Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₂, X₂, X₃) :|: 1 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 1
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ < 1
t₄: l3(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁
t₈: l4(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l5(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l3(X₀, X₁-1, X₂, X₃)
Preprocessing
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 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 ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₂, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 1 ∧ X₀ ≤ X₃
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ < 1 ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀
t₄: l3(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀
t₈: l4(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃ ∧ X₀ ≤ 0
t₇: l5(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
t₆: l6(X₀, X₁, X₂, X₃) → l3(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₂, X₂, X₃) :|: 1 ≤ 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 t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ < 1 ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l5 [X₀-1 ]
l1 [X₀ ]
l6 [X₀ ]
l3 [X₀ ]
MPRF for transition t₇: l5(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l5 [X₀ ]
l1 [X₀ ]
l6 [X₀ ]
l3 [X₀ ]
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 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 ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ for location l3
Time-Bound by TWN-Loops:
TWN-Loops: t₄ 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
TWN-Loops:
entry: t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₂, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
results in twn-loop: twn:Inv: [1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀,X₁-1,X₂,X₃) :|: 1 ≤ X₁
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
X₂: X₂
X₃: X₃
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1
Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
relevant size-bounds w.r.t. t₂:
X₁: 2⋅X₂ {O(n)}
Runtime-bound of t₂: X₃ {O(n)}
Results in: 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
Time-Bound by TWN-Loops:
TWN-Loops: t₆ 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
relevant size-bounds w.r.t. t₂:
X₁: 2⋅X₂ {O(n)}
Runtime-bound of t₂: X₃ {O(n)}
Results in: 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
Analysing control-flow refined program
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___3
Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___1
Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ 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 ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₃ {O(n)} for transition t₇₃: l3(X₀, X₁, X₂, X₃) → n_l6___3(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₃ {O(n)} for transition t₇₅: n_l6___3(X₀, X₁, X₂, X₃) → n_l3___2(X₀, X₁-1, X₂, X₃) :|: X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₇₂: n_l3___2(X₀, X₁, X₂, X₃) → n_l6___1(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂⋅X₃+X₃ {O(n^2)}
MPRF:
l3 [0 ]
n_l6___3 [0 ]
l1 [0 ]
l5 [0 ]
n_l6___1 [X₁ ]
n_l3___2 [X₁+1 ]
MPRF for transition t₇₉: n_l3___2(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ < 1 ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l3 [X₀ ]
l1 [X₀ ]
l5 [X₀-1 ]
n_l6___1 [X₀ ]
n_l6___3 [X₀ ]
n_l3___2 [X₀ ]
MPRF for transition t₇₄: n_l6___1(X₀, X₁, X₂, X₃) → n_l3___2(X₀, X₁-1, X₂, X₃) :|: X₀ ≤ X₃ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂⋅X₃ {O(n^2)}
MPRF:
l3 [0 ]
n_l6___3 [0 ]
l1 [0 ]
l5 [0 ]
n_l6___1 [X₁ ]
n_l3___2 [X₁ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:8⋅X₂⋅X₃+11⋅X₃+4 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₃ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
t₅: X₃ {O(n)}
t₈: 1 {O(1)}
t₇: X₃ {O(n)}
t₆: 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
Costbounds
Overall costbound: 8⋅X₂⋅X₃+11⋅X₃+4 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₃ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
t₅: X₃ {O(n)}
t₈: 1 {O(1)}
t₇: X₃ {O(n)}
t₆: 4⋅X₂⋅X₃+4⋅X₃ {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₂, X₀: X₃ {O(n)}
t₂, X₁: 2⋅X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₃ {O(n)}
t₃, X₁: 4⋅X₂+X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₄, X₀: X₃ {O(n)}
t₄, X₁: 2⋅X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₃ {O(n)}
t₅, X₁: 4⋅X₂ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₈, X₀: 2⋅X₃ {O(n)}
t₈, X₁: 4⋅X₂+X₁ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₇, X₀: X₃ {O(n)}
t₇, X₁: 4⋅X₂ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₆, X₀: X₃ {O(n)}
t₆, X₁: 2⋅X₂ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}