DL0030

matrix_rank_recode_congruences_exists

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

Existing constructive CRT recoding yields all finite congruences at a fixed common-multiple scale; no selector-code bound is assumed.

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 k c b e. (forall mdr_t_recode_common. (exists mdr_h_recode_common. S mdr_t_recode_common + S mdr_h_recode_common = S (k)) -> exists mdr_q_recode_common. c = S mdr_t_recode_common * mdr_q_recode_common) -> exists z. (forall mdr_i_recode_result mdr_a_recode_result. (exists mdr_gap_recode_resulti. mdr_gap_recode_resulti + S (mdr_i_recode_result) = (k)) -> (((exists ff_h_mdr_recode_resulta. ff_h_mdr_recode_resulta + S (mdr_a_recode_result) = S ((S (mdr_i_recode_result)) * e)) /\ exists ff_q_mdr_recode_resulta. b = ff_q_mdr_recode_resulta * S ((S (mdr_i_recode_result)) * e) + (mdr_a_recode_result))) -> (exists mdr_u_recode_resultm mdr_v_recode_resultm. (z) + (S ((S (mdr_i_recode_result)) * (c))) * mdr_u_recode_resultm = (mdr_a_recode_result) + (S ((S (mdr_i_recode_result)) * (c))) * mdr_v_recode_resultm))

Constructive proof overview

Generated structural guide

Existing constructive CRT recoding yields all finite congruences at a fixed common-multiple scale; no selector-code bound is assumed.

The unchanged tactic script uses 2 declared prerequisites and contains 24 exact native proof lines.

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

Proof neighborhood

Direct dependencies

bounded_beta_exclusive_recode_invariant Stable theorem; checked-use authorized le_refl Stable theorem; checked-use authorized

Direct dependents

DL0031 matrix_rank_bounded_recode_in_fixed_box

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

24 script commands · 6 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.

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 assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro c
  3. L3
    intro b
  4. L4
    intro e
  5. L5
    intro hcommon
02Establish hallL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply bounded beta exclusive recode invariant.

  1. L6
    have hall : ∀ n. Le(n,k) → ∃ x. ∃ y. ¬x = 0 ∧ ((∀ z. Lt(z,n) → ∃ m. x = S (S z · c) · m) ∧ ((∀ z. ∀ m. Lt(z,n) → BetaAt(b,e,z,m) → ∃ i. ∃ j. y + S (S z · c) · i = m + S (S z · c) · j) ∧ (∀ z. Le(n,z) → Le(z,k) → ∀ m. (∃ i. x = m · i) → (∃ i. S (S z · c) = m · i) → m = 1)))Definitions: LeLtBetaAt
  2. L7
    specialize bounded_beta_exclusive_recode_invariant (k)
  3. L8
    specialize bounded_beta_exclusive_recode_invariant (c)
  4. L9
    specialize bounded_beta_exclusive_recode_invariant (b)
  5. L10
    specialize bounded_beta_exclusive_recode_invariant (e)
  6. L11
    apply bounded_beta_exclusive_recode_invariant
  7. L12
    exact hcommon
03Establish hinvL13–17

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

  1. L13
    have hinv : ∃ P. ∃ z. ¬P = 0 ∧ ((∀ x. Lt(x,k) → ∃ y. P = S (S x · c) · y) ∧ ((∀ x. ∀ y. Lt(x,k) → BetaAt(b,e,x,y) → ∃ n. ∃ m. z + S (S x · c) · n = y + S (S x · c) · m) ∧ (∀ x. Le(k,x) → Le(x,k) → ∀ y. (∃ n. P = y · n) → (∃ n. S (S x · c) = y · n) → y = 1)))Definitions: LeLtBetaAt
  2. L14
    specialize hall (k)
  3. L15
    apply hall
  4. L16
    specialize le_refl (k)
  5. L17
    apply le_refl
04Separate the logical casesL18–22

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

  1. L18
    cases hinv
  2. L19
    cases hinv_witness
  3. L20
    cases hinv_witness_witness
  4. L21
    cases hinv_witness_witness_right
  5. L22
    cases hinv_witness_witness_right_right
05Construct an explicit witnessL23–23

