TH0002

horner_mod_congruence_successor_step

One genuine Horner value transition preserves balanced congruence at congruent evaluation points.

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

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

Definition DAG

ModEq(m,a,b) · 3

Actual proof prerequisites

mod_eq_mul · checked external prerequisitemod_eq_refl · checked external prerequisitemod_eq_add · checked external prerequisite
Original expanded first-order statement 
forall m x y r s a. (exists hgcrt_mod_left_pth_step_base hgcrt_mod_right_pth_step_base. x + m * hgcrt_mod_left_pth_step_base = y + m * hgcrt_mod_right_pth_step_base) -> (exists hgcrt_mod_left_pth_step_prefix hgcrt_mod_right_pth_step_prefix. r + m * hgcrt_mod_left_pth_step_prefix = s + m * hgcrt_mod_right_pth_step_prefix) -> (exists hgcrt_mod_left_pth_step_result hgcrt_mod_right_pth_step_result. (r * x + a) + m * hgcrt_mod_left_pth_step_result = (s * y + a) + m * hgcrt_mod_right_pth_step_result)

Complete unchanged native tactic proof

All 29 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

29 script commands · 4 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 assumptionsL1–8

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 a
  7. L7
    intro hbase
  8. L8
    intro hprefix
02Establish hproductL9–17

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

  1. L9
    have hproduct : exists hgcrt_mod_left_pth_step_product hgcrt_mod_right_pth_step_product. (r * x) + m * hgcrt_mod_left_pth_step_product = (s * y) + m * hgcrt_mod_right_pth_step_product
  2. L10
    specialize mod_eq_mul m
  3. L11
    specialize mod_eq_mul r
  4. L12
    specialize mod_eq_mul s
  5. L13
    specialize mod_eq_mul x
  6. L14
    specialize mod_eq_mul y
  7. L15
    apply mod_eq_mul
  8. L16
    exact hprefix
  9. L17
    exact hbase
03Establish hcoefficientL18–27

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

  1. L18
    have hcoefficient : exists hgcrt_mod_left_pth_step_coefficient hgcrt_mod_right_pth_step_coefficient. a + m * hgcrt_mod_left_pth_step_coefficient = a + m * hgcrt_mod_right_pth_step_coefficient
  2. L19
    specialize mod_eq_refl m
  3. L20
    specialize mod_eq_refl a
  4. L21
    apply mod_eq_refl
  5. L22
    specialize mod_eq_add m
  6. L23
    specialize mod_eq_add (r * x)
  7. L24
    specialize mod_eq_add (s * y)
  8. L25
    specialize mod_eq_add a
  9. L26
    specialize mod_eq_add a
  10. L27
    apply mod_eq_add
04Use earlier factsL28–29

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

  1. L28
    exact hproduct
  2. L29
    exact hcoefficient

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro m
  2. 0002intro x
  3. 0003intro y
  4. 0004intro r
  5. 0005intro s
  6. 0006intro a
  7. 0007intro hbase
  8. 0008intro hprefix
  9. 0009have hproduct : exists hgcrt_mod_left_pth_step_product hgcrt_mod_right_pth_step_product. (r * x) + m * hgcrt_mod_left_pth_step_product = (s * y) + m * hgcrt_mod_right_pth_step_product
  10. 0010specialize mod_eq_mul m
  11. 0011specialize mod_eq_mul r
  12. 0012specialize mod_eq_mul s
  13. 0013specialize mod_eq_mul x
  14. 0014specialize mod_eq_mul y
  15. 0015apply mod_eq_mul
  16. 0016exact hprefix
  17. 0017exact hbase
  18. 0018have hcoefficient : exists hgcrt_mod_left_pth_step_coefficient hgcrt_mod_right_pth_step_coefficient. a + m * hgcrt_mod_left_pth_step_coefficient = a + m * hgcrt_mod_right_pth_step_coefficient
  19. 0019specialize mod_eq_refl m
  20. 0020specialize mod_eq_refl a
  21. 0021apply mod_eq_refl
  22. 0022specialize mod_eq_add m
  23. 0023specialize mod_eq_add (r * x)
  24. 0024specialize mod_eq_add (s * y)
  25. 0025specialize mod_eq_add a
  26. 0026specialize mod_eq_add a
  27. 0027apply mod_eq_add
  28. 0028exact hproduct
  29. 0029exact hcoefficient