/
Skelet1.v
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Skelet1.v
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(** * Skelet #1 *)
(** Following https://www.sligocki.com/2023/03/13/skelet-1-infinite.html *)
Set Warnings "-abstract-large-number".
From Coq Require Import PeanoNat.
From Coq Require Import List. Import ListNotations.
From Coq Require Import Lia.
From Coq Require Import PArith.BinPos PArith.Pnat.
From Coq Require Import NArith.BinNat NArith.Nnat.
From Coq Require Import Program.Tactics.
From Coq Require Import ZifyBool.
From BusyCoq Require Import Individual52.
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_x_left : forall l r, l <* x <| r -->* l <| x *> r.
Proof. triv. Qed.
Lemma rule_D_left : forall l r, l <* Dl <| r -->* l <| Dr *> r.
Proof. triv. Qed.
Lemma rule_C_left : forall l r, l <* C <| r -->* l <| C *> r.
Proof. triv. Qed.
Lemma rule_x_right : forall l r, l |> x *> r -->* l <* x |> r.
Proof. triv. Qed.
Lemma rule_D_right : forall l r, l |> Dr *> r -->* l <* Dl |> r.
Proof. triv. Qed.
Lemma rule_xR : forall l, l <* x |> R -->* l <| C *> x *> P *> R.
Proof. unfold R, C, x, P. triv. Qed.
Lemma rule_DR : forall l, l <* Dl |> R -->* l <| x *> R.
Proof. unfold R, x. triv. Qed.
Lemma rule_L : forall r, L <| C *> x *> r -->* L <* C1 <* Dl |> P *> r.
Proof. unfold L, C1, Dl, C. triv. Qed.
Lemma rule_C30 : forall l r, l <* x |> C *> r -->* l <* C0 |> r.
Proof. triv. Qed.
Lemma rule_C01 : forall l r, l <* C0 <| r -->* l <* C1 <* x |> r.
Proof. triv. Qed.
Lemma rule_C12 : forall l r, l <* C1 <| r -->* l <* C2 |> r.
Proof. triv. Qed.
Lemma rule_C23 : forall l r, l <* C2 <| r -->* l <* C <* x |> r.
Proof. triv. Qed.
Lemma rule_DC : forall l r, l <* Dl |> C *> r -->* l <* P <* x |> r.
Proof. triv. Qed.
Lemma rule_C2_C : forall l r, l <* C2 |> C *> r -->* l <* F0 |> r.
Proof. triv. Qed.
Lemma rule_F0 : forall l r, l <* F0 <| r -->* l <* F1 <* x |> r.
Proof. triv. Qed.
Lemma rule_F1 : forall l r, l <* F1 <| r -->* l <* F2 |> r.
Proof. triv. Qed.
Lemma rule_F2 : forall l r, l <* F2 <| r -->* l <* F3 <* x |> r.
Proof. triv. Qed.
Lemma rule_F3 : forall l r, l <* x <* F3 <| r -->* l <* P <* C1 <* Dl |> r.
Proof. triv. Qed.
Lemma rule_C03 : forall l r, l <* C0 |> C *> r -->* l <* G0 |> r.
Proof. triv. Qed.
Lemma rule_G0 : forall l r, l <* G0 <| r -->* l <* G1 <* x |> r.
Proof. triv. Qed.
Lemma rule_G1 : forall l r, l <* G1 <| r -->* l <* G2 |> r.
Proof. triv. Qed.
Lemma rule_G2 : forall l r, l <* G2 <| r -->* l <* P <* Dl <* x |> r.
Proof. triv. Qed.
Lemma rule_P_left : forall l r, l <* P <| r -->* l <| P *> r.
Proof. triv. Qed.
Lemma rule_P_P : forall l r, l |> P *> P *> r -->* l <* x |> r.
Proof. triv. Qed.
Lemma rule_P_x : forall l r, l |> P *> x *> r -->* l <* x |> P *> r.
Proof. triv. Qed.
Lemma rule_P_R : forall l, l |> P *> R -->* l <| P *> R.
Proof. unfold P, R. triv. Qed.
Lemma rule_P_Dx : forall l r,
l |> P *> Dr *> x *> r -->* l <* C1 <* Dl |> P *> r.
Proof. triv. Qed.
Lemma rule_P_Cx : forall l r,
l |> P *> C *> x *> r -->* l <| P *> Dr *> P *> r.
Proof. triv. Qed.
