MV0004

mobius_prime_factor_count_unique

Actual unordered prime-factor uniqueness proves literal equality of factor counts; no canonical list is assumed.

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. ∀ b. ∀ c. ∀ l. ∀ d. ∀ e. ∀ m. PrimeFactorList(n,b,c,l) → PrimeFactorList(n,d,e,m) → l = m

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

Definition DAG

PrimeFactorList(n,b,c,l) · 2Lt(a,b) · 1BetaAt(b,c,i,x) · 3BoundedPrefix(b,c,l) · 1InjectivePrefix(b,c,l) · 1SurjectivePrefix(b,c,l) · 1

Actual proof prerequisites

prime_factor_lists_matching_by_length · checked external prerequisite
Original expanded first-order statement 
forall n b c l d e m. ((~(n = 0) /\ ((exists ff_u_fsat_mv_count_first_product ff_v_fsat_mv_count_first_product. ((((exists ff_h_fsat_mv_count_first_product_start. ff_h_fsat_mv_count_first_product_start + S (1) = S ((S (0)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_start. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_start * S ((S (0)) * ff_v_fsat_mv_count_first_product) + (1))) /\ ((((exists ff_h_fsat_mv_count_first_product_terminal. ff_h_fsat_mv_count_first_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_terminal. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_terminal * S ((S (l)) * ff_v_fsat_mv_count_first_product) + (n))) /\ forall ff_i_fsat_mv_count_first_product. (exists ff_lt_fsat_mv_count_first_product_bound. ff_lt_fsat_mv_count_first_product_bound + S ff_i_fsat_mv_count_first_product = l) -> exists ff_p_fsat_mv_count_first_product ff_r_fsat_mv_count_first_product ff_s_fsat_mv_count_first_product. ((((exists ff_h_fsat_mv_count_first_product_factor. ff_h_fsat_mv_count_first_product_factor + S (ff_p_fsat_mv_count_first_product) = S ((S (ff_i_fsat_mv_count_first_product)) * c)) /\ exists ff_q_fsat_mv_count_first_product_factor. b = ff_q_fsat_mv_count_first_product_factor * S ((S (ff_i_fsat_mv_count_first_product)) * c) + (ff_p_fsat_mv_count_first_product))) /\ ((((exists ff_h_fsat_mv_count_first_product_partial. ff_h_fsat_mv_count_first_product_partial + S (ff_r_fsat_mv_count_first_product) = S ((S (ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_partial. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_partial * S ((S (ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product) + (ff_r_fsat_mv_count_first_product))) /\ ((((exists ff_h_fsat_mv_count_first_product_successor. ff_h_fsat_mv_count_first_product_successor + S (ff_s_fsat_mv_count_first_product) = S ((S (S ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_successor. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_successor * S ((S (S ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product) + (ff_s_fsat_mv_count_first_product))) /\ ff_s_fsat_mv_count_first_product = ff_r_fsat_mv_count_first_product * ff_p_fsat_mv_count_first_product)))))) /\ (forall ftsf_index_fsat_mv_count_first_primes. (exists ftsf_gap_fsat_mv_count_first_primes_bound. ftsf_gap_fsat_mv_count_first_primes_bound + S ftsf_index_fsat_mv_count_first_primes = (l)) -> exists ftsf_factor_fsat_mv_count_first_primes. ((((exists ff_h_ftsf_fsat_mv_count_first_primes_entry. ff_h_ftsf_fsat_mv_count_first_primes_entry + S (ftsf_factor_fsat_mv_count_first_primes) = S ((S (ftsf_index_fsat_mv_count_first_primes)) * c)) /\ exists ff_q_ftsf_fsat_mv_count_first_primes_entry. b = ff_q_ftsf_fsat_mv_count_first_primes_entry * S ((S (ftsf_index_fsat_mv_count_first_primes)) * c) + (ftsf_factor_fsat_mv_count_first_primes))) /\ ((~(ftsf_factor_fsat_mv_count_first_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mv_count_first_primes_prime frm_prime_right_ftsf_fsat_mv_count_first_primes_prime. ftsf_factor_fsat_mv_count_first_primes = frm_prime_left_ftsf_fsat_mv_count_first_primes_prime * frm_prime_right_ftsf_fsat_mv_count_first_primes_prime -> frm_prime_left_ftsf_fsat_mv_count_first_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mv_count_first_primes_prime = 1))))))) -> ((~(n = 0) /\ ((exists ff_u_fsat_mv_count_second_product ff_v_fsat_mv_count_second_product. ((((exists ff_h_fsat_mv_count_second_product_start. ff_h_fsat_mv_count_second_product_start + S (1) = S ((S (0)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_start. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_start * S ((S (0)) * ff_v_fsat_mv_count_second_product) + (1))) /\ ((((exists ff_h_fsat_mv_count_second_product_terminal. ff_h_fsat_mv_count_second_product_terminal + S (n) = S ((S (m)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_terminal. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_terminal * S ((S (m)) * ff_v_fsat_mv_count_second_product) + (n))) /\ forall ff_i_fsat_mv_count_second_product. (exists ff_lt_fsat_mv_count_second_product_bound. ff_lt_fsat_mv_count_second_product_bound + S ff_i_fsat_mv_count_second_product = m) -> exists ff_p_fsat_mv_count_second_product ff_r_fsat_mv_count_second_product ff_s_fsat_mv_count_second_product. ((((exists ff_h_fsat_mv_count_second_product_factor. ff_h_fsat_mv_count_second_product_factor + S (ff_p_fsat_mv_count_second_product) = S ((S (ff_i_fsat_mv_count_second_product)) * e)) /\ exists ff_q_fsat_mv_count_second_product_factor. d = ff_q_fsat_mv_count_second_product_factor * S ((S (ff_i_fsat_mv_count_second_product)) * e) + (ff_p_fsat_mv_count_second_product))) /\ ((((exists ff_h_fsat_mv_count_second_product_partial. ff_h_fsat_mv_count_second_product_partial + S (ff_r_fsat_mv_count_second_product) = S ((S (ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_partial. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_partial * S ((S (ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product) + (ff_r_fsat_mv_count_second_product))) /\ ((((exists ff_h_fsat_mv_count_second_product_successor. ff_h_fsat_mv_count_second_product_successor + S (ff_s_fsat_mv_count_second_product) = S ((S (S ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_successor. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_successor * S ((S (S ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product) + (ff_s_fsat_mv_count_second_product))) /\ ff_s_fsat_mv_count_second_product = ff_r_fsat_mv_count_second_product * ff_p_fsat_mv_count_second_product)))))) /\ (forall ftsf_index_fsat_mv_count_second_primes. (exists ftsf_gap_fsat_mv_count_second_primes_bound. ftsf_gap_fsat_mv_count_second_primes_bound + S ftsf_index_fsat_mv_count_second_primes = (m)) -> exists ftsf_factor_fsat_mv_count_second_primes. ((((exists ff_h_ftsf_fsat_mv_count_second_primes_entry. ff_h_ftsf_fsat_mv_count_second_primes_entry + S (ftsf_factor_fsat_mv_count_second_primes) = S ((S (ftsf_index_fsat_mv_count_second_primes)) * e)) /\ exists ff_q_ftsf_fsat_mv_count_second_primes_entry. d = ff_q_ftsf_fsat_mv_count_second_primes_entry * S ((S (ftsf_index_fsat_mv_count_second_primes)) * e) + (ftsf_factor_fsat_mv_count_second_primes))) /\ ((~(ftsf_factor_fsat_mv_count_second_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mv_count_second_primes_prime frm_prime_right_ftsf_fsat_mv_count_second_primes_prime. ftsf_factor_fsat_mv_count_second_primes = frm_prime_left_ftsf_fsat_mv_count_second_primes_prime * frm_prime_right_ftsf_fsat_mv_count_second_primes_prime -> frm_prime_left_ftsf_fsat_mv_count_second_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mv_count_second_primes_prime = 1))))))) -> l = m

