fix(library/tc_multigraph): avoid name collisions

@avigad, @fpvandoorn, @rlewis1988, @dselsam

I changed how transitive instances are named.
The motivation is to avoid a naming collision problem found by Daniel.
Before this commit, we were getting an error on the following file
tests/lean/run/collision_bug.lean.

Now, transitive instances contain the prefix "_trans_".
It makes it clear this is an internal definition and it should not be used
by users.

This change also demonstrates (again) how the `rewrite` tactic is
fragile. The problem is that the matching procedure used by it has
very little support for solving matching constraints that involving type
class instances. Eventually, we will need to reimplement `rewrite`
using the new unification procedure used in blast.

In the meantime, the workaround is to use `krewrite` (as usual).

hott/algebra/category/functor/yoneda.hlean

@@ -17,7 +17,7 @@ namespace yoneda
     However, we don't want to have them globally, because that will unfold the composition g ∘ f
     in a Category to category.category.comp g f
   -/
-  local attribute Category.to.precategory category.to_precategory [constructor]
+  local attribute category.to_precategory [constructor]
 
   -- should this be defined as "yoneda_embedding Cᵒᵖ"?
   definition contravariant_yoneda_embedding [constructor] [reducible]

hott/algebra/category/limits/functor_preserve.hlean

@@ -50,7 +50,7 @@ namespace category
 
   /- yoneda preserves existing limits -/
 
-  local attribute Category.to.precategory category.to_precategory [constructor]
+  local attribute category.to_precategory [constructor]
 
   definition preserves_existing_limits_yoneda_embedding_lemma [constructor] (y : cone_obj F)
     [H : is_terminal y] {G : Cᵒᵖ ⇒ cset} (η : constant_functor I G ⟹ ɏ ∘f F) :

hott/algebra/category/limits/set.hlean

@@ -11,7 +11,6 @@ import .colimits ..constructions.set hit.set_quotient
 open eq functor is_trunc sigma pi sigma.ops trunc set_quotient
 
 namespace category
-  local attribute Category.to.precategory [unfold 1]
   local attribute category.to_precategory [unfold 2]
 
   definition is_complete_set_cone.{u v w} [constructor]

hott/algebra/hott.hlean

@@ -27,7 +27,7 @@ namespace algebra
   -- coercions between them anymore.
   -- So, an application such as (@mul A G g h) in the following definition
   -- is type incorrect if the coercion is not added (explicitly or implicitly).
-  local attribute group.to.has_mul [coercion]
+  definition group.to_has_mul [coercion] {A : Type} (H : group A) : has_mul A := _
   local attribute group.to_has_inv [coercion]
 
   universe variable l

hott/types/int/basic.hlean

@@ -380,7 +380,8 @@ calc
     ... = nat_abs a : abstr_repr
 
 theorem padd_pneg (p : ℕ × ℕ) : padd p (pneg p) ≡ (0, 0) :=
-show pr1 p + pr2 p + 0 = pr2 p + pr1 p + 0, from !nat.add_comm ▸ rfl
+show pr1 p + pr2 p + 0 = pr2 p + pr1 p + 0,
+by rewrite [nat.add_comm (pr1 p)]
 
 theorem padd_padd_pneg (p q : ℕ × ℕ) : padd (padd p q) (pneg q) ≡ p :=
 calc      pr1 p + pr1 q + pr2 q + pr2 p

library/algebra/group_power.lean

@@ -71,7 +71,7 @@ theorem one_pow : ∀ n : ℕ, 1^n = (1:A)
 theorem pow_add (a : A) (m n : ℕ) : a^(m + n) = a^m * a^n :=
 begin
   induction n with n ih,
-    {rewrite [nat.add_zero, pow_zero, mul_one]},
+    {krewrite [nat.add_zero, pow_zero, mul_one]},
   rewrite [add_succ, *pow_succ', ih, mul.assoc]
 end
 
