refactor(library/data/int/sub): rename theorems, add theorems, clean up
This commit is contained in:
Jeremy Avigad
parent
4f519be55e
commit
0dcf06000b
5 changed files with 213 additions and 238 deletions
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@ -45,7 +45,6 @@ prefix `-` := has_neg.neg
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notation 1 := !has_one.one
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notation 0 := !has_zero.zero
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/- semigroup -/
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structure semigroup [class] (A : Type) extends has_mul A :=
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@ -80,7 +79,6 @@ theorem mul.right_cancel [s : right_cancel_semigroup A] {a b c : A} :
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a * b = c * b → a = c :=
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!right_cancel_semigroup.mul_right_cancel
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/- additive semigroup -/
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structure add_semigroup [class] (A : Type) extends has_add A :=
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@ -127,7 +125,6 @@ theorem mul_one [s : monoid A] (a : A) : a * 1 = a := !monoid.mul_one
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structure comm_monoid [class] (A : Type) extends monoid A, comm_semigroup A
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/- additive monoid -/
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structure add_monoid [class] (A : Type) extends add_semigroup A, has_zero A :=
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@ -139,7 +136,6 @@ theorem add_zero [s : add_monoid A] (a : A) : a + 0 = a := !add_monoid.add_zero
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structure add_comm_monoid [class] (A : Type) extends add_monoid A, add_comm_semigroup A
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/- group -/
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structure group [class] (A : Type) extends monoid A, has_inv A :=
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@ -277,7 +273,6 @@ end group
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structure comm_group [class] (A : Type) extends group A, comm_monoid A
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/- additive group -/
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structure add_group [class] (A : Type) extends add_monoid A, has_neg A :=
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@ -489,7 +484,6 @@ include s
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end add_comm_group
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/- bundled structures -/
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structure Semigroup :=
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@ -122,7 +122,7 @@ theorem of_nat_succ (n : ℕ) : of_nat (succ n) = of_nat n + 1 := rfl
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theorem mul_of_nat (n m : ℕ) : of_nat n * of_nat m = n * m := rfl
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theorem sub_nat_nat_of_ge {m n : ℕ} (H : m ≥ n) : sub_nat_nat m n = of_nat (m - n) :=
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have H1 : n - m = 0, from le_imp_sub_eq_zero H,
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have H1 : n - m = 0, from sub_eq_zero_of_le H,
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calc
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sub_nat_nat m n = nat.cases_on 0 (of_nat (m - n)) _ : H1 ▸ rfl
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... = of_nat (m - n) : rfl
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@ -131,7 +131,7 @@ context
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reducible sub_nat_nat
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theorem sub_nat_nat_of_lt {m n : ℕ} (H : m < n) :
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sub_nat_nat m n = neg_succ_of_nat (pred (n - m)) :=
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have H1 : n - m = succ (pred (n - m)), from (succ_pred_of_pos (sub_pos_of_gt H))⁻¹,
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have H1 : n - m = succ (pred (n - m)), from (succ_pred_of_pos (sub_pos_of_lt H))⁻¹,
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calc
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sub_nat_nat m n = nat.cases_on (succ (pred (n - m))) (of_nat (m - n))
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(take k, neg_succ_of_nat k) : H1 ▸ rfl
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@ -218,15 +218,15 @@ or.elim (@le_or_gt n m)
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have H1 : repr (sub_nat_nat m n) = (m - n, 0), from sub_nat_nat_of_ge H ▸ rfl,
