Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ ≤ 99
t₃: l1(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 100 ≤ X₁
t₄: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂
t₅: l2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₂ ≤ X₀
t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂)
t₇: l4(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂)
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l1(0, 0, X₂)

Preprocessing

Found invariant X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

Found invariant 100 ≤ X₁ ∧ 100 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l7

Found invariant 100 ≤ X₁ ∧ 100 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5

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

Found invariant X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l4

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ ≤ 99 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₃: l1(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 100 ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₄: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₇: l4(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: 100 ≤ X₁ ∧ 100 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁: l6(X₀, X₁, X₂) → l1(0, 0, X₂)

MPRF for transition t₄: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l2 [X₂-X₀ ]
l3 [X₂-X₀-1 ]
l4 [X₂-X₀ ]
l1 [X₂-X₀ ]

MPRF for transition t₅: l2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

100 {O(1)}

MPRF:

l2 [100-X₁ ]
l3 [100-X₁ ]
l4 [99-X₁ ]
l1 [100-X₁ ]

MPRF for transition t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l2 [X₂-X₀ ]
l3 [X₂-X₀ ]
l4 [X₂-X₀ ]
l1 [X₂-X₀ ]

MPRF for transition t₇: l4(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

100 {O(1)}

MPRF:

l2 [100-X₁ ]
l3 [100-X₁ ]
l4 [100-X₁ ]
l1 [100-X₁ ]

Found invariant X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ X₀+X₁ ≤ 99 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l2

Found invariant X₁ ≤ 100 ∧ X₁ ≤ 100+X₀ ∧ X₀+X₁ ≤ 100 ∧ 100 ≤ X₁ ∧ 100 ≤ X₀+X₁ ∧ 100+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l7

Found invariant X₁ ≤ 100 ∧ X₁ ≤ 100+X₀ ∧ X₀+X₁ ≤ 100 ∧ 100 ≤ X₁ ∧ 100 ≤ X₀+X₁ ∧ 100+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5

Found invariant X₁ ≤ 100 ∧ X₁ ≤ 100+X₀ ∧ X₀+X₁ ≤ 100 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l1

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 99 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 0 ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ X₀+X₁ ≤ 99 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l4

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ 98+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l3

Found invariant 1 ≤ 0 for location l2

Found invariant 1 ≤ 0 for location l7

Found invariant 1 ≤ 0 for location l5

Found invariant 1 ≤ 0 for location l1

Found invariant 1 ≤ 0 for location l4

Found invariant 1 ≤ 0 for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 401⋅X₂+203 {O(n)}

TWN-Loops:

entry: t₁: l6(X₀, X₁, X₂) → l1(0, 0, X₂)
results in twn-loop: twn:Inv: [0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁+1,X₂) :|: X₁ ≤ 99 ∧ X₂ ≤ X₀
entry: t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
results in twn-loop: twn:Inv: [0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 99 ∧ X₁ ≤ 99+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁+1,X₂) :|: X₁ ≤ 99 ∧ X₂ ≤ X₀
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂

Termination: true
Formula:

X₂ < X₀ ∧ 1 < 0
∨ X₂ < X₀ ∧ X₁ < 99 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 99 ∧ 99 ≤ X₁
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 < 0
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ < 99 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 99 ∧ 99 ≤ X₁

Stabilization-Threshold for: X₁ ≤ 99
alphas_abs: X₁+99
M: 0
N: 1
Bound: 2⋅X₁+200 {O(n)}

relevant size-bounds w.r.t. t₁:
X₁: 0 {O(1)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 203 {O(1)}

order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂

Termination: true
Formula:

X₂ < X₀ ∧ 1 < 0
∨ X₂ < X₀ ∧ X₁ < 99 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 99 ∧ 99 ≤ X₁
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 < 0
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ < 99 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 99 ∧ 99 ≤ X₁

Stabilization-Threshold for: X₁ ≤ 99
alphas_abs: X₁+99
M: 0
N: 1
Bound: 2⋅X₁+200 {O(n)}

relevant size-bounds w.r.t. t₆:
X₁: 99 {O(1)}
Runtime-bound of t₆: X₂ {O(n)}
Results in: 401⋅X₂ {O(n)}

401⋅X₂+203 {O(n)}

All Bounds

Timebounds

Overall timebound:403⋅X₂+407 {O(n)}
t₀: 1 {O(1)}
t₂: 401⋅X₂+203 {O(n)}
t₃: 1 {O(1)}
t₄: X₂ {O(n)}
t₅: 100 {O(1)}
t₆: X₂ {O(n)}
t₇: 100 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 403⋅X₂+407 {O(n)}
t₀: 1 {O(1)}
t₂: 401⋅X₂+203 {O(n)}
t₃: 1 {O(1)}
t₄: X₂ {O(n)}
t₅: 100 {O(1)}
t₆: X₂ {O(n)}
t₇: 100 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: 99 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 2⋅X₂ {O(n)}
t₃, X₁: 199 {O(1)}
t₃, X₂: 2⋅X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: 99 {O(1)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: 99 {O(1)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₂ {O(n)}
t₆, X₁: 99 {O(1)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: 100 {O(1)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: 2⋅X₂ {O(n)}
t₈, X₁: 199 {O(1)}
t₈, X₂: 2⋅X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: 0 {O(1)}
t₁, X₂: X₂ {O(n)}