b ∣ b exact gcd_dvd_right a b case mpr.intro.intro.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ g ∣ gcd a b exact Nat.dvd_gcd ha hb ** Qed
Nat.lcm_comm m n : Nat ⊢ lcm m n = lcm n m rw [lcm, lcm, Nat.mul_comm n m, gcd_comm n m] ** Qed
Nat.lcm_zero_left m : Nat ⊢ lcm 0 m = 0 simp [lcm] ** Qed
Nat.lcm_zero_right m : Nat ⊢ lcm m 0 = 0 simp [lcm] ** Qed
Nat.lcm_one_left m : Nat ⊢ lcm 1 m = m simp [lcm] ** Qed
Nat.lcm_one_right m : Nat ⊢ lcm m 1 = m simp [lcm] ** Qed
Nat.lcm_self m : Nat ⊢ lcm m m = m match eq_zero_or_pos m with \| .inl h => rw [h, lcm_zero_left] \| .inr h => simp [lcm, Nat.mul_div_cancel _ h] m : Nat h : m = 0 ⊢ lcm m m = m rw [h, lcm_zero_left] m : Nat h : m > 0 ⊢ lcm m m = m simp [lcm, Nat.mul_div_cancel _ h] ** Qed
Nat.dvd_lcm_left ** m n : Nat ⊢ lcm m n = m * (n / gcd m n) rw [← Nat.mul_div_assoc m (Nat.gcd_dvd_right m n)] m n : Nat ⊢ lcm m n = m * n / gcd m n rfl Qed
Nat.gcd_mul_lcm ** m n : Nat ⊢ gcd m n * lcm m n = m * n rw [lcm, Nat.mul_div_cancel' (Nat.dvd_trans (gcd_dvd_left m n) (Nat.dvd_mul_right m n))] Qed
Nat.lcm_ne_zero m n : Nat hm : m ≠ 0 hn : n ≠ 0 ⊢ lcm m n ≠ 0 intro h m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 ⊢ False have h1 := gcd_mul_lcm m n ** m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : gcd m n * lcm m n = m * n ⊢ False rw [h, Nat.mul_zero] at h1 m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : 0 = m * n ⊢ False match mul_eq_zero.1 h1.symm with \| .inl hm1 => exact hm hm1 \| .inr hn1 => exact hn hn1 m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : 0 = m * n hm1 : m = 0 ⊢ False exact hm hm1 m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : 0 = m * n hn1 : n = 0 ⊢ False exact hn hn1 Qed
Nat.Coprime.dvd_of_dvd_mul_right ** k n m : Nat H1 : Coprime k n H2 : k ∣ m * n ⊢ k ∣ m let t := dvd_gcd (Nat.dvd_mul_left k m) H2 k n m : Nat H1 : Coprime k n H2 : k ∣ m * n t : k ∣ gcd (m * k) (m * n) := dvd_gcd (Nat.dvd_mul_left k m) H2 ⊢ k ∣ m rwa [gcd_mul_left, H1.gcd_eq_one, Nat.mul_one] at t Qed
Nat.Coprime.dvd_of_dvd_mul_left ** k m n : Nat H1 : Coprime k m H2 : k ∣ m * n ⊢ k ∣ n * m rwa [Nat.mul_comm] Qed
Nat.Coprime.gcd_mul_right_cancel ** k n m : Nat H : Coprime k n ⊢ gcd (m * k) n = gcd m n rw [Nat.mul_comm m k, H.gcd_mul_left_cancel m] Qed
Nat.Coprime.gcd_mul_left_cancel_right ** k m n : Nat H : Coprime k m ⊢ gcd m (k * n) = gcd m n ** rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n] ** Qed
Nat.Coprime.gcd_mul_right_cancel_right ** k m n : Nat H : Coprime k m ⊢ gcd m (n * k) = gcd m n rw [Nat.mul_comm n k, H.gcd_mul_left_cancel_right n] Qed
Nat.coprime_div_gcd_div_gcd m n : Nat H : 0 < gcd m n ⊢ Coprime (m / gcd m n) (n / gcd m n) rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H] ** Qed
Nat.not_coprime_of_dvd_of_dvd d m n : Nat dgt1 : 1 < d Hm : d ∣ m Hn : d ∣ n co : Coprime m n ⊢ d ∣ 1 rw [← co.gcd_eq_one] d m n : Nat dgt1 : 1 < d Hm : d ∣ m Hn : d ∣ n co : Coprime m n ⊢ d ∣ gcd m n exact dvd_gcd Hm Hn ** Qed
Nat.Coprime.coprime_dvd_left m k n : Nat H1 : m ∣ k H2 : Coprime k n ⊢ Coprime m n apply eq_one_of_dvd_one case H m k n : Nat H1 : m ∣ k H2 : Coprime k n ⊢ gcd m n ∣ 1 rw [Coprime] at H2 case H m k n : Nat H1 : m ∣ k H2 : gcd k n = 1 ⊢ gcd m n ∣ 1 have := Nat.gcd_dvd_gcd_of_dvd_left n H1 case H m k n : Nat H1 : m ∣ k H2 : gcd k n = 1 this : gcd m n ∣ gcd k n ⊢ gcd m n ∣ 1 rwa [← H2] ** Qed
Nat.Coprime.coprime_div_left m n a : Nat cmn : Coprime m n dvd : a ∣ m ⊢ Coprime (m / a) n match eq_zero_or_pos a with \| .inl h0 => rw [h0] at dvd rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢ simp; assumption \| .inr hpos => let ⟨k, hk⟩ := dvd rw [hk, Nat.mul_div_cancel_left _ hpos] rw [hk] at cmn exact cmn.coprime_mul_left m n a : Nat cmn : Coprime m n dvd : a ∣ m h0 : a = 0 ⊢ Coprime (m / a) n rw [h0] at dvd m n a : Nat cmn : Coprime m n dvd : 0 ∣ m h0 : a = 0 ⊢ Coprime (m / a) n rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢ m n a : Nat cmn : Coprime 0 n dvd : 0 ∣ m h0 : a = 0 ⊢ Coprime (0 / a) n simp m n a : Nat cmn : Coprime 0 n dvd : 0 ∣ m h0 : a = 0 ⊢ Coprime 0 n assumption m n a : Nat cmn : Coprime m n dvd : a ∣ m hpos : a > 0 ⊢ Coprime (m / a) n let ⟨k, hk⟩ := dvd ** m n a : Nat cmn : Coprime m n dvd : a ∣ m hpos : a > 0 k : Nat hk : m = a * k ⊢ Coprime (m / a) n rw [hk, Nat.mul_div_cancel_left _ hpos] m n a : Nat cmn : Coprime m n dvd : a ∣ m hpos : a > 0 k : Nat hk : m = a * k ⊢ Coprime k n rw [hk] at cmn m n a : Nat dvd : a ∣ m hpos : a > 0 k : Nat cmn : Coprime (a * k) n hk : m = a * k ⊢ Coprime k n exact cmn.coprime_mul_left Qed
Nat.coprime_mul_iff_left ** m n k : Nat x✝ : Coprime m k ∧ Coprime n k h : Coprime m k right✝ : Coprime n k ⊢ Coprime (m * n) k rwa [coprime_iff_gcd_eq_one, h.gcd_mul_left_cancel n] Qed
