KoAT2 Proof Output

Initial Problem

Start: eval_size03_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_size03_bb0_in, eval_size03_bb1_in, eval_size03_bb2_in, eval_size03_bb3_in, eval_size03_bb4_in, eval_size03_bb5_in, eval_size03_start, eval_size03_stop
Transitions:
t₁: eval_size03_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb1_in(X₃, X₆, X₂, X₃, X₄, X₅, X₆)
t₂: eval_size03_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀
t₃: eval_size03_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₄: eval_size03_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb1_in(X₀-1, 1+(X₀)²+X₁-2⋅X₀, X₂, X₃, X₄, X₅, X₆)
t₅: eval_size03_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂
t₆: eval_size03_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₂ ≤ 0
t₇: eval_size03_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅, X₆)
t₈: eval_size03_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_size03_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size03_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Eliminate variables [X₄; X₅] that do not contribute to the problem

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_size03_stop

Found invariant X₀ ≤ X₃ for location eval_size03_bb1_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size03_bb3_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_size03_bb5_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size03_bb4_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location eval_size03_bb2_in

Problem after Preprocessing

Start: eval_size03_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_size03_bb0_in, eval_size03_bb1_in, eval_size03_bb2_in, eval_size03_bb3_in, eval_size03_bb4_in, eval_size03_bb5_in, eval_size03_start, eval_size03_stop
Transitions:
t₁₇: eval_size03_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₈: eval_size03_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
t₁₉: eval_size03_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb3_in(X₀, X₁, X₁, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₂₀: eval_size03_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb1_in(X₀-1, 1+(X₀)²+X₁-2⋅X₀, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃
t₂₁: eval_size03_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁
t₂₂: eval_size03_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁
t₂₃: eval_size03_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb3_in(X₀, X₁, X₂-1, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁
t₂₄: eval_size03_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 0
t₂₅: eval_size03_start(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₁₈: eval_size03_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_size03_bb1_in: [X₀]
• eval_size03_bb2_in: [X₀-1]

MPRF for transition t₂₀: eval_size03_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb1_in(X₀-1, Temp_Int₁₇₀+Temp_Int₁₇₁+X₁-2⋅X₀, X₂, X₃, X₄) :|: 1 ≤ Temp_Int₁₇₁ ∧ X₀ ≤ Temp_Int₁₇₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_size03_bb1_in: [X₀]
• eval_size03_bb2_in: [X₀]

MPRF for transition t₂₁: eval_size03_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}

MPRF:

• eval_size03_bb3_in: [X₂]
• eval_size03_bb4_in: [X₂-1]

MPRF for transition t₂₃: eval_size03_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_size03_bb3_in(X₀, X₁, X₂-1, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}

MPRF:

• eval_size03_bb3_in: [X₂]
• eval_size03_bb4_in: [X₂]

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_size03_stop

Found invariant X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size03_bb3_in_v1

Found invariant X₀ ≤ X₃ for location eval_size03_bb1_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 0 for location eval_size03_bb3_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size03_bb4_in_v1

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_size03_bb5_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location eval_size03_bb2_in

Found invariant X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size03_bb4_in_v2

All Bounds

Timebounds

Overall timebound:28⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+10⋅X₃+4⋅X₄+5 {O(n^3)}
t₁₇: 1 {O(1)}
t₁₈: X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₃ {O(n)}
t₂₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₂: 1 {O(1)}
t₂₃: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 28⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+10⋅X₃+4⋅X₄+5 {O(n^3)}
t₁₇: 1 {O(1)}
t₁₈: X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₃ {O(n)}
t₂₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₂: 1 {O(1)}
t₂₃: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₃ {O(n)}
t₁₇, X₁: X₄ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: X₃ {O(n)}
t₁₈, X₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+4⋅X₃+X₄ {O(n^3)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 2⋅X₃ {O(n)}
t₁₉, X₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₁₉, X₂: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: X₃ {O(n)}
t₂₀, X₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+4⋅X₃+X₄ {O(n^3)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: 2⋅X₃ {O(n)}
t₂₁, X₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₁, X₂: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₁, X₃: 2⋅X₃ {O(n)}
t₂₁, X₄: 2⋅X₄ {O(n)}
t₂₂, X₀: 4⋅X₃ {O(n)}
t₂₂, X₁: 28⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+4⋅X₄+8⋅X₃ {O(n^3)}
t₂₂, X₂: 28⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+4⋅X₄+8⋅X₃ {O(n^3)}
t₂₂, X₃: 4⋅X₃ {O(n)}
t₂₂, X₄: 4⋅X₄ {O(n)}
t₂₃, X₀: 2⋅X₃ {O(n)}
t₂₃, X₁: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₃, X₂: 14⋅X₃⋅X₃⋅X₃+10⋅X₃⋅X₃+2⋅X₄+4⋅X₃ {O(n^3)}
t₂₃, X₃: 2⋅X₃ {O(n)}
t₂₃, X₄: 2⋅X₄ {O(n)}
t₂₄, X₀: 4⋅X₃ {O(n)}
t₂₄, X₁: 28⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+4⋅X₄+8⋅X₃ {O(n^3)}
t₂₄, X₂: 28⋅X₃⋅X₃⋅X₃+20⋅X₃⋅X₃+4⋅X₄+8⋅X₃ {O(n^3)}
t₂₄, X₃: 4⋅X₃ {O(n)}
t₂₄, X₄: 4⋅X₄ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}