(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* System TCG: Especification *)

  (* Set of Labels *)

  Inductive Label : Set :=
    | Approach : Label
    | In : Label
    | Exit : Label
    | Lower : Label
    | Raise : Label
    | Up : Label
    | Down : Label
    | Tick : Label.(* Tick represent temporal transitions *)  

  
  (* Locations: ST (Train), SC (Controller) and SG (Gate) *)
  
  Inductive ST : Set :=
    | Far : ST
    | Near : ST
    | Inside : ST.(* Tick represent temporal transitions *)  

  
  (* Locations: ST (Train), SC (Controller) and SG (Gate) *)
  
  Inductive ST : Set :=
    | Far : ST
    | Near : ST
    | Inside : ST.(* Tick represent temporal transitions *)  

  
  (* Locations: ST (Train), SC (Controller) and SG (Gate) *)
  
  Inductive ST : Set :=
    | Far : ST
    | Near : ST
    | Inside : ST.(* Tick represent temporal transitions *)  

  
  (* Locations: ST (Train), SC (Controller) and SG (Gate) *)
  
  Inductive ST : Set :=
    | Far : ST
    | Near : ST
    | Inside : ST.Inductive SC : Set :=
    | Sc1 : SC
    | Sc2 : SC
    | Sc3 : SC
    | Sc4 : SC.Inductive SC : Set :=
    | Sc1 : SC
    | Sc2 : SC
    | Sc3 : SC
    | Sc4 : SC.Inductive SC : Set :=
    | Sc1 : SC
    | Sc2 : SC
    | Sc3 : SC
    | Sc4 : SC.Inductive SC : Set :=
    | Sc1 : SC
    | Sc2 : SC
    | Sc3 : SC
    | Sc4 : SC.Inductive SC : Set :=
    | Sc1 : SC
    | Sc2 : SC
    | Sc3 : SC
    | Sc4 : SC.Inductive SG : Set :=
    | Open : SG
    | Lowering : SG
    | Closed : SG
    | Raising : SG.Inductive SG : Set :=
    | Open : SG
    | Lowering : SG
    | Closed : SG
    | Raising : SG.Inductive SG : Set :=
    | Open : SG
    | Lowering : SG
    | Closed : SG
    | Raising : SG.Inductive SG : Set :=
    | Open : SG
    | Lowering : SG
    | Closed : SG
    | Raising : SG.Inductive SG : Set :=
    | Open : SG
    | Lowering : SG
    | Closed : SG
    | Raising : SG.(* States with one clock *)

  Definition S_Ck (S : Set) := (S * Clock)%type.(* TCG constants *)

  Definition kt1 : Instant := 8.Definition kt2 : Instant := 20.Definition kc1 : Instant := 4.Definition kc2 : Instant := 4.Definition kg1 : Instant := 4.Definition kg2 : Instant := 4.Definition kg3 : Instant := 8.(* Invariants of locations for the Train *)

  Inductive InvT : S_Ck ST -> Prop :=
    | Ifar : forall x : Clock, InvT (Far, x)
    | Inear : forall x : Clock, lt_Ck x kt2 -> InvT (Near, x)
    | Iinside : forall x : Clock, lt_Ck x kt2 -> InvT (Inside, x).(* Invariants of locations for the Train *)

  Inductive InvT : S_Ck ST -> Prop :=
    | Ifar : forall x : Clock, InvT (Far, x)
    | Inear : forall x : Clock, lt_Ck x kt2 -> InvT (Near, x)
    | Iinside : forall x : Clock, lt_Ck x kt2 -> InvT (Inside, x).(* Invariants of locations for the Train *)

  Inductive InvT : S_Ck ST -> Prop :=
    | Ifar : forall x : Clock, InvT (Far, x)
    | Inear : forall x : Clock, lt_Ck x kt2 -> InvT (Near, x)
    | Iinside : forall x : Clock, lt_Ck x kt2 -> InvT (Inside, x).(* Invariants of locations for the Train *)

