diff --git a/library/data/perm.lean b/library/data/perm.lean
index e0154d585..39756db3e 100644
--- a/library/data/perm.lean
+++ b/library/data/perm.lean
@@ -19,130 +19,133 @@ inductive perm : list A → list A → Prop :=
 | trans : Π {l₁ l₂ l₃ : list A}, perm l₁ l₂ → perm l₂ l₃ → perm l₁ l₃
 
 namespace perm
-  theorem eq_nil_of_perm_nil {l₁ : list A} (p : perm [] l₁) : l₁ = [] :=
-  have gen : ∀ (l₂ : list A) (p : perm l₂ l₁), l₂ = [] → l₁ = [], from
-    take l₂ p, perm.induction_on p
-      (λ h, h)
-      (λ x y l₁ l₂ p₁ r₁, list.no_confusion r₁)
-      (λ x y l e, list.no_confusion e)
-      (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
-  gen [] p rfl
-
-  theorem not_perm_nil_cons (x : A) (l : list A) : ¬ perm [] (x::l) :=
-  have gen : ∀ (l₁ l₂ : list A) (p : perm l₁ l₂), l₁ = [] → l₂ = (x::l) → false, from
-    take l₁ l₂ p, perm.induction_on p
-      (λ e₁ e₂, list.no_confusion e₂)
-      (λ x l₁ l₂ p₁ r₁ e₁ e₂, list.no_confusion e₁)
-      (λ x y l e₁ e₂, list.no_confusion e₁)
-      (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e₁ e₂,
-        begin
-          rewrite [e₂ at *, e₁ at *],
-          have e₃ : l₂ = [], from eq_nil_of_perm_nil p₁,
-          exact (r₂ e₃ rfl)
-        end),
-  assume p, gen [] (x::l) p rfl rfl
-
-  protected theorem refl : ∀ (l : list A), perm l l
-  | []      := nil
-  | (x::xs) := skip x (refl xs)
-
-  protected theorem symm : ∀ {l₁ l₂ : list A}, perm l₁ l₂ → perm l₂ l₁ :=
-  take l₁ l₂ p, perm.induction_on p
-    nil
-    (λ x l₁ l₂ p₁ r₁, skip x r₁)
-    (λ x y l, swap y x l)
-    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₂ r₁)
-
-  theorem is_eqv (A : Type) : equivalence (@perm A) :=
-  mk_equivalence (@perm A) (@refl A) (@symm A) (@trans A)
-
-  protected definition is_setoid [instance] (A : Type) : setoid (list A) :=
-  setoid.mk (@perm A) (is_eqv A)
-
-  theorem mem_perm (a : A) (l₁ l₂ : list A) : perm l₁ l₂ → a ∈ l₁ → a ∈ l₂ :=
-  assume p, perm.induction_on p
+infix ~:50 := perm
+theorem eq_nil_of_perm_nil {l₁ : list A} (p : [] ~ l₁) : l₁ = [] :=
+have gen : ∀ (l₂ : list A) (p : l₂ ~ l₁), l₂ = [] → l₁ = [], from
+  take l₂ p, perm.induction_on p
     (λ h, h)
-    (λ x l₁ l₂ p₁ r₁ i, or.elim i
-      (λ aeqx, by rewrite aeqx; apply !mem_cons)
-      (λ ainl₁ : a ∈ l₁, or.inr (r₁ ainl₁)))
-    (λ x y l ainyxl, or.elim ainyxl
-      (λ aeqy  : a = y, by rewrite aeqy; exact (or.inr !mem_cons))
-      (λ ainxl : a ∈ x::l, or.elim ainxl
-        (λ aeqx : a = x, or.inl aeqx)
-        (λ ainl : a ∈ l, or.inr (or.inr ainl))))
-    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ ainl₁, r₂ (r₁ ainl₁))
+    (λ x y l₁ l₂ p₁ r₁, list.no_confusion r₁)
+    (λ x y l e, list.no_confusion e)
+    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
+gen [] p rfl
 
