SL000R

quadratic_supplement_two_residue_iff_mod_eight_one_or_seven

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The second supplementary law: two is a quadratic residue modulo an odd prime exactly in classes one and seven 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))))

Constructive proof overview

Generated structural guide

The second supplementary law: two is a quadratic residue modulo an odd prime exactly in classes one and seven modulo eight.

The unchanged tactic script uses 1 declared prerequisite and contains 14 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

SL000Q quadratic_supplement_two_half_complete

Direct dependents

SL000T quadratic_supplement_two_complete

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

14 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.

Named ingredients (1)
quadratic supplement two half complete

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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
    cases hodd
03Establish hcompleteL5–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply quadratic supplement two half complete.

  1. L5
    have hcomplete : (QRes(p,2) → Mod8One(p) ∨ Mod8Seven(p)) ∧ (Mod8One(p) ∨ Mod8Seven(p) → QRes(p,2)) ∧ ((¬QRes(p,2) → Mod8Three(p) ∨ Mod8Five(p)) ∧ (Mod8Three(p) ∨ Mod8Five(p) → ¬QRes(p,2)))Definitions: Mod8OneMod8ThreeMod8FiveMod8SevenQRes
  2. L6
    specialize quadratic_supplement_two_half_complete p
  3. L7
    specialize quadratic_supplement_two_half_complete x
  4. L8
    specialize quadratic_supplement_two_half_complete 2
  5. L9
    apply quadratic_supplement_two_half_complete
  6. L10
    exact hodd_witness
  7. L11
    exact hprime
  8. L12
    refl
04Separate the logical casesL13–13

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

  1. L13
    cases hcomplete
05Use earlier factsL14–14

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

  1. L14
    exact hcomplete_left

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003intro hodd
  4. 0004cases hodd
  5. 0005have hcomplete : (((((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))))))
  6. 0006specialize quadratic_supplement_two_half_complete p
  7. 0007specialize quadratic_supplement_two_half_complete x
  8. 0008specialize quadratic_supplement_two_half_complete 2
  9. 0009apply quadratic_supplement_two_half_complete
  10. 0010exact hodd_witness
  11. 0011exact hprime
  12. 0012refl
  13. 0013cases hcomplete
  14. 0014exact hcomplete_left