fix(library/data/nat/order): delete unused material at end of file

library/data/int/order.lean

@@ -481,532 +481,4 @@ calc
   of_nat (nat_abs a) = of_nat (nat_abs (-a)) : nat_abs_neg
                  ... = -a                    : of_nat_nat_abs_of_nonneg H1
 
-exit
-
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-
--- ### interaction with add
-
-theorem le_add_of_nat_right (a : ℤ) (n : ℕ) : a ≤ a + n :=
-le.intro (eq.refl (a + n))
-
-theorem le_add_of_nat_left (a : ℤ) (n : ℕ) : a ≤ n + a :=
-le.intro (add.comm a n)
-
-
-theorem add_le_right {a b : ℤ} (H : a ≤ b) (c : ℤ) : a + c ≤ b + c :=
-add.comm c b ▸ add.comm c a ▸ add_le_add_left H c
-
-theorem add_le {a b c d : ℤ} (H1 : a ≤ b) (H2 : c ≤ d) : a + c ≤ b + d :=
-le_trans (add_le_right H1 c) (add_le_add_left H2 b)
-
-theorem add_le_cancel_right {a b c : ℤ} (H : a + c ≤ b + c) : a ≤ b :=
-have H1 : a + c + -c ≤ b + c + -c, from add_le_right H (-c),
-!add_neg_cancel_right ▸ !add_neg_cancel_right ▸ H1
-
-theorem add_le_cancel_left {a b c : ℤ} (H : c + a ≤ c + b) : a ≤ b :=
-add_le_cancel_right (add.comm c b ▸ add.comm c a ▸ H)
-
-theorem add_le_inv {a b c d : ℤ} (H1 : a + b ≤ c + d) (H2 : c ≤ a) : b ≤ d :=
-obtain (n : ℕ) (Hn : c + n = a), from le.elim H2,
-have H3 : c + (n + b) ≤ c + d, from add.assoc c n b ▸ Hn⁻¹ ▸ H1,
-have H4 : n + b ≤ d, from add_le_cancel_left H3,
-show b ≤ d, from le_trans (le_add_of_nat_left b n) H4
-
-theorem le_add_of_nat_right_trans {a b : ℤ} (H : a ≤ b) (n : ℕ) : a ≤ b + n :=
-le_trans H (le_add_of_nat_right b n)
-
-theorem le_imp_succ_le_or_eq {a b : ℤ} (H : a ≤ b) : a + 1 ≤ b ∨ a = b :=
-obtain (n : ℕ) (Hn : a + n = b), from le.elim H,
-discriminate
-  (assume H2 : n = 0,
-    have H3 : a = b, from
-      calc
-        a = a + 0 : (add_zero a)⁻¹
-          ... = a + n : {H2⁻¹}
-          ... = b : Hn,
-    or.inr H3)
-  (take k : ℕ,
-    assume H2 : n = succ k,
-    have H3 : a + 1 + k = b, from
-      calc
-        a + 1 + k = a + succ k : by simp
-          ... = a + n : by simp
-          ... = b : Hn,
-    or.inl (le.intro H3))
-
--- ### interaction with neg and sub
-
-theorem le_neg {a b : ℤ} (H : a ≤ b) : -b ≤ -a :=
-obtain (n : ℕ) (Hn : a + n = b), from le.elim H,
-have H2 : b - n = a, from (iff.mp !add_eq_iff_eq_add_neg Hn)⁻¹,
-have H3 : -b + n = -a, from
-  calc
-    -b + n = -b + -(-n) : {(neg_neg n)⁻¹}
-      ... = -(b + -n) : (neg_add_distrib b (-n))⁻¹
-      ... = -a : {H2},
-le.intro H3
-
-theorem neg_le_zero {a : ℤ} (H : 0 ≤ a) : -a ≤ 0 :=
-neg_zero ▸ (le_neg H)
-
-theorem zero_le_neg {a : ℤ} (H : a ≤ 0) : 0 ≤ -a :=
-neg_zero ▸ (le_neg H)
-
