TH0006

hensel_add_swap_nested

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

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

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.

Historical partial components only: this chapter proves exact natural polynomial Taylor remainders, bounded corrections, and one-step divisibility lifts. G095 is now closed in the separate Alpha-v27 hensel-lifting branch for integer polynomials, unrestricted input roots, unique canonical representatives, and every positive prime power. Full G095 proof · Alpha v27

Exact theorem in conservative defined notation

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

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

none

Actual proof prerequisites

add_assoc · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order statement 
forall a b c. a + (b + c) = b + (a + c)

Complete unchanged native tactic proof

All 11 lines are the exact independently kernel-checked original script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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