PA00A9 · theorem

nonendpoint_prefix_zero

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

The nonendpoint range invariant is vacuous on the empty 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

∀ b. ∀ c. ∀ n. ∀ x. ∀ y. Lt(x,0) → BetaAt(b,c,x,y) → ¬y = 0 ∧ ¬S y = 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

PD0002 Lt PD0013 BetaAt

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c n. (forall wpo_position_wpoi_zero_nonendpoint wpo_value_wpoi_zero_nonendpoint. (exists wpo_gap_wpoi_zero_nonendpoint_position_bound. wpo_gap_wpoi_zero_nonendpoint_position_bound + S (wpo_position_wpoi_zero_nonendpoint) = 0) -> (((exists wpo_beta_height_wpoi_zero_nonendpoint_entry. wpo_beta_height_wpoi_zero_nonendpoint_entry + S (wpo_value_wpoi_zero_nonendpoint) = S ((S (wpo_position_wpoi_zero_nonendpoint)) * c)) /\ exists wpo_beta_quotient_wpoi_zero_nonendpoint_entry. b = wpo_beta_quotient_wpoi_zero_nonendpoint_entry * S ((S (wpo_position_wpoi_zero_nonendpoint)) * c) + (wpo_value_wpoi_zero_nonendpoint))) -> (~(wpo_value_wpoi_zero_nonendpoint = 0) /\ ~((S wpo_value_wpoi_zero_nonendpoint) = n)))

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

17 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 assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro n
  4. L4
    intro q
  5. L5
    intro s
  6. L6
    intro hq
  7. L7
    intro hentry
02Separate the logical casesL8–9

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

  1. L8
    exfalso
  2. L9
    cases hq
03Establish hsqL10–17

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

  1. L10
    have hsq : S q = 0
  2. L11
    specialize add_eq_zero_right x
  3. L12
    specialize add_eq_zero_right (S q)
  4. L13
    apply add_eq_zero_right
  5. L14
    exact hq_witness
  6. L15
    specialize succ_ne_zero q
  7. L16
    apply succ_ne_zero
  8. L17
    exact hsq

Library-wide reading audit

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