-theorem le_neg_inv {a b : ℤ} (H : -a ≤ -b) : b ≤ a :=
-neg_neg b ▸ neg_neg a ▸ le_neg H
-
-theorem le_sub_of_nat (a : ℤ) (n : ℕ) : a - n ≤ a :=
-le.intro (neg_add_cancel_right a n)
-
-theorem sub_le_right {a b : ℤ} (H : a ≤ b) (c : ℤ) : a - c ≤ b - c :=
-add_le_right H _
-
-theorem sub_le_left {a b : ℤ} (H : a ≤ b) (c : ℤ) : c - b ≤ c - a :=
-add_le_add_left (le_neg H) _
-
-theorem sub_le {a b c d : ℤ} (H1 : a ≤ b) (H2 : d ≤ c) : a - c ≤ b - d :=
-add_le H1 (le_neg H2)
-
-theorem sub_le_right_inv {a b c : ℤ} (H : a - c ≤ b - c) : a ≤ b :=
-add_le_cancel_right H
-
-theorem sub_le_left_inv {a b c : ℤ} (H : c - a ≤ c - b) : b ≤ a :=
-le_neg_inv (add_le_cancel_left H)
-
-theorem le_iff_sub_nonneg (a b : ℤ) : a ≤ b ↔ 0 ≤ b - a :=
-iff.intro
-  (assume H, !sub_self ▸ sub_le_right H a)
-  (assume H,
-     have H1 : a ≤ b - a + a, from zero_add a ▸ add_le_right H a,
-     !neg_add_cancel_right ▸ H1)
-
-
--- Less than, Greater than, Greater than or equal
--- ----------------------------------------------
-
-definition ge (a b : ℤ) := b ≤ a
-notation a >= b := int.ge a b
-notation a ≥ b  := int.ge a b
-
-definition gt (a b : ℤ) := b < a
-notation a > b  := int.gt a b
-
-theorem lt_def (a b : ℤ) : a < b ↔ a + 1 ≤ b :=
-iff.refl (a < b)
-
-theorem gt_def (n m : ℕ) : n > m ↔ m < n :=
-iff.refl (n > m)
-
-theorem ge_def (n m : ℕ) : n ≥ m ↔ m ≤ n :=
-iff.refl (n ≥ m)
-
--- add_rewrite gt_def ge_def --it might be possible to remove this in Lean 0.2
-
-
--- -- ### basic facts
-
-theorem gt_of_nat (n m : ℕ) : (of_nat n > of_nat m) ↔ (n > m) :=
-of_nat_lt_of_nat m n
-
--- ### interaction with le
-
-
-theorem le_imp_lt_or_eq {a b : ℤ} (H : a ≤ b) : a < b ∨ a = b :=
-le_imp_succ_le_or_eq H
-
-theorem le_ne_imp_lt {a b : ℤ} (H1 : a ≤ b)  (H2 : a ≠ b) : a < b :=
-or_resolve_left (le_imp_lt_or_eq H1) H2
-
-theorem le_imp_lt_succ {a b : ℤ} (H : a ≤ b) : a < b + 1 :=
-add_le_right H 1
-
-theorem lt_succ_imp_le {a b : ℤ} (H : a < b + 1) : a ≤ b :=
-add_le_cancel_right H
-
-
--- ### transitivity, antisymmmetry
-
-theorem lt_le_trans {a b c : ℤ} (H1 : a < b) (H2 : b ≤ c) : a < c :=
-le_trans H1 H2
-
-theorem le_lt_trans {a b c : ℤ} (H1 : a ≤ b) (H2 : b < c) : a < c :=
-le_trans (add_le_right H1 1) H2
-
-theorem lt_trans {a b c : ℤ} (H1 : a < b) (H2 : b < c) : a < c :=
-lt_le_trans H1 (le_of_lt H2)
-
-theorem le_imp_not_gt {a b : ℤ} (H : a ≤ b) : ¬ a > b :=
-(assume H2 : a > b, absurd (le_lt_trans H H2) (lt.irrefl a))
-
-theorem lt_imp_not_ge {a b : ℤ} (H : a < b) : ¬ a ≥ b :=
-(assume H2 : a ≥ b, absurd (lt_le_trans H H2) (lt.irrefl a))
-
-theorem lt_antisym {a b : ℤ} (H : a < b) : ¬ b < a :=
