KoAT2 Proof Output

Initial Problem

Start: eval_twn07_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_twn07_bb0_in, eval_twn07_bb1_in, eval_twn07_bb2_in, eval_twn07_bb3_in, eval_twn07_bb4_in, eval_twn07_bb5_in, eval_twn07_start, eval_twn07_stop
Transitions:
t₁: eval_twn07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb1_in(X₃, X₄, X₂, X₃, X₄, X₅)
t₂: eval_twn07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀
t₃: eval_twn07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: X₀ ≤ 0
t₄: eval_twn07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb1_in(X₀-1, X₁+(X₅)³, X₂, X₃, X₄, X₅)
t₅: eval_twn07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂
t₆: eval_twn07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ 0
t₇: eval_twn07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅)
t₈: eval_twn07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_twn07_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_twn07_bb5_in

Found invariant X₀ ≤ X₃ for location eval_twn07_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location eval_twn07_bb2_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0 for location eval_twn07_bb3_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_twn07_bb4_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_twn07_stop

Problem after Preprocessing

Start: eval_twn07_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_twn07_bb0_in, eval_twn07_bb1_in, eval_twn07_bb2_in, eval_twn07_bb3_in, eval_twn07_bb4_in, eval_twn07_bb5_in, eval_twn07_start, eval_twn07_stop
Transitions:
t₁: eval_twn07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb1_in(X₃, X₄, X₂, X₃, X₄, X₅)
t₂: eval_twn07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
t₃: eval_twn07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₄: eval_twn07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb1_in(X₀-1, X₁+(X₅)³, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃
t₅: eval_twn07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁
t₆: eval_twn07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁
t₇: eval_twn07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁
t₈: eval_twn07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 0
t₀: eval_twn07_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_twn07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_twn07_bb1_in: [X₀]
• eval_twn07_bb2_in: [X₀-1]

MPRF for transition t₄: eval_twn07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb1_in(X₀-1, Temp_Int₁₇₆+X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_twn07_bb1_in: [X₀]
• eval_twn07_bb2_in: [X₀]

MPRF for transition t₅: eval_twn07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

2⋅X₃⋅X₅⋅X₅⋅X₅+2⋅X₄ {O(n^4)}

MPRF:

• eval_twn07_bb3_in: [X₂]
• eval_twn07_bb4_in: [X₂-1]

MPRF for transition t₇: eval_twn07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_twn07_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

2⋅X₃⋅X₅⋅X₅⋅X₅+2⋅X₄ {O(n^4)}

MPRF:

• eval_twn07_bb3_in: [X₂]
• eval_twn07_bb4_in: [X₂]

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_twn07_bb5_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 0 for location eval_twn07_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_twn07_bb4_in_v1

Found invariant X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_twn07_bb3_in_v1

Found invariant X₀ ≤ X₃ for location eval_twn07_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location eval_twn07_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_twn07_bb4_in_v2

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_twn07_stop

All Bounds

Timebounds

Overall timebound:4⋅X₃⋅X₅⋅X₅⋅X₅+2⋅X₃+4⋅X₄+5 {O(n^4)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₃ {O(n)}
t₅: 2⋅X₃⋅X₅⋅X₅⋅X₅+2⋅X₄ {O(n^4)}
t₆: 1 {O(1)}
t₇: 2⋅X₃⋅X₅⋅X₅⋅X₅+2⋅X₄ {O(n^4)}
t₈: 1 {O(1)}

Costbounds

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