Complete tactic proof in conservative notation

All 22 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

22 script commands · 5 reading checkpoints · 1 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 assumptionsL1–9

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

  1. L1
    intro n
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro d
  6. L6
    intro e
  7. L7
    intro m
  8. L8
    intro ha
  9. L9
    intro hb
02Establish hL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime factor lists matching by length.

  1. L10
    have h : l = m ∧ (∃ x. ∃ y. BoundedPrefix(x,y,l) ∧ (InjectivePrefix(x,y,l) ∧ SurjectivePrefix(x,y,l)) ∧ (∀ z. ∀ n. ∀ k. Lt(z,l) → BetaAt(x,y,z,n) → BetaAt(b,c,z,k) → BetaAt(d,e,n,k)))Definitions: BoundedPrefix(x,y,l)InjectivePrefix(x,y,l)SurjectivePrefix(x,y,l)Lt(z,l)BetaAt(x,y,z,n)BetaAt(b,c,z,k)BetaAt(d,e,n,k)Original native command in the exact edition
  2. L11
    specialize prime_factor_lists_matching_by_length (l)
  3. L12
    specialize prime_factor_lists_matching_by_length (n)
  4. L13
    specialize prime_factor_lists_matching_by_length (b)
  5. L14
    specialize prime_factor_lists_matching_by_length (c)
  6. L15
    specialize prime_factor_lists_matching_by_length (m)
  7. L16
    specialize prime_factor_lists_matching_by_length (d)
  8. L17
    specialize prime_factor_lists_matching_by_length (e)
  9. L18
    apply prime_factor_lists_matching_by_length
  10. L19
    exact ha
03Use earlier factsL20–20

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

  1. L20
    exact hb
04Separate the logical casesL21–21

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

  1. L21
    cases h
05Use earlier factsL22–22

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

  1. L22
    exact h_left

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro d
  6. 0006intro e
  7. 0007intro m
  8. 0008intro ha
  9. 0009intro hb
  10. 0010have h : l = m ∧ (∃ x. ∃ y. BoundedPrefix(x,y,l) ∧ (InjectivePrefix(x,y,l) ∧ SurjectivePrefix(x,y,l)) ∧ (∀ z. ∀ n. ∀ k. Lt(z,l) → BetaAt(x,y,z,n) → BetaAt(b,c,z,k) → BetaAt(d,e,n,k)))
  11. 0011specialize prime_factor_lists_matching_by_length (l)
  12. 0012specialize prime_factor_lists_matching_by_length (n)
  13. 0013specialize prime_factor_lists_matching_by_length (b)
  14. 0014specialize prime_factor_lists_matching_by_length (c)
  15. 0015specialize prime_factor_lists_matching_by_length (m)
  16. 0016specialize prime_factor_lists_matching_by_length (d)
  17. 0017specialize prime_factor_lists_matching_by_length (e)
  18. 0018apply prime_factor_lists_matching_by_length
  19. 0019exact ha
  20. 0020exact hb
  21. 0021cases h
  22. 0022exact h_left