Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁ < X₀
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: X₁ < X₀
t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁
t₇: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)
Preprocessing
Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5
Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁ < X₀
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: X₁ < X₀
t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁
t₇: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁ of depth 1:
new bound:
X₁+X₂ {O(n)}
MPRF for transition t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁ of depth 1:
new bound:
X₁+X₂+1 {O(n)}
MPRF for transition t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁ < X₀ of depth 1:
new bound:
2⋅X₁⋅X₁+2⋅X₂⋅X₂+4⋅X₁⋅X₂+6⋅X₁+6⋅X₂+5 {O(n^2)}
MPRF for transition t₅: l3(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: X₁ < X₀ of depth 1:
new bound:
2⋅X₁⋅X₁+2⋅X₂⋅X₂+4⋅X₁⋅X₂+5⋅X₁+5⋅X₂+3 {O(n^2)}
Chain transitions t₆: l3→l1 and t₄: l1→l4 to t₇₂: l3→l4
Chain transitions t₅: l3→l1 and t₄: l1→l4 to t₇₃: l3→l4
Chain transitions t₅: l3→l1 and t₃: l1→l3 to t₇₄: l3→l3
Chain transitions t₆: l3→l1 and t₃: l1→l3 to t₇₅: l3→l3
Chain transitions t₁: l2→l1 and t₃: l1→l3 to t₇₆: l2→l3
Chain transitions t₁: l2→l1 and t₄: l1→l4 to t₇₇: l2→l4
Chain transitions t₁: l2→l1 and t₂: l1→l3 to t₇₈: l2→l3
Chain transitions t₅: l3→l1 and t₂: l1→l3 to t₇₉: l3→l3
Chain transitions t₆: l3→l1 and t₂: l1→l3 to t₈₀: l3→l3
Analysing control-flow refined program
Cut unsatisfiable transition t₇₅: l3→l3
Cut unsatisfiable transition t₇₉: l3→l3
Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5
Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4
MPRF for transition t₇₄: l3(X₀, X₁, X₂) -{2}> l3(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₁+1 < X₀ of depth 1:
new bound:
X₁+X₂ {O(n)}
MPRF for transition t₈₀: l3(X₀, X₁, X₂) -{2}> l3(1+X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1+X₀ < X₁ of depth 1:
new bound:
X₁+X₂+1 {O(n)}
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Cut unsatisfiable transition t₁₈₈: n_l1___2→n_l3___4
Cut unsatisfiable transition t₁₈₉: n_l1___5→n_l3___3
Cut unreachable locations [n_l3___3] from the program graph
Found invariant 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ for location n_l3___4
Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location n_l3___6
Found invariant 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ for location n_l1___2
Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ for location n_l1___5
Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1
Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4
Found invariant 2+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location n_l3___1
Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location n_l3___7
MPRF for transition t₁₈₇: n_l1___2(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ of depth 1:
new bound:
X₁+X₂+2 {O(n)}
MPRF for transition t₁₉₃: n_l3___1(X₀, X₁, X₂) → n_l1___2(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₁ < X₀ ∧ 2+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ of depth 1:
