Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₀, X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀

Preprocessing

Found invariant 1 ≤ X₀ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₀, X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀

Found invariant 1 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 2⋅X₀+2⋅X₁+6 {O(n)}

TWN-Loops:

entry: t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
results in twn-loop: twn:Inv: [1 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁+X₀,X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * X₀ * n^1
X₂: X₂

Termination: true
Formula:

1 < X₀ ∧ X₀+1 < X₂ ∧ X₀ < 0
∨ 1 < X₀ ∧ X₀+1 < X₂ ∧ X₁+1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ 1 < X₀ ∧ X₀+1 < X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 < X₀ ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1 ∧ X₀ < 0
∨ 1 < X₀ ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1 ∧ X₁+1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ 1 < X₀ ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀+1 < X₂ ∧ X₀ < 0
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀+1 < X₂ ∧ X₁+1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀+1 < X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1 ∧ X₀ < 0
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1 ∧ X₁+1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1

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

relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₀+2⋅X₁+6 {O(n)}

2⋅X₀+2⋅X₁+6 {O(n)}

Found invariant 1 ≤ 0 for location l1

Found invariant 1 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 10⋅X₀+10⋅X₁+35 {O(n)}

TWN-Loops:

entry: t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₀, X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [1 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁,X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀
entry: t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
results in twn-loop: twn:Inv: [1 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁,X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: [[n == 0]] * X₂ + [[n != 0]] * (X₁-X₀)

Termination: true
Formula:

1 < X₀ ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 < X₀
∨ 1 < X₀ ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 < X₀ ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 < X₀
∨ 1 < X₀ ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 < X₀
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 < X₀
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 2⋅X₀+2⋅X₁+6 {O(n)}
Results in: 10⋅X₀+10⋅X₁+30 {O(n)}

order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: [[n == 0]] * X₂ + [[n != 0]] * (X₁-X₀)

Termination: true
Formula:

1 < X₀ ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 < X₀
∨ 1 < X₀ ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 < X₀ ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 < X₀
∨ 1 < X₀ ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 < X₀
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 < X₁ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 < X₀
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 2⋅X₀+1 ≤ X₁ ∧ X₁ ≤ 2⋅X₀+1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1

relevant size-bounds w.r.t. t₀:
Runtime-bound of t₀: 1 {O(1)}
Results in: 5 {O(1)}

10⋅X₀+10⋅X₁+35 {O(n)}

All Bounds

Timebounds

Overall timebound:12⋅X₀+12⋅X₁+42 {O(n)}
t₀: 1 {O(1)}
t₁: 2⋅X₀+2⋅X₁+6 {O(n)}
t₂: 10⋅X₀+10⋅X₁+35 {O(n)}

Costbounds

Overall costbound: 12⋅X₀+12⋅X₁+42 {O(n)}
t₀: 1 {O(1)}
t₁: 2⋅X₀+2⋅X₁+6 {O(n)}
t₂: 10⋅X₀+10⋅X₁+35 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 4⋅X₀⋅X₀+4⋅X₀⋅X₁+14⋅X₀+X₁ {O(n^2)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 4⋅X₀⋅X₀+4⋅X₀⋅X₁+14⋅X₀+2⋅X₁ {O(n^2)}
t₂, X₂: 4⋅X₀⋅X₀+4⋅X₀⋅X₁+16⋅X₀+2⋅X₁ {O(n^2)}