KoAT2 Proof Output

Initial Problem

Start: evalloopsstart
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: evalloopsbb1in, evalloopsbb3in, evalloopsbb4in, evalloopsbb5in, evalloopsbb6in, evalloopsentryin, evalloopsreturnin, evalloopsstart, evalloopsstop
Transitions:
t₅: evalloopsbb1in(X₀, X₁) → evalloopsbb4in(X₀, 1) :|: 2 ≤ X₀
t₆: evalloopsbb1in(X₀, X₁) → evalloopsbb5in(X₀, C) :|: X₀ ≤ 1
t₉: evalloopsbb3in(X₀, X₁) → evalloopsbb4in(X₀, 2⋅X₁)
t₇: evalloopsbb4in(X₀, X₁) → evalloopsbb3in(X₀, X₁) :|: 1+X₁ ≤ X₀
t₈: evalloopsbb4in(X₀, X₁) → evalloopsbb5in(X₀, X₁) :|: X₀ ≤ X₁
t₁₀: evalloopsbb5in(X₀, X₁) → evalloopsbb6in(X₀-1, X₁)
t₃: evalloopsbb6in(X₀, X₁) → evalloopsbb1in(X₀, X₁) :|: 0 ≤ X₀
t₄: evalloopsbb6in(X₀, X₁) → evalloopsreturnin(X₀, X₁) :|: 1+X₀ ≤ 0
t₁: evalloopsentryin(X₀, X₁) → evalloopsbb6in(X₀, X₁) :|: 0 ≤ X₀
t₂: evalloopsentryin(X₀, X₁) → evalloopsreturnin(X₀, X₁) :|: 1+X₀ ≤ 0
t₁₁: evalloopsreturnin(X₀, X₁) → evalloopsstop(X₀, X₁)
t₀: evalloopsstart(X₀, X₁) → evalloopsentryin(X₀, X₁)

Preprocessing

Found invariant 1+X₀ ≤ 0 for location evalloopsstop

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalloopsbb3in

Found invariant 0 ≤ X₀ for location evalloopsbb1in

Found invariant 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalloopsbb4in

Found invariant 0 ≤ 1+X₀ for location evalloopsbb6in

Found invariant 1+X₀ ≤ 0 for location evalloopsreturnin

Found invariant 0 ≤ X₀ for location evalloopsbb5in

Problem after Preprocessing

Start: evalloopsstart
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: evalloopsbb1in, evalloopsbb3in, evalloopsbb4in, evalloopsbb5in, evalloopsbb6in, evalloopsentryin, evalloopsreturnin, evalloopsstart, evalloopsstop
Transitions:
t₅: evalloopsbb1in(X₀, X₁) → evalloopsbb4in(X₀, 1) :|: 2 ≤ X₀ ∧ 0 ≤ X₀
t₆: evalloopsbb1in(X₀, X₁) → evalloopsbb5in(X₀, C) :|: X₀ ≤ 1 ∧ 0 ≤ X₀
t₉: evalloopsbb3in(X₀, X₁) → evalloopsbb4in(X₀, 2⋅X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁
t₇: evalloopsbb4in(X₀, X₁) → evalloopsbb3in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁
t₈: evalloopsbb4in(X₀, X₁) → evalloopsbb5in(X₀, X₁) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁
t₁₀: evalloopsbb5in(X₀, X₁) → evalloopsbb6in(X₀-1, X₁) :|: 0 ≤ X₀
t₃: evalloopsbb6in(X₀, X₁) → evalloopsbb1in(X₀, X₁) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀
t₄: evalloopsbb6in(X₀, X₁) → evalloopsreturnin(X₀, X₁) :|: 1+X₀ ≤ 0 ∧ 0 ≤ 1+X₀
t₁: evalloopsentryin(X₀, X₁) → evalloopsbb6in(X₀, X₁) :|: 0 ≤ X₀
t₂: evalloopsentryin(X₀, X₁) → evalloopsreturnin(X₀, X₁) :|: 1+X₀ ≤ 0
t₁₁: evalloopsreturnin(X₀, X₁) → evalloopsstop(X₀, X₁) :|: 1+X₀ ≤ 0
t₀: evalloopsstart(X₀, X₁) → evalloopsentryin(X₀, X₁)

MPRF for transition t₃: evalloopsbb6in(X₀, X₁) → evalloopsbb1in(X₀, X₁) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalloopsbb1in: [X₀]
• evalloopsbb3in: [X₀]
• evalloopsbb4in: [X₀]
• evalloopsbb5in: [X₀]
• evalloopsbb6in: [1+X₀]

MPRF for transition t₅: evalloopsbb1in(X₀, X₁) → evalloopsbb4in(X₀, 1) :|: 2 ≤ X₀ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalloopsbb1in: [X₀-1]
• evalloopsbb3in: [X₀-2]
• evalloopsbb4in: [X₀-2]
• evalloopsbb5in: [X₀-2]
• evalloopsbb6in: [X₀-1]