Lemma rule_P_DP : forall l r,
l |> P *> Dr *> P *> r -->* l <* C1 <* Dl |> r.
Proof. triv. Qed.
Lemma rule_P_DDx : forall l r,
l |> P *> Dr *> Dr *> x *> r -->* l <* C2 <* C1 <* Dl |> r.
Proof. triv. Qed.
Lemma rule_P_DCx : forall l r,
l |> P *> Dr *> C *> x *> r -->* l <* G1 <* Dl |> P *> r.
Proof. triv. Qed.
Opaque x Dl Dr C0 C1 C2 C3 P F0 F1 F2 F3 G0 G1 G2.
Lemma rule_xn_left : forall n l r,
l <* x^^n <| r -->* l <| x^^n *> r.
Proof.
induction n; introv.
- finish.
- simpl_tape.
follow rule_x_left.
follow IHn.
simpl_tape. finish.
Qed.
Lemma rule_xn_right : forall n l r,
l |> x^^n *> r -->* l <* x^^n |> r.
Proof.
induction n; introv.
- finish.
- simpl_tape.
follow rule_x_right.
follow IHn.
simpl_tape. finish.
Qed.
Lemma rule_P_xn : forall n l r,
l |> P *> x^^n *> r -->* l <* x^^n |> P *> r.
Proof.
induction n; introv.
- finish.
- simpl_tape.
follow rule_P_x. follow IHn.
simpl_tape. finish.
Qed.
Lemma rule_P_DG : forall l r,
l |> P *> Dr *> Gr *> r -->* l <* Hl |> P *> Dr *> r.
Proof.
introv. unfold Gr.
rewrite (lpow_S 299). autorewrite with tape_post.
follow rule_P_Dx. follow rule_P_xn.
rewrite (lpow_S 30825). autorewrite with tape_post.
follow rule_P_Dx. follow rule_P_xn.
rewrite (lpow_S 72141). autorewrite with tape_post.
follow rule_P_Dx. follow rule_P_xn.
rewrite (lpow_S 3075). autorewrite with tape_post.
follow rule_P_Dx. follow rule_P_xn.
rewrite (lpow_S 1537). autorewrite with tape_post.
follow rule_P_Dx. follow rule_P_xn.
unfold Hl. autorewrite with tape_post.
finish.
Qed.
Lemma rule_P_DGn : forall n l r,
l |> P *> Dr *> Gr^^n *> r -->* l <* Hl^^n |> P *> Dr *> r.
Proof.
induction n; introv.
- finish.
- simpl_tape. follow rule_P_DG. follow IHn.
simpl_tape. finish.
Qed.
Lemma rule_G_right : forall l r, l |> Gr *> r -->* l <* Gl |> r.
Proof.
introv. unfold Gl, Gr.
autorewrite with tape_post.
repeat (follow rule_xn_right; follow rule_D_right).
finish.
Qed.
Lemma rule_Gn_right : forall n l r,
l |> Gr^^n *> r -->* l <* Gl^^n |> r.
Proof.
induction n; introv.
- finish.
- simpl_tape. follow rule_G_right.
follow IHn.
simpl_tape. finish.
Qed.
Lemma rule_G_left : forall l r, l <* Gl <| r -->* l <| Gr *> r.
Proof.
introv. unfold Gl, Gr.
autorewrite with tape_post.
repeat (follow rule_D_left; follow rule_xn_left).
finish.
Qed.
Lemma rule_Gn_left : forall n l r,
l <* Gl^^n <| r -->* l <| Gr^^n *> r.
Proof.
induction n; introv.
- finish.
- simpl_tape. follow rule_G_left.
follow IHn.
simpl_tape. finish.
Qed.
Inductive lsym :=
| l_xs (n : positive)
| l_D
| l_P
| l_C0 | l_C1 | l_C2 | l_C3
| l_F0 | l_F1 | l_F2 | l_F3
| l_G0 | l_G1 | l_G2
| l_Fs (n : positive)
| l_Gs (n : positive)
| l_Hs (n : positive).
Inductive rsym :=
| r_xs (n : positive)
| r_D
| r_C
| r_P
| r_Gs (n : positive).
Notation ltape := (list lsym).
Notation rtape := (list rsym).
Definition F := [l_xs 10344; l_D; l_xs 7640; l_C2].
Definition G := [r_xs 300; r_D; r_xs 30826; r_D; r_xs 72142; r_D;
r_xs 3076; r_D; r_xs 1538; r_D].
