KoAT2 Proof Output

Initial Problem

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval1, eval2, start
Transitions:
t₀: eval1(X₀, X₁) → eval2(X₀, 0) :|: 1 ≤ X₀
t₂: eval2(X₀, X₁) → eval1(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
t₁: eval2(X₀, X₁) → eval2(X₀, 1+X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₃: start(X₀, X₁) → eval1(X₀, X₁)

Preprocessing

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval2

Problem after Preprocessing

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval1, eval2, start
Transitions:
t₀: eval1(X₀, X₁) → eval2(X₀, 0) :|: 1 ≤ X₀
t₂: eval2(X₀, X₁) → eval1(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₁: eval2(X₀, X₁) → eval2(X₀, 1+X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₃: start(X₀, X₁) → eval1(X₀, X₁)

MPRF for transition t₀: eval1(X₀, X₁) → eval2(X₀, 0) :|: 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval1: [X₀]
• eval2: [X₀-1]

MPRF for transition t₂: eval2(X₀, X₁) → eval1(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval1: [1+X₀]
• eval2: [1+X₀]

TWN: t₁: eval2→eval2

cycle: [t₁: eval2→eval2]
original loop: (1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀,(X₀,X₁) -> (X₀,1+X₁))
transformed loop: (1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀,(X₀,X₁) -> (X₀,1+X₁))
loop: (1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀,(X₀,X₁) -> (X₀,1+X₁))
order: [X₁; X₀]
closed-form:

X₁: X₁ + [[n != 0]]⋅n^1
X₀: X₀

Termination: true
Formula:

0 ≤ 1 ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
∨ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₁

Stabilization-Threshold for: X₁ ≤ X₀

alphas_abs: 1+X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}

Stabilization-Threshold for: 1+X₁ ≤ X₀

alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}

TWN - Lifting for [1: eval2->eval2] of 4⋅X₀+4⋅X₁+8 {O(n)}

relevant size-bounds w.r.t. t₀: eval1→eval2:
X₀: X₀ {O(n)}
X₁: 0 {O(1)}
Runtime-bound of t₀: X₀ {O(n)}
Results in: 4⋅X₀⋅X₀+8⋅X₀ {O(n^2)}

Cut unsatisfiable transition [t₂: eval2→eval1; t₂₃: eval2→eval1]

Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval2

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval2_v1

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅X₀+10⋅X₀+2 {O(n^2)}
t₀: X₀ {O(n)}
t₁: 4⋅X₀⋅X₀+8⋅X₀ {O(n^2)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀⋅X₀+10⋅X₀+2 {O(n^2)}
t₀: X₀ {O(n)}
t₁: 4⋅X₀⋅X₀+8⋅X₀ {O(n^2)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}

Sizebounds

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