501 lines
19 KiB
Text
501 lines
19 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Jeremy Avigad
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Subtraction on the natural numbers, as well as min, max, and distance.
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-/
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import .order
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open eq.ops
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namespace nat
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/- subtraction -/
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theorem sub_zero (n : ℕ) : n - 0 = n :=
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rfl
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theorem sub_succ (n m : ℕ) : n - succ m = pred (n - m) :=
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rfl
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theorem zero_sub (n : ℕ) : 0 - n = 0 :=
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nat.induction_on n !sub_zero
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(take k : nat,
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assume IH : 0 - k = 0,
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calc
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0 - succ k = pred (0 - k) : sub_succ
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... = pred 0 : IH
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... = 0 : pred_zero)
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theorem succ_sub_succ (n m : ℕ) : succ n - succ m = n - m :=
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succ_sub_succ_eq_sub n m
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theorem sub_self (n : ℕ) : n - n = 0 :=
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nat.induction_on n !sub_zero (take k IH, !succ_sub_succ ⬝ IH)
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theorem add_sub_add_right (n k m : ℕ) : (n + k) - (m + k) = n - m :=
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nat.induction_on k
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(calc
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(n + 0) - (m + 0) = n - (m + 0) : {!add_zero}
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... = n - m : {!add_zero})
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(take l : nat,
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assume IH : (n + l) - (m + l) = n - m,
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calc
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(n + succ l) - (m + succ l) = succ (n + l) - (m + succ l) : {!add_succ}
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... = succ (n + l) - succ (m + l) : {!add_succ}
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... = (n + l) - (m + l) : !succ_sub_succ
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... = n - m : IH)
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theorem add_sub_add_left (k n m : ℕ) : (k + n) - (k + m) = n - m :=
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!add.comm ▸ !add.comm ▸ !add_sub_add_right
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theorem add_sub_cancel (n m : ℕ) : n + m - m = n :=
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nat.induction_on m
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(begin rewrite add_zero end)
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(take k : ℕ,
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assume IH : n + k - k = n,
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calc
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n + succ k - succ k = succ (n + k) - succ k : add_succ
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... = n + k - k : succ_sub_succ
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... = n : IH)
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theorem add_sub_cancel_left (n m : ℕ) : n + m - n = m :=
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!add.comm ▸ !add_sub_cancel
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theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k) :=
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nat.induction_on k
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(calc
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n - m - 0 = n - m : sub_zero
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... = n - (m + 0) : add_zero)
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(take l : nat,
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assume IH : n - m - l = n - (m + l),
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calc
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n - m - succ l = pred (n - m - l) : !sub_succ
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... = pred (n - (m + l)) : IH
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... = n - succ (m + l) : sub_succ
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... = n - (m + succ l) : by rewrite add_succ)
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theorem succ_sub_sub_succ (n m k : ℕ) : succ n - m - succ k = n - m - k :=
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calc
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succ n - m - succ k = succ n - (m + succ k) : sub_sub
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... = succ n - succ (m + k) : add_succ
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... = n - (m + k) : succ_sub_succ
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... = n - m - k : sub_sub
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theorem sub_self_add (n m : ℕ) : n - (n + m) = 0 :=
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calc
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n - (n + m) = n - n - m : sub_sub
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... = 0 - m : sub_self
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... = 0 : zero_sub
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theorem sub.right_comm (m n k : ℕ) : m - n - k = m - k - n :=
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calc
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m - n - k = m - (n + k) : !sub_sub
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... = m - (k + n) : {!add.comm}
