Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l10, l11, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)
t₁₂: l10(X₀, X₁, X₂, X₃) → l11(X₀, X₁, X₂, X₃)
t₁: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₂: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₄: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₅: l5(X₀, X₁, X₂, X₃) → l6(X₂, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: X₀ < X₃
t₇: l6(X₀, X₁, X₂, X₃) → l8(X₀, X₃, X₂, X₃) :|: X₃ ≤ X₀
t₈: l7(X₀, X₁, X₂, X₃) → l6(X₀+1, X₁, X₂, X₃)
t₁₀: l8(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₉: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃) :|: X₁ < X₀
t₁₁: l9(X₀, X₁, X₂, X₃) → l8(X₀, X₁+1, X₂, X₃)

Preprocessing

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l11

Found invariant X₂ ≤ X₀ for location l6

Found invariant 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location l7

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location l8

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l10

Found invariant X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location l9

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l10, l11, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)
t₁₂: l10(X₀, X₁, X₂, X₃) → l11(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₂: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₄: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₅: l5(X₀, X₁, X₂, X₃) → l6(X₂, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: X₀ < X₃ ∧ X₂ ≤ X₀
t₇: l6(X₀, X₁, X₂, X₃) → l8(X₀, X₃, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀
t₈: l7(X₀, X₁, X₂, X₃) → l6(X₀+1, X₁, X₂, X₃) :|: 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀
t₁₀: l8(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀
t₉: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃) :|: X₁ < X₀ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀
t₁₁: l9(X₀, X₁, X₂, X₃) → l8(X₀, X₁+1, X₂, X₃) :|: X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l11

Found invariant X₂ ≤ X₀ for location l6

Found invariant 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location l7

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location l8

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l10

Found invariant X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location l9

Time-Bound by TWN-Loops:

TWN-Loops: t₆ 2⋅X₂+2⋅X₃+4 {O(n)}

TWN-Loops:

entry: t₅: l5(X₀, X₁, X₂, X₃) → l6(X₂, X₁, X₂, X₃)
results in twn-loop: twn:Inv: [X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+1,X₁,X₂,X₃) :|: X₀ < X₃
order: [X₀; X₂; X₃]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₂: X₂
X₃: X₃

Termination: true
Formula:

1 < 0
∨ X₀ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₀ < X₃
alphas_abs: X₀+X₃
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₃+2 {O(n)}

relevant size-bounds w.r.t. t₅:
X₀: X₂ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₅: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4 {O(n)}

2⋅X₂+2⋅X₃+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₈ 2⋅X₂+2⋅X₃+4 {O(n)}

relevant size-bounds w.r.t. t₅:
X₀: X₂ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₅: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4 {O(n)}

2⋅X₂+2⋅X₃+4 {O(n)}

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l11

Found invariant X₂ ≤ X₀ for location l6

Found invariant 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location l7

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location l8

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l10

Found invariant X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location l9

Time-Bound by TWN-Loops:

TWN-Loops: t₉ 8⋅X₂+8⋅X₃+12 {O(n)}

TWN-Loops:

entry: t₇: l6(X₀, X₁, X₂, X₃) → l8(X₀, X₃, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀
results in twn-loop: twn:Inv: [X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀,X₁+1,X₂,X₃) :|: X₁ < X₀
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂
X₃: X₃

Termination: true
Formula:

1 < 0
∨ X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₇:
X₀: 2⋅X₃+4⋅X₂+4 {O(n)}
X₁: 2⋅X₃ {O(n)}
Runtime-bound of t₇: 1 {O(1)}
Results in: 8⋅X₂+8⋅X₃+12 {O(n)}

8⋅X₂+8⋅X₃+12 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₁ 8⋅X₂+8⋅X₃+12 {O(n)}

relevant size-bounds w.r.t. t₇:
X₀: 2⋅X₃+4⋅X₂+4 {O(n)}
X₁: 2⋅X₃ {O(n)}
Runtime-bound of t₇: 1 {O(1)}
Results in: 8⋅X₂+8⋅X₃+12 {O(n)}

8⋅X₂+8⋅X₃+12 {O(n)}

All Bounds

Timebounds

Overall timebound:20⋅X₂+20⋅X₃+41 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁₂: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 2⋅X₂+2⋅X₃+4 {O(n)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+2⋅X₃+4 {O(n)}
t₉: 8⋅X₂+8⋅X₃+12 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 8⋅X₂+8⋅X₃+12 {O(n)}

Costbounds

Overall costbound: 20⋅X₂+20⋅X₃+41 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁₂: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 2⋅X₂+2⋅X₃+4 {O(n)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+2⋅X₃+4 {O(n)}
t₉: 8⋅X₂+8⋅X₃+12 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 8⋅X₂+8⋅X₃+12 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₁₂, X₀: 4⋅X₃+8⋅X₂+8 {O(n)}
t₁₂, X₁: 12⋅X₃+8⋅X₂+12 {O(n)}
t₁₂, X₂: 4⋅X₂ {O(n)}
t₁₂, X₃: 4⋅X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 2⋅X₃+3⋅X₂+4 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: 2⋅X₃+4⋅X₂+4 {O(n)}
t₇, X₁: 2⋅X₃ {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₈, X₀: 2⋅X₃+3⋅X₂+4 {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: 2⋅X₃+4⋅X₂+4 {O(n)}
t₉, X₁: 10⋅X₃+8⋅X₂+12 {O(n)}
t₉, X₂: 2⋅X₂ {O(n)}
t₉, X₃: 2⋅X₃ {O(n)}
t₁₀, X₀: 4⋅X₃+8⋅X₂+8 {O(n)}
t₁₀, X₁: 12⋅X₃+8⋅X₂+12 {O(n)}
t₁₀, X₂: 4⋅X₂ {O(n)}
t₁₀, X₃: 4⋅X₃ {O(n)}
t₁₁, X₀: 2⋅X₃+4⋅X₂+4 {O(n)}
t₁₁, X₁: 10⋅X₃+8⋅X₂+12 {O(n)}
t₁₁, X₂: 2⋅X₂ {O(n)}
t₁₁, X₃: 2⋅X₃ {O(n)}