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H1⁻¹ ▸
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(calc
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m - n + n = m : add_sub_ge_left H
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m - n + n = m : sub_add_cancel H
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... = 0 + m : zero_add))
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(take H : m < n,
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have H1 : repr (sub_nat_nat m n) = (0, succ (pred (n - m))), from sub_nat_nat_of_lt H ▸ rfl,
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H1⁻¹ ▸
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(calc
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0 + n = n : zero_add
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... = n - m + m : add_sub_ge_left (le_of_lt H)
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... = succ (pred (n - m)) + m : (succ_pred_of_pos (sub_pos_of_gt H))⁻¹))
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... = n - m + m : sub_add_cancel (le_of_lt H)
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... = succ (pred (n - m)) + m : (succ_pred_of_pos (sub_pos_of_lt H))⁻¹))
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theorem repr_abstr (p : ℕ × ℕ) : repr (abstr p) ≡ p :=
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!prod.eta ▸ !repr_sub_nat_nat
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@ -238,28 +238,28 @@ or.elim (equiv_cases Hequiv)
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have H4 : pr1 q ≥ pr2 q, from and.elim_right H2,
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have H5 : pr1 p = pr1 q - pr2 q + pr2 p, from
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calc
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pr1 p = pr1 p + pr2 q - pr2 q : sub_add_left
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pr1 p = pr1 p + pr2 q - pr2 q : add_sub_cancel
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... = pr2 p + pr1 q - pr2 q : Hequiv
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... = pr2 p + (pr1 q - pr2 q) : add_sub_assoc H4
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... = pr1 q - pr2 q + pr2 p : add.comm,
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have H6 : pr1 p - pr2 p = pr1 q - pr2 q, from
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calc
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pr1 p - pr2 p = pr1 q - pr2 q + pr2 p - pr2 p : H5
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... = pr1 q - pr2 q : sub_add_left,
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... = pr1 q - pr2 q : add_sub_cancel,
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abstr_of_ge H3 ⬝ congr_arg of_nat H6 ⬝ (abstr_of_ge H4)⁻¹)
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(assume H2,
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have H3 : pr1 p < pr2 p, from and.elim_left H2,
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have H4 : pr1 q < pr2 q, from and.elim_right H2,
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have H5 : pr2 p = pr2 q - pr1 q + pr1 p, from
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calc
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pr2 p = pr2 p + pr1 q - pr1 q : sub_add_left
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pr2 p = pr2 p + pr1 q - pr1 q : add_sub_cancel
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... = pr1 p + pr2 q - pr1 q : Hequiv
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... = pr1 p + (pr2 q - pr1 q) : add_sub_assoc (le_of_lt H4)
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... = pr2 q - pr1 q + pr1 p : add.comm,
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have H6 : pr2 p - pr1 p = pr2 q - pr1 q, from
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calc
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pr2 p - pr1 p = pr2 q - pr1 q + pr1 p - pr1 p : H5
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... = pr2 q - pr1 q : sub_add_left,
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... = pr2 q - pr1 q : add_sub_cancel,
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abstr_of_lt H3 ⬝ congr_arg neg_succ_of_nat (congr_arg pred H6)⬝ (abstr_of_lt H4)⁻¹)
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theorem equiv_iff (p q : ℕ × ℕ) : (p ≡ q) ↔ ((p ≡ p) ∧ (q ≡ q) ∧ (abstr p = abstr q)) :=
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@ -289,13 +289,13 @@ or.elim (@le_or_gt n m)
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(assume H : m ≥ n,
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calc
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nat_abs (abstr (m, n)) = nat_abs (of_nat (m - n)) : int.abstr_of_ge H
