diff --git a/library/algebra/field.lean b/library/algebra/field.lean
index 5bb640df2..bfc1178f6 100644
--- a/library/algebra/field.lean
+++ b/library/algebra/field.lean
@@ -26,7 +26,7 @@ section division_ring
   include s
 
   definition divide (a b : A) : A := a * b⁻¹
-  infix `/` := divide
+  infix [priority algebra.prio] `/` := divide
 
   -- only in this file
   local attribute divide [reducible]
diff --git a/library/algebra/group.lean b/library/algebra/group.lean
index 169339f98..360df42ab 100644
--- a/library/algebra/group.lean
+++ b/library/algebra/group.lean
@@ -7,7 +7,7 @@ Various multiplicative and additive structures. Partially modeled on Isabelle's
 -/
 
 import logic.eq data.unit data.sigma data.prod
-import algebra.function algebra.binary
+import algebra.function algebra.binary algebra.priority
 
 open eq eq.ops   -- note: ⁻¹ will be overloaded
 open binary
@@ -36,12 +36,12 @@ structure has_inv [class] (A : Type) :=
 structure has_neg [class] (A : Type) :=
 (neg : A → A)
 
-infixl `*`   := has_mul.mul
-infixl `+`   := has_add.add
-postfix `⁻¹` := has_inv.inv
-prefix `-`   := has_neg.neg
-notation 1   := !has_one.one
-notation 0   := !has_zero.zero
+infixl [priority algebra.prio] `*`   := has_mul.mul
+infixl [priority algebra.prio] `+`   := has_add.add
+postfix [priority algebra.prio] `⁻¹` := has_inv.inv
+prefix [priority algebra.prio] `-`   := has_neg.neg
+notation 1  := !has_one.one
+notation 0  := !has_zero.zero
 
 /- semigroup -/
 
@@ -410,7 +410,7 @@ section add_group
   -- TODO: derive corresponding facts for div in a field
   definition sub [reducible] (a b : A) : A := a + -b
 
-  infix `-` := sub
+  infix [priority algebra.prio] `-` := sub
 
   theorem sub_eq_add_neg (a b : A) : a - b = a + -b := rfl
 
diff --git a/library/algebra/group_power.lean b/library/algebra/group_power.lean
index 586d27ae1..9143b6713 100644
--- a/library/algebra/group_power.lean
+++ b/library/algebra/group_power.lean
@@ -28,7 +28,7 @@ definition pow (a : A) : ℕ → A
 | 0     := 1
 | (n+1) := pow n * a
 
-infix `^` := pow
+infix [priority algebra.prio] `^` := pow
 
 theorem pow_zero (a : A) : a^0 = 1 := rfl
 theorem pow_succ (a : A) (n : ℕ) : a^(succ n) = a^n * a := rfl
@@ -92,7 +92,7 @@ theorem pow_inv_comm (a : A) : ∀m n, (a⁻¹)^m * a^n = a^n * (a⁻¹)^m
 
 end nat
 
-open nat int algebra
+open int
 
 definition ipow (a : A) : ℤ → A
 | (of_nat n) := a^n
diff --git a/library/algebra/order.lean b/library/algebra/order.lean
index e20650e31..e549024cc 100644
--- a/library/algebra/order.lean
+++ b/library/algebra/order.lean
@@ -5,7 +5,7 @@ Author: Jeremy Avigad
 
 Weak orders "≤", strict orders "<", and structures that include both.
 -/
-import logic.eq logic.connectives algebra.binary
+import logic.eq logic.connectives algebra.binary algebra.priority
 open eq eq.ops
 
 namespace algebra
@@ -20,16 +20,16 @@ structure has_le [class] (A : Type) :=
 structure has_lt [class] (A : Type) :=
 (lt : A → A → Prop)
 
-infixl `<=`  := has_le.le
-infixl `≤`   := has_le.le
-infixl `<`   := has_lt.lt
+infixl [priority algebra.prio] `<=`  := has_le.le
+infixl [priority algebra.prio] `≤`   := has_le.le
+infixl [priority algebra.prio] `<`   := has_lt.lt
 
 definition has_le.ge [reducible] {A : Type} [s : has_le A] (a b : A) := b ≤ a
-notation a ≥ b := has_le.ge a b
-notation a >= b := has_le.ge a b
+notation [priority algebra.prio] a ≥ b := has_le.ge a b
+notation [priority algebra.prio] a >= b := has_le.ge a b
 
 definition has_lt.gt [reducible] {A : Type} [s : has_lt A] (a b : A) := b < a
-notation a > b := has_lt.gt a b
+notation [priority algebra.prio] a > b := has_lt.gt a b
 
