KoAT2 Proof Output

Initial Problem

Start: eval_twn04_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: eval_twn04_bb0_in, eval_twn04_bb1_in, eval_twn04_bb2_in, eval_twn04_bb3_in, eval_twn04_bb4_in, eval_twn04_bb5_in, eval_twn04_start, eval_twn04_stop
Transitions:
t₁: eval_twn04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb1_in(X₅, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₆
t₂: eval_twn04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₆ ≤ 0
t₃: eval_twn04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb2_in(X₀, X₆, X₇, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀
t₄: eval_twn04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₀ ≤ 0
t₅: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ (X₂)²
t₆: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: (X₂)² ≤ X₁
t₇: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₁ ≤ 0
t₈: eval_twn04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb2_in(X₀, (X₀)²+5⋅X₁, 2⋅X₂, X₃, X₄, X₅, X₆, X₇)
t₉: eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb1_in(X₀-1, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₁₀: eval_twn04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₀: eval_twn04_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_twn04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

Preprocessing

Eliminate variables [X₃; X₄] that do not contribute to the problem

Found invariant X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb4_in

Found invariant 1 ≤ X₄ ∧ X₀ ≤ X₃ for location eval_twn04_bb1_in

Found invariant X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb2_in

Found invariant X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb3_in

Problem after Preprocessing

Start: eval_twn04_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_twn04_bb0_in, eval_twn04_bb1_in, eval_twn04_bb2_in, eval_twn04_bb3_in, eval_twn04_bb4_in, eval_twn04_bb5_in, eval_twn04_start, eval_twn04_stop
Transitions:
t₂₁: eval_twn04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb1_in(X₃, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₄
t₂₂: eval_twn04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ 0
t₂₃: eval_twn04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb2_in(X₀, X₄, X₅, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ X₀ ≤ X₃
t₂₄: eval_twn04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ 1 ≤ X₄ ∧ X₀ ≤ X₃
t₂₅: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ (X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₂₆: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: (X₂)² ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₂₇: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₂₈: eval_twn04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb2_in(X₀, (X₀)²+5⋅X₁, 2⋅X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₂₉: eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb1_in(X₀-1, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₃₀: eval_twn04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₃₁: eval_twn04_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂₃: eval_twn04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb2_in(X₀, X₄, X₅, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_twn04_bb1_in: [X₀]
• eval_twn04_bb2_in: [X₀-1]
• eval_twn04_bb3_in: [X₀-1]
• eval_twn04_bb4_in: [X₀-1]

MPRF for transition t₂₆: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: (X₂)² ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_twn04_bb1_in: [X₀]
• eval_twn04_bb2_in: [X₀]
• eval_twn04_bb3_in: [X₀]
• eval_twn04_bb4_in: [X₀-1]

MPRF for transition t₂₇: eval_twn04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

1 {O(1)}

MPRF:

• eval_twn04_bb1_in: [1]
• eval_twn04_bb2_in: [1]
• eval_twn04_bb3_in: [1]
• eval_twn04_bb4_in: [1]

MPRF for transition t₂₉: eval_twn04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn04_bb1_in(X₀-1, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_twn04_bb1_in: [X₀]
• eval_twn04_bb2_in: [X₀]
• eval_twn04_bb3_in: [X₀]
• eval_twn04_bb4_in: [X₀]

Cut unsatisfiable transition [t₂₇: eval_twn04_bb2_in→eval_twn04_bb4_in]

Found invariant 5+X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 7 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 7 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 6 ≤ X₁ ∧ 7 ≤ X₀+X₁ ∧ 5+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb2_in_v1

Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb2_in

Found invariant 1 ≤ X₄ ∧ X₀ ≤ X₃ for location eval_twn04_bb1_in

Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb3_in_v1

Found invariant 5+X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 7 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 7 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 6 ≤ X₁ ∧ 7 ≤ X₀+X₁ ∧ 5+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_twn04_bb3_in_v2

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: inf {Infinity}
t₂₆: X₃ {O(n)}
t₂₇: 1 {O(1)}
t₂₈: inf {Infinity}
t₂₉: X₃ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: inf {Infinity}
t₂₆: X₃ {O(n)}
t₂₇: 1 {O(1)}
t₂₈: inf {Infinity}
t₂₉: X₃ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Sizebounds

t₂₁, X₀: X₃ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: X₃ {O(n)}
t₂₃, X₁: 2⋅X₄ {O(n)}
t₂₃, X₂: 2⋅X₅ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₃, X₅: X₅ {O(n)}
t₂₄, X₀: 2⋅X₃ {O(n)}
t₂₄, X₃: 2⋅X₃ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₄, X₅: 2⋅X₅ {O(n)}
t₂₅, X₀: X₃ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₆, X₀: X₃ {O(n)}
t₂₆, X₃: X₃ {O(n)}
t₂₆, X₄: X₄ {O(n)}
t₂₆, X₅: X₅ {O(n)}
t₂₇, X₀: 0 {O(1)}
t₂₇, X₁: 0 {O(1)}
t₂₇, X₃: 0 {O(1)}
t₂₇, X₄: 0 {O(1)}
t₂₇, X₅: X₅ {O(n)}
t₂₈, X₀: X₃ {O(n)}
t₂₈, X₃: X₃ {O(n)}
t₂₈, X₄: X₄ {O(n)}
t₂₈, X₅: X₅ {O(n)}
t₂₉, X₀: X₃ {O(n)}
t₂₉, X₃: X₃ {O(n)}
t₂₉, X₄: X₄ {O(n)}
t₂₉, X₅: X₅ {O(n)}
t₃₀, X₀: 2⋅X₃+X₀ {O(n)}
t₃₀, X₃: 3⋅X₃ {O(n)}
t₃₀, X₄: 3⋅X₄ {O(n)}
t₃₀, X₅: 3⋅X₅ {O(n)}
t₃₁, X₀: X₀ {O(n)}
t₃₁, X₁: X₁ {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₁, X₃: X₃ {O(n)}
t₃₁, X₄: X₄ {O(n)}
t₃₁, X₅: X₅ {O(n)}