TS0008 · theorem body

prime_mod_four_one_divisible_two_square_norm_exists

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

A prime equal to one modulo four divides an explicitly witnessed two-square norm r²+1.

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

∀ p. ∀ n. p = S n → Prime(p) → Mod4One(p) → ∃ x. Dvd(p,x · x + 1)

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

PD0003 Dvd PD0004 Prime PD0011 Mod4One

In local proof propositions

PD0021 QRes
Exact expanded first-order statement
forall p n. p = S n -> ((~(p = 1) /\ forall frm_prime_left_fts_prime frm_prime_right_fts_prime. p = frm_prime_left_fts_prime * frm_prime_right_fts_prime -> frm_prime_left_fts_prime = 1 \/ frm_prime_right_fts_prime = 1)) -> (exists fts_four_prime_one. p = 4 * fts_four_prime_one + 1) -> exists r k. r * r + 1 = p * k

Proof neighborhood

Direct theorem prerequisites

TS0005 prime_mod_four_one_minus_one_square_exists TS0007 predecessor_square_congruence_yields_divisible_norm

Direct theorem dependents

none

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

20 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.

Named ingredients (2)
prime mod four one minus one square existspredecessor square congruence yields divisible norm
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro hpredecessor
  4. L4
    intro hprime
  5. L5
    intro hfourone
02Establish hresidueL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime mod four one minus one square exists.

  1. L6
    have hresidue : QRes(p,n)Definitions: QRes(p,n)Original native command in the exact edition
  2. L7
    specialize prime_mod_four_one_minus_one_square_exists p
  3. L8
    specialize prime_mod_four_one_minus_one_square_exists n
  4. L9
    apply prime_mod_four_one_minus_one_square_exists
  5. L10
    exact hpredecessor
  6. L11
    exact hprime
  7. L12
    exact hfourone
03Separate the logical casesL13–13

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

  1. L13
    cases hresidue
04Construct an explicit witnessL14–14

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

  1. L14
    exists x
05Use earlier factsL15–20

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

  1. L15
    specialize predecessor_square_congruence_yields_divisible_norm p
  2. L16
    specialize predecessor_square_congruence_yields_divisible_norm n
  3. L17
    specialize predecessor_square_congruence_yields_divisible_norm x
  4. L18
    apply predecessor_square_congruence_yields_divisible_norm
  5. L19
    exact hpredecessor
  6. L20
    exact hresidue_witness

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hpredecessor
  4. 0004intro hprime
  5. 0005intro hfourone
  6. 0006have hresidue : QRes(p,n)
    Exact native replay linehave hresidue : exists qr_x_fts_predecessor. exists qr_u_fts_predecessor qr_v_fts_predecessor. qr_x_fts_predecessor * qr_x_fts_predecessor + p * qr_u_fts_predecessor = n + p * qr_v_fts_predecessor
  7. 0007specialize prime_mod_four_one_minus_one_square_exists p
  8. 0008specialize prime_mod_four_one_minus_one_square_exists n
  9. 0009apply prime_mod_four_one_minus_one_square_exists
  10. 0010exact hpredecessor
  11. 0011exact hprime
  12. 0012exact hfourone
  13. 0013cases hresidue
  14. 0014exists x
  15. 0015specialize predecessor_square_congruence_yields_divisible_norm p
  16. 0016specialize predecessor_square_congruence_yields_divisible_norm n
  17. 0017specialize predecessor_square_congruence_yields_divisible_norm x
  18. 0018apply predecessor_square_congruence_yields_divisible_norm
  19. 0019exact hpredecessor
  20. 0020exact hresidue_witness