include "arithmetics/nat.ma".
-(*BUG: kind signatures in type declarations*)
+(*FEATURE: kind signatures in type declarations*)
+
+axiom Nat: Type[0].
+axiom S: Nat → Nat.
+axiom O: Nat.
axiom test1: Type[1].
(* not in F_omega ? *)
axiom test4: ∀A:Type[1]. A → ∀B:Type[1]. B → ∀C:Prop. C → Type[1].
+axiom test4': ∀C:Prop. C → C.
+
+axiom test4'': ∀C:Prop. C → Nat.
+
+axiom test4''': ∀C:Type[1]. C.
+
+axiom test4_5: (∀A:Type[0].A) → Nat.
+
axiom test5: (Type[1] → Type[1]) → Type[1].
(* no content *)
axiom test6: Type[1] → Prop.
-definition dtest1: Type[1] ≝ nat → nat.
+definition dtest1: Type[1] ≝ Nat → Nat.
definition dtest2: Type[2] ≝ ∀A:Type[1]. A → A.
definition dtest5: Type[1] → Type[1] ≝ dtest3.
-definition dtest6: Type[1] ≝ dtest3 nat.
+definition dtest6: Type[1] ≝ dtest3 Nat.
(* no content *)
definition dtest7: Prop ≝ True → True.
definition dtest10: Type[1] → Type[1] → Type[1] ≝ λX,Y.X.
-(*BUG: a' vs a''*)
-definition dtest11: Type[1] → nat → Type[1] → Type[1] ≝ λ_:Type[1].λ_:nat.λX:Type[1]. X → nat → test1.
+definition dtest11: Type[1] → Nat → Type[1] → Type[1] ≝ λ_:Type[1].λ_:Nat.λX:Type[1]. X → Nat → test1.
-definition dtest12 ≝ λ_:Type[1].λ_:nat.λX:Type[1]. X → nat → test1.
-definition dtest13 ≝ dtest3 nat → dtest3 True → dtest3 Prop → nat.
+definition dtest12 ≝ λ_:Type[1].λ_:Nat.λX:Type[1]. X → Nat → test1.
+
+definition dtest13 ≝ dtest3 Nat → dtest3 True → dtest3 Prop → Nat.
definition dtest14 ≝ λX:Type[2]. X → X.
(*FEATURE: type the forall bound variables*)
-definition dtest15 ≝ ∀T:Type[1] → Type[1]. T nat → T nat.
+definition dtest15 ≝ ∀T:Type[1] → Type[1]. T Nat → T Nat.
-definition dtest16 ≝ ∀T:Type[1]. T → nat.
+definition dtest16 ≝ ∀T:Type[1]. T → Nat.
-definition dtest17 ≝ ∀T:dtest14 Type[1]. T nat → dtest14 nat → dtest14 nat.
+definition dtest17 ≝ ∀T:dtest14 Type[1]. T Nat → dtest14 Nat → dtest14 Nat.
-definition dtest18 ≝ λA,B:Type[0].λn:nat.λC:Type[0].A.
+definition dtest18 ≝ λA,B:Type[0].λn:Nat.λC:Type[0].A.
-definition dtest19 ≝ dtest18 nat True O nat → dtest18 nat nat O nat.
+definition dtest19 ≝ dtest18 Nat True O Nat → dtest18 Nat Nat O Nat.
definition dtest20 ≝ test5 test2.
to be traced, raises NotInFOmega; why? *)
definition dtest21 ≝ test5 (λX:Type[1].X).
-definition ttest1 ≝ λx:nat.x.
+definition ttest1 ≝ λx:Nat.x.
(*BUG: clash of name due to capitalisation*)
-definition Ttest1 ≝ λx:nat.x.
+(*definition Ttest1 ≝ λx:Nat.x.*)
+
+(*FEATURE: print binders in the l.h.s. without using lambda abstraction*)
+definition ttest2 ≝ λT:Type[1].λx:T.x.
+
+definition ttest3 ≝ λT:Type[1].λx:T.let y ≝ x in y.
+
+definition ttest4 ≝ λT:Type[1].let y ≝ T in λx:y.x.
+
+(*BUG IN HASKELL PRETTY PRINTING: all lets are let rec*)
+(*definition ttest5 ≝ λT:Type[1].λy:T.let y ≝ y in y.*)
+
+definition ttest6 ≝ ttest4 Nat.
+
+definition ttest7 ≝ λf:∀X:Type[1].X. f (Nat → Nat) O.
+
+definition ttest8 ≝ λf:∀X:Type[1].X. f (True → True) I.
+
+definition ttest9 ≝ λf:∀X:Type[1].X. f (True → Nat) I.
+
+definition ttest10 ≝ λf:∀X:Type[1].X. f (True → Nat → Nat) I O.
+
+definition ttest11_aux ≝ λS:Type[1]. S → Nat.
+
+definition ttest11 ≝ λf:ttest11_aux True. f I.
+
+(*BUG here: requires F_omega eat_args*)
+definition ttest12 ≝ λf:True → Nat. f I.
-(*BUG: Failure nth*)
-definition ttest2 ≝ λT:Type[1].λx:T.x.
\ No newline at end of file
+(*GENERAL BUG: name clashes when binders shadow each other in CIC*)