SL000T · theorem body

quadratic_supplement_two_complete

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

Complete constructive second supplementary law: the quadratic residue status of two is classified exactly by the four odd classes modulo eight.

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. Prime(p) → Odd(p) → (QRes(p,2) → (∃ x. p = 8 · x + 1) ∨ (∃ x. p = 8 · x + 7)) ∧ ((∃ x. p = 8 · x + 1) ∨ (∃ x. p = 8 · x + 7) → QRes(p,2)) ∧ ((¬QRes(p,2) → (∃ x. p = 8 · x + 3) ∨ (∃ x. p = 8 · x + 5)) ∧ ((∃ x. p = 8 · x + 3) ∨ (∃ x. p = 8 · x + 5) → ¬QRes(p,2)))

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 PD0021 QRes

In local proof propositions

none
Exact expanded first-order statement
forall p. ((~(p = 1) /\ forall frm_prime_left_qst_endpoint_prime frm_prime_right_qst_endpoint_prime. p = frm_prime_left_qst_endpoint_prime * frm_prime_right_qst_endpoint_prime -> frm_prime_left_qst_endpoint_prime = 1 \/ frm_prime_right_qst_endpoint_prime = 1)) -> (exists qst_odd_modulus. p = 2 * qst_odd_modulus + 1) -> ((((((exists qst_root_two. exists qst_mod_left_two qst_mod_right_two. qst_root_two * qst_root_two + p * qst_mod_left_two = 2 + p * qst_mod_right_two) -> (((exists qst_mod_eight_modulus_one. p = 8 * qst_mod_eight_modulus_one + 1) \/ (exists qst_mod_eight_modulus_seven. p = 8 * qst_mod_eight_modulus_seven + 7)))) /\ ((((exists qst_mod_eight_modulus_one. p = 8 * qst_mod_eight_modulus_one + 1) \/ (exists qst_mod_eight_modulus_seven. p = 8 * qst_mod_eight_modulus_seven + 7))) -> (exists qst_root_two. exists qst_mod_left_two qst_mod_right_two. qst_root_two * qst_root_two + p * qst_mod_left_two = 2 + p * qst_mod_right_two)))) /\ ((((~(exists qst_root_two. exists qst_mod_left_two qst_mod_right_two. qst_root_two * qst_root_two + p * qst_mod_left_two = 2 + p * qst_mod_right_two)) -> (((exists qst_mod_eight_modulus_three. p = 8 * qst_mod_eight_modulus_three + 3) \/ (exists qst_mod_eight_modulus_five. p = 8 * qst_mod_eight_modulus_five + 5)))) /\ ((((exists qst_mod_eight_modulus_three. p = 8 * qst_mod_eight_modulus_three + 3) \/ (exists qst_mod_eight_modulus_five. p = 8 * qst_mod_eight_modulus_five + 5))) -> (~(exists qst_root_two. exists qst_mod_left_two qst_mod_right_two. qst_root_two * qst_root_two + p * qst_mod_left_two = 2 + p * qst_mod_right_two)))))))

Proof neighborhood

Direct theorem prerequisites

SL000R quadratic_supplement_two_residue_iff_mod_eight_one_or_seven SL000S quadratic_supplement_two_nonresidue_iff_mod_eight_three_or_five

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

12 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.

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 two residue iff mod eight one or sevenquadratic supplement two nonresidue iff mod eight three or five
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro hprime
  3. L3
    intro hodd
02Separate the logical casesL4–4

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

  1. L4
    split
03Use earlier factsL5–12

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

  1. L5
    specialize quadratic_supplement_two_residue_iff_mod_eight_one_or_seven p
  2. L6
    apply quadratic_supplement_two_residue_iff_mod_eight_one_or_seven
  3. L7
    exact hprime
  4. L8
    exact hodd
  5. L9
    specialize quadratic_supplement_two_nonresidue_iff_mod_eight_three_or_five p
  6. L10
    apply quadratic_supplement_two_nonresidue_iff_mod_eight_three_or_five
  7. L11
    exact hprime
  8. L12
    exact hodd

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003intro hodd
  4. 0004split
  5. 0005specialize quadratic_supplement_two_residue_iff_mod_eight_one_or_seven p
  6. 0006apply quadratic_supplement_two_residue_iff_mod_eight_one_or_seven
  7. 0007exact hprime
  8. 0008exact hodd
  9. 0009specialize quadratic_supplement_two_nonresidue_iff_mod_eight_three_or_five p
  10. 0010apply quadratic_supplement_two_nonresidue_iff_mod_eight_three_or_five
  11. 0011exact hprime
  12. 0012exact hodd