chore(library/init/data/nat/basic): cleanup
This commit is contained in:
Leonardo de Moura
parent
7da6d5c50d
commit
752e0a134e
1 changed files with 157 additions and 158 deletions
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@ -10,24 +10,27 @@ notation `ℕ` := nat
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namespace nat
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inductive less_than_or_equal (a : ℕ) : ℕ → Prop
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inductive less_than_or_equal (a : nat) : nat → Prop
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| refl : less_than_or_equal a
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| step : Π {b}, less_than_or_equal b → less_than_or_equal (succ b)
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instance : has_le ℕ :=
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instance : has_le nat :=
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⟨nat.less_than_or_equal⟩
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protected def le (n m : ℕ) := nat.less_than_or_equal n m
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protected def lt (n m : ℕ) := nat.less_than_or_equal (succ n) m
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protected def le (n m : nat) :=
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nat.less_than_or_equal n m
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instance : has_lt ℕ :=
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protected def lt (n m : nat) :=
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nat.less_than_or_equal (succ n) m
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instance : has_lt nat :=
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⟨nat.lt⟩
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def pred : ℕ → ℕ
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def pred : nat → nat
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| 0 := 0
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| (a+1) := a
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protected def sub : ℕ → ℕ → ℕ
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protected def sub : nat → nat → nat
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| a 0 := a
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| a (b+1) := pred (sub a b)
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@ -35,13 +38,13 @@ protected def mul : nat → nat → nat
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| a 0 := 0
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| a (b+1) := (mul a b) + a
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instance : has_sub ℕ :=
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instance : has_sub nat :=
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⟨nat.sub⟩
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instance : has_mul ℕ :=
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instance : has_mul nat :=
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⟨nat.mul⟩
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instance : decidable_eq ℕ
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instance : decidable_eq nat
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| zero zero := is_true rfl
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| (succ x) zero := is_false (λ h, nat.no_confusion h)
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| zero (succ y) := is_false (λ h, nat.no_confusion h)
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@ -49,59 +52,58 @@ instance : decidable_eq ℕ
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match decidable_eq x y with
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| is_true xeqy := is_true (xeqy ▸ eq.refl (succ x))
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| is_false xney := is_false (λ h, nat.no_confusion h (λ xeqy, absurd xeqy xney))
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end
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def {u} repeat {α : Type u} (f : ℕ → α → α) : ℕ → α → α
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| 0 a := a
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| (succ n) a := f n (repeat n a)
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def {u} repeat {α : Type u} (f : nat → α → α) : nat → α → α
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| 0 m := m
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| (succ n) m := f n (repeat n m)
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protected def pow (b : ℕ) : ℕ → ℕ
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protected def pow (m : nat) : nat → nat
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| 0 := 1
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| (succ n) := pow n * b
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| (succ n) := pow n * m
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instance : has_pow nat nat :=
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⟨nat.pow⟩
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/- nat.add theorems -/
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protected theorem zero_add : ∀ n : ℕ, 0 + n = n
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protected theorem zero_add : ∀ n : nat, 0 + n = n
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| 0 := rfl
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| (n+1) := congr_arg succ (zero_add n)
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theorem succ_add : ∀ n m : ℕ, (succ n) + m = succ (n + m)
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theorem succ_add : ∀ n m : nat, (succ n) + m = succ (n + m)
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| n 0 := rfl
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| n (m+1) := congr_arg succ (succ_add n m)
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theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
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theorem add_succ (n m : nat) : n + succ m = succ (n + m) :=
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rfl
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protected theorem add_zero (n : ℕ) : n + 0 = n :=
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protected theorem add_zero (n : nat) : n + 0 = n :=
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rfl
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theorem add_one (n : ℕ) : n + 1 = succ n :=
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theorem add_one (n : nat) : n + 1 = succ n :=
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rfl
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theorem succ_eq_add_one (n : ℕ) : succ n = n + 1 :=
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theorem succ_eq_add_one (n : nat) : succ n = n + 1 :=
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rfl
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protected theorem add_comm : ∀ n m : ℕ, n + m = m + n
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protected theorem add_comm : ∀ n m : nat, n + m = m + n
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| n 0 := eq.symm (nat.zero_add n)
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| n (m+1) :=
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suffices succ (n + m) = succ (m + n), from
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eq.symm (succ_add m n) ▸ this,
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congr_arg succ (add_comm n m)
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protected theorem add_assoc : ∀ n m k : ℕ, (n + m) + k = n + (m + k)
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protected theorem add_assoc : ∀ n m k : nat, (n + m) + k = n + (m + k)
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| n m 0 := rfl
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| n m (succ k) := congr_arg succ (add_assoc n m k)
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protected theorem add_left_comm : ∀ (n m k : ℕ), n + (m + k) = m + (n + k) :=
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protected theorem add_left_comm : ∀ (n m k : nat), n + (m + k) = m + (n + k) :=
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left_comm nat.add nat.add_comm nat.add_assoc
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protected theorem add_right_comm : ∀ (n m k : ℕ), (n + m) + k = (n + k) + m :=
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protected theorem add_right_comm : ∀ (n m k : nat), (n + m) + k = (n + k) + m :=
