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)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₀
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1)
t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂)
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 1) :|: 1 ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₀, X₂)

Preprocessing

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

Found invariant X₁ ≤ 0 for location l7

Found invariant X₁ ≤ 0 for location l5

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ 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)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 1) :|: 1 ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₀, X₂)

MPRF for transition t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 1) :|: 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF for transition t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF for transition t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

TWN: t₄: l1→l2

cycle: [t₄: l1→l2; t₆: l2→l1]
loop: (X₂ ≤ X₀,(X₀,X₂) -> (X₀,X₂+1)
order: [X₀; X₂]
closed-form:
X₀: X₀
X₂: X₂ + [[n != 0]] * n^1

Termination: true
Formula:

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

Stabilization-Threshold for: X₂ ≤ X₀
alphas_abs: X₂+X₀
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₂+2 {O(n)}

TWN - Lifting for t₄: l1→l2 of 2⋅X₀+2⋅X₂+4 {O(n)}

relevant size-bounds w.r.t. t₂:
X₀: X₁ {O(n)}
X₂: 1 {O(1)}
Runtime-bound of t₂: X₀ {O(n)}
Results in: 2⋅X₀⋅X₁+6⋅X₀ {O(n^2)}

TWN: t₆: l2→l1

TWN - Lifting for t₆: l2→l1 of 2⋅X₀+2⋅X₂+4 {O(n)}

relevant size-bounds w.r.t. t₂:
X₀: X₁ {O(n)}
X₂: 1 {O(1)}
Runtime-bound of t₂: X₀ {O(n)}
Results in: 2⋅X₀⋅X₁+6⋅X₀ {O(n^2)}

Chain transitions t₂: l4→l1 and t₅: l1→l3 to t₅₃: l4→l3

Chain transitions t₆: l2→l1 and t₅: l1→l3 to t₅₄: l2→l3

Chain transitions t₆: l2→l1 and t₄: l1→l2 to t₅₅: l2→l2

Chain transitions t₂: l4→l1 and t₄: l1→l2 to t₅₆: l4→l2

Chain transitions t₅₃: l4→l3 and t₇: l3→l4 to t₅₇: l4→l4

Chain transitions t₅₄: l2→l3 and t₇: l3→l4 to t₅₈: l2→l4

Analysing control-flow refined program

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

Found invariant X₁ ≤ 0 for location l7

Found invariant X₁ ≤ 0 for location l5

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

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

MPRF for transition t₅₆: l4(X₀, X₁, X₂) -{2}> l2(X₀, X₁, 1) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 0 ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF for transition t₅₇: l4(X₀, X₁, X₂) -{3}> l4(X₀, X₁-1, 1) :|: 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ 0 ∧ 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF for transition t₅₈: l2(X₀, X₁, X₂) -{3}> l4(X₀, X₁-1, 1+X₂) :|: X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

TWN: t₅₅: l2→l2

cycle: [t₅₅: l2→l2]
loop: (1+X₂ ≤ X₀,(X₀,X₂) -> (X₀,1+X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀
X₂: X₂ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ 1+X₂ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂

Stabilization-Threshold for: 1+X₂ ≤ X₀
alphas_abs: 1+X₂+X₀
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₂+4 {O(n)}

TWN - Lifting for t₅₅: l2→l2 of 2⋅X₀+2⋅X₂+6 {O(n)}

relevant size-bounds w.r.t. t₅₆:
X₀: X₁ {O(n)}
X₂: 1 {O(1)}
Runtime-bound of t₅₆: X₀ {O(n)}
Results in: 2⋅X₀⋅X₁+8⋅X₀ {O(n^2)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___2

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l2___1

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l2___3

Found invariant X₁ ≤ 0 for location l7

Found invariant X₁ ≤ 0 for location l5

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

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

knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₁₂₁: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: X₂ ≤ 1 ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁

knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₁₂₃: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: X₂ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀

MPRF for transition t₁₂₀: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: 2 ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀⋅X₁+3⋅X₀ {O(n^2)}

MPRF for transition t₁₂₂: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: 2 ≤ X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₁+3⋅X₀ {O(n^2)}

MPRF for transition t₁₂₇: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅X₁+15⋅X₀+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₀⋅X₁+6⋅X₀ {O(n^2)}
t₅: X₀ {O(n)}
t₆: 2⋅X₀⋅X₁+6⋅X₀ {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀⋅X₁+15⋅X₀+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₀⋅X₁+6⋅X₀ {O(n^2)}
t₅: X₀ {O(n)}
t₆: 2⋅X₀⋅X₁+6⋅X₀ {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 1 {O(1)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: 2⋅X₀⋅X₁+6⋅X₀+X₂+2 {O(n^2)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: 2⋅X₀⋅X₁+6⋅X₀+1 {O(n^2)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 2⋅X₀⋅X₁+6⋅X₀+2 {O(n^2)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 2⋅X₀⋅X₁+6⋅X₀+1 {O(n^2)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 2⋅X₀⋅X₁+6⋅X₀+2 {O(n^2)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₀ {O(n)}
t₈, X₂: 2⋅X₀⋅X₁+6⋅X₀+X₂+2 {O(n^2)}