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... = m - k - n : !sub_sub⁻¹
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theorem sub_one (n : ℕ) : n - 1 = pred n :=
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rfl
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theorem succ_sub_one (n : ℕ) : succ n - 1 = n :=
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rfl
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/- interaction with multiplication -/
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theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m :=
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nat.induction_on n
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(calc
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pred 0 * m = 0 * m : pred_zero
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... = 0 : zero_mul
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... = 0 - m : zero_sub
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... = 0 * m - m : zero_mul)
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(take k : nat,
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assume IH : pred k * m = k * m - m,
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calc
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pred (succ k) * m = k * m : pred_succ
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... = k * m + m - m : add_sub_cancel
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... = succ k * m - m : succ_mul)
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theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n :=
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calc
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n * pred m = pred m * n : mul.comm
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... = m * n - n : mul_pred_left
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... = n * m - n : mul.comm
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theorem mul_sub_right_distrib (n m k : ℕ) : (n - m) * k = n * k - m * k :=
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nat.induction_on m
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(calc
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(n - 0) * k = n * k : sub_zero
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... = n * k - 0 : sub_zero
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... = n * k - 0 * k : zero_mul)
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(take l : nat,
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assume IH : (n - l) * k = n * k - l * k,
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calc
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(n - succ l) * k = pred (n - l) * k : sub_succ
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... = (n - l) * k - k : mul_pred_left
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... = n * k - l * k - k : IH
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... = n * k - (l * k + k) : sub_sub
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... = n * k - (succ l * k) : succ_mul)
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theorem mul_sub_left_distrib (n m k : ℕ) : n * (m - k) = n * m - n * k :=
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calc
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n * (m - k) = (m - k) * n : !mul.comm
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... = m * n - k * n : !mul_sub_right_distrib
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... = n * m - k * n : {!mul.comm}
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... = n * m - n * k : {!mul.comm}
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theorem mul_self_sub_mul_self_eq (a b : nat) : a * a - b * b = (a + b) * (a - b) :=
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by rewrite [mul_sub_left_distrib, *mul.right_distrib, mul.comm b a, add.comm (a*a) (a*b), add_sub_add_left]
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theorem succ_mul_succ_eq (a : nat) : succ a * succ a = a*a + a + a + 1 :=
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calc succ a * succ a = (a+1)*(a+1) : by rewrite [add_one]
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... = a*a + a + a + 1 : by rewrite [mul.right_distrib, mul.left_distrib, one_mul, mul_one]
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/- interaction with inequalities -/
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theorem succ_sub {m n : ℕ} : m ≥ n → succ m - n = succ (m - n) :=
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sub_induction n m
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(take k, assume H : 0 ≤ k, rfl)
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(take k,
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assume H : succ k ≤ 0,
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absurd H !not_succ_le_zero)
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(take k l,
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assume IH : k ≤ l → succ l - k = succ (l - k),
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take H : succ k ≤ succ l,
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calc
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succ (succ l) - succ k = succ l - k : succ_sub_succ
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... = succ (l - k) : IH (le_of_succ_le_succ H)
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... = succ (succ l - succ k) : succ_sub_succ)
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theorem sub_eq_zero_of_le {n m : ℕ} (H : n ≤ m) : n - m = 0 :=
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obtain (k : ℕ) (Hk : n + k = m), from le.elim H, Hk ▸ !sub_self_add
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theorem add_sub_of_le {n m : ℕ} : n ≤ m → n + (m - n) = m :=
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sub_induction n m
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(take k,
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assume H : 0 ≤ k,
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calc
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0 + (k - 0) = k - 0 : zero_add
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... = k : sub_zero)
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(take k, assume H : succ k ≤ 0, absurd H !not_succ_le_zero)