new bound:
X₁+X₂+1 {O(n)}
MPRF for transition t₁₉₀: n_l1___5(X₀, X₁, X₂) → n_l3___4(X₀, X₁, X₂) :|: X₀ ≤ 1+X₁ ∧ X₀ < X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:
new bound:
X₁+X₂+2 {O(n)}
MPRF for transition t₁₉₅: n_l3___4(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ of depth 1:
new bound:
X₁+X₂+2 {O(n)}
CFR did not improve the program. Rolling back
CFR: Improvement to new bound with the following program:
new bound:
4⋅X₁+4⋅X₂+7 {O(n)}
cfr-program:
Start: l0
Program_Vars: X₀, X₁, X₂
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, X₂) → l2(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁₉₁: l1(X₀, X₁, X₂) → n_l3___6(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ < X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁₉₂: l1(X₀, X₁, X₂) → n_l3___7(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₇: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₂₀₈: n_l1___2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀
t₁₈₇: n_l1___2(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀
t₂₀₉: n_l1___5(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁
t₁₉₀: n_l1___5(X₀, X₁, X₂) → n_l3___4(X₀, X₁, X₂) :|: X₀ ≤ 1+X₁ ∧ X₀ < X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁
t₁₉₃: n_l3___1(X₀, X₁, X₂) → n_l1___2(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₁ < X₀ ∧ 2+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀
t₁₉₅: n_l3___4(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₁₉₆: n_l3___6(X₀, X₁, X₂) → n_l1___2(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ < X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀
t₁₉₇: n_l3___7(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁
All Bounds
Timebounds
Overall timebound:4⋅X₁+4⋅X₂+17 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 1 {O(1)}
t₇: 1 {O(1)}
t₁₈₇: X₁+X₂+2 {O(n)}
t₁₉₀: X₁+X₂+2 {O(n)}
t₁₉₁: 1 {O(1)}
t₁₉₂: 1 {O(1)}
t₁₉₃: X₁+X₂+1 {O(n)}
t₁₉₅: X₁+X₂+2 {O(n)}
t₁₉₆: 1 {O(1)}
t₁₉₇: 1 {O(1)}
t₂₀₈: 1 {O(1)}
t₂₀₉: 1 {O(1)}
Costbounds
Overall costbound: 4⋅X₁+4⋅X₂+17 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 1 {O(1)}
t₇: 1 {O(1)}
t₁₈₇: X₁+X₂+2 {O(n)}
t₁₉₀: X₁+X₂+2 {O(n)}
t₁₉₁: 1 {O(1)}
t₁₉₂: 1 {O(1)}
t₁₉₃: X₁+X₂+1 {O(n)}
t₁₉₅: X₁+X₂+2 {O(n)}
t₁₉₆: 1 {O(1)}
t₁₉₇: 1 {O(1)}
t₂₀₈: 1 {O(1)}
t₂₀₉: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₇, X₀: 2⋅X₁+7⋅X₂+7 {O(n)}
t₇, X₁: 5⋅X₁ {O(n)}
t₇, X₂: 5⋅X₂ {O(n)}
t₁₈₇, X₀: 2⋅X₂+X₁+2 {O(n)}
t₁₈₇, X₁: X₁ {O(n)}
t₁₈₇, X₂: X₂ {O(n)}
t₁₉₀, X₀: 2⋅X₂+X₁+3 {O(n)}
t₁₉₀, X₁: X₁ {O(n)}
t₁₉₀, X₂: X₂ {O(n)}
t₁₉₁, X₀: X₂ {O(n)}
t₁₉₁, X₁: X₁ {O(n)}
t₁₉₁, X₂: X₂ {O(n)}
t₁₉₂, X₀: X₂ {O(n)}
t₁₉₂, X₁: X₁ {O(n)}
t₁₉₂, X₂: X₂ {O(n)}
t₁₉₃, X₀: 2⋅X₂+X₁+2 {O(n)}
t₁₉₃, X₁: X₁ {O(n)}
t₁₉₃, X₂: X₂ {O(n)}
t₁₉₅, X₀: 2⋅X₂+X₁+3 {O(n)}
t₁₉₅, X₁: X₁ {O(n)}
t₁₉₅, X₂: X₂ {O(n)}
t₁₉₆, X₀: X₂+1 {O(n)}
t₁₉₆, X₁: X₁ {O(n)}
t₁₉₆, X₂: X₂ {O(n)}
t₁₉₇, X₀: X₂+1 {O(n)}
t₁₉₇, X₁: X₁ {O(n)}
t₁₉₇, X₂: X₂ {O(n)}
t₂₀₈, X₀: 3⋅X₂+X₁+3 {O(n)}
t₂₀₈, X₁: 2⋅X₁ {O(n)}
t₂₀₈, X₂: 2⋅X₂ {O(n)}
t₂₀₉, X₀: 3⋅X₂+X₁+4 {O(n)}
t₂₀₉, X₁: 2⋅X₁ {O(n)}
t₂₀₉, X₂: 2⋅X₂ {O(n)}