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

  1. L23
    exists x1
06Use earlier factsL24–24

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

  1. L24
    exact hinv_witness_witness_right_right_left

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro k
  2. 0002intro c
  3. 0003intro b
  4. 0004intro e
  5. 0005intro hcommon
  6. 0006have hall : forall n. (exists mdr_gap_invariant_bound. mdr_gap_invariant_bound + (n) = (k)) -> exists P z. (((~(P = 0)) /\ ((forall mdr_i_all_invariantdiv. (exists mdr_gap_all_invariantdivi. mdr_gap_all_invariantdivi + S (mdr_i_all_invariantdiv) = (n)) -> (exists mdr_q_all_invariantdivd. P = (S ((S (mdr_i_all_invariantdiv)) * (c))) * mdr_q_all_invariantdivd)) /\ ((forall mdr_i_all_invariantcong mdr_a_all_invariantcong. (exists mdr_gap_all_invariantcongi. mdr_gap_all_invariantcongi + S (mdr_i_all_invariantcong) = (n)) -> (((exists ff_h_mdr_all_invariantconga. ff_h_mdr_all_invariantconga + S (mdr_a_all_invariantcong) = S ((S (mdr_i_all_invariantcong)) * e)) /\ exists ff_q_mdr_all_invariantconga. b = ff_q_mdr_all_invariantconga * S ((S (mdr_i_all_invariantcong)) * e) + (mdr_a_all_invariantcong))) -> (exists mdr_u_all_invariantcongm mdr_v_all_invariantcongm. (z) + (S ((S (mdr_i_all_invariantcong)) * (c))) * mdr_u_all_invariantcongm = (mdr_a_all_invariantcong) + (S ((S (mdr_i_all_invariantcong)) * (c))) * mdr_v_all_invariantcongm)) /\ (forall mdr_j_all_invariant. (exists mdr_gap_all_invariantlow. mdr_gap_all_invariantlow + (n) = (mdr_j_all_invariant)) -> (exists mdr_gap_all_invarianthigh. mdr_gap_all_invarianthigh + (mdr_j_all_invariant) = (k)) -> forall mdr_d_all_invariant. (exists mdr_q_all_invariantfactor. P = (mdr_d_all_invariant) * mdr_q_all_invariantfactor) -> (exists mdr_q_all_invariantmod. S ((S (mdr_j_all_invariant)) * (c)) = (mdr_d_all_invariant) * mdr_q_all_invariantmod) -> mdr_d_all_invariant = 1)))))
  7. 0007specialize bounded_beta_exclusive_recode_invariant (k)
  8. 0008specialize bounded_beta_exclusive_recode_invariant (c)
  9. 0009specialize bounded_beta_exclusive_recode_invariant (b)
  10. 0010specialize bounded_beta_exclusive_recode_invariant (e)
  11. 0011apply bounded_beta_exclusive_recode_invariant
  12. 0012exact hcommon
  13. 0013have hinv : exists P z. (((~(P = 0)) /\ ((forall mdr_i_terminal_invariantdiv. (exists mdr_gap_terminal_invariantdivi. mdr_gap_terminal_invariantdivi + S (mdr_i_terminal_invariantdiv) = (k)) -> (exists mdr_q_terminal_invariantdivd. P = (S ((S (mdr_i_terminal_invariantdiv)) * (c))) * mdr_q_terminal_invariantdivd)) /\ ((forall mdr_i_terminal_invariantcong mdr_a_terminal_invariantcong. (exists mdr_gap_terminal_invariantcongi. mdr_gap_terminal_invariantcongi + S (mdr_i_terminal_invariantcong) = (k)) -> (((exists ff_h_mdr_terminal_invariantconga. ff_h_mdr_terminal_invariantconga + S (mdr_a_terminal_invariantcong) = S ((S (mdr_i_terminal_invariantcong)) * e)) /\ exists ff_q_mdr_terminal_invariantconga. b = ff_q_mdr_terminal_invariantconga * S ((S (mdr_i_terminal_invariantcong)) * e) + (mdr_a_terminal_invariantcong))) -> (exists mdr_u_terminal_invariantcongm mdr_v_terminal_invariantcongm. (z) + (S ((S (mdr_i_terminal_invariantcong)) * (c))) * mdr_u_terminal_invariantcongm = (mdr_a_terminal_invariantcong) + (S ((S (mdr_i_terminal_invariantcong)) * (c))) * mdr_v_terminal_invariantcongm)) /\ (forall mdr_j_terminal_invariant. (exists mdr_gap_terminal_invariantlow. mdr_gap_terminal_invariantlow + (k) = (mdr_j_terminal_invariant)) -> (exists mdr_gap_terminal_invarianthigh. mdr_gap_terminal_invarianthigh + (mdr_j_terminal_invariant) = (k)) -> forall mdr_d_terminal_invariant. (exists mdr_q_terminal_invariantfactor. P = (mdr_d_terminal_invariant) * mdr_q_terminal_invariantfactor) -> (exists mdr_q_terminal_invariantmod. S ((S (mdr_j_terminal_invariant)) * (c)) = (mdr_d_terminal_invariant) * mdr_q_terminal_invariantmod) -> mdr_d_terminal_invariant = 1)))))
  14. 0014specialize hall (k)
  15. 0015apply hall
  16. 0016specialize le_refl (k)
  17. 0017apply le_refl
  18. 0018cases hinv
  19. 0019cases hinv_witness
  20. 0020cases hinv_witness_witness
  21. 0021cases hinv_witness_witness_right
  22. 0022cases hinv_witness_witness_right_right
  23. 0023exists x1
  24. 0024exact hinv_witness_witness_right_right_left