Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₀, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₅, X₁, X₂, X₃, X₄, X₅)
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₁, X₂, X₃, X₄, X₅)
t₅: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ < X₃
t₆: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l6(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₁
t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁+1, X₂, X₃, X₄, X₅)
t₈: l6(X₀, X₁, X₂, X₃, X₄, X₅) → l7(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

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

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l6

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l7

Found invariant 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ X₀ for location l1

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₈: l0(X₀, X₁, X₃, X₄, X₅) → l2(X₀, X₁, X₃, X₄, X₅)
t₁₉: l1(X₀, X₁, X₃, X₄, X₅) → l3(X₀, X₁, X₃, X₄, X₅) :|: X₀ < X₄ ∧ X₅ ≤ X₀
t₂₀: l1(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₀, X₃, X₄, X₅) :|: X₄ ≤ X₀ ∧ X₅ ≤ X₀
t₂₁: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₅, X₁, X₃, X₄, X₅)
t₂₂: l3(X₀, X₁, X₃, X₄, X₅) → l1(X₀+1, X₁, X₃, X₄, X₅) :|: 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄
t₂₃: l4(X₀, X₁, X₃, X₄, X₅) → l5(X₀, X₁, X₃, X₄, X₅) :|: X₁ < X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁
t₂₄: l4(X₀, X₁, X₃, X₄, X₅) → l6(X₀, X₁, X₃, X₄, X₅) :|: X₃ ≤ X₁ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁
t₂₅: l5(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₁+1, X₃, X₄, X₅) :|: 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁
t₂₆: l6(X₀, X₁, X₃, X₄, X₅) → l7(X₀, X₁, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l6

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l7

Found invariant 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ X₀ for location l1

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₉ 2⋅X₄+2⋅X₅+4 {O(n)}

TWN-Loops:

entry: t₂₁: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₅, X₁, X₃, X₄, X₅)
results in twn-loop: twn:Inv: [X₅ ≤ X₀ ∧ 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄] , (X₀,X₁,X₃,X₄,X₅) -> (X₀+1,X₁,X₃,X₄,X₅) :|: X₀ < X₄
order: [X₀; X₄; X₅]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₄: X₄
X₅: X₅

Termination: true
Formula:

1 < 0
∨ X₀ < X₄ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₀ < X₄
alphas_abs: X₀+X₄
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₄+2 {O(n)}

relevant size-bounds w.r.t. t₂₁:
X₀: X₅ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₂₁: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+4 {O(n)}

2⋅X₄+2⋅X₅+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂₂ 2⋅X₄+2⋅X₅+4 {O(n)}

relevant size-bounds w.r.t. t₂₁:
X₀: X₅ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₂₁: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+4 {O(n)}

2⋅X₄+2⋅X₅+4 {O(n)}

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l6

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l7

Found invariant 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ X₀ for location l1

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂₃ 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}

TWN-Loops:

entry: t₂₀: l1(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₀, X₃, X₄, X₅) :|: X₄ ≤ X₀ ∧ X₅ ≤ X₀
results in twn-loop: twn:Inv: [X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁] , (X₀,X₁,X₃,X₄,X₅) -> (X₀,X₁+1,X₃,X₄,X₅) :|: X₁ < X₃
order: [X₀; X₁; X₃; X₄; X₅]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₃: X₃
X₄: X₄
X₅: X₅

Termination: true
Formula:

1 < 0
∨ X₁ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₁ < X₃
alphas_abs: X₁+X₃
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₃+2 {O(n)}

relevant size-bounds w.r.t. t₂₀:
X₁: 2⋅X₄+4⋅X₅+4 {O(n)}
X₃: 2⋅X₃ {O(n)}
Runtime-bound of t₂₀: 1 {O(1)}
Results in: 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}

4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂₅ 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}

relevant size-bounds w.r.t. t₂₀:
X₁: 2⋅X₄+4⋅X₅+4 {O(n)}
X₃: 2⋅X₃ {O(n)}
Runtime-bound of t₂₀: 1 {O(1)}
Results in: 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}

4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}

All Bounds

Timebounds

Overall timebound:12⋅X₄+20⋅X₅+8⋅X₃+37 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₄+2⋅X₅+4 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₄+2⋅X₅+4 {O(n)}
t₂₃: 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}
t₂₆: 1 {O(1)}

Costbounds

Overall costbound: 12⋅X₄+20⋅X₅+8⋅X₃+37 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₄+2⋅X₅+4 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₄+2⋅X₅+4 {O(n)}
t₂₃: 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 4⋅X₃+4⋅X₄+8⋅X₅+12 {O(n)}
t₂₆: 1 {O(1)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₉, X₀: 2⋅X₄+3⋅X₅+4 {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}
t₂₀, X₀: 2⋅X₄+4⋅X₅+4 {O(n)}
t₂₀, X₁: 2⋅X₄+4⋅X₅+4 {O(n)}
t₂₀, X₃: 2⋅X₃ {O(n)}
t₂₀, X₄: 2⋅X₄ {O(n)}
t₂₀, X₅: 2⋅X₅ {O(n)}
t₂₁, X₀: X₅ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₂, X₀: 2⋅X₄+3⋅X₅+4 {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: 2⋅X₄+4⋅X₅+4 {O(n)}
t₂₃, X₁: 12⋅X₅+4⋅X₃+6⋅X₄+16 {O(n)}
t₂₃, X₃: 2⋅X₃ {O(n)}
t₂₃, X₄: 2⋅X₄ {O(n)}
t₂₃, X₅: 2⋅X₅ {O(n)}
t₂₄, X₀: 4⋅X₄+8⋅X₅+8 {O(n)}
t₂₄, X₁: 16⋅X₅+4⋅X₃+8⋅X₄+20 {O(n)}
t₂₄, X₃: 4⋅X₃ {O(n)}
t₂₄, X₄: 4⋅X₄ {O(n)}
t₂₄, X₅: 4⋅X₅ {O(n)}
t₂₅, X₀: 2⋅X₄+4⋅X₅+4 {O(n)}
t₂₅, X₁: 12⋅X₅+4⋅X₃+6⋅X₄+16 {O(n)}
t₂₅, X₃: 2⋅X₃ {O(n)}
t₂₅, X₄: 2⋅X₄ {O(n)}
t₂₅, X₅: 2⋅X₅ {O(n)}
t₂₆, X₀: 4⋅X₄+8⋅X₅+8 {O(n)}
t₂₆, X₁: 16⋅X₅+4⋅X₃+8⋅X₄+20 {O(n)}
t₂₆, X₃: 4⋅X₃ {O(n)}
t₂₆, X₄: 4⋅X₄ {O(n)}
t₂₆, X₅: 4⋅X₅ {O(n)}