Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₂, X₂, X₃, X₄) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₁, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l6(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₁
t₈: l4(X₀, X₁, X₂, X₃, X₄) → l7(X₀, X₁, X₂, X₃, X₄)
t₇: l5(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁, X₂, X₃, X₄)
t₆: l6(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁-1, X₂, X₃, X₄)
Preprocessing
Eliminate variables {X₃,X₄} that do not contribute to the problem
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₀ ≤ X₂ for location l1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ 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:
t₁₇: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₈: l1(X₀, X₁, X₂) → l3(X₀, X₂, X₂) :|: 0 < X₀ ∧ X₀ ≤ X₂
t₁₉: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₀: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₂₁: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂₂: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 0 < X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂₃: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₀ ≤ 0
t₂₄: l5(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂₅: l6(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₁₈: l1(X₀, X₁, X₂) → l3(X₀, X₂, X₂) :|: 0 < X₀ ∧ X₀ ≤ X₂ of depth 1:
new bound:
X₂ {O(n)}
MPRF for transition t₂₁: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
MPRF for transition t₂₄: l5(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
TWN: t₂₂: l3→l6
cycle: [t₂₂: l3→l6; t₂₅: l6→l3]
loop: (0 < X₁,(X₁) -> (X₁-1)
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
TWN - Lifting for t₂₂: l3→l6 of 2⋅X₁+4 {O(n)}
relevant size-bounds w.r.t. t₁₈:
X₁: 2⋅X₂ {O(n)}
Runtime-bound of t₁₈: X₂ {O(n)}
Results in: 4⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
TWN: t₂₅: l6→l3
TWN - Lifting for t₂₅: l6→l3 of 2⋅X₁+4 {O(n)}
relevant size-bounds w.r.t. t₁₈:
X₁: 2⋅X₂ {O(n)}
Runtime-bound of t₁₈: X₂ {O(n)}
Results in: 4⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
Chain transitions t₂₄: l5→l1 and t₁₉: l1→l4 to t₅₃: l5→l4
Chain transitions t₂₀: l2→l1 and t₁₉: l1→l4 to t₅₄: l2→l4
Chain transitions t₂₀: l2→l1 and t₁₈: l1→l3 to t₅₅: l2→l3
Chain transitions t₂₄: l5→l1 and t₁₈: l1→l3 to t₅₆: l5→l3
Chain transitions t₂₅: l6→l3 and t₂₂: l3→l6 to t₅₇: l6→l6
Chain transitions t₅₆: l5→l3 and t₂₂: l3→l6 to t₅₈: l5→l6
Chain transitions t₅₆: l5→l3 and t₂₁: l3→l5 to t₅₉: l5→l5
Chain transitions t₂₅: l6→l3 and t₂₁: l3→l5 to t₆₀: l6→l5
Chain transitions t₅₅: l2→l3 and t₂₁: l3→l5 to t₆₁: l2→l5
Chain transitions t₅₅: l2→l3 and t₂₂: 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 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₀ ≤ X₂ for location l1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
MPRF for transition t₅₈: l5(X₀, X₁, X₂) -{3}> l6(X₀-1, X₂, X₂) :|: 1 < X₀ ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ 2⋅X₂ ∧ 0 ≤ 0 ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂+1 {O(n)}
MPRF for transition t₆₀: l6(X₀, X₁, X₂) -{2}> l5(X₀, X₁-1, X₂) :|: X₁ ≤ 1 ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
TWN: t₅₇: l6→l6
cycle: [t₅₇: l6→l6]
loop: (1 < X₁,(X₁) -> (X₁-1)
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 1 < X₁
alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}
loop: (1 < X₁,(X₁) -> (X₁-1)
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 1 < X₁
alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}
TWN - Lifting for t₅₇: l6→l6 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₅₈:
X₁: X₂ {O(n)}
Runtime-bound of t₅₈: X₂+1 {O(n)}
Results in: 2⋅X₂⋅X₂+8⋅X₂+6 {O(n^2)}
TWN - Lifting for t₅₇: l6→l6 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₆₂:
X₁: X₂ {O(n)}
Runtime-bound of t₆₂: 1 {O(1)}
Results in: 2⋅X₂+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 t₂₁: l3→l5
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l6___3
Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l7
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___2
Found invariant X₀ ≤ X₂ for location l1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₂ {O(n)} for transition t₁₃₃: l3(X₀, X₁, X₂) → n_l6___3(X₀, X₁, X₂) :|: 0 < X₁ ∧ 0 < X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₂ {O(n)} for transition t₁₃₅: n_l6___3(X₀, X₁, X₂) → n_l3___2(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
MPRF for transition t₁₃₂: n_l3___2(X₀, X₁, X₂) → n_l6___1(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂ ∧ 0 < X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂⋅X₂+X₂ {O(n^2)}
MPRF for transition t₁₃₄: n_l6___1(X₀, X₁, X₂) → n_l3___2(X₀, X₁-1, X₂) :|: 1+X₁ ≤ X₂ ∧ 0 < X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂⋅X₂ {O(n^2)}
MPRF for transition t₁₃₉: n_l3___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:8⋅X₂⋅X₂+11⋅X₂+4 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: X₂ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂ {O(n)}
t₂₂: 4⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
t₂₃: 1 {O(1)}
t₂₄: X₂ {O(n)}
t₂₅: 4⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
Costbounds
Overall costbound: 8⋅X₂⋅X₂+11⋅X₂+4 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: X₂ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂ {O(n)}
t₂₂: 4⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
t₂₃: 1 {O(1)}
t₂₄: X₂ {O(n)}
t₂₅: 4⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
Sizebounds
t₁₇, X₀: X₀ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₀: X₂ {O(n)}
t₁₈, X₁: 2⋅X₂ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₀: 2⋅X₂ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: 0 {O(1)}
t₂₁, X₂: X₂ {O(n)}
t₂₂, X₀: X₂ {O(n)}
t₂₂, X₁: 2⋅X₂ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: 2⋅X₂ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: 2⋅X₂ {O(n)}
t₂₄, X₀: X₂ {O(n)}
t₂₄, X₁: 0 {O(1)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: X₂ {O(n)}
t₂₅, X₁: 2⋅X₂ {O(n)}
t₂₅, X₂: X₂ {O(n)}