Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₄, X₂, X₃, X₄)
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁-1, X₂, X₃, X₄) :|: 0 < X₁ ∧ 0 < X₁
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁-1, X₂, X₃, X₄) :|: 0 < X₁ ∧ X₁ ≤ 0
t₆: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₂, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ 0 < X₁
t₇: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₂, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ 0
t₈: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Cut unsatisfiable transition t₅: l3→l1

Cut unsatisfiable transition t₆: l3→l1

Eliminate variables {X₃} that do not contribute to the problem

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l5

Found invariant X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₈: l0(X₀, X₁, X₂, X₄) → l2(X₀, X₁, X₂, X₄)
t₁₉: l1(X₀, X₁, X₂, X₄) → l3(X₀, X₁, X₂, X₄) :|: 0 < X₀ ∧ X₀ ≤ X₂
t₂₀: l1(X₀, X₁, X₂, X₄) → l4(X₀, X₁, X₂, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₁: l2(X₀, X₁, X₂, X₄) → l1(X₂, X₄, X₂, X₄)
t₂₂: l3(X₀, X₁, X₂, X₄) → l1(X₀, X₁-1, X₂, X₄) :|: 0 < X₁ ∧ 0 < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀
t₂₃: l3(X₀, X₁, X₂, X₄) → l1(X₀-1, X₂, X₂, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀
t₂₄: l4(X₀, X₁, X₂, X₄) → l5(X₀, X₁, X₂, X₄) :|: X₀ ≤ X₂ ∧ X₀ ≤ 0

MPRF for transition t₂₃: l3(X₀, X₁, X₂, X₄) → l1(X₀-1, X₂, X₂, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l3 [X₀ ]
l1 [X₀ ]

MPRF for transition t₂₂: l3(X₀, X₁, X₂, X₄) → l1(X₀, X₁-1, X₂, X₄) :|: 0 < X₁ ∧ 0 < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+X₂+X₄+1 {O(n^2)}

MPRF:

l3 [X₁+1 ]
l1 [X₁+1 ]

knowledge_propagation leads to new time bound X₂⋅X₂+2⋅X₂+X₄+2 {O(n^2)} for transition t₁₉: l1(X₀, X₁, X₂, X₄) → l3(X₀, X₁, X₂, X₄) :|: 0 < X₀ ∧ X₀ ≤ X₂

Analysing control-flow refined program

Cut unsatisfiable transition t₇₁: n_l1___2→l4

Cut unsatisfiable transition t₇₃: n_l1___6→l4

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___6

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___4

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___3

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l5

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___5

Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___1

Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location n_l3___7

MPRF for transition t₅₅: n_l1___2(X₀, X₁, X₂, X₄) → n_l3___1(X₀, X₁, X₂, X₄) :|: X₀ ≤ X₂ ∧ 0 < X₀ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

n_l3___1 [X₀-1 ]
n_l3___3 [X₀ ]
n_l1___2 [X₀ ]
n_l1___5 [X₀ ]
n_l3___4 [X₀-1 ]
n_l1___6 [X₀-1 ]

MPRF for transition t₅₆: n_l1___5(X₀, X₁, X₂, X₄) → n_l3___3(X₀, X₁, X₂, X₄) :|: X₀ ≤ X₂ ∧ 0 < X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

n_l3___1 [X₀ ]
n_l3___3 [X₀ ]
n_l1___2 [X₀ ]
n_l1___5 [X₀+1 ]
n_l3___4 [X₀ ]
n_l1___6 [X₀ ]

MPRF for transition t₅₉: n_l3___1(X₀, X₁, X₂, X₄) → n_l1___6(X₀, X₁-1, X₂, X₄) :|: X₀ ≤ X₂ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

n_l3___1 [X₀ ]
n_l3___3 [X₀ ]
n_l1___2 [X₀ ]
n_l1___5 [X₀ ]
n_l3___4 [X₀-1 ]
n_l1___6 [X₀-1 ]

MPRF for transition t₆₀: n_l3___3(X₀, X₁, X₂, X₄) → n_l1___2(X₀, X₁-1, X₂, X₄) :|: 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

n_l3___1 [X₀-1 ]
n_l3___3 [X₀ ]
n_l1___2 [X₀-1 ]
n_l1___5 [X₀ ]
n_l3___4 [X₀-1 ]
n_l1___6 [X₀-1 ]

MPRF for transition t₆₁: n_l3___4(X₀, X₁, X₂, X₄) → n_l1___5(X₀-1, X₂, X₂, X₄) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₂+2 {O(n)}

MPRF:

n_l3___1 [X₀+X₂-X₁ ]
n_l3___3 [X₀+1 ]
n_l1___2 [X₀+X₂-X₁ ]
n_l1___5 [X₀+1 ]
n_l3___4 [X₀+1 ]
n_l1___6 [X₀+1 ]

MPRF for transition t₅₇: n_l1___6(X₀, X₁, X₂, X₄) → n_l3___4(X₀, X₁, X₂, X₄) :|: X₀ ≤ X₂ ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

4⋅X₂⋅X₂+5⋅X₂+X₄+1 {O(n^2)}

MPRF:

n_l1___5 [X₁ ]
n_l3___1 [X₁ ]
n_l3___3 [X₁ ]
n_l1___2 [X₁ ]
n_l3___4 [X₁ ]
n_l1___6 [X₁+1 ]

MPRF for transition t₆₂: n_l3___4(X₀, X₁, X₂, X₄) → n_l1___6(X₀, X₁-1, X₂, X₄) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

4⋅X₂⋅X₂+5⋅X₂+X₄+1 {O(n^2)}

MPRF:

n_l1___5 [X₂ ]
n_l3___1 [X₁ ]
n_l3___3 [X₂ ]
n_l1___2 [X₁ ]
n_l3___4 [X₁+1 ]
n_l1___6 [X₁+1 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+2⋅X₄+4⋅X₂+7 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₂⋅X₂+2⋅X₂+X₄+2 {O(n^2)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₂⋅X₂+X₂+X₄+1 {O(n^2)}
t₂₃: X₂ {O(n)}
t₂₄: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+2⋅X₄+4⋅X₂+7 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₂⋅X₂+2⋅X₂+X₄+2 {O(n^2)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₂⋅X₂+X₂+X₄+1 {O(n^2)}
t₂₃: X₂ {O(n)}
t₂₄: 1 {O(1)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: X₂ {O(n)}
t₁₉, X₁: X₂+X₄ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₂₀, X₀: 2⋅X₂ {O(n)}
t₂₀, X₁: X₂+X₄ {O(n)}
t₂₀, X₂: 2⋅X₂ {O(n)}
t₂₀, X₄: 2⋅X₄ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₄ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: X₂ {O(n)}
t₂₂, X₁: X₂+X₄ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₃, X₀: X₂ {O(n)}
t₂₃, X₁: X₂ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₄, X₀: 2⋅X₂ {O(n)}
t₂₄, X₁: X₂+X₄ {O(n)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}