TS0011

balanced_zero_congruence_implies_multiple

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 17 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

factor_difference Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

TS0017 negative_one_congruent_square_norm_multiple TS0024 prime_divisible_two_square_norm_unit_coordinate_yields_negative_one_root

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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