HL0017

hensel_signed_blend_balance

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

The natural recoding and its negative component balance to the positive component plus a full modulus multiple.

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 h A B. (A + h * B) + B = A + (S h) * B

Constructive proof overview

Generated structural guide

The natural recoding and its negative component balance to the positive component plus a full modulus multiple.

The unchanged tactic script uses 2 declared prerequisites and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

HL0018 hensel_signed_blend_mod_iff

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

14 script commands · 8 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro h
  2. L2
    intro A
  3. L3
    intro B
02Calculate and transport equalitiesL4–4

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

  1. L4
    trans A + (h * B + B)
03Use earlier factsL5–5

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

  1. L5
    apply add_assoc
04Calculate and transport equalitiesL6–9

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

  1. L6
    congr
  2. L7
    refl
  3. L8
    symm
  4. L9
    trans B * S h
05Use earlier factsL10–10

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

  1. L10
    apply mul_comm
06Calculate and transport equalitiesL11–12

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

  1. L11
    rewrite PA6
  2. L12
    congr
07Use earlier factsL13–13

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

  1. L13
    apply mul_comm
08Calculate and transport equalitiesL14–14

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

  1. L14
    refl

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro h
  2. 0002intro A
  3. 0003intro B
  4. 0004trans A + (h * B + B)
  5. 0005apply add_assoc
  6. 0006congr
  7. 0007refl
  8. 0008symm
  9. 0009trans B * S h
  10. 0010apply mul_comm
  11. 0011rewrite PA6
  12. 0012congr
  13. 0013apply mul_comm
  14. 0014refl