SL000B · theorem body

mod_eight_remainder_unique

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ p. ∀ a. ∀ r. ∀ b. ∀ s. p = 8 · a + r → Lt(r,8) → p = 8 · b + s → Lt(s,8) → r = s

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

PD0002 Lt

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

division_remainder_unique · Stable closed

Direct theorem dependents

SL000C mod_eight_good_bad_exclusive

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

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.

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 assumptionsL1–9

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 defined 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