Nat.coprime_mul_iff_right ** k m n : Nat ⊢ Coprime k (m * n) ↔ Coprime k m ∧ Coprime k n rw [@coprime_comm k, @coprime_comm k, @coprime_comm k, coprime_mul_iff_left] Qed
Nat.coprime_zero_left n : Nat ⊢ Coprime 0 n ↔ n = 1 simp [Coprime] ** Qed
Nat.coprime_zero_right n : Nat ⊢ Coprime n 0 ↔ n = 1 simp [Coprime] ** Qed
Nat.coprime_self n : Nat ⊢ Coprime n n ↔ n = 1 simp [Coprime] ** Qed
Nat.Coprime.pow_left m k n : Nat H1 : Coprime m k ⊢ Coprime (m ^ n) k induction n with \| zero => exact coprime_one_left _ \| succ n ih => have hm := H1.mul ih; rwa [Nat.pow_succ, Nat.mul_comm] case zero m k : Nat H1 : Coprime m k ⊢ Coprime (m ^ zero) k exact coprime_one_left _ case succ m k : Nat H1 : Coprime m k n : Nat ih : Coprime (m ^ n) k ⊢ Coprime (m ^ succ n) k have hm := H1.mul ih ** case succ m k : Nat H1 : Coprime m k n : Nat ih : Coprime (m ^ n) k hm : Coprime (m * m ^ n) k ⊢ Coprime (m ^ succ n) k rwa [Nat.pow_succ, Nat.mul_comm] Qed
Nat.Coprime.eq_one_of_dvd k m : Nat H : Coprime k m d : k ∣ m ⊢ k = 1 rw [← H.gcd_eq_one, gcd_eq_left d] ** Qed
Nat.gcd_mul_dvd_mul_gcd ** k m n : Nat ⊢ gcd k (m * n) ∣ gcd k m * gcd k n ** let ⟨⟨⟨m', hm'⟩, ⟨n', hn'⟩⟩, (h : gcd k (m * n) = m' * n')⟩ := prod_dvd_and_dvd_of_dvd_prod <\| gcd_dvd_right k (m * n) ** k m n m' : Nat hm' : m' ∣ m n' : Nat hn' : n' ∣ n h : gcd k (m * n) = m' * n' ⊢ gcd k (m * n) ∣ gcd k m * gcd k n rw [h] k m n m' : Nat hm' : m' ∣ m n' : Nat hn' : n' ∣ n h : gcd k (m * n) = m' * n' ⊢ m' * n' ∣ gcd k m * gcd k n ** have h' : m' * n' ∣ k := h ▸ gcd_dvd_left .. ** k m n m' : Nat hm' : m' ∣ m n' : Nat hn' : n' ∣ n h : gcd k (m * n) = m' * n' h' : m' * n' ∣ k ⊢ m' * n' ∣ gcd k m * gcd k n exact Nat.mul_dvd_mul (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_right m' n') h') hm') (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_left n' m') h') hn') Qed
Rat.normalize.reduced num : Int den g : Nat den_nz : den ≠ 0 e : g = Nat.gcd (Int.natAbs num) den ⊢ Int.natAbs (Int.div num ↑g) = Int.natAbs num / g match num, num.eq_nat_or_neg with \| _, ⟨_, .inl rfl⟩ => rfl \| _, ⟨_, .inr rfl⟩ => rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg]; rfl num : Int den g : Nat den_nz : den ≠ 0 w✝ : Nat e : g = Nat.gcd (Int.natAbs ↑w✝) den ⊢ Int.natAbs (Int.div ↑w✝ ↑g) = Int.natAbs ↑w✝ / g rfl num : Int den g : Nat den_nz : den ≠ 0 w✝ : Nat e : g = Nat.gcd (Int.natAbs (-↑w✝)) den ⊢ Int.natAbs (Int.div (-↑w✝) ↑g) = Int.natAbs (-↑w✝) / g rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg] num : Int den g : Nat den_nz : den ≠ 0 w✝ : Nat e : g = Nat.gcd (Int.natAbs (-↑w✝)) den ⊢ Int.natAbs (Int.div ↑w✝ ↑g) = Int.natAbs ↑w✝ / g rfl ** Qed
Int.ofNat_fdiv x✝ : Nat ⊢ ↑(0 / x✝) = fdiv ↑0 ↑x✝ simp [fdiv] ** Qed
Int.zero_div a✝ : Nat ⊢ ofNat (0 / a✝) = 0 simp a✝ : Nat ⊢ -ofNat (0 / succ a✝) = 0 simp ** Qed
Int.zero_ediv a✝ : Nat ⊢ ofNat (0 / a✝) = 0 simp a✝ : Nat ⊢ -ofNat (0 / succ a✝) = 0 simp ** Qed
Int.zero_fdiv b : Int ⊢ fdiv 0 b = 0 cases b <;> rfl ** Qed
Int.div_zero a✝ : Nat ⊢ ofNat (a✝ / 0) = 0 simp ** Qed
Int.ediv_zero a✝ : Nat ⊢ ofNat (a✝ / 0) = 0 simp ** Qed
Int.fdiv_eq_ediv a✝ : Nat x✝ : 0 ≤ 0 ⊢ fdiv -[a✝+1] 0 = -[a✝+1] / 0 simp ** Qed
Int.div_eq_ediv x✝² : Int x✝¹ : 0 ≤ x✝² x✝ : 0 ≤ 0 ⊢ div x✝² 0 = x✝² / 0 simp ** Qed
Int.div_neg m : Nat ⊢ ofNat (m / 0) = -↑(m / 0) rw [Nat.div_zero] m : Nat ⊢ ofNat 0 = -↑0 rfl ** Qed
Int.ediv_neg m : Nat ⊢ ofNat (m / 0) = -↑(m / 0) rw [Nat.div_zero] m : Nat ⊢ ofNat 0 = -↑0 rfl ** Qed
Int.neg_div n : Int ⊢ div (-0) n = -div 0 n simp [Int.neg_zero] ** Qed
Int.neg_div_neg a b : Int ⊢ div (-a) (-b) = div a b simp [Int.div_neg, Int.neg_div, Int.neg_neg] ** Qed
Int.div_one n : Nat ⊢ div -[n+1] 1 = -[n+1] simp [Int.div, neg_ofNat_succ] ** Qed
Int.add_ediv_of_dvd_right a b c : Int H : c ∣ b h : c = 0 ⊢ (a + b) / c = a / c + b / c simp [h] a b c : Int H : c ∣ b h : ¬c = 0 ⊢ (a + b) / c = a / c + b / c let ⟨k, hk⟩ := H ** a b c : Int H : c ∣ b h : ¬c = 0 k : Int hk : b = c * k ⊢ (a + b) / c = a / c + b / c ** rw [hk, Int.mul_comm c k, Int.add_mul_ediv_right _ _ h, ← Int.zero_add (k * c), Int.add_mul_ediv_right _ _ h, Int.zero_ediv, Int.zero_add] ** Qed
Int.add_ediv_of_dvd_left a b c : Int H : c ∣ a ⊢ (a + b) / c = a / c + b / c rw [Int.add_comm, Int.add_ediv_of_dvd_right H, Int.add_comm] ** Qed
Int.mul_ediv_cancel ** a b : Int H : b ≠ 0 ⊢ a * b / b = a have := Int.add_mul_ediv_right 0 a H a b : Int H : b ≠ 0 this : (0 + a * b) / b = 0 / b + a ⊢ a * b / b = a rwa [Int.zero_add, Int.zero_ediv, Int.zero_add] at this Qed
Int.mul_fdiv_cancel ** a b : Int H : b ≠ 0 b0 : 0 ≤ b ⊢ fdiv (a * b) b = a rw [fdiv_eq_ediv _ b0, mul_ediv_cancel _ H] a b x✝¹ : Int x✝ : x✝¹ < 0 H : x✝¹ ≠ 0 b0 : ¬0 ≤ x✝¹ ⊢ fdiv (0 * x✝¹) x✝¹ = 0 simp [Int.zero_mul] Qed
