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 < 0
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₀
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₁, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂)
t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

Eliminate variables {X₂} that do not contribute to the problem

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

Found invariant X₀ ≤ X₁ for location l1

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 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:
t₁₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁₄: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 0 ∧ X₀ ≤ X₁
t₁₅: l1(X₀, X₁) → l3(X₀, X₁) :|: 0 < X₀ ∧ X₀ ≤ X₁
t₁₆: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₁₇: l2(X₀, X₁) → l1(X₁, X₁)
t₁₈: l3(X₀, X₁) → l1(X₀-1, X₁) :|: X₀ ≤ X₁
t₁₉: l4(X₀, X₁) → l5(X₀, X₁) :|: 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

MPRF for transition t₁₅: l1(X₀, X₁) → l3(X₀, X₁) :|: 0 < X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

Chain transitions t₁₈: l3→l1 and t₁₆: l1→l4 to t₄₉: l3→l4

Chain transitions t₁₇: l2→l1 and t₁₆: l1→l4 to t₅₀: l2→l4

Chain transitions t₁₇: l2→l1 and t₁₅: l1→l3 to t₅₁: l2→l3

Chain transitions t₁₈: l3→l1 and t₁₅: l1→l3 to t₅₂: l3→l3

Chain transitions t₁₇: l2→l1 and t₁₄: l1→l3 to t₅₃: l2→l3

Chain transitions t₁₈: l3→l1 and t₁₄: l1→l3 to t₅₄: l3→l3

Analysing control-flow refined program

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

Found invariant X₀ ≤ X₁ for location l1

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

Found invariant X₀ ≤ X₁ for location l3

MPRF for transition t₅₂: l3(X₀, X₁) -{2}> l3(X₀-1, X₁) :|: 1 < X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1+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 t₁₄₁: n_l1___3→n_l3___2

Cut unsatisfiable transition t₁₅₆: n_l1___5→l4

Cut unreachable locations [n_l3___2] from the program graph

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

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___6

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___3

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

Found invariant 1+X₁ ≤ 0 ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ 2+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 ≤ 0 ∧ 0 ≤ X₀ for location l4

Found invariant 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ 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 t₁₄₀: n_l1___3(X₀, X₁) → n_l3___1(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF for transition t₁₄₅: n_l3___1(X₀, X₁) → n_l1___3(X₀-1, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁ {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}
t₁₃: 1 {O(1)}
t₁₄: inf {Infinity}
t₁₅: X₁ {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: inf {Infinity}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₁₃: 1 {O(1)}
t₁₄: inf {Infinity}
t₁₅: X₁ {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: inf {Infinity}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: X₀ {O(n)}
t₁₃, X₁: X₁ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₆, X₀: 0 {O(1)}
t₁₆, X₁: 2⋅X₁ {O(n)}
t₁₇, X₀: X₁ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₉, X₀: 0 {O(1)}
t₁₉, X₁: 2⋅X₁ {O(n)}