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(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 0 < X₃
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 0 < X₂
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 0 < X₀
Preprocessing
Found invariant X₃ ≤ 0 for location l2
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 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(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 0 < X₃
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 0 < X₂ ∧ X₃ ≤ 0
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 0 < X₀ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀
MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 0 < X₃ of depth 1:
new bound:
X₃ {O(n)}
MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 0 < X₂ ∧ X₃ ≤ 0 of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
TWN: t₄: l3→l3
cycle: [t₄: l3→l3]
loop: (0 < X₀,(X₀,X₁) -> (X₀+X₁,X₁-1)
order: [X₁; X₀]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₁ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < 2⋅X₁+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₁+1 ∧ 2⋅X₁+1 ≤ 0
Stabilization-Threshold for: 0 < X₀
alphas_abs: 1+2⋅X₀+2⋅X₁
M: 0
N: 2
Bound: 4⋅X₀+4⋅X₁+5 {O(n)}
TWN - Lifting for t₄: l3→l3 of 4⋅X₀+4⋅X₁+7 {O(n)}
relevant size-bounds w.r.t. t₃:
X₀: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
X₁: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
Runtime-bound of t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
Results in: 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+78⋅X₃⋅X₃+14⋅X₂+14⋅X₃ {O(n^4)}
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 X₃ ≤ 0 for location l2
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ for location l3
knowledge_propagation leads to new time bound 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+78⋅X₃⋅X₃+14⋅X₂+14⋅X₃ {O(n^4)} for transition t₄₁: l3(X₀, X₁, X₂, X₃) -{2}> l3(X₂-1, X₂-1, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 1 < X₂ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Found invariant X₃ ≤ 0 for location l2
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___3
Found invariant X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ 2+X₁+X₃ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location n_l2___1
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l3___2
MPRF for transition t₈₆: n_l2___1(X₀, X₁, X₂, X₃) → n_l3___3(X₂, X₂, X₂, X₃) :|: 0 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 < X₂ ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ 2+X₁+X₃ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF for transition t₈₈: n_l3___2(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF for transition t₉₀: n_l3___3(X₀, X₁, X₂, X₃) → n_l3___2(X₀+X₁, X₁-1, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
4⋅X₃⋅X₃+4⋅X₂+4⋅X₃+1 {O(n^2)}
TWN: t₈₉: n_l3___2→n_l3___2
cycle: [t₈₉: n_l3___2→n_l3___2]
loop: (0 < X₀,(X₀,X₁) -> (X₀+X₁,X₁-1)
order: [X₁; X₀]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₁ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < 2⋅X₁+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₁+1 ∧ 2⋅X₁+1 ≤ 0
Stabilization-Threshold for: 0 < X₀
alphas_abs: 1+2⋅X₀+2⋅X₁
M: 0
N: 2
Bound: 4⋅X₀+4⋅X₁+5 {O(n)}
TWN - Lifting for t₈₉: n_l3___2→n_l3___2 of 4⋅X₀+4⋅X₁+7 {O(n)}
relevant size-bounds w.r.t. t₉₀:
X₀: 8⋅X₃⋅X₃+8⋅X₂+8⋅X₃ {O(n^2)}
X₁: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
Runtime-bound of t₉₀: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃+1 {O(n^2)}
Results in: 192⋅X₃⋅X₃⋅X₃⋅X₃+384⋅X₂⋅X₃⋅X₃+384⋅X₃⋅X₃⋅X₃+192⋅X₂⋅X₂+268⋅X₃⋅X₃+384⋅X₂⋅X₃+76⋅X₂+76⋅X₃+7 {O(n^4)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+82⋅X₃⋅X₃+18⋅X₂+19⋅X₃+2 {O(n^4)}
t₀: 1 {O(1)}
t₁: X₃ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄: 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+78⋅X₃⋅X₃+14⋅X₂+14⋅X₃ {O(n^4)}
t₅: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
Costbounds
Overall costbound: 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+82⋅X₃⋅X₃+18⋅X₂+19⋅X₃+2 {O(n^4)}
t₀: 1 {O(1)}
t₁: X₃ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄: 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+78⋅X₃⋅X₃+14⋅X₂+14⋅X₃ {O(n^4)}
t₅: 2⋅X₃⋅X₃+2⋅X₂+2⋅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)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₃⋅X₃+2⋅X₃+X₂ {O(n^2)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₂, X₃: 2⋅X₃ {O(n)}
t₃, X₀: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
t₃, X₁: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
t₃, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 4096⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+24576⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+26880⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+49152⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₂⋅X₂⋅X₂⋅X₃⋅X₃+23296⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+49152⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃+56064⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+11380⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₂⋅X₂⋅X₂⋅X₃+30208⋅X₂⋅X₃⋅X₃⋅X₃+31488⋅X₂⋅X₂⋅X₃⋅X₃+4096⋅X₂⋅X₂⋅X₂⋅X₂+2304⋅X₂⋅X₂⋅X₂+3048⋅X₃⋅X₃⋅X₃+6912⋅X₂⋅X₂⋅X₃+7656⋅X₂⋅X₃⋅X₃+372⋅X₂⋅X₂+398⋅X₃⋅X₃+744⋅X₂⋅X₃+26⋅X₂+26⋅X₃ {O(n^8)}
t₄, X₁: 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+82⋅X₃⋅X₃+18⋅X₂+18⋅X₃ {O(n^4)}
t₄, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: 4096⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+24576⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+26880⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+49152⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₂⋅X₂⋅X₂⋅X₃⋅X₃+23296⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+49152⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃+56064⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+11380⋅X₃⋅X₃⋅X₃⋅X₃+16384⋅X₂⋅X₂⋅X₂⋅X₃+30208⋅X₂⋅X₃⋅X₃⋅X₃+31488⋅X₂⋅X₂⋅X₃⋅X₃+4096⋅X₂⋅X₂⋅X₂⋅X₂+2304⋅X₂⋅X₂⋅X₂+3048⋅X₃⋅X₃⋅X₃+6912⋅X₂⋅X₂⋅X₃+7656⋅X₂⋅X₃⋅X₃+372⋅X₂⋅X₂+398⋅X₃⋅X₃+744⋅X₂⋅X₃+26⋅X₂+26⋅X₃ {O(n^8)}
t₅, X₁: 64⋅X₃⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃⋅X₃+128⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₃+64⋅X₂⋅X₂+82⋅X₃⋅X₃+18⋅X₂+18⋅X₃ {O(n^4)}
t₅, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₅, X₃: 2⋅X₃ {O(n)}