Initial Problem

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

Preprocessing

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

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

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

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

Found invariant 0 ≤ X₁ for location l4

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ 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:
t₀: l0(X₀, X₁) → l5(X₀, X₁)
t₇: l1(X₀, X₁) → l4(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₄: l2(X₀, X₁) → l3(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₆: l3(X₀, X₁) → l1(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l4(X₀, X₁) → l2(X₀, X₁) :|: X₁ < X₀ ∧ 0 ≤ X₁
t₃: l4(X₀, X₁) → l6(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₁: l5(X₀, X₁) → l4(X₀, 0)
t₈: l6(X₀, X₁) → l7(X₀, X₁) :|: 0 ≤ X₁ ∧ X₀ ≤ X₁

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

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

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

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

Found invariant 0 ≤ X₁ for location l4

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 2⋅X₀+4 {O(n)}

TWN-Loops:

entry: t₁: l5(X₀, X₁) → l4(X₀, 0)
results in twn-loop: twn:Inv: [0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁) -> (X₀,X₁+1) :|: X₁ < X₀
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₁ < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₀ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₀+4 {O(n)}

2⋅X₀+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₀ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₀+4 {O(n)}

2⋅X₀+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₆ 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₀ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₀+4 {O(n)}

2⋅X₀+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₀ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₀+4 {O(n)}

2⋅X₀+4 {O(n)}

All Bounds

Timebounds

Overall timebound:8⋅X₀+20 {O(n)}
t₀: 1 {O(1)}
t₇: 2⋅X₀+4 {O(n)}
t₄: 2⋅X₀+4 {O(n)}
t₆: 2⋅X₀+4 {O(n)}
t₂: 2⋅X₀+4 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₀+20 {O(n)}
t₀: 1 {O(1)}
t₇: 2⋅X₀+4 {O(n)}
t₄: 2⋅X₀+4 {O(n)}
t₆: 2⋅X₀+4 {O(n)}
t₂: 2⋅X₀+4 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: 2⋅X₀+4 {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: 2⋅X₀+4 {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: 2⋅X₀+4 {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: 2⋅X₀+4 {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₀+4 {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 0 {O(1)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₀+4 {O(n)}