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... = dist m n : dist_ge H)
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... = dist m n : dist_eq_sub_of_ge H)
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(assume H : m < n,
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calc
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nat_abs (abstr (m, n)) = nat_abs (neg_succ_of_nat (pred (n - m))) : int.abstr_of_lt H
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... = succ (pred (n - m)) : rfl
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... = n - m : succ_pred_of_pos (sub_pos_of_gt H)
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... = dist m n : dist_le (le_of_lt H))
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... = n - m : succ_pred_of_pos (sub_pos_of_lt H)
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... = dist m n : dist_eq_sub_of_le (le_of_lt H))
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end
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theorem cases_of_nat (a : ℤ) : (∃n : ℕ, a = of_nat n) ∨ (∃n : ℕ, a = - of_nat n) :=
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@ -418,7 +418,7 @@ calc
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nat_abs (-a) = nat_abs (abstr (repr (-a))) : abstr_repr
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... = nat_abs (abstr (pneg (repr a))) : repr_neg
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... = dist (pr1 (pneg (repr a))) (pr2 (pneg (repr a))) : nat_abs_abstr
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... = dist (pr2 (pneg (repr a))) (pr1 (pneg (repr a))) : dist_comm
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... = dist (pr2 (pneg (repr a))) (pr1 (pneg (repr a))) : dist.comm
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... = nat_abs (abstr (repr a)) : nat_abs_abstr
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... = nat_abs a : abstr_repr
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@ -458,7 +458,8 @@ have H : nat_abs (a + b) = pabs (padd (repr a) (repr b)), from
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... = pabs (padd (repr a) (repr b)) : pabs_congr !repr_add,
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have H1 : nat_abs a = pabs (repr a), from !nat_abs_eq_pabs_repr,
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have H2 : nat_abs b = pabs (repr b), from !nat_abs_eq_pabs_repr,
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have H3 : pabs (padd (repr a) (repr b)) ≤ pabs (repr a) + pabs (repr b), from !dist_add_le_add_dist,
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have H3 : pabs (padd (repr a) (repr b)) ≤ pabs (repr a) + pabs (repr b),
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from !dist_add_add_le_add_dist_dist,
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H⁻¹ ▸ H1⁻¹ ▸ H2⁻¹ ▸ H3
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context
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@ -1,4 +1,4 @@
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/-
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/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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@ -46,10 +46,10 @@ theorem div_rec {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a div b = succ ((a
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divide_def a b ⬝ if_pos (and.intro h₁ h₂)
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theorem div_add_self_right {x z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
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calc (x + z) div z
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= if 0 < z ∧ z ≤ x + z then divide (x + z - z) z + 1 else 0 : !divide_def
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... = divide (x + z - z) z + 1 : if_pos (and.intro H (le_add_left z x))
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... = succ (x div z) : sub_add_left
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calc
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(x + z) div z = if 0 < z ∧ z ≤ x + z then (x + z - z) div z + 1 else 0 : !divide_def
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... = (x + z - z) div z + 1 : if_pos (and.intro H (le_add_left z x))
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... = succ (x div z) : add_sub_cancel
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theorem div_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
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induction_on y
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@ -89,7 +89,7 @@ theorem mod_add_self_right {x z : ℕ} (H : z > 0) : (x + z) mod z = x mod z :=
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calc (x + z) mod z
