Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, 0, X₂, X₃, X₄) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₁, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄) :|: X₁ < X₀
t₈: l4(X₀, X₁, X₂, X₃, X₄) → l7(X₀, X₁, X₂, X₃, X₄)
t₇: l5(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁, X₂, X₃, X₄)
t₆: l6(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁+1, X₂, X₃, X₄)
Preprocessing
Eliminate variables {X₂,X₄} that do not contribute to the problem
Found invariant 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₀ ≤ X₃ for location l1
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₈: l0(X₀, X₁, X₃) → l2(X₀, X₁, X₃)
t₁₉: l1(X₀, X₁, X₃) → l3(X₀, 0, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₃
t₂₀: l1(X₀, X₁, X₃) → l4(X₀, X₁, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₂₁: l2(X₀, X₁, X₃) → l1(X₃, X₁, X₃)
t₂₃: l3(X₀, X₁, X₃) → l5(X₀, X₁, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂₂: l3(X₀, X₁, X₃) → l6(X₀, X₁, X₃) :|: X₁ < X₀ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂₄: l4(X₀, X₁, X₃) → l7(X₀, X₁, X₃) :|: X₀ ≤ X₃ ∧ X₀ ≤ 0
t₂₅: l5(X₀, X₁, X₃) → l1(X₀-1, X₁, X₃) :|: 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₂₆: l6(X₀, X₁, X₃) → l3(X₀, X₁+1, X₃) :|: 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₁₉: l1(X₀, X₁, X₃) → l3(X₀, 0, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₃ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l5 [X₀-1 ]
l1 [X₀ ]
l6 [X₀-1 ]
l3 [X₀-1 ]
MPRF for transition t₂₃: l3(X₀, X₁, X₃) → l5(X₀, X₁, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l5 [X₁-1 ]
l1 [X₀ ]
l6 [X₀ ]
l3 [X₀ ]
MPRF for transition t₂₅: l5(X₀, X₁, X₃) → l1(X₀-1, X₁, X₃) :|: 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l5 [X₁ ]
l1 [X₀ ]
l6 [X₀ ]
l3 [X₀ ]
Found invariant 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₀ ≤ X₃ for location l1
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
Time-Bound by TWN-Loops:
TWN-Loops: t₂₂ 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
TWN-Loops:
entry: t₁₉: l1(X₀, X₁, X₃) → l3(X₀, 0, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₃
results in twn-loop: twn:Inv: [1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁,X₃) -> (X₀,X₁+1,X₃) :|: X₁ < X₀
order: [X₀; X₁; X₃]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₃: X₃
Termination: true
Formula:
1 < 0
∨ X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: X₁ < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
relevant size-bounds w.r.t. t₁₉:
X₀: X₃ {O(n)}
Runtime-bound of t₁₉: X₃ {O(n)}
Results in: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
Time-Bound by TWN-Loops:
TWN-Loops: t₂₆ 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
relevant size-bounds w.r.t. t₁₉:
X₀: X₃ {O(n)}
Runtime-bound of t₁₉: X₃ {O(n)}
Results in: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
Analysing control-flow refined program
Cut unsatisfiable transition t₂₃: l3→l5
Found invariant 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___3
Found invariant 2 ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l6___1
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___2
Found invariant X₀ ≤ X₃ for location l1
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₃ {O(n)} for transition t₇₃: l3(X₀, X₁, X₃) → n_l6___3(X₀, X₁, X₃) :|: X₁ < X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₃ {O(n)} for transition t₇₅: n_l6___3(X₀, X₁, X₃) → n_l3___2(X₀, X₁+1, X₃) :|: X₁ < X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₇₂: n_l3___2(X₀, X₁, X₃) → n_l6___1(X₀, X₁, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃⋅X₃+2⋅X₃ {O(n^2)}
MPRF:
l3 [0 ]
n_l6___3 [0 ]
l1 [0 ]
l5 [X₀-X₁ ]
n_l6___1 [X₀-X₁ ]
n_l3___2 [X₀+1-X₁ ]
MPRF for transition t₇₉: n_l3___2(X₀, X₁, X₃) → l5(X₀, X₁, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃+1 {O(n)}
MPRF:
l3 [X₀+1 ]
l1 [X₀+1 ]
l5 [X₁ ]
n_l6___1 [X₀+1 ]
n_l6___3 [X₀+1 ]
n_l3___2 [X₀+1 ]
MPRF for transition t₇₄: n_l6___1(X₀, X₁, X₃) → n_l3___2(X₀, X₁+1, X₃) :|: X₀ ≤ X₃ ∧ X₁ < X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₃ {O(n^2)}
MPRF:
l3 [X₃ ]
n_l6___3 [X₃ ]
l1 [X₃ ]
l5 [X₀+X₃-X₁ ]
n_l6___1 [X₀+X₃-X₁ ]
n_l3___2 [X₀+X₃-X₁ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₃⋅X₃+11⋅X₃+4 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₃ {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₂₃: X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₃ {O(n)}
t₂₆: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
Costbounds
Overall costbound: 4⋅X₃⋅X₃+11⋅X₃+4 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₃ {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₂₃: X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₃ {O(n)}
t₂₆: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
Sizebounds
t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: X₃ {O(n)}
t₁₉, X₁: 0 {O(1)}
t₁₉, X₃: X₃ {O(n)}
t₂₀, X₀: 2⋅X₃ {O(n)}
t₂₀, X₁: 2⋅X₃⋅X₃+4⋅X₃+X₁ {O(n^2)}
t₂₀, X₃: 2⋅X₃ {O(n)}
t₂₁, X₀: X₃ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₃ {O(n)}
t₂₂, X₁: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: X₃ {O(n)}
t₂₃, X₁: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₂₃, X₃: X₃ {O(n)}
t₂₄, X₀: 2⋅X₃ {O(n)}
t₂₄, X₁: 2⋅X₃⋅X₃+4⋅X₃+X₁ {O(n^2)}
t₂₄, X₃: 2⋅X₃ {O(n)}
t₂₅, X₀: X₃ {O(n)}
t₂₅, X₁: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₂₅, X₃: X₃ {O(n)}
t₂₆, X₀: X₃ {O(n)}
t₂₆, X₁: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₂₆, X₃: X₃ {O(n)}