LU000F · theorem body

lucas_base_p_two_digit_reconstruction

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

Two coherent bounded base-p digits reconstruct the exact natural n = p²*q1 + p*d1 + d0 without introducing exponentiation.

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. ∀ n. ∀ q0. ∀ d0. ∀ q1. ∀ d1. DivRem(n,p,q0,d0) → DivRem(q0,p,q1,d1) → n = p · p · q1 + (p · d1 + d0)

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

PD0007 DivRem

In local proof propositions

none
Exact expanded first-order statement
forall p n q0 d0 q1 d1. (((n) = (p) * (q0) + (d0)) /\ (exists ldc_lt_first_bound. ldc_lt_first_bound + S (d0) = p)) -> (((q0) = (p) * (q1) + (d1)) /\ (exists ldc_lt_second_bound. ldc_lt_second_bound + S (d1) = p)) -> n = (p * p) * q1 + (p * d1 + d0)

Proof neighborhood

Direct theorem prerequisites

mul_add · Stable closed mul_assoc · Stable closed add_assoc · Stable closed

Direct theorem dependents

none

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

25 script commands · 10 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 assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro q0
  4. L4
    intro d0
  5. L5
    intro q1
  6. L6
    intro d1
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Separate the logical casesL9–10

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

  1. L9
    cases hfirst
  2. L10
    cases hsecond
03Calculate and transport equalitiesL11–12

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

  1. L11
    rewrite hsecond_left at hfirst_left
  2. L12
    trans p * (p * q1 + d1) + d0
04Use earlier factsL13–13

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

  1. L13
    exact hfirst_left
05Calculate and transport equalitiesL14–15

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

  1. L14
    trans (p * (p * q1) + p * d1) + d0
  2. L15
    congr
06Use earlier factsL16–16

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

  1. L16
    apply mul_add
07Calculate and transport equalitiesL17–21

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

  1. L17
    refl
  2. L18
    trans ((p * p) * q1 + p * d1) + d0
  3. L19
    congr
  4. L20
    congr
  5. L21
    symm
08Use earlier factsL22–22

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

  1. L22
    apply mul_assoc
09Calculate and transport equalitiesL23–24

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

  1. L23
    refl
  2. L24
    refl
10Use earlier factsL25–25

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

  1. L25
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q0
  4. 0004intro d0
  5. 0005intro q1
  6. 0006intro d1
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009cases hfirst
  10. 0010cases hsecond
  11. 0011rewrite hsecond_left at hfirst_left
  12. 0012trans p * (p * q1 + d1) + d0
  13. 0013exact hfirst_left
  14. 0014trans (p * (p * q1) + p * d1) + d0
  15. 0015congr
  16. 0016apply mul_add
  17. 0017refl
  18. 0018trans ((p * p) * q1 + p * d1) + d0
  19. 0019congr
  20. 0020congr
  21. 0021symm
  22. 0022apply mul_assoc
  23. 0023refl
  24. 0024refl
  25. 0025apply add_assoc