Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₆: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃
t₅: l1(X₀, X₁, X₂, X₃) → l4(X₂, X₃, X₂, X₃) :|: X₃ ≤ 0
t₇: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: E+1 ≤ 0
t₈: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ E
t₉: l2(X₀, X₁, X₂, X₃) → l4(X₂, X₃, X₂, X₃)
t₁₀: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 1, X₃-1)
t₂: l4(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 0, X₁) :|: X₀ ≤ 1 ∧ 1 ≤ X₀
t₃: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₄: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: 2 ≤ X₀
t₁₁: l5(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₁: l6(X₀, X₁, X₂, X₃) → l4(1, X₀, X₂, X₃)
Preprocessing
Found invariant X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location l2
Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location l7
Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5
Found invariant X₃ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location l1
Found invariant X₀ ≤ 1 ∧ 0 ≤ X₀ for location l4
Found invariant X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location l3
Cut unsatisfiable transition t₄: l4→l5
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₆: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₅: l1(X₀, X₁, X₂, X₃) → l4(X₂, X₃, X₂, X₃) :|: X₃ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₇: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: E+1 ≤ 0 ∧ X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₈: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ E ∧ X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₉: l2(X₀, X₁, X₂, X₃) → l4(X₂, X₃, X₂, X₃) :|: X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₁₀: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 1, X₃-1) :|: X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₂: l4(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 0, X₁) :|: X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀
t₃: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 1 ∧ 0 ≤ X₀
t₁₁: l5(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l6(X₀, X₁, X₂, X₃) → l4(1, X₀, X₂, X₃)
MPRF for transition t₆: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀+2 {O(n)}
MPRF for transition t₇: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: E+1 ≤ 0 ∧ X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₈: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ E ∧ X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₉: l2(X₀, X₁, X₂, X₃) → l4(X₂, X₃, X₂, X₃) :|: X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₁₀: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 1, X₃-1) :|: X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 2 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
TWN: t₂: l4→l1
cycle: [t₅: l1→l4; t₂: l4→l1]
loop: (X₃ ≤ 0 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂,(X₂,X₃) -> (0,X₃)
order: [X₂; X₃]
closed-form:
X₂: [[n == 0]] * X₂
X₃: X₃
Termination: true
Formula:
1 < 0 ∧ 0 < 1 ∧ X₃ < 0
∨ 1 < 0 ∧ 0 < 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₃ < 0
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1 ∧ X₃ < 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₃ < 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
loop: (X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₁ ≤ 0,(X₀,X₁) -> (0,X₁)
order: [X₀; X₁]
closed-form:
X₀: [[n == 0]] * X₀
X₁: X₁
Termination: true
Formula:
X₁ < 0 ∧ 1 < 0 ∧ 0 < 1
∨ X₁ < 0 ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₁ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₁ < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 0 < 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
loop: (X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₁ ≤ 0,(X₀,X₁) -> (0,X₁)
order: [X₀; X₁]
closed-form:
X₀: [[n == 0]] * X₀
X₁: X₁
Termination: true
Formula:
X₁ < 0 ∧ 1 < 0 ∧ 0 < 1
∨ X₁ < 0 ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₁ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₁ < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 0 < 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0
TWN - Lifting for t₂: l4→l1 of 5 {O(1)}
relevant size-bounds w.r.t. t₁₀:
Runtime-bound of t₁₀: X₀ {O(n)}
Results in: 5⋅X₀ {O(n)}
TWN - Lifting for t₂: l4→l1 of 5 {O(1)}
relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 5 {O(1)}
TWN - Lifting for t₂: l4→l1 of 5 {O(1)}
relevant size-bounds w.r.t. t₉:
Runtime-bound of t₉: X₀+1 {O(n)}
Results in: 5⋅X₀+5 {O(n)}
TWN: t₅: l1→l4
TWN - Lifting for t₅: l1→l4 of 5 {O(1)}
relevant size-bounds w.r.t. t₁₀:
Runtime-bound of t₁₀: X₀ {O(n)}
Results in: 5⋅X₀ {O(n)}
TWN - Lifting for t₅: l1→l4 of 5 {O(1)}
relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 5 {O(1)}
TWN - Lifting for t₅: l1→l4 of 5 {O(1)}
relevant size-bounds w.r.t. t₉:
Runtime-bound of t₉: X₀+1 {O(n)}
Results in: 5⋅X₀+5 {O(n)}
All Bounds
Timebounds
Overall timebound:26⋅X₀+29 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 10⋅X₀+10 {O(n)}
t₃: 1 {O(1)}
t₅: 10⋅X₀+10 {O(n)}
t₆: 2⋅X₀+2 {O(n)}
t₇: X₀+1 {O(n)}
t₈: X₀+1 {O(n)}
t₉: X₀+1 {O(n)}
t₁₀: X₀ {O(n)}
t₁₁: 1 {O(1)}
Costbounds
Overall costbound: 26⋅X₀+29 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 10⋅X₀+10 {O(n)}
t₃: 1 {O(1)}
t₅: 10⋅X₀+10 {O(n)}
t₆: 2⋅X₀+2 {O(n)}
t₇: X₀+1 {O(n)}
t₈: X₀+1 {O(n)}
t₉: X₀+1 {O(n)}
t₁₀: X₀ {O(n)}
t₁₁: 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: 1 {O(1)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 1 {O(1)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 0 {O(1)}
t₂, X₃: X₀ {O(n)}
t₃, X₀: 0 {O(1)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: 2 {O(1)}
t₃, X₃: 3⋅X₀ {O(n)}
t₅, X₀: 1 {O(1)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 1 {O(1)}
t₅, X₃: 2⋅X₀ {O(n)}
t₆, X₀: 1 {O(1)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 1 {O(1)}
t₆, X₃: X₀ {O(n)}
t₇, X₀: 1 {O(1)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 1 {O(1)}
t₇, X₃: X₀ {O(n)}
t₈, X₀: 1 {O(1)}
t₈, X₁: X₀ {O(n)}
t₈, X₂: 1 {O(1)}
t₈, X₃: X₀ {O(n)}
t₉, X₀: 1 {O(1)}
t₉, X₁: X₀ {O(n)}
t₉, X₂: 1 {O(1)}
t₉, X₃: X₀ {O(n)}
t₁₀, X₀: 1 {O(1)}
t₁₀, X₁: X₀ {O(n)}
t₁₀, X₂: 1 {O(1)}
t₁₀, X₃: X₀ {O(n)}
t₁₁, X₀: 0 {O(1)}
t₁₁, X₁: 2⋅X₀ {O(n)}
t₁₁, X₂: 2 {O(1)}
t₁₁, X₃: 3⋅X₀ {O(n)}