  Inductive InvT : S_Ck ST -> Prop :=
    | Ifar : forall x : Clock, InvT (Far, x)
    | Inear : forall x : Clock, lt_Ck x kt2 -> InvT (Near, x)
    | Iinside : forall x : Clock, lt_Ck x kt2 -> InvT (Inside, x).(* Invariants of locations for the Controller *)

  Inductive InvC : S_Ck SC -> Prop :=
    | Isc1 : forall y : Clock, InvC (Sc1, y)
    | Isc2 : forall y : Clock, le_Ck y kc1 -> InvC (Sc2, y)
    | Isc3 : forall y : Clock, InvC (Sc3, y)
    | Isc4 : forall y : Clock, lt_Ck y kc2 -> InvC (Sc4, y).(* Invariants of locations for the Controller *)

  Inductive InvC : S_Ck SC -> Prop :=
    | Isc1 : forall y : Clock, InvC (Sc1, y)
    | Isc2 : forall y : Clock, le_Ck y kc1 -> InvC (Sc2, y)
    | Isc3 : forall y : Clock, InvC (Sc3, y)
    | Isc4 : forall y : Clock, lt_Ck y kc2 -> InvC (Sc4, y).(* Invariants of locations for the Controller *)

  Inductive InvC : S_Ck SC -> Prop :=
    | Isc1 : forall y : Clock, InvC (Sc1, y)
    | Isc2 : forall y : Clock, le_Ck y kc1 -> InvC (Sc2, y)
    | Isc3 : forall y : Clock, InvC (Sc3, y)
    | Isc4 : forall y : Clock, lt_Ck y kc2 -> InvC (Sc4, y).(* Invariants of locations for the Controller *)

  Inductive InvC : S_Ck SC -> Prop :=
    | Isc1 : forall y : Clock, InvC (Sc1, y)
    | Isc2 : forall y : Clock, le_Ck y kc1 -> InvC (Sc2, y)
    | Isc3 : forall y : Clock, InvC (Sc3, y)
    | Isc4 : forall y : Clock, lt_Ck y kc2 -> InvC (Sc4, y).(* Invariants of locations for the Controller *)

  Inductive InvC : S_Ck SC -> Prop :=
    | Isc1 : forall y : Clock, InvC (Sc1, y)
    | Isc2 : forall y : Clock, le_Ck y kc1 -> InvC (Sc2, y)
    | Isc3 : forall y : Clock, InvC (Sc3, y)
    | Isc4 : forall y : Clock, lt_Ck y kc2 -> InvC (Sc4, y).(* Invariants of locations for the Gate *)

  Inductive InvG : S_Ck SG -> Prop :=
    | Iopen : forall z : Clock, InvG (Open, z)
    | Ilowering : forall z : Clock, lt_Ck z kg1 -> InvG (Lowering, z)
    | Iclosed : forall z : Clock, InvG (Closed, z)
    | Iraising : forall z : Clock, lt_Ck z kg3 -> InvG (Raising, z).(* Invariants of locations for the Gate *)

  Inductive InvG : S_Ck SG -> Prop :=
    | Iopen : forall z : Clock, InvG (Open, z)
    | Ilowering : forall z : Clock, lt_Ck z kg1 -> InvG (Lowering, z)
    | Iclosed : forall z : Clock, InvG (Closed, z)
    | Iraising : forall z : Clock, lt_Ck z kg3 -> InvG (Raising, z).(* Invariants of locations for the Gate *)

  Inductive InvG : S_Ck SG -> Prop :=
    | Iopen : forall z : Clock, InvG (Open, z)
    | Ilowering : forall z : Clock, lt_Ck z kg1 -> InvG (Lowering, z)
    | Iclosed : forall z : Clock, InvG (Closed, z)
    | Iraising : forall z : Clock, lt_Ck z kg3 -> InvG (Raising, z).(* Invariants of locations for the Gate *)