@@ -171,7 +171,7 @@ include s
 theorem pow_pos {a : A} (H : a > 0) (n : ℕ) : a ^ n > 0 :=
   begin
     induction n,
-    rewrite pow_zero,
+    krewrite pow_zero,
     apply zero_lt_one,
     rewrite pow_succ',
     apply mul_pos,
@@ -181,7 +181,7 @@ theorem pow_pos {a : A} (H : a > 0) (n : ℕ) : a ^ n > 0 :=
 theorem pow_ge_one_of_ge_one {a : A} (H : a ≥ 1) (n : ℕ) : a ^ n ≥ 1 :=
   begin
     induction n,
-    rewrite pow_zero,
+    krewrite pow_zero,
     apply le.refl,
     rewrite [pow_succ', -mul_one 1],
     apply mul_le_mul v_0 H zero_le_one,

library/algebra/ring_power.lean

@@ -21,7 +21,7 @@ theorem zero_pow {m : ℕ} (mpos : m > 0) : 0^m = (0 : A) :=
 have h₁ : ∀ m : nat, (0 : A)^(succ m) = (0 : A),
   begin
     intro m, induction m,
-      rewrite pow_one,
+      krewrite pow_one,
       apply zero_mul
   end,
 obtain m' (h₂ : m = succ m'), from exists_eq_succ_of_pos mpos,
@@ -45,7 +45,7 @@ or.elim (eq_zero_or_pos m)
       begin
         intro m,
         induction m with m ih,
-          {rewrite pow_one; intros; assumption},
+          {krewrite pow_one; intros; assumption},
         rewrite pow_succ,
         intro H,
         cases eq_zero_or_eq_zero_of_mul_eq_zero H with h₃ h₄,
@@ -73,7 +73,7 @@ or.elim (eq_zero_or_pos m)
       begin
         intro m,
         induction m with m ih,
-          {rewrite pow_one; assumption},
+          { krewrite pow_one; assumption },
         rewrite pow_succ,
         apply division_ring.mul_ne_zero H ih
       end,
@@ -136,7 +136,7 @@ monoid_has_pow_nat
 theorem abs_pow (a : A) (n : ℕ) : abs (a^n) = abs a^n :=
 begin
   induction n with n ih,
-    rewrite [*pow_zero, (abs_of_nonneg zero_le_one : abs (1 : A) = 1)],
+    krewrite [*pow_zero, (abs_of_nonneg zero_le_one : abs (1 : A) = 1)],
   rewrite [*pow_succ, abs_mul, ih]
 end
 
@@ -149,7 +149,7 @@ include s
 theorem field.div_pow (a : A) {b : A} {n : ℕ} (bnz : b ≠ 0) : (a / b)^n = a^n / b^n :=
 begin
   induction n with n ih,
-    rewrite [*pow_zero, div_one],
+    krewrite [*pow_zero, div_one],
   have bnnz : b^n ≠ 0, from division_ring.pow_ne_zero_of_ne_zero bnz,
   rewrite [*pow_succ, ih, !field.div_mul_div bnz bnnz]
 end
@@ -163,7 +163,7 @@ include s
 theorem div_pow (a : A) {b : A} {n : ℕ} : (a / b)^n = a^n / b^n :=
 begin
   induction n with n ih,
-    rewrite [*pow_zero, div_one],
+    krewrite [*pow_zero, div_one],
   rewrite [*pow_succ, ih, div_mul_div]
 end
 

library/data/complex.lean

@@ -30,10 +30,10 @@ protected proposition eta (z : ℂ) : complex.mk (complex.re z) (complex.im z) =
 by cases z; exact rfl
 
 definition of_real [coercion] (x : ℝ) : ℂ := complex.mk x 0
-definition of_rat [coercion] (q : ℚ) : ℂ := rat.to.complex q
-definition of_int [coercion] (i : ℤ) : ℂ := int.to.complex i
-definition of_nat [coercion] (n : ℕ) : ℂ := nat.to.complex n
-definition of_num [coercion] [reducible] (n : num) : ℂ := num.to.complex n
+definition of_rat [coercion] (q : ℚ) : ℂ := q
+definition of_int [coercion] (i : ℤ) : ℂ := i
+definition of_nat [coercion] (n : ℕ) : ℂ := n
+definition of_num [coercion] [reducible] (n : num) : ℂ := n
 
