MV0006

mobius_zero_has_no_value

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

No canonical signed value is asserted for the excluded input zero.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Mobius(n,z) is defined only for positive n. The canonical signed codes are 0 for zero, 2 for +1, and 1 for −1. The function is defined independently of any divisor-sum identity. These values and prime-step lemmas are prerequisites for G007; divisor-sum cancellation and full Möbius inversion remain open in this checkpoint. No signed-table proof is included here.

Exact theorem in conservative defined notation

∀ z. ¬Mobius(0,z)

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

Definition DAG

Mobius(n,z) · 1

Actual proof prerequisites

mobius_input_positive
Original expanded first-order statement 
forall z. (((~((0) = 0)) /\ ((((exists mv_square_prime_zero_excludedsquare. ((~((mv_square_prime_zero_excludedsquare) = 1) /\ forall pvs_left_zero_excludedsquareprime pvs_right_zero_excludedsquareprime. (mv_square_prime_zero_excludedsquare) = pvs_left_zero_excludedsquareprime * pvs_right_zero_excludedsquareprime -> pvs_left_zero_excludedsquareprime = 1 \/ pvs_right_zero_excludedsquareprime = 1) /\ (exists pvs_factor_zero_excludedsquaredivisor. (0) = (mv_square_prime_zero_excludedsquare * mv_square_prime_zero_excludedsquare) * pvs_factor_zero_excludedsquaredivisor))) /\ ((z) = 0))) \/ (((((~((0) = 0)) /\ (forall sfd_prime_zero_excludedsquarefree. (~((sfd_prime_zero_excludedsquarefree) = 1) /\ forall pvs_left_zero_excludedsquarefreedomain pvs_right_zero_excludedsquarefreedomain. (sfd_prime_zero_excludedsquarefree) = pvs_left_zero_excludedsquarefreedomain * pvs_right_zero_excludedsquarefreedomain -> pvs_left_zero_excludedsquarefreedomain = 1 \/ pvs_right_zero_excludedsquarefreedomain = 1) -> (exists pvs_le_gap_zero_excludedsquarefreebound. pvs_le_gap_zero_excludedsquarefreebound + (sfd_prime_zero_excludedsquarefree) = (0)) -> ~(exists pvs_factor_zero_excludedsquarefreesquare. (0) = (sfd_prime_zero_excludedsquarefree * sfd_prime_zero_excludedsquarefree) * pvs_factor_zero_excludedsquarefreesquare)))) /\ (exists mv_factor_code_zero_excludedfactors mv_factor_scale_zero_excludedfactors mv_factor_count_zero_excludedfactors. (((~(0 = 0) /\ ((exists ff_u_fsat_zero_excludedfactorsfactorization_product ff_v_fsat_zero_excludedfactorsfactorization_product. ((((exists ff_h_fsat_zero_excludedfactorsfactorization_product_start. ff_h_fsat_zero_excludedfactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_zero_excludedfactorsfactorization_product)) /\ exists ff_q_fsat_zero_excludedfactorsfactorization_product_start. ff_u_fsat_zero_excludedfactorsfactorization_product = ff_q_fsat_zero_excludedfactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_zero_excludedfactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_zero_excludedfactorsfactorization_product_terminal. ff_h_fsat_zero_excludedfactorsfactorization_product_terminal + S (0) = S ((S (mv_factor_count_zero_excludedfactors)) * ff_v_fsat_zero_excludedfactorsfactorization_product)) /\ exists ff_q_fsat_zero_excludedfactorsfactorization_product_terminal. ff_u_fsat_zero_excludedfactorsfactorization_product = ff_q_fsat_zero_excludedfactorsfactorization_product_terminal * S ((S (mv_factor_count_zero_excludedfactors)) * ff_v_fsat_zero_excludedfactorsfactorization_product) + (0))) /\ forall ff_i_fsat_zero_excludedfactorsfactorization_product. (exists ff_lt_fsat_zero_excludedfactorsfactorization_product_bound. ff_lt_fsat_zero_excludedfactorsfactorization_product_bound + S ff_i_fsat_zero_excludedfactorsfactorization_product = mv_factor_count_zero_excludedfactors) -> exists ff_p_fsat_zero_excludedfactorsfactorization_product ff_r_fsat_zero_excludedfactorsfactorization_product ff_s_fsat_zero_excludedfactorsfactorization_product. ((((exists ff_h_fsat_zero_excludedfactorsfactorization_product_factor. ff_h_fsat_zero_excludedfactorsfactorization_product_factor + S (ff_p_fsat_zero_excludedfactorsfactorization_product) = S ((S (ff_i_fsat_zero_excludedfactorsfactorization_product)) * mv_factor_scale_zero_excludedfactors)) /\ exists ff_q_fsat_zero_excludedfactorsfactorization_product_factor. mv_factor_code_zero_excludedfactors = ff_q_fsat_zero_excludedfactorsfactorization_product_factor * S ((S (ff_i_fsat_zero_excludedfactorsfactorization_product)) * mv_factor_scale_zero_excludedfactors) + (ff_p_fsat_zero_excludedfactorsfactorization_product))) /\ ((((exists ff_h_fsat_zero_excludedfactorsfactorization_product_partial. ff_h_fsat_zero_excludedfactorsfactorization_product_partial + S (ff_r_fsat_zero_excludedfactorsfactorization_product) = S ((S (ff_i_fsat_zero_excludedfactorsfactorization_product)) * ff_v_fsat_zero_excludedfactorsfactorization_product)) /\ exists ff_q_fsat_zero_excludedfactorsfactorization_product_partial. ff_u_fsat_zero_excludedfactorsfactorization_product = ff_q_fsat_zero_excludedfactorsfactorization_product_partial * S ((S (ff_i_fsat_zero_excludedfactorsfactorization_product)) * ff_v_fsat_zero_excludedfactorsfactorization_product) + (ff_r_fsat_zero_excludedfactorsfactorization_product))) /\ ((((exists ff_h_fsat_zero_excludedfactorsfactorization_product_successor. ff_h_fsat_zero_excludedfactorsfactorization_product_successor + S (ff_s_fsat_zero_excludedfactorsfactorization_product) = S ((S (S ff_i_fsat_zero_excludedfactorsfactorization_product)) * ff_v_fsat_zero_excludedfactorsfactorization_product)) /\ exists ff_q_fsat_zero_excludedfactorsfactorization_product_successor. ff_u_fsat_zero_excludedfactorsfactorization_product = ff_q_fsat_zero_excludedfactorsfactorization_product_successor * S ((S (S ff_i_fsat_zero_excludedfactorsfactorization_product)) * ff_v_fsat_zero_excludedfactorsfactorization_product) + (ff_s_fsat_zero_excludedfactorsfactorization_product))) /\ ff_s_fsat_zero_excludedfactorsfactorization_product = ff_r_fsat_zero_excludedfactorsfactorization_product * ff_p_fsat_zero_excludedfactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_zero_excludedfactorsfactorization_primes. (exists ftsf_gap_fsat_zero_excludedfactorsfactorization_primes_bound. ftsf_gap_fsat_zero_excludedfactorsfactorization_primes_bound + S ftsf_index_fsat_zero_excludedfactorsfactorization_primes = (mv_factor_count_zero_excludedfactors)) -> exists ftsf_factor_fsat_zero_excludedfactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_zero_excludedfactorsfactorization_primes_entry. ff_h_ftsf_fsat_zero_excludedfactorsfactorization_primes_entry + S (ftsf_factor_fsat_zero_excludedfactorsfactorization_primes) = S ((S (ftsf_index_fsat_zero_excludedfactorsfactorization_primes)) * mv_factor_scale_zero_excludedfactors)) /\ exists ff_q_ftsf_fsat_zero_excludedfactorsfactorization_primes_entry. mv_factor_code_zero_excludedfactors = ff_q_ftsf_fsat_zero_excludedfactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_zero_excludedfactorsfactorization_primes)) * mv_factor_scale_zero_excludedfactors) + (ftsf_factor_fsat_zero_excludedfactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_zero_excludedfactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_zero_excludedfactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_zero_excludedfactorsfactorization_primes_prime. ftsf_factor_fsat_zero_excludedfactorsfactorization_primes = frm_prime_left_ftsf_fsat_zero_excludedfactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_zero_excludedfactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_zero_excludedfactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_zero_excludedfactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_zero_excludedfactorsparityeven. (mv_factor_count_zero_excludedfactors) = 2 * mv_even_half_zero_excludedfactorsparityeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_zero_excludedfactorsparityodd. (mv_factor_count_zero_excludedfactors) = 2 * mv_odd_half_zero_excludedfactorsparityodd + 1) /\ ((z) = 1))))))))))) -> false

Complete tactic proof in conservative notation

All 7 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

7 script commands · 3 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.

Named ingredients (1)
mobius input positive
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro h
02Use earlier factsL3–6

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

  1. L3
    specialize mobius_input_positive (0)
  2. L4
    specialize mobius_input_positive (z)
  3. L5
    apply mobius_input_positive
  4. L6
    exact h
03Calculate and transport equalitiesL7–7

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

  1. L7
    refl

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro z
  2. 0002intro h
  3. 0003specialize mobius_input_positive (0)
  4. 0004specialize mobius_input_positive (z)
  5. 0005apply mobius_input_positive
  6. 0006exact h
  7. 0007refl