diff --git a/examples/logic.pg b/examples/logic.pg
index c1da0c2..834a3a3 100644
--- a/examples/logic.pg
+++ b/examples/logic.pg
@@ -11,7 +11,7 @@ def false_elim (A : ★) (contra : false) : A := contra A;
 
 -- True
 
-def true : ★ := forall (A : ★), A → A;
+def true : ★ := forall (A : ★), A -> A;
 
 def true_intro : true := [A : ★][x : A] x;
 
@@ -19,10 +19,10 @@ def true_intro : true := [A : ★][x : A] x;
 
 -- Negation
 
-def not (A : ★) : ★ := A → false;
+def not (A : ★) : ★ := A -> false;
 
 -- introduction rule (kinda just the definition)
-def not_intro (A : ★) (h : A → false) : not A := h;
+def not_intro (A : ★) (h : A -> false) : not A := h;
 
 -- elimination rule
 def not_elim (A B : ★) (a : A) (na : not A) : B := na a B;
@@ -35,11 +35,11 @@ def double_neg_intro (A : ★) (a : A) : not (not A) :=
 
 -- Conjunction
 
-def ∧ (A B : ★) : ★ := {A × B};
+def ∧ (A B : ★) : ★ := A × B;
 infixl 10 ∧;
 
 -- introduction rule
-def and_intro (A B : ★) (a : A) (b : B) : A ∧ B := <a, b>;
+def and_intro (A B : ★) (a : A) (b : B) : A ∧ B := (a, b);
 
 -- left elimination rule
 def and_elim_l (A B : ★) (ab : A ∧ B) : A := π₁ ab;
@@ -52,19 +52,19 @@ def and_elim_r (A B : ★) (ab : A ∧ B) : B := π₂ ab;
 -- Disjunction
 
 -- 2nd order disjunction
-def ∨ (A B : ★) : ★ := forall (C : ★), (A → C) → (B → C) → C;
+def ∨ (A B : ★) : ★ := forall (C : ★), (A -> C) -> (B -> C) -> C;
 infixl 5 ∨;
 
 -- left introduction rule
 def or_intro_l (A B : ★) (a : A) : A ∨ B :=
-    fun (C : ★) (ha : A → C) (hb : B → C) => ha a;
+    fun (C : ★) (ha : A -> C) (hb : B -> C) => ha a;
 
 -- right introduction rule
 def or_intro_r (A B : ★) (b : B) : A ∨ B :=
-    fun (C : ★) (ha : A → C) (hb : B → C) => hb b;
+    fun (C : ★) (ha : A -> C) (hb : B -> C) => hb b;
 
 -- elimination rule (kinda just the definition)
-def or_elim (A B C : ★) (ab : A ∨ B) (ha : A → C) (hb : B → C) : C :=
+def or_elim (A B C : ★) (ab : A ∨ B) (ha : A -> C) (hb : B -> C) : C :=
     ab C ha hb;
 
 -- --------------------------------------------------------------------------------------------------------------
@@ -72,14 +72,14 @@ def or_elim (A B C : ★) (ab : A ∨ B) (ha : A → C) (hb : B → C) : C :=
 -- Existential
 
 -- 2nd order existential
-def exists (A : ★) (P : A → ★) : ★ := forall (C : ★), (forall (x : A), P x → C) → C;
+def exists (A : ★) (P : A -> ★) : ★ := forall (C : ★), (forall (x : A), P x -> C) -> C;
 
 -- introduction rule
-def exists_intro (A : ★) (P : A → ★) (a : A) (h : P a) : exists A P :=
-    fun (C : ★) (g : forall (x : A), P x → C) => g a h;
+def exists_intro (A : ★) (P : A -> ★) (a : A) (h : P a) : exists A P :=
+    fun (C : ★) (g : forall (x : A), P x -> C) => g a h;
 