Definition J := [l_D; l_C2; l_xs 95; l_C0;
l_xs 7713; l_D; l_D; l_xs 1866; l_C1;
l_xs 13231; l_D; l_xs 6197; l_C3;
l_xs 11066; l_D; l_xs 7279; l_C0;
l_xs 10524; l_D; l_xs 7550; l_C2;
l_xs 10389; l_D; l_xs 7618; l_C1;
l_xs 10355; l_D; l_xs 7635; l_C3;
l_xs 10347; l_D; l_xs 7639; l_C3;
l_xs 10345; l_D; l_xs 7640; l_C1].
Definition K : rtape :=
[r_xs 7639; r_D; r_xs 10347; r_C;
r_xs 7635; r_D; r_xs 10355; r_C;
r_xs 7619; r_D; r_xs 10387; r_C;
r_xs 7555; r_D; r_xs 10515; r_C;
r_xs 7299; r_D; r_xs 11027; r_C;
r_xs 6275; r_D; r_xs 13075; r_C;
r_xs 2179; r_D; r_D; r_xs 7088; r_C;
r_xs 1; r_C; r_xs 3849; r_P].
Definition uni_P : positive := 53946.
Definition uni_T : positive := 4 * uni_P - 5.
Definition eqb_l (a b : lsym) : {a = b} + {a <> b}.
Proof.
decide equality; apply Pos.eq_dec.
Defined.
Definition eqb_r (a b : rsym) : {a = b} + {a <> b}.
Proof.
decide equality; apply Pos.eq_dec.
Defined.
Definition eqb_rtape (xs ys : rtape) : {xs = ys} + {xs <> ys}.
Proof.
decide equality; apply eqb_r.
Defined.
Notation left := TM.L.
Notation right := TM.R.
Notation conf := (dir * ltape * rtape)%type.
Fixpoint lift_right (t : rtape) : side :=
match t with
| [] => R
| r_xs n :: t => x^^(Pos.to_nat n) *> lift_right t
| r_D :: t => Dr *> lift_right t
| r_C :: t => C *> lift_right t
| r_P :: t => P *> lift_right t
| r_Gs n :: t => Gr^^(Pos.to_nat n) *> lift_right t
end.
Fixpoint lift_left (t : ltape) : side :=
match t with
| [] => L
| l_xs n :: t => lift_left t <* x^^(Pos.to_nat n)
| l_D :: t => lift_left t <* Dl
| l_P :: t => lift_left t <* P
| l_C0 :: t => lift_left t <* C0
| l_C1 :: t => lift_left t <* C1
| l_C2 :: t => lift_left t <* C2
| l_C3 :: t => lift_left t <* C3
| l_F0 :: t => lift_left t <* F0
| l_F1 :: t => lift_left t <* F1
| l_F2 :: t => lift_left t <* F2
| l_F3 :: t => lift_left t <* F3
| l_G0 :: t => lift_left t <* G0
| l_G1 :: t => lift_left t <* G1
| l_G2 :: t => lift_left t <* G2
| l_Fs n :: t => lift_left t <* Fl^^(Pos.to_nat n)
| l_Gs n :: t => lift_left t <* Gl^^(Pos.to_nat n)
| l_Hs n :: t => lift_left t <* Hl^^(Pos.to_nat n)
end.
Definition lift (c : conf) : Q * tape :=
match c with
| (left, l, r) => lift_left l <| lift_right r
| (right, l, r) => lift_left l |> lift_right r
end.
Definition decr (n : positive) : N :=
N.pred (N.pos n).
Lemma decr_nat : forall n, N.to_nat (decr (Pos.of_succ_nat n)) = n.
Proof. unfold decr. lia. Qed.
Definition lxs (n : N) (l : ltape) : ltape :=
match n with
| N0 => l
| Npos n =>
match l with
| l_xs m :: l => l_xs (n + m) :: l
| _ => l_xs n :: l
end
end.
Definition rxs (n : N) (r : rtape) : rtape :=
match n with
| N0 => r
| Npos n =>
match r with
| r_xs m :: r => r_xs (n + m) :: r
| _ => r_xs n :: r
end
end.
Definition Fls (n : N) (l : ltape) : ltape :=
match n with
| N0 => l
| Npos n =>
match l with
| l_Fs m :: l => l_Fs (n + m) :: l
| _ => l_Fs n :: l
end
end.
Definition Gls (n : N) (l : ltape) : ltape :=
match n with
| N0 => l
| Npos n =>
match l with
| l_Gs m :: l => l_Gs (n + m) :: l
| _ => l_Gs n :: l
end
end.
