TH0006

hensel_add_swap_nested

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

Two adjacent natural summands can be swapped inside a right-associated finite sum.

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 a b c. a + (b + c) = b + (a + c)

Constructive proof overview

Generated structural guide

Two adjacent natural summands can be swapped inside a right-associated finite sum.

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

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

Proof neighborhood

Direct dependencies

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

Direct dependents

TH0007 horner_taylor_successor_identity

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

11 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
02Calculate and transport equalitiesL4–5

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

  1. L4
    trans (a + b) + c
  2. L5
    symm
03Use earlier factsL6–6

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

  1. L6
    apply add_assoc
04Calculate and transport equalitiesL7–8

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

  1. L7
    trans (b + a) + c
  2. L8
    congr
05Use earlier factsL9–9

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

  1. L9
    apply add_comm
06Calculate and transport equalitiesL10–10

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

  1. L10
    refl
07Use earlier factsL11–11

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

  1. L11
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004trans (a + b) + c
  5. 0005symm
  6. 0006apply add_assoc
  7. 0007trans (b + a) + c
  8. 0008congr
  9. 0009apply add_comm
  10. 0010refl
  11. 0011apply add_assoc

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