FS001F · theorem body

four_square_descent_modular_seed_multiplier_nonzero

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

A modular seed x²+y²+1=p·k has a constructively nonzero multiplier because its left side is a successor.

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 p x y k. x * x + y * y + 1 = p * 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 p x y k. x * x + y * y + 1 = p * k -> ~(k = 0)

Proof neighborhood

Direct theorem prerequisites

succ_ne_zero · Stable closed

Direct theorem dependents

FS001I four_square_descent_prime_from_modular_seed_and_step FS0034 four_square_prime_from_bounded_strict_descent_and_seed

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

14 script commands · 5 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 assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro k
  5. L5
    intro hseed
  6. L6
    intro hzero
02Calculate and transport equalitiesL7–8

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

  1. L7
    rewrite hzero at hseed
  2. L8
    rewrite PA5 at hseed
03Use earlier factsL9–10

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

  1. L9
    specialize succ_ne_zero (x * x + y * y)
  2. L10
    apply succ_ne_zero
04Calculate and transport equalitiesL11–13

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

  1. L11
    trans x * x + y * y + 1
  2. L12
    symm
  3. L13
    simp
05Use earlier factsL14–14

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

  1. L14
    exact hseed

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro x
  3. 0003intro y
  4. 0004intro k
  5. 0005intro hseed
  6. 0006intro hzero
  7. 0007rewrite hzero at hseed
  8. 0008rewrite PA5 at hseed
  9. 0009specialize succ_ne_zero (x * x + y * y)
  10. 0010apply succ_ne_zero
  11. 0011trans x * x + y * y + 1
  12. 0012symm
  13. 0013simp
  14. 0014exact hseed