Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁-1, X₂, X₃) :|: 1 ≤ X₁
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₀, X₃) :|: X₁ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₂) :|: 1 ≤ X₂
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₃ ≤ 0 ∧ 1 ≤ X₂
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃-1) :|: 1 ≤ X₃ ∧ 1 ≤ X₂
Preprocessing
Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l2
Found invariant 0 ≤ X₀ for location l1
Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 0 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₀, X₃) :|: X₁ ≤ 0 ∧ 0 ≤ X₀
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₂) :|: 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃-1) :|: 1 ≤ X₃ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₂) :|: 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁+1 {O(n)}
MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
TWN: t₄: l3→l3
cycle: [t₄: l3→l3]
loop: (1 ≤ X₃ ∧ 1 ≤ X₂,(X₂,X₃) -> (X₂,X₃-1)
order: [X₂; X₃]
closed-form:
X₂: X₂
X₃: X₃ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < X₂ ∧ 1 < 0
∨ 1 < X₂ ∧ 1 < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1 < 0
∨ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1 < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
Stabilization-Threshold for: 1 ≤ X₃
alphas_abs: 1+X₃
M: 0
N: 1
Bound: 2⋅X₃+4 {O(n)}
TWN - Lifting for t₄: l3→l3 of 2⋅X₃+7 {O(n)}
relevant size-bounds w.r.t. t₃:
X₃: 2⋅X₁ {O(n)}
Runtime-bound of t₃: X₁+1 {O(n)}
Results in: 4⋅X₁⋅X₁+11⋅X₁+7 {O(n^2)}
Chain transitions t₅: l3→l2 and t₃: l2→l3 to t₄₁: l3→l3
Chain transitions t₂: l1→l2 and t₃: l2→l3 to t₄₂: l1→l3
Analysing control-flow refined program
Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l2
Found invariant 0 ≤ X₀ for location l1
Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound 4⋅X₁⋅X₁+11⋅X₁+7 {O(n^2)} for transition t₄₁: l3(X₀, X₁, X₂, X₃) -{2}> l3(X₀, X₁, X₂-1, X₂-1) :|: X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ 2 ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₀+1 ∧ 1 ≤ X₂ ∧ X₁+1 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Cut unsatisfiable transition t₅: l3→l2
Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l2
Found invariant 0 ≤ X₀ for location l1
Found invariant 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location n_l3___1
Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₁+1 {O(n)} for transition t₇₉: l3(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃-1) :|: 1 ≤ X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 1 ≤ X₃ ∧ X₁ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
MPRF for transition t₇₈: n_l3___1(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃-1) :|: 1+X₃ ≤ X₂ ∧ 1 ≤ X₃ ∧ X₁ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ of depth 1:
new bound:
3⋅X₁⋅X₁+5⋅X₁+1 {O(n^2)}
MPRF for transition t₈₂: n_l3___1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₁⋅X₁+14⋅X₁+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₂: 1 {O(1)}
t₃: X₁+1 {O(n)}
t₄: 4⋅X₁⋅X₁+11⋅X₁+7 {O(n^2)}
t₅: X₁ {O(n)}
Costbounds
Overall costbound: 4⋅X₁⋅X₁+14⋅X₁+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₂: 1 {O(1)}
t₃: X₁+1 {O(n)}
t₄: 4⋅X₁⋅X₁+11⋅X₁+7 {O(n^2)}
t₅: X₁ {O(n)}
Sizebounds
t₀, X₀: 0 {O(1)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: X₁ {O(n)}
t₂, X₃: 2⋅X₃ {O(n)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₁ {O(n)}
t₃, X₃: 2⋅X₁ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: X₁ {O(n)}
t₄, X₃: 2⋅X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: X₁ {O(n)}
t₅, X₃: 0 {O(1)}