-le_imp_not_gt (le_of_lt H)
-
--- ### interaction with addition
-
--- TODO: note: no longer works without the "show"
-theorem add_lt_left {a b : ℤ} (H : a < b) (c : ℤ) : c + a < c + b :=
-show (c + a) + 1 ≤ c + b, from (add.assoc c a 1)⁻¹ ▸ add_le_add_left H c
-
-theorem add_lt_right {a b : ℤ} (H : a < b) (c : ℤ) : a + c < b + c :=
-add.comm c b ▸ add.comm c a ▸ add_lt_left H c
-
-theorem add_le_lt {a b c d : ℤ} (H1 : a ≤ c) (H2 : b < d) : a + b < c + d :=
-le_lt_trans (add_le_right H1 b) (add_lt_left H2 c)
-
-theorem add_lt_le {a b c d : ℤ} (H1 : a < c) (H2 : b ≤ d) : a + b < c + d :=
-lt_le_trans (add_lt_right H1 b) (add_le_add_left H2 c)
-
-theorem add_lt {a b c d : ℤ} (H1 : a < c) (H2 : b < d) : a + b < c + d :=
-add_lt_le H1 (le_of_lt H2)
-
-theorem add_lt_cancel_left {a b c : ℤ} (H : c + a < c + b) : a < b :=
-show a + 1 ≤ b, from add_le_cancel_left (add.assoc c a 1 ▸ H)
-
-theorem add_lt_cancel_right {a b c : ℤ} (H : a + c < b + c) : a < b :=
-add_lt_cancel_left (add.comm b c ▸ add.comm a c ▸ H)
-
--- ### interaction with neg and sub
-
-theorem lt_neg {a b : ℤ} (H : a < b) : -b < -a :=
-have H2 : -(a + 1) + 1 = -a, by simp,
-have H3 : -b ≤ -(a + 1), from le_neg H,
-have H4 : -b + 1 ≤ -(a + 1) + 1, from add_le_right H3 1,
-H2 ▸ H4
-
-theorem neg_lt_zero {a : ℤ} (H : 0 < a) : -a < 0 :=
-neg_zero ▸ lt_neg H
-
-theorem zero_lt_neg {a : ℤ} (H : a < 0) : 0 < -a :=
-neg_zero ▸ lt_neg H
-
-theorem lt_neg_inv {a b : ℤ} (H : -a < -b) : b < a :=
-neg_neg b ▸ neg_neg a ▸ lt_neg H
-
-theorem lt_sub_of_nat_succ (a : ℤ) (n : ℕ) : a - succ n < a :=
-lt.intro (neg_add_cancel_right a (succ n))
-
-theorem sub_lt_right {a b : ℤ} (H : a < b) (c : ℤ) : a - c < b - c :=
-add_lt_right H _
-
-theorem sub_lt_left {a b : ℤ} (H : a < b) (c : ℤ) : c - b < c - a :=
-add_lt_left (lt_neg H) _
-
-theorem sub_lt {a b c d : ℤ} (H1 : a < b) (H2 : d < c) : a - c < b - d :=
-add_lt H1 (lt_neg H2)
-
-theorem sub_lt_right_inv {a b c : ℤ} (H : a - c < b - c) : a < b :=
-add_lt_cancel_right H
-
-theorem sub_lt_left_inv {a b c : ℤ} (H : c - a < c - b) : b < a :=
-lt_neg_inv (add_lt_cancel_left H)
-
--- ### totality of lt and le
-
--- add_rewrite succ_pos zero_le --move some of these to nat.lean
--- add_rewrite le_of_nat lt_of_nat gt_of_nat --remove gt_of_nat in Lean 0.2
--- add_rewrite le_neg lt_neg neg_le_zero zero_le_neg zero_lt_neg neg_lt_zero
-
-theorem neg_le_pos (n m : ℕ) : -n ≤ m :=
-have H1 : of_nat 0 ≤ of_nat m, by simp,
-have H2 : -n ≤ 0, by simp,
-le_trans H2 H1
-
-theorem le_or_gt (a b : ℤ) : a ≤ b ∨ a > b :=
-by_cases_of_nat a
-  (take n : ℕ,
-    by_cases_of_nat_succ b
-      (take m : ℕ,
