Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₀) :|: 0 < X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: l2(X₀, X₁, X₂) → l1(X₀, X₀, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ 0
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 0 < X₂
t₈: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₇: l5(X₀, X₁, X₂) → l1(X₀, X₁-1, X₂)
t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂-1)
Preprocessing
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location l7
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₁ ≤ X₀ for location l1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location l4
Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₀) :|: 0 < X₁ ∧ X₁ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₀, X₀, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 0 < X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₈: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₁ ≤ X₀
t₇: l5(X₀, X₁, X₂) → l1(X₀, X₁-1, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂-1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₀) :|: 0 < X₁ ∧ X₁ ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
MPRF for transition t₇: l5(X₀, X₁, X₂) → l1(X₀, X₁-1, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
TWN: t₄: l3→l6
cycle: [t₄: l3→l6; t₆: l6→l3]
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
TWN - Lifting for t₄: l3→l6 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₂:
X₂: 2⋅X₀ {O(n)}
Runtime-bound of t₂: X₀+1 {O(n)}
Results in: 4⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
TWN: t₆: l6→l3
TWN - Lifting for t₆: l6→l3 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₂:
X₂: 2⋅X₀ {O(n)}
Runtime-bound of t₂: X₀+1 {O(n)}
Results in: 4⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
Chain transitions t₇: l5→l1 and t₃: l1→l4 to t₅₃: l5→l4
Chain transitions t₁: l2→l1 and t₃: l1→l4 to t₅₄: l2→l4
Chain transitions t₁: l2→l1 and t₂: l1→l3 to t₅₅: l2→l3
Chain transitions t₇: l5→l1 and t₂: l1→l3 to t₅₆: l5→l3
Chain transitions t₆: l6→l3 and t₄: l3→l6 to t₅₇: l6→l6
Chain transitions t₅₆: l5→l3 and t₄: l3→l6 to t₅₈: l5→l6
Chain transitions t₅₆: l5→l3 and t₅: l3→l5 to t₅₉: l5→l5
Chain transitions t₆: l6→l3 and t₅: l3→l5 to t₆₀: l6→l5
Chain transitions t₅₅: l2→l3 and t₅: l3→l5 to t₆₁: l2→l5
Chain transitions t₅₅: l2→l3 and t₄: l3→l6 to t₆₂: l2→l6
Analysing control-flow refined program
Cut unsatisfiable transition t₅₉: l5→l5
Cut unsatisfiable transition t₆₁: l2→l5
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location l7
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₁ ≤ X₀ for location l1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location l4
Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
MPRF for transition t₅₈: l5(X₀, X₁, X₂) -{3}> l6(X₀, X₁-1, X₀) :|: 1 < X₁ ∧ 0 < X₀ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀+1 ∧ 0 ≤ 0 ∧ 0 ≤ X₀ ∧ 2 ≤ X₁+X₀ ∧ 1 ≤ 2⋅X₀ ∧ X₁ ≤ X₀+1 ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₆₀: l6(X₀, X₁, X₂) -{2}> l5(X₀, X₁, X₂-1) :|: X₂ ≤ 1 ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₀+1 ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
TWN: t₅₇: l6→l6
cycle: [t₅₇: l6→l6]
loop: (1 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 1 < X₂
alphas_abs: 1+X₂
M: 0
N: 1
Bound: 2⋅X₂+4 {O(n)}
loop: (1 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 1 < X₂
alphas_abs: 1+X₂
M: 0
N: 1
Bound: 2⋅X₂+4 {O(n)}
TWN - Lifting for t₅₇: l6→l6 of 2⋅X₂+6 {O(n)}
relevant size-bounds w.r.t. t₅₈:
X₂: X₀ {O(n)}
Runtime-bound of t₅₈: X₀+1 {O(n)}
Results in: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
TWN - Lifting for t₅₇: l6→l6 of 2⋅X₂+6 {O(n)}
relevant size-bounds w.r.t. t₆₂:
X₂: X₀ {O(n)}
Runtime-bound of t₆₂: 1 {O(1)}
Results in: 2⋅X₀+6 {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 t₅: l3→l5
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___3
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l6___1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location l7
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___2
Found invariant X₁ ≤ X₀ for location l1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location l4
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₁₃₃: l3(X₀, X₁, X₂) → n_l6___3(X₀, X₁, X₂) :|: 0 < X₂ ∧ 0 < X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₁₃₅: n_l6___3(X₀, X₁, X₂) → n_l3___2(X₀, X₁, X₂-1) :|: 0 < X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₁₃₂: n_l3___2(X₀, X₁, X₂) → n_l6___1(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ 0 < X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+3⋅X₀+1 {O(n^2)}
MPRF for transition t₁₃₄: n_l6___1(X₀, X₁, X₂) → n_l3___2(X₀, X₁, X₂-1) :|: 1+X₂ ≤ X₀ ∧ 0 < X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+2⋅X₀ {O(n^2)}
MPRF for transition t₁₃₉: n_l3___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:8⋅X₀⋅X₀+19⋅X₀+13 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
t₄: 4⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
t₅: X₀ {O(n)}
t₆: 4⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}
Costbounds
Overall costbound: 8⋅X₀⋅X₀+19⋅X₀+13 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
t₄: 4⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
t₅: X₀ {O(n)}
t₆: 4⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 2⋅X₀ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: 2⋅X₀ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 0 {O(1)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 2⋅X₀ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 0 {O(1)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₀ {O(n)}
t₈, X₂: X₂ {O(n)}