  Inductive InvG : S_Ck SG -> Prop :=
    | Iopen : forall z : Clock, InvG (Open, z)
    | Ilowering : forall z : Clock, lt_Ck z kg1 -> InvG (Lowering, z)
    | Iclosed : forall z : Clock, InvG (Closed, z)
    | Iraising : forall z : Clock, lt_Ck z kg3 -> InvG (Raising, z).(* Invariants of locations for the Gate *)

  Inductive InvG : S_Ck SG -> Prop :=
    | Iopen : forall z : Clock, InvG (Open, z)
    | Ilowering : forall z : Clock, lt_Ck z kg1 -> InvG (Lowering, z)
    | Iclosed : forall z : Clock, InvG (Closed, z)
    | Iraising : forall z : Clock, lt_Ck z kg3 -> InvG (Raising, z).(* Train transitions *)

  Inductive TrT : S_Ck ST -> Label -> S_Ck ST -> Prop :=
    | ttApproach : forall x : Clock, TrT (Far, x) Approach (Near, Reset)
    | ttIn :
        forall x : Clock,
        gt_Ck x kt1 -> lt_Ck x kt2 -> TrT (Near, x) In (Inside, x)
    | ttExit : forall x : Clock, lt_Ck x kt2 -> TrT (Inside, x) Exit (Far, x)
    | ttTick :
        forall (x : Clock) (s : ST),
        InvT (s, Inc x) -> TrT (s, x) Tick (s, Inc x).(* Train transitions *)

  Inductive TrT : S_Ck ST -> Label -> S_Ck ST -> Prop :=
    | ttApproach : forall x : Clock, TrT (Far, x) Approach (Near, Reset)
    | ttIn :
        forall x : Clock,
        gt_Ck x kt1 -> lt_Ck x kt2 -> TrT (Near, x) In (Inside, x)
    | ttExit : forall x : Clock, lt_Ck x kt2 -> TrT (Inside, x) Exit (Far, x)
    | ttTick :
        forall (x : Clock) (s : ST),
        InvT (s, Inc x) -> TrT (s, x) Tick (s, Inc x).(* Train transitions *)

  Inductive TrT : S_Ck ST -> Label -> S_Ck ST -> Prop :=
    | ttApproach : forall x : Clock, TrT (Far, x) Approach (Near, Reset)
    | ttIn :
        forall x : Clock,
        gt_Ck x kt1 -> lt_Ck x kt2 -> TrT (Near, x) In (Inside, x)
    | ttExit : forall x : Clock, lt_Ck x kt2 -> TrT (Inside, x) Exit (Far, x)
    | ttTick :
        forall (x : Clock) (s : ST),
        InvT (s, Inc x) -> TrT (s, x) Tick (s, Inc x).(* Train transitions *)

  Inductive TrT : S_Ck ST -> Label -> S_Ck ST -> Prop :=
    | ttApproach : forall x : Clock, TrT (Far, x) Approach (Near, Reset)
    | ttIn :
        forall x : Clock,
        gt_Ck x kt1 -> lt_Ck x kt2 -> TrT (Near, x) In (Inside, x)
    | ttExit : forall x : Clock, lt_Ck x kt2 -> TrT (Inside, x) Exit (Far, x)
    | ttTick :
        forall (x : Clock) (s : ST),
        InvT (s, Inc x) -> TrT (s, x) Tick (s, Inc x).(* Train transitions *)

  Inductive TrT : S_Ck ST -> Label -> S_Ck ST -> Prop :=
    | ttApproach : forall x : Clock, TrT (Far, x) Approach (Near, Reset)
    | ttIn :
        forall x : Clock,
        gt_Ck x kt1 -> lt_Ck x kt2 -> TrT (Near, x) In (Inside, x)
    | ttExit : forall x : Clock, lt_Ck x kt2 -> TrT (Inside, x) Exit (Far, x)
    | ttTick :
        forall (x : Clock) (s : ST),
        InvT (s, Inc x) -> TrT (s, x) Tick (s, Inc x).(* Controller transitions *)

