JT0006

jordan_primitive_tuple_divisor_modulus

Primitivity descends from a modulus to a genuine divisor of it.

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

∀ n. ∀ m. ∀ b. ∀ c. ∀ k. Dvd(n,m) → JordanPrimitiveTuple(m,b,c,k) → JordanPrimitiveTuple(n,b,c,k)

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

Definition DAG

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

Actual proof prerequisites

multiple_trans · checked external prerequisite
Original expanded first-order statement 
forall n m b c k. (exists jt_factor_smalldivisor. (m)=(n)*jt_factor_smalldivisor) -> (forall jt_divisor_largemod. (exists jt_factor_largemodmodulus. (m)=(jt_divisor_largemod)*jt_factor_largemodmodulus) -> (forall jt_index_largemodcoordinates jt_value_largemodcoordinates. (exists jt_gap_largemodcoordinatesindex. jt_gap_largemodcoordinatesindex+S (jt_index_largemodcoordinates)=(k)) -> (((exists fs_h_jt_largemodcoordinatesat. fs_h_jt_largemodcoordinatesat + S (jt_value_largemodcoordinates) = S ((S (jt_index_largemodcoordinates)) * c)) /\ exists fs_q_jt_largemodcoordinatesat. b = fs_q_jt_largemodcoordinatesat * S ((S (jt_index_largemodcoordinates)) * c) + (jt_value_largemodcoordinates))) -> (exists jt_factor_largemodcoordinatesdivides. (jt_value_largemodcoordinates)=(jt_divisor_largemod)*jt_factor_largemodcoordinatesdivides)) -> jt_divisor_largemod=1) -> (forall jt_divisor_smallmod. (exists jt_factor_smallmodmodulus. (n)=(jt_divisor_smallmod)*jt_factor_smallmodmodulus) -> (forall jt_index_smallmodcoordinates jt_value_smallmodcoordinates. (exists jt_gap_smallmodcoordinatesindex. jt_gap_smallmodcoordinatesindex+S (jt_index_smallmodcoordinates)=(k)) -> (((exists fs_h_jt_smallmodcoordinatesat. fs_h_jt_smallmodcoordinatesat + S (jt_value_smallmodcoordinates) = S ((S (jt_index_smallmodcoordinates)) * c)) /\ exists fs_q_jt_smallmodcoordinatesat. b = fs_q_jt_smallmodcoordinatesat * S ((S (jt_index_smallmodcoordinates)) * c) + (jt_value_smallmodcoordinates))) -> (exists jt_factor_smallmodcoordinatesdivides. (jt_value_smallmodcoordinates)=(jt_divisor_smallmod)*jt_factor_smallmodcoordinatesdivides)) -> jt_divisor_smallmod=1)

Complete tactic proof in conservative notation

All 19 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

19 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 assumptionsL1–10

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

  1. L1
    intro n
  2. L2
    intro m
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro k
  6. L6
    intro hnm
  7. L7
    intro hp
  8. L8
    intro q
  9. L9
    intro hqn
  10. L10
    intro hall
02Use earlier factsL11–19

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

  1. L11
    specialize hp (q)
  2. L12
    apply hp
  3. L13
    specialize multiple_trans (n)
  4. L14
    specialize multiple_trans (q)
  5. L15
    specialize multiple_trans (m)
  6. L16
    apply multiple_trans
  7. L17
    exact hnm
  8. L18
    exact hqn
  9. L19
    exact hall

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro n
  2. 0002intro m
  3. 0003intro b
  4. 0004intro c
  5. 0005intro k
  6. 0006intro hnm
  7. 0007intro hp
  8. 0008intro q
  9. 0009intro hqn
  10. 0010intro hall
  11. 0011specialize hp (q)
  12. 0012apply hp
  13. 0013specialize multiple_trans (n)
  14. 0014specialize multiple_trans (q)
  15. 0015specialize multiple_trans (m)
  16. 0016apply multiple_trans
  17. 0017exact hnm
  18. 0018exact hqn
  19. 0019exact hall