TS0025 · theorem body

three_mod_four_prime_has_no_negative_one_root

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

For an odd prime congruent to three modulo four, the first supplementary law excludes square roots of minus one.

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) → 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 PD0012 Mod4Three PD0021 QRes

In local proof propositions

PD0012 Mod4Three PD0021 QRes
Exact expanded first-order statement
forall p n. p = S n -> ((~(p = 1) /\ forall frm_prime_left_ftsc_prime frm_prime_right_ftsc_prime. p = frm_prime_left_ftsc_prime * frm_prime_right_ftsc_prime -> frm_prime_left_ftsc_prime = 1 \/ frm_prime_right_ftsc_prime = 1)) -> (exists ftsc_odd_ftsc_prime. (p) = 2 * ftsc_odd_ftsc_prime + 1) -> (exists ftsc_four_three_ftsc_prime. (p) = 4 * ftsc_four_three_ftsc_prime + 3) -> ~(exists qr_x_ftsc_predecessor. exists qr_u_ftsc_predecessor qr_v_ftsc_predecessor. qr_x_ftsc_predecessor * qr_x_ftsc_predecessor + p * qr_u_ftsc_predecessor = n + p * qr_v_ftsc_predecessor)

Proof neighborhood

Direct theorem prerequisites

quadratic_supplement_minus_one_nonresidue_iff_mod_four_three · Alpha closed

Direct theorem dependents

TS0026 three_mod_four_prime_norm_divisor_forces_second_coordinate

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 · 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.

01Fix variables and assumptionsL1–6

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
  6. L6
    intro hthree
02Establish hlawL7–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply quadratic supplement minus one nonresidue iff mod four three.

  1. L7
    have hlaw : (¬QRes(p,n) → Mod4Three(p)) ∧ (Mod4Three(p) → ¬QRes(p,n))Definitions: QRes(p,n)Mod4Three(p)Original native command in the exact edition
  2. L8
    specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three p
  3. L9
    specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three n
  4. L10
    apply quadratic_supplement_minus_one_nonresidue_iff_mod_four_three
  5. L11
    exact hpredecessor
  6. L12
    exact hprime
  7. L13
    exact hodd
03Separate the logical casesL14–14

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

  1. L14
    cases hlaw
04Fix variables and assumptionsL15–15

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

  1. L15
    intro hresidue
05Use earlier factsL16–18

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

  1. L16
    apply hlaw_right
  2. L17
    exact hthree
  3. L18
    exact hresidue

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. 0006intro hthree
  7. 0007have hlaw : (¬QRes(p,n) → Mod4Three(p)) ∧ (Mod4Three(p) → ¬QRes(p,n))
    Exact native replay linehave hlaw : ((~(exists qr_x_ftsc_predecessor. exists qr_u_ftsc_predecessor qr_v_ftsc_predecessor. qr_x_ftsc_predecessor * qr_x_ftsc_predecessor + p * qr_u_ftsc_predecessor = n + p * qr_v_ftsc_predecessor) -> (exists ftsc_four_three_ftsc_prime. (p) = 4 * ftsc_four_three_ftsc_prime + 3)) /\ ((exists ftsc_four_three_ftsc_prime. (p) = 4 * ftsc_four_three_ftsc_prime + 3) -> ~(exists qr_x_ftsc_predecessor. exists qr_u_ftsc_predecessor qr_v_ftsc_predecessor. qr_x_ftsc_predecessor * qr_x_ftsc_predecessor + p * qr_u_ftsc_predecessor = n + p * qr_v_ftsc_predecessor)))
  8. 0008specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three p
  9. 0009specialize quadratic_supplement_minus_one_nonresidue_iff_mod_four_three n
  10. 0010apply quadratic_supplement_minus_one_nonresidue_iff_mod_four_three
  11. 0011exact hpredecessor
  12. 0012exact hprime
  13. 0013exact hodd
  14. 0014cases hlaw
  15. 0015intro hresidue
  16. 0016apply hlaw_right
  17. 0017exact hthree
  18. 0018exact hresidue