refactor(library/data/{int,rat}/*): clean up casts between nat, int, and rat
This commit is contained in:
Jeremy Avigad
parent
20f6b4c6bd
commit
de83a68667
10 changed files with 97 additions and 66 deletions
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@ -87,7 +87,10 @@ infix [priority int.prio] * := int.mul
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theorem of_nat.inj {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
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int.no_confusion H imp.id
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theorem of_nat_eq_of_nat (m n : ℕ) : of_nat m = of_nat n ↔ m = n :=
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theorem eq_of_of_nat_eq_of_nat {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
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of_nat.inj H
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theorem of_nat_eq_of_nat_iff (m n : ℕ) : of_nat m = of_nat n ↔ m = n :=
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iff.intro of_nat.inj !congr_arg
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theorem neg_succ_of_nat.inj {m n : ℕ} (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n :=
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@ -97,7 +100,7 @@ theorem neg_succ_of_nat_eq (n : ℕ) : -[1+ n] = -(n + 1) := rfl
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private definition has_decidable_eq₂ : Π (a b : ℤ), decidable (a = b)
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| (of_nat m) (of_nat n) := decidable_of_decidable_of_iff
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(nat.has_decidable_eq m n) (iff.symm (of_nat_eq_of_nat m n))
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(nat.has_decidable_eq m n) (iff.symm (of_nat_eq_of_nat_iff m n))
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| (of_nat m) -[1+ n] := inr (by contradiction)
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| -[1+ m] (of_nat n) := inr (by contradiction)
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| -[1+ m] -[1+ n] := if H : m = n then
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@ -288,7 +288,7 @@ obtain m (Hm : a = of_nat m), from exists_eq_of_nat H1,
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obtain n (Hn : b = of_nat n), from exists_eq_of_nat (le_of_lt (lt_of_le_of_lt H1 H2)),
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begin
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revert H2,
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rewrite [Hm, Hn, of_nat_mod, of_nat_lt_of_nat, of_nat_eq_of_nat],
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rewrite [Hm, Hn, of_nat_mod, of_nat_lt_of_nat_iff, of_nat_eq_of_nat_iff],
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apply nat.mod_eq_of_lt
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end
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@ -481,21 +481,21 @@ nat.by_cases_zero_pos n
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assume H4 : of_nat n' = of_nat m * c,
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have H5 : c > 0, from pos_of_mul_pos_left (H4 ▸ H2) !of_nat_nonneg,
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obtain k (H6 : c = of_nat k), from exists_eq_of_nat (le_of_lt H5),
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have H7 : n' = (#nat m * k), from (!iff.mp !of_nat_eq_of_nat (H6 ▸ H4)),
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have H7 : n' = (#nat m * k), from (of_nat.inj (H6 ▸ H4)),
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nat.dvd.intro H7⁻¹))
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theorem of_nat_dvd_of_nat_of_dvd {m n : ℕ} (H : #nat m ∣ n) : of_nat m ∣ of_nat n :=
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nat.dvd.elim H
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(take k, assume H1 : #nat n = m * k,
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dvd.intro (!iff.mpr !of_nat_eq_of_nat H1⁻¹))
