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refactor(data/nat/sub): use new policy for marking implicit arguments and '!' operator

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This commit is contained in:
Leonardo de Moura 2014-10-05 12:39:13 -07:00
parent fa96596bf7
commit efaeeb0726
3 changed files with 115 additions and 116 deletions
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14
library/data/int/basic.lean
View file

@ -76,9 +76,9 @@ or.elim le_or_gt
have H3 : pr1 a = pr2 a, from le_antisym H H2,
calc
proj a = pair (pr1 a - pr2 a) 0 : proj_ge H2
... = pair (pr1 a - pr2 a) (pr1 a - pr1 a) : {sub_self⁻¹}
... = pair (pr1 a - pr2 a) (pr1 a - pr1 a) : {!sub_self⁻¹}
... = pair (pr2 a - pr2 a) (pr2 a - pr1 a) : {H3}
... = pair 0 (pr2 a - pr1 a) : {sub_self})
... = pair 0 (pr2 a - pr1 a) : {!sub_self})
(assume H2 : pr1 a < pr2 a, proj_lt H2)
theorem proj_ge_pr1 {a : ℕ × ℕ} (H : pr1 a ≥ pr2 a) : pr1 (proj a) = pr1 a - pr2 a :=
@ -153,10 +153,10 @@ have special : ∀a b, pr2 a ≤ pr1 a → rel a b → proj a = proj b, from
have H5 : pr1 (proj a) = pr1 (proj b), from
calc
pr1 (proj a) = pr1 a - pr2 a : proj_ge_pr1 H2
... = pr1 a + pr2 b - pr2 b - pr2 a : {sub_add_left⁻¹}
... = pr1 a + pr2 b - pr2 b - pr2 a : {!sub_add_left⁻¹}
... = pr2 a + pr1 b - pr2 b - pr2 a : {H}
... = pr2 a + pr1 b - pr2 a - pr2 b : {sub_comm}
... = pr1 b - pr2 b : {sub_add_left2}
... = pr2 a + pr1 b - pr2 a - pr2 b : {!sub_comm}
... = pr1 b - pr2 b : {!sub_add_left2}
... = pr1 (proj b) : (proj_ge_pr1 H4)⁻¹,
have H6 : pr2 (proj a) = pr2 (proj b), from
calc
@ -427,7 +427,7 @@ theorem to_nat_add_le (a b : ℤ) : to_nat (a + b) ≤ to_nat a + to_nat b :=
obtain (xa ya : ℕ) (Ha : a = psub (pair xa ya)), from destruct a,
obtain (xb yb : ℕ) (Hb : b = psub (pair xb yb)), from destruct b,
have H : dist (xa + xb) (ya + yb) ≤ dist xa ya + dist xb yb,
from dist_add_le_add_dist,
from !dist_add_le_add_dist,
by simp
-- TODO: note, we have to add #nat to get the right interpretation
@ -697,7 +697,7 @@ theorem mul_to_nat (a b : ℤ) : (to_nat (a * b)) = #nat (to_nat a) * (to_nat b)
obtain (xa ya : ℕ) (Ha : a = psub (pair xa ya)), from destruct a,
obtain (xb yb : ℕ) (Hb : b = psub (pair xb yb)), from destruct b,
have H : dist xa ya * dist xb yb = dist (xa * xb + ya * yb) (xa * yb + ya * xb),
from dist_mul_dist,
from !dist_mul_dist,
by simp
-- add_rewrite mul_zero_left mul_zero_right mul_one_right mul_one_left

4
library/data/nat/div.lean
View file

@ -467,9 +467,9 @@ have H1 : z * y = x mod y + x div y * y, from
H ⬝ div_mod_eq ⬝ !add.comm,
have H2 : (z - x div y) * y = x mod y, from
calc
(z - x div y) * y = z * y - x div y * y : mul_sub_distr_right
(z - x div y) * y = z * y - x div y * y : !mul_sub_distr_right
... = x mod y + x div y * y - x div y * y : {H1}
... = x mod y : sub_add_left,
... = x mod y : !sub_add_left,
show x mod y = 0, from
by_cases
(assume yz : y = 0,

209
library/data/nat/sub.lean
View file

@ -22,39 +22,39 @@ namespace nat
definition sub (n m : ℕ) : nat := rec n (fun m x, pred x) m
