PX0015

prime_field_polynomial_left_pad_entry

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

Each copied coefficient is identified with the actual source value, without identifying codes.

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 t d e i a r. (((forall pfp_repeat_index_left_pad_entry_graphzeros. (exists pfa_gap_left_pad_entry_graphzerosindex. pfa_gap_left_pad_entry_graphzerosindex + S (pfp_repeat_index_left_pad_entry_graphzeros) = (t)) -> (((exists ff_h_pfp_left_pad_entry_graphzerosentry. ff_h_pfp_left_pad_entry_graphzerosentry + S (0) = S ((S (pfp_repeat_index_left_pad_entry_graphzeros)) * e)) /\ exists ff_q_pfp_left_pad_entry_graphzerosentry. d = ff_q_pfp_left_pad_entry_graphzerosentry * S ((S (pfp_repeat_index_left_pad_entry_graphzeros)) * e) + (0)))) /\ ((forall pfrep_index_left_pad_entry_graph pfrep_value_left_pad_entry_graph. (exists pfa_gap_left_pad_entry_graphbound. pfa_gap_left_pad_entry_graphbound + S (pfrep_index_left_pad_entry_graph) = (L)) -> (((exists ff_h_pfp_left_pad_entry_graphinput. ff_h_pfp_left_pad_entry_graphinput + S (pfrep_value_left_pad_entry_graph) = S ((S (pfrep_index_left_pad_entry_graph)) * c)) /\ exists ff_q_pfp_left_pad_entry_graphinput. b = ff_q_pfp_left_pad_entry_graphinput * S ((S (pfrep_index_left_pad_entry_graph)) * c) + (pfrep_value_left_pad_entry_graph))) -> (((exists ff_h_pfp_left_pad_entry_graphoutput. ff_h_pfp_left_pad_entry_graphoutput + S (pfrep_value_left_pad_entry_graph) = S ((S ((t)+pfrep_index_left_pad_entry_graph)) * e)) /\ exists ff_q_pfp_left_pad_entry_graphoutput. d = ff_q_pfp_left_pad_entry_graphoutput * S ((S ((t)+pfrep_index_left_pad_entry_graph)) * e) + (pfrep_value_left_pad_entry_graph))))))) -> (exists pfa_gap_left_pad_entry_bound. pfa_gap_left_pad_entry_bound + S (i) = (L)) -> (((exists ff_h_pfp_left_pad_entry_source. ff_h_pfp_left_pad_entry_source + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_left_pad_entry_source. b = ff_q_pfp_left_pad_entry_source * S ((S (i)) * c) + (a))) -> (((exists ff_h_pfp_left_pad_entry_target. ff_h_pfp_left_pad_entry_target + S (r) = S ((S (t+i)) * e)) /\ exists ff_q_pfp_left_pad_entry_target. d = ff_q_pfp_left_pad_entry_target * S ((S (t+i)) * e) + (r))) -> r=a

Constructive proof overview

Generated structural guide

Each copied coefficient is identified with the actual source value, without identifying codes.

The unchanged tactic script uses 1 declared prerequisite and contains 26 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_at_unique Alpha theorem; checked-use authorized

Direct dependents

none

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

26 script commands · 5 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.

01Fix variables and assumptionsL1–10

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 t
  5. L5
    intro d
  6. L6
    intro e
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro r
  10. L10
    intro h
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hr
03Separate the logical casesL14–14

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

  1. L14
    cases h
04Use earlier factsL15–24

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

  1. L15
    specialize beta_at_unique (d)
  2. L16
    specialize beta_at_unique (e)
  3. L17
    specialize beta_at_unique (t+i)
  4. L18
    specialize beta_at_unique (r)
  5. L19
    specialize beta_at_unique (a)
  6. L20
    apply beta_at_unique
  7. L21
    exact hr
  8. L22
    specialize h_right (i)
  9. L23
    specialize h_right (a)
  10. L24
    apply h_right
05Use earlier factsL25–26

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

  1. L25
    exact hi
  2. L26
    exact ha

Library-wide reading audit

Original exact command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro d
  6. 0006intro e
  7. 0007intro i
  8. 0008intro a
  9. 0009intro r
  10. 0010intro h
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hr
  14. 0014cases h
  15. 0015specialize beta_at_unique (d)
  16. 0016specialize beta_at_unique (e)
  17. 0017specialize beta_at_unique (t+i)
  18. 0018specialize beta_at_unique (r)
  19. 0019specialize beta_at_unique (a)
  20. 0020apply beta_at_unique
  21. 0021exact hr
  22. 0022specialize h_right (i)
  23. 0023specialize h_right (a)
  24. 0024apply h_right
  25. 0025exact hi
  26. 0026exact ha