KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb2_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ X₀ for location eval_abc_11

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ X₁ for location eval_abc_bb4_in

Found invariant 0 ≤ X₁ for location eval_abc_bb1_in

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₁ for location eval_abc_stop

Found invariant 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb3_in

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₁ for location eval_abc_bb5_in

Problem after Preprocessing

Start: eval_abc_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_abc_0, eval_abc_1, eval_abc_10, eval_abc_11, eval_abc_2, eval_abc_3, eval_abc_4, eval_abc_bb0_in, eval_abc_bb1_in, eval_abc_bb2_in, eval_abc_bb3_in, eval_abc_bb4_in, eval_abc_bb5_in, eval_abc_start, eval_abc_stop
Transitions:
t₂: eval_abc_0(X₀, X₁, X₂, X₃) → eval_abc_1(X₀, X₁, X₂, X₃)
t₃: eval_abc_1(X₀, X₁, X₂, X₃) → eval_abc_2(X₀, X₁, X₂, X₃)
t₁₃: eval_abc_10(X₀, X₁, X₂, X₃) → eval_abc_11(X₀, X₁, X₂, X₃) :|: X₀ ≤ 1+X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₃
t₁₄: eval_abc_11(X₀, X₁, X₂, X₃) → eval_abc_bb1_in(X₀, X₀, X₂, X₃) :|: X₀ ≤ 1+X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₃
t₄: eval_abc_2(X₀, X₁, X₂, X₃) → eval_abc_3(X₀, X₁, X₂, X₃)
t₅: eval_abc_3(X₀, X₁, X₂, X₃) → eval_abc_4(X₀, X₁, X₂, X₃)
t₆: eval_abc_4(X₀, X₁, X₂, X₃) → eval_abc_bb1_in(X₀, 0, X₂, X₃)
t₁: eval_abc_bb0_in(X₀, X₁, X₂, X₃) → eval_abc_0(X₀, X₁, X₂, X₃)
t₇: eval_abc_bb1_in(X₀, X₁, X₂, X₃) → eval_abc_bb2_in(X₀, X₁, 0, X₃) :|: 1+X₁ ≤ X₃ ∧ 0 ≤ X₁
t₈: eval_abc_bb1_in(X₀, X₁, X₂, X₃) → eval_abc_bb5_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ 0 ≤ X₁
t₉: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ X₃
t₁₀: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ X₃
t₁₁: eval_abc_bb3_in(X₀, X₁, X₂, X₃) → eval_abc_bb2_in(X₀, X₁, 1+X₂, X₃) :|: 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₁₂: eval_abc_bb4_in(X₀, X₁, X₂, X₃) → eval_abc_10(1+X₁, X₁, X₂, X₃) :|: X₂ ≤ 1+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₃
t₁₅: eval_abc_bb5_in(X₀, X₁, X₂, X₃) → eval_abc_stop(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ X₃ ≤ X₁
t₀: eval_abc_start(X₀, X₁, X₂, X₃) → eval_abc_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₇: eval_abc_bb1_in(X₀, X₁, X₂, X₃) → eval_abc_bb2_in(X₀, X₁, 0, X₃) :|: 1+X₁ ≤ X₃ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_abc_10: [X₃-1-X₁]
• eval_abc_11: [X₃-X₂]
• eval_abc_bb1_in: [X₃-X₁]
• eval_abc_bb2_in: [X₃-1-X₁]
• eval_abc_bb3_in: [X₃-1-X₁]
• eval_abc_bb4_in: [X₃-1-X₁]

MPRF for transition t₁₀: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_abc_10: [X₃-X₀]
• eval_abc_11: [X₃-X₀]
• eval_abc_bb1_in: [X₃-X₁]
• eval_abc_bb2_in: [X₃-X₁]
• eval_abc_bb3_in: [X₃-X₁]
• eval_abc_bb4_in: [X₃-1-X₁]

MPRF for transition t₁₂: eval_abc_bb4_in(X₀, X₁, X₂, X₃) → eval_abc_10(1+X₁, X₁, X₂, X₃) :|: X₂ ≤ 1+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_abc_10: [X₃-X₀]
• eval_abc_11: [X₃-X₀]
• eval_abc_bb1_in: [X₃-X₁]
• eval_abc_bb2_in: [X₃-X₁]
• eval_abc_bb3_in: [X₃-X₁]
• eval_abc_bb4_in: [X₃-X₁]

MPRF for transition t₁₃: eval_abc_10(X₀, X₁, X₂, X₃) → eval_abc_11(X₀, X₁, X₂, X₃) :|: X₀ ≤ 1+X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_abc_10: [1+X₃-X₁]
• eval_abc_11: [X₃-X₁]
• eval_abc_bb1_in: [1+X₃-X₁]
• eval_abc_bb2_in: [1+X₃-X₁]
• eval_abc_bb3_in: [1+X₃-X₁]
• eval_abc_bb4_in: [1+X₃-X₁]

