Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₂
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₁, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀
t₅: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₀ ≤ 0
t₆: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 0 < X₀
t₇: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0
t₈: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
Preprocessing
Cut unsatisfiable transition t₅: l3→l1
Cut unsatisfiable transition t₆: l3→l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l5
Found invariant X₂ ≤ X₁ for location l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₂ ∧ X₂ ≤ X₁
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₁, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₇: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₈: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁
MPRF for transition t₇: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₁ {O(n)}
TWN: t₂: l1→l3
cycle: [t₂: l1→l3; t₄: l3→l1]
loop: (0 < X₂ ∧ 0 < X₀ ∧ 0 < X₀,(X₀,X₂) -> (X₀-1,X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₂: X₂
Termination: true
Formula:
1 < 0 ∧ 0 < X₂
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₂
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
loop: (0 < X₂ ∧ 0 < X₀ ∧ 0 < X₀,(X₀,X₂) -> (X₀-1,X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₂: X₂
Termination: true
Formula:
1 < 0 ∧ 0 < X₂
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₂
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
TWN - Lifting for t₂: l1→l3 of 2⋅X₀+5 {O(n)}
relevant size-bounds w.r.t. t₇:
X₀: X₁ {O(n)}
Runtime-bound of t₇: X₁ {O(n)}
Results in: 2⋅X₁⋅X₁+5⋅X₁ {O(n^2)}
TWN - Lifting for t₂: l1→l3 of 2⋅X₀+5 {O(n)}
relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+5 {O(n)}
TWN: t₄: l3→l1
TWN - Lifting for t₄: l3→l1 of 2⋅X₀+5 {O(n)}
relevant size-bounds w.r.t. t₇:
X₀: X₁ {O(n)}
Runtime-bound of t₇: X₁ {O(n)}
Results in: 2⋅X₁⋅X₁+5⋅X₁ {O(n^2)}
TWN - Lifting for t₄: l3→l1 of 2⋅X₀+5 {O(n)}
relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+5 {O(n)}
Chain transitions t₇: l3→l1 and t₃: l1→l4 to t₅₃: l3→l4
Chain transitions t₄: l3→l1 and t₃: l1→l4 to t₅₄: l3→l4
Chain transitions t₄: l3→l1 and t₂: l1→l3 to t₅₅: l3→l3
Chain transitions t₇: l3→l1 and t₂: l1→l3 to t₅₆: l3→l3
Chain transitions t₁: l2→l1 and t₂: l1→l3 to t₅₇: l2→l3
Chain transitions t₁: l2→l1 and t₃: l1→l4 to t₅₈: l2→l4
Analysing control-flow refined program
Cut unsatisfiable transition t₅₄: l3→l4
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l5
Found invariant X₂ ≤ X₁ for location l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l3
MPRF for transition t₅₆: l3(X₀, X₁, X₂, X₃) -{2}> l3(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ 1 < X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₁+1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₁ {O(n)}
TWN: t₅₅: l3→l3
cycle: [t₅₅: l3→l3]
loop: (0 < X₀ ∧ 0 < X₀ ∧ 0 < X₂,(X₀,X₂) -> (X₀-1,X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₂: X₂
Termination: true
Formula:
0 < X₂ ∧ 1 < 0
∨ 0 < X₂ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
loop: (0 < X₀ ∧ 0 < X₀ ∧ 0 < X₂,(X₀,X₂) -> (X₀-1,X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₂: X₂
Termination: true
Formula:
0 < X₂ ∧ 1 < 0
∨ 0 < X₂ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
TWN - Lifting for t₅₅: l3→l3 of 2⋅X₀+5 {O(n)}
relevant size-bounds w.r.t. t₅₆:
X₀: 2⋅X₁ {O(n)}
Runtime-bound of t₅₆: X₁ {O(n)}
Results in: 4⋅X₁⋅X₁+5⋅X₁ {O(n^2)}
TWN - Lifting for t₅₅: l3→l3 of 2⋅X₀+5 {O(n)}
relevant size-bounds w.r.t. t₅₇:
X₀: X₃ {O(n)}
Runtime-bound of t₅₇: 1 {O(1)}
Results in: 2⋅X₃+5 {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Cut unsatisfiable transition t₁₄₂: n_l1___2→l4
