PA00A8 · theorem

orbit_closed_prefix_zero

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

Orbit closure is vacuous on the empty decoded prefix.

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.

Statement with defined notation

∀ u. ∀ v. ∀ b. ∀ c. ∀ x. ∀ y. ∀ z. Lt(x,0) → BetaAt(b,c,x,y) → BetaAt(u,v,y,z) → ContainsPrefix(b,c,0,z)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0013 BetaAt PD0027 ContainsPrefix

4 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall u v b c. (forall wpo_position_wpoi_zero_closed wpo_source_wpoi_zero_closed wpo_mate_wpoi_zero_closed. (exists wpo_gap_wpoi_zero_closed_position_bound. wpo_gap_wpoi_zero_closed_position_bound + S (wpo_position_wpoi_zero_closed) = 0) -> (((exists wpo_beta_height_wpoi_zero_closed_source_entry. wpo_beta_height_wpoi_zero_closed_source_entry + S (wpo_source_wpoi_zero_closed) = S ((S (wpo_position_wpoi_zero_closed)) * c)) /\ exists wpo_beta_quotient_wpoi_zero_closed_source_entry. b = wpo_beta_quotient_wpoi_zero_closed_source_entry * S ((S (wpo_position_wpoi_zero_closed)) * c) + (wpo_source_wpoi_zero_closed))) -> (((exists wpo_beta_height_wpoi_zero_closed_inverse_entry. wpo_beta_height_wpoi_zero_closed_inverse_entry + S (wpo_mate_wpoi_zero_closed) = S ((S (wpo_source_wpoi_zero_closed)) * v)) /\ exists wpo_beta_quotient_wpoi_zero_closed_inverse_entry. u = wpo_beta_quotient_wpoi_zero_closed_inverse_entry * S ((S (wpo_source_wpoi_zero_closed)) * v) + (wpo_mate_wpoi_zero_closed))) -> exists wpo_mate_position_wpoi_zero_closed. ((exists wpo_gap_wpoi_zero_closed_mate_bound. wpo_gap_wpoi_zero_closed_mate_bound + S (wpo_mate_position_wpoi_zero_closed) = 0) /\ (((exists wpo_beta_height_wpoi_zero_closed_mate_entry. wpo_beta_height_wpoi_zero_closed_mate_entry + S (wpo_mate_wpoi_zero_closed) = S ((S (wpo_mate_position_wpoi_zero_closed)) * c)) /\ exists wpo_beta_quotient_wpoi_zero_closed_mate_entry. b = wpo_beta_quotient_wpoi_zero_closed_mate_entry * S ((S (wpo_mate_position_wpoi_zero_closed)) * c) + (wpo_mate_wpoi_zero_closed)))))

Proof neighborhood

Direct theorem prerequisites

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Direct theorem dependents

PA00AA pair_order_state_zero

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

20 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
add eq zero rightsucc ne zero
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 q
  6. L6
    intro s
  7. L7
    intro m
  8. L8
    intro hq
  9. L9
    intro hsource
  10. L10
    intro hinverse
02Separate the logical casesL11–12

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

  1. L11
    exfalso
  2. L12
    cases hq
03Establish hsqL13–20

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

  1. L13
    have hsq : S q = 0
  2. L14
    specialize add_eq_zero_right x
  3. L15
    specialize add_eq_zero_right (S q)
  4. L16
    apply add_eq_zero_right
  5. L17
    exact hq_witness
  6. L18
    specialize succ_ne_zero q
  7. L19
    apply succ_ne_zero
  8. L20
    exact hsq

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro u
  2. 0002intro v
  3. 0003intro b
  4. 0004intro c
  5. 0005intro q
  6. 0006intro s
  7. 0007intro m
  8. 0008intro hq
  9. 0009intro hsource
  10. 0010intro hinverse
  11. 0011exfalso
  12. 0012cases hq
  13. 0013have hsq : S q = 0
  14. 0014specialize add_eq_zero_right x
  15. 0015specialize add_eq_zero_right (S q)
  16. 0016apply add_eq_zero_right
  17. 0017exact hq_witness
  18. 0018specialize succ_ne_zero q
  19. 0019apply succ_ne_zero
  20. 0020exact hsq