  Inductive TrC : S_Ck SC -> Label -> S_Ck SC -> Prop :=
    | tcApproach : forall y : Clock, TrC (Sc1, y) Approach (Sc2, Reset)
    | tcLower : forall y : Clock, eq_Ck y kc1 -> TrC (Sc2, y) Lower (Sc3, y)
    | tcExit : forall y : Clock, TrC (Sc3, y) Exit (Sc4, Reset)
    | tcRaise : forall y : Clock, lt_Ck y kc2 -> TrC (Sc4, y) Raise (Sc1, y)
    | tcTick :
        forall (y : Clock) (s : SC),
        InvC (s, Inc y) -> TrC (s, y) Tick (s, Inc y).(* Controller transitions *)

  Inductive TrC : S_Ck SC -> Label -> S_Ck SC -> Prop :=
    | tcApproach : forall y : Clock, TrC (Sc1, y) Approach (Sc2, Reset)
    | tcLower : forall y : Clock, eq_Ck y kc1 -> TrC (Sc2, y) Lower (Sc3, y)
    | tcExit : forall y : Clock, TrC (Sc3, y) Exit (Sc4, Reset)
    | tcRaise : forall y : Clock, lt_Ck y kc2 -> TrC (Sc4, y) Raise (Sc1, y)
    | tcTick :
        forall (y : Clock) (s : SC),
        InvC (s, Inc y) -> TrC (s, y) Tick (s, Inc y).(* Controller transitions *)

  Inductive TrC : S_Ck SC -> Label -> S_Ck SC -> Prop :=
    | tcApproach : forall y : Clock, TrC (Sc1, y) Approach (Sc2, Reset)
    | tcLower : forall y : Clock, eq_Ck y kc1 -> TrC (Sc2, y) Lower (Sc3, y)
    | tcExit : forall y : Clock, TrC (Sc3, y) Exit (Sc4, Reset)
    | tcRaise : forall y : Clock, lt_Ck y kc2 -> TrC (Sc4, y) Raise (Sc1, y)
    | tcTick :
        forall (y : Clock) (s : SC),
        InvC (s, Inc y) -> TrC (s, y) Tick (s, Inc y).(* Controller transitions *)

  Inductive TrC : S_Ck SC -> Label -> S_Ck SC -> Prop :=
    | tcApproach : forall y : Clock, TrC (Sc1, y) Approach (Sc2, Reset)
    | tcLower : forall y : Clock, eq_Ck y kc1 -> TrC (Sc2, y) Lower (Sc3, y)
    | tcExit : forall y : Clock, TrC (Sc3, y) Exit (Sc4, Reset)
    | tcRaise : forall y : Clock, lt_Ck y kc2 -> TrC (Sc4, y) Raise (Sc1, y)
    | tcTick :
        forall (y : Clock) (s : SC),
        InvC (s, Inc y) -> TrC (s, y) Tick (s, Inc y).(* Controller transitions *)

  Inductive TrC : S_Ck SC -> Label -> S_Ck SC -> Prop :=
    | tcApproach : forall y : Clock, TrC (Sc1, y) Approach (Sc2, Reset)
    | tcLower : forall y : Clock, eq_Ck y kc1 -> TrC (Sc2, y) Lower (Sc3, y)
    | tcExit : forall y : Clock, TrC (Sc3, y) Exit (Sc4, Reset)
    | tcRaise : forall y : Clock, lt_Ck y kc2 -> TrC (Sc4, y) Raise (Sc1, y)
    | tcTick :
        forall (y : Clock) (s : SC),
        InvC (s, Inc y) -> TrC (s, y) Tick (s, Inc y).(* Controller transitions *)

