JT005D

jordan_primitive_tuple_avoids_prime_common_divisor

A prime divisor of the modulus cannot divide every coordinate of a primitive 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

∀ p. ∀ n. ∀ b. ∀ c. ∀ k. Prime(p) → Dvd(p,n) → JordanPrimitiveTuple(n,b,c,k) → ¬JordanTupleAllDivisible(p,b,c,k)

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

Definition DAG

JordanTupleAllDivisible(d,b,c,k) · 1JordanPrimitiveTuple(n,b,c,k) · 1Dvd(d,n) · 1Prime(p) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall p n b c k. (~((p) = 1) /\ forall pvs_left_jordan_base pvs_right_jordan_base. (p) = pvs_left_jordan_base * pvs_right_jordan_base -> pvs_left_jordan_base = 1 \/ pvs_right_jordan_base = 1) -> (exists jt_factor_jordan_modulus. (n)=(p)*jt_factor_jordan_modulus) -> (forall jt_divisor_power_primitive. (exists jt_factor_power_primitivemodulus. (n)=(jt_divisor_power_primitive)*jt_factor_power_primitivemodulus) -> (forall jt_index_power_primitivecoordinates jt_value_power_primitivecoordinates. (exists jt_gap_power_primitivecoordinatesindex. jt_gap_power_primitivecoordinatesindex+S (jt_index_power_primitivecoordinates)=(k)) -> (((exists fs_h_jt_power_primitivecoordinatesat. fs_h_jt_power_primitivecoordinatesat + S (jt_value_power_primitivecoordinates) = S ((S (jt_index_power_primitivecoordinates)) * c)) /\ exists fs_q_jt_power_primitivecoordinatesat. b = fs_q_jt_power_primitivecoordinatesat * S ((S (jt_index_power_primitivecoordinates)) * c) + (jt_value_power_primitivecoordinates))) -> (exists jt_factor_power_primitivecoordinatesdivides. (jt_value_power_primitivecoordinates)=(jt_divisor_power_primitive)*jt_factor_power_primitivecoordinatesdivides)) -> jt_divisor_power_primitive=1) -> ~(forall jt_index_power_all_divisible jt_value_power_all_divisible. (exists jt_gap_power_all_divisibleindex. jt_gap_power_all_divisibleindex+S (jt_index_power_all_divisible)=(k)) -> (((exists fs_h_jt_power_all_divisibleat. fs_h_jt_power_all_divisibleat + S (jt_value_power_all_divisible) = S ((S (jt_index_power_all_divisible)) * c)) /\ exists fs_q_jt_power_all_divisibleat. b = fs_q_jt_power_all_divisibleat * S ((S (jt_index_power_all_divisible)) * c) + (jt_value_power_all_divisible))) -> (exists jt_factor_power_all_divisibledivides. (jt_value_power_all_divisible)=(p)*jt_factor_power_all_divisibledivides))

Complete tactic proof in conservative notation

All 15 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

15 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 assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro k
  6. L6
    intro hp
  7. L7
    intro hpn
  8. L8
    intro hprimitive
  9. L9
    intro hall
02Separate the logical casesL10–10

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

  1. L10
    cases hp
03Use earlier factsL11–15

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

  1. L11
    apply hp_left
  2. L12
    specialize hprimitive (p)
  3. L13
    apply hprimitive
  4. L14
    exact hpn
  5. L15
    exact hall

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro b
  4. 0004intro c
  5. 0005intro k
  6. 0006intro hp
  7. 0007intro hpn
  8. 0008intro hprimitive
  9. 0009intro hall
  10. 0010cases hp
  11. 0011apply hp_left
  12. 0012specialize hprimitive (p)
  13. 0013apply hprimitive
  14. 0014exact hpn
  15. 0015exact hall