KoAT2 Proof Output

Initial Problem

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval1, eval2, start
Transitions:
t₀: eval1(X₀, X₁) → eval2(X₀, 1) :|: 0 ≤ X₀
t₂: eval2(X₀, X₁) → eval1(X₀-1, X₁) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₁: eval2(X₀, X₁) → eval2(X₀, 2⋅X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₃: start(X₀, X₁) → eval1(X₀, X₁)

Preprocessing

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ 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, 1) :|: 0 ≤ X₀
t₂: eval2(X₀, X₁) → eval1(X₀-1, X₁) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁
t₁: eval2(X₀, X₁) → eval2(X₀, 2⋅X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₀+X₁
t₃: start(X₀, X₁) → eval1(X₀, X₁)

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

new bound:

X₀+1 {O(n)}

MPRF:

• eval1: [1+X₀]
• eval2: [X₀]

MPRF for transition t₂: eval2(X₀, X₁) → eval1(X₀-1, X₁) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ 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+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₀+X₁,(X₀,X₁) -> (X₀,2⋅X₁))
transformed loop: (1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₀+X₁,(X₀,X₁) -> (X₀,2⋅X₁))
loop: (1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₀+X₁,(X₀,X₁) -> (X₀,2⋅X₁))
order: [X₁; X₀]
closed-form:

X₁: X₁⋅(2)^n
X₀: X₀

Termination: true
Formula:

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

Stabilization-Threshold for: 1 ≤ X₀+X₁

alphas_abs: X₀
M': 1
N: 1
Bound: log(X₀)+2 {O(log(n))}

Stabilization-Threshold for: 1+X₁ ≤ X₀

alphas_abs: X₀
M': 1
N: 1
Bound: log(X₀)+2 {O(log(n))}

TWN - Lifting for [1: eval2->eval2] of 2⋅log(X₀)+6 {O(log(n))}

relevant size-bounds w.r.t. t₀: eval1→eval2:
X₀: X₀+1 {O(n)}
Runtime-bound of t₀: X₀+1 {O(n)}
Results in: 2⋅X₀⋅log(X₀)+6⋅X₀+2⋅log(X₀)+6 {O(log(n)*n)}

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅log(X₀)+8⋅X₀+2⋅log(X₀)+9 {O(log(n)*n)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅log(X₀)+6⋅X₀+2⋅log(X₀)+6 {O(log(n)*n)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅log(X₀)+8⋅X₀+2⋅log(X₀)+9 {O(log(n)*n)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅log(X₀)+6⋅X₀+2⋅log(X₀)+6 {O(log(n)*n)}
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⋅log(X₀))⋅2^(2⋅log(X₀))⋅2^(6⋅X₀)⋅64 {O(EXP)}
t₂, X₀: X₀+1 {O(n)}
t₂, X₁: 2^(2⋅X₀⋅log(X₀))⋅2^(2⋅log(X₀))⋅2^(6⋅X₀)⋅64+1 {O(EXP)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}