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dvd.intro (H1⁻¹ ▸ rfl))
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theorem of_nat_dvd_of_nat (m n : ℕ) : of_nat m ∣ of_nat n ↔ (#nat m ∣ n) :=
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theorem of_nat_dvd_of_nat_iff (m n : ℕ) : of_nat m ∣ of_nat n ↔ (#nat m ∣ n) :=
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iff.intro dvd_of_of_nat_dvd_of_nat of_nat_dvd_of_nat_of_dvd
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theorem dvd.antisymm {a b : ℤ} (H1 : a ≥ 0) (H2 : b ≥ 0) : a ∣ b → b ∣ a → a = b :=
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begin
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rewrite [-abs_of_nonneg H1, -abs_of_nonneg H2, -*of_nat_nat_abs],
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rewrite [*of_nat_dvd_of_nat, *of_nat_eq_of_nat],
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rewrite [*of_nat_dvd_of_nat_iff, *of_nat_eq_of_nat_iff],
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apply nat.dvd.antisymm
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end
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@ -63,7 +63,7 @@ by rewrite [↑gcd, nat.gcd_self, of_nat_nat_abs]
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theorem gcd_dvd_left (a b : ℤ) : gcd a b ∣ a :=
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have gcd a b ∣ abs a,
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by rewrite [↑gcd, -of_nat_nat_abs, of_nat_dvd_of_nat]; apply nat.gcd_dvd_left,
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by rewrite [↑gcd, -of_nat_nat_abs, of_nat_dvd_of_nat_iff]; apply nat.gcd_dvd_left,
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iff.mp !dvd_abs_iff this
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theorem gcd_dvd_right (a b : ℤ) : gcd a b ∣ b :=
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@ -72,7 +72,7 @@ by rewrite gcd.comm; apply gcd_dvd_left
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theorem dvd_gcd {a b c : ℤ} : a ∣ b → a ∣ c → a ∣ gcd b c :=
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begin
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rewrite [↑gcd, -*(abs_dvd_iff a), -(dvd_abs_iff _ b), -(dvd_abs_iff _ c), -*of_nat_nat_abs],
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rewrite [*of_nat_dvd_of_nat] ,
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rewrite [*of_nat_dvd_of_nat_iff] ,
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apply nat.dvd_gcd
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end
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@ -212,21 +212,21 @@ theorem lcm_self (a : ℤ) : lcm a a = abs a :=
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by krewrite [↑lcm, nat.lcm_self, of_nat_nat_abs]
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theorem dvd_lcm_left (a b : ℤ) : a ∣ lcm a b :=
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by rewrite [↑lcm, -abs_dvd_iff, -of_nat_nat_abs, of_nat_dvd_of_nat]; apply nat.dvd_lcm_left
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by rewrite [↑lcm, -abs_dvd_iff, -of_nat_nat_abs, of_nat_dvd_of_nat_iff]; apply nat.dvd_lcm_left
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theorem dvd_lcm_right (a b : ℤ) : b ∣ lcm a b :=
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!lcm.comm ▸ !dvd_lcm_left
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theorem gcd_mul_lcm (a b : ℤ) : gcd a b * lcm a b = abs (a * b) :=
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begin
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rewrite [↑gcd, ↑lcm, -of_nat_nat_abs, -of_nat_mul, of_nat_eq_of_nat, nat_abs_mul],
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rewrite [↑gcd, ↑lcm, -of_nat_nat_abs, -of_nat_mul, of_nat_eq_of_nat_iff, nat_abs_mul],
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apply nat.gcd_mul_lcm
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end
