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, 10, X₂, X₃) :|: X₀ ≤ 10
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 10 < X₀
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1
t₄: l3(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: 1 < X₁
t₈: l4(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l5(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃)
t₆: 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 ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10 for location l6

Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location l7

Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10 for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location l4

Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 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:
t₁₈: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₉: l1(X₀, X₁, X₂) → l3(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀
t₂₀: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 10 < X₀ ∧ X₂ ≤ X₀
t₂₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₂₃: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10
t₂₂: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 1 < X₁ ∧ X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10
t₂₄: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 11 ≤ X₀
t₂₅: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10
t₂₆: l6(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂ ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10

MPRF for transition t₁₉: l1(X₀, X₁, X₂) → l3(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+11 {O(n)}

MPRF:

l5 [10-X₀ ]
l1 [11-X₀ ]
l6 [10-X₀ ]
l3 [10-X₀ ]

MPRF for transition t₂₃: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 of depth 1:

new bound:

X₂+11 {O(n)}

MPRF:

l5 [10-X₀ ]
l1 [11-X₀ ]
l6 [11-X₀ ]
l3 [11-X₀ ]

MPRF for transition t₂₅: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10 of depth 1:

new bound:

X₂+11 {O(n)}

MPRF:

l5 [11-X₀ ]
l1 [11-X₀ ]
l6 [11-X₀ ]
l3 [11-X₀ ]

Found invariant X₂ ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10 for location l6

Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location l7

Found invariant X₂ ≤ 10 ∧ X₂ ≤ 9+X₁ ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ 1 ≤ X₁ ∧ X₀ ≤ 9+X₁ ∧ X₀ ≤ 10 for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location l4

Found invariant X₂ ≤ 10 ∧ X₂ ≤ 9+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 1 ≤ X₁ ∧ X₀ ≤ 9+X₁ ∧ X₀ ≤ 10 for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂₂ 24⋅X₂+264 {O(n)}

TWN-Loops:

entry: t₁₉: l1(X₀, X₁, X₂) → l3(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀
results in twn-loop: twn:Inv: [X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 ∧ X₂ ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10] , (X₀,X₁,X₂) -> (X₀,X₁-1,X₂) :|: 1 < X₁
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
X₂: X₂

Termination: true
Formula:

1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: 1 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₁₉:
X₁: 10 {O(1)}
Runtime-bound of t₁₉: X₂+11 {O(n)}
Results in: 24⋅X₂+264 {O(n)}

24⋅X₂+264 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂₆ 24⋅X₂+264 {O(n)}

relevant size-bounds w.r.t. t₁₉:
X₁: 10 {O(1)}
Runtime-bound of t₁₉: X₂+11 {O(n)}
Results in: 24⋅X₂+264 {O(n)}

24⋅X₂+264 {O(n)}

All Bounds

Timebounds

Overall timebound:51⋅X₂+565 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: X₂+11 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 24⋅X₂+264 {O(n)}
t₂₃: X₂+11 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+11 {O(n)}
t₂₆: 24⋅X₂+264 {O(n)}

Costbounds

Overall costbound: 51⋅X₂+565 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: X₂+11 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 24⋅X₂+264 {O(n)}
t₂₃: X₂+11 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+11 {O(n)}
t₂₆: 24⋅X₂+264 {O(n)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₀: 2⋅X₂+11 {O(n)}
t₁₉, X₁: 10 {O(1)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: 3⋅X₂+11 {O(n)}
t₂₀, X₁: X₁+9 {O(n)}
t₂₀, X₂: 2⋅X₂ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₂, X₀: 2⋅X₂+11 {O(n)}
t₂₂, X₁: 10 {O(1)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: 2⋅X₂+11 {O(n)}
t₂₃, X₁: 9 {O(1)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 3⋅X₂+11 {O(n)}
t₂₄, X₁: X₁+9 {O(n)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₅, X₀: 2⋅X₂+11 {O(n)}
t₂₅, X₁: 9 {O(1)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: 2⋅X₂+11 {O(n)}
t₂₆, X₁: 9 {O(1)}
t₂₆, X₂: X₂ {O(n)}