sort Nat

op 0, 1 : Nat

op __+__, __-__, min : Nat * Nat -> Nat

pred __>=__, __<=__, __>__ : Nat * Nat

free type Boolean ::= True | False

var S, V : Type

type Set : Type -> Type := \ S : Type . S ->? Unit

op emptySet : Set S;

{__} : S -> Set S;

__isIn__ : S * Set S ->? Unit;

__subset__ : Pred (Set (S) * Set (S));

__union__, __intersection__, __\\__ : Set S * Set S -> Set S;

__disjoint__ : Pred (Set (S) * Set (S));

__*__ : Set S * Set V -> Set (S * V);

__disjointUnion__ : Set S * Set S -> Set (S * Boolean);

injl, injr : S -> S * Boolean

var Elem : Type

type MultiSet : Type -> Type := \ Elem : Type . Elem ->? Nat

op __isIn__ : Pred (Elem * MultiSet Elem);

__<=__ : Pred (MultiSet Elem * MultiSet Elem);

{} : MultiSet Elem;

{__} : Elem -> MultiSet Elem;

__+__, __-__, __intersection__

: MultiSet Elem * MultiSet Elem -> MultiSet Elem;

freq : Elem * MultiSet Elem -> Nat;

setToMultiSet : Set Elem -> MultiSet Elem

var Elem : Type

op MultiSetToSet : MultiSet Elem -> Set Elem

forall B : MultiSet Elem; S : Set Elem

. let S = MultiSetToSet B in

forall x : Elem . (x isIn S) <=> ((freq (x, B)) > 0);

var S : Type

type MapMultiSet : Type -> Type

:= \ S : Type . MultiSet S ->? MultiSet S

var a : Type

op sumN : (Nat ->? Nat) -> Nat -> Nat;

sumSet : Set Nat ->? Nat;

sum : (a ->? Nat) -> Pred a ->? Nat

var S, V, U : Type

type Map : Type -> Type := \ S : Type . S ->? S

op dom : (S ->? V) -> Set S;

range : (S ->? V) -> Set V;

image : (S ->? V) -> Set S -> Set V;

emptyMap : (S ->? V);

__::__-->__ : Pred ((S ->? V) * Pred (S) * Pred (V));

__[__/__] : (S ->? V) * S * V -> (S ->? V);

__-__ : (S ->? V) * S -> (S ->? V);

__o__ : (V ->? U) * (S ->? V) -> (S ->? U);

__||__ : (S ->? V) * Set S -> (S ->? V);

undef__ : S ->? V;

ker : (S ->? V) -> Pred (S * S);

injective : Pred (S ->? V);

__intersectionMap__, __unionMap__

: (S ->? V) * (S ->? V) -> (S ->? V);

__restrict__ : (S ->? V) * Set S -> (S ->? V)

var S, V : Type

op __::__-->__ : Pred ((S ->? MultiSet V) * Set S * Set V);

freeMap : Map S -> MapMultiSet S;

linMap : (S ->? MultiSet V) -> (MultiSet S ->? MultiSet V)

op __intersection__

: MultiSet Elem * MultiSet Elem -> MultiSet Elem, assoc, comm, idem

sort P, T

type Net =

{(p, pre, post) : Set P * (T ->? MultiSet P) * (T ->? MultiSet P) .

dom pre = dom post /\

(forall p1 : MultiSet P .

(p1 isIn (range pre)) => ((MultiSetToSet p1) subset p))

/\

(forall p1 : MultiSet P .

(p1 isIn (range pre)) => ((MultiSetToSet p1) subset p))

}

op places : Net -> Set P;

transitions : Net -> Set T;

preMap, postMap : Net -> (T ->? MultiSet P)

type HomNet =

{(n1, hp, ht, n2) : Net * (P ->? P) * (T ->? T) * Net .

(hp :: places n1 --> (places n2)) /\

(ht :: transitions n1 --> (transitions n2))

/\

(forall t : T .