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theorem lcm_dvd {a b c : ℤ} : a ∣ c → b ∣ c → lcm a b ∣ c :=
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begin
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rewrite [↑lcm, -(abs_dvd_iff a), -(abs_dvd_iff b), -*(dvd_abs_iff _ c), -*of_nat_nat_abs],
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rewrite [*of_nat_dvd_of_nat] ,
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rewrite [*of_nat_dvd_of_nat_iff] ,
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apply nat.lcm_dvd
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end
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@ -261,7 +261,7 @@ dvd_of_coprime_of_dvd_mul_right H1 (!mul.comm ▸ H2)
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theorem gcd_mul_left_cancel_of_coprime {c : ℤ} (a : ℤ) {b : ℤ} (H : coprime c b) :
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gcd (c * a) b = gcd a b :=
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begin
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revert H, rewrite [↑coprime, ↑gcd, *of_nat_eq_of_nat, nat_abs_mul],
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revert H, rewrite [↑coprime, ↑gcd, *of_nat_eq_of_nat_iff, nat_abs_mul],
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apply nat.gcd_mul_left_cancel_of_coprime
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end
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@ -45,7 +45,7 @@ theorem le.total (a b : ℤ) : a ≤ b ∨ b ≤ a :=
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or.imp_right
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(assume H : nonneg (-(b - a)),
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have -(b - a) = a - b, from !neg_sub,
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show nonneg (a - b), from this ▸ H) -- too bad: can't do it in one step
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show nonneg (a - b), from this ▸ H)
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(nonneg_or_nonneg_neg (b - a))
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theorem of_nat_le_of_nat_of_le {m n : ℕ} (H : #nat m ≤ n) : of_nat m ≤ of_nat n :=
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@ -57,7 +57,7 @@ obtain (k : ℕ) (Hk : of_nat m + of_nat k = of_nat n), from le.elim H,
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have m + k = n, from of_nat.inj (of_nat_add m k ⬝ Hk),
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nat.le.intro this
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theorem of_nat_le_of_nat (m n : ℕ) : of_nat m ≤ of_nat n ↔ m ≤ n :=
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theorem of_nat_le_of_nat_iff (m n : ℕ) : of_nat m ≤ of_nat n ↔ m ≤ n :=
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iff.intro le_of_of_nat_le_of_nat of_nat_le_of_nat_of_le
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theorem lt_add_succ (a : ℤ) (n : ℕ) : a < a + succ n :=
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@ -78,18 +78,18 @@ have a + succ n = b, from
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... = b : Hn,
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exists.intro n this
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theorem of_nat_lt_of_nat (n m : ℕ) : of_nat n < of_nat m ↔ n < m :=
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theorem of_nat_lt_of_nat_iff (n m : ℕ) : of_nat n < of_nat m ↔ n < m :=
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calc
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of_nat n < of_nat m ↔ of_nat n + 1 ≤ of_nat m : iff.refl
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... ↔ of_nat (nat.succ n) ≤ of_nat m : of_nat_succ n ▸ !iff.refl
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... ↔ nat.succ n ≤ m : of_nat_le_of_nat
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... ↔ nat.succ n ≤ m : of_nat_le_of_nat_iff
