CD0005

finite_bit_count_subset_le

Subset inclusion implies the exact inequality between the two witnessed finite cardinalities.

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. ∀ l. ∀ n. ∀ m. BitCount(b,c,l,n) → BitCount(d,e,l,m) → ModularSetSubset(b,c,d,e,l) → Le(n,m)

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

Definition DAG

ModularSetSubset(b,c,d,e,p) · 1Le(a,b) · 1BitCount(b,c,l,z) · 2

Actual proof prerequisites

finite_bit_subset_pointwise_lebeta_sum_pointwise_le · checked external prerequisite
Original expanded first-order statement 
forall b c d e l n m. (((exists ff_u_fms_count ff_v_fms_count. ((((exists ff_h_fms_count_start. ff_h_fms_count_start + S (0) = S ((S (0)) * ff_v_fms_count)) /\ exists ff_q_fms_count_start. ff_u_fms_count = ff_q_fms_count_start * S ((S (0)) * ff_v_fms_count) + (0))) /\ ((((exists ff_h_fms_count_terminal. ff_h_fms_count_terminal + S ((n)) = S ((S ((l))) * ff_v_fms_count)) /\ exists ff_q_fms_count_terminal. ff_u_fms_count = ff_q_fms_count_terminal * S ((S ((l))) * ff_v_fms_count) + ((n)))) /\ forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (l)) -> exists ff_a_fms_count ff_r_fms_count ff_s_fms_count. ((((exists ff_h_fms_count_summand. ff_h_fms_count_summand + S (ff_a_fms_count) = S ((S (ff_i_fms_count)) * (c))) /\ exists ff_q_fms_count_summand. (b) = ff_q_fms_count_summand * S ((S (ff_i_fms_count)) * (c)) + (ff_a_fms_count))) /\ ((((exists ff_h_fms_count_partial. ff_h_fms_count_partial + S (ff_r_fms_count) = S ((S (ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_partial. ff_u_fms_count = ff_q_fms_count_partial * S ((S (ff_i_fms_count)) * ff_v_fms_count) + (ff_r_fms_count))) /\ ((((exists ff_h_fms_count_successor. ff_h_fms_count_successor + S (ff_s_fms_count) = S ((S (S ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_successor. ff_u_fms_count = ff_q_fms_count_successor * S ((S (S ff_i_fms_count)) * ff_v_fms_count) + (ff_s_fms_count))) /\ ff_s_fms_count = ff_r_fms_count + ff_a_fms_count)))))) /\ (forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (l)) -> exists ff_bit_fms_count. ((((exists ff_h_fms_count_decoded. ff_h_fms_count_decoded + S (ff_bit_fms_count) = S ((S (ff_i_fms_count)) * (c))) /\ exists ff_q_fms_count_decoded. (b) = ff_q_fms_count_decoded * S ((S (ff_i_fms_count)) * (c)) + (ff_bit_fms_count))) /\ (ff_bit_fms_count = 0 \/ ff_bit_fms_count = 1))))) -> (((exists ff_u_fms_count ff_v_fms_count. ((((exists ff_h_fms_count_start. ff_h_fms_count_start + S (0) = S ((S (0)) * ff_v_fms_count)) /\ exists ff_q_fms_count_start. ff_u_fms_count = ff_q_fms_count_start * S ((S (0)) * ff_v_fms_count) + (0))) /\ ((((exists ff_h_fms_count_terminal. ff_h_fms_count_terminal + S ((m)) = S ((S ((l))) * ff_v_fms_count)) /\ exists ff_q_fms_count_terminal. ff_u_fms_count = ff_q_fms_count_terminal * S ((S ((l))) * ff_v_fms_count) + ((m)))) /\ forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (l)) -> exists ff_a_fms_count ff_r_fms_count ff_s_fms_count. ((((exists ff_h_fms_count_summand. ff_h_fms_count_summand + S (ff_a_fms_count) = S ((S (ff_i_fms_count)) * (e))) /\ exists ff_q_fms_count_summand. (d) = ff_q_fms_count_summand * S ((S (ff_i_fms_count)) * (e)) + (ff_a_fms_count))) /\ ((((exists ff_h_fms_count_partial. ff_h_fms_count_partial + S (ff_r_fms_count) = S ((S (ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_partial. ff_u_fms_count = ff_q_fms_count_partial * S ((S (ff_i_fms_count)) * ff_v_fms_count) + (ff_r_fms_count))) /\ ((((exists ff_h_fms_count_successor. ff_h_fms_count_successor + S (ff_s_fms_count) = S ((S (S ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_successor. ff_u_fms_count = ff_q_fms_count_successor * S ((S (S ff_i_fms_count)) * ff_v_fms_count) + (ff_s_fms_count))) /\ ff_s_fms_count = ff_r_fms_count + ff_a_fms_count)))))) /\ (forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (l)) -> exists ff_bit_fms_count. ((((exists ff_h_fms_count_decoded. ff_h_fms_count_decoded + S (ff_bit_fms_count) = S ((S (ff_i_fms_count)) * (e))) /\ exists ff_q_fms_count_decoded. (d) = ff_q_fms_count_decoded * S ((S (ff_i_fms_count)) * (e)) + (ff_bit_fms_count))) /\ (ff_bit_fms_count = 0 \/ ff_bit_fms_count = 1))))) -> (forall fms_i_subset. (exists fms_gap_subset. fms_gap_subset + S (fms_i_subset) = (l)) -> (((exists fs_h_fms_subset_left. fs_h_fms_subset_left + S (1) = S ((S (fms_i_subset)) * c)) /\ exists fs_q_fms_subset_left. b = fs_q_fms_subset_left * S ((S (fms_i_subset)) * c) + (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)))) -> (exists fms_gap_le. fms_gap_le + (n) = (m))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 4 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.

