Initial Problem

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

Preprocessing

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant X₂ ≤ X₀ for location l3

Problem after Preprocessing

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

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant X₂ ≤ X₀ for location l3

Analysing control-flow refined program

Cut unsatisfiable transition t₁₀₁: n_l1___2→l4

Cut unsatisfiable transition t₈₇: n_l1___5→n_l3___3

Cut unreachable locations [n_l3___3] from the program graph

Found invariant 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ for location n_l3___4

Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location n_l3___6

Found invariant 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l1___2

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ for location n_l1___5

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l3___1

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location n_l3___7

Found invariant 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ for location n_l3___4

Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location n_l3___6

Found invariant 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l1___2

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ for location n_l1___5

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l3___1

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location n_l3___7

Found invariant 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ for location n_l3___4

Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location n_l3___6

Found invariant 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l1___2

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ for location n_l1___5

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l3___1

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location n_l3___7

Time-Bound by TWN-Loops:

TWN-Loops: t₈₈ 10⋅X₂+2⋅X₁+16 {O(n)}

TWN-Loops:

entry: t₉₅: n_l3___7(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁
results in twn-loop: twn:Inv: [1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁] , (X₀,X₁,X₂) -> (X₀+1,X₁,X₂) :|: X₂ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ < X₁ ∧ X₀ < X₁ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ X₀
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁
X₂: X₂

Termination: true
Formula:

0 < 1 ∧ 1 < 0
∨ 0 < 1 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ 1+X₂ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ 0 < 1 ∧ 1+X₂ < X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 < 0
∨ 0 < 1 ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 < 1 ∧ 1 < 0
∨ X₂ < X₀ ∧ 0 < 1 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ < X₀ ∧ 1+X₂ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ X₂ < X₀ ∧ 1+X₂ < X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 < 0
∨ X₂ < X₀ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 < 1 ∧ 1 < 0
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 < 1 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₂ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₂ < X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 < 0
∨ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₂ ≤ X₀
alphas_abs: X₀+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₂+2 {O(n)}
Stabilization-Threshold for: 1+X₂ ≤ X₀
alphas_abs: 1+X₀+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₂+4 {O(n)}
Stabilization-Threshold for: X₀ < X₁
alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₉₅:
X₀: X₂+1 {O(n)}
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₉₅: 1 {O(1)}
Results in: 10⋅X₂+2⋅X₁+16 {O(n)}

10⋅X₂+2⋅X₁+16 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₉₃ 10⋅X₂+2⋅X₁+16 {O(n)}

relevant size-bounds w.r.t. t₉₅:
X₀: X₂+1 {O(n)}
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₉₅: 1 {O(1)}
Results in: 10⋅X₂+2⋅X₁+16 {O(n)}

10⋅X₂+2⋅X₁+16 {O(n)}

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₂: inf {Infinity}
t₃: inf {Infinity}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: inf {Infinity}
t₆: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₂: inf {Infinity}
t₃: inf {Infinity}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: inf {Infinity}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₄, X₁: 3⋅X₁ {O(n)}
t₄, X₂: 3⋅X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₆, X₁: 3⋅X₁ {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}