Definition Grs (n : N) (r : rtape) : rtape :=
match n with
| N0 => r
| Npos n =>
match r with
| r_Gs m :: r => r_Gs (n + m) :: r
| _ => r_Gs n :: r
end
end.
Definition Hls (n : N) (l : ltape) : ltape :=
match n with
| N0 => l
| Npos n =>
match l with
| l_Hs m :: l => l_Hs (n + m) :: l
| _ => l_Hs n :: l
end
end.
Ltac prove_liftrule :=
intros n t; destruct n; try reflexivity;
destruct t as [| [] t]; try reflexivity;
simpl; rewrite Pos2Nat.inj_add;
rewrite lpow_add; simpl_tape; reflexivity.
Lemma lift_lxs : forall n l,
lift_left (lxs n l) = lift_left l <* x^^(N.to_nat n).
Proof. prove_liftrule. Qed.
Lemma lift_rxs : forall n r,
lift_right (rxs n r) = x^^(N.to_nat n) *> lift_right r.
Proof. prove_liftrule. Qed.
Lemma lift_Fls : forall n l,
lift_left (Fls n l) = lift_left l <* Fl^^(N.to_nat n).
Proof. prove_liftrule. Qed.
Lemma lift_Gls : forall n l,
lift_left (Gls n l) = lift_left l <* Gl^^(N.to_nat n).
Proof. prove_liftrule. Qed.
Lemma lift_Grs : forall n r,
lift_right (Grs n r) = Gr^^(N.to_nat n) *> lift_right r.
Proof. prove_liftrule. Qed.
Lemma lift_Hls : forall n l,
lift_left (Hls n l) = lift_left l <* Hl^^(N.to_nat n).
Proof. prove_liftrule. Qed.
Ltac pos_nat p :=
let p' := fresh p in
let E := fresh in
destruct (Pos2Nat.is_succ p) as [p' E];
try rewrite E in *;
apply SuccNat2Pos.inv in E; subst p;
rename p' into p.
Ltac handle_decr :=
lazymatch goal with
| |- context [decr ?p] => pos_nat p; try rewrite decr_nat
end.
Ltac simp :=
simpl;
try rewrite lift_lxs;
try rewrite lift_rxs;
try rewrite lift_Fls;
try rewrite lift_Gls;
try rewrite lift_Grs;
try rewrite lift_Hls;
try rewrite Pos2Nat.inj_succ;
try handle_decr;
simpl.
Definition unrxs (r : rtape) : option rtape :=
match r with
| r_xs n :: r => Some (rxs (decr n) r)
| r_Gs n :: r =>
Some (r_xs 299 :: r_D :: r_xs 30826 :: r_D :: r_xs 72142 :: r_D ::
r_xs 3076 :: r_D :: r_xs 1538 :: r_D :: Grs (decr n) r)
| _ => None
end.
Lemma unrxs_spec : forall r r',
unrxs r = Some r' ->
lift_right r = x *> lift_right r'.
Proof.
introv H.
destruct r as [| [] r]; try discriminate; inverts H.
- simp. reflexivity.
- cbn[lift_right].
rewrite lift_Grs.
handle_decr.
cbn[lpow].
unfold Gr at 1.
autorewrite with tape_post.
reflexivity.
Qed.
Arguments unrxs _ : simpl never.
