Initial Problem
Start: l0
Program_Vars: X₀, X₁
Temp_Vars: C, D
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁) → l8(X₀, X₁)
t₈: l1(X₀, X₁) → l2(X₀, X₁)
t₉: l2(X₀, X₁) → l3(X₀, X₁+1)
t₄: l3(X₀, X₁) → l4(X₀, X₁) :|: 1+X₁ ≤ X₀
t₅: l3(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ X₁
t₆: l4(X₀, X₁) → l1(X₀, X₁) :|: D+1 ≤ C
t₇: l4(X₀, X₁) → l2(X₀, X₁) :|: C ≤ D
t₁₀: l5(X₀, X₁) → l6(X₀-1, X₁)
t₂: l6(X₀, X₁) → l3(X₀, 0) :|: 0 ≤ X₀
t₃: l6(X₀, X₁) → l7(X₀, X₁) :|: X₀+1 ≤ 0
t₁₁: l7(X₀, X₁) → l9(X₀, X₁)
t₁: l8(X₀, X₁) → l6(X₀, X₁)
Preprocessing
Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2
Found invariant 1+X₀ ≤ 0 for location l7
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l5
Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l4
Found invariant 1+X₀ ≤ 0 for location l9
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars: C, D
Locations: l0, l1, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁) → l8(X₀, X₁)
t₈: l1(X₀, X₁) → l2(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₉: l2(X₀, X₁) → l3(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₄: l3(X₀, X₁) → l4(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l3(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₆: l4(X₀, X₁) → l1(X₀, X₁) :|: D+1 ≤ C ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₇: l4(X₀, X₁) → l2(X₀, X₁) :|: C ≤ D ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₁₀: l5(X₀, X₁) → l6(X₀-1, X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂: l6(X₀, X₁) → l3(X₀, 0) :|: 0 ≤ X₀
t₃: l6(X₀, X₁) → l7(X₀, X₁) :|: X₀+1 ≤ 0
t₁₁: l7(X₀, X₁) → l9(X₀, X₁) :|: 1+X₀ ≤ 0
t₁: l8(X₀, X₁) → l6(X₀, X₁)
MPRF for transition t₅: l3(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l1 [X₀+1 ]
l4 [X₀+1 ]
l2 [X₀+1 ]
l5 [X₁ ]
l6 [X₀+1 ]
l3 [X₀+1 ]
MPRF for transition t₁₀: l5(X₀, X₁) → l6(X₀-1, X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l1 [X₀+1 ]
l4 [X₀+1 ]
l2 [X₀+1 ]
l5 [X₁+1 ]
l6 [X₀+1 ]
l3 [X₀+1 ]
MPRF for transition t₂: l6(X₀, X₁) → l3(X₀, 0) :|: 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l1 [X₀ ]
l4 [X₀ ]
l2 [X₀ ]
l5 [X₀ ]
l6 [X₀+1 ]
l3 [X₀ ]
MPRF for transition t₈: l1(X₀, X₁) → l2(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+3⋅X₀+2 {O(n^2)}
MPRF:
l3 [X₀+1-X₁ ]
l1 [X₀+1-X₁ ]
l4 [X₀+1-X₁ ]
l2 [X₀-X₁ ]
l5 [1 ]
l6 [1 ]
MPRF for transition t₉: l2(X₀, X₁) → l3(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
MPRF:
l3 [2⋅X₀-X₁ ]
l1 [2⋅X₀-X₁ ]
l4 [2⋅X₀-X₁ ]
l2 [2⋅X₀-X₁ ]
l5 [2⋅X₀-X₁ ]
l6 [X₀+1 ]
MPRF for transition t₄: l3(X₀, X₁) → l4(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+3⋅X₀+2 {O(n^2)}
MPRF:
l3 [X₀+1-X₁ ]
l1 [X₀-X₁ ]
l4 [X₀-X₁ ]
l2 [X₀-X₁ ]
l5 [X₀+1-X₁ ]
l6 [1 ]
MPRF for transition t₆: l4(X₀, X₁) → l1(X₀, X₁) :|: D+1 ≤ C ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+3⋅X₀+2 {O(n^2)}
MPRF:
l3 [X₀+1-X₁ ]
l1 [X₀-X₁ ]
l4 [X₀+1-X₁ ]
l2 [X₀-X₁ ]
l5 [1 ]
l6 [1 ]
MPRF for transition t₇: l4(X₀, X₁) → l2(X₀, X₁) :|: C ≤ D ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+3⋅X₀+2 {O(n^2)}
MPRF:
l3 [X₀+1-X₁ ]
l1 [X₀-X₁ ]
l4 [X₀+1-X₁ ]
l2 [X₀-X₁ ]
l5 [1 ]
l6 [1 ]
Analysing control-flow refined program
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___6
Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___4
Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l4___3
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l2___5
Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l1___2
Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l2___1
Found invariant 1+X₀ ≤ 0 for location l7
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l5
Found invariant 1+X₀ ≤ 0 for location l9
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l3
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l4___7
