CD003A

finite_modular_dyson_lower_subset

The lower Dyson transform is an actual subset of the original second set.

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. ∀ d. ∀ e. ∀ u. ∀ v. ∀ p. ∀ t. (∀ x. Lt(x,p) → (BetaAt(u,v,x,1) → BetaAt(d,e,x,1) ∧ (∃ y. ModularSetMember(b,c,p,y) ∧ ModEq(p,x + t,y))) ∧ (BetaAt(d,e,x,1) ∧ (∃ y. ModularSetMember(b,c,p,y) ∧ ModEq(p,x + t,y)) → BetaAt(u,v,x,1))) → ModularSetSubset(u,v,d,e,p)

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

Definition DAG

ModularSetMember(b,c,p,x) · 5ModularSetSubset(b,c,d,e,p) · 1Lt(a,b) · 1ModEq(m,a,b) · 5BetaAt(b,c,i,x) · 9

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c d e u v p t. (forall cd_output_lower. (exists fms_gap_cd_lower_bound. fms_gap_cd_lower_bound + S (cd_output_lower) = (p)) -> ((((((exists fs_h_cd_lower_result. fs_h_cd_lower_result + S (1) = S ((S (cd_output_lower)) * v)) /\ exists fs_q_cd_lower_result. u = fs_q_cd_lower_result * S ((S (cd_output_lower)) * v) + (1))) -> ((((exists fs_h_cd_lower_old. fs_h_cd_lower_old + S (1) = S ((S (cd_output_lower)) * e)) /\ exists fs_q_cd_lower_old. d = fs_q_cd_lower_old * S ((S (cd_output_lower)) * e) + (1))) /\ (exists cd_source_lower. (((exists fms_gap_cd_lower_member. fms_gap_cd_lower_member + S (cd_source_lower) = (p)) /\ (((exists fs_h_fms_cd_lower_member. fs_h_fms_cd_lower_member + S (1) = S ((S (cd_source_lower)) * c)) /\ exists fs_q_fms_cd_lower_member. b = fs_q_fms_cd_lower_member * S ((S (cd_source_lower)) * c) + (1))))) /\ (exists fms_u_cd_lower_mod fms_v_cd_lower_mod. (cd_output_lower+t) + (p) * fms_u_cd_lower_mod = (cd_source_lower) + (p) * fms_v_cd_lower_mod)))) /\ (((((exists fs_h_cd_lower_old. fs_h_cd_lower_old + S (1) = S ((S (cd_output_lower)) * e)) /\ exists fs_q_cd_lower_old. d = fs_q_cd_lower_old * S ((S (cd_output_lower)) * e) + (1))) /\ (exists cd_source_lower. (((exists fms_gap_cd_lower_member. fms_gap_cd_lower_member + S (cd_source_lower) = (p)) /\ (((exists fs_h_fms_cd_lower_member. fs_h_fms_cd_lower_member + S (1) = S ((S (cd_source_lower)) * c)) /\ exists fs_q_fms_cd_lower_member. b = fs_q_fms_cd_lower_member * S ((S (cd_source_lower)) * c) + (1))))) /\ (exists fms_u_cd_lower_mod fms_v_cd_lower_mod. (cd_output_lower+t) + (p) * fms_u_cd_lower_mod = (cd_source_lower) + (p) * fms_v_cd_lower_mod))) -> (((exists fs_h_cd_lower_result. fs_h_cd_lower_result + S (1) = S ((S (cd_output_lower)) * v)) /\ exists fs_q_cd_lower_result. u = fs_q_cd_lower_result * S ((S (cd_output_lower)) * v) + (1))))))) -> (forall fms_i_subset. (exists fms_gap_subset. fms_gap_subset + S (fms_i_subset) = (p)) -> (((exists fs_h_fms_subset_left. fs_h_fms_subset_left + S (1) = S ((S (fms_i_subset)) * v)) /\ exists fs_q_fms_subset_left. u = fs_q_fms_subset_left * S ((S (fms_i_subset)) * v) + (1))) -> (((exists fs_h_fms_subset_right. fs_h_fms_subset_right + S (1) = S ((S (fms_i_subset)) * e)) /\ exists fs_q_fms_subset_right. d = fs_q_fms_subset_right * S ((S (fms_i_subset)) * e) + (1))))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 7 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.

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 assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro u
  6. L6
    intro v
  7. L7
    intro p
  8. L8
    intro t
  9. L9
    intro hlower
  10. L10
    intro i
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hi
  2. L12
    intro hmember
03Establish heL13–16

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

  1. L13
    have he : (BetaAt(u,v,i,1) → BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x))) ∧ (BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x)) → BetaAt(u,v,i,1))Definitions: BetaAt(u,v,i,1)BetaAt(d,e,i,1)ModularSetMember(b,c,p,x)ModEq(p,i + t,x)Original native command in the exact edition
  2. L14
    specialize hlower i
  3. L15
    apply hlower
  4. L16
    exact hi
04Separate the logical casesL17–17

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

  1. L17
    cases he
05Establish hbothL18–20

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

  1. L18
    have hboth : BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x))Definitions: BetaAt(d,e,i,1)ModularSetMember(b,c,p,x)ModEq(p,i + t,x)Original native command in the exact edition
  2. L19
    apply he_left
  3. L20
    exact hmember
06Separate the logical casesL21–21

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

  1. L21
    cases hboth
07Use earlier factsL22–22

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

  1. L22
    exact hboth_left

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro u
  6. 0006intro v
  7. 0007intro p
  8. 0008intro t
  9. 0009intro hlower
  10. 0010intro i
  11. 0011intro hi
  12. 0012intro hmember
  13. 0013have he : (BetaAt(u,v,i,1) → BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x))) ∧ (BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x)) → BetaAt(u,v,i,1))
  14. 0014specialize hlower i
  15. 0015apply hlower
  16. 0016exact hi
  17. 0017cases he
  18. 0018have hboth : BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x))
  19. 0019apply he_left
  20. 0020exact hmember
  21. 0021cases hboth
  22. 0022exact hboth_left