KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ X₆ ∧ X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_twn03_bb3_in

Found invariant X₂ ≤ X₆ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₀ for location eval_twn03_bb1_in

Found invariant X₂ ≤ X₆ ∧ X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_twn03_bb4_in

Found invariant 1 ≤ X₆ ∧ 2 ≤ X₄+X₆ ∧ 2 ≤ X₂+X₆ ∧ X₂ ≤ X₆ ∧ 2 ≤ X₀+X₆ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location eval_twn03_bb2_in

Problem after Preprocessing

Start: eval_twn03_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_twn03_bb0_in, eval_twn03_bb1_in, eval_twn03_bb2_in, eval_twn03_bb3_in, eval_twn03_bb4_in, eval_twn03_bb5_in, eval_twn03_start, eval_twn03_stop
Transitions:
t₁: eval_twn03_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb1_in(X₄, X₁, X₆, X₃, X₄, X₅, X₆) :|: 1 ≤ X₄
t₂: eval_twn03_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₄ ≤ 0
t₃: eval_twn03_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₄ ≤ X₀ ∧ X₂ ≤ X₆
t₄: eval_twn03_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb3_in(X₀, X₅, X₂, X₀, X₄, X₅, X₆) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₄ ≤ X₀ ∧ X₂ ≤ X₆
t₅: eval_twn03_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb1_in(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₆ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₆ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₂+X₆ ∧ 2 ≤ X₄+X₆ ∧ X₄ ≤ X₀ ∧ X₂ ≤ X₆
t₆: eval_twn03_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₃ ≤ (X₁)² ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆ ∧ X₄ ≤ X₃
t₇: eval_twn03_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: (X₁)² ≤ X₃ ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆ ∧ X₄ ≤ X₃
t₈: eval_twn03_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb3_in(X₀, 2⋅X₁, X₂, (X₂)²+5⋅X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆ ∧ X₄ ≤ X₃
t₉: eval_twn03_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_twn03_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

MPRF for transition t₃: eval_twn03_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₄ ≤ X₀ ∧ X₂ ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• eval_twn03_bb1_in: [X₂]
• eval_twn03_bb2_in: [X₂-1]

MPRF for transition t₅: eval_twn03_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_twn03_bb1_in(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₆ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₆ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₂+X₆ ∧ 2 ≤ X₄+X₆ ∧ X₄ ≤ X₀ ∧ X₂ ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• eval_twn03_bb1_in: [X₂]
• eval_twn03_bb2_in: [X₂]

Found invariant X₂ ≤ X₆ ∧ 4+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 6 ≤ X₃+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 5 ≤ X₃ ∧ 5+X₂ ≤ X₃ ∧ 6 ≤ X₀+X₃ ∧ 4+X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_twn03_bb3_in_v1

Found invariant X₂ ≤ X₆ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₀ for location eval_twn03_bb1_in

Found invariant X₂ ≤ X₆ ∧ 4+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 6 ≤ X₃+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 5 ≤ X₃ ∧ 5+X₂ ≤ X₃ ∧ 6 ≤ X₀+X₃ ∧ 4+X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_twn03_bb4_in_v2

Found invariant X₂ ≤ X₆ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_twn03_bb4_in_v1

Found invariant X₂ ≤ X₆ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_twn03_bb3_in

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₆ {O(n)}
t₄: 1 {O(1)}
t₅: X₆ {O(n)}
t₆: inf {Infinity}
t₇: 1 {O(1)}
t₈: inf {Infinity}
t₉: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₆ {O(n)}
t₄: 1 {O(1)}
t₅: X₆ {O(n)}
t₆: inf {Infinity}
t₇: 1 {O(1)}
t₈: inf {Infinity}
t₉: 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)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₆ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₃, X₀: X₆⋅X₆+X₄+X₆ {O(n^2)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₆ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₃, X₆: X₆ {O(n)}
t₄, X₀: X₆⋅X₆+2⋅X₄+X₆ {O(n^2)}
t₄, X₁: 2⋅X₅ {O(n)}
t₄, X₂: 2⋅X₆ {O(n)}
t₄, X₃: X₆⋅X₆+2⋅X₄+X₆ {O(n^2)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₄, X₆: 2⋅X₆ {O(n)}
t₅, X₀: X₆⋅X₆+X₄+X₆ {O(n^2)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₆ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₅, X₆: X₆ {O(n)}
t₆, X₀: X₆⋅X₆+2⋅X₄+X₆ {O(n^2)}
t₆, X₂: 2⋅X₆ {O(n)}
t₆, X₄: 2⋅X₄ {O(n)}
t₆, X₅: 2⋅X₅ {O(n)}
t₆, X₆: 2⋅X₆ {O(n)}
t₇, X₀: 2⋅X₆⋅X₆+2⋅X₆+4⋅X₄ {O(n^2)}
t₇, X₂: 4⋅X₆ {O(n)}
t₇, X₄: 4⋅X₄ {O(n)}
t₇, X₅: 4⋅X₅ {O(n)}
t₇, X₆: 4⋅X₆ {O(n)}
t₈, X₀: X₆⋅X₆+2⋅X₄+X₆ {O(n^2)}
t₈, X₂: 2⋅X₆ {O(n)}
t₈, X₄: 2⋅X₄ {O(n)}
t₈, X₅: 2⋅X₅ {O(n)}
t₈, X₆: 2⋅X₆ {O(n)}
t₉, X₀: 2⋅X₆⋅X₆+2⋅X₆+4⋅X₄+X₀ {O(n^2)}
t₉, X₂: 4⋅X₆+X₂ {O(n)}
t₉, X₄: 5⋅X₄ {O(n)}
t₉, X₅: 5⋅X₅ {O(n)}
t₉, X₆: 5⋅X₆ {O(n)}