fix(library/data/int,library/data/rat): int and rat

library/algebra/ring_power.lean

@@ -18,6 +18,9 @@ section semiring
 variable [s : semiring A]
 include s
 
+definition semiring_has_pow_nat [reducible] [instance] : has_pow_nat A :=
+monoid_has_pow_nat
+
 theorem zero_pow {m : ℕ} (mpos : m > 0) : 0^m = (0 : A) :=
 have h₁ : ∀ m : nat, (0 : A)^(succ m) = (0 : A),
   begin
@@ -34,6 +37,9 @@ section integral_domain
 variable [s : integral_domain A]
 include s
 
+definition integral_domain_has_pow_nat [reducible] [instance] : has_pow_nat A :=
+monoid_has_pow_nat
+
 theorem eq_zero_of_pow_eq_zero {a : A} {m : ℕ} (H : a^m = 0) : a = 0 :=
 or.elim (eq_zero_or_pos m)
   (suppose m = 0,
@@ -128,6 +134,9 @@ section decidable_linear_ordered_comm_ring
 variable [s : decidable_linear_ordered_comm_ring A]
 include s
 
+definition decidable_linear_ordered_comm_ring_has_pow_nat [reducible] [instance] : has_pow_nat A :=
+monoid_has_pow_nat
+
 theorem abs_pow (a : A) (n : ℕ) : abs (a^n) = abs a^n :=
 begin
   induction n with n ih,

library/data/int/basic.lean

@@ -107,7 +107,7 @@ private definition has_decidable_eq₂ : Π (a b : ℤ), decidable (a = b)
 | -[1+ m]    -[1+ n]    := if H : m = n then
     inl (congr_arg neg_succ_of_nat H) else inr (not.mto neg_succ_of_nat.inj H)
 
-definition has_decidable_eq [instance] : decidable_eq ℤ := has_decidable_eq₂
+definition has_decidable_eq [instance] [priority int.prio] : decidable_eq ℤ := has_decidable_eq₂
 
 theorem of_nat_add (n m : nat) : of_nat (n + m) = of_nat n + of_nat m := rfl
 
@@ -501,7 +501,7 @@ theorem eq_zero_or_eq_zero_of_mul_eq_zero {a b : ℤ} (H : a * b = 0) : a = 0
 or.imp eq_zero_of_nat_abs_eq_zero eq_zero_of_nat_abs_eq_zero
   (eq_zero_or_eq_zero_of_mul_eq_zero (by rewrite [-nat_abs_mul, H]))
 
-protected definition integral_domain [reducible] [instance] : algebra.integral_domain int :=
+protected definition integral_domain [reducible] [trans_instance] : algebra.integral_domain int :=
 ⦃algebra.integral_domain,
   add            := int.add,
   add_assoc      := add.assoc,

library/data/int/order.lean

@@ -233,7 +233,7 @@ theorem lt_of_le_of_lt  {a b c : ℤ} (Hab : a ≤ b) (Hbc : b < c) : a < c :=
   (iff.mpr !lt_iff_le_and_ne) (and.intro Hac
     (assume Heq, not_le_of_gt (Heq⁻¹ ▸ Hbc) Hab))
 
-protected definition linear_ordered_comm_ring [reducible] [instance] :
+protected definition linear_ordered_comm_ring [reducible] [trans_instance] :
     algebra.linear_ordered_comm_ring int :=
 ⦃algebra.linear_ordered_comm_ring, int.integral_domain,
   le               := int.le,

library/data/int/power.lean

@@ -10,7 +10,7 @@ import data.int.basic data.int.order data.int.div algebra.group_power data.nat.p
 namespace int
 open - [notations] algebra
 
-definition int_has_pow_nat : has_pow_nat int :=
+definition int_has_pow_nat [reducible] [instance] [priority int.prio] : has_pow_nat int :=
 has_pow_nat.mk has_pow_nat.pow_nat
 
