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) :|: 0 < X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: l2(X₀, X₁, X₂) → l1(X₂, 1, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀
t₅: l3(X₀, X₁, X₂) → l1(X₀, 1, X₂) :|: 0 < X₀ ∧ X₀ ≤ 0
t₆: l3(X₀, X₁, X₂) → l1(X₀-1, 0, X₂) :|: X₀ ≤ 0 ∧ 0 < X₀
t₇: l3(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0
t₈: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

Cut unsatisfiable transition t₅: l3→l1

Cut unsatisfiable transition t₆: l3→l1

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l5

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁ for location l1

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

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ 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) :|: 0 < X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁
t₁: l2(X₀, X₁, X₂) → l1(X₂, 1, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁
t₇: l3(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁
t₈: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁

MPRF for transition t₄: l3(X₀, X₁, X₂) → l1(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF for transition t₇: l3(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ of depth 1:

new bound:

1 {O(1)}

TWN: t₂: l1→l3

cycle: [t₂: l1→l3; t₇: l3→l1]
loop: (0 < X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ 0,(X₀,X₁) -> (X₀,0)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: [[n == 0]] * X₁

Termination: true
Formula:

loop: (0 < X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ 0,(X₀,X₁) -> (X₀,0)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: [[n == 0]] * X₁

Termination: true
Formula:

TWN - Lifting for t₂: l1→l3 of 3 {O(1)}

relevant size-bounds w.r.t. t₄:
Runtime-bound of t₄: X₂ {O(n)}
Results in: 3⋅X₂ {O(n)}

TWN - Lifting for t₂: l1→l3 of 3 {O(1)}

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 3 {O(1)}

All Bounds

Timebounds

Overall timebound:4⋅X₂+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 3⋅X₂+3 {O(n)}
t₃: 1 {O(1)}
t₄: X₂ {O(n)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 3⋅X₂+3 {O(n)}
t₃: 1 {O(1)}
t₄: X₂ {O(n)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: 1 {O(1)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: 1 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: 0 {O(1)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: 1 {O(1)}
t₄, X₂: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: 0 {O(1)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: 0 {O(1)}
t₈, X₂: X₂ {O(n)}