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= if 0 < z ∧ z ≤ x + z then (x + z - z) mod z else _ : modulo_def
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... = (x + z - z) mod z : if_pos (and.intro H (le_add_left z x))
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... = x mod z : sub_add_left
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... = x mod z : add_sub_cancel
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theorem mod_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
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induction_on y
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@ -167,7 +167,7 @@ by_cases_zero_pos y
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... = ((succ x - y) div y) * y + y + (succ x - y) mod y : H4
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... = ((succ x - y) div y) * y + (succ x - y) mod y + y : add.right_comm
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... = succ x - y + y : {!(IH _ H6)⁻¹}
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... = succ x : add_sub_ge_left H2)⁻¹)))
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... = succ x : sub_add_cancel H2)⁻¹)))
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theorem mod_le {x y : ℕ} : x mod y ≤ x :=
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div_mod_eq⁻¹ ▸ !le_add_left
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@ -272,9 +272,9 @@ have H1 : z * y = x mod y + x div y * y, from
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H ⬝ div_mod_eq ⬝ !add.comm,
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have H2 : (z - x div y) * y = x mod y, from
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calc
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(z - x div y) * y = z * y - x div y * y : mul_sub_distr_right
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(z - x div y) * y = z * y - x div y * y : mul_sub_right_distrib
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... = x mod y + x div y * y - x div y * y : H1
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... = x mod y : sub_add_left,
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... = x mod y : add_sub_cancel,
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show x mod y = 0, from
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by_cases
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(assume yz : y = 0,
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@ -348,10 +348,10 @@ dvd_of_dvd_add_left (!add.comm ▸ H)
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theorem dvd_sub {m n1 n2 : ℕ} (H1 : m | n1) (H2 : m | n2) : m | (n1 - n2) :=
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by_cases
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(assume H3 : n1 ≥ n2,
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have H4 : n1 = n1 - n2 + n2, from (add_sub_ge_left H3)⁻¹,
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have H4 : n1 = n1 - n2 + n2, from (sub_add_cancel H3)⁻¹,
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show m | n1 - n2, from dvd_add_cancel_right (H4 ▸ H1) H2)
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(assume H3 : ¬ (n1 ≥ n2),
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have H4 : n1 - n2 = 0, from le_imp_sub_eq_zero (le_of_lt (lt_of_not_le H3)),
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have H4 : n1 - n2 = 0, from sub_eq_zero_of_le (le_of_lt (lt_of_not_le H3)),
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show m | n1 - n2, from H4⁻¹ ▸ dvd_zero _)
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-- Gcd and lcm
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@ -308,6 +308,11 @@ nat.cases_on n
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assume H : succ n' ≤ m,
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!pred_succ⁻¹ ▸ succ_le_of_le_pred H)
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theorem lt_of_pred_lt_pred {n m : ℕ} (H : pred n < pred m) : n < m :=
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lt_of_not_le
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(take H1 : m ≤ n,
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not_lt_of_le (pred_le_pred_of_le H1) H)
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theorem le_or_eq_succ_of_le_succ {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m :=
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or_of_or_of_imp_left (succ_le_or_eq_of_le H)
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(take H2 : succ n ≤ succ m, show n ≤ m, from le_of_succ_le_succ H2)
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@ -1,11 +1,12 @@