 /-

library/data/nat/basic.lean

@@ -290,9 +290,10 @@ nat.cases_on n
           !succ_ne_zero))
 
 open - [notations] algebra
-protected definition comm_semiring [reducible] [instance] : algebra.comm_semiring nat :=
+protected definition comm_semiring [reducible] [trans_instance] : algebra.comm_semiring nat :=
 ⦃algebra.comm_semiring,
  add            := nat.add,
+
  add_assoc      := add.assoc,
  zero           := nat.zero,
  zero_add       := zero_add,

library/data/nat/order.lean

@@ -136,7 +136,7 @@ else (eq_max_left h) ▸ !le.refl
 
 open - [notations] algebra
 
-protected definition decidable_linear_ordered_semiring [reducible] [instance] :
+protected definition decidable_linear_ordered_semiring [reducible] [trans_instance] :
 algebra.decidable_linear_ordered_semiring nat :=
 ⦃ algebra.decidable_linear_ordered_semiring, nat.comm_semiring,
   add_left_cancel            := @add.cancel_left,

library/data/rat/basic.lean

@@ -535,7 +535,7 @@ decidable.by_cases
       end))
 
 
-protected definition discrete_field [reducible] [instance] : algebra.discrete_field rat :=
+protected definition discrete_field [reducible] [trans_instance] : algebra.discrete_field rat :=
 ⦃algebra.discrete_field,
  add              := rat.add,
  add_assoc        := add.assoc,
@@ -560,10 +560,10 @@ protected definition discrete_field [reducible] [instance] : algebra.discrete_fi
  has_decidable_eq := has_decidable_eq⦄
 
 definition rat_has_division [instance] [reducible] [priority rat.prio] : has_division rat :=
-has_division.mk algebra.division
+has_division.mk has_division.division
 
 definition rat_has_pow_nat [instance] [reducible] [priority rat.prio] : has_pow_nat rat :=
-has_pow_nat.mk pow_nat
+has_pow_nat.mk has_pow_nat.pow_nat
 
 theorem eq_num_div_denom (a : ℚ) : a = num a / denom a :=
 have H : of_int (denom a) ≠ 0, from assume H', ne_of_gt (denom_pos a) (of_int.inj H'),

library/data/rat/order.lean

@@ -302,7 +302,7 @@ let H' := le_of_lt H in
                                   (take Heq, let Heq' := add_left_cancel Heq in
                                    !lt_irrefl (Heq' ▸ H)))
 
-protected definition discrete_linear_ordered_field [reducible] [instance] :
+protected definition discrete_linear_ordered_field [reducible] [trans_instance] :
     algebra.discrete_linear_ordered_field rat :=
 ⦃algebra.discrete_linear_ordered_field,
  rat.discrete_field,
@@ -386,7 +386,7 @@ section
       rewrite [-mul_denom],
       apply mul_neg_of_neg_of_pos H,
       change of_int (denom q) > of_int 0,
-      rewrite [of_int_lt_of_int_iff],
+      xrewrite [of_int_lt_of_int_iff],
       exact !denom_pos
     end,
   show num q < 0, from lt_of_of_int_lt_of_int this
@@ -397,7 +397,7 @@ section
       rewrite [-mul_denom],
       apply mul_nonpos_of_nonpos_of_nonneg H,
       change of_int (denom q) ≥ of_int 0,
-      rewrite [of_int_le_of_int_iff],
+      xrewrite [of_int_le_of_int_iff],
       exact int.le_of_lt !denom_pos
     end,
   show num q ≤ 0, from le_of_of_int_le_of_int this

