513 lines
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17 KiB
Text
513 lines
No EOL
17 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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--- Author: Floris van Doorn
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----------------------------------------------------------------------------------------------------
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import data.nat.order
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using nat eq_proofs tactic
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using helper_tactics
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namespace nat
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-- data.nat.basic2
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-- ===============
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--
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-- More basic operations on the natural numbers.
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-- subtraction
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-- -----------
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definition sub (n m : ℕ) : nat := nat_rec n (fun m x, pred x) m
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infixl `-` : 65 := sub
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theorem sub_zero_right (n : ℕ) : n - 0 = n
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theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m)
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opaque_hint (hiding sub)
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theorem sub_zero_left (n : ℕ) : 0 - n = 0 :=
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induction_on n (sub_zero_right 0)
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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_right 0 k
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... = pred 0 : {IH}
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... = 0 : pred_zero)
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--(
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--theorem sub_succ_left (n m : ℕ) : pred (succ n - m) = n - m
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-- :=
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-- induction_on m
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-- (calc
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-- pred (succ n - 0) = pred (succ n) : {sub_zero_right (succ n)}
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-- ... = n : pred_succ n
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-- ... = n - 0 : symm (sub_zero_right n))
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-- (take k : nat,
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-- assume IH : pred (succ n - k) = n - k,
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-- _)
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--)
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--succ_sub_succ
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theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m :=
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induction_on m
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(calc
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succ n - 1 = pred (succ n - 0) : sub_succ_right (succ n) 0
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... = pred (succ n) : {sub_zero_right (succ n)}
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... = n : pred_succ n
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... = n - 0 : symm (sub_zero_right n))
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(take k : nat,
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assume IH : succ n - succ k = n - k,
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calc
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succ n - succ (succ k) = pred (succ n - succ k) : sub_succ_right (succ n) (succ k)
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... = pred (n - k) : {IH}
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... = n - succ k : symm (sub_succ_right n k))
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theorem sub_self (n : ℕ) : n - n = 0 :=
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induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
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-- TODO: add_sub_add_right
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theorem sub_add_add_right (n k m : ℕ) : (n + k) - (m + k) = n - m :=
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induction_on k
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(calc
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(n + 0) - (m + 0) = n - (m + 0) : {add_zero_right _}
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... = n - m : {add_zero_right _})
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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_right _ _}
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... = succ (n + l) - succ (m + l) : {add_succ_right _ _}
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... = (n + l) - (m + l) : sub_succ_succ _ _
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... = n - m : IH)
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theorem sub_add_add_left (k n m : ℕ) : (k + n) - (k + m) = n - m :=
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add_comm m k ▸ add_comm n k ▸ sub_add_add_right n k m
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-- TODO: add_sub_inv
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theorem sub_add_left (n m : ℕ) : n + m - m = n :=
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induction_on m
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((add_zero_right n)⁻¹ ▸ sub_zero_right n)
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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_right n k}
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... = n + k - k : sub_succ_succ _ _
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... = n : IH)
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-- TODO: add_sub_inv'
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theorem sub_add_left2 (n m : ℕ) : n + m - n = m :=
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add_comm m n ▸ sub_add_left m n
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theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k) :=
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induction_on k
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(calc
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n - m - 0 = n - m : sub_zero_right _
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... = n - (m + 0) : {symm (add_zero_right m)})
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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_right (n - m) l
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... = pred (n - (m + l)) : {IH}
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... = n - succ (m + l) : symm (sub_succ_right n (m + l))
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... = n - (m + succ l) : {symm (add_succ_right m l)})
