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 ]
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
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:
4⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}
MPRF:
l1 [X₀+1 ]
l5 [X₀-X₁ ]
l6 [X₀+1-X₁ ]
l3 [X₀+1-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 t₇₅: 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 t₇₇: 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 t₇₄: 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 t₈₁: 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 t₇₆: 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:6⋅X₂⋅X₂+9⋅X₂+6 {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: 4⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}
Costbounds
Overall costbound: 6⋅X₂⋅X₂+9⋅X₂+6 {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: 4⋅X₂⋅X₂+2⋅X₂+1 {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: 4⋅X₂⋅X₂+2⋅X₂+X₁+1 {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: 4⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: 2⋅X₂ {O(n)}
t₅, X₁: 4⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}
t₅, X₂: X₂ {O(n)}
t₈, X₀: 2⋅X₂ {O(n)}
t₈, X₁: 4⋅X₂⋅X₂+2⋅X₂+X₁+1 {O(n^2)}
t₈, X₂: 2⋅X₂ {O(n)}
t₇, X₀: 2⋅X₂ {O(n)}
t₇, X₁: 4⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}
t₇, X₂: X₂ {O(n)}
t₆, X₀: 2⋅X₂ {O(n)}
t₆, X₁: 4⋅X₂⋅X₂+2⋅X₂+1 {O(n^2)}
t₆, X₂: X₂ {O(n)}