KoAT2 Proof Output

Initial Problem

Start: eval_ex_paper1_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_ex_paper1_bb0_in, eval_ex_paper1_bb1_in, eval_ex_paper1_bb2_in, eval_ex_paper1_bb3_in, eval_ex_paper1_start, eval_ex_paper1_stop
Transitions:
t₁: eval_ex_paper1_bb0_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃
t₃: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₄: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀
t₅: eval_ex_paper1_bb2_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₁
t₆: eval_ex_paper1_bb2_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ 0
t₇: eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_stop(X₀, X₁, X₂, X₃)
t₀: eval_ex_paper1_start(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₀ for location eval_ex_paper1_bb2_in

Problem after Preprocessing

Start: eval_ex_paper1_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_ex_paper1_bb0_in, eval_ex_paper1_bb1_in, eval_ex_paper1_bb2_in, eval_ex_paper1_bb3_in, eval_ex_paper1_start, eval_ex_paper1_stop
Transitions:
t₁: eval_ex_paper1_bb0_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃
t₃: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₄: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀
t₅: eval_ex_paper1_bb2_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₃
t₆: eval_ex_paper1_bb2_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₃
t₇: eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_stop(X₀, X₁, X₂, X₃)
t₀: eval_ex_paper1_start(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb0_in(X₀, X₁, X₂, X₃)

Found invariant 3 ≤ X₃ ∧ 5 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3+X₁ ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ 2+X₀ ≤ X₃ ∧ 2 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_ex_paper1_bb2_in_v3

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_ex_paper1_bb1_in_v2

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_ex_paper1_bb2_in_v1

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 2+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 2+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location eval_ex_paper1_bb1_in_v1

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_ex_paper1_bb1_in

Found invariant 3 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ 2+X₂ ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ 5 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_ex_paper1_bb2_in_v2

Analysing control-flow refined program

MPRF for transition t₅₂: eval_ex_paper1_bb1_in_v1(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in_v3(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 2+X₀ ≤ X₃ ∧ 2+X₁ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_ex_paper1_bb1_in_v1: [1+X₀]
• eval_ex_paper1_bb2_in_v3: [X₀]

MPRF for transition t₅₃: eval_ex_paper1_bb2_in_v3(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in_v1(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2+X₀ ≤ X₃ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ ∧ 3+X₁ ≤ X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ 5 ≤ X₂+X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_ex_paper1_bb1_in_v1: [X₀]
• eval_ex_paper1_bb2_in_v3: [X₀]

MPRF for transition t₄₉: eval_ex_paper1_bb1_in_v2(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in_v2(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₂+X₃+2 {O(n)}

MPRF:

• eval_ex_paper1_bb1_in_v2: [1+X₃-X₀]
• eval_ex_paper1_bb2_in_v2: [X₃-X₀]

MPRF for transition t₅₀: eval_ex_paper1_bb2_in_v2(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in_v2(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₂ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ 4 ≤ X₂+X₃ ∧ 5 ≤ X₀+X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_ex_paper1_bb1_in_v2: [X₃-X₀]
• eval_ex_paper1_bb2_in_v2: [X₃-X₀]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

cfr-program:

Start: eval_ex_paper1_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_ex_paper1_bb0_in, eval_ex_paper1_bb1_in, eval_ex_paper1_bb1_in_v1, eval_ex_paper1_bb1_in_v2, eval_ex_paper1_bb2_in_v1, eval_ex_paper1_bb2_in_v2, eval_ex_paper1_bb2_in_v3, eval_ex_paper1_bb3_in, eval_ex_paper1_start, eval_ex_paper1_stop
Transitions:
t₁: eval_ex_paper1_bb0_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in(X₂, X₁, X₂, X₃)
t₄₅: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in_v1(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₃: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₄: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₄₃: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₄₄: eval_ex_paper1_bb1_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₅₂: eval_ex_paper1_bb1_in_v1(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in_v3(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 2+X₀ ≤ X₃ ∧ 2+X₁ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0
t₅₁: eval_ex_paper1_bb1_in_v1(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 2+X₀ ≤ X₃ ∧ 2+X₁ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0
t₄₉: eval_ex_paper1_bb1_in_v2(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb2_in_v2(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₃
t₄₈: eval_ex_paper1_bb1_in_v2(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₃
t₄₆: eval_ex_paper1_bb2_in_v1(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in_v1(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₄₇: eval_ex_paper1_bb2_in_v1(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in_v2(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₅₀: eval_ex_paper1_bb2_in_v2(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in_v2(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₂ ≤ X₃ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ 4 ≤ X₂+X₃ ∧ 5 ≤ X₀+X₃
t₅₃: eval_ex_paper1_bb2_in_v3(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb1_in_v1(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2+X₀ ≤ X₃ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ ∧ 3+X₁ ≤ X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ 5 ≤ X₂+X₃
t₇: eval_ex_paper1_bb3_in(X₀, X₁, X₂, X₃) → eval_ex_paper1_stop(X₀, X₁, X₂, X₃)
t₀: eval_ex_paper1_start(X₀, X₁, X₂, X₃) → eval_ex_paper1_bb0_in(X₀, X₁, X₂, X₃)

