SL0006 · theorem body

quadratic_supplement_minus_one_complete

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

Complete constructive first supplementary law, including both the one-modulo-four residue and three-modulo-four nonresidue cases.

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) → Odd(p) → (QRes(p,n) → Mod4One(p)) ∧ (Mod4One(p) → QRes(p,n)) ∧ ((¬QRes(p,n) → Mod4Three(p)) ∧ (Mod4Three(p) → ¬QRes(p,n)))

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

PD0004 Prime PD0010 Odd PD0011 Mod4One PD0012 Mod4Three PD0021 QRes

In local proof propositions

none
Exact expanded first-order statement
forall p n. p = S n -> ((~(p = 1) /\ forall frm_prime_left_qsm_prime frm_prime_right_qsm_prime. p = frm_prime_left_qsm_prime * frm_prime_right_qsm_prime -> frm_prime_left_qsm_prime = 1 \/ frm_prime_right_qsm_prime = 1)) -> (exists qsm_odd_modulus. p = 2 * qsm_odd_modulus + 1) -> (((((exists qr_x_qsm_predecessor. exists qr_u_qsm_predecessor qr_v_qsm_predecessor. qr_x_qsm_predecessor * qr_x_qsm_predecessor + p * qr_u_qsm_predecessor = n + p * qr_v_qsm_predecessor) -> (exists qsm_four_one_modulus. p = 4 * qsm_four_one_modulus + 1)) /\ ((exists qsm_four_one_modulus. p = 4 * qsm_four_one_modulus + 1) -> (exists qr_x_qsm_predecessor. exists qr_u_qsm_predecessor qr_v_qsm_predecessor. qr_x_qsm_predecessor * qr_x_qsm_predecessor + p * qr_u_qsm_predecessor = n + p * qr_v_qsm_predecessor)))) /\ ((((~(exists qr_x_qsm_predecessor. exists qr_u_qsm_predecessor qr_v_qsm_predecessor. qr_x_qsm_predecessor * qr_x_qsm_predecessor + p * qr_u_qsm_predecessor = n + p * qr_v_qsm_predecessor)) -> (exists qsm_four_three_modulus. p = 4 * qsm_four_three_modulus + 3)) /\ ((exists qsm_four_three_modulus. p = 4 * qsm_four_three_modulus + 3) -> (~(exists qr_x_qsm_predecessor. exists qr_u_qsm_predecessor qr_v_qsm_predecessor. qr_x_qsm_predecessor * qr_x_qsm_predecessor + p * qr_u_qsm_predecessor = n + p * qr_v_qsm_predecessor))))))

Proof neighborhood

Direct theorem prerequisites

SL0004 quadratic_supplement_minus_one_residue_iff_mod_four_one SL0005 quadratic_supplement_minus_one_nonresidue_iff_mod_four_three

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

18 script commands · 4 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.

Named ingredients (2)
quadratic supplement minus one residue iff mod four onequadratic supplement minus one nonresidue iff mod four three
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 hodd
02Separate the logical casesL6–6

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

  1. L6
    split
03Use earlier factsL7–16

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

  1. L7
    specialize quadratic_supplement_minus_one_residue_iff_mod_four_one p
  2. L8
    specialize quadratic_supplement_minus_one_residue_iff_mod_four_one n
  3. L9
    apply quadratic_supplement_minus_one_residue_iff_mod_four_one
  4. L10
    exact hpredecessor
  5. L11
    exact hprime
  6. L12
    exact hodd
  7. L13
    specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three p
  8. L14
    specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three n
  9. L15
    apply quadratic_supplement_minus_one_nonresidue_iff_mod_four_three
  10. L16
    exact hpredecessor
04Use earlier factsL17–18

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

  1. L17
    exact hprime
  2. L18
    exact hodd

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hpredecessor
  4. 0004intro hprime
  5. 0005intro hodd
  6. 0006split
  7. 0007specialize quadratic_supplement_minus_one_residue_iff_mod_four_one p
  8. 0008specialize quadratic_supplement_minus_one_residue_iff_mod_four_one n
  9. 0009apply quadratic_supplement_minus_one_residue_iff_mod_four_one
  10. 0010exact hpredecessor
  11. 0011exact hprime
  12. 0012exact hodd
  13. 0013specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three p
  14. 0014specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three n
  15. 0015apply quadratic_supplement_minus_one_nonresidue_iff_mod_four_three
  16. 0016exact hpredecessor
  17. 0017exact hprime
  18. 0018exact hodd