FP0051

prime_field_unit_trace_exists

Construct an actual history of n additions of one for every natural n, including the empty history.

Alpha v34 checked-use · first admitted v31 · 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.

This checkpoint constructs prime-order fields (k=1), with genuine finite arithmetic tables, cardinality, and characteristic. Inversion is proved only for nonzero elements; the table's zero entry is a zero-to-zero convention. G091 for every prime power p^k, with an irreducible polynomial of degree k, remains open. No extension-field construction or G091 closure is claimed.

Exact theorem in conservative defined notation

∀ p. ∀ n. Prime(p) → ∃ x. ∃ y. ∃ z. FpUnitTrace(p,x,y,n,z)

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

Definition DAG

FpAdd(p,a,b,c) · 1FpUnitTrace(p,b,c,n,r) · 2FpUnitMultiple(p,n,r) · 1Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisiteprime_field_unit_trace_result_boundedprime_field_add_existsprime_two_le · checked external prerequisiteprime_field_unit_trace_successor
Original expanded first-order statement 
forall p n. (~((p) = 1) /\ forall pfa_factor_left_trace_exists_domain pfa_factor_right_trace_exists_domain. (p) = pfa_factor_left_trace_exists_domain * pfa_factor_right_trace_exists_domain -> pfa_factor_left_trace_exists_domain = 1 \/ pfa_factor_right_trace_exists_domain = 1) -> exists b c r. (((((exists ff_h_pft_trace_exists_resultstart. ff_h_pft_trace_exists_resultstart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_exists_resultstart. b = ff_q_pft_trace_exists_resultstart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_exists_resultterminal. ff_h_pft_trace_exists_resultterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_exists_resultterminal. b = ff_q_pft_trace_exists_resultterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_exists_resultsteps. (exists pfa_gap_trace_exists_resultstepsindex. pfa_gap_trace_exists_resultstepsindex + S (pff_trace_index_trace_exists_resultsteps) = (n)) -> exists pff_trace_before_trace_exists_resultsteps pff_trace_after_trace_exists_resultsteps. ((((exists ff_h_pft_trace_exists_resultstepsbefore. ff_h_pft_trace_exists_resultstepsbefore + S (pff_trace_before_trace_exists_resultsteps) = S ((S (pff_trace_index_trace_exists_resultsteps)) * c)) /\ exists ff_q_pft_trace_exists_resultstepsbefore. b = ff_q_pft_trace_exists_resultstepsbefore * S ((S (pff_trace_index_trace_exists_resultsteps)) * c) + (pff_trace_before_trace_exists_resultsteps))) /\ (((((exists ff_h_pft_trace_exists_resultstepsafter. ff_h_pft_trace_exists_resultstepsafter + S (pff_trace_after_trace_exists_resultsteps) = S ((S (S (pff_trace_index_trace_exists_resultsteps))) * c)) /\ exists ff_q_pft_trace_exists_resultstepsafter. b = ff_q_pft_trace_exists_resultstepsafter * S ((S (S (pff_trace_index_trace_exists_resultsteps))) * c) + (pff_trace_after_trace_exists_resultsteps))) /\ ((((exists pfa_gap_trace_exists_resultstepsadditionleft. pfa_gap_trace_exists_resultstepsadditionleft + S (pff_trace_before_trace_exists_resultsteps) = (p)) /\ (((exists pfa_gap_trace_exists_resultstepsadditionright. pfa_gap_trace_exists_resultstepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_exists_resultstepsadditionresultbound. pfa_gap_trace_exists_resultstepsadditionresultbound + S (pff_trace_after_trace_exists_resultsteps) = (p)) /\ ((exists pfa_offset_left_trace_exists_resultstepsadditionresultcongruence pfa_offset_right_trace_exists_resultstepsadditionresultcongruence. ((pff_trace_before_trace_exists_resultsteps) + (1)) + (p) * pfa_offset_left_trace_exists_resultstepsadditionresultcongruence = (pff_trace_after_trace_exists_resultsteps) + (p) * pfa_offset_right_trace_exists_resultstepsadditionresultcongruence)))))))))))))))))))

Complete tactic proof in conservative notation

All 71 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

71 script commands · 26 reading checkpoints · 4 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 (3)
prime field unit trace result boundedprime field add existsprime field unit trace successor
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro n
02Induction on nL3–4

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L3
    induction n
  2. L4
    intro hp
03Construct an explicit witnessL5–7

Supply the displayed value, then prove that it has the required property.

