PG0030

prime_field_convolution_coefficient_left_unit

The actual residue of the unit-left natural sum is its already bounded right-input value. No polynomial equality is hidden among the premises.

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

All products and sums use actual beta-coded coefficients; equality is formal coefficient equivalence, not equality of evaluations or raw codes. The zero gcd is included. Uniqueness is up to formal equivalence, not unique Bézout coefficients. This proves the polynomial gcd component, not G091 irreducible-polynomial existence or arbitrary prime-power-field construction.

Exact theorem in conservative defined notation

∀ p. ∀ ub. ∀ uc. ∀ ab. ∀ ac. ∀ L. ∀ i. ∀ a. ∀ r. BetaAt(ub,uc,0,1) → Lt(i,L) → BetaAt(ab,ac,i,a) → Lt(a,p) → FpConvolutionCoefficient(p,ub,uc,1,ab,ac,L,i,r) → r = a

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

FpConvolutionCoefficient(p,ab,ac,L,bb,bc,M,i,r) · 1Lt(a,b) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

polynomial_diagonal_left_unit_natural_sumprime_field_residue_bounded_value · checked external prerequisite
Original expanded first-order statement 
forall p ub uc ab ac L i a r. (((exists ff_h_pfp_unit_coefficient_unit. ff_h_pfp_unit_coefficient_unit + S (1) = S ((S (0)) * uc)) /\ exists ff_q_pfp_unit_coefficient_unit. ub = ff_q_pfp_unit_coefficient_unit * S ((S (0)) * uc) + (1))) -> (exists pfa_gap_unit_coefficient_index. pfa_gap_unit_coefficient_index + S (i) = (L)) -> (((exists ff_h_pfp_unit_coefficient_A. ff_h_pfp_unit_coefficient_A + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_unit_coefficient_A. ab = ff_q_pfp_unit_coefficient_A * S ((S (i)) * ac) + (a))) -> (exists pfa_gap_unit_coefficient_bound. pfa_gap_unit_coefficient_bound + S (a) = (p)) -> (exists pfc_terms_code_unit_coefficient_actual pfc_terms_scale_unit_coefficient_actual pfc_natural_sum_unit_coefficient_actual. ((forall pfc_index_unit_coefficient_actualdiagonal. (exists pfa_gap_unit_coefficient_actualdiagonalbound. pfa_gap_unit_coefficient_actualdiagonalbound + S (pfc_index_unit_coefficient_actualdiagonal) = (S (i))) -> exists pfc_value_unit_coefficient_actualdiagonal. ((((exists ff_h_pfp_unit_coefficient_actualdiagonalentry. ff_h_pfp_unit_coefficient_actualdiagonalentry + S (pfc_value_unit_coefficient_actualdiagonal) = S ((S (pfc_index_unit_coefficient_actualdiagonal)) * pfc_terms_scale_unit_coefficient_actual)) /\ exists ff_q_pfp_unit_coefficient_actualdiagonalentry. pfc_terms_code_unit_coefficient_actual = ff_q_pfp_unit_coefficient_actualdiagonalentry * S ((S (pfc_index_unit_coefficient_actualdiagonal)) * pfc_terms_scale_unit_coefficient_actual) + (pfc_value_unit_coefficient_actualdiagonal))) /\ ((exists pfc_complement_unit_coefficient_actualdiagonalterm pfc_left_unit_coefficient_actualdiagonalterm pfc_right_unit_coefficient_actualdiagonalterm. (((pfc_index_unit_coefficient_actualdiagonal)+pfc_complement_unit_coefficient_actualdiagonalterm=(i)) /\ ((((((exists pfa_gap_unit_coefficient_actualdiagonaltermleftinside. pfa_gap_unit_coefficient_actualdiagonaltermleftinside + S (pfc_index_unit_coefficient_actualdiagonal) = (1)) /\ ((((exists ff_h_pfp_unit_coefficient_actualdiagonaltermleftentry. ff_h_pfp_unit_coefficient_actualdiagonaltermleftentry + S (pfc_left_unit_coefficient_actualdiagonalterm) = S ((S (pfc_index_unit_coefficient_actualdiagonal)) * uc)) /\ exists ff_q_pfp_unit_coefficient_actualdiagonaltermleftentry. ub = ff_q_pfp_unit_coefficient_actualdiagonaltermleftentry * S ((S (pfc_index_unit_coefficient_actualdiagonal)) * uc) + (pfc_left_unit_coefficient_actualdiagonalterm)))))) \/ (((exists pfc_gap_unit_coefficient_actualdiagonaltermleftoutside. pfc_gap_unit_coefficient_actualdiagonaltermleftoutside+(1)=(pfc_index_unit_coefficient_actualdiagonal)) /\ (((pfc_left_unit_coefficient_actualdiagonalterm)=0))))) /\ ((((((exists pfa_gap_unit_coefficient_actualdiagonaltermrightinside. pfa_gap_unit_coefficient_actualdiagonaltermrightinside + S (pfc_complement_unit_coefficient_actualdiagonalterm) = (L)) /\ ((((exists ff_h_pfp_unit_coefficient_actualdiagonaltermrightentry. ff_h_pfp_unit_coefficient_actualdiagonaltermrightentry + S (pfc_right_unit_coefficient_actualdiagonalterm) = S ((S (pfc_complement_unit_coefficient_actualdiagonalterm)) * ac)) /\ exists ff_q_pfp_unit_coefficient_actualdiagonaltermrightentry. ab = ff_q_pfp_unit_coefficient_actualdiagonaltermrightentry * S ((S (pfc_complement_unit_coefficient_actualdiagonalterm)) * ac) + (pfc_right_unit_coefficient_actualdiagonalterm)))))) \/ (((exists pfc_gap_unit_coefficient_actualdiagonaltermrightoutside. pfc_gap_unit_coefficient_actualdiagonaltermrightoutside+(L)=(pfc_complement_unit_coefficient_actualdiagonalterm)) /\ (((pfc_right_unit_coefficient_actualdiagonalterm)=0))))) /\ (((pfc_value_unit_coefficient_actualdiagonal)=pfc_left_unit_coefficient_actualdiagonalterm*pfc_right_unit_coefficient_actualdiagonalterm))))))))))) /\ (((exists fs_u_pfc_unit_coefficient_actualsum fs_v_pfc_unit_coefficient_actualsum. ((((exists fs_h_pfc_unit_coefficient_actualsum_body_start. fs_h_pfc_unit_coefficient_actualsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_coefficient_actualsum)) /\ exists fs_q_pfc_unit_coefficient_actualsum_body_start. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_start * S ((S (0)) * fs_v_pfc_unit_coefficient_actualsum) + (0))) /\ ((((exists fs_h_pfc_unit_coefficient_actualsum_body_terminal. fs_h_pfc_unit_coefficient_actualsum_body_terminal + S (pfc_natural_sum_unit_coefficient_actual) = S ((S (S (i))) * fs_v_pfc_unit_coefficient_actualsum)) /\ exists fs_q_pfc_unit_coefficient_actualsum_body_terminal. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_terminal * S ((S (S (i))) * fs_v_pfc_unit_coefficient_actualsum) + (pfc_natural_sum_unit_coefficient_actual))) /\ forall fs_i_pfc_unit_coefficient_actualsum_body_steps. (exists fs_lt_pfc_unit_coefficient_actualsum_body_steps_bound. fs_lt_pfc_unit_coefficient_actualsum_body_steps_bound + S fs_i_pfc_unit_coefficient_actualsum_body_steps = S (i)) -> exists fs_a_pfc_unit_coefficient_actualsum_body_steps fs_r_pfc_unit_coefficient_actualsum_body_steps fs_s_pfc_unit_coefficient_actualsum_body_steps. ((((exists fs_h_pfc_unit_coefficient_actualsum_body_steps_summand. fs_h_pfc_unit_coefficient_actualsum_body_steps_summand + S (fs_a_pfc_unit_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * pfc_terms_scale_unit_coefficient_actual)) /\ exists fs_q_pfc_unit_coefficient_actualsum_body_steps_summand. pfc_terms_code_unit_coefficient_actual = fs_q_pfc_unit_coefficient_actualsum_body_steps_summand * S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * pfc_terms_scale_unit_coefficient_actual) + (fs_a_pfc_unit_coefficient_actualsum_body_steps))) /\ ((((exists fs_h_pfc_unit_coefficient_actualsum_body_steps_partial. fs_h_pfc_unit_coefficient_actualsum_body_steps_partial + S (fs_r_pfc_unit_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum)) /\ exists fs_q_pfc_unit_coefficient_actualsum_body_steps_partial. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_steps_partial * S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum) + (fs_r_pfc_unit_coefficient_actualsum_body_steps))) /\ ((((exists fs_h_pfc_unit_coefficient_actualsum_body_steps_successor. fs_h_pfc_unit_coefficient_actualsum_body_steps_successor + S (fs_s_pfc_unit_coefficient_actualsum_body_steps) = S ((S (S fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum)) /\ exists fs_q_pfc_unit_coefficient_actualsum_body_steps_successor. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_steps_successor * S ((S (S fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum) + (fs_s_pfc_unit_coefficient_actualsum_body_steps))) /\ fs_s_pfc_unit_coefficient_actualsum_body_steps = fs_r_pfc_unit_coefficient_actualsum_body_steps + fs_a_pfc_unit_coefficient_actualsum_body_steps)))))) /\ ((((exists pfa_gap_unit_coefficient_actualresiduebound. pfa_gap_unit_coefficient_actualresiduebound + S (r) = (p)) /\ ((exists pfa_offset_left_unit_coefficient_actualresiduecongruence pfa_offset_right_unit_coefficient_actualresiduecongruence. (pfc_natural_sum_unit_coefficient_actual) + (p) * pfa_offset_left_unit_coefficient_actualresiduecongruence = (r) + (p) * pfa_offset_right_unit_coefficient_actualresiduecongruence))))))))) -> (r=a)

Complete tactic proof in conservative notation

All 43 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

43 script commands · 7 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.

