PA00AS

orbit_closed_unused_mate

Alpha v34 checked-use theorem · independently closed; not Stable

Orbit closure turns omission of one endpoint of a decoded two-cycle into omission of its mate.

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 PA statement

forall u v b c l i j. (forall wpo_position_unused_closed wpo_source_unused_closed wpo_mate_unused_closed. (exists wpo_gap_unused_closed_position_bound. wpo_gap_unused_closed_position_bound + S (wpo_position_unused_closed) = l) -> (((exists wpo_beta_height_unused_closed_source_entry. wpo_beta_height_unused_closed_source_entry + S (wpo_source_unused_closed) = S ((S (wpo_position_unused_closed)) * c)) /\ exists wpo_beta_quotient_unused_closed_source_entry. b = wpo_beta_quotient_unused_closed_source_entry * S ((S (wpo_position_unused_closed)) * c) + (wpo_source_unused_closed))) -> (((exists wpo_beta_height_unused_closed_inverse_entry. wpo_beta_height_unused_closed_inverse_entry + S (wpo_mate_unused_closed) = S ((S (wpo_source_unused_closed)) * v)) /\ exists wpo_beta_quotient_unused_closed_inverse_entry. u = wpo_beta_quotient_unused_closed_inverse_entry * S ((S (wpo_source_unused_closed)) * v) + (wpo_mate_unused_closed))) -> exists wpo_mate_position_unused_closed. ((exists wpo_gap_unused_closed_mate_bound. wpo_gap_unused_closed_mate_bound + S (wpo_mate_position_unused_closed) = l) /\ (((exists wpo_beta_height_unused_closed_mate_entry. wpo_beta_height_unused_closed_mate_entry + S (wpo_mate_unused_closed) = S ((S (wpo_mate_position_unused_closed)) * c)) /\ exists wpo_beta_quotient_unused_closed_mate_entry. b = wpo_beta_quotient_unused_closed_mate_entry * S ((S (wpo_mate_position_unused_closed)) * c) + (wpo_mate_unused_closed))))) -> (~(exists wpo_index_unused_source_omit_contains. ((exists wpo_gap_unused_source_omit_contains_bound. wpo_gap_unused_source_omit_contains_bound + S (wpo_index_unused_source_omit_contains) = l) /\ (((exists wpo_beta_height_unused_source_omit_contains_entry. wpo_beta_height_unused_source_omit_contains_entry + S (i) = S ((S (wpo_index_unused_source_omit_contains)) * c)) /\ exists wpo_beta_quotient_unused_source_omit_contains_entry. b = wpo_beta_quotient_unused_source_omit_contains_entry * S ((S (wpo_index_unused_source_omit_contains)) * c) + (i)))))) -> (((exists wpo_beta_height_unused_back. wpo_beta_height_unused_back + S (i) = S ((S (j)) * v)) /\ exists wpo_beta_quotient_unused_back. u = wpo_beta_quotient_unused_back * S ((S (j)) * v) + (i))) -> (~(exists wpo_index_unused_mate_omit_contains. ((exists wpo_gap_unused_mate_omit_contains_bound. wpo_gap_unused_mate_omit_contains_bound + S (wpo_index_unused_mate_omit_contains) = l) /\ (((exists wpo_beta_height_unused_mate_omit_contains_entry. wpo_beta_height_unused_mate_omit_contains_entry + S (j) = S ((S (wpo_index_unused_mate_omit_contains)) * c)) /\ exists wpo_beta_quotient_unused_mate_omit_contains_entry. b = wpo_beta_quotient_unused_mate_omit_contains_entry * S ((S (wpo_index_unused_mate_omit_contains)) * c) + (j))))))

Structural proof guide

Generated structural guide

Orbit closure turns omission of one endpoint of a decoded two-cycle into omission of its mate.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by case analysis (4), intermediate claims (1).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA00AV prime_pair_order_choose_append

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

28 script commands · 9 reading checkpoints · 1 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro u
  2. L2
    intro v
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro j
  8. L8
    intro hclosed
  9. L9
    intro hiomit
  10. L10
    intro hback
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hjcontains
03Separate the logical casesL12–13

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

  1. L12
    cases hjcontains
  2. L13
    cases hjcontains_witness
04Establish hioccursL14–21

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

  1. L14
    have hioccurs : exists wpo_index_unused_source_occurs. ((exists wpo_gap_unused_source_occurs_bound. wpo_gap_unused_source_occurs_bound + S (wpo_index_unused_source_occurs) = l) /\ (((exists wpo_beta_height_unused_source_occurs_entry. wpo_beta_height_unused_source_occurs_entry + S (i) = S ((S (wpo_index_unused_source_occurs)) * c)) /\ exists wpo_beta_quotient_unused_source_occurs_entry. b = wpo_beta_quotient_unused_source_occurs_entry * S ((S (wpo_index_unused_source_occurs)) * c) + (i))))
  2. L15
    specialize hclosed x
  3. L16
    specialize hclosed j
  4. L17
    specialize hclosed i
  5. L18
    apply hclosed
  6. L19
    exact hjcontains_witness_left
  7. L20
    exact hjcontains_witness_right
  8. L21
    exact hback
05Separate the logical casesL22–23

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

  1. L22
    cases hioccurs
  2. L23
    cases hioccurs_witness
06Use earlier factsL24–24

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

  1. L24
    apply hiomit
07Construct an explicit witnessL25–25

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

  1. L25
    exists x1
08Separate the logical casesL26–26

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

  1. L26
    split
09Use earlier factsL27–28

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

  1. L27
    exact hioccurs_witness_left
  2. L28
    exact hioccurs_witness_right

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro u
  2. 0002intro v
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro i
  7. 0007intro j
  8. 0008intro hclosed
  9. 0009intro hiomit
  10. 0010intro hback
  11. 0011intro hjcontains
  12. 0012cases hjcontains
  13. 0013cases hjcontains_witness
  14. 0014have hioccurs : exists wpo_index_unused_source_occurs. ((exists wpo_gap_unused_source_occurs_bound. wpo_gap_unused_source_occurs_bound + S (wpo_index_unused_source_occurs) = l) /\ (((exists wpo_beta_height_unused_source_occurs_entry. wpo_beta_height_unused_source_occurs_entry + S (i) = S ((S (wpo_index_unused_source_occurs)) * c)) /\ exists wpo_beta_quotient_unused_source_occurs_entry. b = wpo_beta_quotient_unused_source_occurs_entry * S ((S (wpo_index_unused_source_occurs)) * c) + (i))))
  15. 0015specialize hclosed x
  16. 0016specialize hclosed j
  17. 0017specialize hclosed i
  18. 0018apply hclosed
  19. 0019exact hjcontains_witness_left
  20. 0020exact hjcontains_witness_right
  21. 0021exact hback
  22. 0022cases hioccurs
  23. 0023cases hioccurs_witness
  24. 0024apply hiomit
  25. 0025exists x1
  26. 0026split
  27. 0027exact hioccurs_witness_left
  28. 0028exact hioccurs_witness_right