KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location eval_ex_paper2_bb5_in

Found invariant 1 ≤ X₁ for location eval_ex_paper2_bb1_in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_ex_paper2_bb3_in

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_ex_paper2_bb4_in

Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location eval_ex_paper2_stop

Found invariant X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_ex_paper2_bb2_in

Problem after Preprocessing

Start: eval_ex_paper2_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_ex_paper2_bb0_in, eval_ex_paper2_bb1_in, eval_ex_paper2_bb2_in, eval_ex_paper2_bb3_in, eval_ex_paper2_bb4_in, eval_ex_paper2_bb5_in, eval_ex_paper2_start, eval_ex_paper2_stop
Transitions:
t₁: eval_ex_paper2_bb0_in(X₀, X₁, X₂) → eval_ex_paper2_bb1_in(X₀, 1, X₂)
t₂: eval_ex_paper2_bb1_in(X₀, X₁, X₂) → eval_ex_paper2_bb2_in(X₀, X₁, X₁) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁
t₃: eval_ex_paper2_bb1_in(X₀, X₁, X₂) → eval_ex_paper2_bb5_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₁
t₄: eval_ex_paper2_bb2_in(X₀, X₁, X₂) → eval_ex_paper2_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂
t₅: eval_ex_paper2_bb2_in(X₀, X₁, X₂) → eval_ex_paper2_bb4_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂
t₆: eval_ex_paper2_bb3_in(X₀, X₁, X₂) → eval_ex_paper2_bb2_in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂
t₇: eval_ex_paper2_bb4_in(X₀, X₁, X₂) → eval_ex_paper2_bb1_in(X₀, 1+X₁, X₂) :|: X₂ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ X₀
t₈: eval_ex_paper2_bb5_in(X₀, X₁, X₂) → eval_ex_paper2_stop(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₁
t₀: eval_ex_paper2_start(X₀, X₁, X₂) → eval_ex_paper2_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂: eval_ex_paper2_bb1_in(X₀, X₁, X₂) → eval_ex_paper2_bb2_in(X₀, X₁, X₁) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

• eval_ex_paper2_bb1_in: [1+X₀-X₁]
• eval_ex_paper2_bb2_in: [X₀-X₁]
• eval_ex_paper2_bb3_in: [X₀-X₁]
• eval_ex_paper2_bb4_in: [X₀-X₁]

MPRF for transition t₅: eval_ex_paper2_bb2_in(X₀, X₁, X₂) → eval_ex_paper2_bb4_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

• eval_ex_paper2_bb1_in: [1+X₀-X₁]
• eval_ex_paper2_bb2_in: [1+X₀-X₁]
• eval_ex_paper2_bb3_in: [1+X₀-X₁]
• eval_ex_paper2_bb4_in: [X₀-X₁]

MPRF for transition t₇: eval_ex_paper2_bb4_in(X₀, X₁, X₂) → eval_ex_paper2_bb1_in(X₀, 1+X₁, X₂) :|: X₂ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

• eval_ex_paper2_bb1_in: [1+X₀-X₁]
• eval_ex_paper2_bb2_in: [1+X₀-X₁]
• eval_ex_paper2_bb3_in: [1+X₀-X₁]
• eval_ex_paper2_bb4_in: [1+X₀-X₁]

MPRF for transition t₄: eval_ex_paper2_bb2_in(X₀, X₁, X₂) → eval_ex_paper2_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₀⋅X₀+3⋅X₀ {O(n^2)}

MPRF:

• eval_ex_paper2_bb1_in: [X₀]
• eval_ex_paper2_bb2_in: [1+X₀-X₂]
• eval_ex_paper2_bb3_in: [X₀-X₂]
• eval_ex_paper2_bb4_in: [0]

MPRF for transition t₆: eval_ex_paper2_bb3_in(X₀, X₁, X₂) → eval_ex_paper2_bb2_in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₀⋅X₀+3⋅X₀ {O(n^2)}

MPRF:

• eval_ex_paper2_bb1_in: [X₀]
• eval_ex_paper2_bb2_in: [1+X₀-X₂]
• eval_ex_paper2_bb3_in: [1+X₀-X₂]
• eval_ex_paper2_bb4_in: [X₀-X₂]

Cut unsatisfiable transition [t₅: eval_ex_paper2_bb2_in→eval_ex_paper2_bb4_in; t₄₅: eval_ex_paper2_bb2_in→eval_ex_paper2_bb4_in]

Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location eval_ex_paper2_bb5_in

Found invariant 1 ≤ X₁ for location eval_ex_paper2_bb1_in

Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location eval_ex_paper2_stop

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_ex_paper2_bb2_in

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_ex_paper2_bb2_in_v1

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_ex_paper2_bb3_in_v2

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_ex_paper2_bb3_in_v1

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+9⋅X₀+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+2 {O(n)}
t₃: 1 {O(1)}
t₄: X₀⋅X₀+3⋅X₀ {O(n^2)}
t₅: X₀+2 {O(n)}
t₆: X₀⋅X₀+3⋅X₀ {O(n^2)}
t₇: X₀+2 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+9⋅X₀+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+2 {O(n)}
t₃: 1 {O(1)}
t₄: X₀⋅X₀+3⋅X₀ {O(n^2)}
t₅: X₀+2 {O(n)}
t₆: X₀⋅X₀+3⋅X₀ {O(n^2)}
t₇: X₀+2 {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 1 {O(1)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀+3 {O(n)}
t₂, X₂: X₀+4 {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: X₀+4 {O(n)}
t₃, X₂: 3⋅X₀+X₂+10 {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀+3 {O(n)}
t₄, X₂: 3⋅X₀+10 {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀+3 {O(n)}
t₅, X₂: 3⋅X₀+10 {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀+3 {O(n)}
t₆, X₂: 3⋅X₀+10 {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀+3 {O(n)}
t₇, X₂: 3⋅X₀+10 {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: X₀+4 {O(n)}
t₈, X₂: 3⋅X₀+X₂+10 {O(n)}