-  theorem perm_app_left {l₁ l₂ : list A} (t₁ : list A) : perm l₁ l₂ → perm (l₁++t₁) (l₂++t₁) :=
-  assume p, perm.induction_on p
-    !refl
-    (λ x l₁ l₂ p₁ r₁, skip x r₁)
-    (λ x y l, !swap)
-    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
-
-  theorem perm_app_right (l : list A) {t₁ t₂ : list A} : perm t₁ t₂ → perm (l++t₁) (l++t₂) :=
-  list.induction_on l
-    (λ p, p)
-    (λ x xs r p, skip x (r p))
-
-  theorem perm_app {l₁ l₂ t₁ t₂ : list A} : perm l₁ l₂ → perm t₁ t₂ → perm (l₁++t₁) (l₂++t₂) :=
-  assume p₁ p₂, trans (perm_app_left t₁ p₁) (perm_app_right l₂ p₂)
-
-  theorem perm_app_cons (a : A) {h₁ h₂ t₁ t₂ : list A} : perm h₁ h₂ → perm t₁ t₂ → perm (h₁ ++ (a::t₁)) (h₂ ++ (a::t₂)) :=
-  assume p₁ p₂, perm_app p₁ (skip a p₂)
-
-  theorem perm_cons_app (a : A) : ∀ (l : list A), perm (a::l) (l ++ [a])
-  | []      := !refl
-  | (x::xs) :=
-    show perm (a::x::xs) (x::(xs ++ [a])),    from
-    have p₁ : perm (a::xs) (xs++[a]),         from perm_cons_app xs,
-    have p₂ : perm (x::a::xs) (x::(xs++[a])), from skip x p₁,
-    have p₃ : perm (a::x::xs) (x::a::xs),     from swap x a xs,
-    trans p₃ p₂
-
-  theorem perm_app_comm {l₁ l₂ : list A} : perm (l₁++l₂) (l₂++l₁) :=
-  list.induction_on l₁
-    (by rewrite [append_nil_right, append_nil_left]; apply refl)
-    (λ a t r,
-      show perm (a::(t++l₂)) (l₂++(a::t)), from
+theorem not_perm_nil_cons (x : A) (l : list A) : ¬ [] ~ (x::l) :=
+have gen : ∀ (l₁ l₂ : list A) (p : l₁ ~ l₂), l₁ = [] → l₂ = (x::l) → false, from
+  take l₁ l₂ p, perm.induction_on p
+    (λ e₁ e₂, list.no_confusion e₂)
+    (λ x l₁ l₂ p₁ r₁ e₁ e₂, list.no_confusion e₁)
+    (λ x y l e₁ e₂, list.no_confusion e₁)
+    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e₁ e₂,
       begin
-        have p₀ : perm (a::(t++l₂)) (a::(l₂++t)),   from skip a r,
-        have p₁ : perm (a::(l₂++t)) (l₂++t++[a]),   from !perm_cons_app,
-        have p₂ : perm (t++[a]) (a::t),             from symm (perm_cons_app a t),
-        have p₃ : perm (l₂++(t++[a])) (l₂++(a::t)), from perm_app_right l₂ p₂,
-        rewrite [append.assoc at p₁],
-        exact (trans p₀ (trans p₁ p₃))
-      end)
+        rewrite [e₂ at *, e₁ at *],
+        have e₃ : l₂ = [], from eq_nil_of_perm_nil p₁,
+        exact (r₂ e₃ rfl)
+      end),
+assume p, gen [] (x::l) p rfl rfl
 
-  theorem length_eq_lenght_of_perm {l₁ l₂ : list A} : perm l₁ l₂ → length l₁ = length l₂ :=
-  assume p, perm.induction_on p
-    rfl
-    (λ x l₁ l₂ p r, by rewrite [*length_cons, r])
-    (λ x y l, by rewrite *length_cons)
-    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, eq.trans r₁ r₂)
+protected theorem refl : ∀ (l : list A), l ~ l
+| []      := nil
+| (x::xs) := skip x (refl xs)
 
-  theorem eq_singlenton_of_perm_inv (a : A) {l : list A} : perm [a] l → l = [a] :=
-  have gen : ∀ l₂, perm l₂ l → l₂ = [a] → l = [a], from
-     take l₂, assume p, perm.induction_on p
-       (λ e, e)
-       (λ x l₁ l₂ p r e, list.no_confusion e (λ (e₁ : x = a) (e₂ : l₁ = []),
-         begin
-           rewrite [e₁, e₂ at p],
-           have h₁ : l₂ = [], from eq_nil_of_perm_nil p,
-           rewrite h₁
-         end))
-       (λ x y l e, list.no_confusion e (λ e₁ e₂, list.no_confusion e₂))
-       (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
-  assume p, gen [a] p rfl
+protected theorem symm : ∀ {l₁ l₂ : list A}, l₁ ~ l₂ → l₂ ~ l₁ :=
+take l₁ l₂ p, perm.induction_on p
+  nil
+  (λ x l₁ l₂ p₁ r₁, skip x r₁)
+  (λ x y l, swap y x l)
+  (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₂ r₁)
 
-  theorem eq_singlenton_of_perm (a b : A) : perm [a] [b] → a = b :=
-  assume p, list.no_confusion (eq_singlenton_of_perm_inv a p) (λ e₁ e₂, by rewrite e₁)
+theorem eqv (A : Type) : equivalence (@perm A) :=
+mk_equivalence (@perm A) (@perm.refl A) (@perm.symm A) (@perm.trans A)
 
