Initial Problem

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

Preprocessing

Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5

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

Problem after Preprocessing

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

MPRF for transition t₂: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 0 of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [-X₀-1 ]
l1 [-X₀ ]

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

new bound:

X₁+1 {O(n)}

MPRF:

l3 [1-X₀ ]
l1 [1-X₀ ]

MPRF for transition t₃: l1(X₀, X₁) → l3(X₀, X₁) :|: 0 < X₀ of depth 1:

new bound:

X₁⋅X₁+5⋅X₁+5 {O(n^2)}

MPRF:

l3 [X₀+1 ]
l1 [X₀+2 ]

MPRF for transition t₅: l3(X₀, X₁) → l1(X₀-1, X₁) :|: 0 < X₀ of depth 1:

new bound:

X₁⋅X₁+4⋅X₁+3 {O(n^2)}

MPRF:

l3 [X₀+1 ]
l1 [X₀+1 ]

Analysing control-flow refined program

Cut unsatisfiable transition t₇₅: n_l1___2→n_l3___4

Cut unsatisfiable transition t₇₆: n_l1___5→n_l3___3

Cut unreachable locations [n_l3___3] from the program graph

Found invariant 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0 for location n_l3___4

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___6

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___2

Found invariant X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5

Found invariant 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location n_l1___5

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

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

Found invariant 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___1

Found invariant 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location n_l3___7

MPRF for transition t₇₄: n_l1___2(X₀, X₁) → n_l3___1(X₀, X₁) :|: 0 < X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l3___1 [X₀ ]
n_l1___2 [X₀+1 ]

MPRF for transition t₈₀: n_l3___1(X₀, X₁) → n_l1___2(X₀-1, X₁) :|: 0 < X₀ ∧ 0 < X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

n_l3___1 [X₀ ]
n_l1___2 [X₀ ]

MPRF for transition t₇₇: n_l1___5(X₀, X₁) → n_l3___4(X₀, X₁) :|: X₀ ≤ 1 ∧ X₀ < 0 ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l3___4 [-X₀ ]
n_l1___5 [1-X₀ ]

MPRF for transition t₈₂: n_l3___4(X₀, X₁) → n_l1___5(X₀+1, X₁) :|: X₀ < 0 ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l3___4 [1-X₀ ]
n_l1___5 [1-X₀ ]

CFR: Improvement to new bound with the following program:

new bound:

4⋅X₁+3 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l4, l5, n_l1___2, n_l1___5, n_l3___1, n_l3___4, n_l3___6, n_l3___7
Transitions:
t₀: l0(X₀, X₁) → l2(X₀, X₁)
t₄: l1(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₇₈: l1(X₀, X₁) → n_l3___6(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₇₉: l1(X₀, X₁) → n_l3___7(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁: l2(X₀, X₁) → l1(X₁, X₁)
t₇: l4(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₉₅: n_l1___2(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₇₄: n_l1___2(X₀, X₁) → n_l3___1(X₀, X₁) :|: 0 < X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₉₆: n_l1___5(X₀, X₁) → l4(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0
t₇₇: n_l1___5(X₀, X₁) → n_l3___4(X₀, X₁) :|: X₀ ≤ 1 ∧ X₀ < 0 ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0
t₈₀: n_l3___1(X₀, X₁) → n_l1___2(X₀-1, X₁) :|: 0 < X₀ ∧ 0 < X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀
t₈₂: n_l3___4(X₀, X₁) → n_l1___5(X₀+1, X₁) :|: X₀ < 0 ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0
t₈₃: n_l3___6(X₀, X₁) → n_l1___2(X₀-1, X₁) :|: 0 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₈₄: n_l3___7(X₀, X₁) → n_l1___5(X₀+1, X₁) :|: X₀ < 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ 0

All Bounds

Timebounds

Overall timebound:4⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₄: 1 {O(1)}
t₇₈: 1 {O(1)}
t₇₉: 1 {O(1)}
t₁: 1 {O(1)}
t₇: 1 {O(1)}
t₇₄: X₁+1 {O(n)}
t₉₅: 1 {O(1)}
t₇₇: X₁+1 {O(n)}
t₉₆: 1 {O(1)}
t₈₀: X₁ {O(n)}
t₈₂: X₁+1 {O(n)}
t₈₃: 1 {O(1)}
t₈₄: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₄: 1 {O(1)}
t₇₈: 1 {O(1)}
t₇₉: 1 {O(1)}
t₁: 1 {O(1)}
t₇: 1 {O(1)}
t₇₄: X₁+1 {O(n)}
t₉₅: 1 {O(1)}
t₇₇: X₁+1 {O(n)}
t₉₆: 1 {O(1)}
t₈₀: X₁ {O(n)}
t₈₂: X₁+1 {O(n)}
t₈₃: 1 {O(1)}
t₈₄: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₄, X₀: 0 {O(1)}
t₄, X₁: X₁ {O(n)}
t₇₈, X₀: X₁ {O(n)}
t₇₈, X₁: X₁ {O(n)}
t₇₉, X₀: X₁ {O(n)}
t₇₉, X₁: X₁ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₇, X₀: 0 {O(1)}
t₇, X₁: 5⋅X₁ {O(n)}
t₇₄, X₀: X₁ {O(n)}
t₇₄, X₁: X₁ {O(n)}
t₉₅, X₀: 0 {O(1)}
t₉₅, X₁: 2⋅X₁ {O(n)}
t₇₇, X₀: X₁ {O(n)}
t₇₇, X₁: X₁ {O(n)}
t₉₆, X₀: 0 {O(1)}
t₉₆, X₁: 2⋅X₁ {O(n)}
t₈₀, X₀: X₁ {O(n)}
t₈₀, X₁: X₁ {O(n)}
t₈₂, X₀: X₁ {O(n)}
t₈₂, X₁: X₁ {O(n)}
t₈₃, X₀: X₁ {O(n)}
t₈₃, X₁: X₁ {O(n)}
t₈₄, X₀: X₁ {O(n)}
t₈₄, X₁: X₁ {O(n)}