Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁) → l2(X₀, X₁)
t₂: l1(X₀, X₁) → l1(X₀, X₁) :|: 0 < X₀ ∧ X₀ < 1
t₃: l1(X₀, X₁) → l1(X₀-1, X₁) :|: 0 < X₀ ∧ 1 ≤ X₀
t₄: l1(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 0 ∧ X₀ < 0
t₅: l1(X₀, X₁) → l1(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l2(X₀, X₁) → l1(X₁, X₁)

Preprocessing

Cut unsatisfiable transition t₂: l1→l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁) → l2(X₀, X₁)
t₃: l1(X₀, X₁) → l1(X₀-1, X₁) :|: 0 < X₀ ∧ 1 ≤ X₀
t₄: l1(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 0 ∧ X₀ < 0
t₅: l1(X₀, X₁) → l1(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l2(X₀, X₁) → l1(X₁, X₁)

Time-Bound by TWN-Loops:

TWN-Loops: t₃ 4⋅X₁+6 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁) → l1(X₁, X₁)
results in twn-loop: twn: (X₀,X₁) -> (X₀-1,X₁) :|: 0 < X₀ ∧ 1 ≤ X₀
order: [X₀]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 1 < X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₁:
X₀: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₁+6 {O(n)}

4⋅X₁+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 4⋅X₁+6 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁) → l1(X₁, X₁)
results in twn-loop: twn: (X₀,X₁) -> (X₀+1,X₁) :|: X₀ ≤ 0 ∧ X₀ < 0
order: [X₀]
closed-form:
X₀: X₀ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ X₀ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₀ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

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

relevant size-bounds w.r.t. t₁:
X₀: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₁+6 {O(n)}

4⋅X₁+6 {O(n)}

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

Analysing control-flow refined program

Cut unsatisfiable transition t₉₇: n_l1___3→n_l1___4

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___4

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___2

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

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

Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___1

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___4

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___2

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

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

Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___1

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___4

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___2

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

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

Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___1

Time-Bound by TWN-Loops:

TWN-Loops: t₉₆ 2⋅X₁+4 {O(n)}

TWN-Loops:

entry: t₁₀₁: l1(X₀, X₁) → n_l1___3(X₀+1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
results in twn-loop: twn:Inv: [1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0] , (X₀,X₁) -> (X₀+1,X₁) :|: X₀ < 0
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁

Termination: true
Formula:

1 < 0
∨ X₀ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₁₀₁:
X₀: X₁ {O(n)}
Runtime-bound of t₁₀₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___4

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___2

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

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

Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___1

Time-Bound by TWN-Loops:

TWN-Loops: t₉₉ 4⋅X₁+6 {O(n)}

TWN-Loops:

entry: t₁₀₂: l1(X₀, X₁) → n_l1___4(X₀-1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
results in twn-loop: twn:Inv: [1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀-1,X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₀
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 1 < X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0

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

relevant size-bounds w.r.t. t₁₀₂:
X₀: X₁ {O(n)}
Runtime-bound of t₁₀₂: 1 {O(1)}
Results in: 4⋅X₁+6 {O(n)}

4⋅X₁+6 {O(n)}

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___4

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___2

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

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

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___1

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₃: 4⋅X₁+6 {O(n)}
t₄: 4⋅X₁+6 {O(n)}
t₅: inf {Infinity}
t₁: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₃: 4⋅X₁+6 {O(n)}
t₄: 4⋅X₁+6 {O(n)}
t₅: inf {Infinity}
t₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: X₁ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: 0 {O(1)}
t₅, X₁: 3⋅X₁ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}