(t isIn (transitions n1)) =>

freeMap hp (preMap n1 t) = preMap n2 (ht t) /\

freeMap hp (postMap n1 t) = postMap n2 (ht t))

}

op dom : HomNet -> Net;

cod : HomNet -> Net;

placesMap : HomNet -> (P ->? P);

transitionsMap : HomNet -> (T ->? T);

id : Net ->? HomNet;

__o__ : HomNet * HomNet ->? HomNet

pred injective : HomNet

type Marking := MultiSet P

type System =

{(n, m) : Net * Marking .

let (p, pre1, post1) = n in forall x : P . (x isIn m) => (x isIn p)

}

op marking : System -> Marking;

net : System -> Net;

empty : Marking;

__|<__> : System * T -> System;

__|<__> : System * MultiSet T ->? System

type HomSys =

{(sys1, hp, ht, sys2) : System * (P ->? P) * (T ->? T) * System .

((net sys1, hp, ht, net sys2) in HomNet) /\

(forall p : P .

(freq (p, marking sys1)) <= (freq (hp p, marking sys2)))

}

op dom : HomSys -> System;

cod : HomSys -> System;

placesMap : HomSys -> (P ->? P);

transitionsMap : HomSys -> (T ->? T);

id : System ->? HomSys;

__o__ : HomSys * HomSys ->? HomSys

pred injective : HomSys

forall h1, h2 : HomSys

. def (h2 o h1) =>

(h2 o h1) =

((dom h1, (placesMap h2) o (placesMap h1),

(transitionsMap h2) o (transitionsMap h1), cod h2)

as HomSys);

%% Type Constructors -----------------------------------------------------

Boolean

: Type

%[free type Boolean ::=

True : Boolean

False : Boolean]%

HomNet : Type := _t1018 Net (P ->? P) (T ->? T) Net

HomSys : Type := _t1516 System (P ->? P) (T ->? T) System

Map : Type -> Type := \ S : Type . S ->? S

MapMultiSet

: Type -> Type := \ S : Type . MultiSet S ->? MultiSet S

Marking : Type := MultiSet P

MultiSet : Type -> Type := \ Elem : Type . Elem ->? Nat

Nat : Type

Net : Type := _t366 (Set P) (T ->? MultiSet P) (T ->? MultiSet P)

P : Type

Set : Type -> Type := \ S : Type . S ->? Unit

System : Type := _t1088 Net Marking

T : Type

_t1015 : +Type -> Type := _t1018 Net (P ->? P) (T ->? T)

_t1016 : +Type -> +Type -> Type := _t1018 Net (P ->? P)

_t1017 : +Type -> +Type -> +Type -> Type := _t1018 Net

_t1018 : +Type -> +Type -> +Type -> +Type -> Type < __*__*__*__

_t1087 : +Type -> Type := _t1088 Net

_t1088 : +Type -> +Type -> Type < __*__

_t1513 : +Type -> Type := _t1516 System (P ->? P) (T ->? T)

_t1514 : +Type -> +Type -> Type := _t1516 System (P ->? P)

_t1515 : +Type -> +Type -> +Type -> Type := _t1516 System

_t1516 : +Type -> +Type -> +Type -> +Type -> Type < __*__*__*__

_t364 : +Type -> Type := _t366 (Set P) (T ->? MultiSet P)

_t365 : +Type -> +Type -> Type := _t366 (Set P)

_t366 : +Type -> +Type -> +Type -> Type < __*__*__

%% Type Variables --------------------------------------------------------

Elem : Type %(var_6)%

S : Type %(var_62)%

U : Type %(var_60)%

V : Type %(var_63)%

a : Type %(var_57)%

%% Assumptions -----------------------------------------------------------

0 : Nat %(op)%

1 : Nat %(op)%

False : Boolean %(construct Boolean)%

MultiSetToSet

: forall Elem : Type . MultiSet Elem -> Set Elem %(op)%

True : Boolean %(construct Boolean)%

__*__

: forall S : Type; V : Type . Set S * Set V -> Set (S * V) %(op)%

__+__

:

forall Elem : Type . MultiSet Elem * MultiSet Elem -> MultiSet Elem

%(op)%

: Nat * Nat -> Nat %(op)%

__-__

: forall S : Type; V : Type . (S ->? V) * S -> S ->? V %(op)%

:

forall Elem : Type . MultiSet Elem * MultiSet Elem -> MultiSet Elem

%(op)%

: Nat * Nat -> Nat %(op)%

__::__-->__

:

forall S : Type; V : Type .

