LU0011 · theorem body

lucas_prime_block_successor_reassociation

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

A successor prime-block quotient is exactly one prime shift of its predecessor.

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

forall p q a. p * S q + a = p + (p * q + a)

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

none

In local proof propositions

none
Exact expanded first-order statement
forall p q a. p * S q + a = p + (p * q + a)

Proof neighborhood

Direct theorem prerequisites

add_comm · Stable closed add_assoc · Stable closed

Direct theorem dependents

LU000M lucas_prime_block_digit_congruence LU0012 lucas_repeated_prime_shift_below_base

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

12 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 q
  3. L3
    intro a
02Calculate and transport equalitiesL4–5

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

  1. L4
    trans (p * q + p) + a
  2. L5
    congr
03Use earlier factsL6–6

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

  1. L6
    apply PA6
04Calculate and transport equalitiesL7–9

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

  1. L7
    refl
  2. L8
    trans (p + p * q) + a
  3. L9
    congr
05Use earlier factsL10–10

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

  1. L10
    apply add_comm
06Calculate and transport equalitiesL11–11

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

  1. L11
    refl
07Use earlier factsL12–12

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

  1. L12
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro a
  4. 0004trans (p * q + p) + a
  5. 0005congr
  6. 0006apply PA6
  7. 0007refl
  8. 0008trans (p + p * q) + a
  9. 0009congr
  10. 0010apply add_comm
  11. 0011refl
  12. 0012apply add_assoc