From Undecidability.Shared.Libs.DLW
Require Import Vec.pos Vec.vec Code.sss.
Set Implicit Arguments.
Inductive mm_instr (X : Set) : Set :=
| mm_inc : X -> mm_instr X
| mm_dec : X -> nat -> mm_instr X
.
Notation INC := mm_inc.
Notation DEC := mm_dec.
Section Minsky_Machine.
Variable (n : nat).
Definition mm_state := (nat*vec nat n)%type.
Local Notation "e #> x" := (vec_pos e x).
Local Notation "e [ v / x ]" := (vec_change e x v).
Inductive mm_sss : mm_instr (pos n) -> mm_state -> mm_state -> Prop :=
| in_mm_sss_inc : forall i x v, INC x // (i,v) -1> (1+i,v[(S (v#>x))/x])
| in_mm_sss_dec_0 : forall i x k v, v#>x = O -> DEC x k // (i,v) -1> (k,v)
| in_mm_sss_dec_1 : forall i x k v u, v#>x = S u -> DEC x k // (i,v) -1> (1+i,v[u/x])
where "i // s -1> t" := (mm_sss i s t).
Inductive mma_sss : mm_instr (pos n) -> mm_state -> mm_state -> Prop :=
| in_mma_sss_inc : forall i x v, INC x // (i,v) -1> (1+i,v[(S (v#>x))/x])
| in_mma_sss_dec_0 : forall i x k v, v#>x = O -> DEC x k // (i,v) -1> (1+i,v)
| in_mma_sss_dec_1 : forall i x k v u, v#>x = S u -> DEC x k // (i,v) -1> (k,v[u/x])
where "i // s -1> t" := (mma_sss i s t).
End Minsky_Machine.
Section MM_problems.
Notation "P // s ~~> t" := (sss_output (@mm_sss _) P s t).
Notation "P // s ↓" := (sss_terminates (@mm_sss _) P s).
Definition MM_PROBLEM := { n : nat & { P : list (mm_instr (pos n)) & vec nat n } }.
Definition MM_HALTS_ON_ZERO (P : MM_PROBLEM) :=
match P with existT _ n (existT _ P v) => (1,P) // (1,v) ~~> (0,vec_zero) end.
Definition MM_HALTING (P : MM_PROBLEM) :=
match P with existT _ n (existT _ P v) => (1, P) // (1, v) ↓ end.
End MM_problems.
Section MMA_problems.
Notation "P // s ~~> t" := (sss_output (@mma_sss _) P s t).
Notation "P // s ↓" := (sss_terminates (@mma_sss _) P s).
Definition MMA2_PROBLEM := (list (mm_instr (pos 2)) * vec nat 2)%type.
Definition MMA2_HALTS_ON_ZERO (P : MMA2_PROBLEM) := (1,fst P) // (1,snd P) ~~> (0,vec_zero).
Definition MMA2_HALTING (P : MMA2_PROBLEM) := (1,fst P) // (1,snd P) ↓.
End MMA_problems.