HL000F

hensel_positive_power_factor

Every actual positive power of a nonzero base supplies both nonzeroness and an explicit base factor.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

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.

The derivative-nonzero criterion supplies no inverse or power witness: both are constructed. Roots may be arbitrary natural representatives of signed integer polynomials. Singular-root classification and p-adic completion are separate milestones.

Exact theorem in conservative defined notation

∀ p. ∀ k. ∀ m. ¬p = 0 → ¬k = 0 → Pow(p,k,m) → ¬m = 0 ∧ Dvd(p,m)

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

Definition DAG

Dvd(d,n) · 1Pow(a,e,z) · 2

Actual proof prerequisites

pow_nonzero_of_one_le · checked external prerequisiteone_le_of_ne_zero · checked external prerequisitenonzero_is_succ · checked external prerequisitepow_successor_decompose · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order statement 
forall p k m. ~(p = 0) -> ~(k = 0) -> (exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = k) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (m) = S ((S (k)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (k)) * pa_v_hpl_power_product) + (m))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = k) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product)))))))) -> ~(m = 0) /\ exists s. m = p * s

Complete tactic proof in conservative notation

All 36 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

36 script commands · 11 reading checkpoints · 2 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 p
  2. L2
    intro k
  3. L3
    intro m
  4. L4
    intro hp
  5. L5
    intro hk
  6. L6
    intro hpower
02Separate the logical casesL7–7

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

  1. L7
    split
03Fix variables and assumptionsL8–8

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

  1. L8
    intro hzero
04Use earlier factsL9–17

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

  1. L9
    specialize pow_nonzero_of_one_le p
  2. L10
    specialize pow_nonzero_of_one_le k
  3. L11
    specialize pow_nonzero_of_one_le m
  4. L12
    apply pow_nonzero_of_one_le
  5. L13
    specialize one_le_of_ne_zero p
  6. L14
    apply one_le_of_ne_zero
  7. L15
    exact hp
  8. L16
    exact hpower
  9. L17
    exact hzero
05Establish hkpositiveL18–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L18
    have hkpositive : exists e. k = S e
  2. L19
    specialize nonzero_is_succ k
  3. L20
    apply nonzero_is_succ
  4. L21
    exact hk
06Separate the logical casesL22–22

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

  1. L22
    cases hkpositive
07Establish hpreviousL23–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow successor decompose.

  1. L23
    have hprevious : ∃ r. Pow(p,x,r) ∧ m = r · pDefinitions: Pow(p,x,r)Original native command in the exact edition
  2. L24
    specialize pow_successor_decompose p
  3. L25
    specialize pow_successor_decompose x
  4. L26
    specialize pow_successor_decompose k
  5. L27
    specialize pow_successor_decompose m
  6. L28
    apply pow_successor_decompose
  7. L29
    exact hkpositive_witness
  8. L30
    exact hpower
08Separate the logical casesL31–32

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

  1. L31
    cases hprevious
  2. L32
    cases hprevious_witness
09Construct an explicit witnessL33–33

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

  1. L33
    exists x1
10Calculate and transport equalitiesL34–34

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

  1. L34
    trans x1 * p
11Use earlier factsL35–36

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

  1. L35
    exact hprevious_witness_right
  2. L36
    apply mul_comm

Library-wide reading audit

Original defined command ledger · 36 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro m
  4. 0004intro hp
  5. 0005intro hk
  6. 0006intro hpower
  7. 0007split
  8. 0008intro hzero
  9. 0009specialize pow_nonzero_of_one_le p
  10. 0010specialize pow_nonzero_of_one_le k
  11. 0011specialize pow_nonzero_of_one_le m
  12. 0012apply pow_nonzero_of_one_le
  13. 0013specialize one_le_of_ne_zero p
  14. 0014apply one_le_of_ne_zero
  15. 0015exact hp
  16. 0016exact hpower
  17. 0017exact hzero
  18. 0018have hkpositive : exists e. k = S e
  19. 0019specialize nonzero_is_succ k
  20. 0020apply nonzero_is_succ
  21. 0021exact hk
  22. 0022cases hkpositive
  23. 0023have hprevious : ∃ r. Pow(p,x,r) ∧ m = r · p
  24. 0024specialize pow_successor_decompose p
  25. 0025specialize pow_successor_decompose x
  26. 0026specialize pow_successor_decompose k
  27. 0027specialize pow_successor_decompose m
  28. 0028apply pow_successor_decompose
  29. 0029exact hkpositive_witness
  30. 0030exact hpower
  31. 0031cases hprevious
  32. 0032cases hprevious_witness
  33. 0033exists x1
  34. 0034trans x1 * p
  35. 0035exact hprevious_witness_right
  36. 0036apply mul_comm