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right_comm nat.add nat.add_comm nat.add_assoc
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protected theorem add_left_cancel : ∀ {n m k : ℕ}, n + m = n + k → m = k
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protected theorem add_left_cancel : ∀ {n m k : nat}, n + m = n + k → m = k
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| 0 m k h := nat.zero_add m ▸ nat.zero_add k ▸ h
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| (succ n) m k h :=
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have n+m = n+k, from
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@ -109,40 +111,40 @@ protected theorem add_left_cancel : ∀ {n m k : ℕ}, n + m = n + k → m = k
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nat.no_confusion this id,
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add_left_cancel this
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protected theorem add_right_cancel {n m k : ℕ} (h : n + m = k + m) : n = k :=
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protected theorem add_right_cancel {n m k : nat} (h : n + m = k + m) : n = k :=
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have m + n = m + k, from nat.add_comm n m ▸ nat.add_comm k m ▸ h,
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nat.add_left_cancel this
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/- nat.mul theorems -/
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protected theorem mul_zero (n : ℕ) : n * 0 = 0 :=
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protected theorem mul_zero (n : nat) : n * 0 = 0 :=
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rfl
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theorem mul_succ (n m : ℕ) : n * succ m = n * m + n :=
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theorem mul_succ (n m : nat) : n * succ m = n * m + n :=
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rfl
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protected theorem zero_mul : ∀ (n : ℕ), 0 * n = 0
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protected theorem zero_mul : ∀ (n : nat), 0 * n = 0
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| 0 := rfl
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| (succ n) := (mul_succ 0 n).symm ▸ (zero_mul n).symm ▸ rfl
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theorem succ_mul : ∀ (n m : ℕ), (succ n) * m = (n * m) + m
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theorem succ_mul : ∀ (n m : nat), (succ n) * m = (n * m) + m
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| n 0 := rfl
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| n (succ m) :=
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have succ (n * m + m + n) = succ (n * m + n + m), from
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congr_arg succ (nat.add_right_comm _ _ _),
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(mul_succ n m).symm ▸ (mul_succ (succ n) m).symm ▸ (succ_mul n m).symm ▸ this
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protected theorem mul_comm : ∀ (n m : ℕ), n * m = m * n
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protected theorem mul_comm : ∀ (n m : nat), n * m = m * n
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| n 0 := (nat.zero_mul n).symm ▸ (nat.mul_zero n).symm ▸ rfl
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| n (succ m) := (mul_succ n m).symm ▸ (succ_mul m n).symm ▸ (mul_comm n m).symm ▸ rfl
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protected theorem mul_one : ∀ (n : ℕ), n * 1 = n :=
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protected theorem mul_one : ∀ (n : nat), n * 1 = n :=
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nat.zero_add
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protected theorem one_mul (n : ℕ) : 1 * n = n :=
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protected theorem one_mul (n : nat) : 1 * n = n :=
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nat.mul_comm n 1 ▸ nat.mul_one n
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protected theorem left_distrib : ∀ (n m k : ℕ), n * (m + k) = n * m + n * k
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protected theorem left_distrib : ∀ (n m k : nat), n * (m + k) = n * m + n * k
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| 0 m k := (nat.zero_mul (m + k)).symm ▸ (nat.zero_mul m).symm ▸ (nat.zero_mul k).symm ▸ rfl
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| (succ n) m k := calc
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succ n * (m + k)
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@ -154,14 +156,14 @@ protected theorem left_distrib : ∀ (n m k : ℕ), n * (m + k) = n * m + n * k
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... = (n * m + m) + succ n * k : succ_mul n k ▸ rfl
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... = succ n * m + succ n * k : succ_mul n m ▸ rfl
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protected theorem right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
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protected theorem right_distrib (n m k : nat) : (n + m) * k = n * k + m * k :=
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calc (n + m) * k
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= k * (n + m) : nat.mul_comm _ _
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... = k * n + k * m : nat.left_distrib _ _ _
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... = n * k + k * m : nat.mul_comm n k ▸ rfl
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... = n * k + m * k : nat.mul_comm m k ▸ rfl
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protected theorem mul_assoc : ∀ (n m k : ℕ), (n * m) * k = n * (m * k)
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protected theorem mul_assoc : ∀ (n m k : nat), (n * m) * k = n * (m * k)
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| n m 0 := rfl
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| n m (succ k) := calc
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n * m * succ k
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@ -174,160 +176,160 @@ protected theorem mul_assoc : ∀ (n m k : ℕ), (n * m) * k = n * (m * k)
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/- Inequalities -/
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@[refl] protected def le_refl : ∀ a : ℕ, a ≤ a :=
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@[refl] protected def le_refl : ∀ n : nat, n ≤ n :=
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less_than_or_equal.refl
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theorem le_succ (n : ℕ) : n ≤ succ n :=
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theorem le_succ (n : nat) : n ≤ succ n :=
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less_than_or_equal.step (nat.le_refl n)
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theorem succ_le_succ {n m : ℕ} : n ≤ m → succ n ≤ succ m :=
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theorem succ_le_succ {n m : nat} : n ≤ m → succ n ≤ succ m :=
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λ h, less_than_or_equal.rec (nat.le_refl (succ n)) (λ a b, less_than_or_equal.step) h
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theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
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theorem succ_lt_succ {n m : nat} : n < m → succ n < succ m :=
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succ_le_succ
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theorem zero_le : ∀ (n : ℕ), 0 ≤ n
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theorem zero_le : ∀ (n : nat), 0 ≤ n
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| 0 := nat.le_refl 0
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| (n+1) := less_than_or_equal.step (zero_le n)
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theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
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theorem zero_lt_succ (n : nat) : 0 < succ n :=
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succ_le_succ (zero_le n)
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def succ_pos := zero_lt_succ
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theorem not_succ_le_zero : ∀ (n : ℕ), succ n ≤ 0 → false
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theorem not_succ_le_zero : ∀ (n : nat), succ n ≤ 0 → false
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.
