JT0010

jordan_tuple_all_divisible_empty

Every divisor vacuously divides all entries of an empty tuple.

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

∀ d. ∀ b. ∀ c. JordanTupleAllDivisible(d,b,c,0)

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

Definition DAG

JordanTupleAllDivisible(d,b,c,k) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall d b c. forall jt_index_allempty jt_value_allempty. (exists jt_gap_allemptyindex. jt_gap_allemptyindex+S (jt_index_allempty)=(0)) -> (((exists fs_h_jt_allemptyat. fs_h_jt_allemptyat + S (jt_value_allempty) = S ((S (jt_index_allempty)) * c)) /\ exists fs_q_jt_allemptyat. b = fs_q_jt_allemptyat * S ((S (jt_index_allempty)) * c) + (jt_value_allempty))) -> (exists jt_factor_allemptydivides. (jt_value_allempty)=(d)*jt_factor_allemptydivides)

Complete tactic proof in conservative notation

All 14 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

14 script commands · 3 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 assumptionsL1–7

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

  1. L1
    intro d
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro i
  5. L5
    intro a
  6. L6
    intro hi
  7. L7
    intro ha
02Separate the logical casesL8–8

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

  1. L8
    exfalso
03Use earlier factsL9–14

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

  1. L9
    specialize lt_not_le (i)
  2. L10
    specialize lt_not_le (0)
  3. L11
    apply lt_not_le
  4. L12
    exact hi
  5. L13
    specialize zero_le (i)
  6. L14
    apply zero_le

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro d
  2. 0002intro b
  3. 0003intro c
  4. 0004intro i
  5. 0005intro a
  6. 0006intro hi
  7. 0007intro ha
  8. 0008exfalso
  9. 0009specialize lt_not_le (i)
  10. 0010specialize lt_not_le (0)
  11. 0011apply lt_not_le
  12. 0012exact hi
  13. 0013specialize zero_le (i)
  14. 0014apply zero_le