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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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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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-- data.nat.sub
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-- ============
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--
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-- Subtraction on the natural numbers, as well as min, max, and distance.
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Authors: Floris van Doorn, Jeremy Avigad
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Module: data.nat.sub
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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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import tools.fake_simplifier
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@ -15,33 +16,30 @@ open fake_simplifier
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namespace nat
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-- subtraction
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-- -----------
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/- subtraction -/
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theorem sub_zero_right (n : ℕ) : n - 0 = n :=
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theorem sub_zero (n : ℕ) : n - 0 = n :=
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rfl
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theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m) :=
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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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irreducible sub
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theorem sub_zero_left (n : ℕ) : 0 - n = 0 :=
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induction_on n !sub_zero_right
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theorem zero_sub (n : ℕ) : 0 - n = 0 :=
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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_right
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... = pred 0 : {IH}
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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 sub_succ_succ (n m : ℕ) : succ n - succ m = n - m :=
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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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induction_on n !sub_zero_right (take k IH, !sub_succ_succ ⬝ IH)
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induction_on n !sub_zero (take k IH, !succ_sub_succ ⬝ IH)
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theorem sub_add_add_right (n k m : ℕ) : (n + k) - (m + k) = n - m :=
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theorem add_sub_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}
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@ -51,120 +49,112 @@ induction_on k
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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) : !sub_succ_succ
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... = (n + l) - (m + l) : !succ_sub_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 ▸ !add.comm ▸ !sub_add_add_right
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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 sub_add_left (n m : ℕ) : n + m - m = n :=
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theorem add_sub_cancel (n m : ℕ) : n + m - m = n :=
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induction_on m
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(!add_zero⁻¹ ▸ !sub_zero_right)
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(!add_zero⁻¹ ▸ !sub_zero)
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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 : !sub_succ_succ
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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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-- 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 ▸ !sub_add_left
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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) :=
|
||||
induction_on k
|
||||
(calc
|
||||
n - m - 0 = n - m : !sub_zero_right
|
||||
... = n - (m + 0) : {!add_zero⁻¹})
|
||||
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_right
|
||||
... = pred (n - (m + l)) : {IH}
|
||||
... = n - succ (m + l) : !sub_succ_right⁻¹
|
||||
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 (n m k : ℕ) : succ n - m - succ k = n - m - k :=
|
||||
theorem succ_sub_sub_succ (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 - succ (m + k) : {!add_succ}
|
||||
... = n - (m + k) : !sub_succ_succ
|
||||
