Initial Problem

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

Preprocessing

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

Problem after Preprocessing

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

MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l2 [X₀+1 ]
l1 [X₀+1 ]

MPRF for transition t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l2 [X₀+1 ]
l1 [X₀ ]

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

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}

TWN-Loops:

entry: t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 0 ≤ X₀
results in twn-loop: twn:Inv: [X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀,X₁+1) :|: 1+X₁ ≤ X₀
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1

Termination: true
Formula:

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

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

relevant size-bounds w.r.t. t₀:
X₀: X₀+1 {O(n)}
Runtime-bound of t₀: X₀+1 {O(n)}
Results in: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}

2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}

Analysing control-flow refined program

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

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₃₇: l1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₃₆: n_l1___1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 0 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+X₀ {O(n^2)}

MPRF:

l1 [X₀ ]
n_l1___1 [X₀+1-X₁ ]
l2 [X₀-X₁ ]

MPRF for transition t₄₀: n_l1___1(X₀, X₁) → l2(X₀-1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l1 [X₀+1 ]
n_l1___1 [X₀+1 ]
l2 [X₀+1 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+10⋅X₀+9 {O(n^2)}
t₃: 1 {O(1)}
t₁: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
t₂: X₀+1 {O(n)}
t₀: X₀+1 {O(n)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+10⋅X₀+9 {O(n^2)}
t₃: 1 {O(1)}
t₁: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
t₂: X₀+1 {O(n)}
t₀: X₀+1 {O(n)}

Sizebounds

t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₁, X₀: X₀+1 {O(n)}
t₁, X₁: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
t₂, X₀: X₀+1 {O(n)}
t₂, X₁: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
t₀, X₀: X₀+1 {O(n)}
t₀, X₁: 0 {O(1)}