TS0011 · theorem body

balanced_zero_congruence_implies_multiple

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

A balanced congruence to zero yields an actual natural divisibility witness.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

∀ p. ∀ n. ModEq(p,n,0) → Dvd(p,n)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0003 Dvd PD0008 ModEq

In local proof propositions

none
Exact expanded first-order statement
forall p n. (exists ftcn_left_zero_source ftcn_right_zero_source. (n) + (p) * ftcn_left_zero_source = (0) + (p) * ftcn_right_zero_source) -> (exists ftcn_factor_zero_result. (n) = (p) * ftcn_factor_zero_result)

Proof neighborhood

Direct theorem prerequisites

factor_difference · Stable closed zero_add · Stable closed add_comm · Stable closed

Direct theorem dependents

TS0017 negative_one_congruent_square_norm_multiple TS0024 prime_divisible_two_square_norm_unit_coordinate_yields_negative_one_root

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

17 script commands · 7 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 assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro hmod
02Separate the logical casesL4–5

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

  1. L4
    cases hmod
  2. L5
    cases hmod_witness
03Use earlier factsL6–10

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

  1. L6
    specialize factor_difference p
  2. L7
    specialize factor_difference x1
  3. L8
    specialize factor_difference x
  4. L9
    specialize factor_difference n
  5. L10
    apply factor_difference
04Calculate and transport equalitiesL11–12

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

  1. L11
    trans 0 + p * x1
  2. L12
    symm
05Use earlier factsL13–13

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

  1. L13
    apply zero_add
06Calculate and transport equalitiesL14–15

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

  1. L14
    trans n + p * x
  2. L15
    symm
07Use earlier factsL16–17

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

  1. L16
    exact hmod_witness_witness
  2. L17
    apply add_comm

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hmod
  4. 0004cases hmod
  5. 0005cases hmod_witness
  6. 0006specialize factor_difference p
  7. 0007specialize factor_difference x1
  8. 0008specialize factor_difference x
  9. 0009specialize factor_difference n
  10. 0010apply factor_difference
  11. 0011trans 0 + p * x1
  12. 0012symm
  13. 0013apply zero_add
  14. 0014trans n + p * x
  15. 0015symm
  16. 0016exact hmod_witness_witness
  17. 0017apply add_comm