Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l4(X₀, X₁, X₁, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₃, X₆, X₂, X₃, X₄, X₅, X₆)
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₀-1, (X₀)²+X₁+1-2⋅X₀, X₂, X₃, X₄, X₅, X₆)
t₅: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 0 < X₂
t₆: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₂ ≤ 0
t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l4(X₀, X₁, X₂-1, X₃, X₄, X₅, X₆)
t₈: l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l7(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
Preprocessing
Eliminate variables {X₄,X₅} that do not contribute to the problem
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location l6
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l5
Found invariant X₀ ≤ X₃ for location l1
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₇: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₁₈: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀ ∧ X₀ ≤ X₃
t₁₉: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₁, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₂₀: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₂₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, (X₀)²+X₁+1-2⋅X₀, X₂, X₃, X₄) :|: 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀
t₂₂: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₂ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0
t₂₃: l4(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0
t₂₄: l5(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂-1, X₃, X₄) :|: X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0
t₂₅: l6(X₀, X₁, X₂, X₃, X₄) → l7(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0
TWN. Size Bound: t₂₁: l3→l1 for X₁
cycle: [t₂₁: l3→l1; t₁₈: l1→l3]
loop: (1 < X₀,(X₀,X₁) -> (X₀-1,1+(X₀)²+X₁-2⋅X₀)
closed-form: X₁ + [[n != 0]] * (1+(X₀)²-2⋅X₀) * n^1 + [[n != 0, n != 1]] * 1/3 * n^3 + [[n != 0, n != 1]] * 1/2-X₀ * n^2 + [[n != 0, n != 1]] * X₀-5/6 * n^1
runtime bound: X₀+1 {O(n)}
TWN Size Bound - Lifting for t₂₁: l3→l1 and X₁: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+X₄+4 {O(n^3)}
MPRF for transition t₁₈: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀ ∧ X₀ ≤ X₃ of depth 1:
new bound:
X₃ {O(n)}
MPRF for transition t₂₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, Temp_Int₂₃₇+X₁+Temp_Int₂₃₈-2⋅X₀, X₂, X₃, X₄) :|: 0 < Temp_Int₂₃₇ ∧ X₀ ≤ Temp_Int₂₃₇ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃ {O(n)}
MPRF for transition t₂₂: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₂ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
MPRF for transition t₂₄: l5(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂-1, X₃, X₄) :|: X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
Chain transitions t₂₄: l5→l4 and t₂₃: l4→l6 to t₅₇: l5→l6
Chain transitions t₁₉: l1→l4 and t₂₃: l4→l6 to t₅₈: l1→l6
Chain transitions t₁₉: l1→l4 and t₂₂: l4→l5 to t₅₉: l1→l5
Chain transitions t₂₄: l5→l4 and t₂₂: l4→l5 to t₆₀: l5→l5
Analysing control-flow refined program
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location l6
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l5
Found invariant X₀ ≤ X₃ for location l1
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location l3
MPRF for transition t₆₀: l5(X₀, X₁, X₂, X₃, X₄) -{2}> l5(X₀, X₁, X₂-1, X₃, X₄) :|: 1 < X₂ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁+1 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Found invariant X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l5___1
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location l6
Found invariant X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l4___2
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₃ for location l1
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location l3
Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l5___3
MPRF for transition t₁₁₁: n_l4___2(X₀, X₁, X₂, X₃, X₄) → n_l5___1(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 0 < X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+5 {O(n^3)}
MPRF for transition t₁₁₃: n_l5___1(X₀, X₁, X₂, X₃, X₄) → n_l4___2(X₀, X₁, X₂-1, X₃, X₄) :|: 1+X₂ ≤ X₁ ∧ 0 < X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:6⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+24⋅X₃+4⋅X₄+13 {O(n^3)}
t₁₇: 1 {O(1)}
t₁₈: X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₃ {O(n)}
t₂₂: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₃: 1 {O(1)}
t₂₄: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₅: 1 {O(1)}
Costbounds
Overall costbound: 6⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+24⋅X₃+4⋅X₄+13 {O(n^3)}
t₁₇: 1 {O(1)}
t₁₈: X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₃ {O(n)}
t₂₂: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₃: 1 {O(1)}
t₂₄: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₅: 1 {O(1)}
Sizebounds
t₁₇, X₀: X₀ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: X₃ {O(n)}
t₁₈, X₁: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 2⋅X₃ {O(n)}
t₁₉, X₁: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₁₉, X₂: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: X₃ {O(n)}
t₂₀, X₁: X₄ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: X₃ {O(n)}
t₂₁, X₁: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+X₄+4 {O(n^3)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: 2⋅X₃ {O(n)}
t₂₂, X₁: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₂, X₂: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₂, X₃: 2⋅X₃ {O(n)}
t₂₂, X₄: 2⋅X₄ {O(n)}
t₂₃, X₀: 4⋅X₃ {O(n)}
t₂₃, X₁: 6⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+22⋅X₃+4⋅X₄+8 {O(n^3)}
t₂₃, X₂: 6⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+22⋅X₃+4⋅X₄+8 {O(n^3)}
t₂₃, X₃: 4⋅X₃ {O(n)}
t₂₃, X₄: 4⋅X₄ {O(n)}
t₂₄, X₀: 2⋅X₃ {O(n)}
t₂₄, X₁: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₄, X₂: 3⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+11⋅X₃+2⋅X₄+4 {O(n^3)}
t₂₄, X₃: 2⋅X₃ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₅, X₀: 4⋅X₃ {O(n)}
t₂₅, X₁: 6⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+22⋅X₃+4⋅X₄+8 {O(n^3)}
t₂₅, X₂: 6⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+22⋅X₃+4⋅X₄+8 {O(n^3)}
t₂₅, X₃: 4⋅X₃ {O(n)}
t₂₅, X₄: 4⋅X₄ {O(n)}