Definition simple_step (c : conf) : option conf :=
match c with
| (right, l, r) =>
match r with
| [] =>
match l with
(* x > R --> < C x P R *)
| l_xs n :: l =>
Some (left, lxs (decr n) l, [r_C; r_xs 1; r_P])
(* D > R --> < x R *)
| l_D :: l =>
Some (left, l, [r_xs 1])
| _ => None
end
(* > x^n --> x^n > *)
| r_xs n :: r =>
Some (right, lxs (N.pos n) l, r)
(* > D --> D > *)
| r_D :: r =>
Some (right, l_D :: l, r)
| r_C :: r =>
match l with
(* x > C --> C0 > *)
| l_xs n :: l =>
Some (right, l_C0 :: lxs (decr n) l, r)
(* D > C --> P x > *)
| l_D :: l =>
Some (right, l_xs 1 :: l_P :: l, r)
(* C0 > C --> G0 > *)
| l_C0 :: l =>
Some (right, l_G0 :: l, r)
(* C2 > C --> F0 > *)
| l_C2 :: l =>
Some (right, l_F0 :: l, r)
| _ => None
end
(* > P R --> < P R *)
| [r_P] =>
Some (left, l, [r_P])
(* > P x^n --> x^n > P *)
| r_P :: r_xs n :: r =>
Some (right, lxs (N.pos n) l, r_P :: r)
(* > P D x --> C1 D > P *)
(* Note: this rule doesn't use unrxs to let a more specialized rule
trigger on > P D G^n. *)
| r_P :: r_D :: r_xs n :: r =>
Some (right, l_D :: l_C1 :: l, r_P :: rxs (decr n) r)
(* > P DDx --> C2 C1 D > *)
| r_P :: r_D :: r_D :: r =>
match unrxs r with
| Some r => Some (right, l_D :: l_C1 :: l_C2 :: l, r)
| None => None
end
(* > P DCx --> G1 D > P *)
| r_P :: r_D :: r_C :: r =>
match unrxs r with
| Some r => Some (right, l_D :: l_G1 :: l, r_P :: r)
| None => None
end
(* > P D P --> C1 D > *)
| r_P :: r_D :: r_P :: r =>
Some (right, l_D :: l_C1 :: l, r)
(* > P D G^n --> H^n > P D *)
| r_P :: r_D :: r_Gs n :: r =>
Some (right, Hls (N.pos n) l, r_P :: r_D :: r)
(* > P C x --> < P D P *)
| r_P :: r_C :: r =>
match unrxs r with
| Some r => Some (left, l, r_P :: r_D :: r_P :: r)
| None => None
end
(* > P P --> x > *)
| r_P :: r_P :: r =>
Some (right, lxs 1 l, r)
(* > G^n --> G^n > *)
| r_Gs n :: r =>
Some (right, Gls (N.pos n) l, r)
| _ => None
end
| (left, l, r) =>
match l with
(* L < C x --> L C1 D > P *)
| [] =>
match r with
| r_C :: r =>
match unrxs r with
| Some r => Some (right, [l_D; l_C1], r_P :: r)
| None => None
end
| _ => None
end
(* x^n < --> < x^n *)
| l_xs n :: l =>
Some (left, l, rxs (N.pos n) r)
(* D < --> < D *)
| l_D :: l =>
Some (left, l, r_D :: r)
(* P < --> < P *)
| l_P :: l =>
Some (left, l, r_P :: r)
(* C0 < --> C1 x > *)
| l_C0 :: l =>
Some (right, l_xs 1 :: l_C1 :: l, r)
(* C1 < --> C2 > *)
| l_C1 :: l =>
Some (right, l_C2 :: l, r)
(* C2 < --> C3 x > *)
| l_C2 :: l =>
Some (right, l_xs 1 :: l_C3 :: l, r)
(* C < --> < C *)
| l_C3 :: l =>
Some (left, l, r_C :: r)
(* F0 < --> F1 x > *)
| l_F0 :: l =>
Some (right, l_xs 1 :: l_F1 :: l, r)
(* F1 < --> F2 > *)
| l_F1 :: l =>
Some (right, l_F2 :: l, r)
(* F2 < --> F3 x > *)
| l_F2 :: l =>
Some (right, l_xs 1 :: l_F3 :: l, r)
(* x F3 < --> P C1 D > *)
| l_F3 :: l_xs n :: l =>
Some (right, l_D :: l_C1 :: l_P :: lxs (decr n) l, r)
(* G0 < --> G1 x > *)
| l_G0 :: l =>
Some (right, l_xs 1 :: l_G1 :: l, r)
(* G1 < --> G2 > *)
| l_G1 :: l =>
Some (right, l_G2 :: l, r)
(* G2 < --> P D x > *)
| l_G2 :: l =>
Some (right, l_xs 1 :: l_D :: l_P :: l, r)
(* G^n < --> < G^n *)
| l_Gs n :: l =>
Some (left, l, Grs (N.pos n) r)
| _ => None
end
end.
Arguments lxs _ _ : simpl never.
Arguments rxs _ _ : simpl never.
Arguments Fls _ _ : simpl never.
Arguments Gls _ _ : simpl never.
Arguments Grs _ _ : simpl never.
Arguments Hls _ _ : simpl never.
Ltac unrxs :=
lazymatch goal with
| H: context [unrxs ?r] |- _ =>
let E := fresh "E" in
let r' := fresh "r'" in
destruct (unrxs r) as [r' |] eqn:E; try discriminate;
inverts H as H;
apply unrxs_spec in E; simp; simpl in E; try rewrite E
end.