 protected definition prio : num := num.pred real.prio
 

library/data/int/div.lean

@@ -99,7 +99,7 @@ calc
       ... < 0                 : neg_neg_of_pos this
 
 protected theorem zero_div (b : ℤ) : 0 / b = 0 :=
-by rewrite [of_nat_div_eq, nat.zero_div, of_nat_zero, mul_zero]
+by krewrite [of_nat_div_eq, nat.zero_div, of_nat_zero, mul_zero]
 
 protected theorem div_zero (a : ℤ) : a / 0 = 0 :=
 by rewrite [div_def, sign_zero, zero_mul]

library/data/int/power.lean

@@ -23,7 +23,7 @@ theorem of_nat_pow (a n : ℕ) : of_nat (a^n) = (of_nat a)^n :=
 begin
   induction n with n ih,
     apply eq.refl,
-  rewrite [pow_succ, pow_succ, of_nat_mul, ih]
+  krewrite [pow_succ, pow_succ, of_nat_mul, ih]
 end
 
 end int

library/data/nat/bigops.lean

@@ -35,7 +35,7 @@ proposition sum_range_def (m n : ℕ) (f : ℕ → A) :
 
 proposition sum_range_self (m : ℕ) (f : ℕ → A) :
             (∑ i = m...m, f i) = f m :=
-  by rewrite [↑sum_range, succ_sub !le.refl, nat.sub_self, sum_up_to_one, zero_add]
+  by krewrite [↑sum_range, succ_sub !le.refl, nat.sub_self, sum_up_to_one, zero_add]
 
 proposition sum_range_succ {m n : ℕ} (f : ℕ → A) (H : m ≤ succ n) :
             (∑ i = m...succ n, f i) = (∑ i = m...n, f i) + f (succ n) :=
@@ -126,7 +126,7 @@ proposition prod_range_def (m n : ℕ) (f : ℕ → A) :
 
 proposition prod_range_self (m : ℕ) (f : ℕ → A) :
             (∏ i = m...m, f i) = f m :=
-by rewrite [↑prod_range, succ_sub !le.refl, nat.sub_self, prod_up_to_one, zero_add]
+by krewrite [↑prod_range, succ_sub !le.refl, nat.sub_self, prod_up_to_one, zero_add]
 
 proposition prod_range_succ {m n : ℕ} (f : ℕ → A) (H : m ≤ succ n) :
             (∏ i = m...succ n, f i) = (∏ i = m...n, f i) * f (succ n) :=

library/data/nat/power.lean

@@ -24,7 +24,7 @@ or.elim (eq_zero_or_pos m)
       begin
         intro m,
         induction m with m ih,
-          {rewrite pow_one; intros; assumption},
+          {krewrite pow_one; intros; assumption},
         rewrite pow_succ,
         intro H,
         cases eq_zero_or_eq_zero_of_mul_eq_zero H with h₃ h₄,
@@ -53,7 +53,7 @@ theorem le_pow_self {x : ℕ} (H : x > 1) : ∀ i, i ≤ x^i
 
 theorem mul_self_eq_pow_2 (a : nat) : a * a = a ^ 2 :=
 show a * a = a ^ (succ (succ zero)), from
-by rewrite [*pow_succ, *pow_zero, mul_one]
+by krewrite [*pow_succ, *pow_zero, mul_one]
 
 theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a ^ b = a ^ c → b = c
 | a 0        0        h₁ h₂ := rfl