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... ↔ n < m : iff.symm (lt_iff_succ_le _ _)
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theorem lt_of_of_nat_lt_of_nat {m n : ℕ} (H : of_nat m < of_nat n) : #nat m < n :=
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iff.mp !of_nat_lt_of_nat H
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iff.mp !of_nat_lt_of_nat_iff H
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theorem of_nat_lt_of_nat_of_lt {m n : ℕ} (H : #nat m < n) : of_nat m < of_nat n :=
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iff.mpr !of_nat_lt_of_nat H
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iff.mpr !of_nat_lt_of_nat_iff H
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/- show that the integers form an ordered additive group -/
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@ -388,7 +388,6 @@ dvd.elim H'
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(take b,
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suppose 1 = a * b,
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eq_one_of_mul_eq_one_right H this⁻¹)
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--
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theorem ex_smallest_of_bdd {P : ℤ → Prop} [HP : decidable_pred P] (Hbdd : ∃ b : ℤ, ∀ z : ℤ, z ≤ b → ¬ P z)
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(Hinh : ∃ z : ℤ, P z) : ∃ lb : ℤ, P lb ∧ (∀ z : ℤ, z < lb → ¬ P z) :=
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@ -418,7 +417,7 @@ theorem ex_smallest_of_bdd {P : ℤ → Prop} [HP : decidable_pred P] (Hbdd :
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have Hk : nat_abs (z - b) < least (λ n, P (b + of_nat n)) (nat.succ (nat_abs (elt - b))), begin
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let Hz' := iff.mp !int.lt_add_iff_sub_lt_left Hz,
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rewrite [-of_nat_nat_abs_of_nonneg (int.le_of_lt Hpos) at Hz'],
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apply iff.mp !int.of_nat_lt_of_nat Hz'
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apply lt_of_of_nat_lt_of_nat Hz'
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end,
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let Hk' := nat.not_le_of_gt Hk,
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rewrite Hzbk,
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@ -455,7 +454,7 @@ theorem ex_largest_of_bdd {P : ℤ → Prop} [HP : decidable_pred P] (Hbdd : ∃
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have Hk : nat_abs (b - z) < least (λ n, P (b - of_nat n)) (nat.succ (nat_abs (b - elt))), begin
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let Hz' := iff.mp !int.lt_add_iff_sub_lt_left (iff.mpr !int.lt_add_iff_sub_lt_right Hz),
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rewrite [-of_nat_nat_abs_of_nonneg (int.le_of_lt Hpos) at Hz'],
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apply iff.mp !int.of_nat_lt_of_nat Hz'
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apply lt_of_of_nat_lt_of_nat Hz'
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end,
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let Hk' := nat.not_le_of_gt Hk,
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rewrite Hzbk,
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@ -88,16 +88,16 @@ theorem pnat.to_rat_of_nat (n : ℕ+) : rat_of_pnat n = of_nat n~ := rfl
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theorem rat_of_nat_nonneg (n : ℕ) : 0 ≤ of_nat n := trivial
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theorem rat_of_pnat_ge_one (n : ℕ+) : rat_of_pnat n ≥ 1 :=
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(iff.mpr !of_nat_le_of_nat) (pnat_pos n)
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of_nat_le_of_nat_of_le (pnat_pos n)
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theorem rat_of_pnat_is_pos (n : ℕ+) : rat_of_pnat n > 0 :=
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(iff.mpr !of_nat_pos) (pnat_pos n)
