Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < 1 ∧ X₀ < 0
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < 1 ∧ 0 < X₀
t₄: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 < X₀ ∧ X₀ < 0
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 < X₀ ∧ 0 < X₀
t₆: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ 1 ∧ 1 ≤ X₀
t₇: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₈: l3(X₀, X₁, X₂) → l1(X₀-2, X₁, X₂)
t₉: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

Cut unsatisfiable transition t₃: l1→l3

Cut unsatisfiable transition t₄: l1→l3

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

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l5

Found invariant X₀ ≤ X₁ for location l1

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

Found invariant X₀ ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₂₁: l0(X₀, X₁) → l2(X₀, X₁)
t₂₂: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 1 ∧ X₀ < 0 ∧ X₀ ≤ X₁
t₂₃: l1(X₀, X₁) → l3(X₀, X₁) :|: 1 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₁
t₂₄: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁
t₂₅: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₂₆: l2(X₀, X₁) → l1(X₁, X₁)
t₂₇: l3(X₀, X₁) → l1(X₀-2, X₁) :|: X₀ ≤ X₁
t₂₈: l4(X₀, X₁) → l5(X₀, X₁) :|: 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀

MPRF for transition t₂₃: l1(X₀, X₁) → l3(X₀, X₁) :|: 1 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

Chain transitions t₂₇: l3→l1 and t₂₅: l1→l4 to t₆₂: l3→l4

Chain transitions t₂₆: l2→l1 and t₂₅: l1→l4 to t₆₃: l2→l4

Chain transitions t₂₆: l2→l1 and t₂₄: l1→l4 to t₆₄: l2→l4

Chain transitions t₂₇: l3→l1 and t₂₄: l1→l4 to t₆₅: l3→l4

Chain transitions t₂₆: l2→l1 and t₂₃: l1→l3 to t₆₆: l2→l3

Chain transitions t₂₇: l3→l1 and t₂₃: l1→l3 to t₆₇: l3→l3

Chain transitions t₂₆: l2→l1 and t₂₂: l1→l3 to t₆₈: l2→l3

Chain transitions t₂₇: l3→l1 and t₂₂: l1→l3 to t₆₉: l3→l3

Analysing control-flow refined program

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l5

Found invariant X₀ ≤ X₁ for location l1

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

Found invariant X₀ ≤ X₁ for location l3

MPRF for transition t₆₇: l3(X₀, X₁) -{2}> l3(X₀-2, X₁) :|: 3 < X₀ ∧ 2 < X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₁ {O(n)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Cut unsatisfiable transition t₁₆₉: n_l1___3→n_l3___2

Cut unsatisfiable transition t₁₈₅: n_l1___5→l4

Cut unsatisfiable transition t₁₈₇: n_l1___5→l4

Cut unreachable locations [n_l3___2] from the program graph

Found invariant 1+X₁ ≤ 0 ∧ 4+X₀+X₁ ≤ 0 ∧ 2+X₀ ≤ X₁ ∧ 3+X₀ ≤ 0 for location n_l3___4

Found invariant X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location n_l3___6

Found invariant 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___3

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l5

Found invariant 1+X₁ ≤ 0 ∧ 4+X₀+X₁ ≤ 0 ∧ 2+X₀ ≤ X₁ ∧ 3+X₀ ≤ 0 for location n_l1___5

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

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

Found invariant 4 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₀ for location n_l3___1

Found invariant 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location n_l3___7

MPRF for transition t₁₆₈: n_l1___3(X₀, X₁) → n_l3___1(X₀, X₁) :|: 2+X₀ ≤ X₁ ∧ 1 < X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF for transition t₁₇₃: n_l3___1(X₀, X₁) → n_l1___3(X₀-2, X₁) :|: 2+X₀ ≤ X₁ ∧ 1 < X₀ ∧ X₀ ≤ X₁ ∧ 4 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₂₁: 1 {O(1)}
t₂₂: inf {Infinity}
t₂₃: X₁ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}
t₂₆: 1 {O(1)}
t₂₇: inf {Infinity}
t₂₈: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₂₁: 1 {O(1)}
t₂₂: inf {Infinity}
t₂₃: X₁ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}
t₂₆: 1 {O(1)}
t₂₇: inf {Infinity}
t₂₈: 1 {O(1)}

Sizebounds

t₂₁, X₀: X₀ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₄, X₀: 1 {O(1)}
t₂₄, X₁: 2⋅X₁ {O(n)}
t₂₅, X₀: 0 {O(1)}
t₂₅, X₁: 2⋅X₁ {O(n)}
t₂₆, X₀: X₁ {O(n)}
t₂₆, X₁: X₁ {O(n)}
t₂₇, X₁: X₁ {O(n)}
t₂₈, X₀: 1 {O(1)}
t₂₈, X₁: 4⋅X₁ {O(n)}