HL0006

hensel_lift_linear_identity

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

The lifted first-order Taylor term factors exactly by the old modulus.

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 d q t. m * (q + d * t) = m * q + (m * t) * d

Constructive proof overview

Generated structural guide

The lifted first-order Taylor term factors exactly by the old modulus.

The unchanged tactic script uses 3 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

mul_add Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

HL0007 hensel_lift_correction_of_root

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 · 7 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 assumptionsL1–4

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

  1. L1
    intro m
  2. L2
    intro d
  3. L3
    intro q
  4. L4
    intro t
02Calculate and transport equalitiesL5–5

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

  1. L5
    trans m * q + m * (d * t)
03Use earlier factsL6–6

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

  1. L6
    apply mul_add
04Calculate and transport equalitiesL7–11

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

  1. L7
    congr
  2. L8
    refl
  3. L9
    trans m * (t * d)
  4. L10
    congr
  5. L11
    refl
05Use earlier factsL12–12

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

  1. L12
    apply mul_comm
06Calculate and transport equalitiesL13–13

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

  1. L13
    symm
07Use earlier factsL14–14

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

  1. L14
    apply mul_assoc

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro m
  2. 0002intro d
  3. 0003intro q
  4. 0004intro t
  5. 0005trans m * q + m * (d * t)
  6. 0006apply mul_add
  7. 0007congr
  8. 0008refl
  9. 0009trans m * (t * d)
  10. 0010congr
  11. 0011refl
  12. 0012apply mul_comm
  13. 0013symm
  14. 0014apply mul_assoc