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of_nat_lt_of_nat_of_lt (pnat_pos n)
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theorem of_nat_le_of_nat_of_le {m n : ℕ} (H : m ≤ n) : of_nat m ≤ of_nat n :=
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(iff.mpr !of_nat_le_of_nat) H
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of_nat_le_of_nat_of_le H
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theorem of_nat_lt_of_nat_of_lt {m n : ℕ} (H : m < n) : of_nat m < of_nat n :=
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(iff.mpr !of_nat_lt_of_nat) H
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of_nat_lt_of_nat_of_lt H
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theorem rat_of_pnat_le_of_pnat_le {m n : ℕ+} (H : m ≤ n) : rat_of_pnat m ≤ rat_of_pnat n :=
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of_nat_le_of_nat_of_le H
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@ -106,7 +106,7 @@ theorem rat_of_pnat_lt_of_pnat_lt {m n : ℕ+} (H : m < n) : rat_of_pnat m < rat
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of_nat_lt_of_nat_of_lt H
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theorem pnat_le_of_rat_of_pnat_le {m n : ℕ+} (H : rat_of_pnat m ≤ rat_of_pnat n) : m ≤ n :=
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(iff.mp !of_nat_le_of_nat) H
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le_of_of_nat_le_of_nat H
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definition inv (n : ℕ+) : ℚ := (1 : ℚ) / rat_of_pnat n
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postfix `⁻¹` := inv
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@ -349,8 +349,8 @@ namespace rat
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/- operations -/
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definition of_int [coercion] (i : ℤ) : ℚ := ⟦prerat.of_int i⟧
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definition of_nat [coercion] (n : ℕ) : ℚ := ⟦prerat.of_int n⟧
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definition of_num [coercion] [reducible] (n : num) : ℚ := of_int (int.of_num n)
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definition of_nat [coercion] (n : ℕ) : ℚ := nat.to.rat n
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definition of_num [coercion] [reducible] (n : num) : ℚ := num.to.rat n
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definition add : ℚ → ℚ → ℚ :=
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quot.lift₂
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@ -413,16 +413,28 @@ calc
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theorem of_int.inj {a b : ℤ} (H : of_int a = of_int b) : a = b :=
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prerat.of_int.inj (quot.exact H)
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theorem eq_of_of_int_eq_of_int {a b : ℤ} (H : of_int a = of_int b) : a = b :=
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of_int.inj H
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theorem of_nat_eq (a : ℕ) : of_nat a = of_int (int.of_nat a) := rfl
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theorem of_nat_add (a b : ℕ) : of_nat (#nat a + b) = of_nat a + of_nat b :=
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by rewrite [*of_nat_eq, int.of_nat_add, rat.of_int_add]
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by rewrite [of_nat_eq, int.of_nat_add, rat.of_int_add]
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theorem of_nat_mul (a b : ℕ) : of_nat (#nat a * b) = of_nat a * of_nat b :=
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by rewrite [*of_nat_eq, int.of_nat_mul, rat.of_int_mul]
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by rewrite [of_nat_eq, int.of_nat_mul, rat.of_int_mul]
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theorem of_nat_sub {a b : ℕ} (H : #nat a ≥ b) : of_nat (#nat a - b) = of_nat a - of_nat b :=
