KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb2_in

Found invariant 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ 2+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ 2+X₁ ∧ 2 ≤ X₀ for location eval_abc_8

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

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

Found invariant 1+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_2, eval_abc_3, eval_abc_4, eval_abc_8, eval_abc_9, 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)
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)
t₁₃: eval_abc_8(X₀, X₁, X₂, X₃) → eval_abc_9(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₃
t₁₄: eval_abc_9(X₀, X₁, X₂, X₃) → eval_abc_bb1_in(X₀, X₀, X₂, X₃) :|: X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ 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) :|: X₁ ≤ X₃ ∧ 0 ≤ X₁
t₈: eval_abc_bb1_in(X₀, X₁, X₂, X₃) → eval_abc_bb5_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₁ ∧ 0 ≤ X₁
t₉: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₀: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₁: eval_abc_bb3_in(X₀, X₁, X₂, X₃) → eval_abc_bb2_in(X₀, X₁, 2+X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₁₂: eval_abc_bb4_in(X₀, X₁, X₂, X₃) → eval_abc_8(2+X₁, X₁, X₂, X₃) :|: X₂ ≤ 2+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₃
t₁₅: eval_abc_bb5_in(X₀, X₁, X₂, X₃) → eval_abc_stop(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₁ ∧ 0 ≤ 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) :|: X₁ ≤ X₃ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_abc_8: [X₃-X₁]
• eval_abc_9: [1+X₃-X₀]
• eval_abc_bb1_in: [1+X₃-X₁]
• eval_abc_bb2_in: [X₃-X₁]
• eval_abc_bb3_in: [X₃-X₁]
• eval_abc_bb4_in: [X₃-X₁]

MPRF for transition t₁₀: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_abc_8: [X₃-X₁]
• eval_abc_9: [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: [X₃-X₁]

MPRF for transition t₁₂: eval_abc_bb4_in(X₀, X₁, X₂, X₃) → eval_abc_8(2+X₁, X₁, X₂, X₃) :|: X₂ ≤ 2+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_abc_8: [X₃-X₁]
• eval_abc_9: [1+X₃-X₀]
• 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 t₁₃: eval_abc_8(X₀, X₁, X₂, X₃) → eval_abc_9(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+2 {O(n)}

MPRF:

• eval_abc_8: [2+X₃-X₁]
• eval_abc_9: [X₂-1-X₁]
• eval_abc_bb1_in: [2+X₃-X₁]
• eval_abc_bb2_in: [2+X₃-X₁]
• eval_abc_bb3_in: [2+X₃-X₁]
• eval_abc_bb4_in: [2+X₃-X₁]

MPRF for transition t₁₄: eval_abc_9(X₀, X₁, X₂, X₃) → eval_abc_bb1_in(X₀, X₀, X₂, X₃) :|: X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+3 {O(n)}

MPRF:

• eval_abc_8: [2+X₃-X₁]
• eval_abc_9: [2+X₃-X₁]
• eval_abc_bb1_in: [3+X₃-X₁]
• eval_abc_bb2_in: [3+X₃-X₁]
• eval_abc_bb3_in: [3+X₃-X₁]
• eval_abc_bb4_in: [2+X₃-X₁]

MPRF for transition t₉: eval_abc_bb2_in(X₀, X₁, X₂, X₃) → eval_abc_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃⋅X₃+6⋅X₃+8 {O(n^2)}

MPRF:

• eval_abc_8: [0]
• eval_abc_9: [0]
• eval_abc_bb1_in: [2+X₃]
• eval_abc_bb2_in: [2+X₃-X₂]
• eval_abc_bb3_in: [1+X₃-X₂]
• eval_abc_bb4_in: [0]

MPRF for transition t₁₁: eval_abc_bb3_in(X₀, X₁, X₂, X₃) → eval_abc_bb2_in(X₀, X₁, 2+X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃⋅X₃+6⋅X₃+8 {O(n^2)}

MPRF:

• eval_abc_8: [X₃-X₂]
• eval_abc_9: [X₃-X₂]
• eval_abc_bb1_in: [2+X₃]
• eval_abc_bb2_in: [2+X₃-X₂]
• eval_abc_bb3_in: [1+X₃-X₂]
• eval_abc_bb4_in: [X₃-X₂]

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

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

Found invariant 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₂ ∧ 2 ≤ 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 1+X₃ ≤ X₁ ∧ 0 ≤ X₁ for location eval_abc_bb5_in

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

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

Found invariant 2 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_abc_bb3_in_v2

Found invariant 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ 2+X₃ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ 2+X₃ ∧ 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ 2+X₁ ∧ 2 ≤ X₀ for location eval_abc_8

Found invariant 1+X₃ ≤ X₁ ∧ 0 ≤ X₁ for location eval_abc_stop

All Bounds

Timebounds

Overall timebound:2⋅X₃⋅X₃+17⋅X₃+33 {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+1 {O(n)}
t₈: 1 {O(1)}
t₉: X₃⋅X₃+6⋅X₃+8 {O(n^2)}
t₁₀: X₃+1 {O(n)}
t₁₁: X₃⋅X₃+6⋅X₃+8 {O(n^2)}
t₁₂: X₃+1 {O(n)}
t₁₃: X₃+2 {O(n)}
t₁₄: X₃+3 {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃⋅X₃+17⋅X₃+33 {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+1 {O(n)}
t₈: 1 {O(1)}
t₉: X₃⋅X₃+6⋅X₃+8 {O(n^2)}
t₁₀: X₃+1 {O(n)}
t₁₁: X₃⋅X₃+6⋅X₃+8 {O(n^2)}
t₁₂: X₃+1 {O(n)}
t₁₃: X₃+2 {O(n)}
t₁₄: X₃+3 {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: 2⋅X₃+X₀+2 {O(n)}
t₇, X₁: 2⋅X₃+2 {O(n)}
t₇, X₂: 0 {O(1)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: 2⋅X₃+X₀+2 {O(n)}
t₈, X₁: 2⋅X₃+2 {O(n)}
t₈, X₂: 2⋅X₃⋅X₃+12⋅X₃+X₂+16 {O(n^2)}
t₈, X₃: 2⋅X₃ {O(n)}
t₉, X₀: 2⋅X₃+X₀+2 {O(n)}
t₉, X₁: 2⋅X₃+2 {O(n)}
t₉, X₂: 2⋅X₃⋅X₃+12⋅X₃+16 {O(n^2)}
t₉, X₃: X₃ {O(n)}
t₁₀, X₀: 2⋅X₃+X₀+2 {O(n)}
t₁₀, X₁: 2⋅X₃+2 {O(n)}
t₁₀, X₂: 2⋅X₃⋅X₃+12⋅X₃+16 {O(n^2)}
t₁₀, X₃: X₃ {O(n)}
t₁₁, X₀: 2⋅X₃+X₀+2 {O(n)}
t₁₁, X₁: 2⋅X₃+2 {O(n)}
t₁₁, X₂: 2⋅X₃⋅X₃+12⋅X₃+16 {O(n^2)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: 2⋅X₃+2 {O(n)}
t₁₂, X₁: 2⋅X₃+2 {O(n)}
t₁₂, X₂: 2⋅X₃⋅X₃+12⋅X₃+16 {O(n^2)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₃+2 {O(n)}
t₁₃, X₁: 2⋅X₃+2 {O(n)}
t₁₃, X₂: 2⋅X₃⋅X₃+12⋅X₃+16 {O(n^2)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: 2⋅X₃+2 {O(n)}
t₁₄, X₁: 2⋅X₃+2 {O(n)}
t₁₄, X₂: 2⋅X₃⋅X₃+12⋅X₃+16 {O(n^2)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: 2⋅X₃+X₀+2 {O(n)}
t₁₅, X₁: 2⋅X₃+2 {O(n)}
t₁₅, X₂: 2⋅X₃⋅X₃+12⋅X₃+X₂+16 {O(n^2)}
t₁₅, X₃: 2⋅X₃ {O(n)}