KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 2 ≤ X₀ for location eval_perfect1_bb1_in

Found invariant X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb2_in

Found invariant X₃ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb5_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb3_in

Found invariant X₃ ≤ X₀ ∧ X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb6_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb4_in

Problem after Preprocessing

Start: eval_perfect1_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_perfect1_bb0_in, eval_perfect1_bb1_in, eval_perfect1_bb2_in, eval_perfect1_bb3_in, eval_perfect1_bb4_in, eval_perfect1_bb5_in, eval_perfect1_bb6_in, eval_perfect1_bb7_in, eval_perfect1_start, eval_perfect1_stop
Transitions:
t₂: eval_perfect1_bb0_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb1_in(X₀, X₁, X₂, X₃) :|: 2 ≤ X₀
t₁: eval_perfect1_bb0_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb7_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 1
t₃: eval_perfect1_bb1_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₀-1, X₂, X₀) :|: 2 ≤ X₀
t₄: eval_perfect1_bb2_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb3_in(X₀, X₁, X₀, X₃) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁
t₅: eval_perfect1_bb2_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb6_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁
t₆: eval_perfect1_bb3_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₇: eval_perfect1_bb3_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₈: eval_perfect1_bb4_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb3_in(X₀, X₁, X₂-X₁, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂
t₉: eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₁-1, X₂, X₃-X₁) :|: 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 2+X₂ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₃ ≤ X₀
t₁₀: eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1+X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 2+X₂ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₃ ≤ X₀
t₁₁: eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 2+X₂ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₃ ≤ X₀
t₁₂: eval_perfect1_bb6_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb7_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
t₁₃: eval_perfect1_bb6_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb7_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
t₁₄: eval_perfect1_bb6_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb7_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
t₁₅: eval_perfect1_bb7_in(X₀, X₁, X₂, X₃) → eval_perfect1_stop(X₀, X₁, X₂, X₃)
t₀: eval_perfect1_start(X₀, X₁, X₂, X₃) → eval_perfect1_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₄: eval_perfect1_bb2_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb3_in(X₀, X₁, X₀, X₃) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_perfect1_bb2_in: [X₁]
• eval_perfect1_bb3_in: [X₁-1]
• eval_perfect1_bb4_in: [X₁-1]
• eval_perfect1_bb5_in: [X₁-1]

MPRF for transition t₇: eval_perfect1_bb3_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_perfect1_bb2_in: [X₁]
• eval_perfect1_bb3_in: [X₁]
• eval_perfect1_bb4_in: [X₁]
• eval_perfect1_bb5_in: [X₁-1]

MPRF for transition t₉: eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₁-1, X₂, X₃-X₁) :|: 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 2+X₂ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_perfect1_bb2_in: [X₁]
• eval_perfect1_bb3_in: [X₁]
• eval_perfect1_bb4_in: [X₁]
• eval_perfect1_bb5_in: [X₁]

MPRF for transition t₁₀: eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1+X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 2+X₂ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_perfect1_bb2_in: [X₁-1]
• eval_perfect1_bb3_in: [X₁-1]
• eval_perfect1_bb4_in: [X₁-1]
• eval_perfect1_bb5_in: [X₁-1]

MPRF for transition t₁₁: eval_perfect1_bb5_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 2+X₂ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

• eval_perfect1_bb2_in: [2+X₁]
• eval_perfect1_bb3_in: [2+X₁]
• eval_perfect1_bb4_in: [2+X₁]
• eval_perfect1_bb5_in: [2+X₁]

MPRF for transition t₆: eval_perfect1_bb3_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ of depth 1:

new bound:

3⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}

MPRF:

• eval_perfect1_bb2_in: [1+X₀]
• eval_perfect1_bb3_in: [1+X₂-X₁]
• eval_perfect1_bb4_in: [X₂-X₁]
• eval_perfect1_bb5_in: [X₂-X₁]

MPRF for transition t₈: eval_perfect1_bb4_in(X₀, X₁, X₂, X₃) → eval_perfect1_bb3_in(X₀, X₁, X₂-X₁, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

6⋅X₀⋅X₀+8⋅X₀ {O(n^2)}

MPRF:

• eval_perfect1_bb2_in: [X₀+X₁]
• eval_perfect1_bb3_in: [X₁+X₂-1]
• eval_perfect1_bb4_in: [X₂]
• eval_perfect1_bb5_in: [X₁+X₂-1]

Cut unsatisfiable transition [t₇: eval_perfect1_bb3_in→eval_perfect1_bb5_in; t₈₃: eval_perfect1_bb3_in→eval_perfect1_bb5_in]

Found invariant 2 ≤ X₀ for location eval_perfect1_bb1_in

Found invariant X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb2_in

Found invariant X₃ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb4_in_v2

Found invariant X₃ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb5_in

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb3_in

Found invariant X₃ ≤ X₀ ∧ X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb6_in

Found invariant X₃ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb3_in_v1

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_perfect1_bb4_in_v1

Cut unsatisfiable transition [t₁₀: eval_perfect1_bb5_in→eval_perfect1_bb2_in]

All Bounds

Timebounds

Overall timebound:9⋅X₀⋅X₀+20⋅X₀+16 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: X₀ {O(n)}
t₅: 1 {O(1)}
t₆: 3⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 6⋅X₀⋅X₀+8⋅X₀ {O(n^2)}
t₉: X₀ {O(n)}
t₁₀: X₀+1 {O(n)}
t₁₁: X₀+2 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 9⋅X₀⋅X₀+20⋅X₀+16 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: X₀ {O(n)}
t₅: 1 {O(1)}
t₆: 3⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 6⋅X₀⋅X₀+8⋅X₀ {O(n^2)}
t₉: X₀ {O(n)}
t₁₀: X₀+1 {O(n)}
t₁₁: X₀+2 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₀ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₀ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: 4⋅X₀ {O(n)}
t₄, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: 0 {O(1)}
t₅, X₂: 0 {O(1)}
t₅, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 4⋅X₀ {O(n)}
t₆, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 4⋅X₀ {O(n)}
t₇, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₀ {O(n)}
t₈, X₂: 4⋅X₀ {O(n)}
t₈, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀ {O(n)}
t₉, X₂: 0 {O(1)}
t₉, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₀ {O(n)}
t₁₀, X₂: 4⋅X₀ {O(n)}
t₁₀, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₀ {O(n)}
t₁₁, X₂: 4⋅X₀ {O(n)}
t₁₁, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₂, X₀: X₀ {O(n)}
t₁₂, X₁: 0 {O(1)}
t₁₂, X₂: 0 {O(1)}
t₁₂, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₃, X₀: X₀ {O(n)}
t₁₃, X₁: 0 {O(1)}
t₁₃, X₂: 0 {O(1)}
t₁₃, X₃: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: 0 {O(1)}
t₁₄, X₂: 0 {O(1)}
t₁₄, X₃: 0 {O(1)}
t₁₅, X₀: 4⋅X₀ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: 2⋅X₀⋅X₀+4⋅X₀+X₃ {O(n^2)}