Int.div_self a : Int H : a ≠ 0 ⊢ div a a = 1 have := Int.mul_div_cancel 1 H ** a : Int H : a ≠ 0 this : div (1 * a) a = 1 ⊢ div a a = 1 rwa [Int.one_mul] at this Qed
Int.fdiv_self a : Int H : a ≠ 0 ⊢ fdiv a a = 1 have := Int.mul_fdiv_cancel 1 H ** a : Int H : a ≠ 0 this : fdiv (1 * a) a = 1 ⊢ fdiv a a = 1 rwa [Int.one_mul] at this Qed
Int.ediv_self a : Int H : a ≠ 0 ⊢ a / a = 1 have := Int.mul_ediv_cancel 1 H ** a : Int H : a ≠ 0 this : 1 * a / a = 1 ⊢ a / a = 1 rwa [Int.one_mul] at this Qed
Int.ofNat_fmod m n : Nat ⊢ ↑(m % n) = fmod ↑m ↑n cases m <;> simp [fmod] ** Qed
Int.negSucc_emod m : Nat b : Int bpos : 0 < b ⊢ -[m+1] % b = b - 1 - ↑m % b rw [Int.sub_sub, Int.add_comm] m : Nat b : Int bpos : 0 < b ⊢ -[m+1] % b = b - (↑m % b + 1) match b, eq_succ_of_zero_lt bpos with \| _, ⟨n, rfl⟩ => rfl m : Nat b : Int n : Nat bpos : 0 < ↑(succ n) ⊢ -[m+1] % ↑(succ n) = ↑(succ n) - (↑m % ↑(succ n) + 1) rfl ** Qed
Int.zero_mod b : Int ⊢ mod 0 b = 0 cases b <;> simp [mod] ** Qed
Int.zero_fmod b : Int ⊢ fmod 0 b = 0 cases b <;> rfl ** Qed
Int.mod_add_div ** m n : Nat ⊢ mod (ofNat m) -[n+1] + -[n+1] * div (ofNat m) -[n+1] = ofNat m ** show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m ** m n : Nat ⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m rw [Int.neg_mul_neg] m n : Nat ⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m exact congrArg ofNat (Nat.mod_add_div ..) m n : Nat ⊢ mod -[m+1] (ofNat n) + ofNat n * div -[m+1] (ofNat n) = -[m+1] ** show -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m) ** m n : Nat ⊢ -↑(succ m % n) + ↑n * -↑(succ m / n) = -↑(succ m) rw [Int.mul_neg, ← Int.neg_add] m n : Nat ⊢ mod -[m+1] -[n+1] + -[n+1] * div -[m+1] -[n+1] = -[m+1] ** show -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m) ** m n : Nat ⊢ -↑(succ m % succ n) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m) rw [Int.neg_mul, ← Int.neg_add] Qed
Int.emod_add_ediv ** m n : Nat ⊢ ofNat m % -[n+1] + -[n+1] * (ofNat m / -[n+1]) = ofNat m ** show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m ** m n : Nat ⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m rw [Int.neg_mul_neg] m n : Nat ⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m exact congrArg ofNat <\| Nat.mod_add_div .. a✝ : Nat ⊢ -[a✝+1] % 0 + 0 * (-[a✝+1] / 0) = -[a✝+1] rw [emod_zero] a✝ : Nat ⊢ -[a✝+1] + 0 * (-[a✝+1] / 0) = -[a✝+1] rfl x✝¹ x✝ : Int m n : Nat ⊢ ↑n - (↑m % ↑n + 1) - (↑n * (↑m / ↑n) + ↑n) = -[m+1] rw [← ofNat_emod, ← ofNat_ediv, ← Int.sub_sub, negSucc_eq, Int.sub_sub n, ← Int.neg_neg (_-_), Int.neg_sub, Int.sub_sub_self, Int.add_right_comm] x✝¹ x✝ : Int m n : Nat ⊢ -(↑(m % n) + ↑n * ↑(m / n) + 1) = -(↑m + 1) exact congrArg (fun x => -(ofNat x + 1)) (Nat.mod_add_div ..) Qed
Int.mod_def ** a b : Int ⊢ mod a b = a - b * div a b rw [← Int.add_sub_cancel (mod a b), mod_add_div] Qed
Int.fmod_def ** a b : Int ⊢ fmod a b = a - b * fdiv a b rw [← Int.add_sub_cancel (a.fmod b), fmod_add_fdiv] Qed
Int.emod_def ** a b : Int ⊢ a % b = a - b * (a / b) rw [← Int.add_sub_cancel (a % b), emod_add_ediv] Qed
Int.fmod_eq_emod a b : Int hb : 0 ≤ b ⊢ fmod a b = a % b simp [fmod_def, emod_def, fdiv_eq_ediv _ hb] ** Qed
Int.mod_eq_emod a b : Int ha : 0 ≤ a hb : 0 ≤ b ⊢ mod a b = a % b simp [emod_def, mod_def, div_eq_ediv ha hb] ** Qed
Int.mod_neg a b : Int ⊢ mod a (-b) = mod a b rw [mod_def, mod_def, Int.div_neg, Int.neg_mul_neg] ** Qed
Int.emod_neg a b : Int ⊢ a % -b = a % b rw [emod_def, emod_def, Int.ediv_neg, Int.neg_mul_neg] ** Qed
Int.emod_one a : Int ⊢ a % 1 = 0 simp [emod_def, Int.one_mul, Int.sub_self] ** Qed
Int.fmod_one a : Int ⊢ fmod a 1 = 0 simp [fmod_def, Int.one_mul, Int.sub_self] ** Qed
Int.mod_eq_of_lt a b : Int H1 : 0 ≤ a H2 : a < b ⊢ mod a b = a rw [mod_eq_emod H1 (Int.le_trans H1 (Int.le_of_lt H2)), emod_eq_of_lt H1 H2] ** Qed
Int.fmod_eq_of_lt a b : Int H1 : 0 ≤ a H2 : a < b ⊢ fmod a b = a rw [fmod_eq_emod _ (Int.le_trans H1 (Int.le_of_lt H2)), emod_eq_of_lt H1 H2] ** Qed
Int.mod_add_div' ** m k : Int ⊢ mod m k + div m k * k = m rw [Int.mul_comm] m k : Int ⊢ mod m k + k * div m k = m apply mod_add_div Qed
Int.div_add_mod' ** m k : Int ⊢ div m k * k + mod m k = m rw [Int.mul_comm] m k : Int ⊢ k * div m k + mod m k = m apply div_add_mod Qed
Int.emod_add_ediv' ** m k : Int ⊢ m % k + m / k * k = m rw [Int.mul_comm] m k : Int ⊢ m % k + k * (m / k) = m apply emod_add_ediv Qed
Int.add_mul_emod_self ** a b c : Int cz : c = 0 ⊢ (a + b * c) % c = a % c rw [cz, Int.mul_zero, Int.add_zero] a b c : Int cz : ¬c = 0 ⊢ (a + b * c) % c = a % c rw [Int.emod_def, Int.emod_def, Int.add_mul_ediv_right _ _ cz, Int.add_comm _ b, Int.mul_add, Int.mul_comm, ← Int.sub_sub, Int.add_sub_cancel] Qed
