Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l10, l11, l12, l13, 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)
t₈: l10(X₀, X₁, X₂, X₃) → l12(X₀, X₁, X₂, X₃) :|: 2 < X₀ ∧ X₃ ≤ 9
t₉: l10(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2
t₁₀: l10(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: 9 < X₃
t₁₅: l11(X₀, X₁, X₂, X₃) → l13(X₀, X₁, X₂, X₃)
t₁₁: l12(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, X₃+1)
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) → l10(X₀, X₁, X₂, 0) :|: X₀ < 5
t₇: l6(X₀, X₁, X₂, X₃) → l11(X₀, X₁, X₂, X₃) :|: 5 ≤ X₀
t₁₄: l7(X₀, X₁, X₂, X₃) → l6(X₁, X₁, X₂, X₃)
t₁₂: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₀+1, X₂, X₃)
t₁₃: l9(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
Preprocessing
Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location l11
Found invariant X₂ ≤ X₀ for location l6
Found invariant X₃ ≤ 9 ∧ X₂+X₃ ≤ 13 ∧ X₃ ≤ 6+X₀ ∧ X₀+X₃ ≤ 13 ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ 3 ≤ X₀+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location l12
Found invariant 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4 for location l7
Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location l13
Found invariant 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 for location l8
Found invariant 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 for location l10
Found invariant 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4 for location l9
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l10, l11, l12, l13, 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)
t₈: l10(X₀, X₁, X₂, X₃) → l12(X₀, X₁, X₂, X₃) :|: 2 < X₀ ∧ X₃ ≤ 9 ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4
t₉: l10(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2 ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4
t₁₀: l10(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: 9 < X₃ ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4
t₁₅: l11(X₀, X₁, X₂, X₃) → l13(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₀ ∧ 5 ≤ X₀
t₁₁: l12(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, X₃+1) :|: X₃ ≤ 9 ∧ X₂+X₃ ≤ 13 ∧ X₃ ≤ 6+X₀ ∧ X₀+X₃ ≤ 13 ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ 3 ≤ X₀+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀
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) → l10(X₀, X₁, X₂, 0) :|: X₀ < 5 ∧ X₂ ≤ X₀
t₇: l6(X₀, X₁, X₂, X₃) → l11(X₀, X₁, X₂, X₃) :|: 5 ≤ X₀ ∧ X₂ ≤ X₀
t₁₄: l7(X₀, X₁, X₂, X₃) → l6(X₁, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4
t₁₂: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₀+1, X₂, X₃) :|: 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4
t₁₃: l9(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4
MPRF for transition t₆: l6(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, 0) :|: X₀ < 5 ∧ X₂ ≤ X₀ of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₉: l10(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2 ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+3 {O(n)}
MPRF for transition t₁₀: l10(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: 9 < X₃ ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₁₂: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₀+1, X₂, X₃) :|: 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₁₃: l9(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₁₄: l7(X₀, X₁, X₂, X₃) → l6(X₁, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
TWN: t₈: l10→l12
cycle: [t₈: l10→l12; t₁₁: l12→l10]
loop: (2 < X₀ ∧ X₃ ≤ 9,(X₀,X₃) -> (X₀,X₃+1)
order: [X₀; X₃]
closed-form:
X₀: X₀
X₃: X₃ + [[n != 0]] * n^1
Termination: true
Formula:
1 < 0 ∧ 2 < X₀
∨ X₃ < 9 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < X₀
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 9 ∧ 9 ≤ X₃ ∧ 2 < X₀
Stabilization-Threshold for: X₃ ≤ 9
alphas_abs: X₃+9
M: 0
N: 1
Bound: 2⋅X₃+20 {O(n)}
TWN - Lifting for t₈: l10→l12 of 2⋅X₃+23 {O(n)}
relevant size-bounds w.r.t. t₆:
X₃: 0 {O(1)}
Runtime-bound of t₆: X₂+5 {O(n)}
Results in: 23⋅X₂+115 {O(n)}
TWN: t₁₁: l12→l10
TWN - Lifting for t₁₁: l12→l10 of 2⋅X₃+23 {O(n)}
relevant size-bounds w.r.t. t₆:
X₃: 0 {O(1)}
Runtime-bound of t₆: X₂+5 {O(n)}
Results in: 23⋅X₂+115 {O(n)}
All Bounds
Timebounds
Overall timebound:52⋅X₂+266 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: X₂+5 {O(n)}
t₇: 1 {O(1)}
t₈: 23⋅X₂+115 {O(n)}
t₉: X₂+3 {O(n)}
t₁₀: X₂+5 {O(n)}
t₁₁: 23⋅X₂+115 {O(n)}
t₁₂: X₂+5 {O(n)}
t₁₃: X₂+5 {O(n)}
t₁₄: X₂+5 {O(n)}
t₁₅: 1 {O(1)}
Costbounds
Overall costbound: 52⋅X₂+266 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: X₂+5 {O(n)}
t₇: 1 {O(1)}
t₈: 23⋅X₂+115 {O(n)}
t₉: X₂+3 {O(n)}
t₁₀: X₂+5 {O(n)}
t₁₁: 23⋅X₂+115 {O(n)}
t₁₂: X₂+5 {O(n)}
t₁₃: X₂+5 {O(n)}
t₁₄: X₂+5 {O(n)}
t₁₅: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: 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)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, 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⋅X₂+9 {O(n)}
t₆, X₁: 2⋅X₂+X₁+15 {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: 0 {O(1)}
t₇, X₀: 3⋅X₂+9 {O(n)}
t₇, X₁: 2⋅X₂+X₁+15 {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: X₃+10 {O(n)}
t₈, X₀: 4 {O(1)}
t₈, X₁: 2⋅X₂+X₁+15 {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: 9 {O(1)}
t₉, X₀: 2⋅X₂+9 {O(n)}
t₉, X₁: 2⋅X₂+X₁+15 {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: 0 {O(1)}
t₁₀, X₀: 4 {O(1)}
t₁₀, X₁: 2⋅X₂+X₁+15 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: 10 {O(1)}
t₁₁, X₀: 4 {O(1)}
t₁₁, X₁: 2⋅X₂+X₁+15 {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: 10 {O(1)}
t₁₂, X₀: 2⋅X₂+9 {O(n)}
t₁₂, X₁: 2⋅X₂+15 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: 10 {O(1)}
t₁₃, X₀: 2⋅X₂+9 {O(n)}
t₁₃, X₁: 2⋅X₂+15 {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: 10 {O(1)}
t₁₄, X₀: 2⋅X₂+9 {O(n)}
t₁₄, X₁: 2⋅X₂+15 {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: 10 {O(1)}
t₁₅, X₀: 3⋅X₂+9 {O(n)}
t₁₅, X₁: 2⋅X₂+X₁+15 {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: X₃+10 {O(n)}