MV0007

mobius_from_prime_square

A supplied actual prime-square divisor of a positive input constructs its canonical zero Möbius value.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Mobius(n,z) is positive-domain only, independently defined from squarefreeness and actual prime-factor parity. Signed codes 0, 2 and 1 represent zero, +1 and -1. This family proves values and prime-adjunction laws; the separate Möbius-inversion family supplies the complete G007 endpoint.

Exact theorem in conservative defined notation

∀ n. ∀ p. ¬n = 0 → Prime(p) → Dvd(p · p,n) → Mobius(n,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Mobius(n,z) · 1Dvd(d,n) · 1Prime(p) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall n p. ~(n = 0) -> (~((p) = 1) /\ forall pvs_left_zero_prime pvs_right_zero_prime. (p) = pvs_left_zero_prime * pvs_right_zero_prime -> pvs_left_zero_prime = 1 \/ pvs_right_zero_prime = 1) -> (exists pvs_factor_zero_divisor. (n) = (p * p) * pvs_factor_zero_divisor) -> (((~((n) = 0)) /\ ((((exists mv_square_prime_zero_valuesquare. ((~((mv_square_prime_zero_valuesquare) = 1) /\ forall pvs_left_zero_valuesquareprime pvs_right_zero_valuesquareprime. (mv_square_prime_zero_valuesquare) = pvs_left_zero_valuesquareprime * pvs_right_zero_valuesquareprime -> pvs_left_zero_valuesquareprime = 1 \/ pvs_right_zero_valuesquareprime = 1) /\ (exists pvs_factor_zero_valuesquaredivisor. (n) = (mv_square_prime_zero_valuesquare * mv_square_prime_zero_valuesquare) * pvs_factor_zero_valuesquaredivisor))) /\ ((0) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_zero_valuesquarefree. (~((sfd_prime_zero_valuesquarefree) = 1) /\ forall pvs_left_zero_valuesquarefreedomain pvs_right_zero_valuesquarefreedomain. (sfd_prime_zero_valuesquarefree) = pvs_left_zero_valuesquarefreedomain * pvs_right_zero_valuesquarefreedomain -> pvs_left_zero_valuesquarefreedomain = 1 \/ pvs_right_zero_valuesquarefreedomain = 1) -> (exists pvs_le_gap_zero_valuesquarefreebound. pvs_le_gap_zero_valuesquarefreebound + (sfd_prime_zero_valuesquarefree) = (n)) -> ~(exists pvs_factor_zero_valuesquarefreesquare. (n) = (sfd_prime_zero_valuesquarefree * sfd_prime_zero_valuesquarefree) * pvs_factor_zero_valuesquarefreesquare)))) /\ (exists mv_factor_code_zero_valuefactors mv_factor_scale_zero_valuefactors mv_factor_count_zero_valuefactors. (((~(n = 0) /\ ((exists ff_u_fsat_zero_valuefactorsfactorization_product ff_v_fsat_zero_valuefactorsfactorization_product. ((((exists ff_h_fsat_zero_valuefactorsfactorization_product_start. ff_h_fsat_zero_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_zero_valuefactorsfactorization_product)) /\ exists ff_q_fsat_zero_valuefactorsfactorization_product_start. ff_u_fsat_zero_valuefactorsfactorization_product = ff_q_fsat_zero_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_zero_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_zero_valuefactorsfactorization_product_terminal. ff_h_fsat_zero_valuefactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_zero_valuefactors)) * ff_v_fsat_zero_valuefactorsfactorization_product)) /\ exists ff_q_fsat_zero_valuefactorsfactorization_product_terminal. ff_u_fsat_zero_valuefactorsfactorization_product = ff_q_fsat_zero_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_zero_valuefactors)) * ff_v_fsat_zero_valuefactorsfactorization_product) + (n))) /\ forall ff_i_fsat_zero_valuefactorsfactorization_product. (exists ff_lt_fsat_zero_valuefactorsfactorization_product_bound. ff_lt_fsat_zero_valuefactorsfactorization_product_bound + S ff_i_fsat_zero_valuefactorsfactorization_product = mv_factor_count_zero_valuefactors) -> exists ff_p_fsat_zero_valuefactorsfactorization_product ff_r_fsat_zero_valuefactorsfactorization_product ff_s_fsat_zero_valuefactorsfactorization_product. ((((exists ff_h_fsat_zero_valuefactorsfactorization_product_factor. ff_h_fsat_zero_valuefactorsfactorization_product_factor + S (ff_p_fsat_zero_valuefactorsfactorization_product) = S ((S (ff_i_fsat_zero_valuefactorsfactorization_product)) * mv_factor_scale_zero_valuefactors)) /\ exists ff_q_fsat_zero_valuefactorsfactorization_product_factor. mv_factor_code_zero_valuefactors = ff_q_fsat_zero_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_zero_valuefactorsfactorization_product)) * mv_factor_scale_zero_valuefactors) + (ff_p_fsat_zero_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_zero_valuefactorsfactorization_product_partial. ff_h_fsat_zero_valuefactorsfactorization_product_partial + S (ff_r_fsat_zero_valuefactorsfactorization_product) = S ((S (ff_i_fsat_zero_valuefactorsfactorization_product)) * ff_v_fsat_zero_valuefactorsfactorization_product)) /\ exists ff_q_fsat_zero_valuefactorsfactorization_product_partial. ff_u_fsat_zero_valuefactorsfactorization_product = ff_q_fsat_zero_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_zero_valuefactorsfactorization_product)) * ff_v_fsat_zero_valuefactorsfactorization_product) + (ff_r_fsat_zero_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_zero_valuefactorsfactorization_product_successor. ff_h_fsat_zero_valuefactorsfactorization_product_successor + S (ff_s_fsat_zero_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_zero_valuefactorsfactorization_product)) * ff_v_fsat_zero_valuefactorsfactorization_product)) /\ exists ff_q_fsat_zero_valuefactorsfactorization_product_successor. ff_u_fsat_zero_valuefactorsfactorization_product = ff_q_fsat_zero_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_zero_valuefactorsfactorization_product)) * ff_v_fsat_zero_valuefactorsfactorization_product) + (ff_s_fsat_zero_valuefactorsfactorization_product))) /\ ff_s_fsat_zero_valuefactorsfactorization_product = ff_r_fsat_zero_valuefactorsfactorization_product * ff_p_fsat_zero_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_zero_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_zero_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_zero_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_zero_valuefactorsfactorization_primes = (mv_factor_count_zero_valuefactors)) -> exists ftsf_factor_fsat_zero_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_zero_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_zero_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_zero_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_zero_valuefactorsfactorization_primes)) * mv_factor_scale_zero_valuefactors)) /\ exists ff_q_ftsf_fsat_zero_valuefactorsfactorization_primes_entry. mv_factor_code_zero_valuefactors = ff_q_ftsf_fsat_zero_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_zero_valuefactorsfactorization_primes)) * mv_factor_scale_zero_valuefactors) + (ftsf_factor_fsat_zero_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_zero_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_zero_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_zero_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_zero_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_zero_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_zero_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_zero_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_zero_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_zero_valuefactorsparityeven. (mv_factor_count_zero_valuefactors) = 2 * mv_even_half_zero_valuefactorsparityeven) /\ ((0) = 2))) \/ (((exists mv_odd_half_zero_valuefactorsparityodd. (mv_factor_count_zero_valuefactors) = 2 * mv_odd_half_zero_valuefactorsparityodd + 1) /\ ((0) = 1)))))))))))

Complete tactic proof in conservative notation

All 14 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

14 script commands · 8 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
  2. L2
    intro p
  3. L3
    intro hn
  4. L4
    intro hp
  5. L5
    intro hdiv
02Separate the logical casesL6–6

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    split
03Use earlier factsL7–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L7
    exact hn
04Separate the logical casesL8–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L8
    left
  2. L9
    split
05Construct an explicit witnessL10–10

Supply the displayed value, then prove that it has the required property.

  1. L10
    exists p
06Separate the logical casesL11–11

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    split
07Use earlier factsL12–13

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L12
    exact hp
  2. L13
    exact hdiv
08Calculate and transport equalitiesL14–14

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L14
    refl

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro n
  2. 0002intro p
  3. 0003intro hn
  4. 0004intro hp
  5. 0005intro hdiv
  6. 0006split
  7. 0007exact hn
  8. 0008left
  9. 0009split
  10. 0010exists p
  11. 0011split
  12. 0012exact hp
  13. 0013exact hdiv
  14. 0014refl