Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(D, 0, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₁ ≤ 9
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, 0) :|: 10 ≤ X₁
t₂: l2(X₀, X₁, X₂) → l2(X₀, X₁, X₂+1) :|: X₂ ≤ 9
t₃: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 10 ≤ X₂

Preprocessing

Eliminate variables {D,X₀} that do not contribute to the problem

Found invariant X₂ ≤ 10 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 20 ∧ 0 ≤ X₂ ∧ 10 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ for location l2

Found invariant X₁ ≤ 10 ∧ 0 ≤ X₁ for location l1

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

Problem after Preprocessing

Start: l0
Program_Vars: X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₁₂: l0(X₁, X₂) → l1(0, X₂)
t₁₃: l1(X₁, X₂) → l1(X₁+1, X₂) :|: X₁ ≤ 9 ∧ X₁ ≤ 10 ∧ 0 ≤ X₁
t₁₄: l1(X₁, X₂) → l2(X₁, 0) :|: 10 ≤ X₁ ∧ X₁ ≤ 10 ∧ 0 ≤ X₁
t₁₅: l2(X₁, X₂) → l2(X₁, X₂+1) :|: X₂ ≤ 9 ∧ X₂ ≤ 10 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 20 ∧ 0 ≤ X₂ ∧ 10 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁
t₁₆: l2(X₁, X₂) → l3(X₁, X₂) :|: 10 ≤ X₂ ∧ X₂ ≤ 10 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 20 ∧ 0 ≤ X₂ ∧ 10 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁

Found invariant X₂ ≤ 10 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 20 ∧ 0 ≤ X₂ ∧ 10 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ for location l2

Found invariant X₁ ≤ 10 ∧ 0 ≤ X₁ for location l1

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

Time-Bound by TWN-Loops:

TWN-Loops: t₁₃ 22 {O(1)}

TWN-Loops:

entry: t₁₂: l0(X₁, X₂) → l1(0, X₂)
results in twn-loop: twn:Inv: [X₁ ≤ 10 ∧ 0 ≤ X₁] , (X₁,X₂) -> (X₁+1,X₂) :|: X₁ ≤ 9
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₁ < 9 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 9 ∧ 9 ≤ X₁

Stabilization-Threshold for: X₁ ≤ 9
alphas_abs: 9
M: 0
N: 1
Bound: 20 {O(1)}

relevant size-bounds w.r.t. t₁₂:
Runtime-bound of t₁₂: 1 {O(1)}
Results in: 22 {O(1)}

22 {O(1)}

Found invariant X₂ ≤ 10 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 20 ∧ 0 ≤ X₂ ∧ 10 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ for location l2

Found invariant X₁ ≤ 10 ∧ 0 ≤ X₁ for location l1

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

Time-Bound by TWN-Loops:

TWN-Loops: t₁₅ 22 {O(1)}

TWN-Loops:

entry: t₁₄: l1(X₁, X₂) → l2(X₁, 0) :|: 10 ≤ X₁ ∧ X₁ ≤ 10 ∧ 0 ≤ X₁
results in twn-loop: twn:Inv: [X₂ ≤ 10 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 20 ∧ 0 ≤ X₂ ∧ 10 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁] , (X₁,X₂) -> (X₁,X₂+1) :|: X₂ ≤ 9
order: [X₁; X₂]
closed-form:
X₁: X₁
X₂: X₂ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₂ < 9 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₂ ≤ 9 ∧ 9 ≤ X₂

Stabilization-Threshold for: X₂ ≤ 9
alphas_abs: X₂+9
M: 0
N: 1
Bound: 2⋅X₂+20 {O(n)}

relevant size-bounds w.r.t. t₁₄:
X₂: 0 {O(1)}
Runtime-bound of t₁₄: 1 {O(1)}
Results in: 22 {O(1)}

22 {O(1)}

All Bounds

Timebounds

Overall timebound:47 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 22 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 22 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 47 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 22 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 22 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₂, X₁: 0 {O(1)}
t₁₂, X₂: X₂ {O(n)}
t₁₃, X₁: 10 {O(1)}
t₁₃, X₂: X₂ {O(n)}
t₁₄, X₁: 10 {O(1)}
t₁₄, X₂: 0 {O(1)}
t₁₅, X₁: 10 {O(1)}
t₁₅, X₂: 10 {O(1)}
t₁₆, X₁: 10 {O(1)}
t₁₆, X₂: 10 {O(1)}