CD0044

finite_modular_pullback_zero_member

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

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

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.

Sets are complete characteristic-bit codes with actual finite cardinality witnesses. The proof constructs translations and the sumset; no finite-choice oracle, supplied cardinality conclusion, or unproved polynomial-method premise is used.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ u. ∀ v. ∀ p. ∀ t. ¬p = 0 → ModularSetPullback(b,c,u,v,p,t) → ModularSetMember(b,c,p,t) → ModularSetMember(u,v,p,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

ModularSetMember(b,c,p,x) · 2ModularSetPullback(b,c,d,e,p,t) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 4

Actual proof prerequisites

one_le_of_ne_zero · checked external prerequisitezero_add · checked external prerequisitemod_eq_refl · checked external prerequisite
Original expanded first-order 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)))))

Complete tactic proof in conservative notation

All 32 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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
  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(u,v,0,1)BetaAt(b,c,t,1)Original native command in the exact edition
  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 defined 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 : Lt(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 : (BetaAt(u,v,0,1) → BetaAt(b,c,t,1)) ∧ (BetaAt(b,c,t,1) → BetaAt(u,v,0,1))
  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