Int.add_mul_emod_self_left ** a b c : Int ⊢ (a + b * c) % b = a % b rw [Int.mul_comm, Int.add_mul_emod_self] Qed
Int.add_emod_self a b : Int ⊢ (a + b) % b = a % b have := add_mul_emod_self_left a b 1 ** a b : Int this : (a + b * 1) % b = a % b ⊢ (a + b) % b = a % b rwa [Int.mul_one] at this Qed
Int.add_emod_self_left a b : Int ⊢ (a + b) % a = b % a rw [Int.add_comm, Int.add_emod_self] ** Qed
Int.emod_add_emod m n k : Int ⊢ (m % n + k) % n = (m + k) % n have := (add_mul_emod_self_left (m % n + k) n (m / n)).symm ** m n k : Int this : (m % n + k) % n = (m % n + k + n * (m / n)) % n ⊢ (m % n + k) % n = (m + k) % n rwa [Int.add_right_comm, emod_add_ediv] at this Qed
Int.add_emod_emod m n k : Int ⊢ (m + n % k) % k = (m + n) % k rw [Int.add_comm, emod_add_emod, Int.add_comm] ** Qed
Int.add_emod a b n : Int ⊢ (a + b) % n = (a % n + b % n) % n rw [add_emod_emod, emod_add_emod] ** Qed
Int.add_emod_eq_add_emod_right m n k i : Int H : m % n = k % n ⊢ (m + i) % n = (k + i) % n rw [← emod_add_emod, ← emod_add_emod k, H] ** Qed
Int.add_emod_eq_add_emod_left m n k i : Int H : m % n = k % n ⊢ (i + m) % n = (i + k) % n rw [Int.add_comm, add_emod_eq_add_emod_right _ H, Int.add_comm] ** Qed
Int.emod_add_cancel_right m n k i : Int H : (m + i) % n = (k + i) % n ⊢ m % n = k % n have := add_emod_eq_add_emod_right (-i) H m n k i : Int H : (m + i) % n = (k + i) % n this : (m + i + -i) % n = (k + i + -i) % n ⊢ m % n = k % n rwa [Int.add_neg_cancel_right, Int.add_neg_cancel_right] at this ** Qed
Int.emod_add_cancel_left m n k i : Int ⊢ (i + m) % n = (i + k) % n ↔ m % n = k % n rw [Int.add_comm, Int.add_comm i, emod_add_cancel_right] ** Qed
Int.emod_eq_emod_iff_emod_sub_eq_zero m n k : Int ⊢ (m - k) % n = (k - k) % n ↔ (m - k) % n = 0 simp [Int.sub_self] ** Qed
Int.mul_mod_left ** a b : Int h : b = 0 ⊢ mod (a * b) b = 0 simp [h, Int.mul_zero] a b : Int h : ¬b = 0 ⊢ mod (a * b) b = 0 rw [Int.mod_def, Int.mul_div_cancel _ h, Int.mul_comm, Int.sub_self] Qed
Int.mul_fmod_left ** a b : Int h : b = 0 ⊢ fmod (a * b) b = 0 simp [h, Int.mul_zero] a b : Int h : ¬b = 0 ⊢ fmod (a * b) b = 0 rw [Int.fmod_def, Int.mul_fdiv_cancel _ h, Int.mul_comm, Int.sub_self] Qed
Int.mul_emod_left ** a b : Int ⊢ a * b % b = 0 ** rw [← Int.zero_add (a * b), Int.add_mul_emod_self, Int.zero_emod] ** Qed
Int.mul_mod_right ** a b : Int ⊢ mod (a * b) a = 0 rw [Int.mul_comm, mul_mod_left] Qed
Int.mul_fmod_right ** a b : Int ⊢ fmod (a * b) a = 0 rw [Int.mul_comm, mul_fmod_left] Qed
Int.mul_emod_right ** a b : Int ⊢ a * b % a = 0 rw [Int.mul_comm, mul_emod_left] Qed
Int.mod_self a : Int ⊢ mod a a = 0 have := mul_mod_left 1 a ** a : Int this : mod (1 * a) a = 0 ⊢ mod a a = 0 rwa [Int.one_mul] at this Qed
Int.fmod_self a : Int ⊢ fmod a a = 0 have := mul_fmod_left 1 a ** a : Int this : fmod (1 * a) a = 0 ⊢ fmod a a = 0 rwa [Int.one_mul] at this Qed
Int.emod_self a : Int ⊢ a % a = 0 have := mul_emod_left 1 a ** a : Int this : 1 * a % a = 0 ⊢ a % a = 0 rwa [Int.one_mul] at this Qed
Int.emod_emod_of_dvd n m k : Int h : m ∣ k ⊢ n % k % m = n % m conv => rhs; rw [← emod_add_ediv n k] ** n m k : Int h : m ∣ k ⊢ n % k % m = (n % k + k * (n / k)) % m match k, h with \| _, ⟨t, rfl⟩ => rw [Int.mul_assoc, add_mul_emod_self_left] n m k : Int h : m ∣ k t : Int ⊢ n % (m * t) % m = (n % (m * t) + m * t * (n / (m * t))) % m rw [Int.mul_assoc, add_mul_emod_self_left] Qed

formal stringlengths 41 427k	informal stringclasses 1 value
Nat.gcd_eq_left_iff_dvd m n : Nat h : m ∣ n ⊢ gcd m n = m rw [gcd_rec, mod_eq_zero_of_dvd h, gcd_zero_left] ** Qed
Nat.gcd_eq_right_iff_dvd m n : Nat ⊢ m ∣ n ↔ gcd n m = m rw [gcd_comm] m n : Nat ⊢ m ∣ n ↔ gcd m n = m exact gcd_eq_left_iff_dvd ** Qed
Nat.gcd_mul_left ** m n k : Nat ⊢ gcd (m * n) (m * k) = m * gcd n k induction n, k using gcd.induction with \| H0 k => simp \| H1 n k _ IH => rwa [← mul_mod_mul_left, ← gcd_rec, ← gcd_rec] at IH case H0 m k : Nat ⊢ gcd (m * 0) (m * k) = m * gcd 0 k simp case H1 m n k : Nat a✝ : 0 < n IH : gcd (m * (k % n)) (m * n) = m * gcd (k % n) n ⊢ gcd (m * n) (m * k) = m * gcd n k rwa [← mul_mod_mul_left, ← gcd_rec, ← gcd_rec] at IH Qed
Nat.gcd_mul_right ** m n k : Nat ⊢ gcd (m * n) (k * n) = gcd m k * n rw [Nat.mul_comm m n, Nat.mul_comm k n, Nat.mul_comm (gcd m k) n, gcd_mul_left] Qed
Nat.eq_zero_of_gcd_eq_zero_right m n : Nat H : gcd m n = 0 ⊢ n = 0 rw [gcd_comm] at H m n : Nat H : gcd n m = 0 ⊢ n = 0 exact eq_zero_of_gcd_eq_zero_left H ** Qed
Nat.gcd_eq_right m n : Nat H : n ∣ m ⊢ gcd m n = n rw [gcd_comm, gcd_eq_left H] ** Qed
Nat.gcd_mul_left_right ** m n : Nat ⊢ gcd n (m * n) = n rw [gcd_comm, gcd_mul_left_left] Qed