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theorem not_lt_zero (a : ℕ) : ¬ a < 0 := not_succ_le_zero a
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theorem not_lt_zero (n : nat) : ¬ n < 0 :=
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not_succ_le_zero n
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theorem pred_le_pred {n m : ℕ} : n ≤ m → pred n ≤ pred m :=
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theorem pred_le_pred {n m : nat} : n ≤ m → pred n ≤ pred m :=
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λ h, less_than_or_equal.rec_on h
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(nat.le_refl (pred n))
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(λ n, nat.rec (λ a b, b) (λ a b c, less_than_or_equal.step) n)
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theorem le_of_succ_le_succ {n m : ℕ} : succ n ≤ succ m → n ≤ m :=
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theorem le_of_succ_le_succ {n m : nat} : succ n ≤ succ m → n ≤ m :=
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pred_le_pred
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instance decidable_le : ∀ a b : ℕ, decidable (a ≤ b)
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| 0 b := is_true (zero_le b)
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| (a+1) 0 := is_false (not_succ_le_zero a)
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| (a+1) (b+1) :=
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match decidable_le a b with
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instance decidable_le : ∀ n m : nat, decidable (n ≤ m)
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| 0 m := is_true (zero_le m)
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| (n+1) 0 := is_false (not_succ_le_zero n)
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| (n+1) (m+1) :=
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match decidable_le n m with
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| is_true h := is_true (succ_le_succ h)
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| is_false h := is_false (λ a, h (le_of_succ_le_succ a))
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end
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instance decidable_lt : ∀ a b : ℕ, decidable (a < b) :=
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λ a b, nat.decidable_le (succ a) b
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instance decidable_lt (n m : nat) : decidable (n < m) :=
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nat.decidable_le (succ n) m
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protected theorem eq_or_lt_of_le {a b : ℕ} (h : a ≤ b) : a = b ∨ a < b :=
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protected theorem eq_or_lt_of_le {n m: nat} (h : n ≤ m) : n = m ∨ n < m :=
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less_than_or_equal.cases_on h (or.inl rfl) (λ n h, or.inr (succ_le_succ h))
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theorem lt_succ_of_le {a b : ℕ} : a ≤ b → a < succ b :=
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theorem lt_succ_of_le {n m : nat} : n ≤ m → n < succ m :=
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succ_le_succ
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protected theorem sub_zero (n : ℕ) : n - 0 = n :=
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protected theorem sub_zero (n : nat) : n - 0 = n :=
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rfl
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theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
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nat.rec_on b
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(show succ a - succ zero = a - zero, from (eq.refl (succ a - succ zero)))
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(λ b, congr_arg pred)
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theorem succ_sub_succ_eq_sub (n m : nat) : succ n - succ m = n - m :=
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nat.rec_on m
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(show succ n - succ zero = n - zero, from (eq.refl (succ n - succ zero)))
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(λ m, congr_arg pred)
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theorem not_succ_le_self : ∀ n : ℕ, ¬succ n ≤ n :=