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(take k l,
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assume IH : k ≤ l → k + (l - k) = l,
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take H : succ k ≤ succ l,
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calc
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succ k + (succ l - succ k) = succ k + (l - k) : succ_sub_succ
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... = succ (k + (l - k)) : succ_add
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... = succ l : IH (le_of_succ_le_succ H))
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theorem add_sub_of_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n :=
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calc
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n + (m - n) = n + 0 : sub_eq_zero_of_le H
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... = n : add_zero
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theorem sub_add_cancel {n m : ℕ} : n ≥ m → n - m + m = n :=
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!add.comm ▸ !add_sub_of_le
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theorem sub_add_of_le {n m : ℕ} : n ≤ m → n - m + m = m :=
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!add.comm ▸ add_sub_of_ge
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theorem sub.cases {P : ℕ → Prop} {n m : ℕ} (H1 : n ≤ m → P 0) (H2 : ∀k, m + k = n -> P k)
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: P (n - m) :=
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or.elim !le.total
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(assume H3 : n ≤ m, (sub_eq_zero_of_le H3)⁻¹ ▸ (H1 H3))
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(assume H3 : m ≤ n, H2 (n - m) (add_sub_of_le H3))
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theorem exists_sub_eq_of_le {n m : ℕ} (H : n ≤ m) : ∃k, m - k = n :=
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obtain (k : ℕ) (Hk : n + k = m), from le.elim H,
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exists.intro k
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(calc
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m - k = n + k - k : by rewrite Hk
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... = n : add_sub_cancel)
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theorem add_sub_assoc {m k : ℕ} (H : k ≤ m) (n : ℕ) : n + m - k = n + (m - k) :=
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have l1 : k ≤ m → n + m - k = n + (m - k), from
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sub_induction k m
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(take m : ℕ,
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assume H : 0 ≤ m,
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calc
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n + m - 0 = n + m : sub_zero
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... = n + (m - 0) : sub_zero)
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(take k : ℕ, assume H : succ k ≤ 0, absurd H !not_succ_le_zero)
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(take k m,
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assume IH : k ≤ m → n + m - k = n + (m - k),
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take H : succ k ≤ succ m,
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calc
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n + succ m - succ k = succ (n + m) - succ k : add_succ
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... = n + m - k : succ_sub_succ
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... = n + (m - k) : IH (le_of_succ_le_succ H)
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... = n + (succ m - succ k) : succ_sub_succ),
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l1 H
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theorem le_of_sub_eq_zero {n m : ℕ} : n - m = 0 → n ≤ m :=
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sub.cases
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(assume H1 : n ≤ m, assume H2 : 0 = 0, H1)
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(take k : ℕ,
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assume H1 : m + k = n,
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assume H2 : k = 0,
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have H3 : n = m, from !add_zero ▸ H2 ▸ H1⁻¹,
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H3 ▸ !le.refl)
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theorem sub_sub.cases {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0)
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(H2 : ∀k, m = n + k → P 0 k) : P (n - m) (m - n) :=
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or.elim !le.total
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(assume H3 : n ≤ m,
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(sub_eq_zero_of_le H3)⁻¹ ▸ (H2 (m - n) (add_sub_of_le H3)⁻¹))
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(assume H3 : m ≤ n,
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(sub_eq_zero_of_le H3)⁻¹ ▸ (H1 (n - m) (add_sub_of_le H3)⁻¹))
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theorem sub_eq_of_add_eq {n m k : ℕ} (H : n + m = k) : k - n = m :=
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have H2 : k - n + n = m + n, from
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calc
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k - n + n = k : sub_add_cancel (le.intro H)
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... = n + m : H⁻¹
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... = m + n : !add.comm,
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add.cancel_right H2
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theorem eq_sub_of_add_eq {a b c : ℕ} (H : a + c = b) : a = b - c :=
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(sub_eq_of_add_eq (!add.comm ▸ H))⁻¹
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theorem sub_eq_of_eq_add {a b c : ℕ} (H : a = c + b) : a - b = c :=
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sub_eq_of_add_eq (!add.comm ▸ H⁻¹)
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theorem sub_le_sub_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n - k ≤ m - k :=
