PA00B6 · theorem

pair_order_successor_lift_exists

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

Every zero-based pair order has a beta code of successor-valued factors.

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

∀ b. ∀ c. ∀ m. ∃ x. ∃ y. ∀ z. ∀ n. Lt(z,m + m)BetaAt(b,c,z,n)BetaAt(x,y,z,S n)

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

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c m. (exists f g. (forall wsl_index_wsl_lift wsl_value_wsl_lift. (exists wpo_gap_wsl_lift_bound. wpo_gap_wsl_lift_bound + S (wsl_index_wsl_lift) = m + m) -> (((exists wpo_beta_height_wsl_lift_source. wpo_beta_height_wsl_lift_source + S (wsl_value_wsl_lift) = S ((S (wsl_index_wsl_lift)) * c)) /\ exists wpo_beta_quotient_wsl_lift_source. b = wpo_beta_quotient_wsl_lift_source * S ((S (wsl_index_wsl_lift)) * c) + (wsl_value_wsl_lift))) -> (((exists wpo_beta_height_wsl_lift_target. wpo_beta_height_wsl_lift_target + S (S wsl_value_wsl_lift) = S ((S (wsl_index_wsl_lift)) * g)) /\ exists wpo_beta_quotient_wsl_lift_target. f = wpo_beta_quotient_wsl_lift_target * S ((S (wsl_index_wsl_lift)) * g) + (S wsl_value_wsl_lift)))))

Proof neighborhood

Direct theorem prerequisites

Direct theorem dependents

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

7 script commands · 2 reading checkpoints · 0 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 (1)
01Fix variables and assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro m
02Use earlier factsL4–7

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

  1. L4
    specialize beta_successor_lift_exists b
  2. L5
    specialize beta_successor_lift_exists c
  3. L6
    specialize beta_successor_lift_exists (m + m)
  4. L7
    exact beta_successor_lift_exists

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro m
  4. 0004specialize beta_successor_lift_exists b
  5. 0005specialize beta_successor_lift_exists c
  6. 0006specialize beta_successor_lift_exists (m + m)
  7. 0007exact beta_successor_lift_exists