Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₂) :|: X₄ ≤ X₁
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ < X₄
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁, X₂, X₃, X₀, X₅)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ < X₅
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l6(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₃
t₈: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l7(X₀, X₁, X₂, X₃, X₄, X₅)
t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁, X₂, X₃, X₄+1, X₅)
t₆: l6(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅+1)

Preprocessing

Found invariant X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ for location l6

Found invariant 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄ for location l7

Found invariant 1+X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ for location l5

Found invariant X₀ ≤ X₄ for location l1

Found invariant 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄ for location l4

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₂) :|: X₄ ≤ X₁ ∧ X₀ ≤ X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ < X₄ ∧ X₀ ≤ X₄
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁, X₂, X₃, X₀, X₅)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ < X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l6(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁
t₈: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l7(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄
t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁, X₂, X₃, X₄+1, X₅) :|: 1+X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁
t₆: l6(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅+1) :|: X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁

MPRF for transition t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₂) :|: X₄ ≤ X₁ ∧ X₀ ≤ X₄ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

l5 [X₁-X₄ ]
l1 [X₁+1-X₄ ]
l6 [X₁-X₄ ]
l3 [X₁-X₄ ]

MPRF for transition t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ < X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

l5 [X₁-X₄ ]
l1 [X₁+1-X₄ ]
l6 [X₁+1-X₄ ]
l3 [X₁+1-X₄ ]

MPRF for transition t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁, X₂, X₃, X₄+1, X₅) :|: 1+X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

l5 [X₁+1-X₄ ]
l1 [X₁+1-X₄ ]
l6 [X₁+1-X₄ ]
l3 [X₁+1-X₄ ]

Found invariant X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ for location l6

Found invariant 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄ for location l7

Found invariant 1+X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ for location l5

Found invariant X₀ ≤ X₄ for location l1

Found invariant 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄ for location l4

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

TWN-Loops:

entry: t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₂) :|: X₄ ≤ X₁ ∧ X₀ ≤ X₄
results in twn-loop: twn:Inv: [X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ ∧ X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁] , (X₀,X₁,X₂,X₃,X₄,X₅) -> (X₀,X₁,X₂,X₃,X₄,X₅+1) :|: X₅ ≤ X₃
order: [X₀; X₁; X₂; X₃; X₄; X₅]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂
X₃: X₃
X₄: X₄
X₅: X₅ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₅ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₅ ≤ X₃ ∧ X₃ ≤ X₅

Stabilization-Threshold for: X₅ ≤ X₃
alphas_abs: X₅+X₃
M: 0
N: 1
Bound: 2⋅X₃+2⋅X₅+2 {O(n)}

relevant size-bounds w.r.t. t₂:
X₃: X₃ {O(n)}
X₅: 2⋅X₂ {O(n)}
Runtime-bound of t₂: X₀+X₁+1 {O(n)}
Results in: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

Time-Bound by TWN-Loops:

TWN-Loops: t₆ 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

relevant size-bounds w.r.t. t₂:
X₃: X₃ {O(n)}
X₅: 2⋅X₂ {O(n)}
Runtime-bound of t₂: X₀+X₁+1 {O(n)}
Results in: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

Analysing control-flow refined program

Found invariant X₅ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ for location n_l6___3

Found invariant X₅ ≤ X₃ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₁ for location n_l6___1

Found invariant 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄ for location l7

Found invariant 1+X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ 1+X₃ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ for location n_l3___2

Found invariant X₀ ≤ X₄ for location l1

Found invariant 1+X₁ ≤ X₄ ∧ X₀ ≤ X₄ for location l4

Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ for location l3

knowledge_propagation leads to new time bound X₀+X₁+1 {O(n)} for transition t₇₃: l3(X₀, X₁, X₂, X₃, X₄, X₅) → n_l6___3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁

knowledge_propagation leads to new time bound X₀+X₁+1 {O(n)} for transition t₇₅: n_l6___3(X₀, X₁, X₂, X₃, X₄, X₅) → n_l3___2(X₀, X₁, X₂, X₃, X₄, X₅+1) :|: X₂ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁

MPRF for transition t₇₂: n_l3___2(X₀, X₁, X₂, X₃, X₄, X₅) → n_l6___1(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1+X₂ ≤ X₅ ∧ X₅ ≤ 1+X₃ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ 1+X₃ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ of depth 1:

new bound:

3⋅X₀⋅X₃+3⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+3⋅X₀+3⋅X₁+5⋅X₃+6⋅X₂+4 {O(n^2)}

MPRF:

l3 [2⋅X₃+1-2⋅X₅ ]
n_l6___3 [2⋅X₃+1-2⋅X₅ ]
l1 [2⋅X₃+1-2⋅X₂ ]
l5 [2⋅X₃+1-2⋅X₂ ]
n_l6___1 [3⋅X₃+1-2⋅X₂-X₅ ]
n_l3___2 [3⋅X₃+2-2⋅X₂-X₅ ]

MPRF for transition t₇₉: n_l3___2(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ < X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₁ ∧ X₅ ≤ 1+X₃ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

l3 [X₁+1-X₄ ]
l1 [X₁+1-X₄ ]
l5 [X₁-X₄ ]
n_l6___1 [X₁+1-X₄ ]
n_l6___3 [X₁+1-X₄ ]
n_l3___2 [X₁+1-X₄ ]

MPRF for transition t₇₄: n_l6___1(X₀, X₁, X₂, X₃, X₄, X₅) → n_l3___2(X₀, X₁, X₂, X₃, X₄, X₅+1) :|: X₅ ≤ X₃ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₃ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₃ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₀⋅X₀+2⋅X₀⋅X₂+2⋅X₁⋅X₁+2⋅X₁⋅X₂+4⋅X₀⋅X₁+X₀⋅X₃+X₁⋅X₃+2⋅X₂+6⋅X₀+6⋅X₁+X₃+4 {O(n^2)}

MPRF:

l3 [X₁+1-X₄ ]
n_l6___3 [X₁+1-X₄ ]
l1 [X₁+1-X₄ ]
l5 [X₁-X₄ ]
n_l6___1 [X₁+X₃+1-X₄-X₅ ]
n_l3___2 [X₁+X₃+1-X₄-X₅ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅X₃+4⋅X₁⋅X₃+8⋅X₀⋅X₂+8⋅X₁⋅X₂+11⋅X₀+11⋅X₁+4⋅X₃+8⋅X₂+15 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₀+X₁+1 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}
t₅: X₀+X₁+1 {O(n)}
t₈: 1 {O(1)}
t₇: X₀+X₁+1 {O(n)}
t₆: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

Costbounds

Overall costbound: 4⋅X₀⋅X₃+4⋅X₁⋅X₃+8⋅X₀⋅X₂+8⋅X₁⋅X₂+11⋅X₀+11⋅X₁+4⋅X₃+8⋅X₂+15 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₀+X₁+1 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}
t₅: X₀+X₁+1 {O(n)}
t₈: 1 {O(1)}
t₇: X₀+X₁+1 {O(n)}
t₆: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+4⋅X₂+4 {O(n^2)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: 2⋅X₀+X₁+1 {O(n)}
t₂, X₅: 2⋅X₂ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 3⋅X₀+X₁+1 {O(n)}
t₃, X₅: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+8⋅X₂+X₅+4 {O(n^2)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₀ {O(n)}
t₁, X₅: X₅ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: 2⋅X₀+X₁+1 {O(n)}
t₄, X₅: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+6⋅X₂+4 {O(n^2)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: 2⋅X₀+X₁+1 {O(n)}
t₅, X₅: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+8⋅X₂+4 {O(n^2)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₈, X₄: 3⋅X₀+X₁+1 {O(n)}
t₈, X₅: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+8⋅X₂+X₅+4 {O(n^2)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: 2⋅X₀+X₁+1 {O(n)}
t₇, X₅: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+8⋅X₂+4 {O(n^2)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: 2⋅X₀+X₁+1 {O(n)}
t₆, X₅: 2⋅X₀⋅X₃+2⋅X₁⋅X₃+4⋅X₀⋅X₂+4⋅X₁⋅X₂+2⋅X₃+4⋅X₀+4⋅X₁+6⋅X₂+4 {O(n^2)}