PQ001C

prime_field_polynomial_suffix_entry

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

Every actual output decoding equals the input coefficient at the supplied shifted index.

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 t d e M i a r. (forall pftrim_index_suffix_entry pftrim_value_suffix_entry. (exists pfa_gap_suffix_entrybound. pfa_gap_suffix_entrybound + S (pftrim_index_suffix_entry) = (M)) -> (((exists ff_h_pfp_suffix_entrysource. ff_h_pfp_suffix_entrysource + S (pftrim_value_suffix_entry) = S ((S ((t)+pftrim_index_suffix_entry)) * c)) /\ exists ff_q_pfp_suffix_entrysource. b = ff_q_pfp_suffix_entrysource * S ((S ((t)+pftrim_index_suffix_entry)) * c) + (pftrim_value_suffix_entry))) -> (((exists ff_h_pfp_suffix_entryoutput. ff_h_pfp_suffix_entryoutput + S (pftrim_value_suffix_entry) = S ((S (pftrim_index_suffix_entry)) * e)) /\ exists ff_q_pfp_suffix_entryoutput. d = ff_q_pfp_suffix_entryoutput * S ((S (pftrim_index_suffix_entry)) * e) + (pftrim_value_suffix_entry)))) -> (exists pfa_gap_suffix_entry_bound. pfa_gap_suffix_entry_bound + S (i) = (M)) -> (((exists ff_h_pfp_suffix_entry_source. ff_h_pfp_suffix_entry_source + S (a) = S ((S (t+i)) * c)) /\ exists ff_q_pfp_suffix_entry_source. b = ff_q_pfp_suffix_entry_source * S ((S (t+i)) * c) + (a))) -> (((exists ff_h_pfp_suffix_entry_output. ff_h_pfp_suffix_entry_output + S (r) = S ((S (i)) * e)) /\ exists ff_q_pfp_suffix_entry_output. d = ff_q_pfp_suffix_entry_output * S ((S (i)) * e) + (r))) -> r=a

Constructive proof overview

Generated structural guide

Every actual output decoding equals the input coefficient at the supplied shifted index.

The unchanged tactic script uses 1 declared prerequisite and contains 25 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_at_unique Stable 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

25 script commands · 4 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 t
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro M
  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
03Use earlier factsL14–23

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

  1. L14
    specialize beta_at_unique (d)
  2. L15
    specialize beta_at_unique (e)
  3. L16
    specialize beta_at_unique (i)
  4. L17
    specialize beta_at_unique (r)
  5. L18
    specialize beta_at_unique (a)
  6. L19
    apply beta_at_unique
  7. L20
    exact hr
  8. L21
    specialize h (i)
  9. L22
    specialize h (a)
  10. L23
    apply h
04Use earlier factsL24–25

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

  1. L24
    exact hi
  2. L25
    exact ha

Library-wide reading audit

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