TS003U · theorem body

three_mod_four_prime_two_square_norm_valuation_even

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

Every three-modulo-four prime has a constructively even valuation in every explicitly nonzero two-square norm.

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. ∀ a. ∀ b. ∀ e. Prime(p) → Mod4Three(p) → ¬a · a + b · b = 0 → PowerValuation(p,a · a + b · b,e) → ∃ x. e = x + x

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 PD0012 Mod4Three PD0046 PowerValuation

In local proof propositions

none
Exact expanded first-order statement
forall p a b e. ((~(p = 1) /\ forall frm_prime_left_ftsv_prime frm_prime_right_ftsv_prime. p = frm_prime_left_ftsv_prime * frm_prime_right_ftsv_prime -> frm_prime_left_ftsv_prime = 1 \/ frm_prime_right_ftsv_prime = 1)) -> (exists ftsc_four_three_ftsv_prime. (p) = 4 * ftsc_four_three_ftsv_prime + 3) -> ~(a * a + b * b = 0) -> (((exists bpv_gap_ftsv_norm_valuation_exponent_bound. bpv_gap_ftsv_norm_valuation_exponent_bound + e = (a * a + b * b)) /\ (exists bpv_result_ftsv_norm_valuation_selected. ((exists ff_b_ftsv_norm_valuation_selected_power ff_c_ftsv_norm_valuation_selected_power. ((forall ff_i_ftsv_norm_valuation_selected_power_repeat. (exists ff_lt_ftsv_norm_valuation_selected_power_repeat_bound. ff_lt_ftsv_norm_valuation_selected_power_repeat_bound + S ff_i_ftsv_norm_valuation_selected_power_repeat = e) -> (((exists ff_h_ftsv_norm_valuation_selected_power_repeat_decoded. ff_h_ftsv_norm_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_norm_valuation_selected_power_repeat)) * ff_c_ftsv_norm_valuation_selected_power)) /\ exists ff_q_ftsv_norm_valuation_selected_power_repeat_decoded. ff_b_ftsv_norm_valuation_selected_power = ff_q_ftsv_norm_valuation_selected_power_repeat_decoded * S ((S (ff_i_ftsv_norm_valuation_selected_power_repeat)) * ff_c_ftsv_norm_valuation_selected_power) + (p)))) /\ (exists ff_u_ftsv_norm_valuation_selected_power_product ff_v_ftsv_norm_valuation_selected_power_product. ((((exists ff_h_ftsv_norm_valuation_selected_power_product_start. ff_h_ftsv_norm_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_norm_valuation_selected_power_product)) /\ exists ff_q_ftsv_norm_valuation_selected_power_product_start. ff_u_ftsv_norm_valuation_selected_power_product = ff_q_ftsv_norm_valuation_selected_power_product_start * S ((S (0)) * ff_v_ftsv_norm_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_ftsv_norm_valuation_selected_power_product_terminal. ff_h_ftsv_norm_valuation_selected_power_product_terminal + S (bpv_result_ftsv_norm_valuation_selected) = S ((S (e)) * ff_v_ftsv_norm_valuation_selected_power_product)) /\ exists ff_q_ftsv_norm_valuation_selected_power_product_terminal. ff_u_ftsv_norm_valuation_selected_power_product = ff_q_ftsv_norm_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_ftsv_norm_valuation_selected_power_product) + (bpv_result_ftsv_norm_valuation_selected))) /\ forall ff_i_ftsv_norm_valuation_selected_power_product. (exists ff_lt_ftsv_norm_valuation_selected_power_product_bound. ff_lt_ftsv_norm_valuation_selected_power_product_bound + S ff_i_ftsv_norm_valuation_selected_power_product = e) -> exists ff_p_ftsv_norm_valuation_selected_power_product ff_r_ftsv_norm_valuation_selected_power_product ff_s_ftsv_norm_valuation_selected_power_product. ((((exists ff_h_ftsv_norm_valuation_selected_power_product_factor. ff_h_ftsv_norm_valuation_selected_power_product_factor + S (ff_p_ftsv_norm_valuation_selected_power_product) = S ((S (ff_i_ftsv_norm_valuation_selected_power_product)) * ff_c_ftsv_norm_valuation_selected_power)) /\ exists ff_q_ftsv_norm_valuation_selected_power_product_factor. ff_b_ftsv_norm_valuation_selected_power = ff_q_ftsv_norm_valuation_selected_power_product_factor * S ((S (ff_i_ftsv_norm_valuation_selected_power_product)) * ff_c_ftsv_norm_valuation_selected_power) + (ff_p_ftsv_norm_valuation_selected_power_product))) /\ ((((exists ff_h_ftsv_norm_valuation_selected_power_product_partial. ff_h_ftsv_norm_valuation_selected_power_product_partial + S (ff_r_ftsv_norm_valuation_selected_power_product) = S ((S (ff_i_ftsv_norm_valuation_selected_power_product)) * ff_v_ftsv_norm_valuation_selected_power_product)) /\ exists ff_q_ftsv_norm_valuation_selected_power_product_partial. ff_u_ftsv_norm_valuation_selected_power_product = ff_q_ftsv_norm_valuation_selected_power_product_partial * S ((S (ff_i_ftsv_norm_valuation_selected_power_product)) * ff_v_ftsv_norm_valuation_selected_power_product) + (ff_r_ftsv_norm_valuation_selected_power_product))) /\ ((((exists ff_h_ftsv_norm_valuation_selected_power_product_successor. ff_h_ftsv_norm_valuation_selected_power_product_successor + S (ff_s_ftsv_norm_valuation_selected_power_product) = S ((S (S ff_i_ftsv_norm_valuation_selected_power_product)) * ff_v_ftsv_norm_valuation_selected_power_product)) /\ exists ff_q_ftsv_norm_valuation_selected_power_product_successor. ff_u_ftsv_norm_valuation_selected_power_product = ff_q_ftsv_norm_valuation_selected_power_product_successor * S ((S (S ff_i_ftsv_norm_valuation_selected_power_product)) * ff_v_ftsv_norm_valuation_selected_power_product) + (ff_s_ftsv_norm_valuation_selected_power_product))) /\ ff_s_ftsv_norm_valuation_selected_power_product = ff_r_ftsv_norm_valuation_selected_power_product * ff_p_ftsv_norm_valuation_selected_power_product)))))))) /\ (exists bpv_factor_ftsv_norm_valuation_selected_divides. (a * a + b * b) = bpv_result_ftsv_norm_valuation_selected * bpv_factor_ftsv_norm_valuation_selected_divides)))) /\ forall bpv_candidate_ftsv_norm_valuation. (exists bpv_gap_ftsv_norm_valuation_candidate_bound. bpv_gap_ftsv_norm_valuation_candidate_bound + bpv_candidate_ftsv_norm_valuation = (a * a + b * b)) -> (exists bpv_result_ftsv_norm_valuation_candidate. ((exists ff_b_ftsv_norm_valuation_candidate_power ff_c_ftsv_norm_valuation_candidate_power. ((forall ff_i_ftsv_norm_valuation_candidate_power_repeat. (exists ff_lt_ftsv_norm_valuation_candidate_power_repeat_bound. ff_lt_ftsv_norm_valuation_candidate_power_repeat_bound + S ff_i_ftsv_norm_valuation_candidate_power_repeat = bpv_candidate_ftsv_norm_valuation) -> (((exists ff_h_ftsv_norm_valuation_candidate_power_repeat_decoded. ff_h_ftsv_norm_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_norm_valuation_candidate_power_repeat)) * ff_c_ftsv_norm_valuation_candidate_power)) /\ exists ff_q_ftsv_norm_valuation_candidate_power_repeat_decoded. ff_b_ftsv_norm_valuation_candidate_power = ff_q_ftsv_norm_valuation_candidate_power_repeat_decoded * S ((S (ff_i_ftsv_norm_valuation_candidate_power_repeat)) * ff_c_ftsv_norm_valuation_candidate_power) + (p)))) /\ (exists ff_u_ftsv_norm_valuation_candidate_power_product ff_v_ftsv_norm_valuation_candidate_power_product. ((((exists ff_h_ftsv_norm_valuation_candidate_power_product_start. ff_h_ftsv_norm_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_norm_valuation_candidate_power_product)) /\ exists ff_q_ftsv_norm_valuation_candidate_power_product_start. ff_u_ftsv_norm_valuation_candidate_power_product = ff_q_ftsv_norm_valuation_candidate_power_product_start * S ((S (0)) * ff_v_ftsv_norm_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsv_norm_valuation_candidate_power_product_terminal. ff_h_ftsv_norm_valuation_candidate_power_product_terminal + S (bpv_result_ftsv_norm_valuation_candidate) = S ((S (bpv_candidate_ftsv_norm_valuation)) * ff_v_ftsv_norm_valuation_candidate_power_product)) /\ exists ff_q_ftsv_norm_valuation_candidate_power_product_terminal. ff_u_ftsv_norm_valuation_candidate_power_product = ff_q_ftsv_norm_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_ftsv_norm_valuation)) * ff_v_ftsv_norm_valuation_candidate_power_product) + (bpv_result_ftsv_norm_valuation_candidate))) /\ forall ff_i_ftsv_norm_valuation_candidate_power_product. (exists ff_lt_ftsv_norm_valuation_candidate_power_product_bound. ff_lt_ftsv_norm_valuation_candidate_power_product_bound + S ff_i_ftsv_norm_valuation_candidate_power_product = bpv_candidate_ftsv_norm_valuation) -> exists ff_p_ftsv_norm_valuation_candidate_power_product ff_r_ftsv_norm_valuation_candidate_power_product ff_s_ftsv_norm_valuation_candidate_power_product. ((((exists ff_h_ftsv_norm_valuation_candidate_power_product_factor. ff_h_ftsv_norm_valuation_candidate_power_product_factor + S (ff_p_ftsv_norm_valuation_candidate_power_product) = S ((S (ff_i_ftsv_norm_valuation_candidate_power_product)) * ff_c_ftsv_norm_valuation_candidate_power)) /\ exists ff_q_ftsv_norm_valuation_candidate_power_product_factor. ff_b_ftsv_norm_valuation_candidate_power = ff_q_ftsv_norm_valuation_candidate_power_product_factor * S ((S (ff_i_ftsv_norm_valuation_candidate_power_product)) * ff_c_ftsv_norm_valuation_candidate_power) + (ff_p_ftsv_norm_valuation_candidate_power_product))) /\ ((((exists ff_h_ftsv_norm_valuation_candidate_power_product_partial. ff_h_ftsv_norm_valuation_candidate_power_product_partial + S (ff_r_ftsv_norm_valuation_candidate_power_product) = S ((S (ff_i_ftsv_norm_valuation_candidate_power_product)) * ff_v_ftsv_norm_valuation_candidate_power_product)) /\ exists ff_q_ftsv_norm_valuation_candidate_power_product_partial. ff_u_ftsv_norm_valuation_candidate_power_product = ff_q_ftsv_norm_valuation_candidate_power_product_partial * S ((S (ff_i_ftsv_norm_valuation_candidate_power_product)) * ff_v_ftsv_norm_valuation_candidate_power_product) + (ff_r_ftsv_norm_valuation_candidate_power_product))) /\ ((((exists ff_h_ftsv_norm_valuation_candidate_power_product_successor. ff_h_ftsv_norm_valuation_candidate_power_product_successor + S (ff_s_ftsv_norm_valuation_candidate_power_product) = S ((S (S ff_i_ftsv_norm_valuation_candidate_power_product)) * ff_v_ftsv_norm_valuation_candidate_power_product)) /\ exists ff_q_ftsv_norm_valuation_candidate_power_product_successor. ff_u_ftsv_norm_valuation_candidate_power_product = ff_q_ftsv_norm_valuation_candidate_power_product_successor * S ((S (S ff_i_ftsv_norm_valuation_candidate_power_product)) * ff_v_ftsv_norm_valuation_candidate_power_product) + (ff_s_ftsv_norm_valuation_candidate_power_product))) /\ ff_s_ftsv_norm_valuation_candidate_power_product = ff_r_ftsv_norm_valuation_candidate_power_product * ff_p_ftsv_norm_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_ftsv_norm_valuation_candidate_divides. (a * a + b * b) = bpv_result_ftsv_norm_valuation_candidate * bpv_factor_ftsv_norm_valuation_candidate_divides))) -> (exists bpv_gap_ftsv_norm_valuation_maximal. bpv_gap_ftsv_norm_valuation_maximal + bpv_candidate_ftsv_norm_valuation = e)) -> exists h. e = h + h

