Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 0 < X₁
t₄: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂
t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+1, X₃+X₂) :|: X₂ < X₁
t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 0 < X₀

Preprocessing

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

Found invariant 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 0 < X₁ ∧ 0 ≤ X₀
t₄: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 0 ≤ X₀
t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+1, X₃+X₂) :|: X₂ < X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 0 < X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

TWN: t₂: l2→l2

cycle: [t₂: l2→l2]
loop: (X₂ < X₁,(X₁,X₂) -> (X₁,X₂+1)
order: [X₁; X₂]
closed-form:
X₁: X₁
X₂: X₂ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₂ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

TWN - Lifting for t₂: l2→l2 of 2⋅X₁+2⋅X₂+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₁: X₁ {O(n)}
X₂: 0 {O(1)}
Runtime-bound of t₁: X₁ {O(n)}
Results in: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}

Chain transitions t₃: l2→l1 and t₄: l1→l3 to t₄₁: l2→l3

Chain transitions t₀: l0→l1 and t₄: l1→l3 to t₄₂: l0→l3

Chain transitions t₀: l0→l1 and t₁: l1→l2 to t₄₃: l0→l2

Chain transitions t₃: l2→l1 and t₁: l1→l2 to t₄₄: l2→l2

Analysing control-flow refined program

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

Found invariant 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3

knowledge_propagation leads to new time bound 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)} for transition t₄₄: l2(X₀, X₁, X₂, X₃) -{2}> l2(X₀+X₃, X₁-1, 0, 0) :|: X₁ ≤ X₂ ∧ 1 < X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Cut unsatisfiable transition t₃: l2→l1

Found invariant X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ 1+X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l2___1

Found invariant 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3

knowledge_propagation leads to new time bound X₁ {O(n)} for transition t₈₃: l2(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂+1, X₂+X₃) :|: X₂ < X₁ ∧ X₂ < X₁ ∧ X₂ ≤ 1+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₃ ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ 1+X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₈₂: n_l2___1(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂+1, X₂+X₃) :|: X₂ < X₁ ∧ X₂ ≤ 1+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₃ ∧ X₂ ≤ 1+X₃ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₁⋅X₁+3⋅X₁ {O(n^2)}

MPRF for transition t₈₆: n_l2___1(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁+1 {O(EXP)}

Analysing control-flow refined program

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

Found invariant 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location n_l3___1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3

MPRF for transition t₁₂₇: n_l3___1(X₀, X₁, X₂, X₃) → n_l3___1(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁+1 {O(EXP)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁+2⋅X₁⋅X₁+6⋅X₁+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₂: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₃: X₁+1 {O(n)}
t₄: 1 {O(1)}
t₅: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁+1 {O(EXP)}

Costbounds

Overall costbound: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁+2⋅X₁⋅X₁+6⋅X₁+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₂: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₃: X₁+1 {O(n)}
t₄: 1 {O(1)}
t₅: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁+1 {O(EXP)}

Sizebounds

t₀, X₀: 0 {O(1)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: 0 {O(1)}
t₂, X₀: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₂, X₃: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅4⋅X₁ {O(EXP)}
t₃, X₀: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₃, X₃: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅4⋅X₁ {O(EXP)}
t₄, X₀: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 2⋅X₁⋅X₁+4⋅X₁+X₂ {O(n^2)}
t₄, X₃: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅4⋅X₁+X₃ {O(EXP)}
t₅, X₀: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: 2⋅X₁⋅X₁+4⋅X₁+X₂ {O(n^2)}
t₅, X₃: 2⋅2^(2⋅X₁⋅X₁+4⋅X₁)⋅X₁⋅X₁+2^(2⋅X₁⋅X₁+4⋅X₁)⋅4⋅X₁+X₃ {O(EXP)}