  1. L5
    exists 0
  2. L6
    exists 0
  3. L7
    exists 0
04Separate the logical casesL8–9

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

  1. L8
    split
  2. L9
    split
05Construct an explicit witnessL10–10

Supply the displayed value, then prove that it has the required property.

  1. L10
    exists 0
06Calculate and transport equalitiesL11–11

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

  1. L11
    norm_num
07Construct an explicit witnessL12–12

Supply the displayed value, then prove that it has the required property.

  1. L12
    exists 0
08Calculate and transport equalitiesL13–13

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

  1. L13
    norm_num
09Separate the logical casesL14–15

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

  1. L14
    split
  2. L15
    split
10Construct an explicit witnessL16–16

Supply the displayed value, then prove that it has the required property.

  1. L16
    exists 0
11Calculate and transport equalitiesL17–17

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

  1. L17
    norm_num
12Construct an explicit witnessL18–18

Supply the displayed value, then prove that it has the required property.

  1. L18
    exists 0
13Calculate and transport equalitiesL19–19

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

  1. L19
    norm_num
14Fix variables and assumptionsL20–21

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

  1. L20
    intro i
  2. L21
    intro hi
15Separate the logical casesL22–22

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

  1. L22
    exfalso
16Use earlier factsL23–28

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

  1. L23
    specialize lt_not_le (i)
  2. L24
    specialize lt_not_le (0)
  3. L25
    apply lt_not_le
  4. L26
    exact hi
  5. L27
    specialize zero_le (i)
  6. L28
    apply zero_le
17Fix variables and assumptionsL29–29

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

  1. L29
    intro hp
18Establish htL30–32

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply IH.

  1. L30
    have ht : ∃ b. ∃ c. ∃ r. FpUnitTrace(p,b,c,n,r)Definitions: FpUnitTrace(p,b,c,n,r)Original native command in the exact edition
  2. L31
    apply IH
  3. L32
    exact hp
19Separate the logical casesL33–35

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

  1. L33
    cases ht
  2. L34
    cases ht_witness
  3. L35
    cases ht_witness_witness
20Establish hrL36–44

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field unit trace result bounded.

  1. L36
  2. L37
    specialize prime_field_unit_trace_result_bounded (p)
  3. L38
    specialize prime_field_unit_trace_result_bounded (x)
  4. L39
    specialize prime_field_unit_trace_result_bounded (x1)
  5. L40
    specialize prime_field_unit_trace_result_bounded (n)
  6. L41
    specialize prime_field_unit_trace_result_bounded (x2)
  7. L42
    apply prime_field_unit_trace_result_bounded
  8. L43
    exact hp
  9. L44
    exact ht_witness_witness_witness
21Establish hsL45–54

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field add exists.

  1. L45
    have hs : ∃ s. FpAdd(p,x2,1,s)Definitions: FpAdd(p,x2,1,s)Original native command in the exact edition
  2. L46
    specialize prime_field_add_exists (p)
  3. L47
    specialize prime_field_add_exists (x2)
  4. L48
    specialize prime_field_add_exists (1)
  5. L49
    apply prime_field_add_exists
  6. L50
    exact hp
  7. L51
    exact hr
  8. L52
    specialize prime_two_le (p)
  9. L53
    apply prime_two_le
  10. L54
    exact hp
22Separate the logical casesL55–55

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

  1. L55
    cases hs
23Establish hnewL56–65

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field unit trace successor.

