Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₁ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(0, 0, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁+1, X₂, X₃) :|: X₀ ≤ 50
t₅: l3(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁-1, X₂, X₃) :|: 50 < X₀
t₆: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
Preprocessing
Eliminate variables {X₂,X₃} that do not contribute to the problem
Found invariant 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1
Found invariant 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l4
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁₄: l1(X₀, X₁) → l3(X₀, X₁) :|: 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₅: l1(X₀, X₁) → l4(X₀, X₁) :|: X₁ < 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₆: l2(X₀, X₁) → l1(0, 0)
t₁₇: l3(X₀, X₁) → l1(X₀+1, X₁+1) :|: X₀ ≤ 50 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₈: l3(X₀, X₁) → l1(X₀+1, X₁-1) :|: 50 < X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₉: l4(X₀, X₁) → l5(X₀, X₁) :|: 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₁₇: l3(X₀, X₁) → l1(X₀+1, X₁+1) :|: X₀ ≤ 50 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
51 {O(1)}
TWN: t₁₄: l1→l3
cycle: [t₁₄: l1→l3; t₁₈: l3→l1]
loop: (0 ≤ X₁ ∧ 50 < X₀,(X₀,X₁) -> (X₀+1,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
0 < 1 ∧ 1 < 0
∨ 0 < 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 50 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ 50 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 50 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
Stabilization-Threshold for: 50 < X₀
alphas_abs: 50+X₀
M: 0
N: 1
Bound: 2⋅X₀+102 {O(n)}
Stabilization-Threshold for: 0 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
loop: (0 ≤ X₁ ∧ 50 < X₀,(X₀,X₁) -> (X₀+1,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
0 < 1 ∧ 1 < 0
∨ 0 < 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 50 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ 50 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 50 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
Stabilization-Threshold for: 50 < X₀
alphas_abs: 50+X₀
M: 0
N: 1
Bound: 2⋅X₀+102 {O(n)}
Stabilization-Threshold for: 0 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
TWN - Lifting for t₁₄: l1→l3 of 2⋅X₀+2⋅X₁+106 {O(n)}
relevant size-bounds w.r.t. t₁₇:
X₀: 51 {O(1)}
X₁: 51 {O(1)}
Runtime-bound of t₁₇: 51 {O(1)}
Results in: 15810 {O(1)}
TWN - Lifting for t₁₄: l1→l3 of 2⋅X₀+2⋅X₁+106 {O(n)}
relevant size-bounds w.r.t. t₁₆:
X₀: 0 {O(1)}
X₁: 0 {O(1)}
Runtime-bound of t₁₆: 1 {O(1)}
Results in: 106 {O(1)}
TWN: t₁₈: l3→l1
TWN - Lifting for t₁₈: l3→l1 of 2⋅X₀+2⋅X₁+106 {O(n)}
relevant size-bounds w.r.t. t₁₇:
X₀: 51 {O(1)}
X₁: 51 {O(1)}
Runtime-bound of t₁₇: 51 {O(1)}
Results in: 15810 {O(1)}
TWN - Lifting for t₁₈: l3→l1 of 2⋅X₀+2⋅X₁+106 {O(n)}
relevant size-bounds w.r.t. t₁₆:
X₀: 0 {O(1)}
X₁: 0 {O(1)}
Runtime-bound of t₁₆: 1 {O(1)}
Results in: 106 {O(1)}
All Bounds
Timebounds
Overall timebound:31887 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 15916 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 51 {O(1)}
t₁₈: 15916 {O(1)}
t₁₉: 1 {O(1)}
Costbounds
Overall costbound: 31887 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 15916 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 51 {O(1)}
t₁₈: 15916 {O(1)}
t₁₉: 1 {O(1)}
Sizebounds
t₁₃, X₀: X₀ {O(n)}
t₁₃, X₁: X₁ {O(n)}
t₁₄, X₀: 15967 {O(1)}
t₁₄, X₁: 52 {O(1)}
t₁₅, X₀: 15967 {O(1)}
t₁₅, X₁: 1 {O(1)}
t₁₆, X₀: 0 {O(1)}
t₁₆, X₁: 0 {O(1)}
t₁₇, X₀: 51 {O(1)}
t₁₇, X₁: 51 {O(1)}
t₁₈, X₀: 15967 {O(1)}
t₁₈, X₁: 52 {O(1)}
t₁₉, X₀: 15967 {O(1)}
t₁₉, X₁: 1 {O(1)}