GC000D

crt_balanced_bezout_scale

Multiplying a subtraction-free balanced Bezout equation preserves all four witnessed natural coefficients.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

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.

Historical partial components only: this chapter proves canonical solutions under successive-merge compatibility and in the pairwise-compatible dominating-last case. G011 is now closed in the separate Alpha-v27 generalized-crt branch for arbitrary pairwise-compatible finite lists, including noncoprime moduli. Full G011 proof · Alpha v27

Exact theorem in conservative defined notation

∀ k. ∀ a. ∀ b. ∀ g. ∀ xp. ∀ yp. ∀ xn. ∀ yn. a · xp + b · yp = g + (a · xn + b · yn) → k · a · xp + k · b · yp = k · g + (k · a · xn + k · b · yn)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

none

Actual proof prerequisites

mul_add · checked external prerequisitemul_assoc · checked external prerequisite
Original expanded first-order statement 
forall k a b g xp yp xn yn. a * xp + b * yp = g + (a * xn + b * yn) -> (k * a) * xp + (k * b) * yp = k * g + ((k * a) * xn + (k * b) * yn)

Complete unchanged native tactic proof

All 30 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

30 script commands · 15 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 assumptionsL1–9

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro g
  5. L5
    intro xp
  6. L6
    intro yp
  7. L7
    intro xn
  8. L8
    intro yn
  9. L9
    intro hbezout
02Calculate and transport equalitiesL10–12

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

  1. L10
    trans k * (a * xp + b * yp)
  2. L11
    symm
  3. L12
    trans k * (a * xp) + k * (b * yp)
03Use earlier factsL13–13

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

  1. L13
    apply mul_add
04Calculate and transport equalitiesL14–15

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

  1. L14
    congr
  2. L15
    symm
05Use earlier factsL16–16

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

  1. L16
    apply mul_assoc
06Calculate and transport equalitiesL17–17

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

  1. L17
    symm
07Use earlier factsL18–18

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

  1. L18
    apply mul_assoc
08Calculate and transport equalitiesL19–20

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

  1. L19
    rewrite hbezout
  2. L20
    trans k * g + k * (a * xn + b * yn)
09Use earlier factsL21–21

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

  1. L21
    apply mul_add
10Calculate and transport equalitiesL22–24

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

  1. L22
    congr
  2. L23
    refl
  3. L24
    trans k * (a * xn) + k * (b * yn)
11Use earlier factsL25–25

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

  1. L25
    apply mul_add
12Calculate and transport equalitiesL26–27

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

  1. L26
    congr
  2. L27
    symm
13Use earlier factsL28–28

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

  1. L28
    apply mul_assoc
14Calculate and transport equalitiesL29–29

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

  1. L29
    symm
15Use earlier factsL30–30

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

  1. L30
    apply mul_assoc

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro g
  5. 0005intro xp
  6. 0006intro yp
  7. 0007intro xn
  8. 0008intro yn
  9. 0009intro hbezout
  10. 0010trans k * (a * xp + b * yp)
  11. 0011symm
  12. 0012trans k * (a * xp) + k * (b * yp)
  13. 0013apply mul_add
  14. 0014congr
  15. 0015symm
  16. 0016apply mul_assoc
  17. 0017symm
  18. 0018apply mul_assoc
  19. 0019rewrite hbezout
  20. 0020trans k * g + k * (a * xn + b * yn)
  21. 0021apply mul_add
  22. 0022congr
  23. 0023refl
  24. 0024trans k * (a * xn) + k * (b * yn)
  25. 0025apply mul_add
  26. 0026congr
  27. 0027symm
  28. 0028apply mul_assoc
  29. 0029symm
  30. 0030apply mul_assoc