PX0056

prime_field_polynomial_trim_input_transport

Actual trim witnesses transport under equality of the annotated input prefix, including its zero prefix and genuinely shifted suffix.

Alpha v34 checked-use · first admitted v33 · 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.

Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ B. ∀ C. ∀ L. ∀ t. ∀ rb. ∀ rc. ∀ R. BetaPrefixEqual(b,c,B,C,L) → FpPolynomialTrim(p,b,c,L,t,rb,rc,R) → FpPolynomialTrim(p,B,C,L,t,rb,rc,R)

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

Definition DAG

BetaPrefixEqual(b,c,d,e,l) · 2FpPolynomialTrim(p,b,c,L,t,d,e,M) · 2Le(a,b) · 2

Actual proof prerequisites

matrix_rank_prefix_equality_symmetric · checked external prerequisiteprime_field_polynomial_trim_length_bounds · checked external prerequisitematrix_rank_bounded_prefix_transport · checked external prerequisitelt_of_lt_of_le · checked external prerequisitematrix_recursive_lt_add_left · checked external prerequisite
Original expanded first-order statement 
forall p b c B C L t rb rc R. (forall mdr_i_pfp_trim_transport_equal mdr_a_pfp_trim_transport_equal. (exists mdr_gap_pfp_trim_transport_equalb. mdr_gap_pfp_trim_transport_equalb + S (mdr_i_pfp_trim_transport_equal) = (L)) -> (((exists ff_h_mdr_pfp_trim_transport_equalo. ff_h_mdr_pfp_trim_transport_equalo + S (mdr_a_pfp_trim_transport_equal) = S ((S (mdr_i_pfp_trim_transport_equal)) * c)) /\ exists ff_q_mdr_pfp_trim_transport_equalo. b = ff_q_mdr_pfp_trim_transport_equalo * S ((S (mdr_i_pfp_trim_transport_equal)) * c) + (mdr_a_pfp_trim_transport_equal))) -> (((exists ff_h_mdr_pfp_trim_transport_equaln. ff_h_mdr_pfp_trim_transport_equaln + S (mdr_a_pfp_trim_transport_equal) = S ((S (mdr_i_pfp_trim_transport_equal)) * C)) /\ exists ff_q_mdr_pfp_trim_transport_equaln. B = ff_q_mdr_pfp_trim_transport_equaln * S ((S (mdr_i_pfp_trim_transport_equal)) * C) + (mdr_a_pfp_trim_transport_equal)))) -> ((((L)=(t)+(R)) /\ (((forall fom_index_pfp_trim_transport_sourceinput. (exists fom_gap_pfp_trim_transport_sourceinput_index_bound. fom_gap_pfp_trim_transport_sourceinput_index_bound + S (fom_index_pfp_trim_transport_sourceinput) = L) -> exists fom_value_pfp_trim_transport_sourceinput. ((((exists fom_beta_height_pfp_trim_transport_sourceinput_entry. fom_beta_height_pfp_trim_transport_sourceinput_entry + S (fom_value_pfp_trim_transport_sourceinput) = S ((S (fom_index_pfp_trim_transport_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_transport_sourceinput_entry. b = fom_beta_quotient_pfp_trim_transport_sourceinput_entry * S ((S (fom_index_pfp_trim_transport_sourceinput)) * c) + (fom_value_pfp_trim_transport_sourceinput))) /\ (exists fom_gap_pfp_trim_transport_sourceinput_value_bound. fom_gap_pfp_trim_transport_sourceinput_value_bound + S (fom_value_pfp_trim_transport_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_transport_sourceremoved. (exists pfa_gap_trim_transport_sourceremovedindex. pfa_gap_trim_transport_sourceremovedindex + S (pfp_repeat_index_trim_transport_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_transport_sourceremovedentry. ff_h_pfp_trim_transport_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_transport_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_transport_sourceremovedentry. b = ff_q_pfp_trim_transport_sourceremovedentry * S ((S (pfp_repeat_index_trim_transport_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_transport_sourcesuffix pftrim_value_trim_transport_sourcesuffix. (exists pfa_gap_trim_transport_sourcesuffixbound. pfa_gap_trim_transport_sourcesuffixbound + S (pftrim_index_trim_transport_sourcesuffix) = (R)) -> (((exists ff_h_pfp_trim_transport_sourcesuffixsource. ff_h_pfp_trim_transport_sourcesuffixsource + S (pftrim_value_trim_transport_sourcesuffix) = S ((S ((t)+pftrim_index_trim_transport_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_transport_sourcesuffixsource. b = ff_q_pfp_trim_transport_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_transport_sourcesuffix)) * c) + (pftrim_value_trim_transport_sourcesuffix))) -> (((exists