Cut unsatisfiable transition t₁₄₄: n_l1___6→l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l1___6
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___4
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location n_l3___3
Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l5
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___5
Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l4
Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___1
Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ for location n_l3___7
MPRF for transition t₁₂₆: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: 0 < X₂ ∧ 1 ≤ X₂ ∧ 0 < X₀ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 1 ≤ X₂ ∧ 0 < X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF for transition t₁₂₇: n_l1___5(X₀, X₁, X₂, X₃) → n_l3___3(X₀, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 0 < X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF for transition t₁₃₀: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___6(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
4⋅X₁+1 {O(n)}
MPRF for transition t₁₃₁: n_l3___3(X₀, X₁, X₂, X₃) → n_l1___2(X₀-1, X₁, X₂, X₃) :|: 1+X₂ ≤ X₀ ∧ 0 < X₂ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF for transition t₁₃₂: n_l3___4(X₀, X₁, X₂, X₃) → n_l1___5(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₁ {O(n)}
TWN: t₁₂₈: n_l1___6→n_l3___4
cycle: [t₁₃₃: n_l3___4→n_l1___6; t₁₂₈: n_l1___6→n_l3___4]
loop: (0 < X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 < X₂ ∧ 0 < X₀,(X₀,X₂) -> (X₀-1,X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₂: X₂
Termination: true
Formula:
1 < 0 ∧ 0 < X₂ ∧ 1 < X₂
∨ 1 < 0 ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₂ ∧ 1 < X₂
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
loop: (0 < X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 < X₂ ∧ 0 < X₀,(X₀,X₂) -> (X₀-1,X₂)
order: [X₀; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₂: X₂
Termination: true
Formula:
1 < 0 ∧ 0 < X₂ ∧ 1 < X₂
∨ 1 < 0 ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₂ ∧ 1 < X₂
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
TWN - Lifting for t₁₂₈: n_l1___6→n_l3___4 of 2⋅X₀+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₅:
X₀: X₃ {O(n)}
Runtime-bound of t₁₃₅: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}
TWN - Lifting for t₁₂₈: n_l1___6→n_l3___4 of 2⋅X₀+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₀:
X₀: 3⋅X₁ {O(n)}
Runtime-bound of t₁₃₀: 4⋅X₁+1 {O(n)}
Results in: 24⋅X₁⋅X₁+30⋅X₁+6 {O(n^2)}
TWN: t₁₃₃: n_l3___4→n_l1___6
TWN - Lifting for t₁₃₃: n_l3___4→n_l1___6 of 2⋅X₀+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₅:
X₀: X₃ {O(n)}
Runtime-bound of t₁₃₅: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}
TWN - Lifting for t₁₃₃: n_l3___4→n_l1___6 of 2⋅X₀+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₀:
X₀: 3⋅X₁ {O(n)}
Runtime-bound of t₁₃₀: 4⋅X₁+1 {O(n)}
Results in: 24⋅X₁⋅X₁+30⋅X₁+6 {O(n^2)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₁⋅X₁+11⋅X₁+4⋅X₃+14 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₁⋅X₁+2⋅X₃+5⋅X₁+5 {O(n^2)}
t₃: 1 {O(1)}
t₄: 2⋅X₁⋅X₁+2⋅X₃+5⋅X₁+5 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}
Costbounds
Overall costbound: 4⋅X₁⋅X₁+11⋅X₁+4⋅X₃+14 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₁⋅X₁+2⋅X₃+5⋅X₁+5 {O(n^2)}
t₃: 1 {O(1)}
t₄: 2⋅X₁⋅X₁+2⋅X₃+5⋅X₁+5 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₁ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₁+X₃ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₁+X₃ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₁ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₁+X₃ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁ {O(n)}
t₄, X₃: X₃ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₁+X₃ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}