 /- weak orders -/
 
diff --git a/library/algebra/ring.lean b/library/algebra/ring.lean
index 56497c583..93d8b9af6 100644
--- a/library/algebra/ring.lean
+++ b/library/algebra/ring.lean
@@ -73,7 +73,7 @@ section comm_semiring
   include s
 
   definition dvd (a b : A) : Prop := ∃c, b = a * c
-  notation a ∣ b := dvd a b
+  notation [priority algebra.prio] a ∣ b := dvd a b
 
   theorem dvd.intro {a b c : A} (H : a * c = b) : a ∣ b :=
   exists.intro _ H⁻¹
diff --git a/library/data/int/basic.lean b/library/data/int/basic.lean
index aad571abe..12963345c 100644
--- a/library/data/int/basic.lean
+++ b/library/data/int/basic.lean
@@ -86,10 +86,12 @@ definition mul (a b : ℤ) : ℤ :=
 
 /- notation -/
 
+protected definition prio : num := num.pred std.priority.default
+
 notation `-[1+` n `]` := int.neg_succ_of_nat n    -- for pretty-printing output
-prefix - := int.neg
-infix +  := int.add
-infix *  := int.mul
+prefix [priority int.prio] - := int.neg
+infix  [priority int.prio] +  := int.add
+infix  [priority int.prio] *  := int.mul
 
 /- some basic functions and properties -/
 
@@ -627,9 +629,9 @@ section migrate_algebra
 
   local attribute int.integral_domain [instance]
   definition sub (a b : ℤ) : ℤ := algebra.sub a b
-  infix - := int.sub
+  infix [priority int.prio] - := int.sub
   definition dvd (a b : ℤ) : Prop := algebra.dvd a b
-  notation a ∣ b := dvd a b
+  notation [priority int.prio] a ∣ b := dvd a b
 
   migrate from algebra with int
   replacing sub → sub, dvd → dvd
diff --git a/library/data/int/div.lean b/library/data/int/div.lean
index e4dc32492..67a21b9eb 100644
--- a/library/data/int/div.lean
+++ b/library/data/int/div.lean
@@ -23,11 +23,11 @@ sign b *
     | of_nat m := #nat m div (nat_abs b)
     | -[1+m]   := -[1+ (#nat m div (nat_abs b))]
   end)
-notation a div b := divide a b
+notation [priority int.prio] a div b := divide a b
 
 definition modulo (a b : ℤ) : ℤ := a - a div b * b
-notation a mod b := modulo a b
-notation a `≡` b `[mod`:100 c `]`:0 := a mod c = b mod c
+notation [priority int.prio] a mod b := modulo a b
+notation [priority int.prio] a `≡` b `[mod`:100 c `]`:0 := a mod c = b mod c
 
 /- div  -/
 
diff --git a/library/data/int/order.lean b/library/data/int/order.lean
index 636d02d02..0e11aec92 100644
--- a/library/data/int/order.lean
+++ b/library/data/int/order.lean
@@ -18,10 +18,10 @@ private definition nonneg (a : ℤ) : Prop := int.cases_on a (take n, true) (tak
 definition le (a b : ℤ) : Prop := nonneg (sub b a)
 definition lt (a b : ℤ) : Prop := le (add a 1) b
 
-infix - := int.sub
-infix <= := int.le
-infix ≤  := int.le
-infix <  := int.lt
+infix [priority int.prio] - := int.sub
+infix [priority int.prio] <= := int.le
+infix [priority int.prio] ≤  := int.le
+infix [priority int.prio] <  := int.lt
 
 local attribute nonneg [reducible]
 private definition decidable_nonneg [instance] (a : ℤ) : decidable (nonneg a) := int.cases_on a _ _
diff --git a/library/data/int/power.lean b/library/data/int/power.lean
index 930c4749a..dcfd2a837 100644
--- a/library/data/int/power.lean
+++ b/library/data/int/power.lean
@@ -17,7 +17,7 @@ section migrate_algebra
   local attribute int.decidable_linear_ordered_comm_ring [instance]
 
