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

Preprocessing

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l6

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₀ for location l7

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

Found invariant 0 ≤ X₀ for location l1

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

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

MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, 0, X₂) :|: X₀+1 ≤ X₂ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l5 [X₂-X₁ ]
l1 [X₂-X₀ ]
l6 [X₂-X₀-1 ]
l3 [X₂-X₀-1 ]

MPRF for transition t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l5 [X₂-X₀-1 ]
l1 [X₂-X₀ ]
l6 [X₂-X₀ ]
l3 [X₂-X₀ ]

MPRF for transition t₇: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂ {O(n)}

MPRF:

l5 [X₁+2⋅X₂-2⋅X₀-1 ]
l1 [2⋅X₂-X₀ ]
l6 [2⋅X₂-X₀ ]
l3 [2⋅X₂-X₀ ]

MPRF for transition t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂⋅X₂+3⋅X₂+1 {O(n^2)}

MPRF:

l1 [X₂+1 ]
l5 [X₂-X₁ ]
l6 [X₂-X₁ ]
l3 [X₂+1-X₁ ]

MPRF for transition t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁+1, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂⋅X₂+X₂ {O(n^2)}

MPRF:

l1 [X₂ ]
l5 [X₂-X₁ ]
l6 [X₂-X₁ ]
l3 [X₂-X₁ ]

Analysing control-flow refined program

Cut unsatisfiable transition t₅: l3→l5

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l6___3

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___1

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₀ for location l7

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

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___2

Found invariant 0 ≤ X₀ for location l1

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

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

knowledge_propagation leads to new time bound X₂ {O(n)} for transition t₆₁: l3(X₀, X₁, X₂) → n_l6___3(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

knowledge_propagation leads to new time bound X₂ {O(n)} for transition t₆₃: n_l6___3(X₀, X₁, X₂) → n_l3___2(X₀, X₁+1, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₆₀: n_l3___2(X₀, X₁, X₂) → n_l6___1(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}

MPRF:

l3 [-1 ]
n_l6___3 [-1 ]
l1 [-1 ]
l5 [X₀-X₁ ]
n_l6___1 [X₀-X₁ ]
n_l3___2 [X₀+1-X₁ ]

MPRF for transition t₆₇: n_l3___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l3 [X₂-X₀ ]
l1 [X₂-X₀ ]
l5 [X₂-X₀-1 ]
n_l6___1 [X₂-X₀ ]
n_l6___3 [X₂-X₀ ]
n_l3___2 [X₂-X₀ ]

MPRF for transition t₆₂: n_l6___1(X₀, X₁, X₂) → n_l3___2(X₀, X₁+1, X₂) :|: 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₂⋅X₂+2⋅X₂ {O(n^2)}

MPRF:

l3 [0 ]
n_l6___3 [0 ]
l1 [0 ]
l5 [X₀+1-X₁ ]
n_l6___1 [X₀+1-X₁ ]
n_l3___2 [X₀+1-X₁ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:4⋅X₂⋅X₂+8⋅X₂+5 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂⋅X₂+3⋅X₂+1 {O(n^2)}
t₅: X₂ {O(n)}
t₈: 1 {O(1)}
t₇: 2⋅X₂ {O(n)}
t₆: 2⋅X₂⋅X₂+X₂ {O(n^2)}

Costbounds

Overall costbound: 4⋅X₂⋅X₂+8⋅X₂+5 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂⋅X₂+3⋅X₂+1 {O(n^2)}
t₅: X₂ {O(n)}
t₈: 1 {O(1)}
t₇: 2⋅X₂ {O(n)}
t₆: 2⋅X₂⋅X₂+X₂ {O(n^2)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₂ {O(n)}
t₂, X₁: 0 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 2⋅X₂ {O(n)}
t₃, X₁: 2⋅X₂⋅X₂+X₁+X₂ {O(n^2)}
t₃, X₂: 2⋅X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: 2⋅X₂ {O(n)}
t₄, X₁: 2⋅X₂⋅X₂+X₂ {O(n^2)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: 2⋅X₂ {O(n)}
t₅, X₁: 2⋅X₂⋅X₂+X₂ {O(n^2)}
t₅, X₂: X₂ {O(n)}
t₈, X₀: 2⋅X₂ {O(n)}
t₈, X₁: 2⋅X₂⋅X₂+X₁+X₂ {O(n^2)}
t₈, X₂: 2⋅X₂ {O(n)}
t₇, X₀: 2⋅X₂ {O(n)}
t₇, X₁: 2⋅X₂⋅X₂+X₂ {O(n^2)}
t₇, X₂: X₂ {O(n)}
t₆, X₀: 2⋅X₂ {O(n)}
t₆, X₁: 2⋅X₂⋅X₂+X₂ {O(n^2)}
t₆, X₂: X₂ {O(n)}