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theorem succ_sub_sub (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_right m k}
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... = n - (m + k) : sub_succ_succ _ _
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... = n - m - k : symm (sub_sub n m k)
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theorem sub_add_right_eq_zero (n m : ℕ) : n - (n + m) = 0 :=
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calc
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n - (n + m) = n - n - m : symm (sub_sub n n m)
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... = 0 - m : {sub_self n}
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... = 0 : sub_zero_left m
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theorem sub_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 m n k
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... = m - (k + n) : {add_comm n k}
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... = m - k - n : symm (sub_sub m k n)
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theorem sub_one (n : ℕ) : n - 1 = pred n :=
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calc
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n - 1 = pred (n - 0) : sub_succ_right n 0
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... = pred n : {sub_zero_right n}
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theorem succ_sub_one (n : ℕ) : succ n - 1 = n :=
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trans (sub_succ_succ n 0) (sub_zero_right n)
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-- add_rewrite sub_add_left
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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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induction_on n
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(calc
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pred 0 * m = 0 * m : {pred_zero}
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... = 0 : mul_zero_left _
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... = 0 - m : symm (sub_zero_left m)
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... = 0 * m - m : {symm (mul_zero_left m)})
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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 k}
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... = k * m + m - m : symm (sub_add_left _ _)
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... = succ k * m - m : {symm (mul_succ_left k m)})
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theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n :=
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calc n * pred m = pred m * n : mul_comm _ _
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... = m * n - n : mul_pred_left m n
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... = n * m - n : {mul_comm m n}
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theorem mul_sub_distr_right (n m k : ℕ) : (n - m) * k = n * k - m * k :=
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induction_on m
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(calc
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(n - 0) * k = n * k : {sub_zero_right n}
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... = n * k - 0 : symm (sub_zero_right _)
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... = n * k - 0 * k : {symm (mul_zero_left _)})
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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_right n l}
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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) : {symm (mul_succ_left l k)})
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theorem mul_sub_distr_left (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_distr_right _ _ _
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... = n * m - k * n : {mul_comm _ _}
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... = n * m - n * k : {mul_comm _ _}
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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,
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assume H : 0 ≤ k,
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calc
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succ k - 0 = succ k : sub_zero_right (succ k)
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... = succ (k - 0) : {symm (sub_zero_right k)})
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(take k,
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assume H : succ k ≤ 0,
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absurd_elim _ H (not_succ_zero_le k))
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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 : sub_succ_succ (succ l) k
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... = succ (l - k) : IH (succ_le_cancel H)
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... = succ (succ l - succ k) : {symm (sub_succ_succ l k)})
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theorem le_imp_sub_eq_zero {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_add_right_eq_zero n k
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theorem add_sub_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 : add_zero_left (k - 0)
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... = k : sub_zero_right k)
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(take k, assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k))
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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) : {sub_succ_succ l k}
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... = succ (k + (l - k)) : add_succ_left k (l - k)
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... = succ l : {IH (succ_le_cancel H)})
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theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n :=
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add_comm m (n - m) ▸ add_sub_le
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theorem add_sub_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 : {le_imp_sub_eq_zero H}
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... = n : add_zero_right n
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theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m :=
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add_comm m (n - m) ▸ add_sub_ge
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theorem le_add_sub_left (n m : ℕ) : n ≤ n + (m - n) :=
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or_elim (le_total n m)
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(assume H : n ≤ m, (add_sub_le H)⁻¹ ▸ H)
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(assume H : m ≤ n, (add_sub_ge H)⁻¹ ▸ le_refl n)
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theorem le_add_sub_right (n m : ℕ) : m ≤ n + (m - n) :=
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or_elim (le_total n m)
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(assume H : n ≤ m, subst (symm (add_sub_le H)) (le_refl m))
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(assume H : m ≤ n, subst (symm (add_sub_ge H)) H)