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theorem not_succ_le_self : ∀ n : nat, ¬succ n ≤ n :=
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λ n, nat.rec (not_succ_le_zero 0) (λ a b c, b (le_of_succ_le_succ c)) n
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protected theorem lt_irrefl (n : ℕ) : ¬n < n :=
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protected theorem lt_irrefl (n : nat) : ¬n < n :=
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not_succ_le_self n
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protected theorem le_trans {n m k : ℕ} (h1 : n ≤ m) : m ≤ k → n ≤ k :=
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protected theorem le_trans {n m k : nat} (h1 : n ≤ m) : m ≤ k → n ≤ k :=
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less_than_or_equal.rec h1 (λ p h2, less_than_or_equal.step)
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theorem pred_le : ∀ (n : ℕ), pred n ≤ n
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theorem pred_le : ∀ (n : nat), pred n ≤ n
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| 0 := less_than_or_equal.refl 0
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| (succ a) := less_than_or_equal.step (less_than_or_equal.refl a)
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theorem pred_lt : ∀ {n : ℕ}, n ≠ 0 → pred n < n
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theorem pred_lt : ∀ {n : nat}, n ≠ 0 → pred n < n
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| 0 h := absurd rfl h
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| (succ a) h := lt_succ_of_le (less_than_or_equal.refl _)
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theorem sub_le (a b : ℕ) : a - b ≤ a :=
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nat.rec_on b (nat.le_refl (a - 0)) (λ b₁, nat.le_trans (pred_le (a - b₁)))
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theorem sub_le (n m : nat) : n - m ≤ n :=
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nat.rec_on m (nat.le_refl (n - 0)) (λ m, nat.le_trans (pred_le (n - m)))
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theorem sub_lt : ∀ {a b : ℕ}, 0 < a → 0 < b → a - b < a
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| 0 b h1 h2 := absurd h1 (nat.lt_irrefl 0)
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| (a+1) 0 h1 h2 := absurd h2 (nat.lt_irrefl 0)
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| (a+1) (b+1) h1 h2 :=
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eq.symm (succ_sub_succ_eq_sub a b) ▸
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show a - b < succ a, from
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lt_succ_of_le (sub_le a b)
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theorem sub_lt : ∀ {n m : nat}, 0 < n → 0 < m → n - m < n
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| 0 m h1 h2 := absurd h1 (nat.lt_irrefl 0)
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| (n+1) 0 h1 h2 := absurd h2 (nat.lt_irrefl 0)
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| (n+1) (m+1) h1 h2 :=
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eq.symm (succ_sub_succ_eq_sub n m) ▸
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show n - m < succ n, from
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lt_succ_of_le (sub_le n m)
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protected theorem lt_of_lt_of_le {n m k : ℕ} : n < m → m ≤ k → n < k :=
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protected theorem lt_of_lt_of_le {n m k : nat} : n < m → m ≤ k → n < k :=
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nat.le_trans
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protected theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m :=
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protected theorem le_of_eq {n m : nat} (p : n = m) : n ≤ m :=
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p ▸ less_than_or_equal.refl n
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theorem le_succ_of_le {n m : ℕ} (h : n ≤ m) : n ≤ succ m :=
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theorem le_succ_of_le {n m : nat} (h : n ≤ m) : n ≤ succ m :=
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nat.le_trans h (le_succ m)
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theorem le_of_succ_le {n m : ℕ} (h : succ n ≤ m) : n ≤ m :=
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theorem le_of_succ_le {n m : nat} (h : succ n ≤ m) : n ≤ m :=
