Probabilistic Bisimulation - Semantics of Probabilistic Processes: An Operational Approach - page 36

s , t ∈ S . For any s ∈ S , we have that

t ∈ S z s , t

= t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r )

x s , r ·

= r ∈ ʔ 2 ( t ∈ S y r , t )

x s , r
ʔ 2 ( r ) ·

= r ∈ ʔ 2

x s , r
ʔ 2 ( r ) ·

ʔ 2 ( r )

= r ∈ ʔ 2 x s , r
= r ∈ S x s , r
=

ʔ 1 ( s )

∈

and for any t S , we have that

s ∈ S z s , t

= s ∈ S r ∈ ʔ 2 x s , r ·

y r , t
ʔ 2 ( r )

= r ∈ ʔ 2 s ∈ S x s , r ·

y r , t
ʔ 2 ( r )

= r ∈ ʔ 2 ( s ∈ S x s , r )

y r , t
ʔ 2 ( r )

·

= r ∈ ʔ 2

y r , t
ʔ 2 ( r )

ʔ 2 ( r ) ·

= r ∈ ʔ 2 y r , t
= r ∈ S y r , t
= ʔ 3 ( t ) .

Therefore, the real numbers z s , t satisfy the constraints in ( 3.8 ) and we obtain that

m ( ʔ 1 , ʔ 3 )

s , t ∈ S m ( s , t ) · z s , t

≤

s , t ∈ S m ( s , t ) r ∈ ʔ 2

y r , t
ʔ 2 ( r )

= · x s , r ·

s , t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r )

= m ( s , t ) · x s , r ·

s , t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r )

≤ ( m ( s , r ) + m ( r , t )) · x s , r ·

s , t ∈ S r ∈ ʔ 2 s , t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r ) +

y r , t
ʔ 2 ( r )

= m ( s , r ) · x s , r · m ( r , t ) · x s , r ·

t ∈ S y r , t

s ∈ S r ∈ ʔ 2 s , t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r )

= m ( s , r ) · x s , r ·

ʔ 2 ( r ) +

m ( r , t ) · x s , r ·

s ∈ S r ∈ ʔ 2 s , t ∈ S r ∈ ʔ 2

ʔ 2 ( r )
ʔ 2 ( r ) +

y r , t
ʔ 2 ( r )

= m ( s , r ) · x s , r · m ( r , t ) · x s , r ·

s ∈ S r ∈ S m ( s , r ) · x s , r + s , t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r )

= m ( r , t ) · x s , r ·

s , t ∈ S r ∈ ʔ 2

y r , t
ʔ 2 ( r )

m ( ʔ 1 , ʔ 2 )

= + m ( r , t ) · x s , r ·