... = n - m - k : !sub_sub⁻¹
|
||||
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_add_right_eq_zero (n m : ℕ) : n - (n + m) = 0 :=
|
||||
theorem sub_self_add (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
|
||||
... = 0 - m : sub_self
|
||||
... = 0 : zero_sub
|
||||
|
||||
theorem sub_comm (m n k : ℕ) : m - n - k = m - k - n :=
|
||||
theorem sub.right_comm (m n k : ℕ) : 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) : n - 1 = pred n :=
|
||||
calc
|
||||
n - 1 = pred (n - 0) : !sub_succ_right
|
||||
... = pred n : {!sub_zero_right}
|
||||
rfl
|
||||
|
||||
theorem succ_sub_one (n : ℕ) : succ n - 1 = n :=
|
||||
!sub_succ_succ ⬝ !sub_zero_right
|
||||
rfl
|
||||
|
||||
-- add_rewrite sub_add_left
|
||||
|
||||
-- ### interaction with multiplication
|
||||
/- interaction with multiplication -/
|
||||
|
||||
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 : !zero_mul
|
||||
... = 0 - m : !sub_zero_left⁻¹
|
||||
... = 0 * m - m : {!zero_mul⁻¹})
|
||||
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 : !sub_add_left⁻¹
|
||||
... = succ k * m - m : {!succ_mul⁻¹})
|
||||
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) : 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}
|
||||
calc
|
||||
n * pred m = pred m * n : mul.comm
|
||||
... = 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_right_distrib (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 * k - 0 * k : {!zero_mul⁻¹})
|
||||
(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_right}
|
||||
... = (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⁻¹})
|
||||
(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_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_right_distrib
|
||||
... = n * m - k * n : {!mul.comm}
|
||||
... = n * m - n * k : {!mul.comm}
|
||||
|
||||
-- ### interaction with inequalities
|
||||
/- interaction with inequalities -/
|
||||
|
||||
theorem succ_sub {m n : ℕ} : m ≥ n → succ m - n = succ (m - n) :=
|
||||
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})
|
||||
(take k, assume H : 0 ≤ k, rfl)
|
||||
(take k,
|
||||
assume H : succ k ≤ 0,
|
||||
absurd H !not_succ_le_zero)
|
||||
|
|
@ -172,62 +162,52 @@ 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 : succ_sub_succ
|
||||
... = succ (l - k) : IH (le_of_succ_le_succ H)
|
||||
... = succ (succ l - succ k) : {!sub_succ_succ⁻¹})
|
||||
... = succ (succ l - succ k) : succ_sub_succ)
|
||||
|
||||
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
|
||||
theorem sub_eq_zero_of_le {n m : ℕ} (H : n ≤ m) : n - m = 0 :=
|
||||
obtain (k : ℕ) (Hk : n + k = m), from le.elim H, Hk ▸ !sub_self_add
|
||||
|
||||
theorem add_sub_le {n m : ℕ} : n ≤ m → n + (m - n) = m :=
|
||||
theorem add_sub_of_le {n m : ℕ} : 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_right)
|
||||
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) : {!sub_succ_succ}
|
||||
... = succ (k + (l - k)) : !add.succ_left
|
||||
... = succ l : {IH (le_of_succ_le_succ H)})
|
||||
succ k + (succ l - succ k) = succ k + (l - k) : succ_sub_succ
|
||||
... = succ (k + (l - k)) : add.succ_left
|
||||
... = succ l : IH (le_of_succ_le_succ H))
|
||||
|
||||
theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n :=
|
||||
!add.comm ▸ !add_sub_le
|
||||
|
||||
theorem add_sub_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n :=
|
||||
theorem add_sub_of_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n :=
|
||||
calc
|
||||
n + (m - n) = n + 0 : {le_imp_sub_eq_zero H}
|
||||
... = n : !add_zero
|
||||
n + (m - n) = n + 0 : sub_eq_zero_of_le H
|
||||
... = n : add_zero
|
||||
|
||||
theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m :=
|
||||
!add.comm ▸ add_sub_ge
|
||||
theorem sub_add_cancel {n m : ℕ} : n ≥ m → n - m + m = n :=
|
||||
!add.comm ▸ !add_sub_of_le
|
||||
|
||||
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 sub_add_of_le {n m : ℕ} : n ≤ m → n - m + m = m :=
|
||||
!add.comm ▸ add_sub_of_ge
|
||||
|
||||
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)
|
||||
|
||||
theorem sub_split {P : ℕ → Prop} {n m : ℕ} (H1 : n ≤ m → P 0) (H2 : ∀k, m + k = n -> P k)
|
||||
theorem sub.cases {P : ℕ → Prop} {n m : ℕ} (H1 : n ≤ m → P 0) (H2 : ∀k, m + k = n -> P k)
|
||||
: P (n - m) :=
|
||||
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))
|
||||
(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 le_elim_sub {n m : ℕ} (H : n ≤ m) : ∃k, m - k = n :=
|
||||
theorem exists_sub_eq_of_le {n m : ℕ} (H : n ≤ m) : ∃k, m - k = n :=
|
||||
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)
|
||||
m - k = n + k - k : Hk⁻¹
|
||||
... = n : add_sub_cancel)
|
||||
|
||||
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
|
||||
|
|
@ -235,21 +215,21 @@ 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
|
||||
