Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: nondef.0
Locations: l0, l1, l10, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₈: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₇: l1(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₂) :|: 0 < X₀
t₄: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₆: l3(X₀, X₁, X₂, X₃) → l1(nondef.0, X₁, X₂, X₃)
t₉: l4(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂+1, X₃)
t₁₀: l5(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: X₃ < X₁
t₁₁: l5(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₃
t₂: l6(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₂ < X₁
t₃: l6(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₂) :|: X₁ ≤ X₂
t₁: l7(X₀, X₁, X₂, X₃) → l6(X₀, X₁, 0, X₃)
t₁₂: l8(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃+1)
t₁₃: l9(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l2

Found invariant 0 ≤ X₂ for location l6

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂ for location l5

Found invariant 1+X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l8

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ for location l10

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l4

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ for location l9

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: nondef.0
Locations: l0, l1, l10, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₈: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₇: l1(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₂) :|: 0 < X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₄: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₆: l3(X₀, X₁, X₂, X₃) → l1(nondef.0, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₉: l4(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂+1, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0
t₁₀: l5(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃) :|: X₃ < X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂
t₁₁: l5(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂
t₂: l6(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₂ < X₁ ∧ 0 ≤ X₂
t₃: l6(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₂) :|: X₁ ≤ X₂ ∧ 0 ≤ X₂
t₁: l7(X₀, X₁, X₂, X₃) → l6(X₀, X₁, 0, X₃)
t₁₂: l8(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃+1) :|: 1+X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₁₃: l9(X₀, X₁, X₂, X₃) → l10(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂

Analysing control-flow refined program

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l6___5

Found invariant X₂ ≤ 0 ∧ 0 ≤ X₂ for location l6

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l4___6

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location n_l1___7

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l3___3

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location n_l1___2

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location n_l2___9

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location n_l3___8

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l4___1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂ for location l5

Found invariant 1+X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l8

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ for location l10

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ for location l9

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l2___4

CFR did not improve the program. Rolling back

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l2

Found invariant 0 ≤ X₂ for location l6

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂ for location l5

Found invariant 1+X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l8

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ for location l10

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l4

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ for location l9

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₀ 2⋅X₁+4 {O(n)}

TWN-Loops:

entry: t₇: l1(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₂) :|: 0 < X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
results in twn-loop: twn:Inv: [0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁] , (X₀,X₁,X₂,X₃) -> (X₀,X₁,X₂,X₃+1) :|: X₃ < X₁
order: [X₁; X₂; X₃]
closed-form:
X₁: X₁
X₂: X₂
X₃: X₃ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₃ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₃ < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₇:
X₁: X₁ {O(n)}
Runtime-bound of t₇: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₂ 2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₇:
X₁: X₁ {O(n)}
Runtime-bound of t₇: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₇: 1 {O(1)}
t₈: inf {Infinity}
t₄: inf {Infinity}
t₆: inf {Infinity}
t₉: inf {Infinity}
t₁₀: 2⋅X₁+4 {O(n)}
t₁₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₁₂: 2⋅X₁+4 {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₇: 1 {O(1)}
t₈: inf {Infinity}
t₄: inf {Infinity}
t₆: inf {Infinity}
t₉: inf {Infinity}
t₁₀: 2⋅X₁+4 {O(n)}
t₁₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₁₂: 2⋅X₁+4 {O(n)}
t₁₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₇, X₁: X₁ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₃: X₃ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₃: X₃ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₃: X₃ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₃: X₃ {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₁, X₁: 3⋅X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: X₃ {O(n)}
t₁₂, X₁: X₁ {O(n)}
t₁₃, X₁: 3⋅X₁ {O(n)}