-  theorem perm_rev : ∀ (l : list A), perm l (reverse l)
-  | []      := nil
-  | (x::xs) :=
-    begin
-     rewrite [reverse_cons, concat_eq_append],
-     apply (trans (perm_cons_app x xs)),
-     exact (perm_app_left [x] (perm_rev xs))
-    end
+protected definition is_setoid [instance] (A : Type) : setoid (list A) :=
+setoid.mk (@perm A) (perm.eqv A)
+
+calc_refl  perm.refl
+calc_symm  perm.symm
+calc_trans perm.trans
+
+-- TODO: remove this theorems after we improve calc
+theorem perm_of_eq_of_perm (l₁ l₂ l₃ : list A) : l₁ = l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=
+assume e p, eq.rec_on (eq.symm e) p
+theorem perm_of_perm_of_eq (l₁ l₂ l₃ : list A) : l₁ ~ l₂ → l₂ = l₃ → l₁ ~ l₃ :=
+assume p e, eq.rec_on e p
+calc_trans perm_of_perm_of_eq
+calc_trans perm_of_eq_of_perm
+
+theorem mem_perm (a : A) (l₁ l₂ : list A) : l₁ ~ l₂ → a ∈ l₁ → a ∈ l₂ :=
+assume p, perm.induction_on p
+  (λ h, h)
+  (λ x l₁ l₂ p₁ r₁ i, or.elim i
+    (assume aeqx : a = x,   by rewrite aeqx; apply !mem_cons)
+    (assume ainl₁ : a ∈ l₁, or.inr (r₁ ainl₁)))
+  (λ x y l ainyxl, or.elim ainyxl
+    (assume aeqy  : a = y, by rewrite aeqy; exact (or.inr !mem_cons))
+    (assume ainxl : a ∈ x::l, or.elim ainxl
+      (assume aeqx : a = x, or.inl aeqx)
+      (assume ainl : a ∈ l, or.inr (or.inr ainl))))
+  (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ ainl₁, r₂ (r₁ ainl₁))
+
+theorem perm_app_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (l₁++t₁) ~ (l₂++t₁) :=
+assume p, perm.induction_on p
+  !refl
+  (λ x l₁ l₂ p₁ r₁, skip x r₁)
+  (λ x y l, !swap)
+  (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
+
+theorem perm_app_right (l : list A) {t₁ t₂ : list A} : t₁ ~ t₂ → (l++t₁) ~ (l++t₂) :=
+list.induction_on l
+  (λ p, p)
+  (λ x xs r p, skip x (r p))
+
+theorem perm_app {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (l₁++t₁) ~ (l₂++t₂) :=
+assume p₁ p₂, trans (perm_app_left t₁ p₁) (perm_app_right l₂ p₂)
+
+theorem perm_app_cons (a : A) {h₁ h₂ t₁ t₂ : list A} : h₁ ~ h₂ → t₁ ~ t₂ → (h₁ ++ (a::t₁)) ~ (h₂ ++ (a::t₂)) :=
+assume p₁ p₂, perm_app p₁ (skip a p₂)
+
+theorem perm_cons_app (a : A) : ∀ (l : list A), (a::l) ~ (l ++ [a])
+| []      := !refl
+| (x::xs) := calc
+  a::x::xs ~ x::a::xs     : swap x a xs
+       ... ~ x::(xs++[a]) : skip x (perm_cons_app xs)
+
+theorem perm_app_comm {l₁ l₂ : list A} : (l₁++l₂) ~ (l₂++l₁) :=
+list.induction_on l₁
+  (by rewrite [append_nil_right, append_nil_left]; apply refl)
+  (λ a t r, calc
+    a::(t++l₂) ~ a::(l₂++t)   : skip a r
+          ...  ~ l₂++t++[a]   : perm_cons_app
+          ...  = l₂++(t++[a]) : append.assoc
+          ...  ~ l₂++(a::t)   : perm_app_right l₂ (symm (perm_cons_app a t)))
+
+theorem length_eq_length_of_perm {l₁ l₂ : list A} : l₁ ~ l₂ → length l₁ = length l₂ :=
+assume p, perm.induction_on p
+  rfl
+  (λ x l₁ l₂ p r, by rewrite [*length_cons, r])
+  (λ x y l, by rewrite *length_cons)
+  (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, eq.trans r₁ r₂)
+
+theorem eq_singlenton_of_perm_inv (a : A) {l : list A} : [a] ~ l → l = [a] :=
+have gen : ∀ l₂, perm l₂ l → l₂ = [a] → l = [a], from
+  take l₂, assume p, perm.induction_on p
+    (λ e, e)
+    (λ x l₁ l₂ p r e, list.no_confusion e (λ (e₁ : x = a) (e₂ : l₁ = []),
+      begin
+        rewrite [e₁, e₂ at p],
+        have h₁ : l₂ = [], from eq_nil_of_perm_nil p,
+        rewrite h₁
+      end))
+    (λ x y l e, list.no_confusion e (λ e₁ e₂, list.no_confusion e₂))
+    (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
+assume p, gen [a] p rfl
+
+theorem eq_singlenton_of_perm (a b : A) : [a] ~ [b] → a = b :=
+assume p, list.no_confusion (eq_singlenton_of_perm_inv a p) (λ e₁ e₂, by rewrite e₁)
+
+theorem perm_rev : ∀ (l : list A), l ~ (reverse l)
+| []      := nil
+| (x::xs) := calc
+  x::xs ~ xs++[x]           : perm_cons_app x xs
+    ... ~ reverse xs ++ [x] : perm_app_left [x] (perm_rev xs)
+    ... = reverse (x::xs)   : by rewrite [reverse_cons, concat_eq_append]
 end perm