library/theories/number_theory/irrational_roots.lean

@@ -38,8 +38,7 @@ section
   open nat decidable
 
   theorem root_irrational {a b c n : ℕ} (npos : n > 0) (apos : a > 0) (co : coprime a b)
-      (H : a^n = c * b^n) :
-    b = 1 :=
+      (H : a^n = c * b^n) : b = 1 :=
   have bpos : b > 0, from pos_of_ne_zero
     (suppose b = 0,
       have a^n = 0, by krewrite [H, this, zero_pow npos],
@@ -73,8 +72,8 @@ section
   let a := num q, b := denom q in
   have b ≠ 0,                      from ne_of_gt (denom_pos q),
   have bnz : b ≠ (0 : ℚ),          from assume H, `b ≠ 0` (of_int.inj H),
-  have bnnz : ((b : rat)^n ≠ 0), from assume bneqz, bnz (eq_zero_of_pow_eq_zero bneqz),
-  have a^n /[rat] b^n = c, using bnz, begin krewrite [*of_int_pow, -div_pow, -eq_num_div_denom, -H] end,
+  have bnnz : ((b : rat)^n ≠ 0),   from assume bneqz, bnz (algebra.eq_zero_of_pow_eq_zero bneqz),
+  have a^n /[rat] b^n = c,         using bnz, begin rewrite [*of_int_pow, -algebra.div_pow, -eq_num_div_denom, -H] end,
   have (a^n : rat) = c *[rat] b^n, from eq.symm (!mul_eq_of_eq_div bnnz this⁻¹),
   have a^n = c * b^n,  -- int version
     using this, by rewrite [-of_int_pow at this, -of_int_mul at this]; exact of_int.inj this,
@@ -82,7 +81,7 @@ section
     using this, by rewrite [-abs_pow, this, abs_mul, abs_pow],
   have H₁ : (nat_abs a)^n = nat_abs c * (nat_abs b)^n,
     using this,
-      by apply int.of_nat.inj; rewrite [int.of_nat_mul, +int.of_nat_pow, +of_nat_nat_abs]; assumption,
+      begin apply int.of_nat.inj, rewrite [int.of_nat_mul, +int.of_nat_pow, +of_nat_nat_abs], exact this end,
   have H₂ : nat.coprime (nat_abs a) (nat_abs b), from of_nat.inj !coprime_num_denom,
   have nat_abs b = 1, from
     by_cases
@@ -99,13 +98,11 @@ section
         show nat_abs b = 1, from (root_irrational npos (pos_of_ne_zero this) H₂ H₁)),
   show b = 1, using this, by rewrite [-of_nat_nat_abs_of_nonneg (le_of_lt !denom_pos), this]
 
-  exit
-
   theorem eq_num_pow_of_pow_eq {q : ℚ} {n : ℕ} {c : ℤ} (npos : n > 0) (H : q^n = c) :
     c = (num q)^n :=
   have denom q = 1, from denom_eq_one_of_pow_eq npos H,
-  have of_int c = (num q)^n, using this,
-    by rewrite [-H, eq_num_div_denom q at {1}, this, div_one, of_int_pow],
+  have of_int c = of_int ((num q)^n), using this,
+    by krewrite [-H, eq_num_div_denom q at {1}, this, div_one, of_int_pow],
   show c = (num q)^n , from of_int.inj this
 end
 
@@ -121,11 +118,11 @@ section
     (suppose p = 0,
       show false,
         by let H := (pos_of_prime primep); rewrite this at H; exfalso; exact !lt.irrefl H),
-  have agtz : a > 0, from pos_of_ne_zero
+  assert agtz : a > 0, from pos_of_ne_zero
     (suppose a = 0,
-      show false, using npos pnez, by revert peq; rewrite [this, zero_pow npos]; exact pnez),
+      show false, using npos pnez, by revert peq; krewrite [this, zero_pow npos]; exact pnez),
   have n * mult p a = 1, from calc
-    n * mult p a = mult p (a^n) : using agtz, by rewrite [mult_pow n agtz primep]
+    n * mult p a = mult p (a^n) : begin krewrite [mult_pow n agtz primep] end
              ... = mult p p     : peq
              ... = 1            : mult_self (gt_one_of_prime primep),
   have n ∣ 1, from dvd_of_mul_right_eq this,
@@ -163,7 +160,7 @@ section
   have a * a = c * b * b, by rewrite -mul.assoc at H; apply H,
   have a div (gcd a b) = c * b div gcd (c * b) a, from div_gcd_eq_div_gcd this bpos apos,
   have a = c * b div gcd c a,
-    using this, by revert this; rewrite [↑coprime at co, co, div_one, H₁]; intros; assumption,
+    using this, by revert this; krewrite [↑coprime at co, co, int.div_one, H₁]; intros; assumption,
   have a = b * (c div gcd c a),
     using this,
       by revert this; rewrite [mul.comm, !mul_div_assoc !gcd_dvd_left]; intros; assumption,