TH0003

horner_derivative_mod_congruence_successor_step

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

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

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 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 27 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_add Stable theorem; checked-use authorized

Direct dependents

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

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.

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 exact 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

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