   definition pow (a : ℤ) (n : ℕ) : ℤ := algebra.pow a n
-  infix ^ := pow
+  infix [priority int.prio] ^ := pow
 
   migrate from algebra with int
     replacing dvd → dvd, has_le.ge → ge, has_lt.gt → gt, pow → pow
diff --git a/library/data/rat/basic.lean b/library/data/rat/basic.lean
index e8f4992c5..1c97bc994 100644
--- a/library/data/rat/basic.lean
+++ b/library/data/rat/basic.lean
@@ -371,14 +371,16 @@ definition denom (a : ℚ) : ℤ := prerat.denom (reduce a)
 theorem denom_pos (a : ℚ): denom a > 0 :=
 prerat.denom_pos (reduce a)
 
-infix +    := rat.add
-infix *    := rat.mul
-prefix -   := rat.neg
+protected definition prio := num.pred int.prio
+
+infix [priority rat.prio] +    := rat.add
+infix [priority rat.prio] *    := rat.mul
+prefix [priority rat.prio] -   := rat.neg
 
 definition sub [reducible] (a b : rat) : rat := a + (-b)
 
-postfix ⁻¹ := rat.inv
-infix -    := rat.sub
+postfix [priority rat.prio] ⁻¹ := rat.inv
+infix [priority rat.prio] -    := rat.sub
 
 /- properties -/
 
diff --git a/library/data/rat/order.lean b/library/data/rat/order.lean
index ac219acbe..93e98d17b 100644
--- a/library/data/rat/order.lean
+++ b/library/data/rat/order.lean
@@ -146,12 +146,12 @@ definition le (a b : ℚ) : Prop := nonneg (b - a)
 definition gt [reducible] (a b : ℚ) := lt b a
 definition ge [reducible] (a b : ℚ) := le b a
 
-infix <  := rat.lt
-infix <= := rat.le
-infix ≤  := rat.le
-infix >= := rat.ge
-infix ≥  := rat.ge
-infix >  := rat.gt
+infix [priority rat.prio] <  := rat.lt
+infix [priority rat.prio] <= := rat.le
+infix [priority rat.prio] ≤  := rat.le
+infix [priority rat.prio] >= := rat.ge
+infix [priority rat.prio] ≥  := rat.ge
+infix [priority rat.prio] >  := rat.gt
 
 theorem of_int_lt_of_int (a b : ℤ) : of_int a < of_int b ↔ (#int a < b) :=
 iff.symm (calc
diff --git a/library/data/real/basic.lean b/library/data/real/basic.lean
index 1e65bf4c9..91273bfec 100644
--- a/library/data/real/basic.lean
+++ b/library/data/real/basic.lean
@@ -1003,18 +1003,19 @@ definition add (x y : ℝ) : ℝ :=
   (quot.lift_on₂ x y (λ a b, quot.mk (a + b))
                      (take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
                        quot.sound (radd_well_defined Hab Hcd)))
-infix `+` := add
+protected definition prio := num.pred rat.prio
+infix [priority real.prio] `+` := add
 
 definition mul (x y : ℝ) : ℝ :=
   (quot.lift_on₂ x y (λ a b, quot.mk (a * b))
                      (take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
                        quot.sound (rmul_well_defined Hab Hcd)))
-infix `*` := mul
+infix [priority real.prio] `*` := mul
 
 definition neg (x : ℝ) : ℝ :=
   (quot.lift_on x (λ a, quot.mk (-a)) (take a b : reg_seq, take Hab : requiv a b,
                                    quot.sound (rneg_well_defined Hab)))
-prefix `-` := neg
+prefix [priority real.prio] `-` := neg
 
 definition zero : ℝ := quot.mk r_zero
 --notation 0 := zero
diff --git a/library/data/real/division.lean b/library/data/real/division.lean
index 04a27435f..08ade03ee 100644
--- a/library/data/real/division.lean
+++ b/library/data/real/division.lean
@@ -610,7 +610,7 @@ open [classes] s
 
 definition inv (x : ℝ) : ℝ := quot.lift_on x (λ a, quot.mk (s.r_inv a))
            (λ a b H, quot.sound (s.r_inv_well_defined H))
-postfix `⁻¹` := inv
+postfix [priority real.prio] `⁻¹` := inv
 