Lemma simple_step_spec : forall c c',
simple_step c = Some c' ->
lift c -->* lift c'.
Proof.
introv H.
destruct c as [[[] l] r]; simpl in H.
- (* left *)
destruct l as [| [] l]; inverts H as H; simp.
+ (* L < C x --> L C1 D > P *)
destruct r as [| [] r]; try discriminate; unrxs.
apply rule_L.
+ (* x^n < --> < x^n *)
apply rule_xn_left.
+ (* D < --> < D *)
apply rule_D_left.
+ (* P < --> < P *)
apply rule_P_left.
+ (* C0 < --> C1 x > *)
apply rule_C01.
+ (* C1 < --> C2 > *)
apply rule_C12.
+ (* C2 < --> C3 x > *)
apply rule_C23.
+ (* C < --> < C *)
apply rule_C_left.
+ (* F0 < --> F1 x > *)
apply rule_F0.
+ (* F1 < --> F2 > *)
apply rule_F1.
+ (* F2 < --> F3 x > *)
apply rule_F2.
+ (* x F3 < --> P C1 D > *)
destruct l as [| [] l]; inverts H; simp.
apply rule_F3.
+ (* G0 < --> G1 x > *)
apply rule_G0.
+ (* G1 < --> G2 > *)
apply rule_G1.
+ (* G2 < --> P D x > *)
apply rule_G2.
+ (* G^n < --> < G^n *)
apply rule_Gn_left.
- (* right *)
destruct r as [| [] r]; inverts H as H; simp.
+ destruct l as [| [] l]; inverts H; simp.
* (* x > R --> < C x P R *)
apply rule_xR.
* (* D > R --> < x R *)
apply rule_DR.
+ (* > x^n --> x^n > *)
apply rule_xn_right.
+ (* > D --> D > *)
apply rule_D_right.
+ destruct l as [| [] l]; inverts H; simp.
* (* x > C --> C0 > *)
apply rule_C30.
* (* D > C --> P x *)
apply rule_DC.
* (* C0 > C --> G0 > *)
apply rule_C03.
* (* C2 > C --> F0 > *)
apply rule_C2_C.
+ destruct r as [| [] r]; inverts H as H; simp.
* (* > P R --> < P R *)
apply rule_P_R.
* (* > P x^n --> x^n > P *)
apply rule_P_xn.
* destruct r as [| [] r]; (unrxs || inverts H; simp).
** (* > P Dx --> C1 D > P *)
apply rule_P_Dx.
** (* > P DDx --> C2 C1 D > *)
apply rule_P_DDx.
** (* > P DCx --> G1 D > P *)
apply rule_P_DCx.
** (* > P DP --> C1 D *)
apply rule_P_DP.
** (* > P DG^n --> H^n P D *)
apply rule_P_DGn.
* (* > P C x --> < P D P *)
unrxs. apply rule_P_Cx.
* (* > P P --> x > *)
apply rule_P_P.
+ (* > G^n --> G^n > *)
apply rule_Gn_right.
Qed.
(** [max_stride] returns the maximum number of times the stride rule can
be applied to a tape before it becomes no longer possible. If [None]
is returned, that means the rule can be applied an arbitrarily high
amount of times – that happens for the very tail of the tape, without
any [r_C] symbols.
You'll note that [max_stride] is not critical for correctness in any way,
since if it returns a value that's too large, the acceleration will simply
not be applied. As such, we don't really prove anything about [max_stride].
*)
Fixpoint max_stride (xs : N) (t : rtape) : option N :=
match t with
| [r_P] => None
| r_P :: _ => Some 0%N
| [] => Some 0%N
| r_xs xs' :: t => max_stride (xs + N.pos xs') t
| r_D :: t => max_stride 0 t
| r_C :: t =>
match max_stride 0 t with
| Some n' => Some (N.min xs (N.shiftr n' 2))
| None => Some xs
end
| r_Gs _ :: t => max_stride 0 t
end.
Fixpoint stride (xs : N) (n : positive) (t : rtape) : option rtape :=
match t with
| [r_P] => Some (rxs xs [r_P])
| r_P :: _ => None
| [] => None
| r_xs xs' :: t => stride (xs + N.pos xs') n t
| r_D :: t =>
match stride 0 n t with
| Some t => Some (rxs xs (r_D :: t))
| None => None
end
| r_C :: t =>
if (N.pos n <=? xs)%N then
match stride 0 n~0~0 t with
| Some t => Some (rxs (xs - N.pos n) (r_C :: rxs (N.pos n~0) t))
| None => None
end
else
None
| r_Gs gs :: t =>
match stride 0 n t with
| Some t => Some (rxs xs (Grs (N.pos gs) t))
| None => None
end
end.