 -- elimination rule (kinda just the definition)
-def exists_elim (A B : ★) (P : A → ★) (ex_a : exists A P) (h : forall (a : A), P a → B) : B :=
+def exists_elim (A B : ★) (P : A -> ★) (ex_a : exists A P) (h : forall (a : A), P a -> B) : B :=
     ex_a B h;
 
 -- --------------------------------------------------------------------------------------------------------------
@@ -87,53 +87,53 @@ def exists_elim (A B : ★) (P : A → ★) (ex_a : exists A P) (h : forall (a :
 -- Universal
 
 -- 2nd order universal (just ∏, including it for completeness)
-def all (A : ★) (P : A → ★) : ★ := forall (a : A), P a;
+def all (A : ★) (P : A -> ★) : ★ := forall (a : A), P a;
 
 -- introduction rule
-def all_intro (A : ★) (P : A → ★) (h : forall (a : A), P a) : all A P := h;
+def all_intro (A : ★) (P : A -> ★) (h : forall (a : A), P a) : all A P := h;
 
 -- elimination rule
-def all_elim (A : ★) (P : A → ★) (h_all : all A P) (a : A) : P a := h_all a;
+def all_elim (A : ★) (P : A -> ★) (h_all : all A P) (a : A) : P a := h_all a;
 
 -- --------------------------------------------------------------------------------------------------------------
 
 -- Equality
 
 -- 2nd order Leibniz equality
-def eq (A : ★) (x y : A) := forall (P : A → ★), P x → P y;
+def eq (A : ★) (x y : A) := forall (P : A -> ★), P x -> P y;
 
 -- equality is reflexive
-def eq_refl (A : ★) (x : A) : eq A x x := fun (P : A → ★) (Hx : P x) => Hx;
+def eq_refl (A : ★) (x : A) : eq A x x := fun (P : A -> ★) (Hx : P x) => Hx;
 
 -- equality is symmetric
-def eq_sym (A : ★) (x y : A) (Hxy : eq A x y) : eq A y x := fun (P : A → ★) (Hy : P y) =>
-    Hxy (fun (z : A) => P z → P x) (fun (Hx : P x) => Hx) Hy;
+def eq_sym (A : ★) (x y : A) (Hxy : eq A x y) : eq A y x := fun (P : A -> ★) (Hy : P y) =>
+    Hxy (fun (z : A) => P z -> P x) (fun (Hx : P x) => Hx) Hy;
 
 -- equality is transitive
-def eq_trans (A : ★) (x y z : A) (Hxy : eq A x y) (Hyz : eq A y z) : eq A x z := fun (P : A → ★) (Hx : P x) =>
+def eq_trans (A : ★) (x y z : A) (Hxy : eq A x y) (Hyz : eq A y z) : eq A x z := fun (P : A -> ★) (Hx : P x) =>
     Hyz P (Hxy P Hx);
 
 -- equality is a universal congruence
-def eq_cong (A B : ★) (x y : A) (f : A → B) (H : eq A x y) : eq B (f x) (f y) :=
-  fun (P : B → ★) (Hfx : P (f x)) =>
+def eq_cong (A B : ★) (x y : A) (f : A -> B) (H : eq A x y) : eq B (f x) (f y) :=
+  fun (P : B -> ★) (Hfx : P (f x)) =>
       H (fun (a : A) => P (f a)) Hfx;
 
 -- --------------------------------------------------------------------------------------------------------------
 
 -- unique existence
-def exists_uniq (A : ★) (P : A → ★) : ★ :=
-    exists A (fun (x : A) => P x ∧ (forall (y : A), P y → eq A x y));
+def exists_uniq (A : ★) (P : A -> ★) : ★ :=
+    exists A (fun (x : A) => P x ∧ (forall (y : A), P y -> eq A x y));
 
