PQ002F

prime_field_polynomial_trim_nonempty_degree_exists

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

Positive retained length constructs an actual represented degree, with no claim of a degree for empty output.

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 p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_nonempty_inputinput. (exists fom_gap_pfp_trim_nonempty_inputinput_index_bound. fom_gap_pfp_trim_nonempty_inputinput_index_bound + S (fom_index_pfp_trim_nonempty_inputinput) = L) -> exists fom_value_pfp_trim_nonempty_inputinput. ((((exists fom_beta_height_pfp_trim_nonempty_inputinput_entry. fom_beta_height_pfp_trim_nonempty_inputinput_entry + S (fom_value_pfp_trim_nonempty_inputinput) = S ((S (fom_index_pfp_trim_nonempty_inputinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_nonempty_inputinput_entry. b = fom_beta_quotient_pfp_trim_nonempty_inputinput_entry * S ((S (fom_index_pfp_trim_nonempty_inputinput)) * c) + (fom_value_pfp_trim_nonempty_inputinput))) /\ (exists fom_gap_pfp_trim_nonempty_inputinput_value_bound. fom_gap_pfp_trim_nonempty_inputinput_value_bound + S (fom_value_pfp_trim_nonempty_inputinput) = p))) /\ (((forall pfp_repeat_index_trim_nonempty_inputremoved. (exists pfa_gap_trim_nonempty_inputremovedindex. pfa_gap_trim_nonempty_inputremovedindex + S (pfp_repeat_index_trim_nonempty_inputremoved) = (t)) -> (((exists ff_h_pfp_trim_nonempty_inputremovedentry. ff_h_pfp_trim_nonempty_inputremovedentry + S (0) = S ((S (pfp_repeat_index_trim_nonempty_inputremoved)) * c)) /\ exists ff_q_pfp_trim_nonempty_inputremovedentry. b = ff_q_pfp_trim_nonempty_inputremovedentry * S ((S (pfp_repeat_index_trim_nonempty_inputremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_nonempty_inputsuffix pftrim_value_trim_nonempty_inputsuffix. (exists pfa_gap_trim_nonempty_inputsuffixbound. pfa_gap_trim_nonempty_inputsuffixbound + S (pftrim_index_trim_nonempty_inputsuffix) = (M)) -> (((exists ff_h_pfp_trim_nonempty_inputsuffixsource. ff_h_pfp_trim_nonempty_inputsuffixsource + S (pftrim_value_trim_nonempty_inputsuffix) = S ((S ((t)+pftrim_index_trim_nonempty_inputsuffix)) * c)) /\ exists ff_q_pfp_trim_nonempty_inputsuffixsource. b = ff_q_pfp_trim_nonempty_inputsuffixsource * S ((S ((t)+pftrim_index_trim_nonempty_inputsuffix)) * c) + (pftrim_value_trim_nonempty_inputsuffix))) -> (((exists ff_h_pfp_trim_nonempty_inputsuffixoutput. ff_h_pfp_trim_nonempty_inputsuffixoutput + S (pftrim_value_trim_nonempty_inputsuffix) = S ((S (pftrim_index_trim_nonempty_inputsuffix)) * e)) /\ exists ff_q_pfp_trim_nonempty_inputsuffixoutput. d = ff_q_pfp_trim_nonempty_inputsuffixoutput * S ((S (pftrim_index_trim_nonempty_inputsuffix)) * e) + (pftrim_value_trim_nonempty_inputsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_nonempty_inputnormal. ((((exists ff_h_pfp_trim_nonempty_inputnormalentry. ff_h_pfp_trim_nonempty_inputnormalentry + S (pftrim_leading_trim_nonempty_inputnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_nonempty_inputnormalentry. d = ff_q_pfp_trim_nonempty_inputnormalentry * S ((S (0)) * e) + (pftrim_leading_trim_nonempty_inputnormal))) /\ ((~(pftrim_leading_trim_nonempty_inputnormal=0))))))))))))))) -> ~(M=0) -> exists q. ((((M)=S (q)) /\ (((forall fom_index_pfp_trim_nonempty_degreecoefficients. (exists fom_gap_pfp_trim_nonempty_degreecoefficients_index_bound. fom_gap_pfp_trim_nonempty_degreecoefficients_index_bound + S (fom_index_pfp_trim_nonempty_degreecoefficients) = M) -> exists fom_value_pfp_trim_nonempty_degreecoefficients. ((((exists fom_beta_height_pfp_trim_nonempty_degreecoefficients_entry. fom_beta_height_pfp_trim_nonempty_degreecoefficients_entry + S (fom_value_pfp_trim_nonempty_degreecoefficients) = S ((S (fom_index_pfp_trim_nonempty_degreecoefficients)) * e)) /\ exists fom_beta_quotient_pfp_trim_nonempty_degreecoefficients_entry. d = fom_beta_quotient_pfp_trim_nonempty_degreecoefficients_entry * S ((S (fom_index_pfp_trim_nonempty_degreecoefficients)) * e) + (fom_value_pfp_trim_nonempty_degreecoefficients))) /\ (exists fom_gap_pfp_trim_nonempty_degreecoefficients_value_bound. fom_gap_pfp_trim_nonempty_degreecoefficients_value_bound + S (fom_value_pfp_trim_nonempty_degreecoefficients) = p))) /\ ((exists pfd_leading_trim_nonempty_degree. ((((exists ff_h_pfp_trim_nonempty_degreeentry. ff_h_pfp_trim_nonempty_degreeentry + S (pfd_leading_trim_nonempty_degree) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_nonempty_degreeentry. d = ff_q_pfp_trim_nonempty_degreeentry * S ((S (0)) * e) + (pfd_leading_trim_nonempty_degree))) /\ ((~(pfd_leading_trim_nonempty_degree=0))))))))))