Lemma stride_rxs : forall t xs xs' n,
stride xs n (rxs xs' t) = stride (xs + xs') n t.
Proof.
introv.
destruct xs'.
- rewrite N.add_0_r. reflexivity.
- destruct t as [| [] t]; try reflexivity.
simpl. f_equal. lia.
Qed.
Lemma rxs_rxs : forall n m t,
rxs n (rxs m t) = rxs (n + m) t.
Proof.
introv.
destruct n, m; try reflexivity.
destruct t as [| [] t]; try reflexivity.
unfold rxs. simpl.
repeat (lia || f_equal).
Qed.
Lemma Fls_Fls : forall n m t,
Fls n (Fls m t) = Fls (n + m) t.
Proof.
introv.
destruct n, m; try reflexivity.
destruct t as [| [] t]; try reflexivity.
unfold Fls. simpl.
repeat (lia || f_equal).
Qed.
Lemma Grs_Grs : forall n m t,
Grs n (Grs m t) = Grs (n + m) t.
Proof.
introv.
destruct n, m; try reflexivity.
destruct t as [| [] t]; try reflexivity.
unfold Grs. simpl.
repeat (lia || f_equal).
Qed.
(** A "tail recursive" implementation of [stride] that hopefully, perhaps,
possibly, might be better performance-wise when evaluating within Coq.
Actual tail recursion would be a huge pain with smart constructors like
[rxs], so let's try explicit continuations first. *)
Fixpoint stride' (xs : N) (n :positive) (t : rtape)
(k : rtape -> rtape) : option rtape :=
match t with
| [r_P] => Some (k (rxs xs [r_P]))
| r_P :: _ => None
| [] => None
| r_xs xs' :: t => stride' (xs + N.pos xs') n t k
| r_D :: t => stride' 0 n t (fun t => k (rxs xs (r_D :: t)))
| r_C :: t =>
if (N.pos n <=? xs)%N then
stride' 0 n~0~0 t (fun t => k (rxs (xs - N.pos n)
(r_C :: rxs (N.pos n~0) t)))
else
None
| r_Gs gs :: t =>
stride' 0 n t (fun t => k (rxs xs (Grs (N.pos gs) t)))
end.
Lemma stride'_spec : forall t xs n k,
stride' xs n t k = option_map k (stride xs n t).
Proof.
induction t as [| [xs | | | | gs] t IH]; introv.
- reflexivity.
- simpl. rewrite IH. reflexivity.
- simpl. rewrite IH.
destruct (stride 0 n t); reflexivity.
- simpl.
destruct (N.pos n <=? xs)%N; try reflexivity.
rewrite IH. simpl.
destruct (stride 0 n~0~0 t); reflexivity.
- destruct t; reflexivity.
- simpl. rewrite IH.
destruct (stride 0 n t); reflexivity.
Qed.
Lemma stride_more : forall t t' xs xs' n,
stride xs' n t = Some t' ->
stride (xs + xs') n t = Some (rxs xs t').
Proof.
induction t as [| [xs'' | | | | gs] t IH]; introv H; simpl; simpl in H.
- (* [] *) discriminate.
(* simpl. rewrite rxs_rxs. reflexivity. *)
- (* r_xs xs'' :: t *)
simpl in H.
eapply IH in H.
rewrite <- N.add_assoc.
apply H.
- (* r_D :: t *)
simpl in H.
destruct (stride 0 n t) as [t1 |] eqn:E; inverts H.
rewrite rxs_rxs. reflexivity.
- (* r_C :: t *)
destruct (N.leb_spec (N.pos n) xs') as [Hle |]; try discriminate.
destruct (N.leb_spec (N.pos n) (xs + xs')) as [Hle' |]; try lia.
destruct (stride 0 n~0~0 t) as [t1 |] eqn:E; inverts H.
rewrite rxs_rxs.
repeat (lia || f_equal).
- (* r_P :: t *)
destruct t; inverts H.
rewrite rxs_rxs. reflexivity.
- (* r_Gs gs :: t *)
simpl in H.
destruct (stride 0 n t) as [t1 |] eqn:E; inverts H.
rewrite rxs_rxs. reflexivity.
