feat(library): add some spacing hints

library/algebra/field.lean

@@ -25,7 +25,7 @@ section division_ring
   include s
 
   definition divide (a b : A) : A := a * b⁻¹
-  infix [priority algebra.prio] `/` := divide
+  infix [priority algebra.prio] / := divide
 
   -- only in this file
   local attribute divide [reducible]

library/algebra/group.lean

@@ -478,7 +478,7 @@ section add_group
   -- TODO: derive corresponding facts for div in a field
   definition sub [reducible] (a b : A) : A := a + -b
 
-  infix [priority algebra.prio] `-` := sub
+  infix [priority algebra.prio] - := sub
 
   theorem sub_eq_add_neg (a b : A) : a - b = a + -b := rfl
 

library/algebra/order.lean

@@ -20,9 +20,9 @@ structure has_le [class] (A : Type) :=
 structure has_lt [class] (A : Type) :=
 (lt : A → A → Prop)
 
-infixl [priority algebra.prio] `<=`  := has_le.le
-infixl [priority algebra.prio] `≤`   := has_le.le
-infixl [priority algebra.prio] `<`   := 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 [priority algebra.prio] a ≥ b := has_le.ge a b

library/data/equiv.lean

@@ -54,9 +54,9 @@ protected definition trans [trans] {A B C : Type} : A ≃ B → B ≃ C → A
 abbreviation id {A : Type} := equiv.refl A
 
 namespace ops
-  postfix `⁻¹` := equiv.symm
-  postfix `⁻¹` := equiv.inv
-  notation e₁ `∘` e₂  := equiv.trans e₂ e₁
+  postfix ⁻¹ := equiv.symm
+  postfix ⁻¹ := equiv.inv
+  notation e₁ ∘ e₂  := equiv.trans e₂ e₁
 end ops
 open equiv.ops
 

library/data/finset/basic.lean

@@ -74,7 +74,7 @@ quot.lift_on s (λ l, a ∈ elt_of l)
    (λ ainl₁, mem_perm e ainl₁)
    (λ ainl₂, mem_perm (perm.symm e) ainl₂)))
 
-infix [priority finset.prio] `∈` := mem
+infix [priority finset.prio] ∈ := mem
 notation [priority finset.prio] a ∉ b := ¬ mem a b
 
 theorem mem_of_mem_list {a : A} {l : nodup_list A} : a ∈ elt_of l → a ∈ ⟦l⟧ :=
@@ -546,7 +546,7 @@ quot.lift_on₂ s₁ s₂
     (λ s₁ a i, mem_perm p₂ (s₁ a (mem_perm (perm.symm p₁) i)))
     (λ s₂ a i, mem_perm (perm.symm p₂) (s₂ a (mem_perm p₁ i)))))
 
-infix [priority finset.prio] `⊆` := subset
+infix [priority finset.prio] ⊆ := subset
 
 theorem empty_subset (s : finset A) : ∅ ⊆ s :=
 quot.induction_on s (λ l, list.nil_sub (elt_of l))

library/data/finset/comb.lean

@@ -125,7 +125,7 @@ quot.lift_on s
     (list.nodup_filter p (subtype.has_property l)))
   (λ l₁ l₂ u, quot.sound (perm.perm_filter u))
 
-notation [priority finset.prio] `{` binder ∈ s `|` r:(scoped:1 p, sep p s) `}` := r
+notation [priority finset.prio] `{` binder ` ∈ ` s ` | ` r:(scoped:1 p, sep p s) `}` := r
 
 theorem sep_empty : sep p ∅ = ∅ := rfl
 

library/data/hf.lean

@@ -64,7 +64,7 @@ of_finset (finset.insert a (to_finset s))
 definition mem (a : hf) (s : hf) : Prop :=
 finset.mem a (to_finset s)
 
-infix `∈` := mem
+infix ∈ := mem
 notation [priority finset.prio] a ∉ b := ¬ mem a b
 
 lemma insert_lt_of_not_mem {a s : hf} : a ∉ s → s < insert a s :=
@@ -314,7 +314,7 @@ end
 definition subset (s₁ s₂ : hf) : Prop :=
 finset.subset (to_finset s₁) (to_finset s₂)
 
