HL0015

hensel_horner_linear_successor_identity

Both the Horner value and derivative transitions preserve an exact weighted coefficient combination.

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

∀ h. ∀ a. ∀ u. ∀ v. ∀ A. ∀ B. (u + h · v) · a + (A + h · B) = u · a + A + h · (v · a + B)

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

Definition DAG

none

Actual proof prerequisites

add_mul · checked external prerequisitemul_add · checked external prerequisitemul_assoc · checked external prerequisiteadd_shuffle_middle · checked external prerequisite
Original expanded first-order statement 
forall h a u v A B. (u + h * v) * a + (A + h * B) = (u * a + A) + h * (v * a + B)

Complete tactic proof in conservative notation

All 20 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

20 script commands · 9 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 h
  2. L2
    intro a
  3. L3
    intro u
  4. L4
    intro v
  5. L5
    intro A
  6. L6
    intro B
02Calculate and transport equalitiesL7–8

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

  1. L7
    trans (u * a + (h * v) * a) + (A + h * B)
  2. L8
    congr
03Use earlier factsL9–9

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

  1. L9
    apply add_mul
04Calculate and transport equalitiesL10–11

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

  1. L10
    refl
  2. L11
    trans (u * a + A) + ((h * v) * a + h * B)
05Use earlier factsL12–12

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

  1. L12
    apply add_shuffle_middle
06Calculate and transport equalitiesL13–16

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

  1. L13
    congr
  2. L14
    refl
  3. L15
    trans h * (v * a) + h * B
  4. L16
    congr
07Use earlier factsL17–17

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

  1. L17
    apply mul_assoc
08Calculate and transport equalitiesL18–19

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

  1. L18
    refl
  2. L19
    symm
09Use earlier factsL20–20

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

  1. L20
    apply mul_add

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro h
  2. 0002intro a
  3. 0003intro u
  4. 0004intro v
  5. 0005intro A
  6. 0006intro B
  7. 0007trans (u * a + (h * v) * a) + (A + h * B)
  8. 0008congr
  9. 0009apply add_mul
  10. 0010refl
  11. 0011trans (u * a + A) + ((h * v) * a + h * B)
  12. 0012apply add_shuffle_middle
  13. 0013congr
  14. 0014refl
  15. 0015trans h * (v * a) + h * B
  16. 0016congr
  17. 0017apply mul_assoc
  18. 0018refl
  19. 0019symm
  20. 0020apply mul_add