Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, 0, X₂, X₃, X₄) :|: X₀ < X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₁, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₀ < X₁
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₀
t₈: l4(X₀, X₁, X₂, X₃, X₄) → l7(X₀, X₁, X₂, X₃, X₄)
t₇: l5(X₀, X₁, X₂, X₃, X₄) → l1(X₀+1, X₁, X₂, X₃, X₄)
t₆: l6(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁+1, X₂, X₃, X₄)

Preprocessing

Eliminate variables {X₃} that do not contribute to the problem

Found invariant 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l6

Found invariant X₄ ≤ X₀ ∧ X₂ ≤ X₀ for location l7

Found invariant 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₄ ≤ X₀ ∧ X₂ ≤ X₀ for location l4

Found invariant 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₈: l0(X₀, X₁, X₂, X₄) → l2(X₀, X₁, X₂, X₄)
t₁₉: l1(X₀, X₁, X₂, X₄) → l3(X₀, 0, X₂, X₄) :|: X₀ < X₄ ∧ X₂ ≤ X₀
t₂₀: l1(X₀, X₁, X₂, X₄) → l4(X₀, X₁, X₂, X₄) :|: X₄ ≤ X₀ ∧ X₂ ≤ X₀
t₂₁: l2(X₀, X₁, X₂, X₄) → l1(X₂, X₁, X₂, X₄)
t₂₃: l3(X₀, X₁, X₂, X₄) → l5(X₀, X₁, X₂, X₄) :|: X₀ < X₁ ∧ 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₂: l3(X₀, X₁, X₂, X₄) → l6(X₀, X₁, X₂, X₄) :|: X₁ ≤ X₀ ∧ 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₄: l4(X₀, X₁, X₂, X₄) → l7(X₀, X₁, X₂, X₄) :|: X₄ ≤ X₀ ∧ X₂ ≤ X₀
t₂₅: l5(X₀, X₁, X₂, X₄) → l1(X₀+1, X₁, X₂, X₄) :|: 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁
t₂₆: l6(X₀, X₁, X₂, X₄) → l3(X₀, X₁+1, X₂, X₄) :|: 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₁₉: l1(X₀, X₁, X₂, X₄) → l3(X₀, 0, X₂, X₄) :|: X₀ < X₄ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₄ {O(n)}

MPRF:

l5 [X₄-X₀-1 ]
l1 [X₄-X₀ ]
l6 [X₄-X₀-1 ]
l3 [X₄-X₀-1 ]

MPRF for transition t₂₃: l3(X₀, X₁, X₂, X₄) → l5(X₀, X₁, X₂, X₄) :|: X₀ < X₁ ∧ 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+X₄ {O(n)}

MPRF:

l5 [X₄-X₀-1 ]
l1 [X₄-X₀ ]
l6 [X₄-X₀ ]
l3 [X₄-X₀ ]

MPRF for transition t₂₅: l5(X₀, X₁, X₂, X₄) → l1(X₀+1, X₁, X₂, X₄) :|: 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ of depth 1:

new bound:

X₂+X₄ {O(n)}

MPRF:

l5 [X₄-X₀ ]
l1 [X₄-X₀ ]
l6 [X₄-X₀ ]
l3 [X₄-X₀ ]

MPRF for transition t₂₂: l3(X₀, X₁, X₂, X₄) → l6(X₀, X₁, X₂, X₄) :|: X₁ ≤ X₀ ∧ 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

2⋅X₂⋅X₂+3⋅X₂⋅X₄+X₄⋅X₄+2⋅X₂+X₄+1 {O(n^2)}

MPRF:

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

MPRF for transition t₂₆: l6(X₀, X₁, X₂, X₄) → l3(X₀, X₁+1, X₂, X₄) :|: 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}

MPRF:

l1 [X₄ ]
l5 [X₄-X₁ ]
l6 [X₄-X₁ ]
l3 [X₄-X₁ ]

Analysing control-flow refined program

Found invariant 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l6___3

Found invariant 2 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 3 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 3 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___1

Found invariant X₄ ≤ X₀ ∧ X₂ ≤ X₀ for location l7

Found invariant 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l5

Found invariant 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___2

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₄ ≤ X₀ ∧ X₂ ≤ X₀ for location l4