-infix [priority hf.prio] `⊆` := subset
+infix [priority hf.prio] ⊆ := subset
 
 theorem empty_subset (s : hf) : ∅ ⊆ s :=
 begin unfold [empty, subset], rewrite to_finset_of_finset, apply finset.empty_subset (to_finset s) end

library/data/int/basic.lean

@@ -137,7 +137,7 @@ theorem eq_zero_of_nat_abs_eq_zero : Π {a : ℤ}, nat_abs a = 0 → a = 0
 
 protected definition equiv (p q : ℕ × ℕ) : Prop :=  pr1 p + pr2 q = pr2 p + pr1 q
 
-local infix `≡` := int.equiv
+local infix ≡ := int.equiv
 
 protected theorem equiv.refl [refl] {p : ℕ × ℕ} : p ≡ p := !add.comm
 

library/data/int/div.lean

@@ -26,7 +26,7 @@ notation [priority int.prio] a div b := divide a b
 
 definition modulo (a b : ℤ) : ℤ := a - a div b * b
 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
+notation [priority int.prio] a ≡ b `[mod `:100 c `]`:0 := a mod c = b mod c
 
 /- div  -/
 

library/data/int/power.lean

@@ -19,7 +19,7 @@ section migrate_algebra
   definition pow (a : ℤ) (n : ℕ) : ℤ := algebra.pow a n
   infix [priority int.prio] ^ := pow
   definition nmul (n : ℕ) (a : ℤ) : ℤ := algebra.nmul n a
-  infix [priority int.prio] `⬝` := nmul
+  infix [priority int.prio] ⬝ := nmul
   definition imul (i : ℤ) (a : ℤ) : ℤ := algebra.imul i a
 
   migrate from algebra with int

library/data/list/basic.lean

@@ -352,7 +352,7 @@ assume P, and.intro (ne_of_not_mem_cons P) (not_mem_of_not_mem_cons P)
 
 definition sublist (l₁ l₂ : list T) := ∀ ⦃a : T⦄, a ∈ l₁ → a ∈ l₂
 
-infix `⊆` := sublist
+infix ⊆ := sublist
 
 theorem nil_sub [simp] (l : list T) : [] ⊆ l :=
 λ b i, false.elim (iff.mp (mem_nil_iff b) i)

library/data/nat/div.lean

@@ -78,7 +78,7 @@ if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y else x
 
 definition modulo := fix mod.F
 notation a mod b := modulo a b
-notation a `≡` b `[mod`:100 c `]`:0 := a mod c = b mod c
+notation a ≡ b `[mod `:100 c `]`:0 := a mod c = b mod c
 
 theorem modulo_def (x y : nat) : modulo x y = if 0 < y ∧ y ≤ x then modulo (x - y) y else x :=
 congr_fun (fix_eq mod.F x) y

library/data/pnat.lean

@@ -27,18 +27,18 @@ theorem pnat_pos (p : ℕ+) : p~ > 0 := has_property p
 
 definition add (p q : ℕ+) : ℕ+ :=
   tag (p~ + q~) (nat.add_pos (pnat_pos p) (pnat_pos q))
-infix `+` := add
+infix + := add
 
 definition mul (p q : ℕ+) : ℕ+ :=
   tag (p~ * q~) (nat.mul_pos (pnat_pos p) (pnat_pos q))
-infix `*` := mul
+infix * := mul
 
 definition le (p q : ℕ+) := p~ ≤ q~
-infix `≤` := le
-notation p `≥` q := q ≤ p
+infix ≤ := le
+notation p ≥ q := q ≤ p
 
 definition lt (p q : ℕ+) := p~ < q~
-infix `<` := lt
+infix < := lt
 
 protected theorem pnat.eq {p q : ℕ+} : p~ = q~ → p = q :=
   subtype.eq

library/data/rat/basic.lean

@@ -22,7 +22,7 @@ namespace prerat
 
 definition equiv (a b : prerat) : Prop := num a * denom b = num b * denom a
 
-infix `≡` := equiv
+infix ≡ := equiv
 
 theorem equiv.refl [refl] (a : prerat) : a ≡ a := rfl
 
@@ -539,12 +539,12 @@ section migrate_algebra
   local attribute rat.discrete_field [instance]
 
   definition divide (a b : rat) := algebra.divide a b
-  infix [priority rat.prio] `/` := divide
+  infix [priority rat.prio] / := divide
 
