Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, 0, X₂, X₃) :|: X₀ < 5
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 5 ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2
t₆: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: 9 < X₁
t₄: l3(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: 2 < X₀ ∧ X₁ ≤ 9
t₉: l4(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₈: l5(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃)
t₇: l6(X₀, X₁, X₂, X₃) → l3(X₀, X₁+1, X₂, X₃)
Preprocessing
Eliminate variables {X₃} that do not contribute to the problem
Found invariant X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 9 ∧ X₁ ≤ 6+X₀ ∧ X₀+X₁ ≤ 13 ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location l6
Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location l7
Found invariant X₂ ≤ 4 ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 4 for location l5
Found invariant X₂ ≤ X₀ for location l1
Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location l4
Found invariant X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₉: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂₀: l1(X₀, X₁, X₂) → l3(X₀, 0, X₂) :|: X₀ < 5 ∧ X₂ ≤ X₀
t₂₁: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 5 ≤ X₀ ∧ X₂ ≤ X₀
t₂₂: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₂₃: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4
t₂₄: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 9 < X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4
t₂₅: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 2 < X₀ ∧ X₁ ≤ 9 ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4
t₂₆: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 5 ≤ X₀
t₂₇: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₂ ≤ 4 ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 4
t₂₈: l6(X₀, X₁, X₂) → l3(X₀, X₁+1, X₂) :|: X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 9 ∧ X₁ ≤ 6+X₀ ∧ X₀+X₁ ≤ 13 ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 ∧ 3 ≤ X₀
MPRF for transition t₂₀: l1(X₀, X₁, X₂) → l3(X₀, 0, X₂) :|: X₀ < 5 ∧ X₂ ≤ X₀ of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₂₃: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₂₄: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 9 < X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
MPRF for transition t₂₇: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₂ ≤ 4 ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 4 of depth 1:
new bound:
X₂+5 {O(n)}
TWN: t₂₅: l3→l6
cycle: [t₂₅: l3→l6; t₂₈: l6→l3]
loop: (2 < X₀ ∧ X₁ ≤ 9,(X₀,X₁) -> (X₀,X₁+1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
Termination: true
Formula:
1 < 0 ∧ 2 < X₀
∨ X₁ < 9 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < X₀
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 9 ∧ 9 ≤ X₁ ∧ 2 < X₀
Stabilization-Threshold for: X₁ ≤ 9
alphas_abs: X₁+9
M: 0
N: 1
Bound: 2⋅X₁+20 {O(n)}
TWN - Lifting for t₂₅: l3→l6 of 2⋅X₁+23 {O(n)}
relevant size-bounds w.r.t. t₂₀:
X₁: 0 {O(1)}
Runtime-bound of t₂₀: X₂+5 {O(n)}
Results in: 23⋅X₂+115 {O(n)}
TWN: t₂₈: l6→l3
TWN - Lifting for t₂₈: l6→l3 of 2⋅X₁+23 {O(n)}
relevant size-bounds w.r.t. t₂₀:
X₁: 0 {O(1)}
Runtime-bound of t₂₀: X₂+5 {O(n)}
Results in: 23⋅X₂+115 {O(n)}
All Bounds
Timebounds
Overall timebound:50⋅X₂+254 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+5 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: X₂+5 {O(n)}
t₂₄: X₂+5 {O(n)}
t₂₅: 23⋅X₂+115 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₂+5 {O(n)}
t₂₈: 23⋅X₂+115 {O(n)}
Costbounds
Overall costbound: 50⋅X₂+254 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+5 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: X₂+5 {O(n)}
t₂₄: X₂+5 {O(n)}
t₂₅: 23⋅X₂+115 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₂+5 {O(n)}
t₂₈: 23⋅X₂+115 {O(n)}
Sizebounds
t₁₉, X₀: X₀ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: 2⋅X₂+9 {O(n)}
t₂₀, X₁: 0 {O(1)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: 3⋅X₂+9 {O(n)}
t₂₁, X₁: X₁+10 {O(n)}
t₂₁, X₂: 2⋅X₂ {O(n)}
t₂₂, X₀: X₂ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: 2⋅X₂+9 {O(n)}
t₂₃, X₁: 0 {O(1)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 4 {O(1)}
t₂₄, X₁: 10 {O(1)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: 4 {O(1)}
t₂₅, X₁: 9 {O(1)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: 3⋅X₂+9 {O(n)}
t₂₆, X₁: X₁+10 {O(n)}
t₂₆, X₂: 2⋅X₂ {O(n)}
t₂₇, X₀: 2⋅X₂+9 {O(n)}
t₂₇, X₁: 10 {O(1)}
t₂₇, X₂: X₂ {O(n)}
t₂₈, X₀: 4 {O(1)}
t₂₈, X₁: 10 {O(1)}
t₂₈, X₂: X₂ {O(n)}