SK001B

prime_exponent_common_divisor_drop

A common divisor of a successor beta prefix divides every entry of its predecessor.

Alpha v34 checked-use · first admitted v29 · 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.

For n>1 the exponent gcd and a real beta table classify and witness all positive root degrees. The unit n=1 has a separate uniform certificate for every positive degree. Zero is excluded. NaturalSquarefreeDecomposition is deliberately distinct from the unrelated polynomial definition.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ d. (∀ x. ∀ y. Lt(x,S l) → BetaAt(b,c,x,y) → Dvd(d,y)) → ∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → Dvd(d,y)

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

Definition DAG

Lt(a,b) · 2Dvd(d,n) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall b c l d. (forall ppf_index_common_successor ppf_entry_common_successor. (exists pvs_gap_common_successorbound. pvs_gap_common_successorbound + S (ppf_index_common_successor) = (S l)) -> (((exists ff_h_pvs_common_successorentry. ff_h_pvs_common_successorentry + S (ppf_entry_common_successor) = S ((S (ppf_index_common_successor)) * c)) /\ exists ff_q_pvs_common_successorentry. b = ff_q_pvs_common_successorentry * S ((S (ppf_index_common_successor)) * c) + (ppf_entry_common_successor))) -> (exists pvs_factor_common_successordivisor. (ppf_entry_common_successor) = (d) * pvs_factor_common_successordivisor)) -> (forall ppf_index_common_previous ppf_entry_common_previous. (exists pvs_gap_common_previousbound. pvs_gap_common_previousbound + S (ppf_index_common_previous) = (l)) -> (((exists ff_h_pvs_common_previousentry. ff_h_pvs_common_previousentry + S (ppf_entry_common_previous) = S ((S (ppf_index_common_previous)) * c)) /\ exists ff_q_pvs_common_previousentry. b = ff_q_pvs_common_previousentry * S ((S (ppf_index_common_previous)) * c) + (ppf_entry_common_previous))) -> (exists pvs_factor_common_previousdivisor. (ppf_entry_common_previous) = (d) * pvs_factor_common_previousdivisor))

Complete tactic proof in conservative notation

All 17 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

17 script commands · 2 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 assumptionsL1–9

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 d
  5. L5
    intro hcommon
  6. L6
    intro i
  7. L7
    intro e
  8. L8
    intro hi
  9. L9
    intro hat
02Use earlier factsL10–17

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

  1. L10
    specialize hcommon (i)
  2. L11
    specialize hcommon (e)
  3. L12
    apply hcommon
  4. L13
    specialize le_succ (S i)
  5. L14
    specialize le_succ (l)
  6. L15
    apply le_succ
  7. L16
    exact hi
  8. L17
    exact hat

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro d
  5. 0005intro hcommon
  6. 0006intro i
  7. 0007intro e
  8. 0008intro hi
  9. 0009intro hat
  10. 0010specialize hcommon (i)
  11. 0011specialize hcommon (e)
  12. 0012apply hcommon
  13. 0013specialize le_succ (S i)
  14. 0014specialize le_succ (l)
  15. 0015apply le_succ
  16. 0016exact hi
  17. 0017exact hat