KoAT2 Proof Output

Initial Problem

Start: eval_non_linear12_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: eval_non_linear12_0, eval_non_linear12_1, eval_non_linear12_bb0_in, eval_non_linear12_bb1_in, eval_non_linear12_bb2_in, eval_non_linear12_bb3_in, eval_non_linear12_bb4_in, eval_non_linear12_bb5_in, eval_non_linear12_start, eval_non_linear12_stop
Transitions:
t₆: eval_non_linear12_0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₇: eval_non_linear12_1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb1_in(3⋅X₀+2⋅X₁+(X₆)², -5⋅X₀-3⋅X₁, X₂-1, X₃, X₄, X₅, X₆, X₇)
t₁: eval_non_linear12_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb1_in(X₄, X₅, X₇, X₃, X₄, X₅, X₆, X₇)
t₂: eval_non_linear12_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂
t₃: eval_non_linear12_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb3_in(X₀, X₁, X₂, X₀, X₄, X₅, X₆, X₇) :|: X₂ ≤ 0
t₄: eval_non_linear12_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₈: eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₃
t₉: eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃ ≤ 0
t₁₀: eval_non_linear12_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆, X₇)
t₁₁: eval_non_linear12_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₀: eval_non_linear12_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

Preprocessing

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear12_bb3_in

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ ∧ 1 ≤ X₂ for location eval_non_linear12_1

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ ∧ 1 ≤ X₂ for location eval_non_linear12_0

Found invariant X₂ ≤ X₇ for location eval_non_linear12_bb1_in

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ ∧ 1 ≤ X₂ for location eval_non_linear12_bb2_in

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_non_linear12_bb4_in

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear12_bb5_in

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear12_stop

Problem after Preprocessing

Start: eval_non_linear12_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: eval_non_linear12_0, eval_non_linear12_1, eval_non_linear12_bb0_in, eval_non_linear12_bb1_in, eval_non_linear12_bb2_in, eval_non_linear12_bb3_in, eval_non_linear12_bb4_in, eval_non_linear12_bb5_in, eval_non_linear12_start, eval_non_linear12_stop
Transitions:
t₆: eval_non_linear12_0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇
t₇: eval_non_linear12_1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb1_in(3⋅X₀+2⋅X₁+(X₆)², -5⋅X₀-3⋅X₁, X₂-1, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇
t₁: eval_non_linear12_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb1_in(X₄, X₅, X₇, X₃, X₄, X₅, X₆, X₇)
t₂: eval_non_linear12_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ X₂ ≤ X₇
t₃: eval_non_linear12_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb3_in(X₀, X₁, X₂, X₀, X₄, X₅, X₆, X₇) :|: X₂ ≤ 0 ∧ X₂ ≤ X₇
t₄: eval_non_linear12_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇
t₈: eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₇
t₉: eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₇
t₁₀: eval_non_linear12_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₇
t₁₁: eval_non_linear12_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₇ ∧ X₃ ≤ 0
t₀: eval_non_linear12_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

MPRF for transition t₂: eval_non_linear12_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ X₂ ≤ X₇ of depth 1:

new bound:

X₇ {O(n)}

MPRF:

• eval_non_linear12_0: [X₂-1]
• eval_non_linear12_1: [X₂-1]
• eval_non_linear12_bb1_in: [X₂]
• eval_non_linear12_bb2_in: [X₂-1]

MPRF for transition t₄: eval_non_linear12_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ of depth 1:

new bound:

X₇ {O(n)}

MPRF:

• eval_non_linear12_0: [X₂-1]
• eval_non_linear12_1: [X₂-1]
• eval_non_linear12_bb1_in: [X₂]
• eval_non_linear12_bb2_in: [X₂]

MPRF for transition t₆: eval_non_linear12_0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₂ ∧ 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ of depth 1:

new bound:

X₇ {O(n)}

MPRF:

• eval_non_linear12_0: [X₂]
• eval_non_linear12_1: [X₂-1]
• eval_non_linear12_bb1_in: [X₂]
• eval_non_linear12_bb2_in: [X₂]

MPRF for transition t₇: eval_non_linear12_1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb1_in(Temp_Int₅₅₁+3⋅X₀+2⋅X₁, -5⋅X₀-3⋅X₁, X₂-1, X₃, X₄, X₅, X₆, X₇) :|: 0 ≤ Temp_Int₅₅₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ of depth 1:

new bound:

X₇ {O(n)}

MPRF:

• eval_non_linear12_0: [X₂]
• eval_non_linear12_1: [X₂]
• eval_non_linear12_bb1_in: [X₂]
• eval_non_linear12_bb2_in: [X₂]

MPRF for transition t₈: eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₇ of depth 1:

new bound:

8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}

MPRF:

• eval_non_linear12_bb3_in: [X₃]
• eval_non_linear12_bb4_in: [X₃-1]

MPRF for transition t₁₀: eval_non_linear12_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear12_bb3_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₇ of depth 1:

new bound:

8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}

MPRF:

• eval_non_linear12_bb3_in: [X₃]
• eval_non_linear12_bb4_in: [X₃]

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 for location eval_non_linear12_bb3_in

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ ∧ 1 ≤ X₂ for location eval_non_linear12_1

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ ∧ 1 ≤ X₂ for location eval_non_linear12_0

Found invariant X₂ ≤ X₇ for location eval_non_linear12_bb1_in

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₂+X₇ ∧ X₂ ≤ X₇ ∧ 1 ≤ X₂ for location eval_non_linear12_bb2_in

