FS0018 · theorem body

four_square_descent_nonzero_square

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

The square of a nonzero natural multiplier remains nonzero constructively.

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 k. ~(k = 0) -> ~(k * k = 0)

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 k. ~(k = 0) -> ~(k * k = 0)

Proof neighborhood

Direct theorem prerequisites

mul_eq_zero · Stable closed

Direct theorem dependents

FS001B four_square_descent_square_factor_cancel

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

13 script commands · 5 reading checkpoints · 1 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 k
  2. L2
    intro hnonzero
  3. L3
    intro hproduct
02Use earlier factsL4–5

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

  1. L4
    specialize mul_eq_zero k
  2. L5
    specialize mul_eq_zero k
03Establish hcasesL6–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul eq zero.

  1. L6
    have hcases : k = 0 \/ k = 0
  2. L7
    apply mul_eq_zero
  3. L8
    exact hproduct
04Separate the logical casesL9–9

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

  1. L9
    cases hcases
05Use earlier factsL10–13

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

  1. L10
    apply hnonzero
  2. L11
    exact hcases_left
  3. L12
    apply hnonzero
  4. L13
    exact hcases_right

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro k
  2. 0002intro hnonzero
  3. 0003intro hproduct
  4. 0004specialize mul_eq_zero k
  5. 0005specialize mul_eq_zero k
  6. 0006have hcases : k = 0 \/ k = 0
  7. 0007apply mul_eq_zero
  8. 0008exact hproduct
  9. 0009cases hcases
  10. 0010apply hnonzero
  11. 0011exact hcases_left
  12. 0012apply hnonzero
  13. 0013exact hcases_right