infixl `-` := sub
theorem sub_zero_right {n : ℕ} : n - 0 = n
theorem sub_zero_right (n : ℕ) : n - 0 = n
theorem sub_succ_right {n m : ℕ} : n - succ m = pred (n - m)
theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m)
irreducible sub
theorem sub_zero_left {n : ℕ} : 0 - n = 0 :=
induction_on n sub_zero_right
theorem sub_zero_left (n : ℕ) : 0 - n = 0 :=
induction_on n !sub_zero_right
(take k : nat,
assume IH : 0 - k = 0,
calc
0 - succ k = pred (0 - k) : sub_succ_right
0 - succ k = pred (0 - k) : !sub_succ_right
... = pred 0 : {IH}
... = 0 : pred.zero)
theorem sub_succ_succ {n m : ℕ} : succ n - succ m = n - m :=
theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m :=
induction_on m
(calc
succ n - 1 = pred (succ n - 0) : sub_succ_right
... = pred (succ n) : {sub_zero_right}
succ n - 1 = pred (succ n - 0) : !sub_succ_right
... = pred (succ n) : {!sub_zero_right}
... = n : !pred.succ
... = n - 0 : sub_zero_right⁻¹)
... = n - 0 : !sub_zero_right⁻¹)
(take k : nat,
assume IH : succ n - succ k = n - k,
calc
succ n - succ (succ k) = pred (succ n - succ k) : sub_succ_right
succ n - succ (succ k) = pred (succ n - succ k) : !sub_succ_right
... = pred (n - k) : {IH}
... = n - succ k : sub_succ_right⁻¹)
... = n - succ k : !sub_succ_right⁻¹)
theorem sub_self {n : ℕ} : n - n = 0 :=
induction_on n sub_zero_right (take k IH, sub_succ_succ ⬝ IH)
theorem sub_self (n : ℕ) : n - n = 0 :=
induction_on n !sub_zero_right (take k IH, !sub_succ_succ ⬝ IH)
theorem sub_add_add_right {n k m : ℕ} : (n + k) - (m + k) = n - m :=
theorem sub_add_add_right (n k m : ℕ) : (n + k) - (m + k) = n - m :=
induction_on k
(calc
(n + 0) - (m + 0) = n - (m + 0) : {!add.zero_right}
@ -64,108 +64,108 @@ induction_on k
calc
(n + succ l) - (m + succ l) = succ (n + l) - (m + succ l) : {!add.succ_right}
... = succ (n + l) - succ (m + l) : {!add.succ_right}
... = (n + l) - (m + l) : sub_succ_succ
... = (n + l) - (m + l) : !sub_succ_succ
... = n - m : IH)
theorem sub_add_add_left {k n m : ℕ} : (k + n) - (k + m) = n - m :=
!add.comm ▸ !add.comm ▸ sub_add_add_right
theorem sub_add_add_left (k n m : ℕ) : (k + n) - (k + m) = n - m :=
!add.comm ▸ !add.comm ▸ !sub_add_add_right
theorem sub_add_left {n m : ℕ} : n + m - m = n :=
theorem sub_add_left (n m : ℕ) : n + m - m = n :=
induction_on m
(!add.zero_right⁻¹ ▸ sub_zero_right)
(!add.zero_right⁻¹ ▸ !sub_zero_right)
(take k : ℕ,
assume IH : n + k - k = n,
calc
n + succ k - succ k = succ (n + k) - succ k : {!add.succ_right}
... = n + k - k : sub_succ_succ
... = n + k - k : !sub_succ_succ
... = n : IH)
-- TODO: add_sub_inv'
theorem sub_add_left2 {n m : ℕ} : n + m - n = m :=
!add.comm ▸ sub_add_left
theorem sub_add_left2 (n m : ℕ) : n + m - n = m :=
!add.comm ▸ !sub_add_left
theorem sub_sub {n m k : ℕ} : n - m - k = n - (m + k) :=
theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k) :=
induction_on k
(calc
n - m - 0 = n - m : sub_zero_right
n - m - 0 = n - m : !sub_zero_right
... = n - (m + 0) : {!add.zero_right⁻¹})
(take l : nat,
assume IH : n - m - l = n - (m + l),
