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) → l5(X₀, X₁, X₂, X₃)
t₈: l1(X₀, X₁, X₂, X₃) → l4(X₀-1, 1, X₂, X₃) :|: 1 ≤ X₀
t₉: l1(X₀, X₁, X₂, X₃) → l4(X₀-1, 0, X₂, X₃) :|: X₀ < 1
t₅: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₇: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₂: l4(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₁ < 0
t₃: l4(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 0 < X₁
t₄: l4(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 0 ≤ X₁
t₁: l5(X₀, X₁, X₂, X₃) → l4(X₃, X₂, X₂, X₃)
t₁₀: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₀ ≤ X₃ for location l2

Found invariant X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l6

Found invariant X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l7

Found invariant X₀ ≤ X₃ for location l1

Found invariant X₀ ≤ X₃ for location l4

Found invariant X₀ ≤ X₃ for location l3

Problem after Preprocessing

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) → l5(X₀, X₁, X₂, X₃)
t₈: l1(X₀, X₁, X₂, X₃) → l4(X₀-1, 1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
t₉: l1(X₀, X₁, X₂, X₃) → l4(X₀-1, 0, X₂, X₃) :|: X₀ < 1 ∧ X₀ ≤ X₃
t₅: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃
t₇: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃
t₂: l4(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₁ < 0 ∧ X₀ ≤ X₃
t₃: l4(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 0 < X₁ ∧ X₀ ≤ X₃
t₄: l4(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ X₀ ≤ X₃
t₁: l5(X₀, X₁, X₂, X₃) → l4(X₃, X₂, X₂, X₃)
t₁₀: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₂: l4(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₁ < 0 ∧ X₀ ≤ X₃

MPRF for transition t₈: l1(X₀, X₁, X₂, X₃) → l4(X₀-1, 1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF for transition t₉: l1(X₀, X₁, X₂, X₃) → l4(X₀-1, 0, X₂, X₃) :|: X₀ < 1 ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₂+1 {O(n)}

TWN: t₃: l4→l2

cycle: [t₂: l4→l2; t₃: l4→l2; t₅: l2→l3; t₇: l3→l1; t₉: l1→l4]
loop: (X₁ < 0 ∧ X₀ < 1 ∨ 0 < X₁ ∧ X₀ < 1,(X₀,X₁) -> (X₀-1,0)
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: [[n == 0]] * X₁

Termination: true
Formula:

Stabilization-Threshold for: X₀ < 1
alphas_abs: X₀+1
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}
loop: (X₁ < 0 ∧ X₀ < 1 ∨ 0 < X₁ ∧ X₀ < 1,(X₀,X₁) -> (X₀-1,0)
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: [[n == 0]] * X₁

Termination: true
Formula:

Stabilization-Threshold for: X₀ < 1
alphas_abs: X₀+1
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}

TWN - Lifting for t₃: l4→l2 of 2⋅X₀+6 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}

TWN - Lifting for t₃: l4→l2 of 2⋅X₀+6 {O(n)}

relevant size-bounds w.r.t. t₈:
X₀: 2⋅X₃ {O(n)}
Runtime-bound of t₈: X₃ {O(n)}
Results in: 4⋅X₃⋅X₃+6⋅X₃ {O(n^2)}

TWN: t₅: l2→l3

TWN - Lifting for t₅: l2→l3 of 2⋅X₀+6 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}

TWN - Lifting for t₅: l2→l3 of 2⋅X₀+6 {O(n)}

relevant size-bounds w.r.t. t₈:
X₀: 2⋅X₃ {O(n)}
Runtime-bound of t₈: X₃ {O(n)}
Results in: 4⋅X₃⋅X₃+6⋅X₃ {O(n^2)}

TWN: t₇: l3→l1

TWN - Lifting for t₇: l3→l1 of 2⋅X₀+6 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}

TWN - Lifting for t₇: l3→l1 of 2⋅X₀+6 {O(n)}

relevant size-bounds w.r.t. t₈:
X₀: 2⋅X₃ {O(n)}
Runtime-bound of t₈: X₃ {O(n)}
Results in: 4⋅X₃⋅X₃+6⋅X₃ {O(n^2)}

knowledge_propagation leads to new time bound X₃+1 {O(n)} for transition t₃: l4(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 0 < X₁ ∧ X₀ ≤ X₃

knowledge_propagation leads to new time bound X₃+2 {O(n)} for transition t₅: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃

knowledge_propagation leads to new time bound X₃+2 {O(n)} for transition t₇: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃

All Bounds

Timebounds

Overall timebound:4⋅X₃+X₂+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₃+2 {O(n)}
t₇: X₃+2 {O(n)}
t₈: X₃ {O(n)}
t₉: X₂+1 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+X₂+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₃+2 {O(n)}
t₇: X₃+2 {O(n)}
t₈: X₃ {O(n)}
t₉: X₂+1 {O(n)}
t₁₀: 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: X₃ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₃ {O(n)}
t₂, X₁: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₃ {O(n)}
t₃, X₁: X₂+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 3⋅X₃+1 {O(n)}
t₄, X₁: 0 {O(1)}
t₄, X₂: 3⋅X₂ {O(n)}
t₄, X₃: 3⋅X₃ {O(n)}
t₅, X₀: 2⋅X₃ {O(n)}
t₅, X₁: 2⋅X₂+1 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₇, X₀: 2⋅X₃ {O(n)}
t₇, X₁: 2⋅X₂+1 {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₈, X₀: 2⋅X₃ {O(n)}
t₈, X₁: 1 {O(1)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₉, X₀: 2⋅X₃+1 {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₂: 2⋅X₂ {O(n)}
t₉, X₃: 2⋅X₃ {O(n)}
t₁₀, X₀: 3⋅X₃+1 {O(n)}
t₁₀, X₁: 0 {O(1)}
t₁₀, X₂: 3⋅X₂ {O(n)}
t₁₀, X₃: 3⋅X₃ {O(n)}