   definition pow (a : ℚ) (n : ℕ) : ℚ := algebra.pow a n
   infix [priority rat.prio] ^ := pow
   definition nmul (n : ℕ) (a : ℚ) : ℚ := algebra.nmul n a
-  infix [priority rat.prio] `⬝` := nmul
+  infix [priority rat.prio] ⬝ := nmul
   definition imul (i : ℤ) (a : ℚ) : ℚ := algebra.imul i a
 
   migrate from algebra with rat

library/data/real/basic.lean

@@ -936,16 +936,16 @@ definition requiv.trans (s t u : reg_seq) (H : requiv s t) (H2 : requiv t u) : r
 definition radd (s t : reg_seq) : reg_seq :=
   reg_seq.mk (sadd (reg_seq.sq s) (reg_seq.sq t))
              (reg_add_reg (reg_seq.is_reg s) (reg_seq.is_reg t))
-infix `+` := radd
+infix + := radd
 
 definition rmul (s t : reg_seq) : reg_seq :=
   reg_seq.mk (smul (reg_seq.sq s) (reg_seq.sq t))
              (reg_mul_reg (reg_seq.is_reg s) (reg_seq.is_reg t))
-infix `*` := rmul
+infix * := rmul
 
 definition rneg (s : reg_seq) : reg_seq :=
   reg_seq.mk (sneg (reg_seq.sq s)) (reg_neg_reg (reg_seq.is_reg s))
-prefix `-` := rneg
+prefix - := rneg
 
 definition radd_well_defined {s t u v : reg_seq} (H : requiv s u) (H2 : requiv t v) :
            requiv (s + t) (u + v) :=
@@ -1025,13 +1025,13 @@ definition add (x y : ℝ) : ℝ :=
                      (take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
                        quot.sound (radd_well_defined Hab Hcd)))
 protected definition prio := num.pred rat.prio
-infix [priority real.prio] `+` := add
+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 [priority real.prio] `*` := 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,

library/data/set/basic.lean

@@ -15,17 +15,17 @@ variable {X : Type}
 /- membership and subset -/
 
 definition mem [reducible] (x : X) (a : set X) := a x
-infix `∈` := mem
+infix ∈ := mem
 notation a ∉ b := ¬ mem a b
 
 theorem ext {a b : set X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
 funext (take x, propext (H x))
 
 definition subset (a b : set X) := ∀⦃x⦄, x ∈ a → x ∈ b
-infix `⊆` := subset
+infix ⊆ := subset
 
 definition superset [reducible] (s t : set X) : Prop := t ⊆ s
-infix `⊇` := superset
+infix ⊇ := superset
 
 theorem subset.refl (a : set X) : a ⊆ a := take x, assume H, H
 
@@ -239,7 +239,7 @@ notation `{` binder `|` r:(scoped:1 P, set_of P) `}` := r
 
 -- {x ∈ s | P}
 definition sep (P : X → Prop) (s : set X) : set X := λx, x ∈ s ∧ P x
-notation `{` binder ∈ s `|` r:(scoped:1 p, sep p s) `}` := r
+notation `{` binder ` ∈ ` s ` | ` r:(scoped:1 p, sep p s) `}` := r
 
 /- insert -/
 

library/data/sum.lean

@@ -14,7 +14,7 @@ namespace sum
   notation A + B := sum A B
   namespace low_precedence_plus
     reserve infixr ` + `:25  -- conflicts with notation for addition
-    infixr `+` := sum
+    infixr + := sum
   end low_precedence_plus
 
   variables {A B : Type}

library/init/logic.lean

@@ -233,7 +233,7 @@ and.rec H₂ H₁
 
 /- or -/
 
-notation a `\/` b := or a b
+notation a \/ b := or a b
 notation a ∨ b := or a b
 
 namespace or

library/init/nat.lean

@@ -16,15 +16,15 @@ namespace nat
   | refl : le a a
   | step : Π {b}, le a b → le a (succ b)
 
-  infix `≤` := le
+  infix ≤ := le
   attribute le.refl [refl]
 
   definition lt [reducible] (n m : ℕ) := succ n ≤ m
   definition ge [reducible] (n m : ℕ) := m ≤ n
   definition gt [reducible] (n m : ℕ) := succ m ≤ n
-  infix `<` := lt
-  infix `≥` := ge
-  infix `>` := gt
+  infix < := lt
+  infix ≥ := ge
+  infix > := gt
 
   definition pred [unfold 1] (a : nat) : nat :=
   nat.cases_on a zero (λ a₁, a₁)