Opaque bs2nat.
Opaque Cond.
Definition False_e n :=
Comp n (Succ false) [Zero_e n].
Opaque Zero_e One_e False_e Rev_e RemoveLSZ_e.
Definition Normalize_e : Cobham :=
Comp 1 Rev_e [Comp 1 RemoveLSZ_e [Rev_e]].
Opaque Normalize_e.
(** Def 1 of (Rose) *)
Definition Succ_e : Cobham :=
Rec
(Comp 0 (Succ true) [Zero])
(Comp 2 (Succ true) [Proj 2 0])
(Comp 2 (Succ false) [Proj 2 1])
(Comp 1 (Succ true) [Proj 1 0]).
Opaque Succ_e.
(** Def 2 of (Rose) *)
Definition Pred'_e : Cobham :=
Rec
Zero
(Comp 2 (Succ true) [Proj 2 1])
(Comp 2 (Succ false) [Proj 2 0])
(Proj 1 0).
Opaque Pred'_e.
Definition Pred_e : Cobham :=
Comp 1 Pred'_e [Normalize_e].
Opaque Pred_e.
(** Def 3 of (Rose) *)
Definition OneMinus'_e : Cobham :=
Rec2
(One_e 0)
(Zero_e 2)
(One_e 1).
Opaque OneMinus'_e.
Definition OneMinus_e : Cobham :=
Comp 1 OneMinus'_e [Normalize_e].
Opaque OneMinus_e.
(** Def 4 of (Rose) *)
Definition Div2_e : Cobham :=
Rec2
Zero
(Proj 2 0)
(Proj 1 0).
Opaque Div2_e.
(** Def 5 of (Rose) *)
Definition Mod2_e : Cobham :=
Rec
Zero
(False_e 2)
(One_e 2)
(One_e 1).
Opaque Mod2_e.
(** Def 6 of (Rose) *)
Definition Length_e : Cobham :=
Rec2
Zero
(Comp 2 Succ_e [Proj 2 1])
(Proj 1 0).
Opaque Length_e.
(** Def 7 of (Rose) *)
Definition MultOneMinus'_e : Cobham :=
Comp 2 Cond [
Proj 2 1;
Proj 2 0;
Zero_e 2;
Zero_e 2
].
Opaque MultOneMinus'_e.
Definition MultOneMinus_e : Cobham :=
Comp 2 MultOneMinus'_e [Proj 2 0; Comp 2 Normalize_e [Proj 2 1]].
Opaque MultOneMinus_e.
(** Def 8 of (Rose) *)
Definition PlusOneMinus'_e : Cobham :=
Comp 2 Cond [
Proj 2 1;
Comp 2 Succ_e [Proj 2 0];
Proj 2 0;
Proj 2 0
].
Opaque PlusOneMinus'_e.
Definition PlusOneMinus_e : Cobham :=
Comp 2 PlusOneMinus'_e [Proj 2 0; Comp 2 Normalize_e [Proj 2 1] ].
Opaque PlusOneMinus_e.
(** Def 9 of (Rose) *)
Definition OneMinusMultPlus'_e : Cobham :=
Rec
(Proj 1 0)
(Comp 3 (Succ false) [Proj 3 0])
(Comp 3 (Succ true) [Proj 3 0])
(Comp 2 Smash [Comp 2 Succ_e [Proj 2 1]; Comp 2 (Succ true) [Proj 2 0]]).
Opaque OneMinusMultPlus'_e.
Definition OneMinusMultPlus_e : Cobham :=
Comp 2 OneMinusMultPlus'_e [Comp 2 Normalize_e [Proj 2 0]; Proj 2 1].
Opaque OneMinusMultPlus_e.
(** Def 10 of (Rose) *)
Definition PlusMod2_e : Cobham :=
Comp 2 Cond [
Proj 2 1;
Proj 2 0;
Comp 2 Succ_e [Proj 2 0];
Proj 2 0
].
Opaque PlusMod2_e.
(** Def 11 of (Rose) *)
Definition PlusOneMinusMod2_e : Cobham :=
Comp 3 Cond [
Proj 3 2;
Proj 3 0;
Comp 3 PlusOneMinus_e [Proj 3 0; Proj 3 1];
Proj 3 0
].
Opaque PlusOneMinusMod2_e.
(** Def 13 of (Rose) *)
Definition MultTwoPowerLength'_e : Cobham :=
Rec2
(Proj 1 0)
(Comp 3 (Succ false) [Proj 3 1])
(Comp 2 Smash [Comp 2 (Succ true) [Proj 2 0]; Comp 2 (Succ true) [Proj 2 1] ] ).
(* FIXME: According to Rose (Def 13), 'plus 1' in arguments of Smash are not necessary *)
Opaque MultTwoPowerLength'_e.
Definition MultTwoPowerLength_e : Cobham :=
Comp 2 MultTwoPowerLength'_e [Proj 2 1; Proj 2 0].
Opaque MultTwoPowerLength_e.
(** Def 14 of (Rose) *)
Definition DivTwoPower'_e : Cobham :=
Rec2
(Proj 1 0)
(Comp 3 Div2_e [Proj 3 1])
(Proj 2 1).
Opaque DivTwoPower'_e.
Definition DivTwoPower_e : Cobham :=
Comp 2 DivTwoPower'_e [Proj 2 1; Proj 2 0].
Opaque DivTwoPower_e.
(** Def 15 of (Rose) *)
Definition DivDivTwoPower_e : Cobham :=
Comp 3 DivTwoPower_e [Proj 3 0; Comp 3 DivTwoPower_e [Proj 3 1; Proj 3 2] ].
Opaque DivDivTwoPower_e.
(** Def 16 of (Rose) *)
Fixpoint Repeat_e (n l : nat) (b : bool) : Cobham :=
match l with
| 0 => Zero_e n
| S l' => Comp n (Succ b) [Repeat_e n l' b]
end.
Opaque Repeat_e.
Definition RoseS'_e : Cobham :=
Rec2
(Zero_e 2)
(Comp 4 PlusOneMinusMod2_e [
Comp 4 (Succ false) [Proj 4 1];
Comp 4 DivTwoPower_e [Comp 4 DivTwoPower_e [Proj 4 2; Proj 4 0]; Proj 4 3];
Comp 4 DivDivTwoPower_e [Proj 4 2; Proj 4 2; Comp 4 (Succ false) [Proj 4 0]]
])
(Comp 3 Smash [Proj 3 0; Repeat_e 3 2 true]).
Definition arity_RoseS' :
arity RoseS'_e = ok_arity 3.
Definition RoseS_e : Cobham :=
Comp 3 RoseS'_e [Proj 3 2; Proj 3 0; Proj 3 1].
Opaque RoseS'_e.
Transparent RoseS'_e.
Fixpoint RoseS_spec (x y z : bs) :=
match z with
| nil => nil
| _ :: z' =>
if leb (length z) (length x - length y)
then false :: RoseS_spec x y z'
else hd false (skipn (length x - length z) x) :: RoseS_spec x y z'
end.

Lemma RoseS_spec2_false : forall x y z,
length z <= length x - length y ->
bs2nat (RoseS_spec x y z) = 0.