(** * Skelet #1 *)
(** Following https://www.sligocki.com/2023/03/13/skelet-1-infinite.html *)
Set Warnings "-abstract-large-number".
Set Default Goal Selector "!".
Definition tm : TM := fun '(q, s) =>
match q, s with
| A, 0 => Some (1, R, B)  | A, 1 => Some (1, R, D)
| B, 0 => Some (1, L, C)  | B, 1 => Some (0, R, C)
| C, 0 => Some (1, R, A)  | C, 1 => Some (1, L, D)
| D, 0 => Some (0, R, E)  | D, 1 => Some (0, L, B)
| E, 0 => None            | E, 1 => Some (1, R, C)
end.
Notation "c --> c'" := (c -[ tm ]-> c')   (at level 40).
Notation "c -->* c'" := (c -[ tm ]->* c') (at level 40).
Notation "c -->+ c'" := (c -[ tm ]->+ c') (at level 40).
Notation "l <| r" := (l <{{C}} 1 >> 0 >> r)  (at level 30).
Notation "l |> r" := (l {{A}}> r)  (at level 30).
Fixpoint run (n : nat) :=
match n with
| O => [0]
| S n => run n <: 1
end.
Definition x  := run 2 <+ run 2.
Definition Dl := run 2 <+ run 1.
Definition Dr := run 2 +> run 1.
Definition C0 := run 2 <+ run 3 <+ run 2.
Definition C1 := run 2 <+ run 0 <+ run 1.
Definition C2 := run 4 <+ run 2.
Definition C3 := run 1 <+ run 1.
Notation C := C3.
Definition R := const 0.
Definition L := const 0.
Definition P  := run 2.
Definition F0 := run 4 <+ run 3 <+ run 2.
Definition F1 := run 4 <+ run 0 <+ run 1.
Definition F2 := run 6 <+ run 2.
Definition F3 := run 3 <+ run 1.
Definition G0 := run 2 <+ run 3 <+ run 3 <+ run 2.
Definition G1 := run 2 <+ run 3 <+ run 0 <+ run 1.
Definition G2 := run 2 <+ run 5 <+ run 2.
Definition Fl := C2 <+ x^^7640 <+ Dl <+ x^^10344.
Definition Gl :=
x^^300 <+ Dl <+ x^^30826 <+ Dl <+ x^^72142 <+ Dl <+
x^^3076 <+ Dl <+ x^^1538 <+ Dl.
Definition Gr :=
x^^300 +> Dr +> x^^30826 +> Dr +> x^^72142 +> Dr +>
x^^3076 +> Dr +> x^^1538 +> Dr.
Definition Hl :=
C1 <+ Dl <+ x^^299 <+ C1 <+ Dl <+ x^^30825 <+
C1 <+ Dl <+ x^^72141 <+ C1 <+ Dl <+ x^^3075 <+
C1 <+ Dl <+ x^^1537.

Lemma rule_P_xn    : forall n l r,
l |> P *> x^^n *> r -->* l <* x^^n |> P *> r.