PA008Y

adjacent_scaled_orbit_history_zero

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

The explicit adjacent scaled-orbit history is empty at zero pairs.

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. (forall espi_pair_zero_history. (exists wpo_gap_zero_history_pair_bound. wpo_gap_zero_history_pair_bound + S (espi_pair_zero_history) = 0) -> exists espi_left_zero_history espi_right_zero_history. (((((exists wpo_beta_height_zero_history_left_entry. wpo_beta_height_zero_history_left_entry + S (espi_left_zero_history) = S ((S (espi_pair_zero_history + espi_pair_zero_history)) * c)) /\ exists wpo_beta_quotient_zero_history_left_entry. b = wpo_beta_quotient_zero_history_left_entry * S ((S (espi_pair_zero_history + espi_pair_zero_history)) * c) + (espi_left_zero_history))) /\ (((((exists wpo_beta_height_zero_history_right_entry. wpo_beta_height_zero_history_right_entry + S (espi_right_zero_history) = S ((S (S (espi_pair_zero_history + espi_pair_zero_history))) * c)) /\ exists wpo_beta_quotient_zero_history_right_entry. b = wpo_beta_quotient_zero_history_right_entry * S ((S (S (espi_pair_zero_history + espi_pair_zero_history))) * c) + (espi_right_zero_history))) /\ (((exists wpo_beta_height_zero_history_scaled_edge. wpo_beta_height_zero_history_scaled_edge + S (S espi_right_zero_history) = S ((S (espi_left_zero_history)) * v)) /\ exists wpo_beta_quotient_zero_history_scaled_edge. u = wpo_beta_quotient_zero_history_scaled_edge * S ((S (espi_left_zero_history)) * v) + (S espi_right_zero_history))))))))

Structural proof guide

Generated structural guide

The explicit adjacent scaled-orbit history is empty at zero pairs.

Use the direct prerequisites add_eq_zero_right, succ_ne_zero as previously established PA formulas.

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

Referenced ingredients

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Proof neighborhood

Direct dependencies

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Direct dependents

PA009V scaled_inverse_pair_order_paired_iteration

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

16 script commands · 3 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.

Named ingredients (2)
add eq zero rightsucc ne zero
01Fix variables and assumptionsL1–6

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 t
  6. L6
    intro ht
02Separate the logical casesL7–8

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

  1. L7
    exfalso
  2. L8
    cases ht
03Establish hstL9–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add eq zero right.

  1. L9
    have hst : S t = 0
  2. L10
    specialize add_eq_zero_right x
  3. L11
    specialize add_eq_zero_right (S t)
  4. L12
    apply add_eq_zero_right
  5. L13
    exact ht_witness
  6. L14
    specialize succ_ne_zero t
  7. L15
    apply succ_ne_zero
  8. L16
    exact hst

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro u
  2. 0002intro v
  3. 0003intro b
  4. 0004intro c
  5. 0005intro t
  6. 0006intro ht
  7. 0007exfalso
  8. 0008cases ht
  9. 0009have hst : S t = 0
  10. 0010specialize add_eq_zero_right x
  11. 0011specialize add_eq_zero_right (S t)
  12. 0012apply add_eq_zero_right
  13. 0013exact ht_witness
  14. 0014specialize succ_ne_zero t
  15. 0015apply succ_ne_zero
  16. 0016exact hst