KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalfreturnin

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

Found invariant X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalfbb2in

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalfbb3in

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalfstop

Found invariant 1 ≤ X₀ for location evalfbb4in

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 ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁
t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂
t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂
t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ X₁
t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₀) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀
t₃: evalfbb4in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb4in(1, X₁, X₂)
t₈: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

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

new bound:

X₁+2 {O(n)}

MPRF:

• evalfbb1in: [X₁-X₀]
• evalfbb2in: [X₁-X₀]
• evalfbb3in: [X₁-X₀]
• evalfbb4in: [1+X₁-X₀]

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

new bound:

X₁+2 {O(n)}

MPRF:

• evalfbb1in: [1+X₁-X₀]
• evalfbb2in: [1+X₁-X₀]
• evalfbb3in: [X₁-X₀]
• evalfbb4in: [1+X₁-X₀]

MPRF for transition t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• evalfbb1in: [2⋅X₁-X₀]
• evalfbb2in: [2⋅X₁-X₀]
• evalfbb3in: [2⋅X₁-X₀]
• evalfbb4in: [2⋅X₁-X₀]

MPRF for transition t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ of depth 1:

new bound:

2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}

MPRF:

• evalfbb1in: [X₁-X₂]
• evalfbb2in: [1+X₁-X₂]
• evalfbb3in: [X₁-X₂]
• evalfbb4in: [X₁]

MPRF for transition 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 ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}

MPRF:

• evalfbb1in: [1+X₁-X₂]
• evalfbb2in: [1+X₁-X₂]
• evalfbb3in: [X₁-X₂]
• evalfbb4in: [X₁]

Cut unsatisfiable transition [t₅: evalfbb2in→evalfbb3in; t₄₇: evalfbb2in→evalfbb3in]

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalfreturnin

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalfbb2in

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

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

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalfstop

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

Found invariant 1 ≤ X₀ for location evalfbb4in

All Bounds

Timebounds

Overall timebound:4⋅X₁⋅X₁+8⋅X₁+9 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+2 {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: 2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}
t₇: 2⋅X₁+1 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁⋅X₁+8⋅X₁+9 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+2 {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: 2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}
t₇: 2⋅X₁+1 {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 1 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₁+2 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₁+3 {O(n)}
t₃, X₀: 2⋅X₁+3 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₁⋅X₁+4⋅X₁+X₂+3 {O(n^2)}
t₄, X₀: 2⋅X₁+2 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 2⋅X₁⋅X₁+4⋅X₁+3 {O(n^2)}
t₅, X₀: 2⋅X₁+2 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁⋅X₁+4⋅X₁+3 {O(n^2)}
t₆, X₀: 2⋅X₁+2 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 2⋅X₁⋅X₁+4⋅X₁+3 {O(n^2)}
t₇, X₀: 2⋅X₁+2 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 2⋅X₁⋅X₁+4⋅X₁+3 {O(n^2)}
t₈, X₀: 2⋅X₁+3 {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁⋅X₁+4⋅X₁+X₂+3 {O(n^2)}