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obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
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or.elim !le.total
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(assume H2 : n ≤ k, (sub_eq_zero_of_le H2)⁻¹ ▸ !zero_le)
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(assume H2 : k ≤ n,
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have H3 : n - k + l = m - k, from
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calc
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n - k + l = l + (n - k) : add.comm
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... = l + n - k : add_sub_assoc H2 l
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... = n + l - k : add.comm
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... = m - k : Hl,
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le.intro H3)
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theorem sub_le_sub_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k - m ≤ k - n :=
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obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
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sub.cases
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(assume H2 : k ≤ m, !zero_le)
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(take m' : ℕ,
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assume Hm : m + m' = k,
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have H3 : n ≤ k, from le.trans H (le.intro Hm),
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have H4 : m' + l + n = k - n + n, from
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calc
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m' + l + n = n + (m' + l) : add.comm
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... = n + (l + m') : add.comm
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... = n + l + m' : add.assoc
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... = m + m' : Hl
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... = k : Hm
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... = k - n + n : sub_add_cancel H3,
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le.intro (add.cancel_right H4))
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open algebra
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theorem sub_pos_of_lt {m n : ℕ} (H : m < n) : n - m > 0 :=
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assert H1 : n = n - m + m, from (sub_add_cancel (le_of_lt H))⁻¹,
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have H2 : 0 + m < n - m + m, begin rewrite [zero_add, -H1], exact H end,
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!lt_of_add_lt_add_right H2
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theorem lt_of_sub_pos {m n : ℕ} (H : n - m > 0) : m < n :=
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lt_of_not_ge
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(take H1 : m ≥ n,
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have H2 : n - m = 0, from sub_eq_zero_of_le H1,
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!lt.irrefl (H2 ▸ H))
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theorem lt_of_sub_lt_sub_right {n m k : ℕ} (H : n - k < m - k) : n < m :=
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lt_of_not_ge
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(assume H1 : m ≤ n,
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have H2 : m - k ≤ n - k, from sub_le_sub_right H1 _,
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not_le_of_gt H H2)
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theorem lt_of_sub_lt_sub_left {n m k : ℕ} (H : n - m < n - k) : k < m :=
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lt_of_not_ge
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(assume H1 : m ≤ k,
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have H2 : n - k ≤ n - m, from sub_le_sub_left H1 _,
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not_le_of_gt H H2)
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theorem sub_lt_sub_add_sub (n m k : ℕ) : n - k ≤ (n - m) + (m - k) :=
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sub.cases
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(assume H : n ≤ m, !zero_add⁻¹ ▸ sub_le_sub_right H k)
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(take mn : ℕ,
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assume Hmn : m + mn = n,
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sub.cases
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(assume H : m ≤ k,
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have H2 : n - k ≤ n - m, from sub_le_sub_left H n,
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assert H3 : n - k ≤ mn, from sub_eq_of_add_eq Hmn ▸ H2,
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show n - k ≤ mn + 0, begin rewrite add_zero, assumption end)
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(take km : ℕ,
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assume Hkm : k + km = m,
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have H : k + (mn + km) = n, from
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calc
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k + (mn + km) = k + (km + mn): add.comm
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... = k + km + mn : add.assoc
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... = m + mn : Hkm
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... = n : Hmn,
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have H2 : n - k = mn + km, from sub_eq_of_add_eq H,
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H2 ▸ !le.refl))
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theorem sub_lt_self {m n : ℕ} (H1 : m > 0) (H2 : n > 0) : m - n < m :=
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calc
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m - n = succ (pred m) - n : succ_pred_of_pos H1
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... = succ (pred m) - succ (pred n) : succ_pred_of_pos H2
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... = pred m - pred n : succ_sub_succ
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... ≤ pred m : sub_le