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by rewrite [*of_nat_eq, int.of_nat_sub H, rat.of_int_sub]
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by rewrite [of_nat_eq, int.of_nat_sub H, rat.of_int_sub]
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theorem of_nat.inj {a b : ℕ} (H : of_nat a = of_nat b) : a = b :=
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int.of_nat.inj (of_int.inj H)
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theorem eq_of_of_nat_eq_of_nat {a b : ℕ} (H : of_nat a = of_nat b) : a = b :=
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of_nat.inj H
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theorem of_nat_eq_of_nat_iff (a b : ℕ) : of_nat a = of_nat b ↔ a = b :=
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iff.intro of_nat.inj !congr_arg
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theorem add.comm (a b : ℚ) : a + b = b + a :=
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quot.induction_on₂ a b (take u v, quot.sound !prerat.add.comm)
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@ -152,35 +152,52 @@ infix [priority rat.prio] >= := rat.ge
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infix [priority rat.prio] ≥ := rat.ge
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infix [priority rat.prio] > := rat.gt
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theorem of_int_lt_of_int (a b : ℤ) : of_int a < of_int b ↔ (#int a < b) :=
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theorem of_int_lt_of_int_iff (a b : ℤ) : of_int a < of_int b ↔ (#int a < b) :=
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iff.symm (calc
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(#int a < b) ↔ (#int b - a > 0) : iff.symm !int.sub_pos_iff_lt
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... ↔ pos (of_int (#int b - a)) : iff.symm !pos_of_int
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... ↔ pos (of_int b - of_int a) : !of_int_sub ▸ iff.rfl
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... ↔ of_int a < of_int b : iff.rfl)
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theorem of_int_le_of_int (a b : ℤ) : of_int a ≤ of_int b ↔ (#int a ≤ b) :=
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theorem of_int_lt_of_int_of_lt {a b : ℤ} (H : (#int a < b)) : of_int a < of_int b :=
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iff.mpr !of_int_lt_of_int_iff H
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theorem lt_of_of_int_lt_of_int {a b : ℤ} (H : of_int a < of_int b) : (#int a < b) :=
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iff.mp !of_int_lt_of_int_iff H
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theorem of_int_le_of_int_iff (a b : ℤ) : of_int a ≤ of_int b ↔ (#int a ≤ b) :=
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iff.symm (calc
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(#int a ≤ b) ↔ (#int b - a ≥ 0) : iff.symm !int.sub_nonneg_iff_le
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... ↔ nonneg (of_int (#int b - a)) : iff.symm !nonneg_of_int
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... ↔ nonneg (of_int b - of_int a) : !of_int_sub ▸ iff.rfl
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... ↔ of_int a ≤ of_int b : iff.rfl)
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theorem of_int_pos (a : ℤ) : (of_int a > 0) ↔ (#int a > 0) := !of_int_lt_of_int
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theorem of_int_le_of_int_of_le {a b : ℤ} (H : (#int a ≤ b)) : of_int a ≤ of_int b :=
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iff.mpr !of_int_le_of_int_iff H
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theorem of_int_nonneg (a : ℤ) : (of_int a ≥ 0) ↔ (#int a ≥ 0) := !of_int_le_of_int
|
||||
theorem le_of_of_int_le_of_int {a b : ℤ} (H : of_int a ≤ of_int b) : (#int a ≤ b) :=
|
||||
iff.mp !of_int_le_of_int_iff H
|
||||
|
||||
theorem of_nat_lt_of_nat (a b : ℕ) : of_nat a < of_nat b ↔ (#nat a < b) :=