  Inductive TrC : S_Ck SC -> Label -> S_Ck SC -> Prop :=
    | tcApproach : forall y : Clock, TrC (Sc1, y) Approach (Sc2, Reset)
    | tcLower : forall y : Clock, eq_Ck y kc1 -> TrC (Sc2, y) Lower (Sc3, y)
    | tcExit : forall y : Clock, TrC (Sc3, y) Exit (Sc4, Reset)
    | tcRaise : forall y : Clock, lt_Ck y kc2 -> TrC (Sc4, y) Raise (Sc1, y)
    | tcTick :
        forall (y : Clock) (s : SC),
        InvC (s, Inc y) -> TrC (s, y) Tick (s, Inc y).(* Gate transitions *)

  Inductive TrG : S_Ck SG -> Label -> S_Ck SG -> Prop :=
    | tgLower : forall z : Clock, TrG (Open, z) Lower (Lowering, Reset)
    | tgDown :
        forall z : Clock, lt_Ck z kg1 -> TrG (Lowering, z) Down (Closed, z)
    | tgRaise : forall z : Clock, TrG (Closed, z) Raise (Raising, Reset)
    | tgUp :
        forall z : Clock,
        ge_Ck z kg2 -> lt_Ck z kg3 -> TrG (Raising, z) Up (Open, z)
    | tgTick :
        forall (z : Clock) (s : SG),
        InvG (s, Inc z) -> TrG (s, z) Tick (s, Inc z).(* Gate transitions *)

  Inductive TrG : S_Ck SG -> Label -> S_Ck SG -> Prop :=
    | tgLower : forall z : Clock, TrG (Open, z) Lower (Lowering, Reset)
    | tgDown :
        forall z : Clock, lt_Ck z kg1 -> TrG (Lowering, z) Down (Closed, z)
    | tgRaise : forall z : Clock, TrG (Closed, z) Raise (Raising, Reset)
    | tgUp :
        forall z : Clock,
        ge_Ck z kg2 -> lt_Ck z kg3 -> TrG (Raising, z) Up (Open, z)
    | tgTick :
        forall (z : Clock) (s : SG),
        InvG (s, Inc z) -> TrG (s, z) Tick (s, Inc z).(* Gate transitions *)

  Inductive TrG : S_Ck SG -> Label -> S_Ck SG -> Prop :=
    | tgLower : forall z : Clock, TrG (Open, z) Lower (Lowering, Reset)
    | tgDown :
        forall z : Clock, lt_Ck z kg1 -> TrG (Lowering, z) Down (Closed, z)
    | tgRaise : forall z : Clock, TrG (Closed, z) Raise (Raising, Reset)
    | tgUp :
        forall z : Clock,
        ge_Ck z kg2 -> lt_Ck z kg3 -> TrG (Raising, z) Up (Open, z)
    | tgTick :
        forall (z : Clock) (s : SG),
        InvG (s, Inc z) -> TrG (s, z) Tick (s, Inc z).(* Gate transitions *)

  Inductive TrG : S_Ck SG -> Label -> S_Ck SG -> Prop :=
    | tgLower : forall z : Clock, TrG (Open, z) Lower (Lowering, Reset)
    | tgDown :
        forall z : Clock, lt_Ck z kg1 -> TrG (Lowering, z) Down (Closed, z)
    | tgRaise : forall z : Clock, TrG (Closed, z) Raise (Raising, Reset)
    | tgUp :
        forall z : Clock,
        ge_Ck z kg2 -> lt_Ck z kg3 -> TrG (Raising, z) Up (Open, z)
    | tgTick :
        forall (z : Clock) (s : SG),
        InvG (s, Inc z) -> TrG (s, z) Tick (s, Inc z).(* Gate transitions *)