Pred ((S ->? MultiSet V) * Set S * Set V)

%(op)%

: forall S : Type; V : Type . Pred ((S ->? V) * Pred S * Pred V)

%(op)%

__<=__

: forall Elem : Type . Pred (MultiSet Elem * MultiSet Elem) %(op)%

: Nat * Nat ->? Unit %(pred)%

__>__ : Nat * Nat ->? Unit %(pred)%

__>=__ : Nat * Nat ->? Unit %(pred)%

__[__/__]

: forall S : Type; V : Type . (S ->? V) * S * V -> S ->? V %(op)%

__\\__ : forall S : Type . Set S * Set S -> Set S %(op)%

__disjoint__ : forall S : Type . Pred (Set S * Set S) %(op)%

__disjointUnion__

: forall S : Type . Set S * Set S -> Set (S * Boolean) %(op)%

__intersection__

:

forall Elem : Type .

MultiSet Elem * MultiSet Elem -> MultiSet Elem,

assoc, comm, idem %(op)%

: forall S : Type . Set S * Set S -> Set S %(op)%

__intersectionMap__

: forall S : Type; V : Type . (S ->? V) * (S ->? V) -> S ->? V

%(op)%

__isIn__

: forall Elem : Type . Pred (Elem * MultiSet Elem) %(op)%

: forall S : Type . S * Set S ->? Unit %(op)%

__o__

: HomSys * HomSys ->? HomSys %(op)%

: HomNet * HomNet ->? HomNet %(op)%

:

forall S : Type; V : Type; U : Type .

(V ->? U) * (S ->? V) -> S ->? U

%(op)%

__restrict__

: forall S : Type; V : Type . (S ->? V) * Set S -> S ->? V %(op)%

__subset__ : forall S : Type . Pred (Set S * Set S) %(op)%

__union__ : forall S : Type . Set S * Set S -> Set S %(op)%

__unionMap__

: forall S : Type; V : Type . (S ->? V) * (S ->? V) -> S ->? V

%(op)%

__|<__>

: System * MultiSet T ->? System %(op)%

: System * T -> System %(op)%

__||__

: forall S : Type; V : Type . (S ->? V) * Set S -> S ->? V %(op)%

cod

: HomSys -> System %(op)%

: HomNet -> Net %(op)%

dom

: HomSys -> System %(op)%

: HomNet -> Net %(op)%

: forall S : Type; V : Type . (S ->? V) -> Set S %(op)%

empty : Marking %(op)%

emptyMap : forall S : Type; V : Type . S ->? V %(op)%

emptySet : forall S : Type . Set S %(op)%

freeMap : forall S : Type . Map S -> MapMultiSet S %(op)%

freq : forall Elem : Type . Elem * MultiSet Elem -> Nat %(op)%

id

: System ->? HomSys %(op)%

: Net ->? HomNet %(op)%

image

: forall S : Type; V : Type . (S ->? V) -> Set S -> Set V %(op)%

injective

: HomSys ->? Unit %(pred)%

: HomNet ->? Unit %(pred)%

: forall S : Type; V : Type . Pred (S ->? V) %(op)%

injl : forall S : Type . S -> S * Boolean %(op)%

injr : forall S : Type . S -> S * Boolean %(op)%

ker : forall S : Type; V : Type . (S ->? V) -> Pred (S * S) %(op)%

linMap

:

forall S : Type; V : Type .