-        show of_nat n ≤ m ∨ of_nat n > m, from
-        proof
-          or.elim (@nat.le_or_gt n m)
-            (assume H : n ≤ m, or.inl (iff.mp' !of_nat_le_of_nat H))
-            (assume H : n > m, or.inr (iff.mp' !of_nat_lt_of_nat H))
-        qed)
-      (take m : ℕ,
-        show n ≤ -succ m ∨ n > -succ m, from
-          have H0 : -succ m < -m, from lt_neg ((of_nat_succ m)⁻¹ ▸ lt_succ m),
-          have H : -succ m < n, from lt_le_trans H0 (neg_le_pos m n),
-          or.inr H))
-  (take n : ℕ,
-    by_cases_of_nat_succ b
-      (take m : ℕ,
-        show -n ≤ m ∨ -n > m, from
-          or.inl (neg_le_pos n m))
-      (take m : ℕ,
-         show -n ≤ -succ m ∨ -n > -succ m, from
-          or_of_or_of_imp_of_imp le_or_gt
-            (assume H : succ m ≤ n,
-              le_neg (iff.elim_left (iff.symm (of_nat_le_of_nat (succ m) n)) H))
-            (assume H : succ m > n,
-              lt_neg (iff.elim_left (iff.symm (of_nat_lt_of_nat n (succ m))) H))))
-
-theorem trichotomy_alt (a b : ℤ) : (a < b ∨ a = b) ∨ a > b :=
-or_of_or_of_imp_left (le_or_gt a b) (assume H : a ≤ b, le_imp_lt_or_eq H)
-
-theorem trichotomy (a b : ℤ) : a < b ∨ a = b ∨ a > b :=
-iff.elim_left or.assoc (trichotomy_alt a b)
-
-theorem le_total (a b : ℤ) : a ≤ b ∨ b ≤ a :=
-or_of_or_of_imp_right (le_or_gt a b) (assume H : b < a, le_of_lt H)
-
-theorem not_le_of_lt {a b : ℤ} (H : ¬ a < b) : b ≤ a :=
-or_resolve_left (le_or_gt b a) H
-
-theorem not_le_imp_lt {a b : ℤ} (H : ¬ a ≤ b) : b < a :=
-or_resolve_right (le_or_gt a b) H
-
--- (non)positivity and (non)negativity
--- -------------------------------------
-
--- ### basic
-
--- see also "int_by_cases" and similar theorems
-
-theorem pos_imp_exists_nat {a : ℤ} (H : a ≥ 0) : ∃n : ℕ, a = n :=
-obtain (n : ℕ) (Hn : of_nat 0 + n = a), from le.elim H,
-exists.intro n (Hn⁻¹ ⬝ zero_add n)
-
-theorem neg_imp_exists_nat {a : ℤ} (H : a ≤ 0) : ∃n : ℕ, a = -n :=
-have H2 : -a ≥ 0, from zero_le_neg H,
-obtain (n : ℕ) (Hn : -a = n), from pos_imp_exists_nat H2,
-have H3 : a = -n, from (eq_neg_of_eq_neg (Hn⁻¹)),
-exists.intro n H3
-
-theorem nat_abs_nonneg_eq {a : ℤ} (H : a ≥ 0) : (nat_abs a) = a :=
-obtain (n : ℕ) (Hn : a = n), from pos_imp_exists_nat H,
-Hn⁻¹ ▸ congr_arg of_nat (nat_abs_of_nat n)
-
-theorem of_nat_nonneg (n : ℕ) : of_nat n ≥ 0 :=
-iff.mp (iff.symm !of_nat_le_of_nat) !zero_le
-
-definition ge_decidable [instance] {a b : ℤ} : decidable (a ≥ b) := _
-definition lt_decidable [instance] {a b : ℤ} : decidable (a < b) := _
-definition gt_decidable [instance] {a b : ℤ} : decidable (a > b) := _
-
---nat_abs_neg is already taken... rename?