-def exists_uniq_elim (A B : ★) (P : A → ★) (ex_a : exists_uniq A P) (h : forall (a : A), P a → (forall (y : A), P y → eq A a y) → B) : B :=
-    exists_elim A B (fun (x : A) => P x ∧ (forall (y : A), P y → eq A x y)) ex_a
-    (fun (a : A) (h2 : P a ∧ (forall (y : A), P y → eq A a y)) =>
-        h a (and_elim_l (P a) (forall (y : A), P y → eq A a y) h2)
-            (and_elim_r (P a) (forall (y : A), P y → eq A a y) h2));
+def exists_uniq_elim (A B : ★) (P : A -> ★) (ex_a : exists_uniq A P) (h : forall (a : A), P a -> (forall (y : A), P y -> eq A a y) -> B) : B :=
+    exists_elim A B (fun (x : A) => P x ∧ (forall (y : A), P y -> eq A x y)) ex_a
+    (fun (a : A) (h2 : P a ∧ (forall (y : A), P y -> eq A a y)) =>
+        h a (and_elim_l (P a) (forall (y : A), P y -> eq A a y) h2)
+            (and_elim_r (P a) (forall (y : A), P y -> eq A a y) h2));
 
 def exists_uniq_t (A : ★) : ★ :=
     exists A (fun (x : A) => forall (y : A), eq A x y);
 
-def exists_uniq_t_elim (A B : ★) (ex_a : exists_uniq_t A) (h : forall (a : A), (forall (y : A), eq A a y) → B) : B :=
+def exists_uniq_t_elim (A B : ★) (ex_a : exists_uniq_t A) (h : forall (a : A), (forall (y : A), eq A a y) -> B) : B :=
     exists_elim A B (fun (x : A) => forall (y : A), eq A x y) ex_a (fun (a : A) (h2 : forall (y : A), eq A a y) => h a h2);
 
 -- --------------------------------------------------------------------------------------------------------------
@@ -146,8 +146,8 @@ section Theorems
 
     -- ~(A ∨ B) => ~A ∧ ~B
     def de_morgan1 (h : not (A ∨ B)) : not A ∧ not B :=
-        <[a : A] h (or_intro_l A B a)
-        ,[b : B] h (or_intro_r A B b)>;
+        ( [a : A] h (or_intro_l A B a)
+        , [b : B] h (or_intro_r A B b));
 
     -- ~A ∧ ~B => ~(A ∨ B)
     def de_morgan2 (h : not A ∧ not B) : not (A ∨ B) :=
@@ -164,7 +164,7 @@ section Theorems
     -- the last one (~(A ∧ B) => ~A ∨ ~B) is not possible constructively
 
     -- A ∧ B => B ∧ A
-    def and_comm (h : A ∧ B) : B ∧ A := <π₂ h, π₁ h>;
+    def and_comm (h : A ∧ B) : B ∧ A := (π₂ h, π₁ h);
 
     -- A ∨ B => B ∨ A
     def or_comm (h : A ∨ B) : B ∨ A :=
@@ -174,11 +174,11 @@ section Theorems
 
     -- A ∧ (B ∧ C) => (A ∧ B) ∧ C
     def and_assoc_l (h : A ∧ (B ∧ C)) : (A ∧ B) ∧ C :=
-        <<π₁ h, π₁ (π₂ h)>, π₂ (π₂ h)>;
+        ((π₁ h, π₁ (π₂ h)), π₂ (π₂ h));
 
     -- (A ∧ B) ∧ C => A ∧ (B ∧ C)
     def and_assoc_r (h : (A ∧ B) ∧ C) : A ∧ (B ∧ C) :=
-        <π₁ (π₁ h), <π₂ (π₁ h), π₂ h>>;
+        (π₁ (π₁ h), (π₂ (π₁ h), π₂ h));
 
