Initial Problem

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

Preprocessing

Found invariant 1+X₀ ≤ 0 for location l5

Found invariant 1+X₀ ≤ 0 for location l4

Found invariant 0 ≤ X₀ for location l3

Problem after Preprocessing

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

MPRF for transition t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂+X₃+1 {O(n)}

MPRF:

l3 [2⋅X₀+X₁+1 ]
l1 [2⋅X₀+X₁+1 ]

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

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

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ for location l1

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location l4

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

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 13⋅X₃+26⋅X₂+26 {O(n)}

TWN-Loops:

entry: t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀
results in twn-loop: twn:Inv: [0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+X₁,-X₁,X₂,X₃) :|: 0 ≤ X₀ ∧ X₁ < 0
entry: t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
results in twn-loop: twn:Inv: [0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+X₁,-X₁,X₂,X₃) :|: 0 ≤ X₀ ∧ X₁ < 0
order: [X₁; X₀]
closed-form:
X₁: X₁
X₀: X₀

Termination: true
Formula:

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

relevant size-bounds w.r.t. t₄:
Runtime-bound of t₄: 2⋅X₂+X₃+1 {O(n)}
Results in: 13⋅X₃+26⋅X₂+13 {O(n)}

order: [X₁; X₀; X₂; X₃]
closed-form:
X₁: X₁
X₀: X₀
X₂: X₂
X₃: X₃

Termination: true
Formula:

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

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 13 {O(1)}

13⋅X₃+26⋅X₂+26 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 13⋅X₃+26⋅X₂+26 {O(n)}

relevant size-bounds w.r.t. t₄:
Runtime-bound of t₄: 2⋅X₂+X₃+1 {O(n)}
Results in: 13⋅X₃+26⋅X₂+13 {O(n)}

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 13 {O(1)}

13⋅X₃+26⋅X₂+26 {O(n)}

All Bounds

Timebounds

Overall timebound:27⋅X₃+54⋅X₂+57 {O(n)}
t₀: 1 {O(1)}
t₂: 13⋅X₃+26⋅X₂+26 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂+X₃+1 {O(n)}
t₅: 13⋅X₃+26⋅X₂+26 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 27⋅X₃+54⋅X₂+57 {O(n)}
t₀: 1 {O(1)}
t₂: 13⋅X₃+26⋅X₂+26 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂+X₃+1 {O(n)}
t₅: 13⋅X₃+26⋅X₂+26 {O(n)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₂, X₀: 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+7⋅X₃+9⋅X₂+3 {O(n^2)}
t₂, X₁: 2⋅X₂+2⋅X₃+1 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+10⋅X₂+7⋅X₃+3 {O(n^2)}
t₃, X₁: 2⋅X₂+3⋅X₃+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₄, X₀: 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+7⋅X₃+9⋅X₂+3 {O(n^2)}
t₄, X₁: 2⋅X₂+2⋅X₃+1 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+7⋅X₃+9⋅X₂+3 {O(n^2)}
t₅, X₁: 2⋅X₂+2⋅X₃+1 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+10⋅X₂+7⋅X₃+3 {O(n^2)}
t₆, X₁: 2⋅X₂+3⋅X₃+1 {O(n)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}