Nat.gcd_mul_right_left ** m n : Nat ⊢ gcd (n * m) n = n rw [Nat.mul_comm, gcd_mul_left_left] Qed
Nat.gcd_add_mul_self ** m n k : Nat ⊢ gcd m (n + k * m) = gcd m n ** simp [gcd_rec m (n + k * m), gcd_rec m n] ** Qed
Nat.gcd_eq_zero_iff i j : Nat h : i = 0 ∧ j = 0 ⊢ gcd i j = 0 simp [h] ** Qed
Nat.gcd_eq_iff g a b : Nat ⊢ gcd a b = g ↔ g ∣ a ∧ g ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g constructor case mp g a b : Nat ⊢ gcd a b = g → g ∣ a ∧ g ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g rintro rfl case mp a b : Nat ⊢ gcd a b ∣ a ∧ gcd a b ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ gcd a b exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _, fun _ => Nat.dvd_gcd⟩ case mpr g a b : Nat ⊢ (g ∣ a ∧ g ∣ b ∧ ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g) → gcd a b = g rintro ⟨ha, hb, hc⟩ case mpr.intro.intro g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b = g apply Nat.dvd_antisymm case mpr.intro.intro.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b ∣ g apply hc case mpr.intro.intro.a.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b ∣ a exact gcd_dvd_left a b case mpr.intro.intro.a.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ gcd a b ∣ b exact gcd_dvd_right a b case mpr.intro.intro.a g a b : Nat ha : g ∣ a hb : g ∣ b hc : ∀ (c : Nat), c ∣ a → c ∣ b → c ∣ g ⊢ g ∣ gcd a b exact Nat.dvd_gcd ha hb ** Qed
Nat.lcm_comm m n : Nat ⊢ lcm m n = lcm n m rw [lcm, lcm, Nat.mul_comm n m, gcd_comm n m] ** Qed
Nat.lcm_zero_left m : Nat ⊢ lcm 0 m = 0 simp [lcm] ** Qed
Nat.lcm_zero_right m : Nat ⊢ lcm m 0 = 0 simp [lcm] ** Qed
Nat.lcm_one_left m : Nat ⊢ lcm 1 m = m simp [lcm] ** Qed
Nat.lcm_one_right m : Nat ⊢ lcm m 1 = m simp [lcm] ** Qed
Nat.lcm_self m : Nat ⊢ lcm m m = m match eq_zero_or_pos m with \| .inl h => rw [h, lcm_zero_left] \| .inr h => simp [lcm, Nat.mul_div_cancel _ h] m : Nat h : m = 0 ⊢ lcm m m = m rw [h, lcm_zero_left] m : Nat h : m > 0 ⊢ lcm m m = m simp [lcm, Nat.mul_div_cancel _ h] ** Qed
Nat.dvd_lcm_left ** m n : Nat ⊢ lcm m n = m * (n / gcd m n) rw [← Nat.mul_div_assoc m (Nat.gcd_dvd_right m n)] m n : Nat ⊢ lcm m n = m * n / gcd m n rfl Qed
Nat.gcd_mul_lcm ** m n : Nat ⊢ gcd m n * lcm m n = m * n rw [lcm, Nat.mul_div_cancel' (Nat.dvd_trans (gcd_dvd_left m n) (Nat.dvd_mul_right m n))] Qed
Nat.lcm_ne_zero m n : Nat hm : m ≠ 0 hn : n ≠ 0 ⊢ lcm m n ≠ 0 intro h m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 ⊢ False have h1 := gcd_mul_lcm m n ** m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : gcd m n * lcm m n = m * n ⊢ False rw [h, Nat.mul_zero] at h1 m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : 0 = m * n ⊢ False match mul_eq_zero.1 h1.symm with \| .inl hm1 => exact hm hm1 \| .inr hn1 => exact hn hn1 m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : 0 = m * n hm1 : m = 0 ⊢ False exact hm hm1 m n : Nat hm : m ≠ 0 hn : n ≠ 0 h : lcm m n = 0 h1 : 0 = m * n hn1 : n = 0 ⊢ False exact hn hn1 Qed
Nat.Coprime.dvd_of_dvd_mul_right ** k n m : Nat H1 : Coprime k n H2 : k ∣ m * n ⊢ k ∣ m let t := dvd_gcd (Nat.dvd_mul_left k m) H2 k n m : Nat H1 : Coprime k n H2 : k ∣ m * n t : k ∣ gcd (m * k) (m * n) := dvd_gcd (Nat.dvd_mul_left k m) H2 ⊢ k ∣ m rwa [gcd_mul_left, H1.gcd_eq_one, Nat.mul_one] at t Qed
Nat.Coprime.dvd_of_dvd_mul_left ** k m n : Nat H1 : Coprime k m H2 : k ∣ m * n ⊢ k ∣ n * m rwa [Nat.mul_comm] Qed
Nat.Coprime.gcd_mul_right_cancel ** k n m : Nat H : Coprime k n ⊢ gcd (m * k) n = gcd m n rw [Nat.mul_comm m k, H.gcd_mul_left_cancel m] Qed
Nat.Coprime.gcd_mul_left_cancel_right ** k m n : Nat H : Coprime k m ⊢ gcd m (k * n) = gcd m n ** rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n] ** Qed
Nat.Coprime.gcd_mul_right_cancel_right ** k m n : Nat H : Coprime k m ⊢ gcd m (n * k) = gcd m n rw [Nat.mul_comm n k, H.gcd_mul_left_cancel_right n] Qed
Nat.coprime_div_gcd_div_gcd m n : Nat H : 0 < gcd m n ⊢ Coprime (m / gcd m n) (n / gcd m n) rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H] ** Qed
Nat.not_coprime_of_dvd_of_dvd d m n : Nat dgt1 : 1 < d Hm : d ∣ m Hn : d ∣ n co : Coprime m n ⊢ d ∣ 1 rw [← co.gcd_eq_one] d m n : Nat dgt1 : 1 < d Hm : d ∣ m Hn : d ∣ n co : Coprime m n ⊢ d ∣ gcd m n exact dvd_gcd Hm Hn ** Qed
Nat.Coprime.coprime_dvd_left m k n : Nat H1 : m ∣ k H2 : Coprime k n ⊢ Coprime m n apply eq_one_of_dvd_one case H m k n : Nat H1 : m ∣ k H2 : Coprime k n ⊢ gcd m n ∣ 1 rw [Coprime] at H2 case H m k n : Nat H1 : m ∣ k H2 : gcd k n = 1 ⊢ gcd m n ∣ 1 have := Nat.gcd_dvd_gcd_of_dvd_left n H1 case H m k n : Nat H1 : m ∣ k H2 : gcd k n = 1 this : gcd m n ∣ gcd k n ⊢ gcd m n ∣ 1 rwa [← H2] ** Qed
