Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₀, X₂, 0, X₄) :|: 0 < X₂
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₄, X₁, 1, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₁, X₁, X₃, X₃, X₄) :|: X₁ ≤ 0
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₁
t₇: l4(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄)
t₆: l5(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁-1, X₂, 1, X₄)

Preprocessing

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location l6

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂ for location l1

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location l4

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₀, X₂, 0, X₄) :|: 0 < X₂ ∧ X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₄, X₁, 1, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₁, X₁, X₃, X₃, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₁ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀
t₇: l4(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂
t₆: l5(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁-1, X₂, 1, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀

MPRF for transition t₄: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₁ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l1 [X₀ ]
l5 [X₁-1 ]
l3 [X₁ ]

MPRF for transition t₆: l5(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁-1, X₂, 1, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l1 [X₀ ]
l5 [X₁ ]
l3 [X₁ ]

Found invariant X₄ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location l6

Found invariant X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 1+X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5

Found invariant X₄ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂ for location l1

Found invariant X₄ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location l4

Found invariant X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l3

Found invariant 1 ≤ 0 for location l6

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 3⋅X₄+3 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₄, X₁, 1, X₃, X₄)
results in twn-loop: twn:Inv: [X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ 1 ∧ 0 ≤ X₂ ∧ X₂ ≤ 2 ∧ 0 ≤ 0 ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ 0 ≤ 0] , (X₀,X₁,X₂,X₃,X₄) -> (X₀,X₀,0,0,X₄) :|: 0 < X₂ ∧ X₀ ≤ 0
entry: t₆: l5(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁-1, X₂, 1, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₃ ≤ 1 ∧ 0 ≤ X₃] , (X₀,X₁,X₂,X₃,X₄) -> (X₁,X₁,X₃,0,X₄) :|: X₁ ≤ 0 ∧ 0 < X₃
order: [X₀; X₂; X₄]
closed-form:
X₀: X₀
X₂: [[n == 0]] * X₂
X₄: X₄

Termination: true
Formula:

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 3 {O(1)}

order: [X₁; X₀; X₃; X₂; X₄]
closed-form:
X₁: X₁
X₀: [[n == 0]] * X₀ + [[n != 0]] * X₁
X₃: [[n == 0]] * X₃
X₂: [[n == 0]] * X₂ + [[n != 0, n == 1]] * X₃
X₄: X₄

Termination: true
Formula:

relevant size-bounds w.r.t. t₆:
Runtime-bound of t₆: X₄ {O(n)}
Results in: 3⋅X₄ {O(n)}

3⋅X₄+3 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 3⋅X₄+3 {O(n)}

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 3 {O(1)}

relevant size-bounds w.r.t. t₆:
Runtime-bound of t₆: X₄ {O(n)}
Results in: 3⋅X₄ {O(n)}

3⋅X₄+3 {O(n)}

All Bounds

Timebounds

Overall timebound:8⋅X₄+10 {O(n)}
t₀: 1 {O(1)}
t₂: 3⋅X₄+3 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: 3⋅X₄+3 {O(n)}
t₇: 1 {O(1)}
t₆: X₄ {O(n)}

Costbounds

Overall costbound: 8⋅X₄+10 {O(n)}
t₀: 1 {O(1)}
t₂: 3⋅X₄+3 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: 3⋅X₄+3 {O(n)}
t₇: 1 {O(1)}
t₆: X₄ {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₄ {O(n)}
t₂, X₁: X₄ {O(n)}
t₂, X₂: 1 {O(1)}
t₂, X₃: 0 {O(1)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: X₄ {O(n)}
t₃, X₁: 2⋅X₄ {O(n)}
t₃, X₂: 0 {O(1)}
t₃, X₃: 1 {O(1)}
t₃, X₄: X₄ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 1 {O(1)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₄, X₀: 2⋅X₄ {O(n)}
t₄, X₁: X₄ {O(n)}
t₄, X₂: 1 {O(1)}
t₄, X₃: 1 {O(1)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₄ {O(n)}
t₅, X₁: 2⋅X₄ {O(n)}
t₅, X₂: 1 {O(1)}
t₅, X₃: 1 {O(1)}
t₅, X₄: X₄ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: 2⋅X₄ {O(n)}
t₇, X₂: 0 {O(1)}
t₇, X₃: 1 {O(1)}
t₇, X₄: X₄ {O(n)}
t₆, X₀: 2⋅X₄ {O(n)}
t₆, X₁: X₄ {O(n)}
t₆, X₂: 1 {O(1)}
t₆, X₃: 1 {O(1)}
t₆, X₄: X₄ {O(n)}