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theorem sub_split {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 n m)
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(assume H3 : n ≤ m, subst (symm (le_imp_sub_eq_zero H3)) (H1 H3))
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(assume H3 : m ≤ n, H2 (n - m) (add_sub_le H3))
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theorem sub_le_self (n m : ℕ) : n - m ≤ n :=
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sub_split
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(assume H : n ≤ m, zero_le n)
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(take k : ℕ, assume H : m + k = n, le_intro (subst (add_comm m k) H))
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theorem le_elim_sub (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 : {symm Hk}
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... = n : sub_add_left n k)
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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_right (n + m)
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... = n + (m - 0) : {symm (sub_zero_right m)})
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(take k : ℕ, assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k))
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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_right n m}
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... = n + m - k : sub_succ_succ (n + m) k
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... = n + (m - k) : IH (succ_le_cancel H)
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... = n + (succ m - succ k) : {symm (sub_succ_succ m k)}),
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l1 H
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theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m :=
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sub_split
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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 subst (add_zero_right m) (subst H2 (symm H1)),
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subst H3 (le_refl n))
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theorem sub_sub_split {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 n m)
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(assume H3 : n ≤ m,
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le_imp_sub_eq_zero H3⁻¹ ▸ (H2 (m - n) (add_sub_le H3⁻¹)))
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(assume H3 : m ≤ n,
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le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹)))
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theorem sub_intro {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 : add_sub_ge_left (le_intro H)
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... = n + m : symm H
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... = m + n : add_comm n m,
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add_cancel_right H2
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theorem sub_lt {x y : ℕ} (xpos : x > 0) (ypos : y > 0) : x - y < x :=
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obtain (x' : ℕ) (xeq : x = succ x'), from pos_imp_eq_succ xpos,
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obtain (y' : ℕ) (yeq : y = succ y'), from pos_imp_eq_succ ypos,
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have xsuby_eq : x - y = x' - y', from
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calc
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x - y = succ x' - y : {xeq}
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... = succ x' - succ y' : {yeq}
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... = x' - y' : sub_succ_succ _ _,
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have H1 : x' - y' ≤ x', from sub_le_self _ _,
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have H2 : x' < succ x', from self_lt_succ _,
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show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt_trans H1 H2
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theorem sub_le_right {n m : ℕ} (H : n ≤ m) (k : nat) : 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 n k)
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(assume H2 : n ≤ k, (le_imp_sub_eq_zero H2)⁻¹ ▸ zero_le (m - k))
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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) : by simp
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... = l + n - k : symm (add_sub_assoc H2 l)
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... = n + l - k : {add_comm l n}
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... = m - k : {Hl},
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le_intro H3)
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theorem sub_le_left {n m : ℕ} (H : n ≤ m) (k : nat) : k - m ≤ k - n :=
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obtain (l : ℕ) (Hl : n + l = m), from le_elim H,
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sub_split
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(assume H2 : k ≤ m, zero_le (k - n))
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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 + l + m' : by simp
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... = m + m' : {Hl}
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... = k : Hm
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... = k - n + n : symm (add_sub_ge_left H3),
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le_intro (add_cancel_right H4))
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-- theorem sub_lt_cancel_right {n m k : ℕ) (H : n - k < m - k) : n < m
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-- :=
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-- _
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-- theorem sub_lt_cancel_left {n m k : ℕ) (H : n - m < n - k) : k < m
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-- :=
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-- _
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theorem sub_triangle_inequality (n m k : ℕ) : n - k ≤ (n - m) + (m - k) :=
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sub_split
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(assume H : n ≤ m, (add_zero_left (m - k))⁻¹ ▸ sub_le_right H k)
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(take mn : ℕ,
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assume Hmn : m + mn = n,
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sub_split
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(assume H : m ≤ k,
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have H2 : n - k ≤ n - m, from sub_le_left H n,
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have H3 : n - k ≤ mn, from sub_intro Hmn ▸ H2,
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show n - k ≤ mn + 0, from (add_zero_right mn)⁻¹ ▸ H3)
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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 : by simp
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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_intro H,
