(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).(** * Determinisation of the transition relation *)

(** We now define a deterministic variant [dtransition] of the
  transition relation [transition].  [dtransition] is deterministic
  in the sense that at most one transition applies to a given state. *)

Inductive dtransition: state -> state -> Prop :=
  | dtr_nop: forall r mu tau,
      dtransition (State ((r, r) :: mu) nil tau)
                  (State mu nil tau)
  | dtr_start: forall s d mu tau,
      s <> d ->
      dtransition (State ((s, d) :: mu) nil tau)
                  (State mu ((s, d) :: nil) tau)
  | dtr_push: forall mu1 d r mu2 s sigma tau,
      no_read mu1 d ->
      dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
                  (State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
  | dtr_loop_pop: forall mu s r0 d  sigma tau,
      no_read mu r0 ->
      dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
                  (State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
  | dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
      no_read mu d1 -> d1 <> s0 ->
      dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
                  (State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
  | dtr_last: forall mu s d tau,
      no_read mu d ->
      dtransition (State mu ((s, d) :: nil) tau)
                  (State mu nil ((s, d) :: tau)).Definition dtransitions: state -> state -> Prop :=
  clos_refl_trans state dtransition.

(* Specifications: *)
Lemma transition_determ: forall st st', dtransition st st' -> state_wf st -> transitions st st'.
Lemma transitions_determ: forall st st', dtransitions st st' -> state_wf st -> transitions st st'.
Theorem dtransitions_correctness: forall mu tau, move_no_temp mu -> is_mill mu -> dtransitions (State mu nil nil) (State nil nil tau) -> forall e, env_equiv (exec_seq (List.rev tau) e) (exec_par mu e).
Lemma transition_determ: forall st st', dtransition st st' -> state_wf st -> transitions st st'.
Lemma transitions_determ: forall st st', dtransitions st st' -> state_wf st -> transitions st st'.
Theorem dtransitions_correctness: forall mu tau, move_no_temp mu -> is_mill mu -> dtransitions (State mu nil nil) (State nil nil tau) -> forall e, env_equiv (exec_seq (List.rev tau) e) (exec_par mu e).