Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8
Transitions:
t₀: l0(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ < 0
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 0 ≤ X₁
t₁₁: l2(X₀, X₁, X₂) → l8(X₀, X₁, X₂)
t₁: l3(X₀, X₁, X₂) → l1(X₀, X₀, X₂) :|: 0 ≤ X₀
t₂: l3(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ < 0
t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1
t₅: l4(X₀, X₁, X₂) → l6(X₀, X₁, 1) :|: 1 < X₁
t₁₀: l5(X₀, X₁, X₂) → l1(X₀, X₁-1, X₂)
t₈: l6(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₂
t₇: l6(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ < X₁
t₉: l7(X₀, X₁, X₂) → l6(X₀, X₁, 2⋅X₂)
Preprocessing
Found invariant 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l6
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l7
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5
Found invariant X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location l1
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l4
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8
Transitions:
t₀: l0(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ < 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀
t₁₁: l2(X₀, X₁, X₂) → l8(X₀, X₁, X₂)
t₁: l3(X₀, X₁, X₂) → l1(X₀, X₀, X₂) :|: 0 ≤ X₀
t₂: l3(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ < 0
t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l4(X₀, X₁, X₂) → l6(X₀, X₁, 1) :|: 1 < X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₀: l5(X₀, X₁, X₂) → l1(X₀, X₁-1, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₈: l6(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₇: l6(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ < X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₉: l7(X₀, X₁, X₂) → l6(X₀, X₁, 2⋅X₂) :|: 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀
MPRF for transition t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l4 [X₁ ]
l1 [X₁+1 ]
l5 [X₁ ]
l7 [X₁ ]
l6 [X₁ ]
MPRF for transition t₅: l4(X₀, X₁, X₂) → l6(X₀, X₁, 1) :|: 1 < X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l4 [X₁+1 ]
l1 [X₁+1 ]
l5 [X₁ ]
l7 [X₁ ]
l6 [X₁ ]
MPRF for transition t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l4 [X₁+1 ]
l1 [X₁+1 ]
l5 [X₁ ]
l7 [X₁ ]
l6 [X₁ ]
MPRF for transition t₁₀: l5(X₀, X₁, X₂) → l1(X₀, X₁-1, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l4 [X₁+1 ]
l1 [X₁+1 ]
l5 [X₁+1 ]
l7 [X₁+1 ]
l6 [X₁+1 ]
MPRF for transition t₈: l6(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l4 [X₁+1 ]
l1 [X₁+1 ]
l5 [X₁ ]
l7 [X₁+1 ]
l6 [X₁+1 ]
MPRF for transition t₇: l6(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ < X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+2⋅X₀ {O(n^2)}
MPRF:
l4 [X₀ ]
l5 [X₀ ]
l1 [X₀ ]
l7 [X₀-2⋅X₂ ]
l6 [X₀-X₂ ]
MPRF for transition t₉: l7(X₀, X₁, X₂) → l6(X₀, X₁, 2⋅X₂) :|: 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+6⋅X₀+2 {O(n^2)}
MPRF:
l4 [X₀+X₁ ]
l5 [X₀+X₁ ]
l1 [X₀+X₁ ]
l7 [X₀+X₁-2⋅X₂-1 ]
l6 [X₀+X₁-X₂-2 ]
Analysing control-flow refined program
Cut unsatisfiable transition t₈: l6→l5
Found invariant 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l6___2
Found invariant X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l6
Found invariant X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l7___3
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5
Found invariant X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location l1
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l4
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 5 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location n_l7___1
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₈₀: l6(X₀, X₁, X₂) → n_l7___3(X₀, X₁, X₂) :|: X₂ < X₁ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ X₂ < X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₈₂: n_l7___3(X₀, X₁, X₂) → n_l6___2(X₀, X₁, 2⋅X₂) :|: 2 ≤ X₁ ∧ X₂ < X₁ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀
MPRF for transition t₇₉: n_l6___2(X₀, X₁, X₂) → n_l7___1(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₂ ∧ 2+X₂ ≤ 2⋅X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ X₂ < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀ {O(n^2)}
MPRF:
l1 [2⋅X₀ ]
l4 [2⋅X₀ ]
l5 [0 ]
l6 [2⋅X₀ ]
n_l7___1 [2⋅X₁-2⋅X₂-1 ]
n_l7___3 [2⋅X₀ ]
n_l6___2 [2⋅X₁-X₂-1 ]
MPRF for transition t₈₆: n_l6___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l4 [X₁-1 ]
l1 [X₁-1 ]
l6 [X₁-1 ]
l5 [X₁-2 ]
n_l7___1 [X₁-1 ]
n_l7___3 [X₁-X₂ ]
n_l6___2 [X₁-1 ]
MPRF for transition t₈₁: n_l7___1(X₀, X₁, X₂) → n_l6___2(X₀, X₁, 2⋅X₂) :|: X₁ ≤ X₀ ∧ X₂ < X₁ ∧ 2 ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 5 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 3 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+2⋅X₀ {O(n^2)}
MPRF:
l1 [X₀ ]
l4 [X₀ ]
l5 [X₀ ]
l6 [X₀ ]
n_l7___1 [X₀-X₂ ]
n_l7___3 [X₀ ]
n_l6___2 [X₀-X₂ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:3⋅X₀⋅X₀+13⋅X₀+12 {O(n^2)}
t₀: 1 {O(1)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₁₀: X₀+1 {O(n)}
t₇: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₈: X₀+1 {O(n)}
t₉: 2⋅X₀⋅X₀+6⋅X₀+2 {O(n^2)}
Costbounds
Overall costbound: 3⋅X₀⋅X₀+13⋅X₀+12 {O(n^2)}
t₀: 1 {O(1)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₁₀: X₀+1 {O(n)}
t₇: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₈: X₀+1 {O(n)}
t₉: 2⋅X₀⋅X₀+6⋅X₀+2 {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₀+2 {O(n)}
t₃, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2)+X₂ {O(EXP)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: 1 {O(1)}
t₄, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2)+X₂ {O(EXP)}
t₁₁, X₀: 2⋅X₀ {O(n)}
t₁₁, X₁: X₁+1 {O(n)}
t₁₁, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2)+2⋅X₂ {O(EXP)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀+2 {O(n)}
t₅, X₂: 1 {O(1)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: 1 {O(1)}
t₆, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2)+X₂ {O(EXP)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₀+2 {O(n)}
t₁₀, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2)+X₂ {O(EXP)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀+2 {O(n)}
t₇, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2) {O(EXP)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₀+2 {O(n)}
t₈, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2) {O(EXP)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀+2 {O(n)}
t₉, X₂: 2^(2⋅X₀⋅X₀+6⋅X₀+2) {O(EXP)}