library/data/rat/basic.lean

@@ -353,8 +353,8 @@ namespace rat
 /- operations -/
 
 definition of_int [coercion] (i : ℤ) : ℚ := ⟦prerat.of_int i⟧
-definition of_nat [coercion] (n : ℕ) : ℚ := nat.to.rat n
-definition of_num [coercion] [reducible] (n : num) : ℚ := num.to.rat n
+definition of_nat [coercion] (n : ℕ) : ℚ := n
+definition of_num [coercion] [reducible] (n : num) : ℚ := n
 
 protected definition prio := num.pred int.prio
 

library/data/real/basic.lean

@@ -1092,8 +1092,8 @@ has_sub.mk real.sub
 open rat -- no coercions before
 
 definition of_rat [coercion] (a : ℚ) : ℝ := quot.mk (r_const a)
-definition of_int [coercion] (i : ℤ) : ℝ := int.to.real i
-definition of_nat [coercion] (n : ℕ) : ℝ := nat.to.real n
+definition of_int [coercion] (i : ℤ) : ℝ := i
+definition of_nat [coercion] (n : ℕ) : ℝ := n
 definition of_num [coercion] [reducible] (n : num) : ℝ := of_rat (rat.of_num n)
 
 definition real_has_zero [reducible] [instance] [priority real.prio] : has_zero real :=

library/theories/analysis/real_limit.lean

@@ -367,11 +367,11 @@ proposition mul_converges_to_seq (HX : X ⟶ x in ℕ) (HY : Y ⟶ y in ℕ) :
       apply converges_to_seq_constant,
       rewrite -{0}zero_add,
       apply add_converges_to_seq,
-      rewrite -(mul_zero K),
+      krewrite -(mul_zero K),
       apply mul_left_converges_to_seq,
       apply abs_sub_converges_to_seq_of_converges_to_seq,
       exact HY,
-      rewrite -(mul_zero (abs y)),
+      krewrite -(mul_zero (abs y)),
       apply mul_left_converges_to_seq,
       apply abs_sub_converges_to_seq_of_converges_to_seq,
       exact HX

library/theories/combinatorics/choose.lean

@@ -58,7 +58,7 @@ begin
   induction n with [n, ih],
     {apply rfl},
   change choose (succ n) (succ 0) = succ n,
-  rewrite [choose_succ_succ, ih, choose_zero_right]
+  krewrite [choose_succ_succ, ih, choose_zero_right]
 end
 
 theorem choose_pos {n : ℕ} : ∀ {k : ℕ}, k ≤ n → choose n k > 0 :=
@@ -87,7 +87,7 @@ begin
          krewrite [one_mul, choose_zero_succ, choose_eq_zero_of_lt H, zero_mul]}},
   intro k,
   cases k with k',
-    {rewrite [choose_zero_right, choose_one_right]},
+    {krewrite [choose_zero_right, choose_one_right]},
   rewrite [choose_succ_succ (succ n), right_distrib, -ih (succ k')],
   rewrite [choose_succ_succ at {1}, left_distrib, *succ_mul (succ n), mul_succ, -ih k'],
   rewrite [*add.assoc, add.left_comm (choose n _)]

library/theories/number_theory/prime_factorization.lean

@@ -246,7 +246,7 @@ theorem mult_pow_eq_zero_of_prime_of_ne {p q : ℕ} (primep : prime p) (primeq :
   (pneq : p ≠ q) (i : ℕ) : mult p (q^i) = 0 :=
 begin
   induction i with i ih,
-    {rewrite [pow_zero, mult_one_right]},
+    {krewrite [pow_zero, mult_one_right]},
   have qpos : q > 0, from pos_of_prime primeq,
   have qipos : q^i > 0, from !pow_pos_of_pos qpos,
   rewrite [pow_succ', mult_mul primep qipos qpos, ih, mult_eq_zero_of_prime_of_ne primep

src/library/tc_multigraph.cpp

@@ -59,30 +59,8 @@ struct add_edge_fn {
         m_graph.m_edges.insert(edge, src);
     }
 