Found invariant 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l3

knowledge_propagation leads to new time bound X₂+X₄ {O(n)} for transition t₆₁: l3(X₀, X₁, X₂, X₄) → n_l6___3(X₀, X₁, X₂, X₄) :|: X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁

knowledge_propagation leads to new time bound X₂+X₄ {O(n)} for transition t₆₃: n_l6___3(X₀, X₁, X₂, X₄) → n_l3___2(X₀, X₁+1, X₂, X₄) :|: X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₆₀: n_l3___2(X₀, X₁, X₂, X₄) → n_l6___1(X₀, X₁, X₂, X₄) :|: X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂⋅X₂+2⋅X₄⋅X₄+4⋅X₂⋅X₄+2⋅X₂+3⋅X₄ {O(n^2)}

MPRF:

l3 [X₄ ]
n_l6___3 [X₄ ]
l1 [X₄ ]
l5 [X₄ ]
n_l6___1 [X₀+X₄-X₁ ]
n_l3___2 [X₀+X₄+1-X₁ ]

MPRF for transition t₆₇: n_l3___2(X₀, X₁, X₂, X₄) → l5(X₀, X₁, X₂, X₄) :|: X₀ < X₁ ∧ 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂+X₄+1 {O(n)}

MPRF:

l3 [X₄+1-X₀ ]
l1 [X₄+1-X₀ ]
l5 [X₄-X₀ ]
n_l6___1 [X₄+1-X₀ ]
n_l6___3 [X₄+1-X₀ ]
n_l3___2 [X₄+1-X₀ ]

MPRF for transition t₆₂: n_l6___1(X₀, X₁, X₂, X₄) → n_l3___2(X₀, X₁+1, X₂, X₄) :|: X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₄ ∧ 2 ≤ X₄ ∧ 1+X₂ ≤ X₄ ∧ 3 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 3 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₄⋅X₄+3⋅X₂⋅X₂+5⋅X₂⋅X₄+3⋅X₂+3⋅X₄ {O(n^2)}

MPRF:

l3 [X₄-X₂ ]
n_l6___3 [X₄-X₂ ]
l1 [X₄-X₂ ]
l5 [X₄-X₂ ]
n_l6___1 [X₀+X₄+1-X₁-X₂ ]
n_l3___2 [X₀+X₄+1-X₁-X₂ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+2⋅X₄⋅X₄+4⋅X₂⋅X₄+5⋅X₂+5⋅X₄+5 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₂+X₄ {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₂⋅X₂+3⋅X₂⋅X₄+X₄⋅X₄+2⋅X₂+X₄+1 {O(n^2)}
t₂₃: X₂+X₄ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+X₄ {O(n)}
t₂₆: X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+2⋅X₄⋅X₄+4⋅X₂⋅X₄+5⋅X₂+5⋅X₄+5 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₂+X₄ {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₂⋅X₂+3⋅X₂⋅X₄+X₄⋅X₄+2⋅X₂+X₄+1 {O(n^2)}
t₂₃: X₂+X₄ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+X₄ {O(n)}
t₂₆: X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 2⋅X₂+X₄ {O(n)}
t₁₉, X₁: 0 {O(1)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₂₀, X₀: 3⋅X₂+X₄ {O(n)}
t₂₀, X₁: X₂⋅X₄+X₄⋅X₄+X₁+X₄ {O(n^2)}
t₂₀, X₂: 2⋅X₂ {O(n)}
t₂₀, X₄: 2⋅X₄ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: 2⋅X₂+X₄ {O(n)}
t₂₂, X₁: X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₃, X₀: 2⋅X₂+X₄ {O(n)}
t₂₃, X₁: X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₄, X₀: 3⋅X₂+X₄ {O(n)}
t₂₄, X₁: X₂⋅X₄+X₄⋅X₄+X₁+X₄ {O(n^2)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₅, X₀: 2⋅X₂+X₄ {O(n)}
t₂₅, X₁: X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₆, X₀: 2⋅X₂+X₄ {O(n)}
t₂₆, X₁: X₂⋅X₄+X₄⋅X₄+X₄ {O(n^2)}
t₂₆, X₂: X₂ {O(n)}
t₂₆, X₄: X₄ {O(n)}