PX000A

prime_field_polynomial_left_pad_index_cases

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

Every index in an actual left-padded window is in its zero block or has an actual bounded source 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 t L i. (exists pfa_gap_index_cases_bound. pfa_gap_index_cases_bound + S (i) = (t+L)) -> (exists pfa_gap_index_cases_zero. pfa_gap_index_cases_zero + S (i) = (t)) \/ exists j. (((exists pfa_gap_index_cases_copy. pfa_gap_index_cases_copy + S (j) = (L)) /\ ((i=t+j))))

Constructive proof overview

Generated structural guide

Every index in an actual left-padded window is in its zero block or has an actual bounded source index.

The unchanged tactic script uses 4 declared prerequisites and contains 31 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 add_left_cancel Alpha theorem; checked-use authorized add_comm Alpha theorem; checked-use authorized add_assoc Alpha theorem; checked-use authorized

Direct dependents

PX0016 prime_field_polynomial_left_pad_bounded PX0017 prime_field_polynomial_left_pad_functional PX001E prime_field_polynomial_add_left_pad_transport PX001F prime_field_polynomial_subtract_left_pad_transport PX0020 prime_field_polynomial_scale_left_pad_transport PX005D polynomial_left_pad_zero_prefix

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

31 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 t
  2. L2
    intro L
  3. L3
    intro i
  4. L4
    intro hi
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_index_case_ge. pfrep_gap_index_case_ge+(t)=(i)) \/ (exists pfa_gap_index_case_lt. pfa_gap_index_case_lt + S (i) = (t))
  2. L6
    specialize le_or_lt (t)
  3. L7
    specialize le_or_lt (i)
  4. L8
    apply le_or_lt
03Separate the logical casesL9–10

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

  1. L9
    cases ho
  2. L10
    cases ho_left
04Establish heqL11–14

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

  1. L11
    have heq : t+x=i
  2. L12
    trans x+t
  3. L13
    apply add_comm
  4. L14
    exact ho_left_witness
05Separate the logical casesL15–15

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

  1. L15
    right
06Construct an explicit witnessL16–16

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

  1. L16
    exists x
07Separate the logical casesL17–18

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

  1. L17
    split
  2. L18
    cases hi
08Construct an explicit witnessL19–19

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

  1. L19
    exists x1
09Use earlier factsL20–23

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

  1. L20
    specialize add_left_cancel (t)
  2. L21
    specialize add_left_cancel (x1+S x)
  3. L22
    specialize add_left_cancel (L)
  4. L23
    apply add_left_cancel
10Calculate and transport equalitiesL24–26

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

  1. L24
    trans x1+S (t+x)
  2. L25
    simp [add_comm,add_assoc]
  3. L26
    rewrite heq
11Use earlier factsL27–27

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

  1. L27
    exact hi_witness
12Calculate and transport equalitiesL28–28

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

  1. L28
    symm
13Use earlier factsL29–29

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

  1. L29
    exact heq
14Separate the logical casesL30–30

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

  1. L30
    left
15Use earlier factsL31–31

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

  1. L31
    exact ho_right

Library-wide reading audit

Original exact command ledger · 31 lines
  1. 0001intro t
  2. 0002intro L
  3. 0003intro i
  4. 0004intro hi
  5. 0005have ho : (exists pfrep_gap_index_case_ge. pfrep_gap_index_case_ge+(t)=(i)) \/ (exists pfa_gap_index_case_lt. pfa_gap_index_case_lt + S (i) = (t))
  6. 0006specialize le_or_lt (t)
  7. 0007specialize le_or_lt (i)
  8. 0008apply le_or_lt
  9. 0009cases ho
  10. 0010cases ho_left
  11. 0011have heq : t+x=i
  12. 0012trans x+t
  13. 0013apply add_comm
  14. 0014exact ho_left_witness
  15. 0015right
  16. 0016exists x
  17. 0017split
  18. 0018cases hi
  19. 0019exists x1
  20. 0020specialize add_left_cancel (t)
  21. 0021specialize add_left_cancel (x1+S x)
  22. 0022specialize add_left_cancel (L)
  23. 0023apply add_left_cancel
  24. 0024trans x1+S (t+x)
  25. 0025simp [add_comm,add_assoc]
  26. 0026rewrite heq
  27. 0027exact hi_witness
  28. 0028symm
  29. 0029exact heq
  30. 0030left
  31. 0031exact ho_right