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

Named ingredients (1)
finite bit subset pointwise le
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 l
  6. L6
    intro n
  7. L7
    intro m
  8. L8
    intro hn
  9. L9
    intro hm
  10. L10
    intro hsub
02Separate the logical casesL11–12

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

  1. L11
    cases hn
  2. L12
    cases hm
03Use earlier factsL13–22

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

  1. L13
    specialize beta_sum_pointwise_le b
  2. L14
    specialize beta_sum_pointwise_le c
  3. L15
    specialize beta_sum_pointwise_le d
  4. L16
    specialize beta_sum_pointwise_le e
  5. L17
    specialize beta_sum_pointwise_le l
  6. L18
    specialize beta_sum_pointwise_le n
  7. L19
    specialize beta_sum_pointwise_le m
  8. L20
    apply beta_sum_pointwise_le
  9. L21
    specialize finite_bit_subset_pointwise_le b
  10. L22
    specialize finite_bit_subset_pointwise_le c
04Use earlier factsL23–30

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

  1. L23
    specialize finite_bit_subset_pointwise_le d
  2. L24
    specialize finite_bit_subset_pointwise_le e
  3. L25
    specialize finite_bit_subset_pointwise_le l
  4. L26
    apply finite_bit_subset_pointwise_le
  5. L27
    exact hn_right
  6. L28
    exact hsub
  7. L29
    exact hn_left
  8. L30
    exact hm_left

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro l
  6. 0006intro n
  7. 0007intro m
  8. 0008intro hn
  9. 0009intro hm
  10. 0010intro hsub
  11. 0011cases hn
  12. 0012cases hm
  13. 0013specialize beta_sum_pointwise_le b
  14. 0014specialize beta_sum_pointwise_le c
  15. 0015specialize beta_sum_pointwise_le d
  16. 0016specialize beta_sum_pointwise_le e
  17. 0017specialize beta_sum_pointwise_le l
  18. 0018specialize beta_sum_pointwise_le n
  19. 0019specialize beta_sum_pointwise_le m
  20. 0020apply beta_sum_pointwise_le
  21. 0021specialize finite_bit_subset_pointwise_le b
  22. 0022specialize finite_bit_subset_pointwise_le c
  23. 0023specialize finite_bit_subset_pointwise_le d
  24. 0024specialize finite_bit_subset_pointwise_le e
  25. 0025specialize finite_bit_subset_pointwise_le l
  26. 0026apply finite_bit_subset_pointwise_le
  27. 0027exact hn_right
  28. 0028exact hsub
  29. 0029exact hn_left
  30. 0030exact hm_left