Initial Problem

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

Preprocessing

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

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

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

Found invariant 0 ≤ X₀ for location l4

Problem after Preprocessing

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

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

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

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

Found invariant 0 ≤ X₀ for location l4

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₁+4 {O(n)}

TWN-Loops:

entry: t₁: l3(X₀, X₁) → l4(0, X₁)
results in twn-loop: twn:Inv: [0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀+1,X₁) :|: X₀+1 ≤ X₁
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁

Termination: true
Formula:

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

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

relevant size-bounds w.r.t. t₁:
X₁: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₁: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+12 {O(n)}
t₀: 1 {O(1)}
t₄: 2⋅X₁+4 {O(n)}
t₅: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₁+4 {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+12 {O(n)}
t₀: 1 {O(1)}
t₄: 2⋅X₁+4 {O(n)}
t₅: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₁+4 {O(n)}
t₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₄, X₀: 2⋅X₁+4 {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: 2⋅X₁+4 {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: 2⋅X₁+4 {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: 2⋅X₁+4 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}