KoAT2 Proof Output

Initial Problem

Start: evalfstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalfbb1in, evalfbb2in, evalfbb3in, evalfbb4in, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₆: evalfbb1in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 1+X₂)
t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₂ ≤ X₀
t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂
t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(X₀, X₁-1, X₂)
t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 1) :|: 1 ≤ X₁
t₃: evalfbb4in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb4in(X₁, X₀, X₂)
t₈: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂)
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₁ ≤ 0 for location evalfreturnin

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb1in

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location evalfbb2in

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ for location evalfbb3in

Found invariant X₁ ≤ 0 for location evalfstop

Problem after Preprocessing

Start: evalfstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalfbb1in, evalfbb2in, evalfbb3in, evalfbb4in, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₆: evalfbb1in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀
t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂
t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂
t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(X₀, X₁-1, X₂) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂
t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 1) :|: 1 ≤ X₁
t₃: evalfbb4in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb4in(X₁, X₀, X₂)
t₈: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂) :|: X₁ ≤ 0
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

MPRF for transition t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 1) :|: 1 ≤ X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁]
• evalfbb4in: [1+X₁]

MPRF for transition t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁-1]
• evalfbb4in: [X₁]

MPRF for transition t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(X₀, X₁-1, X₂) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁]
• evalfbb4in: [X₁]

TWN: t₄: evalfbb2in→evalfbb1in

cycle: [t₄: evalfbb2in→evalfbb1in; t₆: evalfbb1in→evalfbb2in]
original loop: (X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀,(X₀,X₁,X₂) -> (X₀,X₁,1+X₂))
transformed loop: (X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀,(X₀,X₁,X₂) -> (X₀,X₁,1+X₂))
loop: (X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀,(X₀,X₁,X₂) -> (X₀,X₁,1+X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂ + [[n != 0]]⋅n^1
X₁: X₁
X₀: X₀

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
∨ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂

Stabilization-Threshold for: X₂ ≤ X₀

alphas_abs: 1+X₀+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₂+4 {O(n)}

TWN - Lifting for [4: evalfbb2in->evalfbb1in; 6: evalfbb1in->evalfbb2in] of 2⋅X₀+2⋅X₂+6 {O(n)}

relevant size-bounds w.r.t. t₂: evalfbb4in→evalfbb2in:
X₀: X₁ {O(n)}
X₂: 1 {O(1)}
Runtime-bound of t₂: X₀+1 {O(n)}
Results in: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+8 {O(n^2)}

Found invariant X₁ ≤ 0 for location evalfreturnin

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location evalfbb2in

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ for location evalfbb3in

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb2in_v1

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb1in_v1

Found invariant X₁ ≤ 0 for location evalfstop

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalfbb1in_v2

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅X₁+19⋅X₀+4⋅X₁+21 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+8 {O(n^2)}
t₅: X₀ {O(n)}
t₆: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+8 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀⋅X₁+19⋅X₀+4⋅X₁+21 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+8 {O(n^2)}
t₅: X₀ {O(n)}
t₆: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+8 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 1 {O(1)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+X₂+10 {O(n^2)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+9 {O(n^2)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+10 {O(n^2)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+9 {O(n^2)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+10 {O(n^2)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₀ {O(n)}
t₈, X₂: 2⋅X₀⋅X₁+2⋅X₁+8⋅X₀+X₂+10 {O(n^2)}