EL000E

lte_nondivisor_add_multiple

Adding any actual p-multiple preserves nondivisibility; natural differences are witnessed by balanced equality.

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.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ p. ∀ R. ∀ D. ∀ u. ¬Dvd(p,R) → Dvd(p,D) → ¬Dvd(p,R + D · u)

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

Definition DAG

Dvd(d,n) · 3

Actual proof prerequisites

balanced_zero_congruence_implies_multiple · checked external prerequisitemul_assoc · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall p R D u. ~(exists olte_factor_summand. (R) = (p) * olte_factor_summand) -> (exists olte_factor_multiple. (D) = (p) * olte_factor_multiple) -> ~(exists olte_factor_sum. (R + D * u) = (p) * olte_factor_sum)

Complete tactic proof in conservative notation

All 27 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

27 script commands · 7 reading checkpoints · 1 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 p
  2. L2
    intro R
  3. L3
    intro D
  4. L4
    intro u
  5. L5
    intro hnot
  6. L6
    intro hD
  7. L7
    intro hsum
02Separate the logical casesL8–9

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

  1. L8
    cases hD
  2. L9
    cases hsum
03Use earlier factsL10–13

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

  1. L10
    apply hnot
  2. L11
    specialize balanced_zero_congruence_implies_multiple (p)
  3. L12
    specialize balanced_zero_congruence_implies_multiple (R)
  4. L13
    apply balanced_zero_congruence_implies_multiple
04Construct an explicit witnessL14–15

Supply the displayed value, then prove that it has the required property.

  1. L14
    exists x * u
  2. L15
    exists x1
05Calculate and transport equalitiesL16–19

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

  1. L16
    trans R + (p * x) * u
  2. L17
    congr
  3. L18
    refl
  4. L19
    symm
06Use earlier factsL20–20

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

  1. L20
    apply mul_assoc
07Establish hsumcopyL21–27

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

  1. L21
    have hsumcopy : R + (p * x) * u = p * x1
  2. L22
    rewrite hD_witness at hsum_witness
  3. L23
    exact hsum_witness
  4. L24
    trans p * x1
  5. L25
    exact hsumcopy
  6. L26
    symm
  7. L27
    apply zero_add

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro p
  2. 0002intro R
  3. 0003intro D
  4. 0004intro u
  5. 0005intro hnot
  6. 0006intro hD
  7. 0007intro hsum
  8. 0008cases hD
  9. 0009cases hsum
  10. 0010apply hnot
  11. 0011specialize balanced_zero_congruence_implies_multiple (p)
  12. 0012specialize balanced_zero_congruence_implies_multiple (R)
  13. 0013apply balanced_zero_congruence_implies_multiple
  14. 0014exists x * u
  15. 0015exists x1
  16. 0016trans R + (p * x) * u
  17. 0017congr
  18. 0018refl
  19. 0019symm
  20. 0020apply mul_assoc
  21. 0021have hsumcopy : R + (p * x) * u = p * x1
  22. 0022rewrite hD_witness at hsum_witness
  23. 0023exact hsum_witness
  24. 0024trans p * x1
  25. 0025exact hsumcopy
  26. 0026symm
  27. 0027apply zero_add