ff_h_pfp_trim_transport_sourcesuffixoutput. ff_h_pfp_trim_transport_sourcesuffixoutput + S (pftrim_value_trim_transport_sourcesuffix) = S ((S (pftrim_index_trim_transport_sourcesuffix)) * rc)) /\ exists ff_q_pfp_trim_transport_sourcesuffixoutput. rb = ff_q_pfp_trim_transport_sourcesuffixoutput * S ((S (pftrim_index_trim_transport_sourcesuffix)) * rc) + (pftrim_value_trim_transport_sourcesuffix)))) /\ (((R)=0 \/ (exists pftrim_leading_trim_transport_sourcenormal. ((((exists ff_h_pfp_trim_transport_sourcenormalentry. ff_h_pfp_trim_transport_sourcenormalentry + S (pftrim_leading_trim_transport_sourcenormal) = S ((S (0)) * rc)) /\ exists ff_q_pfp_trim_transport_sourcenormalentry. rb = ff_q_pfp_trim_transport_sourcenormalentry * S ((S (0)) * rc) + (pftrim_leading_trim_transport_sourcenormal))) /\ ((~(pftrim_leading_trim_transport_sourcenormal=0))))))))))))))) -> ((((L)=(t)+(R)) /\ (((forall fom_index_pfp_trim_transport_resultinput. (exists fom_gap_pfp_trim_transport_resultinput_index_bound. fom_gap_pfp_trim_transport_resultinput_index_bound + S (fom_index_pfp_trim_transport_resultinput) = L) -> exists fom_value_pfp_trim_transport_resultinput. ((((exists fom_beta_height_pfp_trim_transport_resultinput_entry. fom_beta_height_pfp_trim_transport_resultinput_entry + S (fom_value_pfp_trim_transport_resultinput) = S ((S (fom_index_pfp_trim_transport_resultinput)) * C)) /\ exists fom_beta_quotient_pfp_trim_transport_resultinput_entry. B = fom_beta_quotient_pfp_trim_transport_resultinput_entry * S ((S (fom_index_pfp_trim_transport_resultinput)) * C) + (fom_value_pfp_trim_transport_resultinput))) /\ (exists fom_gap_pfp_trim_transport_resultinput_value_bound. fom_gap_pfp_trim_transport_resultinput_value_bound + S (fom_value_pfp_trim_transport_resultinput) = p))) /\ (((forall pfp_repeat_index_trim_transport_resultremoved. (exists pfa_gap_trim_transport_resultremovedindex. pfa_gap_trim_transport_resultremovedindex + S (pfp_repeat_index_trim_transport_resultremoved) = (t)) -> (((exists ff_h_pfp_trim_transport_resultremovedentry. ff_h_pfp_trim_transport_resultremovedentry + S (0) = S ((S (pfp_repeat_index_trim_transport_resultremoved)) * C)) /\ exists ff_q_pfp_trim_transport_resultremovedentry. B = ff_q_pfp_trim_transport_resultremovedentry * S ((S (pfp_repeat_index_trim_transport_resultremoved)) * C) + (0)))) /\ (((forall pftrim_index_trim_transport_resultsuffix pftrim_value_trim_transport_resultsuffix. (exists pfa_gap_trim_transport_resultsuffixbound. pfa_gap_trim_transport_resultsuffixbound + S (pftrim_index_trim_transport_resultsuffix) = (R)) -> (((exists ff_h_pfp_trim_transport_resultsuffixsource. ff_h_pfp_trim_transport_resultsuffixsource + S (pftrim_value_trim_transport_resultsuffix) = S ((S ((t)+pftrim_index_trim_transport_resultsuffix)) * C)) /\ exists ff_q_pfp_trim_transport_resultsuffixsource. B = ff_q_pfp_trim_transport_resultsuffixsource * S ((S ((t)+pftrim_index_trim_transport_resultsuffix)) * C) + (pftrim_value_trim_transport_resultsuffix))) -> (((exists ff_h_pfp_trim_transport_resultsuffixoutput. ff_h_pfp_trim_transport_resultsuffixoutput + S (pftrim_value_trim_transport_resultsuffix) = S ((S (pftrim_index_trim_transport_resultsuffix)) * rc)) /\ exists ff_q_pfp_trim_transport_resultsuffixoutput. rb = ff_q_pfp_trim_transport_resultsuffixoutput * S ((S (pftrim_index_trim_transport_resultsuffix)) * rc) + (pftrim_value_trim_transport_resultsuffix)))) /\ (((R)=0 \/ (exists pftrim_leading_trim_transport_resultnormal. ((((exists ff_h_pfp_trim_transport_resultnormalentry. ff_h_pfp_trim_transport_resultnormalentry + S (pftrim_leading_trim_transport_resultnormal) = S ((S (0)) * rc)) /\ exists ff_q_pfp_trim_transport_resultnormalentry. rb = ff_q_pfp_trim_transport_resultnormalentry * S ((S (0)) * rc) + (pftrim_leading_trim_transport_resultnormal))) /\ ((~(pftrim_leading_trim_transport_resultnormal=0)))))))))))))))

Complete tactic proof in conservative notation

All 83 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

83 script commands · 18 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.