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nat.le_trans (le_succ n) h
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protected theorem le_of_lt {n m : ℕ} (h : n < m) : n ≤ m :=
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protected theorem le_of_lt {n m : nat} (h : n < m) : n ≤ m :=
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le_of_succ_le h
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def lt.step {n m : ℕ} : n < m → n < succ m := less_than_or_equal.step
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def lt.step {n m : nat} : n < m → n < succ m := less_than_or_equal.step
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|
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theorem eq_zero_or_pos : ∀ (n : ℕ), n = 0 ∨ n > 0
|
||||
theorem eq_zero_or_pos : ∀ (n : nat), n = 0 ∨ n > 0
|
||||
| 0 := or.inl rfl
|
||||
| (n+1) := or.inr (succ_pos _)
|
||||
|
||||
protected theorem lt_trans {n m k : ℕ} (h₁ : n < m) : m < k → n < k :=
|
||||
protected theorem lt_trans {n m k : nat} (h₁ : n < m) : m < k → n < k :=
|
||||
nat.le_trans (less_than_or_equal.step h₁)
|
||||
|
||||
protected theorem lt_of_le_of_lt {n m k : ℕ} (h₁ : n ≤ m) : m < k → n < k :=
|
||||
protected theorem lt_of_le_of_lt {n m k : nat} (h₁ : n ≤ m) : m < k → n < k :=
|
||||
nat.le_trans (succ_le_succ h₁)
|
||||
|
||||
def lt.base (n : ℕ) : n < succ n := nat.le_refl (succ n)
|
||||
def lt.base (n : nat) : n < succ n := nat.le_refl (succ n)
|
||||
|
||||
theorem lt_succ_self (n : ℕ) : n < succ n := lt.base n
|
||||
theorem lt_succ_self (n : nat) : n < succ n := lt.base n
|
||||
|
||||
protected theorem le_antisymm {n m : ℕ} (h₁ : n ≤ m) : m ≤ n → n = m :=
|
||||
protected theorem le_antisymm {n m : nat} (h₁ : n ≤ m) : m ≤ n → n = m :=
|
||||
less_than_or_equal.cases_on h₁ (λ a, rfl) (λ a b c, absurd (nat.lt_of_le_of_lt b c) (nat.lt_irrefl n))
|
||||
|
||||
protected theorem lt_or_ge : ∀ (a b : ℕ), a < b ∨ a ≥ b
|
||||
| a 0 := or.inr (zero_le a)
|
||||
| a (b+1) :=
|
||||
match lt_or_ge a b with
|
||||
protected theorem lt_or_ge : ∀ (n m : nat), n < m ∨ n ≥ m
|
||||
| n 0 := or.inr (zero_le n)
|
||||
| n (m+1) :=
|
||||
match lt_or_ge n m with
|
||||
| or.inl h := or.inl (le_succ_of_le h)
|
||||
| or.inr h :=
|
||||
match nat.eq_or_lt_of_le h with
|
||||
| or.inl h1 := or.inl (h1 ▸ lt_succ_self b)
|
||||
| or.inl h1 := or.inl (h1 ▸ lt_succ_self m)
|
||||
| or.inr h1 := or.inr h1
|
||||
end
|
||||
end
|
||||
|
||||
protected theorem le_total (m n : ℕ) : m ≤ n ∨ n ≤ m :=
|
||||
protected theorem le_total (m n : nat) : m ≤ n ∨ n ≤ m :=
|
||||
or.elim (nat.lt_or_ge m n)
|
||||
(λ h, or.inl (nat.le_of_lt h))
|
||||
or.inr
|
||||
|
||||
protected theorem lt_of_le_and_ne {m n : ℕ} (h1 : m ≤ n) : m ≠ n → m < n :=
|
||||
protected theorem lt_of_le_and_ne {m n : nat} (h1 : m ≤ n) : m ≠ n → m < n :=
|
||||
resolve_right (or.swap (nat.eq_or_lt_of_le h1))
|
||||
|
||||
theorem eq_zero_of_le_zero {n : nat} (h : n ≤ 0) : n = 0 :=
|
||||
nat.le_antisymm h (zero_le _)
|
||||
|
||||
theorem lt_of_succ_lt {a b : ℕ} : succ a < b → a < b :=
|
||||
theorem lt_of_succ_lt {n m : nat} : succ n < m → n < m :=
|
||||
le_of_succ_le
|
||||
|
||||
theorem lt_of_succ_lt_succ {a b : ℕ} : succ a < succ b → a < b :=
|
||||
theorem lt_of_succ_lt_succ {n m : nat} : succ n < succ m → n < m :=
|
||||
le_of_succ_le_succ
|
||||
|
||||
theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
|
||||
theorem lt_of_succ_le {n m : nat} (h : succ n ≤ m) : n < m :=
|
||||
h
|
||||
|
||||
theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
|
||||
theorem succ_le_of_lt {n m : nat} (h : n < m) : succ n ≤ m :=
|
||||
h
|
||||
|
||||
theorem lt_or_eq_or_le_succ {m n : nat} (h : m ≤ succ n) : m ≤ n ∨ m = succ n :=
|
||||
decidable.by_cases
|
||||
|
|
@ -337,58 +339,55 @@ decidable.by_cases
|
|||
have succ m ≤ succ n, from succ_le_of_lt this,
|
||||
or.inl (le_of_succ_le_succ this))
|
||||
|
||||
theorem le_add_right : ∀ (n k : ℕ), n ≤ n + k
|
||||
theorem le_add_right : ∀ (n k : nat), n ≤ n + k
|
||||
| n 0 := nat.le_refl n
|
||||
| n (k+1) := le_succ_of_le (le_add_right n k)
|
||||
|
||||
theorem le_add_left (n m : ℕ): n ≤ m + n :=
|
||||
theorem le_add_left (n m : nat): n ≤ m + n :=
|
||||
nat.add_comm n m ▸ le_add_right n m
|
||||
|
||||
theorem le.dest : ∀ {n m : ℕ}, n ≤ m → ∃ k, n + k = m
|
||||
theorem le.dest : ∀ {n m : nat}, n ≤ m → ∃ k, n + k = m
|
||||
