Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8
Transitions:
t₀: l0(X₀, X₁) → l7(X₀, X₁)
t₈: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ X₁
t₇: l1(X₀, X₁) → l5(X₀, X₁) :|: X₁+1 ≤ X₀
t₁₀: l2(X₀, X₁) → l3(X₀-1, X₁)
t₃: l3(X₀, X₁) → l4(X₀, X₁) :|: 0 ≤ X₀
t₄: l3(X₀, X₁) → l6(X₀, X₁) :|: X₀+1 ≤ 0
t₅: l4(X₀, X₁) → l1(X₀, 1) :|: 2 ≤ X₀
t₆: l4(X₀, X₁) → l2(X₀, C) :|: X₀ ≤ 1
t₉: l5(X₀, X₁) → l1(X₀, 2⋅X₁)
t₁₁: l6(X₀, X₁) → l8(X₀, X₁)
t₁: l7(X₀, X₁) → l3(X₀, X₁) :|: 0 ≤ X₀
t₂: l7(X₀, X₁) → l6(X₀, X₁) :|: X₀+1 ≤ 0

Preprocessing

Found invariant 0 ≤ X₀ for location l2

Found invariant 1+X₀ ≤ 0 for location l6

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l5

Found invariant 1+X₀ ≤ 0 for location l8

Found invariant 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

Found invariant 0 ≤ 1+X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8
Transitions:
t₀: l0(X₀, X₁) → l7(X₀, X₁)
t₈: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₇: l1(X₀, X₁) → l5(X₀, X₁) :|: X₁+1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₁₀: l2(X₀, X₁) → l3(X₀-1, X₁) :|: 0 ≤ X₀
t₃: l3(X₀, X₁) → l4(X₀, X₁) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀
t₄: l3(X₀, X₁) → l6(X₀, X₁) :|: X₀+1 ≤ 0 ∧ 0 ≤ 1+X₀
t₅: l4(X₀, X₁) → l1(X₀, 1) :|: 2 ≤ X₀ ∧ 0 ≤ X₀
t₆: l4(X₀, X₁) → l2(X₀, C) :|: X₀ ≤ 1 ∧ 0 ≤ X₀
t₉: l5(X₀, X₁) → l1(X₀, 2⋅X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₁₁: l6(X₀, X₁) → l8(X₀, X₁) :|: 1+X₀ ≤ 0
t₁: l7(X₀, X₁) → l3(X₀, X₁) :|: 0 ≤ X₀
t₂: l7(X₀, X₁) → l6(X₀, X₁) :|: X₀+1 ≤ 0

MPRF for transition t₃: l3(X₀, X₁) → l4(X₀, X₁) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₅: l4(X₀, X₁) → l1(X₀, 1) :|: 2 ≤ X₀ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₆: l4(X₀, X₁) → l2(X₀, C) :|: X₀ ≤ 1 ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₈: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₁₀: l2(X₀, X₁) → l3(X₀-1, X₁) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₇: l1(X₀, X₁) → l5(X₀, X₁) :|: X₁+1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+4⋅X₀+2 {O(n^2)}

MPRF for transition t₉: l5(X₀, X₁) → l1(X₀, 2⋅X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+4⋅X₀+2 {O(n^2)}

Chain transitions t₉: l5→l1 and t₇: l1→l5 to t₇₂: l5→l5

Chain transitions t₅: l4→l1 and t₇: l1→l5 to t₇₃: l4→l5

Chain transitions t₅: l4→l1 and t₈: l1→l2 to t₇₄: l4→l2

Chain transitions t₉: l5→l1 and t₈: l1→l2 to t₇₅: l5→l2

Chain transitions t₇₅: l5→l2 and t₁₀: l2→l3 to t₇₆: l5→l3

Chain transitions t₇₄: l4→l2 and t₁₀: l2→l3 to t₇₇: l4→l3

Chain transitions t₆: l4→l2 and t₁₀: l2→l3 to t₇₈: l4→l3

Chain transitions t₁: l7→l3 and t₄: l3→l6 to t₇₉: l7→l6

Chain transitions t₇₆: l5→l3 and t₄: l3→l6 to t₈₀: l5→l6

Chain transitions t₇₆: l5→l3 and t₃: l3→l4 to t₈₁: l5→l4

Chain transitions t₁: l7→l3 and t₃: l3→l4 to t₈₂: l7→l4

Chain transitions t₇₈: l4→l3 and t₃: l3→l4 to t₈₃: l4→l4

Chain transitions t₇₈: l4→l3 and t₄: l3→l6 to t₈₄: l4→l6

Chain transitions t₇₇: l4→l3 and t₃: l3→l4 to t₈₅: l4→l4

Chain transitions t₇₇: l4→l3 and t₄: l3→l6 to t₈₆: l4→l6

Analysing control-flow refined program

Cut unsatisfiable transition t₇₄: l4→l2

Cut unsatisfiable transition t₇₇: l4→l3

Cut unsatisfiable transition t₇₉: l7→l6

Cut unsatisfiable transition t₈₀: l5→l6

Cut unsatisfiable transition t₈₅: l4→l4

Cut unsatisfiable transition t₈₆: l4→l6

Found invariant 0 ≤ X₀ for location l2

Found invariant 1+X₀ ≤ 0 for location l6

Found invariant 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l5

Found invariant 1+X₀ ≤ 0 for location l8

Found invariant 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

Found invariant 0 ≤ 1+X₀ for location l3

MPRF for transition t₇₃: l4(X₀, X₁) -{2}> l5(X₀, 1) :|: 2 ≤ X₀ ∧ 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ 0 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₈₁: l5(X₀, X₁) -{4}> l4(X₀-1, 2⋅X₁) :|: X₀ ≤ 2⋅X₁ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ ∧ 1 ≤ 2⋅X₁ ∧ 3 ≤ X₀+2⋅X₁ ∧ 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₈₃: l4(X₀, X₁) -{3}> l4(X₀-1, C) :|: X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₇₂: l5(X₀, X₁) -{2}> l5(X₀, 2⋅X₁) :|: 2⋅X₁+1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ ∧ 1 ≤ 2⋅X₁ ∧ 3 ≤ X₀+2⋅X₁ ∧ 2 ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+2⋅X₀ {O(n^2)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Cut unsatisfiable transition t₈: l1→l2

Found invariant 0 ≤ X₀ for location l2

Found invariant 1+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location n_l5___1

Found invariant 1+X₀ ≤ 0 for location l6

Found invariant 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l1___2

Found invariant 1+X₀ ≤ 0 for location l8

Found invariant X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

Found invariant 0 ≤ 1+X₀ for location l3

Found invariant X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l5___3

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₁₈₉: l1(X₀, X₁) → n_l5___3(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₁₉₁: n_l5___3(X₀, X₁) → n_l1___2(X₀, 2⋅X₁) :|: X₁ ≤ 1 ∧ 2 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀

MPRF for transition t₁₈₈: n_l1___2(X₀, X₁) → n_l5___1(X₀, X₁) :|: 2 ≤ X₁ ∧ 2+X₁ ≤ 2⋅X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+4⋅X₀+2 {O(n^2)}

MPRF for transition t₁₉₀: n_l5___1(X₀, X₁) → n_l1___2(X₀, 2⋅X₁) :|: 2 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 3 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+4⋅X₀+2 {O(n^2)}

MPRF for transition t₁₉₅: n_l1___2(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+13⋅X₀+14 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₇: X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₈: X₀+1 {O(n)}
t₉: X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₁₀: X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+13⋅X₀+14 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₇: X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₈: X₀+1 {O(n)}
t₉: X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₁₀: X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

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