(S ->? MultiSet V) -> MultiSet S ->? MultiSet V

%(op)%

marking : System -> Marking %(op)%

min : Nat * Nat -> Nat %(op)%

net : System -> Net %(op)%

places : Net -> Set P %(op)%

placesMap

: HomSys -> P ->? P %(op)%

: HomNet -> P ->? P %(op)%

postMap : Net -> T ->? MultiSet P %(op)%

preMap : Net -> T ->? MultiSet P %(op)%

range : forall S : Type; V : Type . (S ->? V) -> Set V %(op)%

setToMultiSet

: forall Elem : Type . Set Elem -> MultiSet Elem %(op)%

sum : forall a : Type . (a ->? Nat) -> Pred a ->? Nat %(op)%

sumN : (Nat ->? Nat) -> Nat -> Nat %(op)%

sumSet : Set Nat ->? Nat %(op)%

transitions : Net -> Set T %(op)%

transitionsMap

: HomSys -> T ->? T %(op)%

: HomNet -> T ->? T %(op)%

undef__ : forall S : Type; V : Type . S ->? V %(op)%

{__}

: forall Elem : Type . Elem -> MultiSet Elem %(op)%

: forall S : Type . S -> Set S %(op)%

{} : forall Elem : Type . MultiSet Elem %(op)%

%% Variables -------------------------------------------------------------

B : MultiSet Elem

S : Set Elem

h1 : HomSys

h2 : HomSys

%% Sentences -------------------------------------------------------------

free type Boolean ::=

True : Boolean

False : Boolean %(ga_Boolean)%

let S = MultiSetToSet B in

forall x : Elem .

(__isIn__ : Elem * Set Elem ->? Unit) (x, S) <=>

((freq (x, B)) > 0)

def (op __o__ : HomSys * HomSys ->? HomSys) (h2, h1) =>

(op __o__ : HomSys * HomSys ->? HomSys) (h2, h1) =

(((op dom : HomSys -> System) h1,

(__o__ : (P ->? P) * (P ->? P) -> P ->? P)

((op placesMap : HomSys -> P ->? P) h2,

(op placesMap : HomSys -> P ->? P) h1),

(__o__ : (T ->? T) * (T ->? T) -> T ->? T)

((op transitionsMap : HomSys -> T ->? T) h2,

(op transitionsMap : HomSys -> T ->? T) h1),

(op cod : HomSys -> System) h2)

as HomSys)

%% Diagnostics -----------------------------------------------------------

### Hint 7.7, is type variable 'S'

### Hint 7.9, is type variable 'V'

### Hint 8.12, rebound type variable 'S'

### Hint 19.7, is type variable 'Elem'

### Hint 20.18, rebound type variable 'Elem'

### Hint 31.6, is type variable 'Elem'

### Hint 31.6, rebound type variable 'Elem'

### Hint 33.10, not a kind 'MultiSet Elem'

### Hint 33.27, not a kind 'Set Elem'

### Hint 35.17, not a class 'Elem'

### Hint 34.12, rebound variable 'S'

### Hint 34.12, rebound variable 'S'

### Hint, rejected 'Unit < Nat' of '((var x : Elem), (var S : Set Elem))'

### Hint 37.7, is type variable 'S'

### Hint 37.7, rebound type variable 'S'

### Hint 38.21, rebound type variable 'S'

### Hint 40.7, is type variable 'a'

### Hint 45.7, is type variable 'S'

### Hint 45.7, rebound type variable 'S'

### Hint 45.9, is type variable 'V'

### Hint 45.9, rebound type variable 'V'

### Hint 45.11, is type variable 'U'

### Hint 46.12, rebound type variable 'S'

### Hint 62.7, is type variable 'S'

### Hint 62.7, rebound type variable 'S'

### Hint 62.10, is type variable 'V'

### Hint 62.10, rebound type variable 'V'

### Hint 72.23, not a kind 'MultiSet P'

### Hint 73.26, not a kind 'MultiSet P'

### Hint 72.45, rejected 'Unit < Nat' of '((var p1 : MultiSet P),

(op range[_v161; _v173 _v174 _v172] :

forall S : Type; V : Type . (S ->? V) -> Set V)

(var pre : T ->? MultiSet P))'

### Hint 73.48, rejected 'Unit < Nat' of '((var p1 : MultiSet P),

(op range[_v276; _v288 _v289 _v287] :

forall S : Type; V : Type . (S ->? V) -> Set V)

(var pre : T ->? MultiSet P))'