| n ._ (less_than_or_equal.refl ._) := ⟨0, rfl⟩
|
||||
| n ._ (@less_than_or_equal.step ._ m h) :=
|
||||
match le.dest h with
|
||||
| ⟨w, hw⟩ := ⟨succ w, hw ▸ add_succ n w⟩
|
||||
end
|
||||
|
||||
theorem le.intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
|
||||
theorem le.intro {n m k : nat} (h : n + k = m) : n ≤ m :=
|
||||
h ▸ le_add_right n k
|
||||
|
||||
protected theorem not_le_of_gt {a b : nat} (h : a > b) : ¬ a ≤ b :=
|
||||
λ (h₁ : a ≤ b),
|
||||
or.elim (nat.lt_or_ge a b)
|
||||
protected theorem not_le_of_gt {n m : nat} (h : n > m) : ¬ n ≤ m :=
|
||||
λ h₁, or.elim (nat.lt_or_ge n m)
|
||||
(λ h₂, absurd (nat.lt_trans h h₂) (nat.lt_irrefl _))
|
||||
(λ h₂, have heq : a = b, from nat.le_antisymm h₁ h₂, absurd (@eq.subst _ _ _ _ heq h) (nat.lt_irrefl b))
|
||||
(λ h₂, have heq : n = m, from nat.le_antisymm h₁ h₂, absurd (@eq.subst _ _ _ _ heq h) (nat.lt_irrefl m))
|
||||
|
||||
theorem gt_of_not_le {a b : nat} (h : ¬ a ≤ b) : a > b :=
|
||||
or.elim (nat.lt_or_ge b a)
|
||||
theorem gt_of_not_le {n m : nat} (h : ¬ n ≤ m) : n > m :=
|
||||
or.elim (nat.lt_or_ge m n)
|
||||
(λ h₁, h₁)
|
||||
(λ h₁, absurd h₁ h)
|
||||
|
||||
protected theorem lt_of_le_of_ne {a b : nat} (h₁ : a ≤ b) (h₂ : a ≠ b) : a < b :=
|
||||
or.elim (nat.lt_or_ge a b)
|
||||
protected theorem lt_of_le_of_ne {n m : nat} (h₁ : n ≤ m) (h₂ : n ≠ m) : n < m :=
|
||||
or.elim (nat.lt_or_ge n m)
|
||||
(λ h₃, h₃)
|
||||
(λ h₃, absurd (nat.le_antisymm h₁ h₃) h₂)
|
||||
|
||||
protected theorem add_le_add_left {n m : ℕ} (h : n ≤ m) (k : ℕ) : k + n ≤ k + m :=
|
||||
protected theorem add_le_add_left {n m : nat} (h : n ≤ m) (k : nat) : k + n ≤ k + m :=
|
||||
match le.dest h with
|
||||
| ⟨w, hw⟩ :=
|
||||
have k + n + w = k + m, from
|
||||
calc k + n + w = k + (n + w) : nat.add_assoc _ _ _
|
||||
... = k + m : congr_arg _ hw,
|
||||
le.intro this
|
||||
end
|
||||
|
||||
protected theorem add_le_add_right {n m : ℕ} (h : n ≤ m) (k : ℕ) : n + k ≤ m + k :=
|
||||
protected theorem add_le_add_right {n m : nat} (h : n ≤ m) (k : nat) : n + k ≤ m + k :=
|
||||
calc
|
||||
n + k = k + n : nat.add_comm _ _
|
||||
... ≤ k + m : nat.add_le_add_left h k
|
||||
... = m + k : nat.add_comm _ _
|
||||
|
||||
protected theorem add_lt_add_left {n m : ℕ} (h : n < m) (k : ℕ) : k + n < k + m :=
|
||||
protected theorem add_lt_add_left {n m : nat} (h : n < m) (k : nat) : k + n < k + m :=
|
||||
lt_of_succ_le (add_succ k n ▸ nat.add_le_add_left (succ_le_of_lt h) k)
|
||||
|
||||
protected theorem add_lt_add_right {n m : ℕ} (h : n < m) (k : ℕ) : n + k < m + k :=
|
||||
protected theorem add_lt_add_right {n m : nat} (h : n < m) (k : nat) : n + k < m + k :=
|
||||
nat.add_comm k m ▸ nat.add_comm k n ▸ nat.add_lt_add_left h k
|
||||
|
||||
protected theorem zero_lt_one : 0 < (1:nat) :=
|
||||
|
|
@ -408,16 +407,16 @@ nat.lt_trans (nat.add_lt_add_right h₁ c) (nat.add_lt_add_left h₂ b)
|
|||
theorem nat_zero_eq_zero : nat.zero = 0 :=
|
||||
rfl
|
||||
|
||||
protected theorem one_ne_zero : 1 ≠ (0 : ℕ) :=
|
||||
protected theorem one_ne_zero : 1 ≠ (0 : nat) :=
|
||||
assume h, nat.no_confusion h
|
||||
|
||||
protected theorem zero_ne_one : 0 ≠ (1 : ℕ) :=
|
||||
protected theorem zero_ne_one : 0 ≠ (1 : nat) :=
|
||||
assume h, nat.no_confusion h
|
||||
|
||||
theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
|
||||
theorem succ_ne_zero (n : nat) : succ n ≠ 0 :=
|
||||
assume h, nat.no_confusion h
|
||||
|
||||
protected theorem bit0_succ_eq (n : ℕ) : bit0 (succ n) = succ (succ (bit0 n)) :=
|
||||
protected theorem bit0_succ_eq (n : nat) : bit0 (succ n) = succ (succ (bit0 n)) :=
|
||||
show succ (succ n + n) = succ (succ (n + n)), from
|
||||
congr_arg succ (succ_add n n)
|
||||
|
||||
|
|
@ -430,41 +429,41 @@ protected theorem zero_lt_bit0 : ∀ {n : nat}, n ≠ 0 → 0 < bit0 n
|
|||
protected theorem zero_lt_bit1 (n : nat) : 0 < bit1 n :=
|
||||
zero_lt_succ _
|
||||
|
||||
protected theorem bit0_ne_zero : ∀ {n : ℕ}, n ≠ 0 → bit0 n ≠ 0
|
||||
protected theorem bit0_ne_zero : ∀ {n : nat}, n ≠ 0 → bit0 n ≠ 0
|
||||
| 0 h := absurd rfl h
|
||||
| (n+1) h :=
|
||||
suffices (n+1) + (n+1) ≠ 0, from this,
|
||||
suffices succ ((n+1) + n) ≠ 0, from this,
|
||||
λ h, nat.no_confusion h
|
||||
|
||||
protected theorem bit1_ne_zero (n : ℕ) : bit1 n ≠ 0 :=
|
||||
protected theorem bit1_ne_zero (n : nat) : bit1 n ≠ 0 :=
|
||||
show succ (n + n) ≠ 0, from
|
||||
λ h, nat.no_confusion h
|
||||
|
||||