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear12_bb5_in

Found invariant X₂ ≤ X₇ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_non_linear12_bb3_in_v1

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_non_linear12_bb4_in_v1

Found invariant X₂ ≤ X₇ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 2+X₂ ≤ X₀ ∧ 2 ≤ X₀ for location eval_non_linear12_bb4_in_v2

Found invariant X₂ ≤ X₇ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear12_stop

All Bounds

Timebounds

Overall timebound:16⋅X₆⋅X₆+10⋅X₄+4⋅X₅+4⋅X₇+5 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₇ {O(n)}
t₃: 1 {O(1)}
t₄: X₇ {O(n)}
t₆: X₇ {O(n)}
t₇: X₇ {O(n)}
t₈: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₉: 1 {O(1)}
t₁₀: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₆⋅X₆+10⋅X₄+4⋅X₅+4⋅X₇+5 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₇ {O(n)}
t₃: 1 {O(1)}
t₄: X₇ {O(n)}
t₆: X₇ {O(n)}
t₇: X₇ {O(n)}
t₈: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₉: 1 {O(1)}
t₁₀: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₀, X₆: X₆ {O(n)}
t₀, X₇: X₇ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₇ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₁, X₇: X₇ {O(n)}
t₂, X₀: 8⋅X₆⋅X₆+2⋅X₅+4⋅X₄ {O(n^2)}
t₂, X₁: 8⋅X₆⋅X₆+4⋅X₅+6⋅X₄ {O(n^2)}
t₂, X₂: X₇ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₂, X₇: X₇ {O(n)}
t₃, X₀: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₃, X₁: 8⋅X₆⋅X₆+5⋅X₅+6⋅X₄ {O(n^2)}
t₃, X₂: 2⋅X₇ {O(n)}
t₃, X₃: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₃, X₆: 2⋅X₆ {O(n)}
t₃, X₇: 2⋅X₇ {O(n)}
t₄, X₀: 8⋅X₆⋅X₆+2⋅X₅+4⋅X₄ {O(n^2)}
t₄, X₁: 8⋅X₆⋅X₆+4⋅X₅+6⋅X₄ {O(n^2)}
t₄, X₂: X₇ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₄, X₆: X₆ {O(n)}
t₄, X₇: X₇ {O(n)}
t₆, X₀: 8⋅X₆⋅X₆+2⋅X₅+4⋅X₄ {O(n^2)}
t₆, X₁: 8⋅X₆⋅X₆+4⋅X₅+6⋅X₄ {O(n^2)}
t₆, X₂: X₇ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₆, X₆: X₆ {O(n)}
t₆, X₇: X₇ {O(n)}
t₇, X₀: 8⋅X₆⋅X₆+2⋅X₅+4⋅X₄ {O(n^2)}
t₇, X₁: 8⋅X₆⋅X₆+4⋅X₅+6⋅X₄ {O(n^2)}
t₇, X₂: X₇ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₇, X₆: X₆ {O(n)}
t₇, X₇: X₇ {O(n)}
t₈, X₀: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₈, X₁: 8⋅X₆⋅X₆+5⋅X₅+6⋅X₄ {O(n^2)}
t₈, X₂: 2⋅X₇ {O(n)}
t₈, X₃: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₈, X₄: 2⋅X₄ {O(n)}
t₈, X₅: 2⋅X₅ {O(n)}
t₈, X₆: 2⋅X₆ {O(n)}
t₈, X₇: 2⋅X₇ {O(n)}
t₉, X₀: 16⋅X₆⋅X₆+10⋅X₄+4⋅X₅ {O(n^2)}
t₉, X₁: 16⋅X₆⋅X₆+10⋅X₅+12⋅X₄ {O(n^2)}
t₉, X₂: 4⋅X₇ {O(n)}
t₉, X₃: 16⋅X₆⋅X₆+10⋅X₄+4⋅X₅ {O(n^2)}
t₉, X₄: 4⋅X₄ {O(n)}
t₉, X₅: 4⋅X₅ {O(n)}
t₉, X₆: 4⋅X₆ {O(n)}
t₉, X₇: 4⋅X₇ {O(n)}
t₁₀, X₀: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₁₀, X₁: 8⋅X₆⋅X₆+5⋅X₅+6⋅X₄ {O(n^2)}
t₁₀, X₂: 2⋅X₇ {O(n)}
t₁₀, X₃: 8⋅X₆⋅X₆+2⋅X₅+5⋅X₄ {O(n^2)}
t₁₀, X₄: 2⋅X₄ {O(n)}
t₁₀, X₅: 2⋅X₅ {O(n)}
t₁₀, X₆: 2⋅X₆ {O(n)}
t₁₀, X₇: 2⋅X₇ {O(n)}
t₁₁, X₀: 16⋅X₆⋅X₆+10⋅X₄+4⋅X₅ {O(n^2)}
t₁₁, X₁: 16⋅X₆⋅X₆+10⋅X₅+12⋅X₄ {O(n^2)}
t₁₁, X₂: 4⋅X₇ {O(n)}
t₁₁, X₃: 16⋅X₆⋅X₆+10⋅X₄+4⋅X₅ {O(n^2)}
t₁₁, X₄: 4⋅X₄ {O(n)}
t₁₁, X₅: 4⋅X₅ {O(n)}
t₁₁, X₆: 4⋅X₆ {O(n)}
t₁₁, X₇: 4⋅X₇ {O(n)}