MPRF for transition t₁₄: eval_abc_11(X₀, X₁, X₂, X₃) → eval_abc_bb1_in(X₀, X₀, X₂, X₃) :|: X₀ ≤ 1+X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_abc_10: [X₃-X₁]
• eval_abc_11: [X₃-X₁]
• eval_abc_bb1_in: [X₃-X₁]
• eval_abc_bb2_in: [X₃-X₁]
• eval_abc_bb3_in: [X₃-X₁]
• eval_abc_bb4_in: [X₃-X₁]

MPRF for transition t₉: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₃⋅X₃+2⋅X₃+1 {O(n^2)}

MPRF:

• eval_abc_10: [X₃-X₂]
• eval_abc_11: [X₃-X₂]
• eval_abc_bb1_in: [1+X₃]
• eval_abc_bb2_in: [1+X₃-X₂]
• eval_abc_bb3_in: [X₃-X₂]
• eval_abc_bb4_in: [X₃-X₂]

MPRF for transition t₁₁: eval_abc_bb3_in(X₀, X₁, X₂, X₃) → eval_abc_bb2_in(X₀, X₁, 1+X₂, X₃) :|: 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₃⋅X₃+X₃+1 {O(n^2)}

MPRF:

• eval_abc_10: [1+X₁-X₂]
• eval_abc_11: [X₂-1-X₁]
• eval_abc_bb1_in: [1+X₁]
• eval_abc_bb2_in: [1+X₁-X₂]
• eval_abc_bb3_in: [1+X₁-X₂]
• eval_abc_bb4_in: [1+X₁-X₂]

Cut unsatisfiable transition [t₁₀: eval_abc_bb2_in→eval_abc_bb4_in; t₈₀: eval_abc_bb2_in→eval_abc_bb4_in]

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb2_in_v1

Found invariant 0 ≤ X₁ for location eval_abc_bb1_in

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₁ for location eval_abc_bb5_in

Found invariant 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb2_in

Found invariant 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb3_in_v1

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_abc_bb3_in_v2

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₁ for location eval_abc_stop

All Bounds

Timebounds

Overall timebound:2⋅X₃⋅X₃+8⋅X₃+12 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₃ {O(n)}
t₈: 1 {O(1)}
t₉: X₃⋅X₃+2⋅X₃+1 {O(n^2)}
t₁₀: X₃ {O(n)}
t₁₁: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₂: X₃ {O(n)}
t₁₃: X₃+1 {O(n)}
t₁₄: X₃ {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃⋅X₃+8⋅X₃+12 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₃ {O(n)}
t₈: 1 {O(1)}
t₉: X₃⋅X₃+2⋅X₃+1 {O(n^2)}
t₁₀: X₃ {O(n)}
t₁₁: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₂: X₃ {O(n)}
t₁₃: X₃+1 {O(n)}
t₁₄: X₃ {O(n)}
t₁₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: 0 {O(1)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₀+X₃ {O(n)}
t₇, X₁: X₃ {O(n)}
t₇, X₂: 0 {O(1)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₀+X₃ {O(n)}
t₈, X₁: X₃ {O(n)}
t₈, X₂: X₃⋅X₃+X₂+X₃+1 {O(n^2)}
t₈, X₃: 2⋅X₃ {O(n)}
t₉, X₀: X₀+X₃ {O(n)}
t₉, X₁: X₃ {O(n)}
t₉, X₂: X₃⋅X₃+X₃+1 {O(n^2)}
t₉, X₃: X₃ {O(n)}
t₁₀, X₀: X₀+X₃ {O(n)}
t₁₀, X₁: X₃ {O(n)}
t₁₀, X₂: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₀, X₃: X₃ {O(n)}
t₁₁, X₀: X₀+X₃ {O(n)}
t₁₁, X₁: X₃ {O(n)}
t₁₁, X₂: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: X₃ {O(n)}
t₁₂, X₁: X₃ {O(n)}
t₁₂, X₂: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: X₃ {O(n)}
t₁₃, X₁: X₃ {O(n)}
t₁₃, X₂: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: X₃ {O(n)}
t₁₄, X₁: X₃ {O(n)}
t₁₄, X₂: X₃⋅X₃+X₃+1 {O(n^2)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: X₀+X₃ {O(n)}
t₁₅, X₁: X₃ {O(n)}
t₁₅, X₂: X₃⋅X₃+X₂+X₃+1 {O(n^2)}
t₁₅, X₃: 2⋅X₃ {O(n)}