 theorem le_total (x y : ℝ) : x ≤ y ∨ y ≤ x :=
   quot.induction_on₂ x y (λ s t, s.r_le_total s t)
diff --git a/library/data/real/order.lean b/library/data/real/order.lean
index 4e621eb61..6a2099a78 100644
--- a/library/data/real/order.lean
+++ b/library/data/real/order.lean
@@ -1018,18 +1018,18 @@ open [classes] s
 namespace real
 
 definition lt (x y : ℝ) := quot.lift_on₂ x y (λ a b, s.r_lt a b) s.r_lt_well_defined
-infix `<` := lt
+infix [priority real.prio] `<` := lt
 
 definition le (x y : ℝ) := quot.lift_on₂ x y (λ a b, s.r_le a b) s.r_le_well_defined
-infix `≤` := le
-infix `<=` := le
+infix [priority real.prio] `≤` := le
+infix [priority real.prio] `<=` := le
 
 definition gt [reducible] (a b : ℝ) := lt b a
 definition ge [reducible] (a b : ℝ) := le b a
 
-infix >= := real.ge
-infix ≥  := real.ge
-infix >  := real.gt
+infix [priority real.prio] >= := real.ge
+infix [priority real.prio] ≥  := real.ge
+infix [priority real.prio] >  := real.gt
 
 definition sep (x y : ℝ) := quot.lift_on₂ x y (λ a b, s.r_sep a b) s.r_sep_well_defined
 infix `≢` : 50 := sep
@@ -1114,9 +1114,9 @@ section migrate_algebra
   local attribute real.ordered_ring [instance]
 
   definition sub (a b : ℝ) : ℝ := algebra.sub a b
-  infix - := real.sub
+  infix [priority real.prio] - := real.sub
   definition dvd (a b : ℝ) : Prop := algebra.dvd a b
-  notation a ∣ b := real.dvd a b
+  notation [priority real.prio] a ∣ b := real.dvd a b
 