Named ingredients (1)
polynomial diagonal left unit natural sum
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro ub
  3. L3
    intro uc
  4. L4
    intro ab
  5. L5
    intro ac
  6. L6
    intro L
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro r
  10. L10
    intro hu
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hb
  4. L14
    intro hr
03Separate the logical casesL15–19

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

  1. L15
    cases hr
  2. L16
    cases hr_witness
  3. L17
    cases hr_witness_witness
  4. L18
    cases hr_witness_witness_witness
  5. L19
    cases hr_witness_witness_witness_right
04Establish hnL20–29

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

  1. L20
    have hn : x2=a
  2. L21
    specialize polynomial_diagonal_left_unit_natural_sum (ub)
  3. L22
    specialize polynomial_diagonal_left_unit_natural_sum (uc)
  4. L23
    specialize polynomial_diagonal_left_unit_natural_sum (ab)
  5. L24
    specialize polynomial_diagonal_left_unit_natural_sum (ac)
  6. L25
    specialize polynomial_diagonal_left_unit_natural_sum (L)
  7. L26
    specialize polynomial_diagonal_left_unit_natural_sum (i)
  8. L27
    specialize polynomial_diagonal_left_unit_natural_sum (a)
  9. L28
    specialize polynomial_diagonal_left_unit_natural_sum (x)
  10. L29
    specialize polynomial_diagonal_left_unit_natural_sum (x1)
05Use earlier factsL30–36

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

  1. L30
    specialize polynomial_diagonal_left_unit_natural_sum (x2)
  2. L31
    apply polynomial_diagonal_left_unit_natural_sum
  3. L32
    exact hu
  4. L33
    exact hi
  5. L34
    exact ha
  6. L35
    exact hr_witness_witness_witness_left
  7. L36
    exact hr_witness_witness_witness_right_left
06Calculate and transport equalitiesL37–37

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L37
    rewrite hn at hr_witness_witness_witness_right_right
07Use earlier factsL38–43

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

  1. L38
    specialize prime_field_residue_bounded_value (p)
  2. L39
    specialize prime_field_residue_bounded_value (a)
  3. L40
    specialize prime_field_residue_bounded_value (r)
  4. L41
    apply prime_field_residue_bounded_value
  5. L42
    exact hb
  6. L43
    exact hr_witness_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 43 lines
  1. 0001intro p
  2. 0002intro ub
  3. 0003intro uc
  4. 0004intro ab
  5. 0005intro ac
  6. 0006intro L
  7. 0007intro i
  8. 0008intro a
  9. 0009intro r
  10. 0010intro hu
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hb
  14. 0014intro hr
  15. 0015cases hr
  16. 0016cases hr_witness
  17. 0017cases hr_witness_witness
  18. 0018cases hr_witness_witness_witness
  19. 0019cases hr_witness_witness_witness_right
  20. 0020have hn : x2=a
  21. 0021specialize polynomial_diagonal_left_unit_natural_sum (ub)
  22. 0022specialize polynomial_diagonal_left_unit_natural_sum (uc)
  23. 0023specialize polynomial_diagonal_left_unit_natural_sum (ab)
  24. 0024specialize polynomial_diagonal_left_unit_natural_sum (ac)
  25. 0025specialize polynomial_diagonal_left_unit_natural_sum (L)
  26. 0026specialize polynomial_diagonal_left_unit_natural_sum (i)
  27. 0027specialize polynomial_diagonal_left_unit_natural_sum (a)
  28. 0028specialize polynomial_diagonal_left_unit_natural_sum (x)
  29. 0029specialize polynomial_diagonal_left_unit_natural_sum (x1)
  30. 0030specialize polynomial_diagonal_left_unit_natural_sum (x2)
  31. 0031apply polynomial_diagonal_left_unit_natural_sum
  32. 0032exact hu
  33. 0033exact hi
  34. 0034exact ha
  35. 0035exact hr_witness_witness_witness_left
  36. 0036exact hr_witness_witness_witness_right_left
  37. 0037rewrite hn at hr_witness_witness_witness_right_right
  38. 0038specialize prime_field_residue_bounded_value (p)
  39. 0039specialize prime_field_residue_bounded_value (a)
  40. 0040specialize prime_field_residue_bounded_value (r)
  41. 0041apply prime_field_residue_bounded_value
  42. 0042exact hb
  43. 0043exact hr_witness_witness_witness_right_right