|
||||
by rewrite [*of_nat_eq, propext !of_int_lt_of_int]; apply int.of_nat_lt_of_nat
|
||||
theorem of_nat_lt_of_nat_iff (a b : ℕ) : of_nat a < of_nat b ↔ (#nat a < b) :=
|
||||
by rewrite [*of_nat_eq, of_int_lt_of_int_iff, int.of_nat_lt_of_nat_iff]
|
||||
|
||||
theorem of_nat_le_of_nat (a b : ℕ) : of_nat a ≤ of_nat b ↔ (#nat a ≤ b) :=
|
||||
by rewrite [*of_nat_eq, propext !of_int_le_of_int]; apply int.of_nat_le_of_nat
|
||||
theorem of_nat_lt_of_nat_of_lt {a b : ℕ} (H : (#nat a < b)) : of_nat a < of_nat b :=
|
||||
iff.mpr !of_nat_lt_of_nat_iff H
|
||||
|
||||
theorem of_nat_pos (a : ℕ) : (of_nat a > 0) ↔ (#nat a > nat.zero) :=
|
||||
!of_nat_lt_of_nat
|
||||
theorem lt_of_of_nat_lt_of_nat {a b : ℕ} (H : of_nat a < of_nat b) : (#nat a < b) :=
|
||||
iff.mp !of_nat_lt_of_nat_iff H
|
||||
|
||||
theorem of_nat_le_of_nat_iff (a b : ℕ) : of_nat a ≤ of_nat b ↔ (#nat a ≤ b) :=
|
||||
by rewrite [*of_nat_eq, of_int_le_of_int_iff, int.of_nat_le_of_nat_iff]
|
||||
|
||||
theorem of_nat_le_of_nat_of_le {a b : ℕ} (H : (#nat a ≤ b)) : of_nat a ≤ of_nat b :=
|
||||
iff.mpr !of_nat_le_of_nat_iff H
|
||||
|
||||
theorem le_of_of_nat_le_of_nat {a b : ℕ} (H : of_nat a ≤ of_nat b) : (#nat a ≤ b) :=
|
||||
iff.mp !of_nat_le_of_nat_iff H
|
||||
|
||||
theorem of_nat_nonneg (a : ℕ) : (of_nat a ≥ 0) :=
|
||||
iff.mpr !of_nat_le_of_nat !nat.zero_le
|
||||
of_nat_le_of_nat_of_le !nat.zero_le
|
||||
|
||||
theorem le.refl (a : ℚ) : a ≤ a :=
|
||||
by rewrite [↑rat.le, sub_self]; apply nonneg_zero
|
||||
|
|
@ -328,8 +345,8 @@ end migrate_algebra
|
|||
theorem rat_of_nat_abs (a : ℤ) : abs (of_int a) = of_nat (int.nat_abs a) :=
|
||||
assert ∀ n : ℕ, of_int (int.neg_succ_of_nat n) = - of_nat (nat.succ n), from λ n, rfl,
|
||||
int.induction_on a
|
||||
(take b, abs_of_nonneg (!of_nat_nonneg))
|
||||
(take b, by rewrite [this, abs_neg, abs_of_nonneg (!of_nat_nonneg)])
|
||||
(take b, abs_of_nonneg !of_nat_nonneg)
|
||||
(take b, by rewrite [this, abs_neg, abs_of_nonneg !of_nat_nonneg])
|
||||
|
||||
section
|
||||
open int
|
||||
|
|
@ -341,40 +358,40 @@ section
|
|||
begin
|
||||
rewrite [-mul_denom],
|
||||
apply mul_nonneg H,
|
||||
rewrite [of_int_le_of_int],
|
||||
rewrite [of_int_le_of_int_iff],
|
||||
exact int.le_of_lt !denom_pos
|
||||
end,
|
||||
show num q ≥ 0, from iff.mp !of_int_le_of_int this
|
||||
show num q ≥ 0, from le_of_of_int_le_of_int this
|
||||
|
||||
theorem num_pos_of_pos {q : ℚ} (H : q > 0) : num q > 0 :=
|
||||
have of_int (num q) > of_int 0,
|
||||
begin
|
||||
rewrite [-mul_denom],
|
||||
apply mul_pos H,
|
||||
rewrite [of_int_lt_of_int],
|
||||
rewrite [of_int_lt_of_int_iff],
|
||||
exact !denom_pos
|
||||
end,
|
||||
show num q > 0, from iff.mp !of_int_lt_of_int this
|
||||
show num q > 0, from lt_of_of_int_lt_of_int this
|
||||
|
||||
theorem num_neg_of_neg {q : ℚ} (H : q < 0) : num q < 0 :=
|
||||
have of_int (num q) < of_int 0,
|
||||
begin
|
||||
rewrite [-mul_denom],
|
||||
apply mul_neg_of_neg_of_pos H,
|
||||
rewrite [of_int_lt_of_int],
|
||||
rewrite [of_int_lt_of_int_iff],
|
||||
exact !denom_pos
|
||||
end,
|
||||
show num q < 0, from iff.mp !of_int_lt_of_int this
|
||||
show num q < 0, from lt_of_of_int_lt_of_int this
|
||||
|
||||
theorem num_nonpos_of_nonpos {q : ℚ} (H : q ≤ 0) : num q ≤ 0 :=
|
||||
have of_int (num q) ≤ of_int 0,
|
||||
begin
|
||||
rewrite [-mul_denom],
|
||||
apply mul_nonpos_of_nonpos_of_nonneg H,
|
||||
rewrite [of_int_le_of_int],
|
||||
rewrite [of_int_le_of_int_iff],
|
||||
exact int.le_of_lt !denom_pos
|
||||
end,
|
||||
show num q ≤ 0, from iff.mp !of_int_le_of_int this
|
||||
show num q ≤ 0, from le_of_of_int_le_of_int this
|
||||
end
|
||||
|
||||
definition ubound : ℚ → ℕ := λ a : ℚ, nat.succ (int.nat_abs (num a))
|
||||
|
|
@ -382,10 +399,10 @@ definition ubound : ℚ → ℕ := λ a : ℚ, nat.succ (int.nat_abs (num a))
|
|||
theorem ubound_ge (a : ℚ) : of_nat (ubound a) ≥ a :=
|
||||
have h : abs a * abs (of_int (denom a)) = abs (of_int (num a)), from
|
||||
