MC000D

divisor_prime_toggle_prefix_lookup

Every actual decoded output of the constructed finite map obeys the independent positive-divisor toggle 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 input n is positive. Signed code 2 denotes +1, so the actual sum is +1 at n=1 and zero for n>1. The positive-values result permits arbitrary F(0), which the divisor mask excludes. Prime-square multiples contribute zero. Full G007 inversion is established in its separate family.

Exact theorem in conservative defined notation

∀ n. ∀ p. ∀ b. ∀ c. ∀ l. ∀ i. ∀ q. DivisorPrimeTogglePrefix(n,p,b,c,l) → Lt(i,l) → BetaAt(b,c,i,q) → DivisorPrimeToggle(n,p,i,q)

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

Definition DAG

DivisorPrimeToggle(n,p,d,e) · 2DivisorPrimeTogglePrefix(n,p,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 p b c l i q. (forall dvi_index_prefix_lookup_source. (exists pvs_gap_prefix_lookup_sourcedomain. pvs_gap_prefix_lookup_sourcedomain + S (dvi_index_prefix_lookup_source) = (l)) -> exists dvi_value_prefix_lookup_source. ((((exists ff_h_pvs_prefix_lookup_sourceentry. ff_h_pvs_prefix_lookup_sourceentry + S (dvi_value_prefix_lookup_source) = S ((S (dvi_index_prefix_lookup_source)) * c)) /\ exists ff_q_pvs_prefix_lookup_sourceentry. b = ff_q_pvs_prefix_lookup_sourceentry * S ((S (dvi_index_prefix_lookup_source)) * c) + (dvi_value_prefix_lookup_source))) /\ ((((~((dvi_index_prefix_lookup_source)=0)) /\ (((exists pvs_factor_prefix_lookup_sourcegraphdivisor. (n) = (dvi_index_prefix_lookup_source) * pvs_factor_prefix_lookup_sourcegraphdivisor) /\ ((((~(exists pvs_factor_prefix_lookup_sourcegraphtogglefresh_input. (dvi_index_prefix_lookup_source) = (p) * pvs_factor_prefix_lookup_sourcegraphtogglefresh_input)) /\ ((dvi_value_prefix_lookup_source)=(p)*(dvi_index_prefix_lookup_source)))) \/ (((((dvi_index_prefix_lookup_source)=(p)*(dvi_value_prefix_lookup_source)) /\ (~(exists pvs_factor_prefix_lookup_sourcegraphtogglefresh_output. (dvi_value_prefix_lookup_source) = (p) * pvs_factor_prefix_lookup_sourcegraphtogglefresh_output)))) \/ (((exists pvs_factor_prefix_lookup_sourcegraphtogglesquare. (dvi_index_prefix_lookup_source) = ((p)*(p)) * pvs_factor_prefix_lookup_sourcegraphtogglesquare) /\ ((dvi_value_prefix_lookup_source)=(dvi_index_prefix_lookup_source)))))))))) \/ ((((dvi_index_prefix_lookup_source)=0 \/ ~(exists pvs_factor_prefix_lookup_sourcegraphnondivisor. (n) = (dvi_index_prefix_lookup_source) * pvs_factor_prefix_lookup_sourcegraphnondivisor)) /\ ((dvi_value_prefix_lookup_source)=(dvi_index_prefix_lookup_source))))))) -> (exists pvs_gap_prefix_lookup_bound. pvs_gap_prefix_lookup_bound + S (i) = (l)) -> (((exists ff_h_pvs_prefix_lookup_entry. ff_h_pvs_prefix_lookup_entry + S (q) = S ((S (i)) * c)) /\ exists ff_q_pvs_prefix_lookup_entry. b = ff_q_pvs_prefix_lookup_entry * S ((S (i)) * c) + (q))) -> ((((~((i)=0)) /\ (((exists pvs_factor_prefix_lookup_resultdivisor. (n) = (i) * pvs_factor_prefix_lookup_resultdivisor) /\ ((((~(exists pvs_factor_prefix_lookup_resulttogglefresh_input. (i) = (p) * pvs_factor_prefix_lookup_resulttogglefresh_input)) /\ ((q)=(p)*(i)))) \/ (((((i)=(p)*(q)) /\ (~(exists pvs_factor_prefix_lookup_resulttogglefresh_output. (q) = (p) * pvs_factor_prefix_lookup_resulttogglefresh_output)))) \/ (((exists pvs_factor_prefix_lookup_resulttogglesquare. (i) = ((p)*(p)) * pvs_factor_prefix_lookup_resulttogglesquare) /\ ((q)=(i)))))))))) \/ ((((i)=0 \/ ~(exists pvs_factor_prefix_lookup_resultnondivisor. (n) = (i) * pvs_factor_prefix_lookup_resultnondivisor)) /\ ((q)=(i)))))

Complete tactic proof in conservative notation

All 31 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

31 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 n
  2. L2
    intro p
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro q
  8. L8
    intro hp
  9. L9
    intro hi
  10. L10
    intro hat
02Establish hvL11–14

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

  1. L11
    have hv : ∃ v. BetaAt(b,c,i,v) ∧ DivisorPrimeToggle(n,p,i,v)Definitions: BetaAt(b,c,i,v)DivisorPrimeToggle(n,p,i,v)Original native command in the exact edition
  2. L12
    specialize hp (i)
  3. L13
    apply hp
  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 heqL17–26

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

  1. L17
    have heq : q=x
  2. L18
    specialize beta_at_unique (b)
  3. L19
    specialize beta_at_unique (c)
  4. L20
    specialize beta_at_unique (i)
  5. L21
    specialize beta_at_unique (q)
  6. L22
    specialize beta_at_unique (x)
  7. L23
    apply beta_at_unique
  8. L24
    exact hat
  9. L25
    exact hv_witness_left
  10. L26
    rewrite heq
05Calculate and transport equalitiesL27–30

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

  1. L27
    rewrite heq
  2. L28
    rewrite heq
  3. L29
    rewrite heq
  4. L30
    rewrite heq
06Use earlier factsL31–31

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

  1. L31
    exact hv_witness_right

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro n
  2. 0002intro p
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro i
  7. 0007intro q
  8. 0008intro hp
  9. 0009intro hi
  10. 0010intro hat
  11. 0011have hv : ∃ v. BetaAt(b,c,i,v) ∧ DivisorPrimeToggle(n,p,i,v)
  12. 0012specialize hp (i)
  13. 0013apply hp
  14. 0014exact hi
  15. 0015cases hv
  16. 0016cases hv_witness
  17. 0017have heq : q=x
  18. 0018specialize beta_at_unique (b)
  19. 0019specialize beta_at_unique (c)
  20. 0020specialize beta_at_unique (i)
  21. 0021specialize beta_at_unique (q)
  22. 0022specialize beta_at_unique (x)
  23. 0023apply beta_at_unique
  24. 0024exact hat
  25. 0025exact hv_witness_left
  26. 0026rewrite heq
  27. 0027rewrite heq
  28. 0028rewrite heq
  29. 0029rewrite heq
  30. 0030rewrite heq
  31. 0031exact hv_witness_right