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H2 ▸ (le_refl (n - k))))
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-- add_rewrite sub_self mul_sub_distr_left mul_sub_distr_right
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-- Max, min, iteration, and absolute difference
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-- --------------------------------------------
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definition max (n m : ℕ) : ℕ := n + (m - n)
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definition min (n m : ℕ) : ℕ := m - (m - n)
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theorem max_le {n m : ℕ} (H : n ≤ m) : n + (m - n) = m := add_sub_le H
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theorem max_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n := add_sub_ge H
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theorem left_le_max (n m : ℕ) : n ≤ n + (m - n) := le_add_sub_left n m
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theorem right_le_max (n m : ℕ) : m ≤ max n m := le_add_sub_right n m
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-- ### absolute difference
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-- This section is still incomplete
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definition dist (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 (n - m) (m - n)
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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 + 0 : by simp
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... = 0 : by simp
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theorem dist_eq_zero {n m : ℕ} (H : dist n m = 0) : n = m :=
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have H2 : n - m = 0, from add_eq_zero_left H,
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have H3 : n ≤ m, from sub_eq_zero_imp_le H2,
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have H4 : m - n = 0, from add_eq_zero_right H,
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have H5 : m ≤ n, from sub_eq_zero_imp_le H4,
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le_antisym H3 H5
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theorem dist_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) : {le_imp_sub_eq_zero H}
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... = m - n : add_zero_left (m - n)
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|
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theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m :=
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dist_comm m n ▸ dist_le H
|
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|
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theorem dist_zero_right (n : ℕ) : dist n 0 = n :=
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trans (dist_ge (zero_le n)) (sub_zero_right n)
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|
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theorem dist_zero_left (n : ℕ) : dist 0 n = n :=
|
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trans (dist_le (zero_le n)) (sub_zero_right n)
|
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|
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theorem dist_intro {n m k : ℕ} (H : n + m = k) : dist k n = m :=
|
||
calc
|
||
dist k n = k - n : dist_ge (le_intro H)
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... = m : sub_intro H
|
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|
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theorem dist_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m :=
|
||
calc
|
||
dist (n + k) (m + k) = ((n+k) - (m+k)) + ((m+k)-(n+k)) : refl _
|
||
... = (n - m) + ((m + k) - (n + k)) : {sub_add_add_right _ _ _}
|
||
... = (n - m) + (m - n) : {sub_add_add_right _ _ _}
|
||
|
||
theorem dist_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m :=
|
||
add_comm m k ▸ add_comm n k ▸ dist_add_right n k m
|
||
|
||
-- add_rewrite dist_self dist_add_right dist_add_left dist_zero_left dist_zero_right
|
||
|
||
theorem dist_ge_add_right {n m : ℕ} (H : n ≥ m) : dist n m + m = n :=
|
||
calc
|
||
dist n m + m = n - m + m : {dist_ge H}
|
||
... = n : add_sub_ge_left H
|
||
|
||
theorem dist_eq_intro {n m k l : ℕ} (H : n + m = k + l) : dist n k = dist l m :=
|
||
calc
|
||
dist n k = dist (n + m) (k + m) : symm (dist_add_right n m k)
|
||
... = dist (k + l) (k + m) : {H}
|
||
... = dist l m : dist_add_left k l m
|
||
|
||
theorem dist_sub_move_add {n m : ℕ} (H : n ≥ m) (k : ℕ) : dist (n - m) k = dist n (k + m) :=
|
||
have H2 : n - m + (k + m) = k + n, from
|
||
calc
|
||
n - m + (k + m) = n - m + m + k : by simp
|
||
... = n + k : {add_sub_ge_left H}
|
||
... = k + n : by simp,
|
||
dist_eq_intro H2
|
||
|
||
theorem dist_sub_move_add' {k m : ℕ} (H : k ≥ m) (n : ℕ) : dist n (k - m) = dist (n + m) k :=
|
||
subst (subst (dist_sub_move_add H n) (dist_comm (k - m) n)) (dist_comm k (n + m))
|
||
|
||
--triangle inequality formulated with dist
|
||
theorem triangle_inequality (n m k : ℕ) : dist n k ≤ dist n m + dist m k :=
|
||
have H : (n - m) + (m - k) + ((k - m) + (m - n)) = (n - m) + (m - n) + ((m - k) + (k - m)),
|
||
by simp,
|
||
H ▸ add_le (sub_triangle_inequality n m k) (sub_triangle_inequality k m n)
|
||
|
||
theorem dist_add_le_add_dist (n m k l : ℕ) : dist (n + m) (k + l) ≤ dist n k + dist m l :=
|
||
have H : dist (n + m) (k + m) + dist (k + m) (k + l) = dist n k + dist m l, from
|
||
calc
|
||
_ = dist n k + dist (k + m) (k + l) : {dist_add_right n m k}
|
||
... = _ : {dist_add_left k m l},
|
||
H ▸ (triangle_inequality (n + m) (k + m) (k + l))
|
||
|
||
--interaction with multiplication
|
||
|
||
theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m :=
|
||
have H : ∀n m, dist n m = n - m + (m - n), from take n m, refl _,
|
||
by simp
|
||
|
||
theorem dist_mul_right (n k m : ℕ) : dist (n * k) (m * k) = dist n m * k :=
|
||
have H : ∀n m, dist n m = n - m + (m - n), from take n m, refl _,
|
||
by simp
|
||
|
||
-- add_rewrite dist_mul_right dist_mul_left dist_comm
|
||
|
||
--needed to prove |a| * |b| = |a * b| in int
|
||
theorem dist_mul_dist (n m k l : ℕ) : dist n m * dist k l = dist (n * k + m * l) (n * l + m * k) :=
|
||
have aux : ∀k l, k ≥ l → dist n m * dist k l = dist (n * k + m * l) (n * l + m * k), from
|
||
take k l : ℕ,
|
||
assume H : k ≥ l,
|
||
have H2 : m * k ≥ m * l, from mul_le_left H m,
|
||
have H3 : n * l + m * k ≥ m * l, from le_trans H2 (le_add_left _ _),
|
||
calc
|
||
dist n m * dist k l = dist n m * (k - l) : {dist_ge H}
|
||
... = dist (n * (k - l)) (m * (k - l)) : symm (dist_mul_right n (k - l) m)
|
||
... = dist (n * k - n * l) (m * k - m * l) : by simp
|
||
... = dist (n * k) (m * k - m * l + n * l) : dist_sub_move_add (mul_le_left H n) _
|
||
... = dist (n * k) (n * l + (m * k - m * l)) : {add_comm _ _}
|
||
... = dist (n * k) (n * l + m * k - m * l) : {symm (add_sub_assoc H2 (n * l))}
|
||
... = dist (n * k + m * l) (n * l + m * k) : dist_sub_move_add' H3 _,
|
||
or_elim (le_total k l)
|
||
(assume H : k ≤ l, dist_comm l k ▸ dist_comm _ _ ▸ aux l k H)
|
||
(assume H : l ≤ k, aux k l H)
|
||
|
||
opaque_hint (hiding dist)
|
||
|
||
end -- namespace nat |