Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, 3, X₂, X₃, X₄) :|: X₀ < 10
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: 10 ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(0, X₁, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: 12 ≤ X₁
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄) :|: X₁ < 12
t₈: l4(X₀, X₁, X₂, X₃, X₄) → l7(X₀, X₁, X₂, X₃, X₄)
t₇: l5(X₀, X₁, X₂, X₃, X₄) → l1(X₀+1, X₁, X₂, X₃, X₄)
t₆: l6(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁+1, X₂, X₃, X₄)

Preprocessing

Eliminate variables {X₂,X₃,X₄} that do not contribute to the problem

Found invariant X₁ ≤ 11 ∧ X₁ ≤ 11+X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l6

Found invariant 10 ≤ X₀ for location l7

Found invariant 12 ≤ X₁ ∧ 12 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5

Found invariant 0 ≤ X₀ for location l1

Found invariant 10 ≤ X₀ for location l4

Found invariant 3 ≤ X₁ ∧ 3 ≤ 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, l6, l7
Transitions:
t₁₈: l0(X₀, X₁) → l2(X₀, X₁)
t₁₉: l1(X₀, X₁) → l3(X₀, 3) :|: X₀ < 10 ∧ 0 ≤ X₀
t₂₀: l1(X₀, X₁) → l4(X₀, X₁) :|: 10 ≤ X₀ ∧ 0 ≤ X₀
t₂₁: l2(X₀, X₁) → l1(0, X₁)
t₂₃: l3(X₀, X₁) → l5(X₀, X₁) :|: 12 ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂₂: l3(X₀, X₁) → l6(X₀, X₁) :|: X₁ < 12 ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂₄: l4(X₀, X₁) → l7(X₀, X₁) :|: 10 ≤ X₀
t₂₅: l5(X₀, X₁) → l1(X₀+1, X₁) :|: 12 ≤ X₁ ∧ 12 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂₆: l6(X₀, X₁) → l3(X₀, X₁+1) :|: X₁ ≤ 11 ∧ X₁ ≤ 11+X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₁₉: l1(X₀, X₁) → l3(X₀, 3) :|: X₀ < 10 ∧ 0 ≤ X₀ of depth 1:

new bound:

10 {O(1)}

MPRF:

l5 [9-X₀ ]
l1 [10-X₀ ]
l6 [9-X₀ ]
l3 [9-X₀ ]

Found invariant X₁ ≤ 11 ∧ X₁ ≤ 11+X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l6

Found invariant 10 ≤ X₀ for location l7

Found invariant X₁ ≤ 12 ∧ X₁ ≤ 12+X₀ ∧ 12 ≤ X₁ ∧ 12 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5

Found invariant 0 ≤ X₀ for location l1

Found invariant 10 ≤ X₀ for location l4

Found invariant X₁ ≤ 12 ∧ X₁ ≤ 12+X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂₂ 280 {O(1)}

TWN-Loops:

entry: t₁₉: l1(X₀, X₁) → l3(X₀, 3) :|: X₀ < 10 ∧ 0 ≤ X₀
results in twn-loop: twn:Inv: [3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ 11 ∧ X₁ ≤ 11+X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀,X₁+1) :|: X₁ < 12
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₁ < 12 ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₁ < 12
alphas_abs: 12
M: 0
N: 1
Bound: 26 {O(1)}

relevant size-bounds w.r.t. t₁₉:
Runtime-bound of t₁₉: 10 {O(1)}
Results in: 280 {O(1)}

280 {O(1)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂₆ 280 {O(1)}

relevant size-bounds w.r.t. t₁₉:
Runtime-bound of t₁₉: 10 {O(1)}
Results in: 280 {O(1)}

280 {O(1)}

knowledge_propagation leads to new time bound 280 {O(1)} for transition t₂₃: l3(X₀, X₁) → l5(X₀, X₁) :|: 12 ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀

knowledge_propagation leads to new time bound 280 {O(1)} for transition t₂₅: l5(X₀, X₁) → l1(X₀+1, X₁) :|: 12 ≤ X₁ ∧ 12 ≤ X₀+X₁ ∧ 0 ≤ X₀

All Bounds

Timebounds

Overall timebound:1134 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 10 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 280 {O(1)}
t₂₃: 280 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 280 {O(1)}
t₂₆: 280 {O(1)}

Costbounds

Overall costbound: 1134 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 10 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 280 {O(1)}
t₂₃: 280 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 280 {O(1)}
t₂₆: 280 {O(1)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₉, X₀: 9 {O(1)}
t₁₉, X₁: 3 {O(1)}
t₂₀, X₀: 10 {O(1)}
t₂₀, X₁: 12 {O(1)}
t₂₁, X₀: 0 {O(1)}
t₂₁, X₁: X₁ {O(n)}
t₂₂, X₀: 9 {O(1)}
t₂₂, X₁: 11 {O(1)}
t₂₃, X₀: 9 {O(1)}
t₂₃, X₁: 12 {O(1)}
t₂₄, X₀: 10 {O(1)}
t₂₄, X₁: 12 {O(1)}
t₂₅, X₀: 10 {O(1)}
t₂₅, X₁: 12 {O(1)}
t₂₆, X₀: 9 {O(1)}
t₂₆, X₁: 12 {O(1)}