protected theorem bit1_eq_succ_bit0 (n : ℕ) : bit1 n = succ (bit0 n) :=
|
||||
protected theorem bit1_eq_succ_bit0 (n : nat) : bit1 n = succ (bit0 n) :=
|
||||
rfl
|
||||
|
||||
protected theorem bit1_succ_eq (n : ℕ) : bit1 (succ n) = succ (succ (bit1 n)) :=
|
||||
protected theorem bit1_succ_eq (n : nat) : bit1 (succ n) = succ (succ (bit1 n)) :=
|
||||
eq.trans (nat.bit1_eq_succ_bit0 (succ n)) (congr_arg succ (nat.bit0_succ_eq n))
|
||||
|
||||
protected theorem bit1_ne_one : ∀ {n : ℕ}, n ≠ 0 → bit1 n ≠ 1
|
||||
protected theorem bit1_ne_one : ∀ {n : nat}, n ≠ 0 → bit1 n ≠ 1
|
||||
| 0 h h1 := absurd rfl h
|
||||
| (n+1) h h1 := nat.no_confusion h1 (λ h2, absurd h2 (succ_ne_zero _))
|
||||
|
||||
protected theorem bit0_ne_one : ∀ n : ℕ, bit0 n ≠ 1
|
||||
protected theorem bit0_ne_one : ∀ n : nat, bit0 n ≠ 1
|
||||
| 0 h := absurd h (ne.symm nat.one_ne_zero)
|
||||
| (n+1) h :=
|
||||
have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
|
||||
nat.no_confusion h1
|
||||
(λ h2, absurd h2 (succ_ne_zero (n + n)))
|
||||
|
||||
protected theorem add_self_ne_one : ∀ (n : ℕ), n + n ≠ 1
|
||||
protected theorem add_self_ne_one : ∀ (n : nat), n + n ≠ 1
|
||||
| 0 h := nat.no_confusion h
|
||||
| (n+1) h :=
|
||||
have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
|
||||
nat.no_confusion h1 (λ h2, absurd h2 (nat.succ_ne_zero (n + n)))
|
||||
|
||||
protected theorem bit1_ne_bit0 : ∀ (n m : ℕ), bit1 n ≠ bit0 m
|
||||
protected theorem bit1_ne_bit0 : ∀ (n m : nat), bit1 n ≠ bit0 m
|
||||
| 0 m h := absurd h (ne.symm (nat.add_self_ne_one m))
|
||||
| (n+1) 0 h :=
|
||||
have h1 : succ (bit0 (succ n)) = 0, from h,
|
||||
|
|
@ -476,10 +475,10 @@ protected theorem bit1_ne_bit0 : ∀ (n m : ℕ), bit1 n ≠ bit0 m
|
|||
nat.no_confusion h1 (λ h2', nat.no_confusion h2' (λ h2'', h2'')),
|
||||
absurd h2 (bit1_ne_bit0 n m)
|
||||
|
||||
protected theorem bit0_ne_bit1 : ∀ (n m : ℕ), bit0 n ≠ bit1 m :=
|
||||
protected theorem bit0_ne_bit1 : ∀ (n m : nat), bit0 n ≠ bit1 m :=
|
||||
λ n m : nat, ne.symm (nat.bit1_ne_bit0 m n)
|
||||
|
||||
protected theorem bit0_inj : ∀ {n m : ℕ}, bit0 n = bit0 m → n = m
|
||||
protected theorem bit0_inj : ∀ {n m : nat}, bit0 n = bit0 m → n = m
|
||||
| 0 0 h := rfl
|
||||
| 0 (m+1) h := absurd h.symm (succ_ne_zero _)
|
||||
| (n+1) 0 h := absurd h (succ_ne_zero _)
|
||||
|
|
@ -491,28 +490,28 @@ protected theorem bit0_inj : ∀ {n m : ℕ}, bit0 n = bit0 m → n = m
|
|||
have n = m, from bit0_inj this,
|
||||
congr_arg (+1) this
|
||||
|
||||
protected theorem bit1_inj : ∀ {n m : ℕ}, bit1 n = bit1 m → n = m :=
|
||||
protected theorem bit1_inj : ∀ {n m : nat}, bit1 n = bit1 m → n = m :=
|
||||
λ n m h,
|
||||
have succ (bit0 n) = succ (bit0 m), from nat.bit1_eq_succ_bit0 n ▸ nat.bit1_eq_succ_bit0 m ▸ h,
|
||||
have bit0 n = bit0 m, from nat.no_confusion this id,
|
||||
nat.bit0_inj this
|
||||
|
||||
protected theorem bit0_ne {n m : ℕ} : n ≠ m → bit0 n ≠ bit0 m :=
|
||||
protected theorem bit0_ne {n m : nat} : n ≠ m → bit0 n ≠ bit0 m :=
|
||||
λ h₁ h₂, absurd (nat.bit0_inj h₂) h₁
|
||||
|
||||
protected theorem bit1_ne {n m : ℕ} : n ≠ m → bit1 n ≠ bit1 m :=
|
||||
protected theorem bit1_ne {n m : nat} : n ≠ m → bit1 n ≠ bit1 m :=
|
||||
λ h₁ h₂, absurd (nat.bit1_inj h₂) h₁
|
||||
|
||||
protected theorem zero_ne_bit0 {n : ℕ} : n ≠ 0 → 0 ≠ bit0 n :=
|
||||
protected theorem zero_ne_bit0 {n : nat} : n ≠ 0 → 0 ≠ bit0 n :=
|
||||
λ h, ne.symm (nat.bit0_ne_zero h)
|
||||
|
||||
protected theorem zero_ne_bit1 (n : ℕ) : 0 ≠ bit1 n :=
|
||||
protected theorem zero_ne_bit1 (n : nat) : 0 ≠ bit1 n :=
|
||||
ne.symm (nat.bit1_ne_zero n)
|
||||
|
||||
protected theorem one_ne_bit0 (n : ℕ) : 1 ≠ bit0 n :=
|
||||
protected theorem one_ne_bit0 (n : nat) : 1 ≠ bit0 n :=
|
||||
ne.symm (nat.bit0_ne_one n)
|
||||
|
||||
protected theorem one_ne_bit1 {n : ℕ} : n ≠ 0 → 1 ≠ bit1 n :=
|
||||
protected theorem one_ne_bit1 {n : nat} : n ≠ 0 → 1 ≠ bit1 n :=
|
||||
λ h, ne.symm (nat.bit1_ne_one h)
|
||||
|
||||
protected theorem one_lt_bit1 : ∀ {n : nat}, n ≠ 0 → 1 < bit1 n
|
||||
|
|
@ -546,11 +545,11 @@ protected theorem bit1_lt_bit0 : ∀ {n m : nat}, n < m → bit1 n < bit0 m
|
|||
have succ (n + n) ≤ succ m + m, from (succ_add m m).symm ▸ this,
|
||||
show succ (n + n) < succ (succ m + m), from lt_succ_of_le this
|
||||
|
||||
protected theorem one_le_bit1 (n : ℕ) : 1 ≤ bit1 n :=
|
||||
protected theorem one_le_bit1 (n : nat) : 1 ≤ bit1 n :=
|
||||
show 1 ≤ succ (bit0 n), from
|
||||
succ_le_succ (zero_le (bit0 n))