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... < succ (pred m) : lt_succ_self
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... = m : succ_pred_of_pos H1
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theorem le_sub_of_add_le {m n k : ℕ} (H : m + k ≤ n) : m ≤ n - k :=
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calc
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m = m + k - k : add_sub_cancel
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... ≤ n - k : sub_le_sub_right H k
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theorem lt_sub_of_add_lt {m n k : ℕ} (H : m + k < n) (H2 : k ≤ n) : m < n - k :=
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lt_of_succ_le (le_sub_of_add_le (calc
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succ m + k = succ (m + k) : succ_add_eq_succ_add
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... ≤ n : succ_le_of_lt H))
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theorem sub_lt_of_lt_add {v n m : nat} (h₁ : v < n + m) (h₂ : n ≤ v) : v - n < m :=
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have succ v ≤ n + m, from succ_le_of_lt h₁,
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have succ (v - n) ≤ m, from
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calc succ (v - n) = succ v - n : succ_sub h₂
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... ≤ n + m - n : sub_le_sub_right this n
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... = m : add_sub_cancel_left,
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lt_of_succ_le this
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/- distance -/
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definition dist [reducible] (n m : ℕ) := (n - m) + (m - n)
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theorem dist.comm (n m : ℕ) : dist n m = dist m n :=
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!add.comm
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theorem dist_self (n : ℕ) : dist n n = 0 :=
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calc
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(n - n) + (n - n) = 0 + (n - n) : sub_self
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... = 0 + 0 : sub_self
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... = 0 : rfl
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theorem eq_of_dist_eq_zero {n m : ℕ} (H : dist n m = 0) : n = m :=
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have H2 : n - m = 0, from eq_zero_of_add_eq_zero_right H,
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have H3 : n ≤ m, from le_of_sub_eq_zero H2,
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have H4 : m - n = 0, from eq_zero_of_add_eq_zero_left H,
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have H5 : m ≤ n, from le_of_sub_eq_zero H4,
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le.antisymm H3 H5
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theorem dist_eq_zero {n m : ℕ} (H : n = m) : dist n m = 0 :=
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by substvars; rewrite [↑dist, *sub_self, add_zero]
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theorem dist_eq_sub_of_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n :=
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calc
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dist n m = 0 + (m - n) : {sub_eq_zero_of_le H}
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... = m - n : zero_add
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theorem dist_eq_sub_of_lt {n m : ℕ} (H : n < m) : dist n m = m - n :=
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dist_eq_sub_of_le (le_of_lt H)
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theorem dist_eq_sub_of_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m :=
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!dist.comm ▸ dist_eq_sub_of_le H
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theorem dist_eq_sub_of_gt {n m : ℕ} (H : n > m) : dist n m = n - m :=
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dist_eq_sub_of_ge (le_of_lt H)
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theorem dist_zero_right (n : ℕ) : dist n 0 = n :=
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dist_eq_sub_of_ge !zero_le ⬝ !sub_zero
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theorem dist_zero_left (n : ℕ) : dist 0 n = n :=
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dist_eq_sub_of_le !zero_le ⬝ !sub_zero
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theorem dist.intro {n m k : ℕ} (H : n + m = k) : dist k n = m :=
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calc
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dist k n = k - n : dist_eq_sub_of_ge (le.intro H)
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... = m : sub_eq_of_add_eq H
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theorem dist_add_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m :=
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calc
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dist (n + k) (m + k) = ((n+k) - (m+k)) + ((m+k)-(n+k)) : rfl
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... = (n - m) + ((m + k) - (n + k)) : add_sub_add_right
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... = (n - m) + (m - n) : add_sub_add_right
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theorem dist_add_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m :=
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!add.comm ▸ !add.comm ▸ !dist_add_add_right
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theorem dist_add_eq_of_ge {n m : ℕ} (H : n ≥ m) : dist n m + m = n :=
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calc
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dist n m + m = n - m + m : {dist_eq_sub_of_ge H}
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... = n : sub_add_cancel H
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theorem dist_eq_intro {n m k l : ℕ} (H : n + m = k + l) : dist n k = dist l m :=
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calc
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dist n k = dist (n + m) (k + m) : dist_add_add_right
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... = dist (k + l) (k + m) : H
