KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalfreturnin

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalfbb1in

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalfbb2in

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalfbb3in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalfstop

Found invariant 0 ≤ 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+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂
t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 0) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₃: evalfbb4in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb4in(0, X₁, X₂)
t₈: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₀
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

MPRF for transition t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 0) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalfbb1in: [X₁-1-X₀]
• evalfbb2in: [X₁-1-X₀]
• evalfbb3in: [X₁-X₂]
• evalfbb4in: [X₁-X₀]

MPRF for transition t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

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

MPRF for transition t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

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

MPRF for transition t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁⋅X₁+2⋅X₁+1 {O(n^2)}

MPRF:

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

MPRF for transition t₆: evalfbb1in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁⋅X₁+X₁+1 {O(n^2)}

MPRF:

• evalfbb1in: [1+X₀-X₂]
• evalfbb2in: [1+X₀-X₂]
• evalfbb3in: [X₀-X₂]
• evalfbb4in: [1+X₀]

Cut unsatisfiable transition [t₅: evalfbb2in→evalfbb3in; t₄₆: evalfbb2in→evalfbb3in]

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalfreturnin

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalfbb2in

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalfbb2in_v1

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalfbb1in_v1

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalfstop

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

Found invariant 0 ≤ X₀ for location evalfbb4in

All Bounds

Timebounds

Overall timebound:2⋅X₁⋅X₁+6⋅X₁+6 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: X₁⋅X₁+2⋅X₁+1 {O(n^2)}
t₅: X₁ {O(n)}
t₆: X₁⋅X₁+X₁+1 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+6⋅X₁+6 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: X₁⋅X₁+2⋅X₁+1 {O(n^2)}
t₅: X₁ {O(n)}
t₆: X₁⋅X₁+X₁+1 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 0 {O(1)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₁⋅X₁+X₁+X₂+1 {O(n^2)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁⋅X₁+X₁+1 {O(n^2)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₁⋅X₁+X₁+1 {O(n^2)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₁⋅X₁+X₁+1 {O(n^2)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁⋅X₁+X₁+1 {O(n^2)}
t₈, X₀: X₁ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: X₁⋅X₁+X₁+X₂+1 {O(n^2)}