JT0007

jordan_tuple_divisor_downward

A divisor of a common coordinate divisor is again a common divisor.

Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable

95 new Alpha admissions come from 96 source lemmas: tuple equality reflexivity reuses an already-admitted theorem and is not counted twice. All counts use actual finite beta-coded enumerations. G008 multiplicativity is proved; the general prime-power count and distinct-prime product formula are further goals. General prime-power fields (G091) remain open. Stable is unchanged.

Exact theorem in conservative defined notation

∀ q. ∀ r. ∀ b. ∀ c. ∀ k. Dvd(q,r) → JordanTupleAllDivisible(r,b,c,k) → JordanTupleAllDivisible(q,b,c,k)

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

Definition DAG

JordanTupleAllDivisible(d,b,c,k) · 2Dvd(d,n) · 1

Actual proof prerequisites

multiple_trans · checked external prerequisite
Original expanded first-order statement 
forall q r b c k. (exists jt_factor_downfactor. (r)=(q)*jt_factor_downfactor) -> (forall jt_index_downsource jt_value_downsource. (exists jt_gap_downsourceindex. jt_gap_downsourceindex+S (jt_index_downsource)=(k)) -> (((exists fs_h_jt_downsourceat. fs_h_jt_downsourceat + S (jt_value_downsource) = S ((S (jt_index_downsource)) * c)) /\ exists fs_q_jt_downsourceat. b = fs_q_jt_downsourceat * S ((S (jt_index_downsource)) * c) + (jt_value_downsource))) -> (exists jt_factor_downsourcedivides. (jt_value_downsource)=(r)*jt_factor_downsourcedivides)) -> (forall jt_index_downtarget jt_value_downtarget. (exists jt_gap_downtargetindex. jt_gap_downtargetindex+S (jt_index_downtarget)=(k)) -> (((exists fs_h_jt_downtargetat. fs_h_jt_downtargetat + S (jt_value_downtarget) = S ((S (jt_index_downtarget)) * c)) /\ exists fs_q_jt_downtargetat. b = fs_q_jt_downtargetat * S ((S (jt_index_downtarget)) * c) + (jt_value_downtarget))) -> (exists jt_factor_downtargetdivides. (jt_value_downtarget)=(q)*jt_factor_downtargetdivides))

Complete tactic proof in conservative notation

All 21 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

21 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro q
  2. L2
    intro r
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro k
  6. L6
    intro hqr
  7. L7
    intro hall
  8. L8
    intro i
  9. L9
    intro a
  10. L10
    intro hi
02Fix variables and assumptionsL11–11

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

  1. L11
    intro ha
03Use earlier factsL12–21

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

  1. L12
    specialize multiple_trans (r)
  2. L13
    specialize multiple_trans (q)
  3. L14
    specialize multiple_trans (a)
  4. L15
    apply multiple_trans
  5. L16
    specialize hall (i)
  6. L17
    specialize hall (a)
  7. L18
    apply hall
  8. L19
    exact hi
  9. L20
    exact ha
  10. L21
    exact hqr

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro q
  2. 0002intro r
  3. 0003intro b
  4. 0004intro c
  5. 0005intro k
  6. 0006intro hqr
  7. 0007intro hall
  8. 0008intro i
  9. 0009intro a
  10. 0010intro hi
  11. 0011intro ha
  12. 0012specialize multiple_trans (r)
  13. 0013specialize multiple_trans (q)
  14. 0014specialize multiple_trans (a)
  15. 0015apply multiple_trans
  16. 0016specialize hall (i)
  17. 0017specialize hall (a)
  18. 0018apply hall
  19. 0019exact hi
  20. 0020exact ha
  21. 0021exact hqr