lean2/library/data/nat/sub.lean

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