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... = dist l m : dist_add_add_left
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theorem dist_sub_eq_dist_add_left {n m : ℕ} (H : n ≥ m) (k : ℕ) :
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dist (n - m) k = dist n (k + m) :=
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have H2 : n - m + (k + m) = k + n, from
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calc
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n - m + (k + m) = n - m + (m + k) : add.comm
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... = n - m + m + k : add.assoc
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... = n + k : sub_add_cancel H
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... = k + n : add.comm,
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dist_eq_intro H2
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theorem dist_sub_eq_dist_add_right {k m : ℕ} (H : k ≥ m) (n : ℕ) :
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dist n (k - m) = dist (n + m) k :=
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(dist_sub_eq_dist_add_left H n ▸ !dist.comm) ▸ !dist.comm
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theorem dist.triangle_inequality (n m k : ℕ) : dist n k ≤ dist n m + dist m k :=
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have (n - m) + (m - k) + ((k - m) + (m - n)) = (n - m) + (m - n) + ((m - k) + (k - m)),
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begin rewrite [add.comm (k - m) (m - n),
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{n - m + _ + _}add.assoc,
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{m - k + _}add.left_comm, -add.assoc] end,
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this ▸ add_le_add !sub_lt_sub_add_sub !sub_lt_sub_add_sub
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theorem dist_add_add_le_add_dist_dist (n m k l : ℕ) : dist (n + m) (k + l) ≤ dist n k + dist m l :=
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have H : dist (n + m) (k + m) + dist (k + m) (k + l) = dist n k + dist m l, from
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!dist_add_add_left ▸ !dist_add_add_right ▸ rfl,
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H ▸ !dist.triangle_inequality
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theorem dist_mul_right (n k m : ℕ) : dist (n * k) (m * k) = dist n m * k :=
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assert ∀ n m, dist n m = n - m + (m - n), from take n m, rfl,
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by rewrite [this, this n m, mul.right_distrib, *mul_sub_right_distrib]
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theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m :=
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begin rewrite [mul.comm k n, mul.comm k m, dist_mul_right, mul.comm] end
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theorem dist_mul_dist (n m k l : ℕ) : dist n m * dist k l = dist (n * k + m * l) (n * l + m * k) :=
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have aux : ∀k l, k ≥ l → dist n m * dist k l = dist (n * k + m * l) (n * l + m * k), from
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take k l : ℕ,
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assume H : k ≥ l,
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have H2 : m * k ≥ m * l, from !mul_le_mul_left H,
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have H3 : n * l + m * k ≥ m * l, from le.trans H2 !le_add_left,
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calc
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dist n m * dist k l = dist n m * (k - l) : dist_eq_sub_of_ge H
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... = dist (n * (k - l)) (m * (k - l)) : dist_mul_right
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... = dist (n * k - n * l) (m * k - m * l) : by rewrite [*mul_sub_left_distrib]
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... = dist (n * k) (m * k - m * l + n * l) : dist_sub_eq_dist_add_left (!mul_le_mul_left H)
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... = dist (n * k) (n * l + (m * k - m * l)) : add.comm
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... = dist (n * k) (n * l + m * k - m * l) : add_sub_assoc H2 (n * l)
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... = dist (n * k + m * l) (n * l + m * k) : dist_sub_eq_dist_add_right H3 _,
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or.elim !le.total
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(assume H : k ≤ l, !dist.comm ▸ !dist.comm ▸ aux l k H)
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(assume H : l ≤ k, aux k l H)
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lemma dist_eq_max_sub_min {i j : nat} : dist i j = (max i j) - min i j :=
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or.elim (lt_or_ge i j)
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(suppose i < j, begin rewrite [max_eq_right_of_lt this, min_eq_left_of_lt this, dist_eq_sub_of_lt this] end)
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(suppose i ≥ j, begin rewrite [max_eq_left this , min_eq_right this, dist_eq_sub_of_ge this] end)
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lemma dist_succ {i j : nat} : dist (succ i) (succ j) = dist i j :=
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by rewrite [↑dist, *succ_sub_succ]
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lemma dist_le_max {i j : nat} : dist i j ≤ max i j :=
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begin rewrite dist_eq_max_sub_min, apply sub_le end
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lemma dist_pos_of_ne {i j : nat} : i ≠ j → dist i j > 0 :=
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assume Pne, lt.by_cases
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(suppose i < j, begin rewrite [dist_eq_sub_of_lt this], apply sub_pos_of_lt this end)
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(suppose i = j, by contradiction)
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(suppose i > j, begin rewrite [dist_eq_sub_of_gt this], apply sub_pos_of_lt this end)
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end nat
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