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:
t₁₄: l0(X₀, X₁) → l2(X₀, X₁)
t₁₅: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 0 ∧ X₀ ≤ X₁
t₁₆: l1(X₀, X₁) → l3(X₀, X₁) :|: 0 < X₀ ∧ X₀ ≤ X₁
t₁₇: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₁₈: l2(X₀, X₁) → l1(X₁, X₁)
t₁₉: l3(X₀, X₁) → l1(X₀-1, X₁) :|: X₀ ≤ X₁
t₂₀: l4(X₀, X₁) → l5(X₀, X₁) :|: 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

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

Analysing control-flow refined program

Cut unsatisfiable transition t₈₇: n_l1___3→n_l3___2

Cut unsatisfiable transition t₁₀₂: 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

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

Time-Bound by TWN-Loops:

TWN-Loops: t₈₆ 10⋅X₁+10 {O(n)}

TWN-Loops:

entry: t₉₄: n_l3___6(X₀, X₁) → n_l1___3(X₀-1, X₁) :|: 0 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀] , (X₀,X₁) -> (X₀-1,X₁) :|: X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₁
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁

Termination: true
Formula:

1 < 0 ∧ 0 < 1
∨ 0 < 1 ∧ 1 < 0 ∧ 1+X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ 0 < 1 ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ 0 < 1 ∧ 0 < X₀ ∧ 1+X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ 0 < 1 ∧ 0 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0 ∧ 0 < 1
∨ X₀ < X₁ ∧ 1 < 0 ∧ 1+X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀ < X₁ ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ X₀ < X₁ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₀ < X₁ ∧ 0 < X₀ ∧ 1+X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀ < X₁ ∧ 0 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < 0 ∧ 0 < 1
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < 0 ∧ 1+X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ 1+X₀ < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀

Stabilization-Threshold for: X₀ ≤ X₁
alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
Stabilization-Threshold for: 1+X₀ ≤ X₁
alphas_abs: 1+X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₉₄:
X₀: X₁ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₉₄: 1 {O(1)}
Results in: 10⋅X₁+10 {O(n)}

10⋅X₁+10 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₉₁ 10⋅X₁+10 {O(n)}

relevant size-bounds w.r.t. t₉₄:
X₀: X₁ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₉₄: 1 {O(1)}
Results in: 10⋅X₁+10 {O(n)}

10⋅X₁+10 {O(n)}

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

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₁₄: 1 {O(1)}
t₁₅: inf {Infinity}
t₁₆: inf {Infinity}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: inf {Infinity}
t₂₀: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₁₄: 1 {O(1)}
t₁₅: inf {Infinity}
t₁₆: inf {Infinity}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: inf {Infinity}
t₂₀: 1 {O(1)}

Sizebounds

t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₅, X₁: 2⋅X₁ {O(n)}
t₁₆, X₁: 2⋅X₁ {O(n)}
t₁₇, X₀: 0 {O(1)}
t₁₇, X₁: 3⋅X₁ {O(n)}
t₁₈, X₀: X₁ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₉, X₁: 2⋅X₁ {O(n)}
t₂₀, X₀: 0 {O(1)}
t₂₀, X₁: 3⋅X₁ {O(n)}