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 assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro B
  5. L5
    intro C
  6. L6
    intro L
  7. L7
    intro t
  8. L8
    intro rb
  9. L9
    intro rc
  10. L10
    intro R
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hequal
  2. L12
    intro htrim
03Establish hreverseL13–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix rank prefix equality symmetric.

  1. L13
    have hreverse : BetaPrefixEqual(B,C,b,c,L)Definitions: BetaPrefixEqual(B,C,b,c,L)Original native command in the exact edition
  2. L14
    specialize matrix_rank_prefix_equality_symmetric (b)
  3. L15
    specialize matrix_rank_prefix_equality_symmetric (c)
  4. L16
    specialize matrix_rank_prefix_equality_symmetric (B)
  5. L17
    specialize matrix_rank_prefix_equality_symmetric (C)
  6. L18
    specialize matrix_rank_prefix_equality_symmetric (L)
  7. L19
    apply matrix_rank_prefix_equality_symmetric
  8. L20
    exact hequal
04Establish hboundsL21–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial trim length bounds.

  1. L21
    have hbounds : Le(t,L) ∧ Le(R,L)Definitions: Le(t,L)Le(R,L)Original native command in the exact edition
  2. L22
    specialize prime_field_polynomial_trim_length_bounds (p)
  3. L23
    specialize prime_field_polynomial_trim_length_bounds (b)
  4. L24
    specialize prime_field_polynomial_trim_length_bounds (c)
  5. L25
    specialize prime_field_polynomial_trim_length_bounds (L)
  6. L26
    specialize prime_field_polynomial_trim_length_bounds (t)
  7. L27
    specialize prime_field_polynomial_trim_length_bounds (rb)
  8. L28
    specialize prime_field_polynomial_trim_length_bounds (rc)
  9. L29
    specialize prime_field_polynomial_trim_length_bounds (R)
  10. L30
    apply prime_field_polynomial_trim_length_bounds
05Use earlier factsL31–31

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

  1. L31
    exact htrim
06Separate the logical casesL32–37

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

  1. L32
    cases hbounds
  2. L33
    cases htrim
  3. L34
    cases htrim_right
  4. L35
    cases htrim_right_right
  5. L36
    cases htrim_right_right_right
  6. L37
    split
07Use earlier factsL38–38

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

  1. L38
    exact htrim_left
08Separate the logical casesL39–39

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

  1. L39
    split
09Use earlier factsL40–48

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

  1. L40
    specialize matrix_rank_bounded_prefix_transport (b)
  2. L41
    specialize matrix_rank_bounded_prefix_transport (c)
  3. L42
    specialize matrix_rank_bounded_prefix_transport (B)
  4. L43
    specialize matrix_rank_bounded_prefix_transport (C)
  5. L44
    specialize matrix_rank_bounded_prefix_transport (L)
  6. L45
    specialize matrix_rank_bounded_prefix_transport (p)
  7. L46
    apply matrix_rank_bounded_prefix_transport
  8. L47
    exact hequal
  9. L48
    exact htrim_right_left
10Separate the logical casesL49–49

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

  1. L49
    split
11Fix variables and assumptionsL50–51

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

  1. L50
    intro i
  2. L51
    intro hindex
12Use earlier factsL52–61

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

  1. L52
    specialize hequal (i)
  2. L53
    specialize hequal (0)
  3. L54
    apply hequal
  4. L55
    specialize lt_of_lt_of_le (i)
  5. L56
    specialize lt_of_lt_of_le (t)
  6. L57
    specialize lt_of_lt_of_le (L)
  7. L58
    apply lt_of_lt_of_le
  8. L59
    exact hindex
  9. L60
    exact hbounds_left
  10. L61
    specialize htrim_right_right_left (i)
13Use earlier factsL62–63

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

  1. L62
    apply htrim_right_right_left
  2. L63
    exact hindex
14Separate the logical casesL64–64

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

  1. L64
    split
15Fix variables and assumptionsL65–68

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

  1. L65
    intro i
  2. L66
    intro a
  3. L67
    intro hindex
  4. L68
    intro hvalue
16Use earlier factsL69–75

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

  1. L69
    specialize htrim_right_right_right_left (i)
  2. L70
    specialize htrim_right_right_right_left (a)
  3. L71
    apply htrim_right_right_right_left
  4. L72
    exact hindex
  5. L73
    specialize hreverse (t+i)
  6. L74
    specialize hreverse (a)
  7. L75
    apply hreverse
17Calculate and transport equalitiesL76–76