-
-    static name compose_name_core(name const & src, name const & tgt) {
-        return src + name("to") + tgt;
-    }
-
-    static name compose_name(name const & p, name const & src, name const & tgt) {
-        if (is_prefix_of(p, tgt))
-            return compose_name_core(src, tgt.replace_prefix(p, name()));
-        else if (p.is_atomic())
-            return compose_name_core(src, tgt);
-        else
-            return compose_name(p.get_prefix(), src, tgt);
-    }
-
-    static name compose_name(name const & src, name const & tgt) {
-        if (src.is_atomic())
-            return compose_name_core(src, tgt);
-        else
-            return compose_name(src.get_prefix(), src, tgt);
-    }
-
-    name compose(name const & src, name const & e1, name const & e2, name const & tgt) {
-        name n = compose_name(src, tgt);
-        pair<environment, name> env_e = ::lean::compose(m_env, *m_tc, e2, e1, optional<name>(n));
+    name compose(name const & base_name, name const & e1, name const & e2) {
+        pair<environment, name> env_e = ::lean::compose(m_env, *m_tc, e2, e1, optional<name>(base_name));
         m_env = env_e.first;
         return env_e.second;
     }
@@ -90,6 +68,7 @@ struct add_edge_fn {
     pair<environment, list<tc_edge>> operator()(name const & src, name const & edge, name const & tgt) {
         buffer<tc_edge> new_edges;
         if (auto preds = m_graph.m_predecessors.find(src)) {
+            name base_name = edge.append_before("_trans_to_");
             for (name const & pred : *preds) {
                 if (pred == tgt)
                     continue; // avoid loops
@@ -97,7 +76,7 @@ struct add_edge_fn {
                     for (pair<name, name> const & p : *pred_succ) {
                         if (p.second != src)
                             continue;
-                        name new_e = compose(pred, p.first, edge, tgt);
+                        name new_e = compose(base_name, p.first, edge);
                         new_edges.emplace_back(pred, new_e, tgt);
                     }
                 }
@@ -107,15 +86,16 @@ struct add_edge_fn {
         buffer<tc_edge> new_back_edges;
         new_back_edges.append(new_edges);
         if (auto succs = m_graph.m_successors.find(tgt)) {
+            name base_name = edge.append_before("_trans_of_");
             for (pair<name, name> const & p : *succs) {
                 if (src == p.second)
                     continue; // avoid loops
-                name new_e = compose(src, edge, p.first, p.second);
+                name new_e = compose(base_name, edge, p.first);
                 new_edges.emplace_back(src, new_e, p.second);
                 for (auto const & back_edge : new_back_edges) {
                     if (back_edge.m_from != p.second)
                         continue;
-                    name new_e = compose(back_edge.m_from, back_edge.m_cnst, p.first, p.second);
+                    name new_e = compose(base_name, back_edge.m_cnst, p.first);
                     new_edges.emplace_back(back_edge.m_from, new_e, p.second);
                 }
             }

tests/lean/852.hlean

@@ -2,4 +2,4 @@ import algebra.category.functor.equivalence
 
 -- print prefix category.equivalence
 
-check @category.equivalence.to._Fun
+-- check @category.equivalence.to._Fun

tests/lean/852.hlean.expected.out

@@ -1,6 +0,0 @@
-category.equivalence.to._Fun :
-  Π {C : category.Precategory} {D : category.Precategory} (c : category.equivalence C D)
-  {a b : category.Precategory.carrier C},
-    category.hom a b →
-    category.hom (functor.to_fun_ob (category.equivalence.to_functor c) a)
-      (functor.to_fun_ob (category.equivalence.to_functor c) b)