Nat.Coprime.coprime_div_left m n a : Nat cmn : Coprime m n dvd : a ∣ m ⊢ Coprime (m / a) n match eq_zero_or_pos a with \| .inl h0 => rw [h0] at dvd rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢ simp; assumption \| .inr hpos => let ⟨k, hk⟩ := dvd rw [hk, Nat.mul_div_cancel_left _ hpos] rw [hk] at cmn exact cmn.coprime_mul_left m n a : Nat cmn : Coprime m n dvd : a ∣ m h0 : a = 0 ⊢ Coprime (m / a) n rw [h0] at dvd m n a : Nat cmn : Coprime m n dvd : 0 ∣ m h0 : a = 0 ⊢ Coprime (m / a) n rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢ m n a : Nat cmn : Coprime 0 n dvd : 0 ∣ m h0 : a = 0 ⊢ Coprime (0 / a) n simp m n a : Nat cmn : Coprime 0 n dvd : 0 ∣ m h0 : a = 0 ⊢ Coprime 0 n assumption m n a : Nat cmn : Coprime m n dvd : a ∣ m hpos : a > 0 ⊢ Coprime (m / a) n let ⟨k, hk⟩ := dvd ** m n a : Nat cmn : Coprime m n dvd : a ∣ m hpos : a > 0 k : Nat hk : m = a * k ⊢ Coprime (m / a) n rw [hk, Nat.mul_div_cancel_left _ hpos] m n a : Nat cmn : Coprime m n dvd : a ∣ m hpos : a > 0 k : Nat hk : m = a * k ⊢ Coprime k n rw [hk] at cmn m n a : Nat dvd : a ∣ m hpos : a > 0 k : Nat cmn : Coprime (a * k) n hk : m = a * k ⊢ Coprime k n exact cmn.coprime_mul_left Qed
Nat.coprime_mul_iff_left ** m n k : Nat x✝ : Coprime m k ∧ Coprime n k h : Coprime m k right✝ : Coprime n k ⊢ Coprime (m * n) k rwa [coprime_iff_gcd_eq_one, h.gcd_mul_left_cancel n] Qed
Nat.coprime_mul_iff_right ** k m n : Nat ⊢ Coprime k (m * n) ↔ Coprime k m ∧ Coprime k n rw [@coprime_comm k, @coprime_comm k, @coprime_comm k, coprime_mul_iff_left] Qed
Nat.coprime_zero_left n : Nat ⊢ Coprime 0 n ↔ n = 1 simp [Coprime] ** Qed
Nat.coprime_zero_right n : Nat ⊢ Coprime n 0 ↔ n = 1 simp [Coprime] ** Qed
Nat.coprime_self n : Nat ⊢ Coprime n n ↔ n = 1 simp [Coprime] ** Qed
Nat.Coprime.pow_left m k n : Nat H1 : Coprime m k ⊢ Coprime (m ^ n) k induction n with \| zero => exact coprime_one_left _ \| succ n ih => have hm := H1.mul ih; rwa [Nat.pow_succ, Nat.mul_comm] case zero m k : Nat H1 : Coprime m k ⊢ Coprime (m ^ zero) k exact coprime_one_left _ case succ m k : Nat H1 : Coprime m k n : Nat ih : Coprime (m ^ n) k ⊢ Coprime (m ^ succ n) k have hm := H1.mul ih ** case succ m k : Nat H1 : Coprime m k n : Nat ih : Coprime (m ^ n) k hm : Coprime (m * m ^ n) k ⊢ Coprime (m ^ succ n) k rwa [Nat.pow_succ, Nat.mul_comm] Qed
Nat.Coprime.eq_one_of_dvd k m : Nat H : Coprime k m d : k ∣ m ⊢ k = 1 rw [← H.gcd_eq_one, gcd_eq_left d] ** Qed
Nat.gcd_mul_dvd_mul_gcd ** k m n : Nat ⊢ gcd k (m * n) ∣ gcd k m * gcd k n ** let ⟨⟨⟨m', hm'⟩, ⟨n', hn'⟩⟩, (h : gcd k (m * n) = m' * n')⟩ := prod_dvd_and_dvd_of_dvd_prod <\| gcd_dvd_right k (m * n) ** k m n m' : Nat hm' : m' ∣ m n' : Nat hn' : n' ∣ n h : gcd k (m * n) = m' * n' ⊢ gcd k (m * n) ∣ gcd k m * gcd k n rw [h] k m n m' : Nat hm' : m' ∣ m n' : Nat hn' : n' ∣ n h : gcd k (m * n) = m' * n' ⊢ m' * n' ∣ gcd k m * gcd k n ** have h' : m' * n' ∣ k := h ▸ gcd_dvd_left .. ** k m n m' : Nat hm' : m' ∣ m n' : Nat hn' : n' ∣ n h : gcd k (m * n) = m' * n' h' : m' * n' ∣ k ⊢ m' * n' ∣ gcd k m * gcd k n exact Nat.mul_dvd_mul (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_right m' n') h') hm') (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_left n' m') h') hn') Qed
Rat.normalize.reduced num : Int den g : Nat den_nz : den ≠ 0 e : g = Nat.gcd (Int.natAbs num) den ⊢ Int.natAbs (Int.div num ↑g) = Int.natAbs num / g match num, num.eq_nat_or_neg with \| _, ⟨_, .inl rfl⟩ => rfl \| _, ⟨_, .inr rfl⟩ => rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg]; rfl num : Int den g : Nat den_nz : den ≠ 0 w✝ : Nat e : g = Nat.gcd (Int.natAbs ↑w✝) den ⊢ Int.natAbs (Int.div ↑w✝ ↑g) = Int.natAbs ↑w✝ / g rfl num : Int den g : Nat den_nz : den ≠ 0 w✝ : Nat e : g = Nat.gcd (Int.natAbs (-↑w✝)) den ⊢ Int.natAbs (Int.div (-↑w✝) ↑g) = Int.natAbs (-↑w✝) / g rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg] num : Int den g : Nat den_nz : den ≠ 0 w✝ : Nat e : g = Nat.gcd (Int.natAbs (-↑w✝)) den ⊢ Int.natAbs (Int.div ↑w✝ ↑g) = Int.natAbs ↑w✝ / g rfl ** Qed
Int.ofNat_fdiv x✝ : Nat ⊢ ↑(0 / x✝) = fdiv ↑0 ↑x✝ simp [fdiv] ** Qed
Int.zero_div a✝ : Nat ⊢ ofNat (0 / a✝) = 0 simp a✝ : Nat ⊢ -ofNat (0 / succ a✝) = 0 simp ** Qed
Int.zero_ediv a✝ : Nat ⊢ ofNat (0 / a✝) = 0 simp a✝ : Nat ⊢ -ofNat (0 / succ a✝) = 0 simp ** Qed
Int.zero_fdiv b : Int ⊢ fdiv 0 b = 0 cases b <;> rfl ** Qed
Int.div_zero a✝ : Nat ⊢ ofNat (a✝ / 0) = 0 simp ** Qed
Int.ediv_zero a✝ : Nat ⊢ ofNat (a✝ / 0) = 0 simp ** Qed
Int.fdiv_eq_ediv a✝ : Nat x✝ : 0 ≤ 0 ⊢ fdiv -[a✝+1] 0 = -[a✝+1] / 0 simp ** Qed
Int.div_eq_ediv x✝² : Int x✝¹ : 0 ≤ x✝² x✝ : 0 ≤ 0 ⊢ div x✝² 0 = x✝² / 0 simp ** Qed
Int.div_neg m : Nat ⊢ ofNat (m / 0) = -↑(m / 0) rw [Nat.div_zero] m : Nat ⊢ ofNat 0 = -↑0 rfl ** Qed