All Bounds

Timebounds

Overall timebound:2⋅X₃+4⋅X₂+16 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₇: 1 {O(1)}
t₄₃: 1 {O(1)}
t₄₄: 1 {O(1)}
t₄₅: 1 {O(1)}
t₄₆: 1 {O(1)}
t₄₇: 1 {O(1)}
t₄₈: 1 {O(1)}
t₄₉: X₂+X₃+2 {O(n)}
t₅₀: X₂+X₃+1 {O(n)}
t₅₁: 1 {O(1)}
t₅₂: X₂+1 {O(n)}
t₅₃: X₂ {O(n)}

Costbounds

Overall costbound: 2⋅X₃+4⋅X₂+16 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₇: 1 {O(1)}
t₄₃: 1 {O(1)}
t₄₄: 1 {O(1)}
t₄₅: 1 {O(1)}
t₄₆: 1 {O(1)}
t₄₇: 1 {O(1)}
t₄₈: 1 {O(1)}
t₄₉: X₂+X₃+2 {O(n)}
t₅₀: X₂+X₃+1 {O(n)}
t₅₁: 1 {O(1)}
t₅₂: X₂+1 {O(n)}
t₅₃: X₂ {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₇, X₀: 7⋅X₂+X₃+4 {O(n)}
t₇, X₁: 8⋅X₁ {O(n)}
t₇, X₂: 8⋅X₂ {O(n)}
t₇, X₃: 8⋅X₃ {O(n)}
t₄₃, X₀: X₂ {O(n)}
t₄₃, X₁: X₁ {O(n)}
t₄₃, X₂: X₂ {O(n)}
t₄₃, X₃: X₃ {O(n)}
t₄₄, X₀: X₂ {O(n)}
t₄₄, X₁: X₁ {O(n)}
t₄₄, X₂: X₂ {O(n)}
t₄₄, X₃: X₃ {O(n)}
t₄₅, X₀: X₂ {O(n)}
t₄₅, X₁: X₁ {O(n)}
t₄₅, X₂: X₂ {O(n)}
t₄₅, X₃: X₃ {O(n)}
t₄₆, X₀: X₂ {O(n)}
t₄₆, X₁: X₁ {O(n)}
t₄₆, X₂: X₂ {O(n)}
t₄₆, X₃: X₃ {O(n)}
t₄₇, X₀: X₂+1 {O(n)}
t₄₇, X₁: X₁ {O(n)}
t₄₇, X₂: X₂ {O(n)}
t₄₇, X₃: X₃ {O(n)}
t₄₈, X₀: 3⋅X₂+X₃+4 {O(n)}
t₄₈, X₁: 2⋅X₁ {O(n)}
t₄₈, X₂: 2⋅X₂ {O(n)}
t₄₈, X₃: 2⋅X₃ {O(n)}
t₄₉, X₀: 2⋅X₂+X₃+4 {O(n)}
t₄₉, X₁: X₁ {O(n)}
t₄₉, X₂: X₂ {O(n)}
t₄₉, X₃: X₃ {O(n)}
t₅₀, X₀: 2⋅X₂+X₃+3 {O(n)}
t₅₀, X₁: X₁ {O(n)}
t₅₀, X₂: X₂ {O(n)}
t₅₀, X₃: X₃ {O(n)}
t₅₁, X₀: 0 {O(1)}
t₅₁, X₁: 2⋅X₁ {O(n)}
t₅₁, X₂: 2⋅X₂ {O(n)}
t₅₁, X₃: 2⋅X₃ {O(n)}
t₅₂, X₀: X₂ {O(n)}
t₅₂, X₁: X₁ {O(n)}
t₅₂, X₂: X₂ {O(n)}
t₅₂, X₃: X₃ {O(n)}
t₅₃, X₀: X₂ {O(n)}
t₅₃, X₁: X₁ {O(n)}
t₅₃, X₂: X₂ {O(n)}
t₅₃, X₃: X₃ {O(n)}