-theorem nat_abs_negative {a : ℤ} (H : a ≤ 0) : (nat_abs a) = -a :=
-obtain (n : ℕ) (Hn : a = -n), from neg_imp_exists_nat H,
-calc
-  (nat_abs a) = (nat_abs (-n)) : {Hn}
-  ... = (nat_abs n) : nat_abs_neg
-  ... = n : {nat_abs_of_nat n}
-  ... = -a : (eq_neg_of_eq_neg Hn)⁻¹
-
-theorem nat_abs_cases (a : ℤ) : a = (nat_abs a) ∨ a = - (nat_abs a) :=
-or_of_or_of_imp_of_imp (le_total 0 a)
-  (assume H : a ≥ 0, (nat_abs_nonneg_eq H)⁻¹)
-  (assume H : a ≤ 0, (eq_neg_of_eq_neg (nat_abs_negative H)))
-
--- ### interaction of mul with le and lt
-
-theorem mul_le_left_nonneg {a b c : ℤ} (Ha : a ≥ 0) (H : b ≤ c) : a * b ≤ a * c :=
-obtain (n : ℕ) (Hn : b + n = c), from le.elim H,
-have H2 : a * b + of_nat ((nat_abs a) * n) = a * c, from
-  calc
-    a * b + of_nat ((nat_abs a) * n) = a * b + (nat_abs a) * of_nat n : by simp
-      ... = a * b + a * n : {nat_abs_nonneg_eq Ha}
-      ... = a * (b + n) : by simp
-      ... = a * c : by simp,
-le.intro H2
-
-theorem mul_le_right_nonneg {a b c : ℤ} (Hb : b ≥ 0) (H : a ≤ c) : a * b ≤ c * b :=
-!mul.comm ▸ !mul.comm ▸ mul_le_left_nonneg Hb H
-
-theorem mul_le_left_nonpos {a b c : ℤ} (Ha : a ≤ 0) (H : b ≤ c) : a * c ≤ a * b :=
-have H2 : -a * b ≤ -a * c, from mul_le_left_nonneg (zero_le_neg Ha) H,
-have H3 : -(a * b) ≤ -(a * c), from !neg_mul_eq_neg_mul⁻¹ ▸ !neg_mul_eq_neg_mul⁻¹ ▸ H2,
-le_neg_inv H3
-
-theorem mul_le_right_nonpos {a b c : ℤ} (Hb : b ≤ 0) (H : c ≤ a) : a * b ≤ c * b :=
-!mul.comm ▸ !mul.comm ▸ mul_le_left_nonpos Hb H
-
----this theorem can be made more general by replacing either Ha with 0 ≤ a or Hb with 0 ≤ d...
-theorem mul_le_nonneg {a b c d : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) (Hc : a ≤ c) (Hd : b ≤ d)
-  : a * b ≤ c * d :=
-le_trans (mul_le_right_nonneg Hb Hc) (mul_le_left_nonneg (le_trans Ha Hc) Hd)
-
-theorem mul_le_nonpos {a b c d : ℤ} (Ha : a ≤ 0) (Hb :b ≤ 0) (Hc : c ≤ a) (Hd : d ≤ b)
-  : a * b ≤ c * d :=
-le_trans (mul_le_right_nonpos Hb Hc) (mul_le_left_nonpos (le_trans Hc Ha) Hd)
-
-theorem mul_lt_left_pos {a b c : ℤ} (Ha : a > 0) (H : b < c) : a * b < a * c :=
-have H2 : a * b < a * b + a, from add_zero (a * b) ▸ add_lt_left Ha (a * b),
-have H3 : a * b + a ≤ a * c, from (by simp) ▸ mul_le_left_nonneg (le_of_lt Ha) H,
-lt_le_trans H2 H3
-
-theorem mul_lt_right_pos {a b c : ℤ} (Hb : b > 0) (H : a < c) : a * b < c * b :=
-mul.comm b c ▸ mul.comm b a ▸ mul_lt_left_pos Hb H
-
-theorem mul_lt_left_neg {a b c : ℤ} (Ha : a < 0) (H : b < c) : a * c < a * b :=
-have H2 : -a * b < -a * c, from mul_lt_left_pos (zero_lt_neg Ha) H,
-have H3 : -(a * b) < -(a * c), from !neg_mul_eq_neg_mul⁻¹ ▸ !neg_mul_eq_neg_mul⁻¹ ▸ H2,
-lt_neg_inv H3
-
-theorem mul_lt_right_neg {a b c : ℤ} (Hb : b < 0) (H : c < a) : a * b < c * b :=
-!mul.comm ▸ !mul.comm ▸ mul_lt_left_neg Hb H
-
-theorem mul_le_lt_pos {a b c d : ℤ} (Ha : a > 0) (Hb : b ≥ 0) (Hc : a ≤ c) (Hd : b < d)
-  : a * b < c * d :=
-le_lt_trans (mul_le_right_nonneg Hb Hc) (mul_lt_left_pos (lt_le_trans Ha Hc) Hd)
-
-theorem mul_lt_le_pos {a b c d : ℤ} (Ha : a ≥ 0) (Hb : b > 0) (Hc : a < c) (Hd : b ≤ d)
-  : a * b < c * d :=
-lt_le_trans (mul_lt_right_pos Hb Hc) (mul_le_left_nonneg (le_trans Ha (le_of_lt Hc)) Hd)
-
-theorem mul_lt_pos {a b c d : ℤ} (Ha : a > 0) (Hb : b > 0) (Hc : a < c) (Hd : b < d)
-  : a * b < c * d :=
-mul_lt_le_pos (le_of_lt Ha) Hb Hc (le_of_lt Hd)
-
-theorem mul_lt_neg {a b c d : ℤ} (Ha : a < 0) (Hb : b < 0) (Hc : c < a) (Hd : d < b)
-  : a * b < c * d :=
-lt_trans (mul_lt_right_neg Hb Hc) (mul_lt_left_neg (lt_trans Hc Ha) Hd)
-
--- theorem mul_le_lt_neg and mul_lt_le_neg?
