TH0002

horner_mod_congruence_successor_step

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 29 exact native proof lines.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mod_eq_mul Stable theorem; checked-use authorized mod_eq_refl Stable theorem; checked-use authorized mod_eq_add Stable theorem; checked-use authorized

Direct dependents

TH0004 beta_horner_eval_mod_congruence TH0005 beta_horner_derivative_mod_congruence

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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

Separate complete second-wave branches: Full G095 proof · Alpha v27.