JT000D

jordan_divisibility_congruence_transport

A genuine common modulus divisor survives balanced congruence, including divisor zero.

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. ∀ d. ∀ a. ∀ z. Dvd(d,n) → Dvd(d,a) → ModEq(n,a,z) → Dvd(d,z)

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

Definition DAG

Dvd(d,n) · 3ModEq(m,a,b) · 1

Actual proof prerequisites

linear_congruence_zero_residue_divides · checked external prerequisitemod_eq_trans · checked external prerequisitemod_eq_symm · checked external prerequisitemod_eq_of_mod_eq_multiple · checked external prerequisitedvd_to_mod_zero · checked external prerequisite
Original expanded first-order statement 
forall n d a z. (exists jt_factor_congdivisor. (n)=(d)*jt_factor_congdivisor) -> (exists jt_factor_congsource. (a)=(d)*jt_factor_congsource) -> (exists jt_left_congmod jt_right_congmod. (a)+(n)*jt_left_congmod=(z)+(n)*jt_right_congmod) -> (exists jt_factor_congtarget. (z)=(d)*jt_factor_congtarget)

Complete tactic proof in conservative notation

All 30 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

30 script commands · 4 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 assumptionsL1–7

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

  1. L1
    intro n
  2. L2
    intro d
  3. L3
    intro a
  4. L4
    intro z
  5. L5
    intro hdn
  6. L6
    intro hda
  7. L7
    intro hm
02Use earlier factsL8–17

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

  1. L8
    specialize linear_congruence_zero_residue_divides (d)
  2. L9
    specialize linear_congruence_zero_residue_divides (z)
  3. L10
    apply linear_congruence_zero_residue_divides
  4. L11
    specialize mod_eq_trans (d)
  5. L12
    specialize mod_eq_trans (z)
  6. L13
    specialize mod_eq_trans (a)
  7. L14
    specialize mod_eq_trans (0)
  8. L15
    apply mod_eq_trans
  9. L16
    specialize mod_eq_symm (d)
  10. L17
    specialize mod_eq_symm (a)
03Use earlier factsL18–27

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

  1. L18
    specialize mod_eq_symm (z)
  2. L19
    apply mod_eq_symm
  3. L20
    specialize mod_eq_of_mod_eq_multiple (d)
  4. L21
    specialize mod_eq_of_mod_eq_multiple (n)
  5. L22
    specialize mod_eq_of_mod_eq_multiple (a)
  6. L23
    specialize mod_eq_of_mod_eq_multiple (z)
  7. L24
    apply mod_eq_of_mod_eq_multiple
  8. L25
    exact hdn
  9. L26
    exact hm
  10. L27
    specialize dvd_to_mod_zero (d)
04Use earlier factsL28–30

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

  1. L28
    specialize dvd_to_mod_zero (a)
  2. L29
    apply dvd_to_mod_zero
  3. L30
    exact hda

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro n
  2. 0002intro d
  3. 0003intro a
  4. 0004intro z
  5. 0005intro hdn
  6. 0006intro hda
  7. 0007intro hm
  8. 0008specialize linear_congruence_zero_residue_divides (d)
  9. 0009specialize linear_congruence_zero_residue_divides (z)
  10. 0010apply linear_congruence_zero_residue_divides
  11. 0011specialize mod_eq_trans (d)
  12. 0012specialize mod_eq_trans (z)
  13. 0013specialize mod_eq_trans (a)
  14. 0014specialize mod_eq_trans (0)
  15. 0015apply mod_eq_trans
  16. 0016specialize mod_eq_symm (d)
  17. 0017specialize mod_eq_symm (a)
  18. 0018specialize mod_eq_symm (z)
  19. 0019apply mod_eq_symm
  20. 0020specialize mod_eq_of_mod_eq_multiple (d)
  21. 0021specialize mod_eq_of_mod_eq_multiple (n)
  22. 0022specialize mod_eq_of_mod_eq_multiple (a)
  23. 0023specialize mod_eq_of_mod_eq_multiple (z)
  24. 0024apply mod_eq_of_mod_eq_multiple
  25. 0025exact hdn
  26. 0026exact hm
  27. 0027specialize dvd_to_mod_zero (d)
  28. 0028specialize dvd_to_mod_zero (a)
  29. 0029apply dvd_to_mod_zero
  30. 0030exact hda