GC0010

crt_product_witness

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic statement

forall a b. exists p. p = a * b

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 0 declared prerequisites and contains 4 exact native proof lines.

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

Proof neighborhood

Direct dependencies

none

Direct dependents

GC0011 crt_is_gcd_coprime_product GC0013 crt_gcd_scaled_coprime_component

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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

Separate complete second-wave branches: Full G011 proof · Alpha v27.