SL000T

quadratic_supplement_two_complete

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 12 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

SL000R quadratic_supplement_two_residue_iff_mod_eight_one_or_seven SL000S quadratic_supplement_two_nonresidue_iff_mod_eight_three_or_five

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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