MPRF for transition t₆: evalloopsbb1in(X₀, X₁) → evalloopsbb5in(X₀, C) :|: X₀ ≤ 1 ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalloopsbb1in: [1+X₀]
• evalloopsbb3in: [X₀]
• evalloopsbb4in: [X₀]
• evalloopsbb5in: [X₀]
• evalloopsbb6in: [1+X₀]

MPRF for transition t₈: evalloopsbb4in(X₀, X₁) → evalloopsbb5in(X₀, X₁) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalloopsbb1in: [X₀-1]
• evalloopsbb3in: [X₀-1]
• evalloopsbb4in: [X₀-1]
• evalloopsbb5in: [X₀-2]
• evalloopsbb6in: [X₀-1]

MPRF for transition t₁₀: evalloopsbb5in(X₀, X₁) → evalloopsbb6in(X₀-1, X₁) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalloopsbb1in: [1+X₀]
• evalloopsbb3in: [1+X₀]
• evalloopsbb4in: [1+X₀]
• evalloopsbb5in: [1+X₀]
• evalloopsbb6in: [1+X₀]

TWN: t₇: evalloopsbb4in→evalloopsbb3in

cycle: [t₇: evalloopsbb4in→evalloopsbb3in; t₉: evalloopsbb3in→evalloopsbb4in]
original loop: (1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁,(X₀,X₁) -> (X₀,2⋅X₁))
transformed loop: (1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁,(X₀,X₁) -> (X₀,2⋅X₁))
loop: (1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₁,(X₀,X₁) -> (X₀,2⋅X₁))
order: [X₁; X₀]
closed-form:

X₁: X₁⋅(2)^n
X₀: X₀

Termination: true
Formula:

X₀ ≤ 3 ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 3 ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ 0 ∧ 2 ≤ X₀ ∧ 3 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 3 ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 4 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₁ ∧ 1+X₁ ≤ 0 ∧ 2 ≤ X₀
∨ 1 ≤ X₁ ∧ 1+X₁ ≤ 0 ∧ 2 ≤ X₀ ∧ 4 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 4 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0

Stabilization-Threshold for: 3 ≤ X₀+X₁

alphas_abs: 2+X₀
M': 1
N: 1
Bound: log(X₀)+4 {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 [7: evalloopsbb4in->evalloopsbb3in; 9: evalloopsbb3in->evalloopsbb4in] of 2⋅log(X₀)+8 {O(log(n))}

relevant size-bounds w.r.t. t₅: evalloopsbb1in→evalloopsbb4in:
X₀: X₀+2 {O(n)}
Runtime-bound of t₅: X₀+1 {O(n)}
Results in: 2⋅X₀⋅log(X₀)+10⋅X₀+2⋅log(X₀)+10 {O(log(n)*n)}

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅log(X₀)+25⋅X₀+4⋅log(X₀)+30 {O(log(n)*n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₇: 2⋅X₀⋅log(X₀)+10⋅X₀+2⋅log(X₀)+10 {O(log(n)*n)}
t₈: X₀+1 {O(n)}
t₉: 2⋅X₀⋅log(X₀)+10⋅X₀+2⋅log(X₀)+10 {O(log(n)*n)}
t₁₀: X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀⋅log(X₀)+25⋅X₀+4⋅log(X₀)+30 {O(log(n)*n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₇: 2⋅X₀⋅log(X₀)+10⋅X₀+2⋅log(X₀)+10 {O(log(n)*n)}
t₈: X₀+1 {O(n)}
t₉: 2⋅X₀⋅log(X₀)+10⋅X₀+2⋅log(X₀)+10 {O(log(n)*n)}
t₁₀: X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: X₀+2 {O(n)}
t₄, X₀: 1 {O(1)}
t₅, X₀: X₀+2 {O(n)}
t₅, X₁: 1 {O(1)}
t₆, X₀: 1 {O(1)}
t₇, X₀: X₀+2 {O(n)}
t₇, X₁: 1024⋅2^(10⋅X₀)⋅2^(2⋅X₀⋅log(X₀))⋅2^(2⋅log(X₀)) {O(EXP)}
t₈, X₀: X₀+2 {O(n)}
t₈, X₁: 1024⋅2^(10⋅X₀)⋅2^(2⋅X₀⋅log(X₀))⋅2^(2⋅log(X₀)) {O(EXP)}
t₉, X₀: X₀+2 {O(n)}
t₉, X₁: 1024⋅2^(10⋅X₀)⋅2^(2⋅X₀⋅log(X₀))⋅2^(2⋅log(X₀)) {O(EXP)}
t₁₀, X₀: X₀+2 {O(n)}
t₁₁, X₀: X₀+1 {O(n)}