Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8
Transitions:
t₀: l0(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₅: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂+1 ≤ X₁
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₂
t₆: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: D+1 ≤ 0
t₇: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 ≤ D
t₈: l2(X₀, X₁, X₂) → l4(X₀, X₁, X₂)
t₉: l3(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1)
t₁₀: l4(X₀, X₁, X₂) → l5(X₂+1, X₁, X₂)
t₂: l5(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: X₀+1 ≤ X₁
t₃: l5(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ ≤ X₀
t₁₁: l6(X₀, X₁, X₂) → l8(X₀, X₁, X₂)
t₁: l7(X₀, X₁, X₂) → l5(0, X₁, X₂)

Preprocessing

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l2

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l6

Found invariant 0 ≤ X₀ for location l5

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l8

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l4

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8
Transitions:
t₀: l0(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₅: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂+1 ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₆: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: D+1 ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₇: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 ≤ D ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₈: l2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₉: l3(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₁₀: l4(X₀, X₁, X₂) → l5(X₂+1, X₁, X₂) :|: X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂: l5(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: X₀+1 ≤ X₁ ∧ 0 ≤ X₀
t₃: l5(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₁₁: l6(X₀, X₁, X₂) → l8(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₁: l7(X₀, X₁, X₂) → l5(0, X₁, X₂)

MPRF for transition t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l2 [X₁+1-X₀ ]
l3 [X₁+1-X₀ ]
l4 [X₁-X₂ ]
l5 [X₁+1-X₀ ]
l1 [X₁+1-X₀ ]

MPRF for transition t₅: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂+1 ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l2 [X₁-X₂ ]
l3 [X₁-X₂ ]
l4 [X₁-X₂ ]
l5 [X₁+1-X₀ ]
l1 [X₁+1-X₂ ]

MPRF for transition t₆: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: D+1 ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₂ ]
l3 [X₁-X₂-1 ]
l4 [X₁-X₂ ]
l5 [X₁-X₀ ]
l1 [X₁-X₂ ]

MPRF for transition t₇: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 ≤ D ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₂ ]
l3 [X₁-X₂-1 ]
l4 [X₁-X₂ ]
l5 [X₁-X₀ ]
l1 [X₁-X₂ ]

MPRF for transition t₈: l2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₂ ]
l3 [X₁-X₂ ]
l4 [X₁-X₂-1 ]
l5 [X₁-X₀ ]
l1 [X₁-X₂ ]

MPRF for transition t₉: l3(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₂ ]
l3 [X₁-X₂ ]
l4 [X₁-X₂ ]
l5 [X₁-X₀ ]
l1 [X₁-X₂ ]

MPRF for transition t₁₀: l4(X₀, X₁, X₂) → l5(X₂+1, X₁, X₂) :|: X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₀ ]
l3 [X₁-X₀ ]
l4 [X₁-X₀ ]
l5 [X₁-X₀ ]
l1 [X₁-X₀ ]

MPRF for transition t₂: l5(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: X₀+1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₂-1 ]
l3 [X₁-X₂-1 ]
l4 [X₁-X₂-1 ]
l5 [X₁-X₀ ]
l1 [X₁-X₂-1 ]

All Bounds

Timebounds

Overall timebound:8⋅X₁+6 {O(n)}
t₀: 1 {O(1)}
t₄: X₁+1 {O(n)}
t₅: X₁+1 {O(n)}
t₆: X₁ {O(n)}
t₇: X₁ {O(n)}
t₈: X₁ {O(n)}
t₉: X₁ {O(n)}
t₁₀: X₁ {O(n)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₁+6 {O(n)}
t₀: 1 {O(1)}
t₄: X₁+1 {O(n)}
t₅: X₁+1 {O(n)}
t₆: X₁ {O(n)}
t₇: X₁ {O(n)}
t₈: X₁ {O(n)}
t₉: X₁ {O(n)}
t₁₀: X₁ {O(n)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₄, X₀: 2⋅X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 2⋅X₁ {O(n)}
t₅, X₀: 2⋅X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁ {O(n)}
t₆, X₀: 2⋅X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 2⋅X₁ {O(n)}
t₇, X₀: 2⋅X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 2⋅X₁ {O(n)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 2⋅X₁ {O(n)}
t₉, X₀: 2⋅X₁ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: 2⋅X₁ {O(n)}
t₁₀, X₀: 2⋅X₁ {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₀, X₂: 4⋅X₁ {O(n)}
t₂, X₀: 2⋅X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₁ {O(n)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 4⋅X₁+X₂ {O(n)}
t₁₁, X₀: 2⋅X₁ {O(n)}
t₁₁, X₁: 2⋅X₁ {O(n)}
t₁₁, X₂: 4⋅X₁+X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}