### Hint 81.19, not a class 'T'

### Hint 80.14-80.35, in type of '((var hp : P ->? P), (op places : Net -> Set P) (var n1 : Net),

(op places : Net -> Set P) (var n2 : Net))'

typename 'P' (79.35)

is not unifiable with type '_v381 ->? Nat' (63.47)

### Hint 80.47-80.78, in type of '((var ht : T ->? T),

(op transitions : Net -> Set T) (var n1 : Net),

(op transitions : Net -> Set T) (var n2 : Net))'

typename 'T' (79.45)

is not unifiable with type '_v555 ->? Nat' (63.47)

### Hint 81.31, rejected 'Unit < Nat' of '((var t : T), (op transitions : Net -> Set T) (var n1 : Net))'

### Hint 95.38, not a class 'P'

### Hint, rejected 'Nat < Unit' of '((var x : P), (var m : Marking))'

### Hint 94.29-95.62, rejected 'Unit < Nat' of 'let ((var p : _v1024 _v1025 _v1023),

(var pre1 : _v1028 _v1029 (_v1032 _v1033 _v1031)),

(var post1 : _v1036 _v1037 (_v1040 _v1041 _v1039)))

= (var n : Net)

in

forall x : P .

((op __isIn__[P] :

forall Elem : Type . Pred (Elem * MultiSet Elem))

((var x : P), (var m : Marking)))

=>

((op __isIn__[_v1069] :

forall Elem : Type . Pred (Elem * MultiSet Elem))

((var x : P), (var p : _v1024 _v1025 _v1023)))'

### Hint 104.19, not a class 'P'

### Hint 104.24-104.74, in type of '((op freq[P] : forall Elem : Type . Elem * MultiSet Elem -> Nat)

((var p : P),

(op marking : System -> Marking) (var sys1 : System)),

(op freq[P] : forall Elem : Type . Elem * MultiSet Elem -> Nat)

((var hp : P ->? P) (var p : P),

(op marking : System -> Marking) (var sys2 : System)))'

typename 'Nat' (28.38)

is not unifiable with type '_v1288 ->? Nat' (23.33)

### Hint 112.12, not a class 'HomSys'

### Hint 112.16, not a class 'HomSys'

### Hint, in type of '((var h2 : HomSys), (var h1 : HomSys))'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '((var h2 : HomSys), (var h1 : HomSys))'

typename '__->?__'

is not unifiable with type '_t1516 System (P ->? P)' (102.54)

### Hint, in type of '((var h2 : HomSys), (var h1 : HomSys))'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '((var h2 : HomSys), (var h1 : HomSys))'

typename '__->?__'

is not unifiable with type '_t1516 System (P ->? P)' (102.54)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '__->?__'

is not unifiable with type '_t1516 System (P ->? P)' (102.54)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint 114.15-114.40, in type of '((op placesMap : HomSys -> P ->? P) (var h2 : HomSys),

(op placesMap : HomSys -> P ->? P) (var h1 : HomSys))'

typename '__->?__'

is not unifiable with type '_t1516 System (P ->? P)' (102.54)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint 114.15-114.40, in type of '((op placesMap : HomSys -> P ->? P) (var h2 : HomSys),

(op placesMap : HomSys -> P ->? P) (var h1 : HomSys))'

typename '__->?__'

is not unifiable with type '_t1018 Net (P ->? P)' (79.35)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint 114.44-114.79, in type of '((op transitionsMap : HomSys -> T ->? T) (var h2 : HomSys),

(op transitionsMap : HomSys -> T ->? T) (var h1 : HomSys))'

typename '__->?__'

is not unifiable with type '_t1516 System (P ->? P)' (102.54)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint 114.44-114.79, in type of '((op transitionsMap : HomSys -> T ->? T) (var h2 : HomSys),

(op transitionsMap : HomSys -> T ->? T) (var h1 : HomSys))'

typename '__->?__'

is not unifiable with type '_t1018 Net (P ->? P)' (79.35)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h1 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)

### Hint, in type of '(var h2 : HomSys)'

typename '_t1088' (93.8)

is not unifiable with type '_t366 (Set P)' (71.34)