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₂₁₂: l3(X₀, X₁) → n_l4___7(X₀, X₁) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₂₁₅: n_l4___7(X₀, X₁) → n_l1___6(X₀, Arg1_P) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+Arg1_P ≤ X₀ ∧ 0 ≤ Arg1_P ∧ X₁ ≤ Arg1_P ∧ Arg1_P ≤ X₁ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₂₁₆: n_l4___7(X₀, X₁) → n_l2___5(X₀, Arg1_P) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+Arg1_P ≤ X₀ ∧ 0 ≤ Arg1_P ∧ X₁ ≤ Arg1_P ∧ Arg1_P ≤ X₁ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₂₀₈: n_l1___6(X₀, X₁) → n_l2___5(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound 2⋅X₀+2 {O(n)} for transition t₂₁₀: n_l2___5(X₀, X₁) → n_l3___4(X₀, X₁+1) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₂₀₇: n_l1___2(X₀, X₁) → n_l2___1(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
MPRF:
n_l4___7 [0 ]
l6 [0 ]
l3 [0 ]
n_l1___6 [X₀ ]
n_l2___5 [X₀ ]
n_l3___4 [X₀-X₁ ]
l5 [0 ]
n_l1___2 [X₀-X₁ ]
n_l4___3 [X₀-X₁ ]
n_l2___1 [X₀-X₁-1 ]
MPRF for transition t₂₀₉: n_l2___1(X₀, X₁) → n_l3___4(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
4⋅X₀⋅X₀+9⋅X₀+4 {O(n^2)}
MPRF:
n_l4___7 [X₀ ]
l6 [X₀ ]
l3 [X₀ ]
n_l1___6 [2⋅X₀ ]
n_l2___5 [2⋅X₀ ]
n_l3___4 [2⋅X₀-X₁ ]
l5 [X₀ ]
n_l1___2 [2⋅X₀-X₁ ]
n_l4___3 [2⋅X₀-X₁ ]
n_l2___1 [2⋅X₀-X₁ ]
MPRF for transition t₂₁₁: n_l3___4(X₀, X₁) → n_l4___3(X₀, X₁) :|: 0 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
MPRF:
n_l4___7 [0 ]
l6 [0 ]
l3 [0 ]
n_l1___6 [X₀ ]
n_l2___5 [X₀ ]
n_l3___4 [X₀+1-X₁ ]
l5 [0 ]
n_l1___2 [X₀-X₁ ]
n_l4___3 [X₀-X₁ ]
n_l2___1 [X₀-X₁ ]
MPRF for transition t₂₂₄: n_l3___4(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l6 [X₀+1 ]
l3 [X₀+1 ]
n_l3___4 [X₀+1 ]
l5 [X₁ ]
n_l1___2 [X₀+1 ]
n_l4___3 [X₀+1 ]
n_l2___1 [X₀+1 ]
n_l1___6 [X₀+1 ]
n_l4___7 [X₀+1 ]
n_l2___5 [X₀+1 ]
MPRF for transition t₂₁₃: n_l4___3(X₀, X₁) → n_l1___2(X₀, Arg1_P) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+Arg1_P ≤ X₀ ∧ 0 ≤ Arg1_P ∧ X₁ ≤ Arg1_P ∧ Arg1_P ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
MPRF:
n_l4___7 [0 ]
l6 [0 ]
l3 [0 ]
n_l1___6 [X₀ ]
n_l2___5 [X₀ ]
n_l3___4 [X₀-X₁ ]
l5 [0 ]
n_l1___2 [X₀-X₁-1 ]
n_l4___3 [X₀-X₁ ]
n_l2___1 [X₀-X₁-1 ]
MPRF for transition t₂₁₄: n_l4___3(X₀, X₁) → n_l2___1(X₀, Arg1_P) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+Arg1_P ≤ X₀ ∧ 0 ≤ Arg1_P ∧ X₁ ≤ Arg1_P ∧ Arg1_P ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
MPRF:
n_l4___7 [0 ]
l6 [0 ]
l3 [0 ]
n_l1___6 [X₀ ]
n_l2___5 [X₀ ]
n_l3___4 [X₀-X₁ ]
l5 [0 ]
n_l1___2 [X₀-X₁ ]
n_l4___3 [X₀-X₁ ]
n_l2___1 [X₀-X₁-1 ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:6⋅X₀⋅X₀+19⋅X₀+17 {O(n^2)}
t₀: 1 {O(1)}
t₈: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₉: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₄: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₅: X₀+1 {O(n)}
t₆: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₇: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₁₀: X₀+1 {O(n)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
Costbounds
Overall costbound: 6⋅X₀⋅X₀+19⋅X₀+17 {O(n^2)}
t₀: 1 {O(1)}
t₈: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₉: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₄: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₅: X₀+1 {O(n)}
t₆: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₇: X₀⋅X₀+3⋅X₀+2 {O(n^2)}
t₁₀: X₀+1 {O(n)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₈, X₀: X₀+1 {O(n)}
t₈, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₉, X₀: X₀+1 {O(n)}
t₉, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₄, X₀: X₀+1 {O(n)}
t₄, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₅, X₀: X₀+1 {O(n)}
t₅, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₆, X₀: X₀+1 {O(n)}
t₆, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₇, X₀: X₀+1 {O(n)}
t₇, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₁₀, X₀: X₀+1 {O(n)}
t₁₀, X₁: 2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
t₂, X₀: X₀+1 {O(n)}
t₂, X₁: 0 {O(1)}
t₃, X₀: 2⋅X₀+1 {O(n)}
t₃, X₁: 2⋅X₀⋅X₀+4⋅X₀+X₁+2 {O(n^2)}
t₁₁, X₀: 2⋅X₀+1 {O(n)}
t₁₁, X₁: 2⋅X₀⋅X₀+4⋅X₀+X₁+2 {O(n^2)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}