  Inductive TrG : S_Ck SG -> Label -> S_Ck SG -> Prop :=
    | tgLower : forall z : Clock, TrG (Open, z) Lower (Lowering, Reset)
    | tgDown :
        forall z : Clock, lt_Ck z kg1 -> TrG (Lowering, z) Down (Closed, z)
    | tgRaise : forall z : Clock, TrG (Closed, z) Raise (Raising, Reset)
    | tgUp :
        forall z : Clock,
        ge_Ck z kg2 -> lt_Ck z kg3 -> TrG (Raising, z) Up (Open, z)
    | tgTick :
        forall (z : Clock) (s : SG),
        InvG (s, Inc z) -> TrG (s, z) Tick (s, Inc z).(* Gate transitions *)

  Inductive TrG : S_Ck SG -> Label -> S_Ck SG -> Prop :=
    | tgLower : forall z : Clock, TrG (Open, z) Lower (Lowering, Reset)
    | tgDown :
        forall z : Clock, lt_Ck z kg1 -> TrG (Lowering, z) Down (Closed, z)
    | tgRaise : forall z : Clock, TrG (Closed, z) Raise (Raising, Reset)
    | tgUp :
        forall z : Clock,
        ge_Ck z kg2 -> lt_Ck z kg3 -> TrG (Raising, z) Up (Open, z)
    | tgTick :
        forall (z : Clock) (s : SG),
        InvG (s, Inc z) -> TrG (s, z) Tick (s, Inc z).(* States for the system: train||controller||gate *)

  Definition StGlobal := (S_Ck ST * (S_Ck SC * S_Ck SG))%type.(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Transitions for the system: train||controller||gate *)

  Inductive TrGlobal : StGlobal -> Label -> StGlobal -> Prop :=
    | tGl_In :
        forall st1 st2 : S_Ck ST,
        TrT st1 In st2 ->
        forall (sc : S_Ck SC) (sg : S_Ck SG),
        TrGlobal (st1, (sc, sg)) In (st2, (sc, sg))
    | tGl_Down :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Down sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Down (st, (sc, sg2))
    | tGl_Up :
        forall sg1 sg2 : S_Ck SG,
        TrG sg1 Up sg2 ->
        forall (st : S_Ck ST) (sc : S_Ck SC),
        TrGlobal (st, (sc, sg1)) Up (st, (sc, sg2))
    | tGl_Approach :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Approach st2 ->
        TrC sc1 Approach sc2 ->
        forall sg : S_Ck SG,
        TrGlobal (st1, (sc1, sg)) Approach (st2, (sc2, sg))
    | tGl_Exit :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC),
        TrT st1 Exit st2 ->
        TrC sc1 Exit sc2 ->
        forall sg : S_Ck SG, TrGlobal (st1, (sc1, sg)) Exit (st2, (sc2, sg))
    | tGl_Lower :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Lower sc2 ->
        TrG sg1 Lower sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Lower (st, (sc2, sg2))
    | tGl_Raise :
        forall (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrC sc1 Raise sc2 ->
        TrG sg1 Raise sg2 ->
        forall st : S_Ck ST, TrGlobal (st, (sc1, sg1)) Raise (st, (sc2, sg2))
    | tGl_Tick :
        forall (st1 st2 : S_Ck ST) (sc1 sc2 : S_Ck SC) (sg1 sg2 : S_Ck SG),
        TrT st1 Tick st2 ->
        TrC sc1 Tick sc2 ->
        TrG sg1 Tick sg2 -> TrGlobal (st1, (sc1, sg1)) Tick (st2, (sc2, sg2)).(* Initial states *) 

  Definition ini_CkT := 0.Definition ini_CkC := 0.Definition ini_CkG := 0.Definition SiniT : S_Ck ST := (Far, ini_CkT).Definition SiniC : S_Ck SC := (Sc1, ini_CkC).Definition SiniG : S_Ck SG := (Open, ini_CkG).Definition SiniTCG : StGlobal := (SiniT, (SiniC, SiniG)).

Lemma Trivial4 : forall x y z : Clock, eq_Ck x (plus_Ck y z) -> eq_Ck (Inc x) (plus_Ck (Inc y) z).