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

  1. L76
    rewrite htrim_left
18Use earlier factsL77–83

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

  1. L77
    specialize matrix_recursive_lt_add_left (i)
  2. L78
    specialize matrix_recursive_lt_add_left (R)
  3. L79
    specialize matrix_recursive_lt_add_left (t)
  4. L80
    apply matrix_recursive_lt_add_left
  5. L81
    exact hindex
  6. L82
    exact hvalue
  7. L83
    exact htrim_right_right_right_right

Library-wide reading audit

Original defined command ledger · 83 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro B
  5. 0005intro C
  6. 0006intro L
  7. 0007intro t
  8. 0008intro rb
  9. 0009intro rc
  10. 0010intro R
  11. 0011intro hequal
  12. 0012intro htrim
  13. 0013have hreverse : BetaPrefixEqual(B,C,b,c,L)
  14. 0014specialize matrix_rank_prefix_equality_symmetric (b)
  15. 0015specialize matrix_rank_prefix_equality_symmetric (c)
  16. 0016specialize matrix_rank_prefix_equality_symmetric (B)
  17. 0017specialize matrix_rank_prefix_equality_symmetric (C)
  18. 0018specialize matrix_rank_prefix_equality_symmetric (L)
  19. 0019apply matrix_rank_prefix_equality_symmetric
  20. 0020exact hequal
  21. 0021have hbounds : Le(t,L) ∧ Le(R,L)
  22. 0022specialize prime_field_polynomial_trim_length_bounds (p)
  23. 0023specialize prime_field_polynomial_trim_length_bounds (b)
  24. 0024specialize prime_field_polynomial_trim_length_bounds (c)
  25. 0025specialize prime_field_polynomial_trim_length_bounds (L)
  26. 0026specialize prime_field_polynomial_trim_length_bounds (t)
  27. 0027specialize prime_field_polynomial_trim_length_bounds (rb)
  28. 0028specialize prime_field_polynomial_trim_length_bounds (rc)
  29. 0029specialize prime_field_polynomial_trim_length_bounds (R)
  30. 0030apply prime_field_polynomial_trim_length_bounds
  31. 0031exact htrim
  32. 0032cases hbounds
  33. 0033cases htrim
  34. 0034cases htrim_right
  35. 0035cases htrim_right_right
  36. 0036cases htrim_right_right_right
  37. 0037split
  38. 0038exact htrim_left
  39. 0039split
  40. 0040specialize matrix_rank_bounded_prefix_transport (b)
  41. 0041specialize matrix_rank_bounded_prefix_transport (c)
  42. 0042specialize matrix_rank_bounded_prefix_transport (B)
  43. 0043specialize matrix_rank_bounded_prefix_transport (C)
  44. 0044specialize matrix_rank_bounded_prefix_transport (L)
  45. 0045specialize matrix_rank_bounded_prefix_transport (p)
  46. 0046apply matrix_rank_bounded_prefix_transport
  47. 0047exact hequal
  48. 0048exact htrim_right_left
  49. 0049split
  50. 0050intro i
  51. 0051intro hindex
  52. 0052specialize hequal (i)
  53. 0053specialize hequal (0)
  54. 0054apply hequal
  55. 0055specialize lt_of_lt_of_le (i)
  56. 0056specialize lt_of_lt_of_le (t)
  57. 0057specialize lt_of_lt_of_le (L)
  58. 0058apply lt_of_lt_of_le
  59. 0059exact hindex
  60. 0060exact hbounds_left
  61. 0061specialize htrim_right_right_left (i)
  62. 0062apply htrim_right_right_left
  63. 0063exact hindex
  64. 0064split
  65. 0065intro i
  66. 0066intro a
  67. 0067intro hindex
  68. 0068intro hvalue
  69. 0069specialize htrim_right_right_right_left (i)
  70. 0070specialize htrim_right_right_right_left (a)
  71. 0071apply htrim_right_right_right_left
  72. 0072exact hindex
  73. 0073specialize hreverse (t+i)
  74. 0074specialize hreverse (a)
  75. 0075apply hreverse
  76. 0076rewrite htrim_left
  77. 0077specialize matrix_recursive_lt_add_left (i)
  78. 0078specialize matrix_recursive_lt_add_left (R)
  79. 0079specialize matrix_recursive_lt_add_left (t)
  80. 0080apply matrix_recursive_lt_add_left
  81. 0081exact hindex
  82. 0082exact hvalue
  83. 0083exact htrim_right_right_right_right