Constructive proof overview

Generated structural guide

Positive retained length constructs an actual represented degree, with no claim of a degree for empty output.

The unchanged tactic script uses 2 declared prerequisites and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

nonzero_is_succ Stable theorem; checked-use authorized PQ002E prime_field_polynomial_trim_represented_degree

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

28 script commands · 6 reading checkpoints · 1 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.

Named ingredients (1)
prime field polynomial trim represented degree
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 L
  5. L5
    intro t
  6. L6
    intro d
  7. L7
    intro e
  8. L8
    intro M
  9. L9
    intro h
  10. L10
    intro hM
02Establish hqL11–14

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

  1. L11
    have hq : exists q. M=S q
  2. L12
    specialize nonzero_is_succ (M)
  3. L13
    apply nonzero_is_succ
  4. L14
    exact hM
03Separate the logical casesL15–15

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

  1. L15
    cases hq
04Construct an explicit witnessL16–16

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

  1. L16
    exists x
05Use earlier factsL17–26

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

  1. L17
    specialize prime_field_polynomial_trim_represented_degree (p)
  2. L18
    specialize prime_field_polynomial_trim_represented_degree (b)
  3. L19
    specialize prime_field_polynomial_trim_represented_degree (c)
  4. L20
    specialize prime_field_polynomial_trim_represented_degree (L)
  5. L21
    specialize prime_field_polynomial_trim_represented_degree (t)
  6. L22
    specialize prime_field_polynomial_trim_represented_degree (d)
  7. L23
    specialize prime_field_polynomial_trim_represented_degree (e)
  8. L24
    specialize prime_field_polynomial_trim_represented_degree (M)
  9. L25
    specialize prime_field_polynomial_trim_represented_degree (x)
  10. L26
    apply prime_field_polynomial_trim_represented_degree
06Use earlier factsL27–28

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

  1. L27
    exact h
  2. L28
    exact hq_witness

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro L
  5. 0005intro t
  6. 0006intro d
  7. 0007intro e
  8. 0008intro M
  9. 0009intro h
  10. 0010intro hM
  11. 0011have hq : exists q. M=S q
  12. 0012specialize nonzero_is_succ (M)
  13. 0013apply nonzero_is_succ
  14. 0014exact hM
  15. 0015cases hq
  16. 0016exists x
  17. 0017specialize prime_field_polynomial_trim_represented_degree (p)
  18. 0018specialize prime_field_polynomial_trim_represented_degree (b)
  19. 0019specialize prime_field_polynomial_trim_represented_degree (c)
  20. 0020specialize prime_field_polynomial_trim_represented_degree (L)
  21. 0021specialize prime_field_polynomial_trim_represented_degree (t)
  22. 0022specialize prime_field_polynomial_trim_represented_degree (d)
  23. 0023specialize prime_field_polynomial_trim_represented_degree (e)
  24. 0024specialize prime_field_polynomial_trim_represented_degree (M)
  25. 0025specialize prime_field_polynomial_trim_represented_degree (x)
  26. 0026apply prime_field_polynomial_trim_represented_degree
  27. 0027exact h
  28. 0028exact hq_witness