JT0058

jordan_zero_tuple_bounded_one

The literal beta tuple (0,0) has every coordinate below one, at every length.

Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable

95 new Alpha admissions come from 96 source lemmas: tuple equality reflexivity reuses an already-admitted theorem and is not counted twice. All counts use actual finite beta-coded enumerations. G008 multiplicativity is proved; the general prime-power count and distinct-prime product formula are further goals. General prime-power fields (G091) remain open. Stable is unchanged.

Exact theorem in conservative defined notation

∀ k. BetaPrefixInto(0,0,k,1)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

BetaPrefixInto(b,c,l,B) · 1

Actual proof prerequisites

finite_beta_zero_code · checked external prerequisite
Original expanded first-order statement 
forall k. (forall jt_index_unit_zero_tuple. (exists jt_gap_unit_zero_tupleindex. jt_gap_unit_zero_tupleindex+S (jt_index_unit_zero_tuple)=(k)) -> exists jt_value_unit_zero_tuple. ((((exists fs_h_jt_unit_zero_tupleat. fs_h_jt_unit_zero_tupleat + S (jt_value_unit_zero_tuple) = S ((S (jt_index_unit_zero_tuple)) * 0)) /\ exists fs_q_jt_unit_zero_tupleat. 0 = fs_q_jt_unit_zero_tupleat * S ((S (jt_index_unit_zero_tuple)) * 0) + (jt_value_unit_zero_tuple))) /\ (exists jt_gap_unit_zero_tuplevalue. jt_gap_unit_zero_tuplevalue+S (jt_value_unit_zero_tuple)=(1))))

Complete tactic proof in conservative notation

All 9 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

9 script commands · 6 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro k
  2. L2
    intro i
  3. L3
    intro hi
02Construct an explicit witnessL4–4

Supply the displayed value, then prove that it has the required property.

  1. L4
    exists 0
03Separate the logical casesL5–5

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

  1. L5
    split
04Use earlier factsL6–7

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

  1. L6
    specialize finite_beta_zero_code (i)
  2. L7
    apply finite_beta_zero_code
05Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists 0
06Calculate and transport equalitiesL9–9

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    simp

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro k
  2. 0002intro i
  3. 0003intro hi
  4. 0004exists 0
  5. 0005split
  6. 0006specialize finite_beta_zero_code (i)
  7. 0007apply finite_beta_zero_code
  8. 0008exists 0
  9. 0009simp