   migrate from algebra with real
     replacing has_le.ge → ge, has_lt.gt → gt, sub → sub, dvd → dvd, divide → divide
diff --git a/tests/lean/notation_priority.lean b/tests/lean/notation_priority.lean
new file mode 100644
index 000000000..dd3d098ab
--- /dev/null
+++ b/tests/lean/notation_priority.lean
@@ -0,0 +1,26 @@
+import data.real
+
+open real int nat
+
+variables a b : nat
+variables i j : int
+
+set_option pp.all true
+check a + b
+check i + j
+
+example : a + b = nat.add a b :=
+rfl
+example : i + j = int.add i j :=
+rfl
+
+open nat real int
+
+example : a + b = nat.add a b :=
+rfl
+example : i + j = int.add i j :=
+rfl
+
+set_option pp.all true
+check a + b
+check i + j
diff --git a/tests/lean/notation_priority.lean.expected.out b/tests/lean/notation_priority.lean.expected.out
new file mode 100644
index 000000000..049aace77
--- /dev/null
+++ b/tests/lean/notation_priority.lean.expected.out
@@ -0,0 +1,4 @@
+nat.add a b : nat
+int.add i j : int
+nat.add a b : nat
+int.add i j : int
diff --git a/tests/lua/parse_table.lua b/tests/lua/parse_table.lua
deleted file mode 100644
index 676cccd97..000000000
--- a/tests/lua/parse_table.lua
+++ /dev/null
@@ -1,119 +0,0 @@
-function display_parse_table(t)
-   t:for_each(function(ts, es)
-                 if not t:is_nud() then
-                    io.write("_ ")
-                 end
-                 for i = 1, #ts do
-                    io.write("'" .. tostring(ts[i][1]) .. "'")
-                    local a = ts[i][2]
-                    local k = a:kind()
-                    if k == notation_action_kind.Skip then
-                    elseif k == notation_action_kind.Binder then
-                       io.write(" binder")
-                    elseif k == notation_action_kind.Binders then
-                       io.write(" binders")
-                    elseif k == notation_action_kind.Ext then
-                       io.write(" external")
-                    elseif k == notation_action_kind.ScopedExpr then
-                       io.write(" [_@" .. tostring(a:rbp()) .. ":" .. tostring(a:rec()) .. "]")
-                    elseif k == notation_action_kind.Expr then
-                       if a:rbp() == 0 then
-                          io.write(" _")
-                       else
-                          io.write(" _@" .. tostring(a:rbp()))
-                       end
-                    elseif k == notation_action_kind.Exprs then
-                       io.write(" [" .. tostring(a:rbp()) .. ", " .. tostring(a:rec()) .. ", " .. tostring(a:ini()) .. ", " .. tostring(a:is_fold_right()) .. "]")
-                    end
-                    io.write(" ")
-                 end
-                 print(":= ")
-                 while (not es:is_nil()) do
-                    print("  " .. tostring(es:head()))
-                    es = es:tail()
-                 end
-   end)
-end
-
-function parse_table_size(t)
-   local r = 0
-   t:for_each(function(ts, es) r = r + #es end)
-   return r
-end
-
-local p = parse_table()
-assert(is_parse_table(p))
-local ite  = Const("ite")
-local ite2 = Const("ite2")
-local If   = Const("if")
-p = p:add({"if", "then", "else"}, ite(Var(0), Var(1), Var(2)))
-p = p:add({"if", "then", "else"}, ite2(Var(0), Var(1), Var(2)))
-p = p:add({"if", "then"}, If(Var(0), Var(1)))
-display_parse_table(p)
-assert(parse_table_size(p) == 3)
-p = p:add({"if", "then", "else"}, ite2(Var(0), Var(1), Var(2)), false)
-print("======")
-display_parse_table(p)
-assert(parse_table_size(p) == 2)
-local f = Const("f")
-p = p:add({{"with", Skip}, "value"}, f(Var(0)))
-local Exists = Const("Exists")
-local ExistUnique = Const("ExistUnique")
-p = p:add({{"exists", Binders}, {",", scoped_expr_notation_action(Exists(Var(0)))}}, Var(0))
-p = p:add({{"exist!", Binder}, {",", scoped_expr_notation_action(ExistUnique(Var(0)))}}, Var(0))
-print("======")
-display_parse_table(p)
-check_error(function() p = p:add({{"with", Skip}, "value"}, f(Var(1))) end)
-check_error(function()
-               p = p:add({{",", scoped_expr_notation_action(Exists(Var(0)))}}, Var(0))
-end)
-local nat_add = Const({"nat", "add"})
-local int_add = Const({"int", "add"})
-check_error(function() p = p:add({{"+", 10}}, nat_add(Var(0), Var(1))) end)
-p = p:add({"if", "then", "else"}, ite(Var(0), Var(1), Var(2)))
-local a, p2 = p:find("if")
-assert(is_notation_action(a))
-assert(a:kind() == notation_action_kind.Expr)
-assert(a:rbp() == 0)
-print("======")
-display_parse_table(p2)
-assert(parse_table_size(p2) == 3)
-local _, p2 = p2:find("then")
-local _, p2 = p2:find("else")
-print("======")
-assert(parse_table_size(p2) == 2)
-display_parse_table(p2)
-local es = p2:is_accepting()
-assert(es)
-assert(#es == 2)
-
-local p = parse_table(false) -- Create led table
-assert(not p:is_nud())
-p = p:add({{"+", 10}}, nat_add(Var(0), Var(1)))
-p = p:add({{"+", 10}}, int_add(Var(0), Var(1)))
-print("======")
-display_parse_table(p)
-
-local nat_mul = Const({"nat", "mul"})
-local int_mul = Const({"int", "mul"})
-local p2 = parse_table(false) -- Create led table
-p2 = p2:add({{"*", 20}}, nat_mul(Var(0), Var(1)))
-p2 = p2:add({{"*", 20}}, int_mul(Var(0), Var(1)))
-print("======")
-display_parse_table(p2)
-
-p = p:merge(p2)
-print("======")
-display_parse_table(p)
-assert(parse_table_size(p) == 4)
-
-local p3 = parse_table()
-check_error(function() p:merge(p3) end)
-
-local a = scoped_expr_notation_action(Var(0), 10)
-assert(a:use_lambda_abstraction())
-local a = scoped_expr_notation_action(Var(0), 10, false)
-assert(not a:use_lambda_abstraction())
-local a = scoped_expr_notation_action(Var(0), 10, true)
-assert(a:use_lambda_abstraction())
-assert(a:rbp() == 10)