Qed.
Lemma stride_Grs : forall t t' xs gs n,
stride 0 n t = Some t' ->
stride xs n (Grs gs t) = Some (rxs xs (Grs gs t')).
Proof.
introv H.
destruct gs.
- eapply stride_more in H. rewrite N.add_0_r in H. exact H.
- destruct t as [| [] t]; try discriminate;
try (unfold Grs at 1; simpl; simpl in H; rewrite H; reflexivity).
simpl. simpl in H. unfold rxs in H.
destruct (stride 0 n t) as [t1|]; inverts H.
rewrite Grs_Grs.
repeat (lia || f_equal).
Qed.
Lemma stride_add : forall t t2 xs n m,
stride xs (n + m) t = Some t2 ->
exists t1, stride xs n t = Some t1 /\ stride 0 m t1 = Some t2.
Proof.
induction t as [| [xs' | | | | gs] t IH]; introv H.
- (* [] *) discriminate.
- (* r_xs xs' :: t *)
simpl in H.
apply IH in H. destruct H as [t1 [H1 H2]].
exists t1. intuition.
- (* r_D :: t *)
simpl in H.
destruct (stride 0 (n + m) t) as [t2' |] eqn:E; inverts H.
apply IH in E. destruct E as [t1 [H1 H2]].
exists (rxs xs (r_D :: t1)).
simpl. rewrite H1.
repeat split.
rewrite stride_rxs. simpl.
rewrite H2. reflexivity.
- (* r_C :: t *)
simpl in H. simpl.
destruct (N.leb_spec (N.pos (n + m)) xs) as [Hle |]; try discriminate.
destruct (stride 0 (n + m)~0~0 t) as [t2' |] eqn:E; inverts H.
destruct (N.leb_spec (N.pos n) xs) as [Hle' |]; try lia.
replace (n + m)~0~0%positive with (n~0~0 + m~0~0)%positive in E by lia.
apply IH in E. destruct E as [t1 [H1 H2]].
rewrite H1.
eexists. repeat split.
rewrite stride_rxs. simpl.
destruct (N.leb_spec (N.pos m) (xs - N.pos n)) as [Hle'' |]; try lia.
rewrite stride_rxs. simpl.
eapply stride_more in H2.
rewrite N.add_0_r in H2. rewrite H2.
rewrite rxs_rxs.
repeat (lia || f_equal).
- (* r_P :: t *)
simpl in H.
destruct t; inverts H.
eexists. repeat split.
rewrite stride_rxs. reflexivity.
- (* r_Gs gs :: t *)
simpl in H.
destruct (stride 0 (n + m) t) as [t2' |] eqn:E; inverts H.
apply IH in E. destruct E as [t1 [H1 H2]].
exists (rxs xs (Grs (N.pos gs) t1)).
simpl. rewrite H1.
repeat split.
rewrite stride_rxs.
eapply stride_Grs in H2. rewrite H2.
reflexivity.
Qed.
Fixpoint stride_level (t : rtape) : nat :=
match t with
| [] => 0
| r_C :: t => S (stride_level t)
| _ :: t => stride_level t
end.
Lemma stride_level_rxs : forall xs t,
stride_level (rxs xs t) = stride_level t.
Proof.
introv. destruct xs; try reflexivity.
destruct t as [| [] t]; reflexivity.
Qed.
Lemma stride_level_Grs : forall xs t,
stride_level (Grs xs t) = stride_level t.
Proof.
introv. destruct xs; try reflexivity.
destruct t as [| [] t]; reflexivity.
Qed.
Lemma stride_same_level : forall t t' xs n,
stride xs n t = Some t' ->
stride_level t = stride_level t'.
Proof.
induction t as [| [] t IH]; introv H.
- (* [] *) discriminate.
- (* r_xs n :: t *)
eapply IH, H.
- (* r_D :: t *)
simpl in H.
destruct (stride 0 n t) as [t1 |] eqn:E; inverts H.
rewrite stride_level_rxs. simpl.
eapply IH, E.
- (* r_C :: t *)
simpl in H.
destruct (N.leb_spec (N.pos n) xs) as [Hle |]; try discriminate.
destruct (stride 0 n~0~0 t) as [t1 |] eqn:E; inverts H.
repeat (rewrite stride_level_rxs; simpl).
f_equal.
eapply IH, E.
- (* r_P :: t *)
simpl in H.
destruct t; inverts H.
rewrite stride_level_rxs.