calc
n - m - succ l = pred (n - m - l) : sub_succ_right
n - m - succ l = pred (n - m - l) : !sub_succ_right
... = pred (n - (m + l)) : {IH}
... = n - succ (m + l) : sub_succ_right⁻¹
... = n - succ (m + l) : !sub_succ_right⁻¹
... = n - (m + succ l) : {!add.succ_right⁻¹})
theorem succ_sub_sub {n m k : ℕ} : succ n - m - succ k = n - m - k :=
theorem succ_sub_sub (n m k : ℕ) : succ n - m - succ k = n - m - k :=
calc
succ n - m - succ k = succ n - (m + succ k) : sub_sub
succ n - m - succ k = succ n - (m + succ k) : !sub_sub
... = succ n - succ (m + k) : {!add.succ_right}
... = n - (m + k) : sub_succ_succ
... = n - m - k : sub_sub⁻¹
... = n - (m + k) : !sub_succ_succ
... = n - m - k : !sub_sub⁻¹
theorem sub_add_right_eq_zero {n m : ℕ} : n - (n + m) = 0 :=
theorem sub_add_right_eq_zero (n m : ℕ) : n - (n + m) = 0 :=
calc
n - (n + m) = n - n - m : sub_sub⁻¹
... = 0 - m : {sub_self}
... = 0 : sub_zero_left
n - (n + m) = n - n - m : !sub_sub⁻¹
... = 0 - m : {!sub_self}
... = 0 : !sub_zero_left
theorem sub_comm {m n k : ℕ} : m - n - k = m - k - n :=
theorem sub_comm (m n k : ℕ) : m - n - k = m - k - n :=
calc
m - n - k = m - (n + k) : sub_sub
m - n - k = m - (n + k) : !sub_sub
... = m - (k + n) : {!add.comm}
... = m - k - n : sub_sub⁻¹
... = m - k - n : !sub_sub⁻¹
theorem sub_one {n : ℕ} : n - 1 = pred n :=
theorem sub_one (n : ℕ) : n - 1 = pred n :=
calc
n - 1 = pred (n - 0) : sub_succ_right
... = pred n : {sub_zero_right}
n - 1 = pred (n - 0) : !sub_succ_right
... = pred n : {!sub_zero_right}
theorem succ_sub_one {n : ℕ} : succ n - 1 = n :=
sub_succ_succ ⬝ sub_zero_right
theorem succ_sub_one (n : ℕ) : succ n - 1 = n :=
!sub_succ_succ ⬝ !sub_zero_right
-- add_rewrite sub_add_left
-- ### interaction with multiplication
theorem mul_pred_left {n m : ℕ} : pred n * m = n * m - m :=
theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m :=
induction_on n
(calc
pred 0 * m = 0 * m : {pred.zero}
... = 0 : !mul.zero_left
... = 0 - m : sub_zero_left⁻¹
... = 0 - m : !sub_zero_left⁻¹
... = 0 * m - m : {!mul.zero_left⁻¹})
(take k : nat,
assume IH : pred k * m = k * m - m,
calc
pred (succ k) * m = k * m : {!pred.succ}
... = k * m + m - m : sub_add_left⁻¹
... = k * m + m - m : !sub_add_left⁻¹
... = succ k * m - m : {!mul.succ_left⁻¹})
theorem mul_pred_right {n m : ℕ} : n * pred m = n * m - n :=
theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n :=
calc n * pred m = pred m * n : !mul.comm
... = m * n - n : mul_pred_left
... = m * n - n : !mul_pred_left
... = n * m - n : {!mul.comm}
theorem mul_sub_distr_right {n m k : ℕ} : (n - m) * k = n * k - m * k :=
theorem mul_sub_distr_right (n m k : ℕ) : (n - m) * k = n * k - m * k :=
induction_on m
(calc
(n - 0) * k = n * k : {sub_zero_right}
... = n * k - 0 : sub_zero_right⁻¹
(n - 0) * k = n * k : {!sub_zero_right}
... = n * k - 0 : !sub_zero_right⁻¹
... = n * k - 0 * k : {!mul.zero_left⁻¹})
(take l : nat,
assume IH : (n - l) * k = n * k - l * k,
calc
(n - succ l) * k = pred (n - l) * k : {sub_succ_right}
... = (n - l) * k - k : mul_pred_left
(n - succ l) * k = pred (n - l) * k : {!sub_succ_right}
... = (n - l) * k - k : !mul_pred_left
... = n * k - l * k - k : {IH}
... = n * k - (l * k + k) : sub_sub
... = n * k - (l * k + k) : !sub_sub
... = n * k - (succ l * k) : {!mul.succ_left⁻¹})
theorem mul_sub_distr_left {n m k : ℕ} : n * (m - k) = n * m - n * k :=
theorem mul_sub_distr_left (n m k : ℕ) : n * (m - k) = n * m - n * k :=
calc
n * (m - k) = (m - k) * n : !mul.comm
... = m * n - k * n : mul_sub_distr_right
... = m * n - k * n : !mul_sub_distr_right
... = n * m - k * n : {!mul.comm}
... = n * m - n * k : {!mul.comm}
@ -176,8 +176,8 @@ sub_induction n m
(take k,
assume H : 0 ≤ k,
calc
succ k - 0 = succ k : sub_zero_right
... = succ (k - 0) : {sub_zero_right⁻¹})
succ k - 0 = succ k : !sub_zero_right
... = succ (k - 0) : {!sub_zero_right⁻¹})
(take k,
assume H : succ k ≤ 0,
absurd H !not_succ_zero_le)
@ -185,12 +185,12 @@ sub_induction n m
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 : sub_succ_succ
succ (succ l) - succ k = succ l - k : !sub_succ_succ
... = succ (l - k) : IH (succ_le_cancel H)
... = succ (succ l - succ k) : {sub_succ_succ⁻¹})
... = succ (succ l - succ k) : {!sub_succ_succ⁻¹})
theorem le_imp_sub_eq_zero {n m : ℕ} (H : n ≤ m) : n - m = 0 :=
obtain (k : ℕ) (Hk : n + k = m), from le_elim H, Hk ▸ sub_add_right_eq_zero
obtain (k : ℕ) (Hk : n + k = m), from le_elim H, Hk ▸ !sub_add_right_eq_zero
theorem add_sub_le {n m : ℕ} : n ≤ m → n + (m - n) = m :=
sub_induction n m
@ -198,13 +198,13 @@ sub_induction n m
assume H : 0 ≤ k,
calc
0 + (k - 0) = k - 0 : !add.zero_left
... = k : sub_zero_right)
... = k : !sub_zero_right)
(take k, assume H : succ k ≤ 0, absurd H !not_succ_zero_le)
(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) : {sub_succ_succ}
succ k + (succ l - succ k) = succ k + (l - k) : {!sub_succ_succ}
... = succ (k + (l - k)) : !add.succ_left
... = succ l : {IH (succ_le_cancel H)})
@ -219,12 +219,12 @@ calc
theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m :=
!add.comm ▸ add_sub_ge
theorem le_add_sub_left {n m : ℕ} : n ≤ n + (m - n) :=
theorem le_add_sub_left (n m : ℕ) : n ≤ n + (m - n) :=
or.elim !le_total
(assume H : n ≤ m, (add_sub_le H)⁻¹ ▸ H)
(assume H : m ≤ n, (add_sub_ge H)⁻¹ ▸ !le_refl)
theorem le_add_sub_right {n m : ℕ} : m ≤ n + (m - n) :=
theorem le_add_sub_right (n m : ℕ) : m ≤ n + (m - n) :=
or.elim !le_total
(assume H : n ≤ m, (add_sub_le H)⁻¹ ▸ !le_refl)
(assume H : m ≤ n, (add_sub_ge H)⁻¹ ▸ H)
@ -235,7 +235,7 @@ or.elim !le_total
(assume H3 : n ≤ m, (le_imp_sub_eq_zero H3)⁻¹ ▸ (H1 H3))
(assume H3 : m ≤ n, H2 (n - m) (add_sub_le H3))
theorem sub_le_self {n m : ℕ} : n - m ≤ n :=
theorem sub_le_self (n m : ℕ) : n - m ≤ n :=
sub_split
(assume H : n ≤ m, !zero_le)
(take k : ℕ, assume H : m + k = n, le_intro (!add.comm ▸ H))
@ -245,7 +245,7 @@ obtain (k : ℕ) (Hk : n + k = m), from le_elim H,
exists_intro k
(calc
m - k = n + k - k : {Hk⁻¹}
... = n : sub_add_left)
... = n : !sub_add_left)
theorem add_sub_assoc {m k : ℕ} (H : k ≤ m) (n : ℕ) : n + m - k = n + (m - k) :=
have l1 : k ≤ m → n + m - k = n + (m - k), from
@ -253,17 +253,17 @@ have l1 : k ≤ m → n + m - k = n + (m - k), from
(take m : ℕ,
assume H : 0 ≤ m,
calc
n + m - 0 = n + m : sub_zero_right
... = n + (m - 0) : {sub_zero_right⁻¹})
n + m - 0 = n + m : !sub_zero_right
... = n + (m - 0) : {!sub_zero_right⁻¹})
(take k : ℕ, assume H : succ k ≤ 0, absurd H !not_succ_zero_le)
(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_right}
... = n + m - k : sub_succ_succ
... = n + m - k : !sub_succ_succ
... = n + (m - k) : IH (succ_le_cancel H)
... = n + (succ m - succ k) : {sub_succ_succ⁻¹}),
... = n + (succ m - succ k) : {!sub_succ_succ⁻¹}),
l1 H
theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m :=
@ -293,15 +293,14 @@ add.cancel_right H2
theorem sub_lt {x y : ℕ} (xpos : x > 0) (ypos : y > 0) : x - y < x :=
obtain (x' : ℕ) (xeq : x = succ x'), from pos_imp_eq_succ xpos,
obtain (y' : ℕ) (yeq : y = succ y'), from pos_imp_eq_succ ypos,
have xsuby_eq : x - y = x' - y', from
calc
obtain (y' : ℕ) (yeq : y = succ y'), from pos_imp_eq_succ ypos,
have xsuby_eq : x - y = x' - y', from calc
x - y = succ x' - y : {xeq}
... = succ x' - succ y' : {yeq}
... = x' - y' : sub_succ_succ,
have H1 : x' - y' ≤ x', from sub_le_self,
have H2 : x' < succ x', from !self_lt_succ,
show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt_trans H1 H2
... = x' - y' : !sub_succ_succ,
have H1 : x' - y' ≤ x', from !sub_le_self,
have H2 : x' < succ x', from !self_lt_succ,
show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt_trans H1 H2
theorem sub_le_right {n m : ℕ} (H : n ≤ m) (k : nat) : n - k ≤ m - k :=
obtain (l : ℕ) (Hl : n + l = m), from le_elim H,
@ -339,7 +338,7 @@ sub_split
-- :=
-- _
theorem sub_triangle_inequality {n m k : ℕ} : n - k ≤ (n - m) + (m - k) :=
theorem sub_triangle_inequality (n m k : ℕ) : n - k ≤ (n - m) + (m - k) :=
sub_split
(assume H : n ≤ m, !add.zero_left⁻¹ ▸ sub_le_right H k)
(take mn : ℕ,
@ -373,9 +372,9 @@ theorem max_le {n m : ℕ} (H : n ≤ m) : n + (m - n) = m := add_sub_le H
theorem max_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n := add_sub_ge H
theorem left_le_max {n m : ℕ} : n ≤ n + (m - n) := le_add_sub_left
theorem left_le_max (n m : ℕ) : n ≤ n + (m - n) := !le_add_sub_left
theorem right_le_max {n m : ℕ} : m ≤ max n m := le_add_sub_right
theorem right_le_max (n m : ℕ) : m ≤ max n m := !le_add_sub_right
-- ### absolute difference
@ -383,10 +382,10 @@ theorem right_le_max {n m : ℕ} : m ≤ max n m := le_add_sub_right
definition dist (n m : ℕ) := (n - m) + (m - n)
theorem dist_comm {n m : ℕ} : dist n m = dist m n :=
theorem dist_comm (n m : ℕ) : dist n m = dist m n :=
!add.comm
theorem dist_self {n : ℕ} : dist n n = 0 :=
theorem dist_self (n : ℕ) : dist n n = 0 :=
calc
(n - n) + (n - n) = 0 + 0 : by simp
... = 0 : by simp
@ -404,27 +403,27 @@ calc
... = m - n : !add.zero_left
theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m :=
dist_comm ▸ dist_le H
!dist_comm ▸ dist_le H
theorem dist_zero_right {n : ℕ} : dist n 0 = n :=
dist_ge !zero_le ⬝ sub_zero_right
theorem dist_zero_right (n : ℕ) : dist n 0 = n :=
dist_ge !zero_le ⬝ !sub_zero_right
theorem dist_zero_left {n : ℕ} : dist 0 n = n :=
dist_le !zero_le ⬝ sub_zero_right
theorem dist_zero_left (n : ℕ) : dist 0 n = n :=
dist_le !zero_le ⬝ !sub_zero_right
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)
... = m : sub_intro H
theorem dist_add_right {n k m : ℕ} : dist (n + k) (m + k) = dist n m :=
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)) : rfl
... = (n - m) + ((m + k) - (n + k)) : {sub_add_add_right}
... = (n - m) + (m - n) : {sub_add_add_right}
... = (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 ▸ !add.comm ▸ dist_add_right
theorem dist_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m :=
!add.comm ▸ !add.comm ▸ !dist_add_right
-- add_rewrite dist_self dist_add_right dist_add_left dist_zero_left dist_zero_right
@ -435,9 +434,9 @@ calc
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) : dist_add_right⁻¹
dist n k = dist (n + m) (k + m) : !dist_add_right⁻¹
... = dist (k + l) (k + m) : {H}
... = dist l m : dist_add_left
... = dist l m : !dist_add_left
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
@ -448,33 +447,33 @@ have H2 : n - m + (k + m) = k + n, from
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 :=
(dist_sub_move_add H n ▸ dist_comm) ▸ dist_comm
(dist_sub_move_add H n ▸ !dist_comm) ▸ !dist_comm
--triangle inequality formulated with dist
theorem triangle_inequality {n m k : ℕ} : dist n k ≤ dist n m + dist m k :=
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 sub_triangle_inequality
H ▸ add_le !sub_triangle_inequality !sub_triangle_inequality
theorem dist_add_le_add_dist {n m k l : ℕ} : dist (n + m) (k + l) ≤ dist n k + dist m l :=
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
dist_add_left ▸ dist_add_right ▸ rfl,
H ▸ triangle_inequality
!dist_add_left ▸ !dist_add_right ▸ rfl,
H ▸ !triangle_inequality
--interaction with multiplication
theorem dist_mul_left {k n m : ℕ} : dist (k * n) (k * m) = k * dist n m :=
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, rfl,
by simp
theorem dist_mul_right {n k m : ℕ} : dist (n * k) (m * k) = dist n m * k :=
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, rfl,
by simp
-- add_rewrite dist_mul_right dist_mul_left dist_comm
--needed to prove of_nat a * of_nat b = of_nat (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) :=
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,
@ -482,14 +481,14 @@ have aux : ∀k l, k ≥ l → dist n m * dist k l = dist (n * k + m * l) (n * l
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)) : dist_mul_right⁻¹
... = dist (n * (k - l)) (m * (k - l)) : !dist_mul_right⁻¹
... = 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) : {(add_sub_assoc H2 (n * l))⁻¹}
... = dist (n * k + m * l) (n * l + m * k) : dist_sub_move_add' H3 _,
or.elim !le_total
(assume H : k ≤ l, dist_comm ▸ dist_comm ▸ aux l k H)
(assume H : k ≤ l, !dist_comm ▸ !dist_comm ▸ aux l k H)
(assume H : l ≤ k, aux k l H)
end nat