Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < 20
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 20 ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(0, X₁, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ 9
t₅: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 9 < X₀
t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

Eliminate variables {X₁,X₂} that do not contribute to the problem

Found invariant 20 ≤ X₀ for location l5

Found invariant 0 ≤ X₀ for location l1

Found invariant 20 ≤ X₀ for location l4

Found invariant X₀ ≤ 19 ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₄: l0(X₀) → l2(X₀)
t₁₅: l1(X₀) → l3(X₀) :|: X₀ < 20 ∧ 0 ≤ X₀
t₁₆: l1(X₀) → l4(X₀) :|: 20 ≤ X₀ ∧ 0 ≤ X₀
t₁₇: l2(X₀) → l1(0)
t₁₈: l3(X₀) → l1(X₀+1) :|: X₀ ≤ 9 ∧ X₀ ≤ 19 ∧ 0 ≤ X₀
t₁₉: l3(X₀) → l1(X₀+1) :|: 9 < X₀ ∧ X₀ ≤ 19 ∧ 0 ≤ X₀
t₂₀: l4(X₀) → l5(X₀) :|: 20 ≤ X₀

Found invariant X₀ ≤ 20 ∧ 20 ≤ X₀ for location l5

Found invariant X₀ ≤ 20 ∧ 0 ≤ X₀ for location l1

Found invariant X₀ ≤ 20 ∧ 20 ≤ X₀ for location l4

Found invariant X₀ ≤ 19 ∧ 0 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₅ 84 {O(1)}

TWN-Loops:

entry: t₁₇: l2(X₀) → l1(0)
results in twn-loop: twn:Inv: [0 ≤ X₀ ∧ X₀ ≤ 19 ∧ 0 ≤ X₀ ∧ X₀ ≤ 19 ∧ 0 ≤ X₀] , (X₀) -> (X₀+1) :|: X₀ < 20 ∧ X₀ ≤ 9 ∨ X₀ < 20 ∧ 9 < X₀
order: [X₀]
closed-form:
X₀: X₀ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ X₀ < 20 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ < 9 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₀ < 9 ∧ X₀ < 20 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ 9 ∧ 9 ≤ X₀ ∧ 1 < 0
∨ X₀ ≤ 9 ∧ 9 ≤ X₀ ∧ X₀ < 20 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ 1 < 0
∨ 0 < 1 ∧ X₀ < 20 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 9 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ 9 < X₀ ∧ X₀ < 20 ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: 9 < X₀
alphas_abs: 9+X₀
M: 0
N: 1
Bound: 2⋅X₀+20 {O(n)}
Stabilization-Threshold for: X₀ < 20
alphas_abs: 20
M: 0
N: 1
Bound: 42 {O(1)}
Stabilization-Threshold for: X₀ ≤ 9
alphas_abs: 9
M: 0
N: 1
Bound: 20 {O(1)}

relevant size-bounds w.r.t. t₁₇:
X₀: 0 {O(1)}
Runtime-bound of t₁₇: 1 {O(1)}
Results in: 84 {O(1)}

84 {O(1)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₈ 84 {O(1)}

relevant size-bounds w.r.t. t₁₇:
X₀: 0 {O(1)}
Runtime-bound of t₁₇: 1 {O(1)}
Results in: 84 {O(1)}

84 {O(1)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₉ 84 {O(1)}

relevant size-bounds w.r.t. t₁₇:
X₀: 0 {O(1)}
Runtime-bound of t₁₇: 1 {O(1)}
Results in: 84 {O(1)}

84 {O(1)}

All Bounds

Timebounds

Overall timebound:256 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 84 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 84 {O(1)}
t₁₉: 84 {O(1)}
t₂₀: 1 {O(1)}

Costbounds

Overall costbound: 256 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 84 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 84 {O(1)}
t₁₉: 84 {O(1)}
t₂₀: 1 {O(1)}

Sizebounds

t₁₄, X₀: X₀ {O(n)}
t₁₅, X₀: 19 {O(1)}
t₁₆, X₀: 20 {O(1)}
t₁₇, X₀: 0 {O(1)}
t₁₈, X₀: 10 {O(1)}
t₁₉, X₀: 20 {O(1)}
t₂₀, X₀: 20 {O(1)}