TH0003

horner_derivative_mod_congruence_successor_step

One coupled formal-derivative Horner transition preserves balanced value and derivative congruence.

Alpha v34 checked-use · first admitted v25 · 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.

Historical partial components only: this chapter proves exact natural polynomial Taylor remainders, bounded corrections, and one-step divisibility lifts. G095 is now closed in the separate Alpha-v27 hensel-lifting branch for integer polynomials, unrestricted input roots, unique canonical representatives, and every positive prime power. Full G095 proof · Alpha v27

Exact theorem in conservative defined notation

∀ m. ∀ x. ∀ y. ∀ r. ∀ s. ∀ d. ∀ e. ModEq(m,x,y) → ModEq(m,r,s) → ModEq(m,d,e) → ModEq(m,d · x + r,e · y + s)

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

Definition DAG

ModEq(m,a,b) · 4

Actual proof prerequisites

mod_eq_mul · checked external prerequisitemod_eq_add · checked external prerequisite
Original expanded first-order statement 
forall m x y r s d e. (exists hgcrt_mod_left_pth_dstep_base hgcrt_mod_right_pth_dstep_base. x + m * hgcrt_mod_left_pth_dstep_base = y + m * hgcrt_mod_right_pth_dstep_base) -> (exists hgcrt_mod_left_pth_dstep_value hgcrt_mod_right_pth_dstep_value. r + m * hgcrt_mod_left_pth_dstep_value = s + m * hgcrt_mod_right_pth_dstep_value) -> (exists hgcrt_mod_left_pth_dstep_derivative hgcrt_mod_right_pth_dstep_derivative. d + m * hgcrt_mod_left_pth_dstep_derivative = e + m * hgcrt_mod_right_pth_dstep_derivative) -> (exists hgcrt_mod_left_pth_dstep_result hgcrt_mod_right_pth_dstep_result. (d * x + r) + m * hgcrt_mod_left_pth_dstep_result = (e * y + s) + m * hgcrt_mod_right_pth_dstep_result)

Complete unchanged native tactic proof

All 27 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

27 script commands · 3 reading checkpoints · 1 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 assumptionsL1–10

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

  1. L1
    intro m
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro r
  5. L5
    intro s
  6. L6
    intro d
  7. L7
    intro e
  8. L8
    intro hbase
  9. L9
    intro hvalue
  10. L10
    intro hderivative
02Establish hproductL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq mul.

  1. L11
    have hproduct : exists hgcrt_mod_left_pth_dstep_product hgcrt_mod_right_pth_dstep_product. (d * x) + m * hgcrt_mod_left_pth_dstep_product = (e * y) + m * hgcrt_mod_right_pth_dstep_product
  2. L12
    specialize mod_eq_mul m
  3. L13
    specialize mod_eq_mul d
  4. L14
    specialize mod_eq_mul e
  5. L15
    specialize mod_eq_mul x
  6. L16
    specialize mod_eq_mul y
  7. L17
    apply mod_eq_mul
  8. L18
    exact hderivative
  9. L19
    exact hbase
  10. L20
    specialize mod_eq_add m
03Use earlier factsL21–27

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

  1. L21
    specialize mod_eq_add (d * x)
  2. L22
    specialize mod_eq_add (e * y)
  3. L23
    specialize mod_eq_add r
  4. L24
    specialize mod_eq_add s
  5. L25
    apply mod_eq_add
  6. L26
    exact hproduct
  7. L27
    exact hvalue

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro m
  2. 0002intro x
  3. 0003intro y
  4. 0004intro r
  5. 0005intro s
  6. 0006intro d
  7. 0007intro e
  8. 0008intro hbase
  9. 0009intro hvalue
  10. 0010intro hderivative
  11. 0011have hproduct : exists hgcrt_mod_left_pth_dstep_product hgcrt_mod_right_pth_dstep_product. (d * x) + m * hgcrt_mod_left_pth_dstep_product = (e * y) + m * hgcrt_mod_right_pth_dstep_product
  12. 0012specialize mod_eq_mul m
  13. 0013specialize mod_eq_mul d
  14. 0014specialize mod_eq_mul e
  15. 0015specialize mod_eq_mul x
  16. 0016specialize mod_eq_mul y
  17. 0017apply mod_eq_mul
  18. 0018exact hderivative
  19. 0019exact hbase
  20. 0020specialize mod_eq_add m
  21. 0021specialize mod_eq_add (d * x)
  22. 0022specialize mod_eq_add (e * y)
  23. 0023specialize mod_eq_add r
  24. 0024specialize mod_eq_add s
  25. 0025apply mod_eq_add
  26. 0026exact hproduct
  27. 0027exact hvalue