Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁) → l2(X₀, X₁)
t₂: l1(X₀, X₁) → l3(X₀, X₁) :|: 3 ≤ X₀
t₃: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ < 3
t₁: l2(X₀, X₁) → l1(3, X₁)
t₄: l3(X₀, X₁) → l5(X₀, X₁) :|: 5 < X₀
t₅: l3(X₀, X₁) → l6(X₀, X₁) :|: X₀ ≤ 5
t₉: l4(X₀, X₁) → l7(X₀, X₁)
t₆: l5(X₀, X₁) → l1(X₀+3, X₁)
t₇: l6(X₀, X₁) → l1(X₀-2, X₁) :|: 10 < X₀
t₈: l6(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 10

Preprocessing

Cut unsatisfiable transition t₇: l6→l1

Eliminate variables {X₁} that do not contribute to the problem

Found invariant X₀ ≤ 5 ∧ 3 ≤ X₀ for location l6

Found invariant 1 ≤ 0 for location l7

Found invariant 6 ≤ X₀ for location l5

Found invariant 3 ≤ X₀ for location l1

Found invariant 1 ≤ 0 for location l4

Found invariant 3 ≤ X₀ for location l3

Cut unsatisfiable transition t₂₂: l1→l4

Cut unsatisfiable transition t₂₆: l4→l7

Cut unreachable locations [l4; l7] from the program graph

Problem after Preprocessing

Start: l0
Program_Vars: X₀
Temp_Vars:
Locations: l0, l1, l2, l3, l5, l6
Transitions:
t₂₀: l0(X₀) → l2(X₀)
t₂₁: l1(X₀) → l3(X₀) :|: 3 ≤ X₀ ∧ 3 ≤ X₀
t₂₃: l2(X₀) → l1(3)
t₂₄: l3(X₀) → l5(X₀) :|: 5 < X₀ ∧ 3 ≤ X₀
t₂₅: l3(X₀) → l6(X₀) :|: X₀ ≤ 5 ∧ 3 ≤ X₀
t₂₇: l5(X₀) → l1(X₀+3) :|: 6 ≤ X₀
t₂₈: l6(X₀) → l1(X₀+1) :|: X₀ ≤ 10 ∧ X₀ ≤ 5 ∧ 3 ≤ X₀

MPRF for transition t₂₅: l3(X₀) → l6(X₀) :|: X₀ ≤ 5 ∧ 3 ≤ X₀ of depth 1:

new bound:

9 {O(1)}

MPRF:

l3 [6-X₀ ]
l5 [3-X₀ ]
l6 [5-X₀ ]
l1 [6-X₀ ]

MPRF for transition t₂₈: l6(X₀) → l1(X₀+1) :|: X₀ ≤ 10 ∧ X₀ ≤ 5 ∧ 3 ≤ X₀ of depth 1:

new bound:

9 {O(1)}

MPRF:

l3 [6-X₀ ]
l5 [3-X₀ ]
l6 [6-X₀ ]
l1 [6-X₀ ]

Analysing control-flow refined program

Found invariant X₀ ≤ 6 ∧ 5 ≤ X₀ for location n_l1___6

Found invariant 9 ≤ X₀ for location n_l5___1

Found invariant X₀ ≤ 4 ∧ 4 ≤ X₀ for location n_l1___9

Found invariant X₀ ≤ 3 ∧ 3 ≤ X₀ for location n_l6___10

Found invariant X₀ ≤ 6 ∧ 5 ≤ X₀ for location n_l3___5

Found invariant 9 ≤ X₀ for location n_l1___3

Found invariant X₀ ≤ 6 ∧ 6 ≤ X₀ for location n_l5___4

Found invariant X₀ ≤ 3 ∧ 3 ≤ X₀ for location n_l3___11

Found invariant X₀ ≤ 4 ∧ 4 ≤ X₀ for location n_l3___8

Found invariant 9 ≤ X₀ for location n_l3___2

Found invariant X₀ ≤ 5 ∧ 4 ≤ X₀ for location n_l6___7

Found invariant X₀ ≤ 3 ∧ 3 ≤ X₀ for location l1

MPRF for transition t₇₄: n_l1___6(X₀) → n_l3___5(X₀) :|: 3 ≤ X₀ ∧ 3 ≤ X₀ ∧ 4 ≤ X₀ ∧ X₀ ≤ 6 ∧ 3 ≤ X₀ ∧ X₀ ≤ 6 ∧ 5 ≤ X₀ of depth 1:

new bound:

10 {O(1)}

MPRF:

n_l3___5 [6-X₀ ]
n_l6___7 [6-X₀ ]
n_l1___6 [7-X₀ ]

MPRF for transition t₇₉: n_l3___5(X₀) → n_l6___7(X₀) :|: X₀ ≤ 6 ∧ 4 ≤ X₀ ∧ 3 ≤ X₀ ∧ X₀ ≤ 5 ∧ X₀ ≤ 6 ∧ 5 ≤ X₀ of depth 1:

new bound:

10 {O(1)}

MPRF:

n_l3___5 [7-X₀ ]
n_l6___7 [6-X₀ ]
n_l1___6 [7-X₀ ]

MPRF for transition t₈₄: n_l6___7(X₀) → n_l1___6(X₀+1) :|: X₀ ≤ 5 ∧ 4 ≤ X₀ ∧ 3 ≤ X₀ ∧ X₀ ≤ 5 ∧ X₀ ≤ 5 ∧ 4 ≤ X₀ of depth 1:

new bound:

10 {O(1)}

MPRF:

n_l3___5 [6-X₀ ]
n_l6___7 [6-X₀ ]
n_l1___6 [6-X₀ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₂₀: 1 {O(1)}
t₂₁: inf {Infinity}
t₂₃: 1 {O(1)}
t₂₄: inf {Infinity}
t₂₅: 9 {O(1)}
t₂₇: inf {Infinity}
t₂₈: 9 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₂₀: 1 {O(1)}
t₂₁: inf {Infinity}
t₂₃: 1 {O(1)}
t₂₄: inf {Infinity}
t₂₅: 9 {O(1)}
t₂₇: inf {Infinity}
t₂₈: 9 {O(1)}

Sizebounds

t₂₀, X₀: X₀ {O(n)}
t₂₃, X₀: 3 {O(1)}
t₂₅, X₀: 5 {O(1)}
t₂₈, X₀: 6 {O(1)}