CD001E

finite_modular_translation_index_entry

Every actual decoded modular-translation index has the canonical bound and the required congruence.

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

∀ p. ∀ t. ∀ r. ∀ s. ∀ l. ∀ i. ∀ j. (∀ x. Lt(x,l) → ∃ y. BetaAt(r,s,x,y) ∧ (Lt(y,p) ∧ ModEq(p,x + t,y))) → Lt(i,l) → BetaAt(r,s,i,j) → Lt(j,p) ∧ ModEq(p,i + t,j)

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

Definition DAG

Lt(a,b) · 5ModEq(m,a,b) · 3BetaAt(b,c,i,x) · 3

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p t r s l i j. (forall fms_i_indices. (exists fms_gap_indices. fms_gap_indices + S (fms_i_indices) = (l)) -> exists fms_j_indices. (((exists fs_h_fms_indices_entry. fs_h_fms_indices_entry + S (fms_j_indices) = S ((S (fms_i_indices)) * s)) /\ exists fs_q_fms_indices_entry. r = fs_q_fms_indices_entry * S ((S (fms_i_indices)) * s) + (fms_j_indices))) /\ ((exists fms_gap_indices_bound. fms_gap_indices_bound + S (fms_j_indices) = (p)) /\ (exists fms_u_indices fms_v_indices. (fms_i_indices+t) + (p) * fms_u_indices = (fms_j_indices) + (p) * fms_v_indices))) -> (exists fms_gap_lt. fms_gap_lt + S (i) = (l)) -> (((exists fs_h_fms_index_entry. fs_h_fms_index_entry + S (j) = S ((S (i)) * s)) /\ exists fs_q_fms_index_entry. r = fs_q_fms_index_entry * S ((S (i)) * s) + (j))) -> (exists fms_gap_lt. fms_gap_lt + S (j) = (p)) /\ (exists fms_u_mod fms_v_mod. (i+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)

Complete tactic proof in conservative notation

All 28 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

28 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.

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 p
  2. L2
    intro t
  3. L3
    intro r
  4. L4
    intro s
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro j
  8. L8
    intro hindices
  9. L9
    intro hi
  10. L10
    intro hj
02Establish hvL11–14

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

  1. L11
    have hv : ∃ v. BetaAt(r,s,i,v) ∧ (Lt(v,p) ∧ ModEq(p,i + t,v))Definitions: BetaAt(r,s,i,v)Lt(v,p)ModEq(p,i + t,v)Original native command in the exact edition
  2. L12
    specialize hindices i
  3. L13
    apply hindices
  4. L14
    exact hi
03Separate the logical casesL15–16

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

  1. L15
    cases hv
  2. L16
    cases hv_witness
04Establish heL17–26

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

  1. L17
    have he : x=j
  2. L18
    specialize beta_at_unique r
  3. L19
    specialize beta_at_unique s
  4. L20
    specialize beta_at_unique i
  5. L21
    specialize beta_at_unique x
  6. L22
    specialize beta_at_unique j
  7. L23
    apply beta_at_unique
  8. L24
    exact hv_witness_left
  9. L25
    exact hj
  10. L26
    rewrite he at hv_witness_right
05Calculate and transport equalitiesL27–27

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L27
    rewrite he at hv_witness_right
06Use earlier factsL28–28

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

  1. L28
    exact hv_witness_right

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro t
  3. 0003intro r
  4. 0004intro s
  5. 0005intro l
  6. 0006intro i
  7. 0007intro j
  8. 0008intro hindices
  9. 0009intro hi
  10. 0010intro hj
  11. 0011have hv : ∃ v. BetaAt(r,s,i,v) ∧ (Lt(v,p) ∧ ModEq(p,i + t,v))
  12. 0012specialize hindices i
  13. 0013apply hindices
  14. 0014exact hi
  15. 0015cases hv
  16. 0016cases hv_witness
  17. 0017have he : x=j
  18. 0018specialize beta_at_unique r
  19. 0019specialize beta_at_unique s
  20. 0020specialize beta_at_unique i
  21. 0021specialize beta_at_unique x
  22. 0022specialize beta_at_unique j
  23. 0023apply beta_at_unique
  24. 0024exact hv_witness_left
  25. 0025exact hj
  26. 0026rewrite he at hv_witness_right
  27. 0027rewrite he at hv_witness_right
  28. 0028exact hv_witness_right