     -- A ∨ (B ∨ C) => (A ∨ B) ∨ C
     def or_assoc_l (h : A ∨ (B ∨ C)) : (A ∨ B) ∨ C :=
@@ -201,14 +201,14 @@ section Theorems
     -- A ∧ (B ∨ C) => A ∧ B ∨ A ∧ C
     def and_distrib_l_or (h : A ∧ (B ∨ C)) : A ∧ B ∨ A ∧ C :=
         or_elim B C (A ∧ B ∨ A ∧ C) (π₂ h)
-            (fun (b : B) => or_intro_l (A ∧ B) (A ∧ C) <π₁ h, b>)
-            (fun (c : C) => or_intro_r (A ∧ B) (A ∧ C) <π₁ h, c>);
+            (fun (b : B) => or_intro_l (A ∧ B) (A ∧ C) (π₁ h, b))
+            (fun (c : C) => or_intro_r (A ∧ B) (A ∧ C) (π₁ h, c));
 
     -- A ∧ B ∨ A ∧ C => A ∧ (B ∨ C)
     def and_factor_l_or (h : A ∧ B ∨ A ∧ C) : A ∧ (B ∨ C) :=
         or_elim (A ∧ B) (A ∧ C) (A ∧ (B ∨ C)) h
-            (fun (ab : A ∧ B) => <π₁ ab, or_intro_l B C (π₂ ab)>)
-            (fun (ac : A ∧ C) => <π₁ ac, or_intro_r B C (π₂ ac)>);
+            (fun (ab : A ∧ B) => (π₁ ab, or_intro_l B C (π₂ ab)))
+            (fun (ac : A ∧ C) => (π₁ ac, or_intro_r B C (π₂ ac)));
 
     -- Thanks Quinn!
     -- A ∨ B => ~B => A
@@ -216,7 +216,7 @@ section Theorems
         or_elim A B A hor ([a : A] a) ([b : B] nb b A);
 
     -- (A => B) => ~B => ~A
-    def contrapositive (f : A → B) (nb : not B) : not A :=
+    def contrapositive (f : A -> B) (nb : not B) : not A :=
         fun (a : A) => nb (f a);
 
 end Theorems
diff --git a/lib/Parser.hs b/lib/Parser.hs
index e0605ef..6b7b303 100644
--- a/lib/Parser.hs
+++ b/lib/Parser.hs
@@ -17,12 +17,25 @@ import qualified Text.Megaparsec.Char.Lexer as L
 newtype TypeError = TE Error
     deriving (Eq, Ord)
 
+data InfixDef = InfixDef
+    { infixFixity :: Fixity
+    , infixOp :: Text -> IRExpr -> IRExpr -> IRExpr
+    }
+
 data Fixity
     = InfixL Int
     | InfixR Int
     deriving (Eq, Show)
 
-type Operators = Map Text Fixity
+type Operators = Map Text InfixDef
+
+initialOps :: Operators
+initialOps =
+    M.fromAscList
+        [ ("→", InfixDef (InfixR 2) (const $ Pi ""))
+        , ("->", InfixDef (InfixR 2) (const $ Pi ""))
+        , ("×", InfixDef (InfixL 10) (const Prod))
+        ]
 
 type Parser = ParsecT TypeError Text (State Operators)
 
@@ -46,7 +59,7 @@ symbol :: Text -> Parser ()
 symbol = void . L.symbol skipSpace
 
 symbols :: String
-symbols = "!@#$%^&*-+=<>,./?[]{}\\|`~'\"∧∨⊙×≅"
+symbols = "→!@#$%^&*-+=<>,./?[]{}\\|`~'\"∧∨⊙×≅"
 
 pKeyword :: Text -> Parser ()
 pKeyword keyword = void $ lexeme (string keyword <* notFollowedBy alphaNumChar)
@@ -174,16 +187,11 @@ pSort = lexeme $ pStar <|> pSquare
 pOpSection :: Parser IRExpr
 pOpSection = lexeme $ parens $ Var <$> pSymbol
 
-pProd :: Parser IRExpr
-pProd = lexeme $ between (char '{') (char '}') $ do
-    left <- pIRExpr
-    _ <- symbol "×"
-    Prod left <$> pIRExpr
-
 pPair :: Parser IRExpr
-pPair = lexeme $ between (char '<') (char '>') $ do
+pPair = lexeme $ between (char '(') (char ')') $ do
+    skipSpace
     left <- pIRExpr
-    _ <- symbol ","
+    _ <- lexeme $ symbol ","
     Pair left <$> pIRExpr
 
 pPi1 :: Parser IRExpr
@@ -193,7 +201,7 @@ pPi2 :: Parser IRExpr
 pPi2 = lexeme $ symbol "π₂" >> Pi2 <$> pIRExpr
 
 pTerm :: Parser IRExpr
-pTerm = lexeme $ label "term" $ choice [pSort, pPi1, pPi2, pPureVar, pVar, pProd, pPair, try pOpSection, parens pIRExpr]
+pTerm = lexeme $ label "term" $ choice [pSort, pPi1, pPi2, pPureVar, pVar, try pPair, try pOpSection, parens pIRExpr]
 
 pInfix :: Parser IRExpr
 pInfix = parseWithPrec 0
@@ -206,7 +214,7 @@ pInfix = parseWithPrec 0
         op <- lookAhead pSymbol
         operators <- get
         case M.lookup op operators of
-            Just fixity -> do
+            Just (InfixDef fixity opFun) -> do
                 let (opPrec, nextPrec) = case fixity of
                         InfixL p -> (p, p)
                         InfixR p -> (p, p + 1)
@@ -215,16 +223,19 @@ pInfix = parseWithPrec 0
                     else do
                         _ <- pSymbol
                         rhs <- parseWithPrec nextPrec
-                        continue prec (App (App (Var op) lhs) rhs)
+                        continue prec $ opFun op lhs rhs
             Nothing -> fail $ "unknown operator '" ++ toString op ++ "'"
 
-pAppTerm :: Parser IRExpr
-pAppTerm = lexeme $ choice [pLAbs, pALAbs, pPAbs, pLet, pInfix]
-
 pIRExpr :: Parser IRExpr
-pIRExpr = lexeme $ do
-    e <- pAppTerm
-    option e $ (symbol "->" <|> symbol "→") >> Pi "" e <$> pIRExpr
+pIRExpr = lexeme $ choice [pLAbs, pALAbs, pPAbs, pLet, pInfix]
+
+-- pAppTerm :: Parser IRExpr
+-- pAppTerm = lexeme $ choice [pLAbs, pALAbs, pPAbs, pLet, pInfix]
+--
+-- pIRExpr :: Parser IRExpr
+-- pIRExpr = lexeme $ do
+--     e <- pAppTerm
+--     option e $ (symbol "->" <|> symbol "→") >> Pi "" e <$> pIRExpr
 
 pAscription :: Parser IRExpr
 pAscription = lexeme $ try $ symbol ":" >> label "type" pIRExpr
@@ -232,7 +243,7 @@ pAscription = lexeme $ try $ symbol ":" >> label "type" pIRExpr
 pAxiom :: Parser IRDef
 pAxiom = lexeme $ label "axiom" $ do
     pKeyword "axiom"
-    ident <- pIdentifier
+    ident <- pIdentifier <|> pSymbol
     params <- pManyParams
     ascription <- fmap (flip (foldr mkPi) params) pAscription
     symbol ";"
@@ -265,7 +276,7 @@ pFixityDec = do
             , InfixR <$> (lexeme (char 'r') >> lexeme L.decimal)
             ]
     ident <- pSymbol
-    modify (M.insert ident fixity)
+    modify $ M.insert ident $ InfixDef fixity $ (App .) . App . Var
     symbol ";"
 
 pSection :: Parser IRSectionDef
@@ -284,7 +295,7 @@ pIRProgram :: Parser IRProgram
 pIRProgram = skipSpace >> concat <$> some pIRDef
 
 parserWrapper :: Parser a -> String -> Text -> Either String a
-parserWrapper p filename input = first errorBundlePretty $ evalState (runParserT p filename input) M.empty
+parserWrapper p filename input = first errorBundlePretty $ evalState (runParserT p filename input) initialOps
 
 parseProgram :: String -> Text -> Either String IRProgram
 parseProgram = parserWrapper pIRProgram