JT0008

jordan_primitive_tuple_modulus_one

Every finite tuple is primitive modulo one, including the zero 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

∀ b. ∀ c. ∀ k. JordanPrimitiveTuple(1,b,c,k)

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

Definition DAG

JordanPrimitiveTuple(n,b,c,k) · 1

Actual proof prerequisites

divisor_one · checked external prerequisite
Original expanded first-order statement 
forall b c k. forall jt_divisor_one. (exists jt_factor_onemodulus. (1)=(jt_divisor_one)*jt_factor_onemodulus) -> (forall jt_index_onecoordinates jt_value_onecoordinates. (exists jt_gap_onecoordinatesindex. jt_gap_onecoordinatesindex+S (jt_index_onecoordinates)=(k)) -> (((exists fs_h_jt_onecoordinatesat. fs_h_jt_onecoordinatesat + S (jt_value_onecoordinates) = S ((S (jt_index_onecoordinates)) * c)) /\ exists fs_q_jt_onecoordinatesat. b = fs_q_jt_onecoordinatesat * S ((S (jt_index_onecoordinates)) * c) + (jt_value_onecoordinates))) -> (exists jt_factor_onecoordinatesdivides. (jt_value_onecoordinates)=(jt_divisor_one)*jt_factor_onecoordinatesdivides)) -> jt_divisor_one=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 · 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.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro k
  4. L4
    intro q
  5. L5
    intro hq
  6. L6
    intro hall
02Use earlier factsL7–9

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

  1. L7
    specialize divisor_one (q)
  2. L8
    apply divisor_one
  3. L9
    exact hq

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro k
  4. 0004intro q
  5. 0005intro hq
  6. 0006intro hall
  7. 0007specialize divisor_one (q)
  8. 0008apply divisor_one
  9. 0009exact hq