SL000B

mod_eight_remainder_unique

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

Two bounded decompositions modulo eight have the same remainder.

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 a r b s. p = 8 * a + r -> (exists u. u + S r = 8) -> p = 8 * b + s -> (exists v. v + S s = 8) -> r = s

Constructive proof overview

Generated structural guide

Two bounded decompositions modulo eight have the same remainder.

The unchanged tactic script uses 1 declared prerequisite and contains 23 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

Proof neighborhood

Direct dependencies

division_remainder_unique Stable theorem; checked-use authorized

Direct dependents

SL000C mod_eight_good_bad_exclusive

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

23 script commands · 5 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.

01Fix variables and assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro r
  4. L4
    intro b
  5. L5
    intro s
  6. L6
    intro hfirst
  7. L7
    intro hrbound
  8. L8
    intro hsecond
  9. L9
    intro hsbound
02Establish huniqueL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder unique.

  1. L10
    have hunique : a = b /\ r = s
  2. L11
    specialize division_remainder_unique 8
  3. L12
    specialize division_remainder_unique p
  4. L13
    specialize division_remainder_unique a
  5. L14
    specialize division_remainder_unique r
  6. L15
    specialize division_remainder_unique b
  7. L16
    specialize division_remainder_unique s
  8. L17
    apply division_remainder_unique
  9. L18
    exact hfirst
  10. L19
    exact hrbound
03Use earlier factsL20–21

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

  1. L20
    exact hsecond
  2. L21
    exact hsbound
04Separate the logical casesL22–22

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

  1. L22
    cases hunique
05Use earlier factsL23–23

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

  1. L23
    exact hunique_right

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro r
  4. 0004intro b
  5. 0005intro s
  6. 0006intro hfirst
  7. 0007intro hrbound
  8. 0008intro hsecond
  9. 0009intro hsbound
  10. 0010have hunique : a = b /\ r = s
  11. 0011specialize division_remainder_unique 8
  12. 0012specialize division_remainder_unique p
  13. 0013specialize division_remainder_unique a
  14. 0014specialize division_remainder_unique r
  15. 0015specialize division_remainder_unique b
  16. 0016specialize division_remainder_unique s
  17. 0017apply division_remainder_unique
  18. 0018exact hfirst
  19. 0019exact hrbound
  20. 0020exact hsecond
  21. 0021exact hsbound
  22. 0022cases hunique
  23. 0023exact hunique_right