Int.ediv_neg m : Nat ⊢ ofNat (m / 0) = -↑(m / 0) rw [Nat.div_zero] m : Nat ⊢ ofNat 0 = -↑0 rfl ** Qed
Int.neg_div n : Int ⊢ div (-0) n = -div 0 n simp [Int.neg_zero] ** Qed
Int.neg_div_neg a b : Int ⊢ div (-a) (-b) = div a b simp [Int.div_neg, Int.neg_div, Int.neg_neg] ** Qed
Int.div_one n : Nat ⊢ div -[n+1] 1 = -[n+1] simp [Int.div, neg_ofNat_succ] ** Qed
Int.add_ediv_of_dvd_right a b c : Int H : c ∣ b h : c = 0 ⊢ (a + b) / c = a / c + b / c simp [h] a b c : Int H : c ∣ b h : ¬c = 0 ⊢ (a + b) / c = a / c + b / c let ⟨k, hk⟩ := H ** a b c : Int H : c ∣ b h : ¬c = 0 k : Int hk : b = c * k ⊢ (a + b) / c = a / c + b / c ** rw [hk, Int.mul_comm c k, Int.add_mul_ediv_right _ _ h, ← Int.zero_add (k * c), Int.add_mul_ediv_right _ _ h, Int.zero_ediv, Int.zero_add] ** Qed
Int.add_ediv_of_dvd_left a b c : Int H : c ∣ a ⊢ (a + b) / c = a / c + b / c rw [Int.add_comm, Int.add_ediv_of_dvd_right H, Int.add_comm] ** Qed
Int.mul_ediv_cancel ** a b : Int H : b ≠ 0 ⊢ a * b / b = a have := Int.add_mul_ediv_right 0 a H a b : Int H : b ≠ 0 this : (0 + a * b) / b = 0 / b + a ⊢ a * b / b = a rwa [Int.zero_add, Int.zero_ediv, Int.zero_add] at this Qed
Int.mul_fdiv_cancel ** a b : Int H : b ≠ 0 b0 : 0 ≤ b ⊢ fdiv (a * b) b = a rw [fdiv_eq_ediv _ b0, mul_ediv_cancel _ H] a b x✝¹ : Int x✝ : x✝¹ < 0 H : x✝¹ ≠ 0 b0 : ¬0 ≤ x✝¹ ⊢ fdiv (0 * x✝¹) x✝¹ = 0 simp [Int.zero_mul] Qed
Int.div_self a : Int H : a ≠ 0 ⊢ div a a = 1 have := Int.mul_div_cancel 1 H ** a : Int H : a ≠ 0 this : div (1 * a) a = 1 ⊢ div a a = 1 rwa [Int.one_mul] at this Qed
Int.fdiv_self a : Int H : a ≠ 0 ⊢ fdiv a a = 1 have := Int.mul_fdiv_cancel 1 H ** a : Int H : a ≠ 0 this : fdiv (1 * a) a = 1 ⊢ fdiv a a = 1 rwa [Int.one_mul] at this Qed
Int.ediv_self a : Int H : a ≠ 0 ⊢ a / a = 1 have := Int.mul_ediv_cancel 1 H ** a : Int H : a ≠ 0 this : 1 * a / a = 1 ⊢ a / a = 1 rwa [Int.one_mul] at this Qed
Int.ofNat_fmod m n : Nat ⊢ ↑(m % n) = fmod ↑m ↑n cases m <;> simp [fmod] ** Qed
Int.negSucc_emod m : Nat b : Int bpos : 0 < b ⊢ -[m+1] % b = b - 1 - ↑m % b rw [Int.sub_sub, Int.add_comm] m : Nat b : Int bpos : 0 < b ⊢ -[m+1] % b = b - (↑m % b + 1) match b, eq_succ_of_zero_lt bpos with \| _, ⟨n, rfl⟩ => rfl m : Nat b : Int n : Nat bpos : 0 < ↑(succ n) ⊢ -[m+1] % ↑(succ n) = ↑(succ n) - (↑m % ↑(succ n) + 1) rfl ** Qed
Int.zero_mod b : Int ⊢ mod 0 b = 0 cases b <;> simp [mod] ** Qed
Int.zero_fmod b : Int ⊢ fmod 0 b = 0 cases b <;> rfl ** Qed
Int.mod_add_div ** m n : Nat ⊢ mod (ofNat m) -[n+1] + -[n+1] * div (ofNat m) -[n+1] = ofNat m ** show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m ** m n : Nat ⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m rw [Int.neg_mul_neg] m n : Nat ⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m exact congrArg ofNat (Nat.mod_add_div ..) m n : Nat ⊢ mod -[m+1] (ofNat n) + ofNat n * div -[m+1] (ofNat n) = -[m+1] ** show -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m) ** m n : Nat ⊢ -↑(succ m % n) + ↑n * -↑(succ m / n) = -↑(succ m) rw [Int.mul_neg, ← Int.neg_add] m n : Nat ⊢ mod -[m+1] -[n+1] + -[n+1] * div -[m+1] -[n+1] = -[m+1] ** show -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m) ** m n : Nat ⊢ -↑(succ m % succ n) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m) rw [Int.neg_mul, ← Int.neg_add] Qed
Int.emod_add_ediv ** m n : Nat ⊢ ofNat m % -[n+1] + -[n+1] * (ofNat m / -[n+1]) = ofNat m ** show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m ** m n : Nat ⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m rw [Int.neg_mul_neg] m n : Nat ⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m exact congrArg ofNat <\| Nat.mod_add_div .. a✝ : Nat ⊢ -[a✝+1] % 0 + 0 * (-[a✝+1] / 0) = -[a✝+1] rw [emod_zero] a✝ : Nat ⊢ -[a✝+1] + 0 * (-[a✝+1] / 0) = -[a✝+1] rfl x✝¹ x✝ : Int m n : Nat ⊢ ↑n - (↑m % ↑n + 1) - (↑n * (↑m / ↑n) + ↑n) = -[m+1] rw [← ofNat_emod, ← ofNat_ediv, ← Int.sub_sub, negSucc_eq, Int.sub_sub n, ← Int.neg_neg (_-_), Int.neg_sub, Int.sub_sub_self, Int.add_right_comm] x✝¹ x✝ : Int m n : Nat ⊢ -(↑(m % n) + ↑n * ↑(m / n) + 1) = -(↑m + 1) exact congrArg (fun x => -(ofNat x + 1)) (Nat.mod_add_div ..) Qed
Int.mod_def ** a b : Int ⊢ mod a b = a - b * div a b rw [← Int.add_sub_cancel (mod a b), mod_add_div] Qed
Int.fmod_def ** a b : Int ⊢ fmod a b = a - b * fdiv a b rw [← Int.add_sub_cancel (a.fmod b), fmod_add_fdiv] Qed
Int.emod_def ** a b : Int ⊢ a % b = a - b * (a / b) rw [← Int.add_sub_cancel (a % b), emod_add_ediv] Qed
Int.fmod_eq_emod a b : Int hb : 0 ≤ b ⊢ fmod a b = a % b simp [fmod_def, emod_def, fdiv_eq_ediv _ hb] ** Qed
Int.mod_eq_emod a b : Int ha : 0 ≤ a hb : 0 ≤ b ⊢ mod a b = a % b simp [emod_def, mod_def, div_eq_ediv ha hb] ** Qed
Int.mod_neg a b : Int ⊢ mod a (-b) = mod a b rw [mod_def, mod_def, Int.div_neg, Int.neg_mul_neg] ** Qed
