PA00AB · theorem

paired_inverse_witness_zero

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

The adjacent inverse-pair witness invariant is vacuous at zero.

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. Lt(x,0) → ∃ y. ∃ z. BetaAt(b,c,x + x,y) ∧ (BetaAt(b,c,S (x + x),z) ∧ BetaAt(u,v,y,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

4 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall u v b c. (forall wpop_pair_wpop_zero_pairs. (exists wpo_gap_wpop_zero_pairs_pair_bound. wpo_gap_wpop_zero_pairs_pair_bound + S (wpop_pair_wpop_zero_pairs) = 0) -> exists wpop_left_wpop_zero_pairs wpop_right_wpop_zero_pairs. ((((exists wpo_beta_height_wpop_zero_pairs_left_entry. wpo_beta_height_wpop_zero_pairs_left_entry + S (wpop_left_wpop_zero_pairs) = S ((S (wpop_pair_wpop_zero_pairs + wpop_pair_wpop_zero_pairs)) * c)) /\ exists wpo_beta_quotient_wpop_zero_pairs_left_entry. b = wpo_beta_quotient_wpop_zero_pairs_left_entry * S ((S (wpop_pair_wpop_zero_pairs + wpop_pair_wpop_zero_pairs)) * c) + (wpop_left_wpop_zero_pairs))) /\ ((((exists wpo_beta_height_wpop_zero_pairs_right_entry. wpo_beta_height_wpop_zero_pairs_right_entry + S (wpop_right_wpop_zero_pairs) = S ((S (S (wpop_pair_wpop_zero_pairs + wpop_pair_wpop_zero_pairs))) * c)) /\ exists wpo_beta_quotient_wpop_zero_pairs_right_entry. b = wpo_beta_quotient_wpop_zero_pairs_right_entry * S ((S (S (wpop_pair_wpop_zero_pairs + wpop_pair_wpop_zero_pairs))) * c) + (wpop_right_wpop_zero_pairs))) /\ (((exists wpo_beta_height_wpop_zero_pairs_inverse_entry. wpo_beta_height_wpop_zero_pairs_inverse_entry + S (wpop_right_wpop_zero_pairs) = S ((S (wpop_left_wpop_zero_pairs)) * v)) /\ exists wpo_beta_quotient_wpop_zero_pairs_inverse_entry. u = wpo_beta_quotient_wpop_zero_pairs_inverse_entry * S ((S (wpop_left_wpop_zero_pairs)) * v) + (wpop_right_wpop_zero_pairs))))))

Proof neighborhood

Direct theorem prerequisites

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Direct theorem dependents

PA00B1 prime_pair_order_paired_iteration

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

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.

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 assumptionsL1–6

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 defined 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