AF0007

factor_permutation_all_prime_entry

Every actual decoded entry of an all-prime prefix is prime, without a supplied choice of matching factor.

Alpha v34 checked-use · first admitted v28 · 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. Exact original first-admission records.

The original division, gcd, cancellation, and factor-existence foundations are exposed through checked wrappers. Unordered uniqueness adds an actual bounded, injective, surjective index map matching repeated prime occurrences. The empty factor list represents one, not zero.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ i. ∀ p. AllPrime(b,c,l) → Lt(i,l) → BetaAt(b,c,i,p) → Prime(p)

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

Definition DAG

Lt(a,b) · 1Prime(p) · 2BetaAt(b,c,i,x) · 2AllPrime(b,c,l) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c l i p. (forall ftsf_index_fsat_entry_primes. (exists ftsf_gap_fsat_entry_primes_bound. ftsf_gap_fsat_entry_primes_bound + S ftsf_index_fsat_entry_primes = (l)) -> exists ftsf_factor_fsat_entry_primes. ((((exists ff_h_ftsf_fsat_entry_primes_entry. ff_h_ftsf_fsat_entry_primes_entry + S (ftsf_factor_fsat_entry_primes) = S ((S (ftsf_index_fsat_entry_primes)) * c)) /\ exists ff_q_ftsf_fsat_entry_primes_entry. b = ff_q_ftsf_fsat_entry_primes_entry * S ((S (ftsf_index_fsat_entry_primes)) * c) + (ftsf_factor_fsat_entry_primes))) /\ ((~(ftsf_factor_fsat_entry_primes = 1) /\ forall frm_prime_left_ftsf_fsat_entry_primes_prime frm_prime_right_ftsf_fsat_entry_primes_prime. ftsf_factor_fsat_entry_primes = frm_prime_left_ftsf_fsat_entry_primes_prime * frm_prime_right_ftsf_fsat_entry_primes_prime -> frm_prime_left_ftsf_fsat_entry_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_entry_primes_prime = 1)))) -> (exists pfp_gap_entry_bound. pfp_gap_entry_bound + S (i) = (l)) -> (((exists ff_h_pfp_entry. ff_h_pfp_entry + S (p) = S ((S (i)) * c)) /\ exists ff_q_pfp_entry. b = ff_q_pfp_entry * S ((S (i)) * c) + (p))) -> ((~(p = 1) /\ forall frm_prime_left_pfp_entry_prime frm_prime_right_pfp_entry_prime. p = frm_prime_left_pfp_entry_prime * frm_prime_right_pfp_entry_prime -> frm_prime_left_pfp_entry_prime = 1 \/ frm_prime_right_pfp_entry_prime = 1))

Complete tactic proof in conservative notation

All 26 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

26 script commands · 6 reading checkpoints · 2 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 assumptionsL1–8

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro i
  5. L5
    intro p
  6. L6
    intro hprimes
  7. L7
    intro hi
  8. L8
    intro hp
02Establish hexL9–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprimes.

  1. L9
    have hex : ∃ q. BetaAt(b,c,i,q) ∧ Prime(q)Definitions: BetaAt(b,c,i,q)Prime(q)Original native command in the exact edition
  2. L10
    specialize hprimes (i)
  3. L11
    apply hprimes
  4. L12
    exact hi
03Separate the logical casesL13–14

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

  1. L13
    cases hex
  2. L14
    cases hex_witness
04Establish heqL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L15
    have heq : p = x
  2. L16
    specialize beta_at_unique (b)
  3. L17
    specialize beta_at_unique (c)
  4. L18
    specialize beta_at_unique (i)
  5. L19
    specialize beta_at_unique (p)
  6. L20
    specialize beta_at_unique (x)
  7. L21
    apply beta_at_unique
  8. L22
    exact hp
  9. L23
    exact hex_witness_left
  10. L24
    rewrite heq
05Calculate and transport equalitiesL25–25

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

  1. L25
    rewrite heq
06Use earlier factsL26–26

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

  1. L26
    exact hex_witness_right

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro p
  6. 0006intro hprimes
  7. 0007intro hi
  8. 0008intro hp
  9. 0009have hex : ∃ q. BetaAt(b,c,i,q) ∧ Prime(q)
  10. 0010specialize hprimes (i)
  11. 0011apply hprimes
  12. 0012exact hi
  13. 0013cases hex
  14. 0014cases hex_witness
  15. 0015have heq : p = x
  16. 0016specialize beta_at_unique (b)
  17. 0017specialize beta_at_unique (c)
  18. 0018specialize beta_at_unique (i)
  19. 0019specialize beta_at_unique (p)
  20. 0020specialize beta_at_unique (x)
  21. 0021apply beta_at_unique
  22. 0022exact hp
  23. 0023exact hex_witness_left
  24. 0024rewrite heq
  25. 0025rewrite heq
  26. 0026exact hex_witness_right