PX000C

prime_field_polynomial_power_coefficient_exists

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

Every natural power has an actual decoded coefficient or the proved exterior zero, for arbitrary beta encodings.

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 expanded first-order arithmetic statement

forall b c L k. exists a. ((exists pfrep_position_power_exists. ((pfrep_position_power_exists+S (k)=(L)) /\ ((((exists ff_h_pfp_power_existsentry. ff_h_pfp_power_existsentry + S (a) = S ((S (pfrep_position_power_exists)) * c)) /\ exists ff_q_pfp_power_existsentry. b = ff_q_pfp_power_existsentry * S ((S (pfrep_position_power_exists)) * c) + (a)))))) \/ (((exists pfrep_gap_power_existsoutside. pfrep_gap_power_existsoutside+(L)=(k)) /\ (((a)=0)))))

Constructive proof overview

Generated structural guide

Every natural power has an actual decoded coefficient or the proved exterior zero, for arbitrary beta encodings.

The unchanged tactic script uses 2 declared prerequisites and contains 27 exact native proof lines.

Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

le_or_lt Alpha theorem; checked-use authorized beta_at_exists Alpha theorem; checked-use authorized

Direct dependents

PX0010 prime_field_polynomial_equivalent_transitive

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

27 script commands · 15 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro L
  4. L4
    intro k
02Establish hoL5–8

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

  1. L5
    have ho : (exists pfrep_gap_power_exists_outside. pfrep_gap_power_exists_outside+(L)=(k)) \/ (exists pfa_gap_power_exists_inside. pfa_gap_power_exists_inside + S (k) = (L))
  2. L6
    specialize le_or_lt (L)
  3. L7
    specialize le_or_lt (k)
  4. L8
    apply le_or_lt
03Separate the logical casesL9–9

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

  1. L9
    cases ho
04Construct an explicit witnessL10–10

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

  1. L10
    exists 0
05Separate the logical casesL11–12

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

  1. L11
    right
  2. L12
    split
06Use earlier factsL13–13

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

  1. L13
    exact ho_left
07Calculate and transport equalitiesL14–14

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

  1. L14
    refl
08Separate the logical casesL15–15

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

  1. L15
    cases ho_right
09Establish hvL16–20

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

  1. L16
    have hv : exists a. (((exists ff_h_pfp_power_exists_value. ff_h_pfp_power_exists_value + S (a) = S ((S (x)) * c)) /\ exists ff_q_pfp_power_exists_value. b = ff_q_pfp_power_exists_value * S ((S (x)) * c) + (a)))
  2. L17
    specialize beta_at_exists (b)
  3. L18
    specialize beta_at_exists (c)
  4. L19
    specialize beta_at_exists (x)
  5. L20
    apply beta_at_exists
10Separate the logical casesL21–21

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

  1. L21
    cases hv
11Construct an explicit witnessL22–22

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

  1. L22
    exists x1
12Separate the logical casesL23–23

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

  1. L23
    left
13Construct an explicit witnessL24–24

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

  1. L24
    exists x
14Separate the logical casesL25–25

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

  1. L25
    split
15Use earlier factsL26–27

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

  1. L26
    exact ho_right_witness
  2. L27
    exact hv_witness

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro k
  5. 0005have ho : (exists pfrep_gap_power_exists_outside. pfrep_gap_power_exists_outside+(L)=(k)) \/ (exists pfa_gap_power_exists_inside. pfa_gap_power_exists_inside + S (k) = (L))
  6. 0006specialize le_or_lt (L)
  7. 0007specialize le_or_lt (k)
  8. 0008apply le_or_lt
  9. 0009cases ho
  10. 0010exists 0
  11. 0011right
  12. 0012split
  13. 0013exact ho_left
  14. 0014refl
  15. 0015cases ho_right
  16. 0016have hv : exists a. (((exists ff_h_pfp_power_exists_value. ff_h_pfp_power_exists_value + S (a) = S ((S (x)) * c)) /\ exists ff_q_pfp_power_exists_value. b = ff_q_pfp_power_exists_value * S ((S (x)) * c) + (a)))
  17. 0017specialize beta_at_exists (b)
  18. 0018specialize beta_at_exists (c)
  19. 0019specialize beta_at_exists (x)
  20. 0020apply beta_at_exists
  21. 0021cases hv
  22. 0022exists x1
  23. 0023left
  24. 0024exists x
  25. 0025split
  26. 0026exact ho_right_witness
  27. 0027exact hv_witness