Initial Problem

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

Preprocessing

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

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

Problem after Preprocessing

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

MPRF for transition t₂: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 0 of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [-X₀-1 ]
l1 [-X₀ ]

MPRF for transition t₆: l3(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l3 [1-X₀ ]
l1 [1-X₀ ]

Found invariant 1 ≤ 0 for location l5

Found invariant 1 ≤ 0 for location l1

Found invariant 1 ≤ 0 for location l4

Found invariant 1 ≤ 0 for location l3

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

Time-Bound by TWN-Loops:

TWN-Loops: t₃ 2⋅X₁⋅X₁+10⋅X₁+10 {O(n^2)}

TWN-Loops:

entry: t₆: l3(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 0
results in twn-loop: twn: (X₀,X₁) -> (X₀-1,X₁) :|: 0 < X₀ ∧ 0 < X₀
entry: t₁: l2(X₀, X₁) → l1(X₁, X₁)
results in twn-loop: twn: (X₀,X₁) -> (X₀-1,X₁) :|: 0 < X₀ ∧ 0 < X₀
order: [X₀]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1

Termination: true
Formula:

1 < 0
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₆:
X₀: X₁+1 {O(n)}
Runtime-bound of t₆: X₁+1 {O(n)}
Results in: 2⋅X₁⋅X₁+8⋅X₁+6 {O(n^2)}

order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁

Termination: true
Formula:

1 < 0
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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: 2⋅X₁+4 {O(n)}

2⋅X₁⋅X₁+10⋅X₁+10 {O(n^2)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 2⋅X₁⋅X₁+10⋅X₁+10 {O(n^2)}

relevant size-bounds w.r.t. t₆:
X₀: X₁+1 {O(n)}
Runtime-bound of t₆: X₁+1 {O(n)}
Results in: 2⋅X₁⋅X₁+8⋅X₁+6 {O(n^2)}

relevant size-bounds w.r.t. t₁:
X₀: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁⋅X₁+10⋅X₁+10 {O(n^2)}

Analysing control-flow refined program

Cut unsatisfiable transition t₉₁: n_l1___2→n_l3___4

Cut unsatisfiable transition t₉₂: n_l1___5→n_l3___3

Cut unreachable locations [n_l3___3] from the program graph

Found invariant 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 1+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___2

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

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

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

Found invariant 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 t₉₀: n_l1___2(X₀, X₁) → n_l3___1(X₀, X₁) :|: 0 < X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l3___1 [X₀ ]
n_l1___2 [X₀+1 ]

MPRF for transition t₉₆: n_l3___1(X₀, X₁) → n_l1___2(X₀-1, X₁) :|: 0 < X₀ ∧ 0 < X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

n_l3___1 [X₀ ]
n_l1___2 [X₀ ]

MPRF for transition t₉₃: n_l1___5(X₀, X₁) → n_l3___4(X₀, X₁) :|: X₀ ≤ 1 ∧ X₀ < 0 ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l3___4 [-X₀ ]
n_l1___5 [1-X₀ ]

MPRF for transition t₉₈: n_l3___4(X₀, X₁) → n_l1___5(X₀+1, X₁) :|: X₀ < 0 ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l3___4 [1-X₀ ]
n_l1___5 [1-X₀ ]

CFR: Improvement to new bound with the following program:

new bound:

4⋅X₁+3 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l4, l5, n_l1___2, n_l1___5, n_l3___1, n_l3___4, n_l3___6, n_l3___7
Transitions:
t₀: l0(X₀, X₁) → l2(X₀, X₁)
t₄: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₉₄: l1(X₀, X₁) → n_l3___6(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₉₅: l1(X₀, X₁) → n_l3___7(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁: l2(X₀, X₁) → l1(X₁, X₁)
t₇: l4(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₁₁₁: n_l1___2(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₉₀: n_l1___2(X₀, X₁) → n_l3___1(X₀, X₁) :|: 0 < X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₁₁₂: n_l1___5(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0
t₉₃: n_l1___5(X₀, X₁) → n_l3___4(X₀, X₁) :|: X₀ ≤ 1 ∧ X₀ < 0 ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0
t₉₆: n_l3___1(X₀, X₁) → n_l1___2(X₀-1, X₁) :|: 0 < X₀ ∧ 0 < X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀
t₉₈: n_l3___4(X₀, X₁) → n_l1___5(X₀+1, X₁) :|: X₀ < 0 ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0
t₉₉: n_l3___6(X₀, X₁) → n_l1___2(X₀-1, X₁) :|: 0 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₁₀₀: n_l3___7(X₀, X₁) → n_l1___5(X₀+1, X₁) :|: X₀ < 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ 0

All Bounds

Timebounds

Overall timebound:4⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₄: 1 {O(1)}
t₉₄: 1 {O(1)}
t₉₅: 1 {O(1)}
t₁: 1 {O(1)}
t₇: 1 {O(1)}
t₉₀: X₁+1 {O(n)}
t₁₁₁: 1 {O(1)}
t₉₃: X₁+1 {O(n)}
t₁₁₂: 1 {O(1)}
t₉₆: X₁ {O(n)}
t₉₈: X₁+1 {O(n)}
t₉₉: 1 {O(1)}
t₁₀₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₄: 1 {O(1)}
t₉₄: 1 {O(1)}
t₉₅: 1 {O(1)}
t₁: 1 {O(1)}
t₇: 1 {O(1)}
t₉₀: X₁+1 {O(n)}
t₁₁₁: 1 {O(1)}
t₉₃: X₁+1 {O(n)}
t₁₁₂: 1 {O(1)}
t₉₆: X₁ {O(n)}
t₉₈: X₁+1 {O(n)}
t₉₉: 1 {O(1)}
t₁₀₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₄, X₀: 0 {O(1)}
t₄, X₁: X₁ {O(n)}
t₉₄, X₀: X₁ {O(n)}
t₉₄, X₁: X₁ {O(n)}
t₉₅, X₀: X₁ {O(n)}
t₉₅, X₁: X₁ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₇, X₀: 0 {O(1)}
t₇, X₁: 5⋅X₁ {O(n)}
t₉₀, X₀: X₁ {O(n)}
t₉₀, X₁: X₁ {O(n)}
t₁₁₁, X₀: 0 {O(1)}
t₁₁₁, X₁: 2⋅X₁ {O(n)}
t₉₃, X₀: X₁ {O(n)}
t₉₃, X₁: X₁ {O(n)}
t₁₁₂, X₀: 0 {O(1)}
t₁₁₂, X₁: 2⋅X₁ {O(n)}
t₉₆, X₀: X₁ {O(n)}
t₉₆, X₁: X₁ {O(n)}
t₉₈, X₀: X₁ {O(n)}
t₉₈, X₁: X₁ {O(n)}
t₉₉, X₀: X₁ {O(n)}
t₉₉, X₁: X₁ {O(n)}
t₁₀₀, X₀: X₁ {O(n)}
t₁₀₀, X₁: X₁ {O(n)}