PX005B

polynomial_zero_extended_left_pad_shift

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

Leading-zero padding preserves every shifted zero-extended coefficient, including indices outside the original finite prefix.

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 B C i a. (((forall pfp_repeat_index_shift_paddingzeros. (exists pfa_gap_shift_paddingzerosindex. pfa_gap_shift_paddingzerosindex + S (pfp_repeat_index_shift_paddingzeros) = (t)) -> (((exists ff_h_pfp_shift_paddingzerosentry. ff_h_pfp_shift_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_shift_paddingzeros)) * C)) /\ exists ff_q_pfp_shift_paddingzerosentry. B = ff_q_pfp_shift_paddingzerosentry * S ((S (pfp_repeat_index_shift_paddingzeros)) * C) + (0)))) /\ ((forall pfrep_index_shift_padding pfrep_value_shift_padding. (exists pfa_gap_shift_paddingbound. pfa_gap_shift_paddingbound + S (pfrep_index_shift_padding) = (L)) -> (((exists ff_h_pfp_shift_paddinginput. ff_h_pfp_shift_paddinginput + S (pfrep_value_shift_padding) = S ((S (pfrep_index_shift_padding)) * c)) /\ exists ff_q_pfp_shift_paddinginput. b = ff_q_pfp_shift_paddinginput * S ((S (pfrep_index_shift_padding)) * c) + (pfrep_value_shift_padding))) -> (((exists ff_h_pfp_shift_paddingoutput. ff_h_pfp_shift_paddingoutput + S (pfrep_value_shift_padding) = S ((S ((t)+pfrep_index_shift_padding)) * C)) /\ exists ff_q_pfp_shift_paddingoutput. B = ff_q_pfp_shift_paddingoutput * S ((S ((t)+pfrep_index_shift_padding)) * C) + (pfrep_value_shift_padding))))))) -> ((((exists pfa_gap_shift_originalinside. pfa_gap_shift_originalinside + S (i) = (L)) /\ ((((exists ff_h_pfp_shift_originalentry. ff_h_pfp_shift_originalentry + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_shift_originalentry. b = ff_q_pfp_shift_originalentry * S ((S (i)) * c) + (a)))))) \/ (((exists pfc_gap_shift_originaloutside. pfc_gap_shift_originaloutside+(L)=(i)) /\ (((a)=0))))) -> ((((exists pfa_gap_shift_resultinside. pfa_gap_shift_resultinside + S (t+i) = (t+L)) /\ ((((exists ff_h_pfp_shift_resultentry. ff_h_pfp_shift_resultentry + S (a) = S ((S (t+i)) * C)) /\ exists ff_q_pfp_shift_resultentry. B = ff_q_pfp_shift_resultentry * S ((S (t+i)) * C) + (a)))))) \/ (((exists pfc_gap_shift_resultoutside. pfc_gap_shift_resultoutside+(t+L)=(t+i)) /\ (((a)=0)))))

Constructive proof overview

Generated structural guide

Leading-zero padding preserves every shifted zero-extended coefficient, including indices outside the original finite prefix.

The unchanged tactic script uses 2 declared prerequisites and contains 34 exact native proof lines.

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

Proof neighborhood

Direct dependencies

matrix_recursive_lt_add_left Alpha theorem; checked-use authorized add_le_add_left Alpha theorem; checked-use authorized

Direct dependents

PX0060 polynomial_diagonal_term_left_padding_left PX0061 polynomial_diagonal_term_left_padding_right

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

34 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 B
  6. L6
    intro C
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro hpad
  10. L10
    intro hvalue
02Separate the logical casesL11–15

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

  1. L11
    cases hpad
  2. L12
    cases hvalue
  3. L13
    cases hvalue_left
  4. L14
    left
  5. L15
    split
03Use earlier factsL16–25

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

  1. L16
    specialize matrix_recursive_lt_add_left (i)
  2. L17
    specialize matrix_recursive_lt_add_left (L)
  3. L18
    specialize matrix_recursive_lt_add_left (t)
  4. L19
    apply matrix_recursive_lt_add_left
  5. L20
    exact hvalue_left_left
  6. L21
    specialize hpad_right (i)
  7. L22
    specialize hpad_right (a)
  8. L23
    apply hpad_right
  9. L24
    exact hvalue_left_left
  10. L25
    exact hvalue_left_right
04Separate the logical casesL26–28

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

  1. L26
    cases hvalue_right
  2. L27
    right
  3. L28
    split
05Use earlier factsL29–34

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

  1. L29
    specialize add_le_add_left (L)
  2. L30
    specialize add_le_add_left (i)
  3. L31
    specialize add_le_add_left (t)
  4. L32
    apply add_le_add_left
  5. L33
    exact hvalue_right_left
  6. L34
    exact hvalue_right_right

Library-wide reading audit

Original exact command ledger · 34 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro B
  6. 0006intro C
  7. 0007intro i
  8. 0008intro a
  9. 0009intro hpad
  10. 0010intro hvalue
  11. 0011cases hpad
  12. 0012cases hvalue
  13. 0013cases hvalue_left
  14. 0014left
  15. 0015split
  16. 0016specialize matrix_recursive_lt_add_left (i)
  17. 0017specialize matrix_recursive_lt_add_left (L)
  18. 0018specialize matrix_recursive_lt_add_left (t)
  19. 0019apply matrix_recursive_lt_add_left
  20. 0020exact hvalue_left_left
  21. 0021specialize hpad_right (i)
  22. 0022specialize hpad_right (a)
  23. 0023apply hpad_right
  24. 0024exact hvalue_left_left
  25. 0025exact hvalue_left_right
  26. 0026cases hvalue_right
  27. 0027right
  28. 0028split
  29. 0029specialize add_le_add_left (L)
  30. 0030specialize add_le_add_left (i)
  31. 0031specialize add_le_add_left (t)
  32. 0032apply add_le_add_left
  33. 0033exact hvalue_right_left
  34. 0034exact hvalue_right_right