Int.emod_neg a b : Int ⊢ a % -b = a % b rw [emod_def, emod_def, Int.ediv_neg, Int.neg_mul_neg] ** Qed
Int.emod_one a : Int ⊢ a % 1 = 0 simp [emod_def, Int.one_mul, Int.sub_self] ** Qed
Int.fmod_one a : Int ⊢ fmod a 1 = 0 simp [fmod_def, Int.one_mul, Int.sub_self] ** Qed
Int.mod_eq_of_lt a b : Int H1 : 0 ≤ a H2 : a < b ⊢ mod a b = a rw [mod_eq_emod H1 (Int.le_trans H1 (Int.le_of_lt H2)), emod_eq_of_lt H1 H2] ** Qed
Int.fmod_eq_of_lt a b : Int H1 : 0 ≤ a H2 : a < b ⊢ fmod a b = a rw [fmod_eq_emod _ (Int.le_trans H1 (Int.le_of_lt H2)), emod_eq_of_lt H1 H2] ** Qed
Int.mod_add_div' ** m k : Int ⊢ mod m k + div m k * k = m rw [Int.mul_comm] m k : Int ⊢ mod m k + k * div m k = m apply mod_add_div Qed
Int.div_add_mod' ** m k : Int ⊢ div m k * k + mod m k = m rw [Int.mul_comm] m k : Int ⊢ k * div m k + mod m k = m apply div_add_mod Qed
Int.emod_add_ediv' ** m k : Int ⊢ m % k + m / k * k = m rw [Int.mul_comm] m k : Int ⊢ m % k + k * (m / k) = m apply emod_add_ediv Qed
Int.add_mul_emod_self ** a b c : Int cz : c = 0 ⊢ (a + b * c) % c = a % c rw [cz, Int.mul_zero, Int.add_zero] a b c : Int cz : ¬c = 0 ⊢ (a + b * c) % c = a % c rw [Int.emod_def, Int.emod_def, Int.add_mul_ediv_right _ _ cz, Int.add_comm _ b, Int.mul_add, Int.mul_comm, ← Int.sub_sub, Int.add_sub_cancel] Qed
Int.add_mul_emod_self_left ** a b c : Int ⊢ (a + b * c) % b = a % b rw [Int.mul_comm, Int.add_mul_emod_self] Qed
Int.add_emod_self a b : Int ⊢ (a + b) % b = a % b have := add_mul_emod_self_left a b 1 ** a b : Int this : (a + b * 1) % b = a % b ⊢ (a + b) % b = a % b rwa [Int.mul_one] at this Qed
Int.add_emod_self_left a b : Int ⊢ (a + b) % a = b % a rw [Int.add_comm, Int.add_emod_self] ** Qed
Int.emod_add_emod m n k : Int ⊢ (m % n + k) % n = (m + k) % n have := (add_mul_emod_self_left (m % n + k) n (m / n)).symm ** m n k : Int this : (m % n + k) % n = (m % n + k + n * (m / n)) % n ⊢ (m % n + k) % n = (m + k) % n rwa [Int.add_right_comm, emod_add_ediv] at this Qed
Int.add_emod_emod m n k : Int ⊢ (m + n % k) % k = (m + n) % k rw [Int.add_comm, emod_add_emod, Int.add_comm] ** Qed
Int.add_emod a b n : Int ⊢ (a + b) % n = (a % n + b % n) % n rw [add_emod_emod, emod_add_emod] ** Qed
Int.add_emod_eq_add_emod_right m n k i : Int H : m % n = k % n ⊢ (m + i) % n = (k + i) % n rw [← emod_add_emod, ← emod_add_emod k, H] ** Qed
Int.add_emod_eq_add_emod_left m n k i : Int H : m % n = k % n ⊢ (i + m) % n = (i + k) % n rw [Int.add_comm, add_emod_eq_add_emod_right _ H, Int.add_comm] ** Qed
Int.emod_add_cancel_right m n k i : Int H : (m + i) % n = (k + i) % n ⊢ m % n = k % n have := add_emod_eq_add_emod_right (-i) H m n k i : Int H : (m + i) % n = (k + i) % n this : (m + i + -i) % n = (k + i + -i) % n ⊢ m % n = k % n rwa [Int.add_neg_cancel_right, Int.add_neg_cancel_right] at this ** Qed
Int.emod_add_cancel_left m n k i : Int ⊢ (i + m) % n = (i + k) % n ↔ m % n = k % n rw [Int.add_comm, Int.add_comm i, emod_add_cancel_right] ** Qed
Int.emod_eq_emod_iff_emod_sub_eq_zero m n k : Int ⊢ (m - k) % n = (k - k) % n ↔ (m - k) % n = 0 simp [Int.sub_self] ** Qed
Int.mul_mod_left ** a b : Int h : b = 0 ⊢ mod (a * b) b = 0 simp [h, Int.mul_zero] a b : Int h : ¬b = 0 ⊢ mod (a * b) b = 0 rw [Int.mod_def, Int.mul_div_cancel _ h, Int.mul_comm, Int.sub_self] Qed
Int.mul_fmod_left ** a b : Int h : b = 0 ⊢ fmod (a * b) b = 0 simp [h, Int.mul_zero] a b : Int h : ¬b = 0 ⊢ fmod (a * b) b = 0 rw [Int.fmod_def, Int.mul_fdiv_cancel _ h, Int.mul_comm, Int.sub_self] Qed
Int.mul_emod_left ** a b : Int ⊢ a * b % b = 0 ** rw [← Int.zero_add (a * b), Int.add_mul_emod_self, Int.zero_emod] ** Qed
Int.mul_mod_right ** a b : Int ⊢ mod (a * b) a = 0 rw [Int.mul_comm, mul_mod_left] Qed
Int.mul_fmod_right ** a b : Int ⊢ fmod (a * b) a = 0 rw [Int.mul_comm, mul_fmod_left] Qed
Int.mul_emod_right ** a b : Int ⊢ a * b % a = 0 rw [Int.mul_comm, mul_emod_left] Qed
Int.mod_self a : Int ⊢ mod a a = 0 have := mul_mod_left 1 a ** a : Int this : mod (1 * a) a = 0 ⊢ mod a a = 0 rwa [Int.one_mul] at this Qed
Int.fmod_self a : Int ⊢ fmod a a = 0 have := mul_fmod_left 1 a ** a : Int this : fmod (1 * a) a = 0 ⊢ fmod a a = 0 rwa [Int.one_mul] at this Qed
Int.emod_self a : Int ⊢ a % a = 0 have := mul_emod_left 1 a ** a : Int this : 1 * a % a = 0 ⊢ a % a = 0 rwa [Int.one_mul] at this Qed
Int.emod_emod_of_dvd n m k : Int h : m ∣ k ⊢ n % k % m = n % m conv => rhs; rw [← emod_add_ediv n k] ** n m k : Int h : m ∣ k ⊢ n % k % m = (n % k + k * (n / k)) % m match k, h with \| _, ⟨t, rfl⟩ => rw [Int.mul_assoc, add_mul_emod_self_left] n m k : Int h : m ∣ k t : Int ⊢ n % (m * t) % m = (n % (m * t) + m * t * (n / (m * t))) % m rw [Int.mul_assoc, add_mul_emod_self_left] Qed