AF0006

factor_permutation_prefix_reflect

Actual beta functionality turns a finite forward entry preservation into reverse preservation at every original index.

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. ∀ d. ∀ e. ∀ l. ∀ i. ∀ a. (∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(d,e,x,y)) → Lt(i,l) → BetaAt(d,e,i,a) → BetaAt(b,c,i,a)

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

Definition DAG

Lt(a,b) · 2BetaAt(b,c,i,x) · 6

Actual proof prerequisites

beta_at_exists · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c d e l i a. (forall pfp_i_reflect pfp_a_reflect. (exists pfp_gap_reflectbound. pfp_gap_reflectbound + S (pfp_i_reflect) = (l)) -> (((exists ff_h_pfp_reflectold. ff_h_pfp_reflectold + S (pfp_a_reflect) = S ((S (pfp_i_reflect)) * c)) /\ exists ff_q_pfp_reflectold. b = ff_q_pfp_reflectold * S ((S (pfp_i_reflect)) * c) + (pfp_a_reflect))) -> (((exists ff_h_pfp_reflectnew. ff_h_pfp_reflectnew + S (pfp_a_reflect) = S ((S (pfp_i_reflect)) * e)) /\ exists ff_q_pfp_reflectnew. d = ff_q_pfp_reflectnew * S ((S (pfp_i_reflect)) * e) + (pfp_a_reflect)))) -> (exists pfp_gap_reflect_index. pfp_gap_reflect_index + S (i) = (l)) -> (((exists ff_h_pfp_reflect_new. ff_h_pfp_reflect_new + S (a) = S ((S (i)) * e)) /\ exists ff_q_pfp_reflect_new. d = ff_q_pfp_reflect_new * S ((S (i)) * e) + (a))) -> (((exists ff_h_pfp_reflect_old. ff_h_pfp_reflect_old + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_reflect_old. b = ff_q_pfp_reflect_old * S ((S (i)) * c) + (a)))

Complete tactic proof in conservative notation

All 34 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

34 script commands · 7 reading checkpoints · 3 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 assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro a
  8. L8
    intro hp
  9. L9
    intro hi
  10. L10
    intro hnew
02Establish holdL11–15

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

  1. L11
    have hold : ∃ z. BetaAt(b,c,i,z)Definitions: BetaAt(b,c,i,z)Original native command in the exact edition
  2. L12
    specialize beta_at_exists (b)
  3. L13
    specialize beta_at_exists (c)
  4. L14
    specialize beta_at_exists (i)
  5. L15
    apply beta_at_exists
03Separate the logical casesL16–16

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

  1. L16
    cases hold
04Establish hmovedL17–22

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

  1. L17
    have hmoved : BetaAt(d,e,i,x)Definitions: BetaAt(d,e,i,x)Original native command in the exact edition
  2. L18
    specialize hp (i)
  3. L19
    specialize hp (x)
  4. L20
    apply hp
  5. L21
    exact hi
  6. L22
    exact hold_witness
05Establish heqL23–32

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

  1. L23
    have heq : a = x
  2. L24
    specialize beta_at_unique (d)
  3. L25
    specialize beta_at_unique (e)
  4. L26
    specialize beta_at_unique (i)
  5. L27
    specialize beta_at_unique (a)
  6. L28
    specialize beta_at_unique (x)
  7. L29
    apply beta_at_unique
  8. L30
    exact hnew
  9. L31
    exact hmoved
  10. L32
    rewrite heq
06Calculate and transport equalitiesL33–33

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

  1. L33
    rewrite heq
07Use earlier factsL34–34

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

  1. L34
    exact hold_witness

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro l
  6. 0006intro i
  7. 0007intro a
  8. 0008intro hp
  9. 0009intro hi
  10. 0010intro hnew
  11. 0011have hold : ∃ z. BetaAt(b,c,i,z)
  12. 0012specialize beta_at_exists (b)
  13. 0013specialize beta_at_exists (c)
  14. 0014specialize beta_at_exists (i)
  15. 0015apply beta_at_exists
  16. 0016cases hold
  17. 0017have hmoved : BetaAt(d,e,i,x)
  18. 0018specialize hp (i)
  19. 0019specialize hp (x)
  20. 0020apply hp
  21. 0021exact hi
  22. 0022exact hold_witness
  23. 0023have heq : a = x
  24. 0024specialize beta_at_unique (d)
  25. 0025specialize beta_at_unique (e)
  26. 0026specialize beta_at_unique (i)
  27. 0027specialize beta_at_unique (a)
  28. 0028specialize beta_at_unique (x)
  29. 0029apply beta_at_unique
  30. 0030exact hnew
  31. 0031exact hmoved
  32. 0032rewrite heq
  33. 0033rewrite heq
  34. 0034exact hold_witness