Proof neighborhood

Direct theorem prerequisites

le_refl · Stable closed TS003T three_mod_four_prime_two_square_norm_valuation_even_bounded

Direct theorem dependents

TS003V three_mod_four_prime_represented_nonzero_valuation_even

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

20 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 (1)
three mod four prime two square norm valuation even bounded
01Fix variables and assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro e
  5. L5
    intro hprime
  6. L6
    intro hthree
  7. L7
    intro hnonzero
  8. L8
    intro hvaluation
02Use earlier factsL9–18

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

  1. L9
    specialize three_mod_four_prime_two_square_norm_valuation_even_bounded (a * a + b * b)
  2. L10
    specialize three_mod_four_prime_two_square_norm_valuation_even_bounded p
  3. L11
    specialize three_mod_four_prime_two_square_norm_valuation_even_bounded a
  4. L12
    specialize three_mod_four_prime_two_square_norm_valuation_even_bounded b
  5. L13
    specialize three_mod_four_prime_two_square_norm_valuation_even_bounded e
  6. L14
    apply three_mod_four_prime_two_square_norm_valuation_even_bounded
  7. L15
    specialize le_refl (a * a + b * b)
  8. L16
    exact le_refl
  9. L17
    exact hprime
  10. L18
    exact hthree
03Use earlier factsL19–20

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

  1. L19
    exact hnonzero
  2. L20
    exact hvaluation

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro e
  5. 0005intro hprime
  6. 0006intro hthree
  7. 0007intro hnonzero
  8. 0008intro hvaluation
  9. 0009specialize three_mod_four_prime_two_square_norm_valuation_even_bounded (a * a + b * b)
  10. 0010specialize three_mod_four_prime_two_square_norm_valuation_even_bounded p
  11. 0011specialize three_mod_four_prime_two_square_norm_valuation_even_bounded a
  12. 0012specialize three_mod_four_prime_two_square_norm_valuation_even_bounded b
  13. 0013specialize three_mod_four_prime_two_square_norm_valuation_even_bounded e
  14. 0014apply three_mod_four_prime_two_square_norm_valuation_even_bounded
  15. 0015specialize le_refl (a * a + b * b)
  16. 0016exact le_refl
  17. 0017exact hprime
  18. 0018exact hthree
  19. 0019exact hnonzero
  20. 0020exact hvaluation