tests/lean/binder_overload.lean.expected.out

@@ -1,8 +1,15 @@
 {x : ℕ ∈ S | x > 0} : set ℕ
 {x : ℕ ∈ s | x > 0} : finset ℕ
-@set.sep.{1} nat (λ (x : nat), @gt.{1} nat _source.to.has_lt x (@zero.{1} nat _source.to.has_zero)) S : set.{1} nat
+@set.sep.{1} nat
+  (λ (x : nat),
+     @gt.{1} nat nat._trans_of_decidable_linear_ordered_semiring_13 x
+       (@zero.{1} nat nat._trans_of_decidable_linear_ordered_semiring_6))
+  S :
+  set.{1} nat
 @finset.sep.{1} nat (λ (a b : nat), nat.has_decidable_eq a b)
-  (λ (x : nat), @gt.{1} nat _source.to.has_lt x (@zero.{1} nat _source.to.has_zero))
-  (λ (a : nat), nat.decidable_lt (@zero.{1} nat _source.to.has_zero) a)
+  (λ (x : nat),
+     @gt.{1} nat nat._trans_of_decidable_linear_ordered_semiring_13 x
+       (@zero.{1} nat nat._trans_of_decidable_linear_ordered_semiring_6))
+  (λ (a : nat), nat.decidable_lt (@zero.{1} nat nat._trans_of_decidable_linear_ordered_semiring_6) a)
   s :
   finset.{1} nat

tests/lean/extra/show_goal.lean

@@ -13,7 +13,7 @@ begin
     have ceq : c = 0, begin
       rewrite aeq0 at h₃,
       rewrite add_zero at h₃,
-      rewrite add_succ at h₃,
+      krewrite add_succ at h₃,
       krewrite add_zero at h₃,
       injection h₃, exact a_1
     end,

tests/lean/notation_priority.lean.expected.out

@@ -1,4 +1,4 @@
-@add.{1} nat _source.to.has_add a b : nat
-@add.{1} int _source.to.has_add_1 i j : int
-@add.{1} nat _source.to.has_add a b : nat
-@add.{1} int _source.to.has_add_1 i j : int
+@add.{1} nat nat._trans_of_decidable_linear_ordered_semiring_2 a b : nat
+@add.{1} int int._trans_of_linear_ordered_comm_ring_16 i j : int
+@add.{1} nat nat._trans_of_decidable_linear_ordered_semiring_2 a b : nat
+@add.{1} int int._trans_of_linear_ordered_comm_ring_16 i j : int

tests/lean/pp_all.lean.expected.out

@@ -1,8 +1,9 @@
 a + of_num b = 10 : Prop
 @eq.{1} nat
-  (@add.{1} nat _source.to.has_add ((λ (x : nat), x) a)
+  (@add.{1} nat nat._trans_of_decidable_linear_ordered_semiring_2 ((λ (x : nat), x) a)
      (nat.of_num (@bit0.{1} num num_has_add (@one.{1} num num_has_one))))
-  (@bit0.{1} nat _source.to.has_add
-     (@bit1.{1} nat _source.to.has_one _source.to.has_add
-        (@bit0.{1} nat _source.to.has_add (@one.{1} nat _source.to.has_one)))) :
+  (@bit0.{1} nat nat._trans_of_decidable_linear_ordered_semiring_2
+     (@bit1.{1} nat nat._trans_of_decidable_linear_ordered_semiring_22 nat._trans_of_decidable_linear_ordered_semiring_2
+        (@bit0.{1} nat nat._trans_of_decidable_linear_ordered_semiring_2
+           (@one.{1} nat nat._trans_of_decidable_linear_ordered_semiring_22)))) :
   Prop

tests/lean/run/collision_bug.lean

@@ -0,0 +1,2 @@
+import theories.measure_theory.sigma_algebra
+import data.int.order

tests/lean/run/inj_tac.lean

@@ -38,6 +38,6 @@ example (a₁ a₂ a₃ b₁ b₂ b₃ : nat) : (a₁+2, a₂+3, a₃+1) = (b₁
 begin
   intro H, injection H with a₁b₁ sa₂b₂ a₃sb₃,
   esimp at *,
-  rewrite [a₁b₁, a₃sb₃, sa₂b₂],
+  krewrite [a₁b₁, a₃sb₃, -sa₂b₂],
   repeat (split | esimp)
 end