FS000U

four_square_bounded_strict_descent_from_odd_signed_quaternion

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

Parity case distinction discharges the entire below-prime strict-descent obligation except for the single explicitly stated odd signed quaternion representation.

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 fsbr_modulus_branch fsbr_multiplier_branch fsbr_half_branch fsbr_coordinate_branch_0 fsbr_coordinate_branch_1 fsbr_coordinate_branch_2 fsbr_coordinate_branch_3 fsbr_center_branch_0 fsbr_center_branch_1 fsbr_center_branch_2 fsbr_center_branch_3 fsbr_quotient_branch. ~(fsbr_multiplier_branch = 0) -> fsbr_multiplier_branch = 2 * fsbr_half_branch + 1 -> fsbr_modulus_branch * fsbr_multiplier_branch = fsbr_coordinate_branch_0 * fsbr_coordinate_branch_0 + fsbr_coordinate_branch_1 * fsbr_coordinate_branch_1 + fsbr_coordinate_branch_2 * fsbr_coordinate_branch_2 + fsbr_coordinate_branch_3 * fsbr_coordinate_branch_3 -> (((exists fsd_center_bound_branch_0. fsd_center_bound_branch_0 + (fsbr_center_branch_0 + fsbr_center_branch_0) = fsbr_multiplier_branch) /\ ((exists fsd_center_lower_branch_0. fsbr_coordinate_branch_0 = fsbr_multiplier_branch * fsd_center_lower_branch_0 + fsbr_center_branch_0) \/ (exists fsd_center_upper_branch_0. fsbr_coordinate_branch_0 + fsbr_center_branch_0 = fsbr_multiplier_branch * fsd_center_upper_branch_0)))) -> (((exists fsd_center_bound_branch_1. fsd_center_bound_branch_1 + (fsbr_center_branch_1 + fsbr_center_branch_1) = fsbr_multiplier_branch) /\ ((exists fsd_center_lower_branch_1. fsbr_coordinate_branch_1 = fsbr_multiplier_branch * fsd_center_lower_branch_1 + fsbr_center_branch_1) \/ (exists fsd_center_upper_branch_1. fsbr_coordinate_branch_1 + fsbr_center_branch_1 = fsbr_multiplier_branch * fsd_center_upper_branch_1)))) -> (((exists fsd_center_bound_branch_2. fsd_center_bound_branch_2 + (fsbr_center_branch_2 + fsbr_center_branch_2) = fsbr_multiplier_branch) /\ ((exists fsd_center_lower_branch_2. fsbr_coordinate_branch_2 = fsbr_multiplier_branch * fsd_center_lower_branch_2 + fsbr_center_branch_2) \/ (exists fsd_center_upper_branch_2. fsbr_coordinate_branch_2 + fsbr_center_branch_2 = fsbr_multiplier_branch * fsd_center_upper_branch_2)))) -> (((exists fsd_center_bound_branch_3. fsd_center_bound_branch_3 + (fsbr_center_branch_3 + fsbr_center_branch_3) = fsbr_multiplier_branch) /\ ((exists fsd_center_lower_branch_3. fsbr_coordinate_branch_3 = fsbr_multiplier_branch * fsd_center_lower_branch_3 + fsbr_center_branch_3) \/ (exists fsd_center_upper_branch_3. fsbr_coordinate_branch_3 + fsbr_center_branch_3 = fsbr_multiplier_branch * fsd_center_upper_branch_3)))) -> fsbr_multiplier_branch * fsbr_quotient_branch = fsbr_center_branch_0 * fsbr_center_branch_0 + fsbr_center_branch_1 * fsbr_center_branch_1 + fsbr_center_branch_2 * fsbr_center_branch_2 + fsbr_center_branch_3 * fsbr_center_branch_3 -> (exists fsl_a_fsbr_signed_branch fsl_b_fsbr_signed_branch fsl_c_fsbr_signed_branch fsl_d_fsbr_signed_branch. (fsbr_modulus_branch * fsbr_quotient_branch) = fsl_a_fsbr_signed_branch * fsl_a_fsbr_signed_branch + fsl_b_fsbr_signed_branch * fsl_b_fsbr_signed_branch + fsl_c_fsbr_signed_branch * fsl_c_fsbr_signed_branch + fsl_d_fsbr_signed_branch * fsl_d_fsbr_signed_branch)) -> (forall fslb_bounded_prime_branch fslb_bounded_multiplier_branch. ((~(fslb_bounded_prime_branch = 1) /\ forall frm_prime_left_fslb_bounded_prime_branch frm_prime_right_fslb_bounded_prime_branch. fslb_bounded_prime_branch = frm_prime_left_fslb_bounded_prime_branch * frm_prime_right_fslb_bounded_prime_branch -> frm_prime_left_fslb_bounded_prime_branch = 1 \/ frm_prime_right_fslb_bounded_prime_branch = 1)) -> ~(fslb_bounded_multiplier_branch = 0) -> ~(fslb_bounded_multiplier_branch = 1) -> (exists fslb_bounded_upper_gap_branch. fslb_bounded_upper_gap_branch + S fslb_bounded_multiplier_branch = fslb_bounded_prime_branch) -> (exists fsl_a_fslb_bounded_source_branch fsl_b_fslb_bounded_source_branch fsl_c_fslb_bounded_source_branch fsl_d_fslb_bounded_source_branch. (fslb_bounded_prime_branch * fslb_bounded_multiplier_branch) = fsl_a_fslb_bounded_source_branch * fsl_a_fslb_bounded_source_branch + fsl_b_fslb_bounded_source_branch * fsl_b_fslb_bounded_source_branch + fsl_c_fslb_bounded_source_branch * fsl_c_fslb_bounded_source_branch + fsl_d_fslb_bounded_source_branch * fsl_d_fslb_bounded_source_branch) -> exists fslb_bounded_smaller_branch. (~(fslb_bounded_smaller_branch = 0) /\ ((exists fslb_bounded_lower_gap_branch. fslb_bounded_lower_gap_branch + S fslb_bounded_smaller_branch = fslb_bounded_multiplier_branch) /\ (exists fsl_a_fslb_bounded_target_branch fsl_b_fslb_bounded_target_branch fsl_c_fslb_bounded_target_branch fsl_d_fslb_bounded_target_branch. (fslb_bounded_prime_branch * fslb_bounded_smaller_branch) = fsl_a_fslb_bounded_target_branch * fsl_a_fslb_bounded_target_branch + fsl_b_fslb_bounded_target_branch * fsl_b_fslb_bounded_target_branch + fsl_c_fslb_bounded_target_branch * fsl_c_fslb_bounded_target_branch + fsl_d_fslb_bounded_target_branch * fsl_d_fslb_bounded_target_branch))))