-
-theorem mul_lt_cancel_left_nonneg {a b c : ℤ} (Hc : c ≥ 0) (H : c * a < c * b) : a < b :=
-or.elim (le_or_gt b a)
-  (assume H2 : b ≤ a,
-    have H3 : c * b ≤ c * a, from mul_le_left_nonneg Hc H2,
-    absurd H3 (lt_imp_not_ge H))
-  (assume H2 : a < b, H2)
-
-theorem mul_lt_cancel_right_nonneg {a b c : ℤ} (Hc : c ≥ 0) (H : a * c < b * c) : a < b :=
-mul_lt_cancel_left_nonneg Hc (mul.comm b c ▸ mul.comm a c ▸ H)
-
-theorem mul_lt_cancel_left_nonpos {a b c : ℤ} (Hc : c ≤ 0) (H : c * b < c * a) : a < b :=
-have H2 : -(c * a) < -(c * b), from lt_neg H,
-have H3 : -c * a < -c * b, from !neg_mul_eq_neg_mul ▸ !neg_mul_eq_neg_mul ▸ H2,
-have H4 : -c ≥ 0, from zero_le_neg Hc,
-mul_lt_cancel_left_nonneg H4 H3
-
-theorem mul_lt_cancel_right_nonpos {a b c : ℤ} (Hc : c ≤ 0) (H : b * c < a * c) : a < b :=
-mul_lt_cancel_left_nonpos Hc (!mul.comm ▸ !mul.comm ▸ H)
-
-theorem mul_le_cancel_left_pos {a b c : ℤ} (Hc : c > 0) (H : c * a ≤ c * b) : a ≤ b :=
-or.elim (le_or_gt a b)
-  (assume H2 : a ≤ b, H2)
-  (assume H2 : a > b,
-    have H3 : c * a > c * b, from mul_lt_left_pos Hc H2,
-    absurd H3 (le_imp_not_gt H))
-
-theorem mul_le_cancel_right_pos {a b c : ℤ} (Hc : c > 0) (H : a * c ≤ b * c) : a ≤ b :=
-mul_le_cancel_left_pos Hc (!mul.comm ▸ !mul.comm ▸ H)
-
-theorem mul_le_cancel_left_neg {a b c : ℤ} (Hc : c < 0) (H : c * b ≤ c * a) : a ≤ b :=
-have H2 : -(c * a) ≤ -(c * b), from le_neg H,
-have H3 : -c * a ≤ -c * b,
-  from neg_mul_eq_neg_mul c b ▸ neg_mul_eq_neg_mul c a ▸ H2,
-have H4 : -c > 0, from zero_lt_neg Hc,
-mul_le_cancel_left_pos H4 H3
-
-theorem mul_le_cancel_right_neg {a b c : ℤ} (Hc : c < 0) (H : b * c ≤ a * c) : a ≤ b :=
-mul_le_cancel_left_neg Hc (!mul.comm ▸ !mul.comm ▸ H)
-
-theorem mul_eq_one_left {a b : ℤ} (H : a * b = 1) : a = 1 ∨ a = - 1 :=
-have H2 : (nat_abs a) * (nat_abs b) = 1, from
-  calc
-    (nat_abs a) * (nat_abs b) = (nat_abs (a * b)) : !mul_nat_abs⁻¹
-                        ... = (nat_abs 1)       : {H}
-                        ... = 1                : nat_abs_of_nat 1,
-have H3 : (nat_abs a) = 1, from mul_eq_one_left H2,
-or_of_or_of_imp_of_imp (nat_abs_cases a)
-  (assume H4 : a = (nat_abs a), H3 ▸ H4)
-  (assume H4 : a = - (nat_abs a), H3 ▸ H4)
-
-theorem mul_eq_one_right {a b : ℤ} (H : a * b = 1) : b = 1 ∨ b = - 1 :=
-mul_eq_one_left (!mul.comm ▸ H)
-
-
--- sign function
--- -------------
-
-definition sign (a : ℤ) : ℤ := if a > 0 then 1 else (if a < 0 then - 1 else 0)
-
-theorem sign_pos {a : ℤ} (H : a > 0) : sign a = 1 :=
-if_pos H
-
-theorem sign_negative {a : ℤ} (H : a < 0) : sign a = - 1 :=
-if_neg (lt_antisym H) ⬝ if_pos H
-
