PA0017 · theorem

common_divisor_beta_moduli_divides_gap_times_c

Stable checked-use theorem · independently closed

A common divisor of two ordered beta moduli divides the index gap times c.

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

∀ c. ∀ i. ∀ j. ∀ gap. ∀ d. j = i + gap → Dvd(d,S (S i · c)) → Dvd(d,S (S j · c)) → Dvd(d,gap · c)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0003 Dvd

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall c i j gap d. j = i + gap -> (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> exists w. gap * c = d * w

Proof neighborhood

Direct theorem prerequisites

PA0014 divides_remainder PA000E add_succ_left PA0016 add_mul PA0001 zero_add

Direct theorem dependents

PA001Q beta_moduli_coprime_of_gap_dvd

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

27 script commands · 3 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.

Named ingredients (2)
divides remainderadd succ left
01Fix variables and assumptionsL1–8

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

  1. L1
    intro c
  2. L2
    intro i
  3. L3
    intro j
  4. L4
    intro gap
  5. L5
    intro d
  6. L6
    intro hij
  7. L7
    intro hmi
  8. L8
    intro hmj
02Establish hstepL9–18

Establish this local claim before using it. It is not an additional assumption.

  1. L9
    have hstep : S ((S j) * c) = S ((S i) * c) * 1 + gap * c
  2. L10
    rewrite hij
  3. L11
    specialize add_succ_left i
  4. L12
    specialize add_succ_left gap
  5. L13
    rewrite <- add_succ_left
  6. L14
    simp [add_mul, zero_add]
  7. L15
    symm
  8. L16
    specialize add_succ_left_before (S i * c)
  9. L17
    specialize add_succ_left_before (gap * c)
  10. L18
    exact add_succ_left_before
03Use earlier factsL19–27

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

  1. L19
    specialize divides_remainder d
  2. L20
    specialize divides_remainder (S ((S j) * c))
  3. L21
    specialize divides_remainder (S ((S i) * c))
  4. L22
    specialize divides_remainder 1
  5. L23
    specialize divides_remainder (gap * c)
  6. L24
    apply divides_remainder
  7. L25
    exact hmj
  8. L26
    exact hmi
  9. L27
    exact hstep

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro c
  2. 0002intro i
  3. 0003intro j
  4. 0004intro gap
  5. 0005intro d
  6. 0006intro hij
  7. 0007intro hmi
  8. 0008intro hmj
  9. 0009have hstep : S ((S j) * c) = S ((S i) * c) * 1 + gap * c
  10. 0010rewrite hij
  11. 0011specialize add_succ_left i
  12. 0012specialize add_succ_left gap
  13. 0013rewrite <- add_succ_left
  14. 0014simp [add_mul, zero_add]
  15. 0015symm
  16. 0016specialize add_succ_left_before (S i * c)
  17. 0017specialize add_succ_left_before (gap * c)
  18. 0018exact add_succ_left_before
  19. 0019specialize divides_remainder d
  20. 0020specialize divides_remainder (S ((S j) * c))
  21. 0021specialize divides_remainder (S ((S i) * c))
  22. 0022specialize divides_remainder 1
  23. 0023specialize divides_remainder (gap * c)
  24. 0024apply divides_remainder
  25. 0025exact hmj
  26. 0026exact hmi
  27. 0027exact hstep