Constructive proof overview

Generated structural guide

Parity case distinction discharges the entire below-prime strict-descent obligation except for the single explicitly stated odd signed quaternion representation.

The unchanged tactic script uses 3 declared prerequisites and contains 33 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

parity_cases Stable theorem; checked-use authorized FS000S four_square_branch_even_represented_strict_step FS000T four_square_branch_odd_represented_strict_step

Direct dependents

FS003M four_square_prime_from_odd_signed_quaternion

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

33 script commands · 6 reading checkpoints · 2 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)
four square branch even represented strict stepfour square branch odd represented strict step

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 assumptionsL1–8

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

  1. L1
    intro hsigned
  2. L2
    intro p
  3. L3
    intro k
  4. L4
    intro hprime
  5. L5
    intro hnonzero
  6. L6
    intro hnonunit
  7. L7
    intro hproper
  8. L8
    intro hrepresented
02Establish hparityL9–11

Establish this local claim before using it. It is not an additional assumption.

  1. L9
    have hparity : exists h. k = 2 * h \/ k = 2 * h + 1
  2. L10
    specialize parity_cases k
  3. L11
    exact parity_cases
03Separate the logical casesL12–13

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

  1. L12
    cases hparity
  2. L13
    cases hparity_witness
04Use earlier factsL14–20

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

  1. L14
    specialize four_square_branch_even_represented_strict_step p
  2. L15
    specialize four_square_branch_even_represented_strict_step k
  3. L16
    specialize four_square_branch_even_represented_strict_step x
  4. L17
    apply four_square_branch_even_represented_strict_step
  5. L18
    exact hnonzero
  6. L19
    exact hparity_witness_left
  7. L20
    exact hrepresented
05Establish hoddstepL21–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square branch odd represented strict step.

