DI0008

divisor_complement_prefix_lookup

Every actual beta lookup in the constructed finite window has the literal complementary-divisor graph.

Alpha v34 checked-use · first admitted v31 · 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.

The complementary quotient is witnessed by n=d*q at positive divisors. The actual permutation covers indices zero through n, fixing zero and nondivisors. This is the involution foundation for the separate cancellation and full G007 inversion proofs, not an assumed divisor bijection.

Exact theorem in conservative defined notation

∀ n. ∀ b. ∀ c. ∀ l. ∀ i. ∀ q. DivisorComplementPrefix(n,b,c,l) → Lt(i,l) → BetaAt(b,c,i,q) → DivisorComplement(n,i,q)

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

Definition DAG

DivisorComplement(n,d,q) · 2DivisorComplementPrefix(n,b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall n b c l i q. (forall dvi_index_lookup_prefix. (exists pvs_gap_lookup_prefixdomain. pvs_gap_lookup_prefixdomain + S (dvi_index_lookup_prefix) = (l)) -> exists dvi_value_lookup_prefix. ((((exists ff_h_pvs_lookup_prefixentry. ff_h_pvs_lookup_prefixentry + S (dvi_value_lookup_prefix) = S ((S (dvi_index_lookup_prefix)) * c)) /\ exists ff_q_pvs_lookup_prefixentry. b = ff_q_pvs_lookup_prefixentry * S ((S (dvi_index_lookup_prefix)) * c) + (dvi_value_lookup_prefix))) /\ ((((~((dvi_index_lookup_prefix)=0)) /\ ((n)=(dvi_index_lookup_prefix)*(dvi_value_lookup_prefix)))) \/ ((((dvi_index_lookup_prefix)=0 \/ ~(exists pvs_factor_lookup_prefixgraphnondivisor. (n) = (dvi_index_lookup_prefix) * pvs_factor_lookup_prefixgraphnondivisor)) /\ ((dvi_value_lookup_prefix)=(dvi_index_lookup_prefix))))))) -> (exists pvs_gap_lookup_index. pvs_gap_lookup_index + S (i) = (l)) -> (((exists ff_h_pvs_lookup_beta. ff_h_pvs_lookup_beta + S (q) = S ((S (i)) * c)) /\ exists ff_q_pvs_lookup_beta. b = ff_q_pvs_lookup_beta * S ((S (i)) * c) + (q))) -> ((((~((i)=0)) /\ ((n)=(i)*(q)))) \/ ((((i)=0 \/ ~(exists pvs_factor_lookup_resultnondivisor. (n) = (i) * pvs_factor_lookup_resultnondivisor)) /\ ((q)=(i)))))

Complete tactic proof in conservative notation

All 27 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

27 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 assumptionsL1–9

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

  1. L1
    intro n
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro i
  6. L6
    intro q
  7. L7
    intro hp
  8. L8
    intro hi
  9. L9
    intro hat
02Establish hvL10–13

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

  1. L10
    have hv : ∃ v. BetaAt(b,c,i,v) ∧ DivisorComplement(n,i,v)Definitions: BetaAt(b,c,i,v)DivisorComplement(n,i,v)Original native command in the exact edition
  2. L11
    specialize hp (i)
  3. L12
    apply hp
  4. L13
    exact hi
03Separate the logical casesL14–15

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

  1. L14
    cases hv
  2. L15
    cases hv_witness
04Establish heqL16–25

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

  1. L16
    have heq : q=x
  2. L17
    specialize beta_at_unique (b)
  3. L18
    specialize beta_at_unique (c)
  4. L19
    specialize beta_at_unique (i)
  5. L20
    specialize beta_at_unique (q)
  6. L21
    specialize beta_at_unique (x)
  7. L22
    apply beta_at_unique
  8. L23
    exact hat
  9. L24
    exact hv_witness_left
  10. L25
    rewrite heq
05Calculate and transport equalitiesL26–26

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

  1. L26
    rewrite heq
06Use earlier factsL27–27

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

  1. L27
    exact hv_witness_right

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro i
  6. 0006intro q
  7. 0007intro hp
  8. 0008intro hi
  9. 0009intro hat
  10. 0010have hv : ∃ v. BetaAt(b,c,i,v) ∧ DivisorComplement(n,i,v)
  11. 0011specialize hp (i)
  12. 0012apply hp
  13. 0013exact hi
  14. 0014cases hv
  15. 0015cases hv_witness
  16. 0016have heq : q=x
  17. 0017specialize beta_at_unique (b)
  18. 0018specialize beta_at_unique (c)
  19. 0019specialize beta_at_unique (i)
  20. 0020specialize beta_at_unique (q)
  21. 0021specialize beta_at_unique (x)
  22. 0022apply beta_at_unique
  23. 0023exact hat
  24. 0024exact hv_witness_left
  25. 0025rewrite heq
  26. 0026rewrite heq
  27. 0027exact hv_witness_right