... = n + (m - 0) : sub_zero)
|
||||
(take k : ℕ, 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 : !sub_succ_succ
|
||||
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) : {!sub_succ_succ⁻¹}),
|
||||
... = n + (succ m - succ k) : succ_sub_succ),
|
||||
l1 H
|
||||
|
||||
theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m :=
|
||||
sub_split
|
||||
theorem le_of_sub_eq_zero {n m : ℕ} : n - m = 0 → n ≤ m :=
|
||||
sub.cases
|
||||
(assume H1 : n ≤ m, assume H2 : 0 = 0, H1)
|
||||
(take k : ℕ,
|
||||
assume H1 : m + k = n,
|
||||
|
|
@ -257,38 +237,38 @@ sub_split
|
|||
have H3 : n = m, from !add_zero ▸ H2 ▸ H1⁻¹,
|
||||
H3 ▸ !le.refl)
|
||||
|
||||
theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0)
|
||||
theorem sub_sub.cases {P : ℕ → ℕ → Prop} {n m : ℕ} (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,
|
||||
le_imp_sub_eq_zero H3⁻¹ ▸ (H2 (m - n) (add_sub_le H3⁻¹)))
|
||||
sub_eq_zero_of_le H3⁻¹ ▸ (H2 (m - n) (add_sub_of_le H3⁻¹)))
|
||||
(assume H3 : m ≤ n,
|
||||
le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹)))
|
||||
sub_eq_zero_of_le H3⁻¹ ▸ (H1 (n - m) (add_sub_of_le H3⁻¹)))
|
||||
|
||||
theorem sub_intro {n m k : ℕ} (H : n + m = k) : k - n = m :=
|
||||
theorem sub_eq_of_add_eq {n m k : ℕ} (H : n + m = k) : k - n = m :=
|
||||
have H2 : k - n + n = m + n, from
|
||||
calc
|
||||
k - n + n = k : add_sub_ge_left (le.intro H)
|
||||
k - n + n = k : sub_add_cancel (le.intro H)
|
||||
... = n + m : H⁻¹
|
||||
... = m + n : !add.comm,
|
||||
add.cancel_right H2
|
||||
|
||||
theorem sub_le_right {n m : ℕ} (H : n ≤ m) (k : nat) : n - k ≤ m - k :=
|
||||
theorem sub_le_sub_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n - k ≤ m - k :=
|
||||
obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
|
||||
or.elim !le.total
|
||||
(assume H2 : n ≤ k, (le_imp_sub_eq_zero H2)⁻¹ ▸ !zero_le)
|
||||
(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)⁻¹
|
||||
... = n + l - k : {!add.comm}
|
||||
... = m - k : {Hl},
|
||||
... = l + n - k : add_sub_assoc H2 l
|
||||
... = n + l - k : add.comm
|
||||
... = m - k : Hl,
|
||||
le.intro H3)
|
||||
|
||||
theorem sub_le_left {n m : ℕ} (H : n ≤ m) (k : nat) : k - m ≤ k - n :=
|
||||
theorem sub_le_sub_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k - m ≤ k - n :=
|
||||
obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
|
||||
sub_split
|
||||
sub.cases
|
||||
(assume H2 : k ≤ m, !zero_le)
|
||||
(take m' : ℕ,
|
||||
assume Hm : m + m' = k,
|
||||
|
|
@ -298,33 +278,43 @@ sub_split
|
|||
m' + l + n = n + (m' + l) : add.comm
|
||||
... = n + (l + m') : add.comm
|
||||
... = n + l + m' : add.assoc
|
||||
... = m + m' : {Hl}
|
||||
... = m + m' : Hl
|
||||
... = k : Hm
|
||||
... = k - n + n : (add_sub_ge_left H3)⁻¹,
|
||||
... = k - n + n : sub_add_cancel H3,
|
||||
le.intro (add.cancel_right H4))
|
||||
|
||||
theorem sub_pos_of_gt {m n : ℕ} (H : n > m) : n - m > 0 :=
|
||||
have H1 : n = n - m + m, from (add_sub_ge_left (le_of_lt H))⁻¹,
|
||||
theorem sub_pos_of_lt {m n : 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 sub_lt_cancel_right {n m k : ℕ) (H : n - k < m - k) : n < m
|
||||
-- :=
|
||||
-- _
|
||||
theorem lt_of_sub_pos {m n : ℕ} (H : n - m > 0) : m < n :=
|
||||
lt_of_not_le
|
||||
(take H1 : m ≥ n,
|
||||
have H2 : n - m = 0, from sub_eq_zero_of_le H1,
|
||||
!lt.irrefl (H2 ▸ H))
|
||||
|
||||
-- theorem sub_lt_cancel_left {n m k : ℕ) (H : n - m < n - k) : k < m
|
||||
-- :=
|
||||
-- _
|
||||
theorem lt_of_sub_lt_sub_right {n m k : ℕ} (H : n - k < m - k) : n < m :=
|
||||
lt_of_not_le
|
||||
(assume H1 : m ≤ n,
|
||||
have H2 : m - k ≤ n - k, from sub_le_sub_right H1 _,
|
||||
not_le_of_lt H H2)
|
||||
|
||||
theorem sub_triangle_inequality (n m k : ℕ) : n - k ≤ (n - m) + (m - k) :=
|
||||
sub_split
|
||||
(assume H : n ≤ m, !zero_add⁻¹ ▸ sub_le_right H k)
|
||||
theorem lt_of_sub_lt_sub_left {n m k : ℕ} (H : n - m < n - k) : k < m :=
|
||||
lt_of_not_le
|
||||
(assume H1 : m ≤ k,
|
||||
have H2 : n - k ≤ n - m, from sub_le_sub_left H1 _,
|
||||
not_le_of_lt H H2)
|
||||
|
||||
theorem sub_lt_sub_add_sub (n m k : ℕ) : n - k ≤ (n - m) + (m - k) :=
|
||||
sub.cases
|
||||
(assume H : n ≤ m, !zero_add⁻¹ ▸ sub_le_sub_right H k)
|
||||
(take mn : ℕ,
|
||||
assume Hmn : m + mn = n,
|
||||
sub_split
|
||||
sub.cases
|
||||
(assume H : m ≤ k,
|
||||
have H2 : n - k ≤ n - m, from sub_le_left H n,
|
||||
have H3 : n - k ≤ mn, from sub_intro Hmn ▸ H2,
|
||||
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 : ℕ,
|
||||
assume Hkm : k + km = m,
|
||||
|
|
@ -334,23 +324,14 @@ sub_split
|
|||
... = k + km + mn : add.assoc
|
||||
... = m + mn : Hkm
|
||||
... = n : Hmn,
|
||||
have H2 : n - k = mn + km, from sub_intro H,
|
||||
have H2 : n - k = mn + km, from sub_eq_of_add_eq H,
|
||||
H2 ▸ !le.refl))
|
||||
|
||||
|
||||
-- add_rewrite sub_self mul_sub_distr_left mul_sub_distr_right
|
||||
|
||||
|
||||
-- absolute difference
|
||||
-- --------------------------------------------
|
||||
|
||||
-- ### absolute difference
|
||||
|
||||
-- This section is still incomplete
|
||||
/- distance -/
|
||||
|
||||
definition dist [reducible] (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 :=
|
||||
|
|
@ -359,78 +340,75 @@ calc
|
|||
... = 0 + 0 : sub_self
|
||||
... = 0 : rfl
|
||||
|
||||
theorem dist_eq_zero {n m : ℕ} (H : dist n m = 0) : n = m :=
|
||||
theorem eq_of_dist_eq_zero {n m : ℕ} (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 sub_eq_zero_imp_le H2,
|
||||
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 sub_eq_zero_imp_le H4,
|
||||
have H5 : m ≤ n, from le_of_sub_eq_zero H4,
|
||||
le.antisymm H3 H5
|
||||
|
||||
theorem dist_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n :=
|
||||
theorem dist_eq_sub_of_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n :=
|
||||
calc
|
||||
dist n m = 0 + (m - n) : {le_imp_sub_eq_zero H}
|
||||
... = m - n : !zero_add
|
||||
dist n m = 0 + (m - n) : {sub_eq_zero_of_le H}
|
||||
... = m - n : zero_add
|
||||
|
||||
theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m :=
|
||||
!dist_comm ▸ dist_le H
|
||||
theorem dist_eq_sub_of_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m :=
|
||||
!dist.comm ▸ dist_eq_sub_of_le H
|
||||
|
||||
theorem dist_zero_right (n : ℕ) : dist n 0 = n :=
|
||||
dist_ge !zero_le ⬝ !sub_zero_right
|
||||
dist_eq_sub_of_ge !zero_le ⬝ !sub_zero
|
||||
|
||||
theorem dist_zero_left (n : ℕ) : dist 0 n = n :=
|
||||
dist_le !zero_le ⬝ !sub_zero_right
|
||||
dist_eq_sub_of_le !zero_le ⬝ !sub_zero
|
||||
|
||||
theorem dist_intro {n m k : ℕ} (H : n + m = k) : dist k n = m :=
|
||||
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
|
||||
dist k n = k - n : dist_eq_sub_of_ge (le.intro H)
|
||||
... = m : sub_eq_of_add_eq H
|
||||
|
||||
theorem dist_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m :=
|
||||
theorem dist_add_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)) : add_sub_add_right
|
||||
... = (n - m) + (m - n) : add_sub_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_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m :=
|
||||
!add.comm ▸ !add.comm ▸ !dist_add_add_right
|
||||
|
||||
-- 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 :=
|
||||
theorem dist_add_eq_of_ge {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
|
||||
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 : 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 (k + l) (k + m) : {H}
|
||||
... = dist l m : !dist_add_left
|
||||
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_move_add {n m : ℕ} (H : n ≥ m) (k : ℕ) : dist (n - m) k = dist n (k + m) :=
|
||||
theorem dist_sub_eq_dist_add_left {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) : add.comm
|
||||
... = n - m + m + k : add.assoc
|
||||
... = n + k : {add_sub_ge_left H}
|
||||
... = n + k : sub_add_cancel H
|
||||
... = k + n : add.comm,
|
||||
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
|
||||
theorem dist_sub_eq_dist_add_right {k m : ℕ} (H : k ≥ m) (n : ℕ) :
|
||||
dist n (k - m) = dist (n + m) k :=
|
||||
(dist_sub_eq_dist_add_left 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 dist.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_add !sub_triangle_inequality !sub_triangle_inequality
|
||||
H ▸ add_le_add !sub_lt_sub_add_sub !sub_lt_sub_add_sub
|
||||
|
||||
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_add_le_add_dist_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
|
||||
|
||||
--interaction with multiplication
|
||||
!dist_add_add_left ▸ !dist_add_add_right ▸ rfl,
|
||||
H ▸ !dist.triangle_inequality
|
||||
|
||||
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,
|
||||
|
|
@ -440,9 +418,6 @@ 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) :=
|
||||
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,
|
||||
|
|
@ -450,15 +425,15 @@ have aux : ∀k l, k ≥ l → dist n m * dist k l = dist (n * k + m * l) (n * l
|
|||
have H2 : m * k ≥ m * l, from mul_le_mul_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)) : !dist_mul_right⁻¹
|
||||
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
|
||||
... = 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_mul_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 _,
|
||||
... = dist (n * k) (m * k - m * l + n * l) : dist_sub_eq_dist_add_left (mul_le_mul_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)
|
||||
... = 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 : k ≤ l, !dist.comm ▸ !dist.comm ▸ aux l k H)
|
||||
(assume H : l ≤ k, aux k l H)
|
||||
|
||||
end nat
|
||||
|
|
|
|||
Loading…
Reference in a new issue