  1. L56
    have hnew : FpUnitMultiple(p,S n,x3)Definitions: FpUnitMultiple(p,S n,x3)Original native command in the exact edition
  2. L57
    specialize prime_field_unit_trace_successor (p)
  3. L58
    specialize prime_field_unit_trace_successor (x)
  4. L59
    specialize prime_field_unit_trace_successor (x1)
  5. L60
    specialize prime_field_unit_trace_successor (n)
  6. L61
    specialize prime_field_unit_trace_successor (x2)
  7. L62
    specialize prime_field_unit_trace_successor (x3)
  8. L63
    apply prime_field_unit_trace_successor
  9. L64
    exact ht_witness_witness_witness
  10. L65
    exact hs_witness
24Separate the logical casesL66–67

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

  1. L66
    cases hnew
  2. L67
    cases hnew_witness
25Construct an explicit witnessL68–70

Supply the displayed value, then prove that it has the required property.

  1. L68
    exists x4
  2. L69
    exists x5
  3. L70
    exists x3
26Use earlier factsL71–71

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

  1. L71
    exact hnew_witness_witness

Library-wide reading audit

Original defined command ledger · 71 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003induction n
  4. 0004intro hp
  5. 0005exists 0
  6. 0006exists 0
  7. 0007exists 0
  8. 0008split
  9. 0009split
  10. 0010exists 0
  11. 0011norm_num
  12. 0012exists 0
  13. 0013norm_num
  14. 0014split
  15. 0015split
  16. 0016exists 0
  17. 0017norm_num
  18. 0018exists 0
  19. 0019norm_num
  20. 0020intro i
  21. 0021intro hi
  22. 0022exfalso
  23. 0023specialize lt_not_le (i)
  24. 0024specialize lt_not_le (0)
  25. 0025apply lt_not_le
  26. 0026exact hi
  27. 0027specialize zero_le (i)
  28. 0028apply zero_le
  29. 0029intro hp
  30. 0030have ht : ∃ b. ∃ c. ∃ r. FpUnitTrace(p,b,c,n,r)
  31. 0031apply IH
  32. 0032exact hp
  33. 0033cases ht
  34. 0034cases ht_witness
  35. 0035cases ht_witness_witness
  36. 0036have hr : Lt(x2,p)
  37. 0037specialize prime_field_unit_trace_result_bounded (p)
  38. 0038specialize prime_field_unit_trace_result_bounded (x)
  39. 0039specialize prime_field_unit_trace_result_bounded (x1)
  40. 0040specialize prime_field_unit_trace_result_bounded (n)
  41. 0041specialize prime_field_unit_trace_result_bounded (x2)
  42. 0042apply prime_field_unit_trace_result_bounded
  43. 0043exact hp
  44. 0044exact ht_witness_witness_witness
  45. 0045have hs : ∃ s. FpAdd(p,x2,1,s)
  46. 0046specialize prime_field_add_exists (p)
  47. 0047specialize prime_field_add_exists (x2)
  48. 0048specialize prime_field_add_exists (1)
  49. 0049apply prime_field_add_exists
  50. 0050exact hp
  51. 0051exact hr
  52. 0052specialize prime_two_le (p)
  53. 0053apply prime_two_le
  54. 0054exact hp
  55. 0055cases hs
  56. 0056have hnew : FpUnitMultiple(p,S n,x3)
  57. 0057specialize prime_field_unit_trace_successor (p)
  58. 0058specialize prime_field_unit_trace_successor (x)
  59. 0059specialize prime_field_unit_trace_successor (x1)
  60. 0060specialize prime_field_unit_trace_successor (n)
  61. 0061specialize prime_field_unit_trace_successor (x2)
  62. 0062specialize prime_field_unit_trace_successor (x3)
  63. 0063apply prime_field_unit_trace_successor
  64. 0064exact ht_witness_witness_witness
  65. 0065exact hs_witness
  66. 0066cases hnew
  67. 0067cases hnew_witness
  68. 0068exists x4
  69. 0069exists x5
  70. 0070exists x3
  71. 0071exact hnew_witness_witness