!abs_mul ▸ !mul_denom ▸ rfl,
|
||||
assert of_int (denom a) > 0, from (iff.mpr !of_int_pos) !denom_pos,
|
||||
assert of_int (denom a) > 0, from of_int_lt_of_int_of_lt !denom_pos,
|
||||
have 1 ≤ abs (of_int (denom a)), begin
|
||||
rewrite (abs_of_pos this),
|
||||
apply iff.mpr !of_int_le_of_int,
|
||||
apply of_int_le_of_int_of_le,
|
||||
apply denom_pos
|
||||
end,
|
||||
have abs a ≤ abs (of_int (num a)), from
|
||||
|
|
|
|||
|
|
@ -1061,11 +1061,11 @@ theorem one_mul (x : ℝ) : 1 * x = x :=
|
|||
theorem mul_one (x : ℝ) : x * 1 = x :=
|
||||
quot.induction_on x (λ s, quot.sound (r_mul_one s))
|
||||
|
||||
theorem distrib (x y z : ℝ) : x * (y + z) = x * y + x * z :=
|
||||
theorem left_distrib (x y z : ℝ) : x * (y + z) = x * y + x * z :=
|
||||
quot.induction_on₃ x y z (λ s t u, quot.sound (r_distrib s t u))
|
||||
|
||||
theorem distrib_l (x y z : ℝ) : (x + y) * z = x * z + y * z :=
|
||||
by rewrite [mul_comm, distrib, {x * _}mul_comm, {y * _}mul_comm] -- this shouldn't be necessary
|
||||
theorem right_distrib (x y z : ℝ) : (x + y) * z = x * z + y * z :=
|
||||
by rewrite [mul_comm, left_distrib, {x * _}mul_comm, {y * _}mul_comm] -- this shouldn't be necessary
|
||||
|
||||
theorem zero_ne_one : ¬ (0 : ℝ) = 1 :=
|
||||
take H : 0 = 1,
|
||||
|
|
@ -1087,8 +1087,8 @@ protected definition comm_ring [reducible] : algebra.comm_ring ℝ :=
|
|||
apply of_num 1,
|
||||
apply one_mul,
|
||||
apply mul_one,
|
||||
apply distrib,
|
||||
apply distrib_l,
|
||||
apply left_distrib,
|
||||
apply right_distrib,
|
||||
apply mul_comm
|
||||
end
|
||||
|
||||
|
|
|
|||
|
|
@ -399,7 +399,7 @@ theorem archimedean_upper_strict (x : ℝ) : ∃ z : ℤ, x < of_rat (of_int z)
|
|||
apply lt_of_le_of_lt,
|
||||
apply Hz,
|
||||
apply of_rat_lt_of_rat_of_lt,
|
||||
apply iff.mpr !of_int_lt_of_int,
|
||||
apply of_int_lt_of_int_of_lt,
|
||||
apply int.lt_add_of_pos_right,
|
||||
apply dec_trivial
|
||||
end
|
||||
|
|
@ -431,7 +431,7 @@ definition ex_floor (x : ℝ) :=
|
|||
apply Har,
|
||||
have H : of_rat (of_int (some (archimedean_upper_strict x))) ≤ of_rat (of_int z), begin
|
||||
apply of_rat_le_of_rat_of_le,
|
||||
apply iff.mpr !of_int_le_of_int,
|
||||
apply of_int_le_of_int_of_le,
|
||||
apply Hz
|
||||
end,
|
||||
exact H
|
||||
|
|
@ -553,7 +553,7 @@ noncomputable definition under : ℚ := of_int (floor (elt - 1))
|
|||
theorem under_spec1 : of_rat under < elt :=
|
||||
have H : of_rat under < of_rat (of_int (floor elt)), begin
|
||||
apply of_rat_lt_of_rat_of_lt,
|
||||
apply iff.mpr !of_int_lt_of_int,
|
||||
apply of_int_lt_of_int_of_lt,
|
||||
apply floor_succ_lt
|
||||
end,
|
||||
lt_of_lt_of_le H !floor_spec
|
||||
|
|
@ -574,7 +574,7 @@ noncomputable definition over : ℚ := of_int (ceil (bound + 1)) -- b
|
|||
theorem over_spec1 : bound < of_rat over :=
|
||||
have H : of_rat (of_int (ceil bound)) < of_rat over, begin
|
||||
apply of_rat_lt_of_rat_of_lt,
|
||||
apply iff.mpr !of_int_lt_of_int,
|
||||
apply of_int_lt_of_int_of_lt,
|
||||
apply ceil_succ
|
||||
end,
|
||||
lt_of_le_of_lt !ceil_spec H
|
||||
|
|
|
|||
|
|
@ -24,7 +24,7 @@ section
|
|||
have even b, from even_of_even_pow this,
|
||||
have 2 ∣ gcd a b, from dvd_gcd (dvd_of_even `even a`) (dvd_of_even `even b`),
|
||||
have 2 ∣ 1, from co ▸ this,
|
||||
absurd `2 ∣ 1` dec_trivial
|
||||
show false, from absurd `2 ∣ 1` dec_trivial
|
||||
end
|
||||
|
||||
/-
|
||||
|
|
@ -81,7 +81,7 @@ section
|
|||
using this, by rewrite [-int.abs_pow, this, int.abs_mul, int.abs_pow],
|
||||
have H₁ : (nat_abs a)^n = nat_abs c * (nat_abs b)^n,
|
||||
using this,
|
||||
by apply of_nat.inj; rewrite [int.of_nat_mul, +of_nat_pow, +of_nat_nat_abs]; assumption,
|
||||
by apply int.of_nat.inj; rewrite [int.of_nat_mul, +of_nat_pow, +of_nat_nat_abs]; assumption,
|
||||
have H₂ : nat.coprime (nat_abs a) (nat_abs b), from of_nat.inj !coprime_num_denom,
|
||||
have nat_abs b = 1, from
|
||||
by_cases
|
||||
|
|
|
|||
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Reference in a new issue