|
||||
|
||||
protected theorem one_le_bit0 : ∀ (n : ℕ), n ≠ 0 → 1 ≤ bit0 n
|
||||
protected theorem one_le_bit0 : ∀ (n : nat), n ≠ 0 → 1 ≤ bit0 n
|
||||
| 0 h := absurd rfl h
|
||||
| (n+1) h :=
|
||||
suffices 1 ≤ succ (succ (bit0 n)), from
|
||||
|
|
@ -559,51 +558,51 @@ protected theorem one_le_bit0 : ∀ (n : ℕ), n ≠ 0 → 1 ≤ bit0 n
|
|||
|
||||
/- mul + order -/
|
||||
|
||||
theorem mul_le_mul_left {n m : ℕ} (k : ℕ) (h : n ≤ m) : k * n ≤ k * m :=
|
||||
theorem mul_le_mul_left {n m : nat} (k : nat) (h : n ≤ m) : k * n ≤ k * m :=
|
||||
match le.dest h with
|
||||
| ⟨l, hl⟩ :=
|
||||
have k * n + k * l = k * m, from nat.left_distrib k n l ▸ hl.symm ▸ rfl,
|
||||
le.intro this
|
||||
end
|
||||
|
||||
theorem mul_le_mul_right {n m : ℕ} (k : ℕ) (h : n ≤ m) : n * k ≤ m * k :=
|
||||
theorem mul_le_mul_right {n m : nat} (k : nat) (h : n ≤ m) : n * k ≤ m * k :=
|
||||
nat.mul_comm k m ▸ nat.mul_comm k n ▸ mul_le_mul_left k h
|
||||
|
||||
protected theorem mul_le_mul {n₁ m₁ n₂ m₂ : nat} (h₁ : n₁ ≤ n₂) (h₂ : m₁ ≤ m₂) : n₁ * m₁ ≤ n₂ * m₂ :=
|
||||
nat.le_trans (mul_le_mul_right _ h₁) (mul_le_mul_left _ h₂)
|
||||
|
||||
protected theorem mul_lt_mul_of_pos_left {n m k : ℕ} (h : n < m) (hk : k > 0) : k * n < k * m :=
|
||||
protected theorem mul_lt_mul_of_pos_left {n m k : nat} (h : n < m) (hk : k > 0) : k * n < k * m :=
|
||||
nat.lt_of_lt_of_le (nat.add_lt_add_left hk _) (nat.mul_succ k n ▸ nat.mul_le_mul_left k (succ_le_of_lt h))
|
||||
|
||||
protected theorem mul_lt_mul_of_pos_right {n m k : ℕ} (h : n < m) (hk : k > 0) : n * k < m * k :=
|
||||
protected theorem mul_lt_mul_of_pos_right {n m k : nat} (h : n < m) (hk : k > 0) : n * k < m * k :=
|
||||
nat.mul_comm k m ▸ nat.mul_comm k n ▸ nat.mul_lt_mul_of_pos_left h hk
|
||||
|
||||
protected theorem mul_pos {a b : nat} (ha : a > 0) (hb : b > 0) : a * b > 0 :=
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have h : 0 * b < a * b, from nat.mul_lt_mul_of_pos_right ha hb,
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nat.zero_mul b ▸ h
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protected theorem mul_pos {n m : nat} (ha : n > 0) (hb : m > 0) : n * m > 0 :=
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have h : 0 * m < n * m, from nat.mul_lt_mul_of_pos_right ha hb,
|
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nat.zero_mul m ▸ h
|
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/- power -/
|
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|
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theorem pow_succ (b n : ℕ) : b^(succ n) = b^n * b := rfl
|
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theorem pow_succ (n m : nat) : n^(succ m) = n^m * n :=
|
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rfl
|
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|
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theorem pow_zero (b : ℕ) : b^0 = 1 := rfl
|
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theorem pow_zero (n : nat) : n^0 = 1 := rfl
|
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|
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theorem pow_le_pow_of_le_left {x y : ℕ} (h : x ≤ y) : ∀ i : ℕ, x^i ≤ y^i
|
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theorem pow_le_pow_of_le_left {n m : nat} (h : n ≤ m) : ∀ i : nat, n^i ≤ m^i
|
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| 0 := nat.le_refl _
|
||||
| (succ i) := nat.mul_le_mul (pow_le_pow_of_le_left i) h
|
||||
|
||||
theorem pow_le_pow_of_le_right {x : ℕ} (hx : x > 0) {i : ℕ} : ∀ {j}, i ≤ j → x^i ≤ x^j
|
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theorem pow_le_pow_of_le_right {n : nat} (hx : n > 0) {i : nat} : ∀ {j}, i ≤ j → n^i ≤ n^j
|
||||
| 0 h :=
|
||||
have i = 0, from eq_zero_of_le_zero h,
|
||||
this.symm ▸ nat.le_refl _
|
||||
| (succ j) h :=
|
||||
or.elim (lt_or_eq_or_le_succ h)
|
||||
(λ h, show x^i ≤ x^j * x, from
|
||||
suffices x^i * 1 ≤ x^j * x, from nat.mul_one (x^i) ▸ this,
|
||||
(λ h, show n^i ≤ n^j * n, from
|
||||
suffices n^i * 1 ≤ n^j * n, from nat.mul_one (n^i) ▸ this,
|
||||
nat.mul_le_mul (pow_le_pow_of_le_right h) hx)
|
||||
(λ h, h.symm ▸ nat.le_refl _)
|
||||
|
||||
theorem pos_pow_of_pos {b : ℕ} (n : ℕ) (h : 0 < b) : 0 < b^n :=
|
||||
theorem pos_pow_of_pos {n : nat} (m : nat) (h : 0 < n) : 0 < n^m :=
|
||||
pow_le_pow_of_le_right h (nat.zero_le _)
|
||||
|
||||
end nat
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue