coq: change notation brackets for terms

coq/subtype.v

@@ -61,10 +61,14 @@ Open Scope ladder_type_scope.
 Open Scope ladder_expr_scope.
 
 Example sub0 :
-  [   < $"Seq"$ < $"Digit"$ $"10"$ > >
+  [<
-    ~ < $"Seq"$ $"Char"$ > ]
+       < $"Seq"$ < $"Digit"$ $"10"$ > >
+     ~ < $"Seq"$ $"Char"$ >
+  >]
 :<=
-  [   < $"Seq"$ $"Char"$ > ]
+  [<
+       < $"Seq"$ $"Char"$ >
+  >]
 .
 Proof.
   apply TSubRepr_Ladder.
@@ -74,13 +78,13 @@ Qed.
 
 
 Example sub1 :
-    [ < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > ]
+    [< < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > >]
-:<= [ < $"Seq"$ $"Char"$ > ]
+:<= [< < $"Seq"$ $"Char"$ > >]
 .
 Proof.
-  set [ < $"Seq"$ < $"Digit"$ $"10"$ > > ~ < $"Seq"$ $"Char"$ > ].
+  set [< < $"Seq"$ < $"Digit"$ $"10"$ > > ~ < $"Seq"$ $"Char"$ > >].
-	set [ < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > ].
+	set [< < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > >].
-	set [ < $"Seq"$ $"Char"$ > ].
+	set [< < $"Seq"$ $"Char"$ > >].
 	set (t0 === t).
 	set (t :<= t0).
 	set (t :<= t1).

coq/terms.v

@@ -50,7 +50,7 @@ Declare Scope ladder_expr_scope.
 Declare Custom Entry ladder_type.
 Declare Custom Entry ladder_expr.
 
-Notation "[ t ]" := t
+Notation "[< t >]" := t
   (t custom ladder_type at level 80) : ladder_type_scope.
 Notation "'∀' x ',' t" := (type_univ x t)
   (t custom ladder_type at level 80, in custom ladder_type at level 80, x constr).
@@ -69,7 +69,7 @@ Notation "'$' x '$'" := (type_id x%string)
 Notation "'%' x '%'" := (type_var x%string)
   (in custom ladder_type at level 0, x constr) : ladder_type_scope.
 
-Notation "[[ e ]]" := e
+Notation "[{ e }]" := e
   (e custom ladder_expr at level 80) : ladder_expr_scope.
 Notation "'%' x '%'" := (expr_var x%string)
   (in custom ladder_expr at level 0, x constr) : ladder_expr_scope.
@@ -84,10 +84,10 @@ Notation "'Λ' t '↦' e" := (expr_ty_abs t e)
 Open Scope ladder_type_scope.
 Open Scope ladder_expr_scope.
 
-Check [ ∀"α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) ].
+Check [< ∀"α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) >].
 
-Definition polymorphic_identity1 : expr_term := [[ Λ"T" ↦ λ"x"%"T"% ↦ %"x"% ]].
+Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x"%"T"% ↦ %"x"% }].
-Definition polymorphic_identity2 : expr_term := [[ Λ"T" ↦ λ"y"%"T"% ↦ %"y"% ]].
+Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y"%"T"% ↦ %"y"% }].
 
 Compute polymorphic_identity1.
 

coq/typing.v

@@ -121,8 +121,8 @@ Definition is_well_typed (e:expr_term) : Prop :=
 
 Example typing1 :
   forall Γ,
-  (context_contains Γ "x" [ %"T"% ]) ->
+  (context_contains Γ "x" [< %"T"% >]) ->
-  Γ |- [[ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% ]] \is [ ∀"T", %"T"% -> %"T"% ].
+  Γ |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
    (* Γ |- [ ΛT ↦ λx:T ↦ x ] : ∀T.(T->T) *)
 Proof.
   intros.
@@ -135,11 +135,11 @@ Qed.
 
 Example typing2 :
   forall Γ,
-  (context_contains Γ "x" [ %"T"% ]) ->
+  (context_contains Γ "x" [< %"T"% >]) ->
-  Γ |- [[ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% ]] \is [ ∀"U", %"U"% -> %"U"% ].
+  Γ |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
 Proof.
   intros.
-  apply T_Sub with (τ:=[∀"T",(%"T"% -> %"T"%)]).
+  apply T_Sub with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]).
   apply T_TypeAbs.
   apply T_Abs.
   apply H.
@@ -156,12 +156,16 @@ Qed.
 
 Example typing3 :
   forall Γ,
-  (context_contains Γ "x" [ %"T"% ]) ->
+  (context_contains Γ "x" [< %"T"% >]) ->
-  (context_contains Γ "y" [ %"U"% ]) ->
+  (context_contains Γ "y" [< %"U"% >]) ->
-  Γ |- [[ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"% ]] \is [ ∀"S",∀"T",(%"S"%->%"T"%->%"T"%) ].
+  Γ |- [{
+    Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"%
+  }] \is [<
+       ∀"S",∀"T",(%"S"%->%"T"%->%"T"%)
+  >].
 Proof.
   intros.
-  apply T_Sub with (τ:=[∀"T",∀"U",(%"T"%->%"U"%->%"U"%)]) (τ':=[∀"S",∀"T",(%"S"%->%"T"%->%"T"%)]).
+  apply T_Sub with (τ:=[< ∀"T",∀"U",(%"T"%->%"U"%->%"U"%) >]) (τ':=[< ∀"S",∀"T",(%"S"%->%"T"%->%"T"%) >]).
   apply T_TypeAbs, T_TypeAbs, T_Abs.
   apply H.
   apply T_Abs.
@@ -169,7 +173,7 @@ Proof.
   apply T_Var, H0.
 
   apply TSubRepr_Refl.
-  apply TEq_Trans with (y:= [∀"S",∀"U",(%"S"%->%"U"%->%"U"%)] ).
+  apply TEq_Trans with (y:= [< ∀"S",∀"U",(%"S"%->%"U"%->%"U"%) >] ).
   apply TEq_Alpha.
   apply TAlpha_Rename.
   apply TSubst_UnivReplace. discriminate.
@@ -192,13 +196,13 @@ Qed.
 
 
 Example typing4 : (is_well_typed
-  [[ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% ]]
+  [{ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% }]
 ).
 Proof.
   unfold is_well_typed.
-  exists (ctx_assign "x" [%"T"%]
+  exists (ctx_assign "x" [< %"T"% >]
-            (ctx_assign "y" [%"U"%] ctx_empty)).
+            (ctx_assign "y" [< %"U"% >] ctx_empty)).
-  exists [ ∀"T",∀"U",%"T"%->%"U"%->%"T"% ].
+  exists [< ∀"T",∀"U",%"T"%->%"U"%->%"T"% >].
   apply T_CompatTypeAbs.
   apply T_CompatTypeAbs.
   apply T_CompatAbs.