  1. L21
    have hoddstep : ∀ q. ∀ j. ∀ t. Prime(q) → ¬j = 0 → ¬j = 1 → Lt(j,q) → j = 2 · t + 1 → (∃ x. ∃ y. ∃ z. ∃ n. FourSquareNorm(q · j,x,y,z,n)) → ∃ x. ¬x = 0 ∧ (Lt(x,j) ∧ (∃ y. ∃ z. ∃ n. ∃ m. FourSquareNorm(q · x,y,z,n,m)))Definitions: FourSquareNormLtPrime
  2. L22
    apply four_square_branch_odd_represented_strict_step
  3. L23
    exact hsigned
  4. L24
    specialize hoddstep p
  5. L25
    specialize hoddstep k
  6. L26
    specialize hoddstep x
  7. L27
    apply hoddstep
  8. L28
    exact hprime
  9. L29
    exact hnonzero
  10. L30
    exact hnonunit
06Use earlier factsL31–33

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

  1. L31
    exact hproper
  2. L32
    exact hparity_witness_right
  3. L33
    exact hrepresented

Library-wide reading audit

Original exact command ledger · 33 lines
  1. 0001intro hsigned
  2. 0002intro p
  3. 0003intro k
  4. 0004intro hprime
  5. 0005intro hnonzero
  6. 0006intro hnonunit
  7. 0007intro hproper
  8. 0008intro hrepresented
  9. 0009have hparity : exists h. k = 2 * h \/ k = 2 * h + 1
  10. 0010specialize parity_cases k
  11. 0011exact parity_cases
  12. 0012cases hparity
  13. 0013cases hparity_witness
  14. 0014specialize four_square_branch_even_represented_strict_step p
  15. 0015specialize four_square_branch_even_represented_strict_step k
  16. 0016specialize four_square_branch_even_represented_strict_step x
  17. 0017apply four_square_branch_even_represented_strict_step
  18. 0018exact hnonzero
  19. 0019exact hparity_witness_left
  20. 0020exact hrepresented
  21. 0021have hoddstep : forall q j t. ((~(q = 1) /\ forall frm_prime_left_fsbr_local_prime frm_prime_right_fsbr_local_prime. q = frm_prime_left_fsbr_local_prime * frm_prime_right_fsbr_local_prime -> frm_prime_left_fsbr_local_prime = 1 \/ frm_prime_right_fsbr_local_prime = 1)) -> ~(j = 0) -> ~(j = 1) -> (exists gap. gap + S j = q) -> j = 2 * t + 1 -> (exists fsl_a_fsbr_local_source fsl_b_fsbr_local_source fsl_c_fsbr_local_source fsl_d_fsbr_local_source. (q * j) = fsl_a_fsbr_local_source * fsl_a_fsbr_local_source + fsl_b_fsbr_local_source * fsl_b_fsbr_local_source + fsl_c_fsbr_local_source * fsl_c_fsbr_local_source + fsl_d_fsbr_local_source * fsl_d_fsbr_local_source) -> exists r. (~(r = 0) /\ ((exists gap. gap + S r = j) /\ (exists fsl_a_fsbr_local_result fsl_b_fsbr_local_result fsl_c_fsbr_local_result fsl_d_fsbr_local_result. (q * r) = fsl_a_fsbr_local_result * fsl_a_fsbr_local_result + fsl_b_fsbr_local_result * fsl_b_fsbr_local_result + fsl_c_fsbr_local_result * fsl_c_fsbr_local_result + fsl_d_fsbr_local_result * fsl_d_fsbr_local_result)))
  22. 0022apply four_square_branch_odd_represented_strict_step
  23. 0023exact hsigned
  24. 0024specialize hoddstep p
  25. 0025specialize hoddstep k
  26. 0026specialize hoddstep x
  27. 0027apply hoddstep
  28. 0028exact hprime
  29. 0029exact hnonzero
  30. 0030exact hnonunit
  31. 0031exact hproper
  32. 0032exact hparity_witness_right
  33. 0033exact hrepresented