-theorem sign_zero : sign 0 = 0 :=
-if_neg (lt.irrefl 0) ⬝ if_neg (lt.irrefl 0)
-
--- add_rewrite sign_negative sign_pos nat_abs_negative nat_abs_nonneg_eq sign_zero mul_nat_abs
-
-theorem mul_sign_nat_abs (a : ℤ) : sign a * (nat_abs a) = a :=
-have temp1 : ∀a : ℤ, a < 0 → a ≤ 0, from take a, le_of_lt,
-have temp2 : ∀a : ℤ, a > 0 → a ≥ 0, from take a, le_of_lt,
-or.elim3 (trichotomy a 0)
-  (assume H : a < 0, by simp)
-  (assume H : a = 0, by simp)
-  (assume H : a > 0, by simp)
-
--- TODO: show decidable for equality (and avoid classical library)
-theorem sign_mul (a b : ℤ) : sign (a * b) = sign a * sign b :=
-or.elim (em (a = 0))
-  (assume Ha : a = 0, by simp)
-  (assume Ha : a ≠ 0,
-    or.elim (em (b = 0))
-      (assume Hb : b = 0, by simp)
-      (assume Hb : b ≠ 0,
-        have H : sign (a * b) * (nat_abs (a * b)) = sign a * sign b  * (nat_abs (a * b)), from
-          calc
-            sign (a * b) * (nat_abs (a * b)) = a * b : mul_sign_nat_abs (a * b)
-              ... = sign a * (nat_abs a) * b : {(mul_sign_nat_abs a)⁻¹}
-              ... = sign a * (nat_abs a) * (sign b * (nat_abs b)) : {(mul_sign_nat_abs b)⁻¹}
-              ... = sign a * sign b  * (nat_abs (a * b)) : by simp,
-        have H2 : (nat_abs (a * b)) ≠ 0, from
-          take H2', mul_ne_zero Ha Hb (nat_abs_eq_zero H2'),
-        have H3 : (nat_abs (a * b)) ≠ of_nat 0, from mt of_nat_inj H2,
-        mul.cancel_right H3 H))
-
-theorem sign_idempotent (a : ℤ) : sign (sign a) = sign a :=
-have temp : of_nat 1 > 0, from iff.elim_left (iff.symm (of_nat_lt_of_nat 0 1)) !succ_pos,
-    --this should be done with simp
-or.elim3 (trichotomy a 0) sorry sorry sorry
---  (by simp)
---  (by simp)
---  (by simp)
-
-theorem sign_succ (n : ℕ) : sign (succ n) = 1 :=
-sign_pos (iff.elim_left (iff.symm (of_nat_lt_of_nat 0 (succ n))) !succ_pos)
-  --this should be done with simp
-
-theorem sign_neg (a : ℤ) : sign (-a) = - sign a :=
-have temp1 : a > 0 → -a < 0, from neg_lt_zero,
-have temp2 : a < 0 → -a > 0, from zero_lt_neg,
-or.elim3 (trichotomy a 0) sorry sorry sorry
---  (by simp)
---  (by simp)
---  (by simp)
-
--- add_rewrite sign_neg
-
-theorem nat_abs_sign_ne_zero {a : ℤ} (H : a ≠ 0) : (nat_abs (sign a)) = 1 :=
-or.elim3 (trichotomy a 0) sorry
---  (by simp)
-  (assume H2 : a = 0, absurd H2 H)
-  sorry
---  (by simp)
-
-theorem sign_nat_abs (a : ℤ) : sign (nat_abs a) = nat_abs (sign a) :=
-have temp1 : ∀a : ℤ, a < 0 → a ≤ 0, from take a, le_of_lt,
-have temp2 : ∀a : ℤ, a > 0 → a ≥ 0, from take a, le_of_lt,
-or.elim3 (trichotomy a 0) sorry sorry sorry
---  (by simp)
---  (by simp)
---  (by simp)
-
 end int