GC0010

crt_product_witness

Every natural product has an explicit first-order value witness without adding multiplication as a new operation.

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

∀ a. ∀ b. ∃ p. p = a · b

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

Definition DAG

none

Actual proof prerequisites

none
Original expanded first-order statement 
forall a b. exists p. p = a * b

Complete unchanged native tactic proof

All 4 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

4 script commands · 3 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 assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Construct an explicit witnessL3–3

Supply the displayed value, then prove that it has the required property.

  1. L3
    exists a * b
03Calculate and transport equalitiesL4–4

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

  1. L4
    refl

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003exists a * b
  4. 0004refl