Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, 2⋅X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁+1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁
t₀: l2(X₀, X₁) → l1(X₀, 1) :|: 0 ≤ X₀

Preprocessing

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, 2⋅X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁+1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀, 1) :|: 0 ≤ X₀

MPRF for transition t₀: l2(X₀, X₁) → l1(X₀, 1) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

TWN: t₁: l1→l1

cycle: [t₁: l1→l1]
loop: (0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁+1 ≤ X₀,(X₀,X₁) -> (X₀,2⋅X₁)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ * 2^n

Termination: true
Formula:

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

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

TWN - Lifting for t₁: l1→l1 of 2⋅X₀+11 {O(n)}

relevant size-bounds w.r.t. t₀:
X₀: X₀+1 {O(n)}
Runtime-bound of t₀: X₀+1 {O(n)}
Results in: 2⋅X₀⋅X₀+15⋅X₀+13 {O(n^2)}

Chain transitions t₂: l1→l2 and t₀: l2→l1 to t₂₅: l1→l1

Chain transitions t₃: l0→l2 and t₀: l2→l1 to t₂₆: l0→l1

Analysing control-flow refined program

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1

MPRF for transition t₂₅: l1(X₀, X₁) -{2}> l1(X₀-1, 1) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₀⋅X₀+16⋅X₀+14 {O(n^2)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Found invariant X₁ ≤ 1 ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l1___1

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₄₉: l1(X₀, X₁) → n_l1___1(X₀, 2⋅X₁) :|: X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₄₈: n_l1___1(X₀, X₁) → n_l1___1(X₀, 2⋅X₁) :|: 2 ≤ X₁ ∧ 2+X₁ ≤ 2⋅X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+2⋅X₀+1 {O(n^2)}

MPRF for transition t₅₂: n_l1___1(X₀, X₁) → l2(X₀-1, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+17⋅X₀+16 {O(n^2)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅X₀+15⋅X₀+13 {O(n^2)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+17⋅X₀+16 {O(n^2)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅X₀+15⋅X₀+13 {O(n^2)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀+1 {O(n)}
t₀, X₁: 1 {O(1)}
t₁, X₀: X₀+1 {O(n)}
t₁, X₁: 2^(2⋅X₀⋅X₀+15⋅X₀+13) {O(EXP)}
t₂, X₀: X₀+1 {O(n)}
t₂, X₁: 2^(2⋅X₀⋅X₀+15⋅X₀+13)+1 {O(EXP)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}