CD0044

finite_modular_pullback_zero_member

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

Pulling back by an actual source member produces an actual set containing zero.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall b c u v p t. ~(p=0) -> (forall fms_i_pullback fms_j_pullback. (exists fms_gap_pullback_i. fms_gap_pullback_i + S (fms_i_pullback) = (p)) -> (exists fms_gap_pullback_j. fms_gap_pullback_j + S (fms_j_pullback) = (p)) -> (exists fms_u_pullback fms_v_pullback. (fms_i_pullback + t) + (p) * fms_u_pullback = (fms_j_pullback) + (p) * fms_v_pullback) -> ((((((exists fs_h_fms_pullback_target. fs_h_fms_pullback_target + S (1) = S ((S (fms_i_pullback)) * v)) /\ exists fs_q_fms_pullback_target. u = fs_q_fms_pullback_target * S ((S (fms_i_pullback)) * v) + (1))) -> (((exists fs_h_fms_pullback_source. fs_h_fms_pullback_source + S (1) = S ((S (fms_j_pullback)) * c)) /\ exists fs_q_fms_pullback_source. b = fs_q_fms_pullback_source * S ((S (fms_j_pullback)) * c) + (1)))) /\ ((((exists fs_h_fms_pullback_source. fs_h_fms_pullback_source + S (1) = S ((S (fms_j_pullback)) * c)) /\ exists fs_q_fms_pullback_source. b = fs_q_fms_pullback_source * S ((S (fms_j_pullback)) * c) + (1))) -> (((exists fs_h_fms_pullback_target. fs_h_fms_pullback_target + S (1) = S ((S (fms_i_pullback)) * v)) /\ exists fs_q_fms_pullback_target. u = fs_q_fms_pullback_target * S ((S (fms_i_pullback)) * v) + (1))))))) -> (((exists fms_gap_member. fms_gap_member + S (t) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (t)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (t)) * c) + (1))))) -> (((exists fms_gap_member. fms_gap_member + S (0) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (0)) * v)) /\ exists fs_q_fms_member. u = fs_q_fms_member * S ((S (0)) * v) + (1)))))

Constructive proof overview

Generated structural guide

Pulling back by an actual source member produces an actual set containing zero.

The unchanged tactic script uses 3 declared prerequisites and contains 32 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

one_le_of_ne_zero Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized mod_eq_refl Stable theorem; checked-use authorized

Direct dependents

CD0046 prime_cauchy_davenport_cover_bound

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

32 script commands · 9 reading checkpoints · 3 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–9

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro u
  4. L4
    intro v
  5. L5
    intro p
  6. L6
    intro t
  7. L7
    intro hp
  8. L8
    intro hpull
  9. L9
    intro hmember
02Separate the logical casesL10–10

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

  1. L10
    cases hmember
03Establish hzL11–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply one le of ne zero.

  1. L11
    have hz : exists fms_gap_lt. fms_gap_lt + S (0) = (p)
  2. L12
    specialize one_le_of_ne_zero p
  3. L13
    apply one_le_of_ne_zero
  4. L14
    exact hp
04Separate the logical casesL15–15

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

  1. L15
    split
05Use earlier factsL16–16

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

  1. L16
    exact hz
06Establish heL17–22

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

  1. L17
    have he : (BetaAt(u,v,0,1) → BetaAt(b,c,t,1)) ∧ (BetaAt(b,c,t,1) → BetaAt(u,v,0,1))Definitions: BetaAt
  2. L18
    specialize hpull 0
  3. L19
    specialize hpull t
  4. L20
    apply hpull
  5. L21
    exact hz
  6. L22
    exact hmember_left
07Establish hzeroL23–29

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

  1. L23
    have hzero : 0+t=t
  2. L24
    specialize zero_add t
  3. L25
    apply zero_add
  4. L26
    rewrite hzero
  5. L27
    specialize mod_eq_refl p
  6. L28
    specialize mod_eq_refl t
  7. L29
    apply mod_eq_refl
08Separate the logical casesL30–30

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

  1. L30
    cases he
09Use earlier factsL31–32

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

  1. L31
    apply he_right
  2. L32
    exact hmember_right

Library-wide reading audit

Original exact command ledger · 32 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro u
  4. 0004intro v
  5. 0005intro p
  6. 0006intro t
  7. 0007intro hp
  8. 0008intro hpull
  9. 0009intro hmember
  10. 0010cases hmember
  11. 0011have hz : exists fms_gap_lt. fms_gap_lt + S (0) = (p)
  12. 0012specialize one_le_of_ne_zero p
  13. 0013apply one_le_of_ne_zero
  14. 0014exact hp
  15. 0015split
  16. 0016exact hz
  17. 0017have he : (((((exists fs_h_cd_norm_zero_target. fs_h_cd_norm_zero_target + S (1) = S ((S (0)) * v)) /\ exists fs_q_cd_norm_zero_target. u = fs_q_cd_norm_zero_target * S ((S (0)) * v) + (1))) -> (((exists fs_h_cd_norm_zero_source. fs_h_cd_norm_zero_source + S (1) = S ((S (t)) * c)) /\ exists fs_q_cd_norm_zero_source. b = fs_q_cd_norm_zero_source * S ((S (t)) * c) + (1)))) /\ ((((exists fs_h_cd_norm_zero_source. fs_h_cd_norm_zero_source + S (1) = S ((S (t)) * c)) /\ exists fs_q_cd_norm_zero_source. b = fs_q_cd_norm_zero_source * S ((S (t)) * c) + (1))) -> (((exists fs_h_cd_norm_zero_target. fs_h_cd_norm_zero_target + S (1) = S ((S (0)) * v)) /\ exists fs_q_cd_norm_zero_target. u = fs_q_cd_norm_zero_target * S ((S (0)) * v) + (1)))))
  18. 0018specialize hpull 0
  19. 0019specialize hpull t
  20. 0020apply hpull
  21. 0021exact hz
  22. 0022exact hmember_left
  23. 0023have hzero : 0+t=t
  24. 0024specialize zero_add t
  25. 0025apply zero_add
  26. 0026rewrite hzero
  27. 0027specialize mod_eq_refl p
  28. 0028specialize mod_eq_refl t
  29. 0029apply mod_eq_refl
  30. 0030cases he
  31. 0031apply he_right
  32. 0032exact hmember_right