FS0006 · theorem body

four_square_add_swap_right_tail

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

Two adjacent natural summands exchange positions without disturbing their shared right tail.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall a b c. a + (b + c) = b + (a + c)

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed

Direct theorem dependents

FS0007 four_square_additive_gap_reorder FS000V four_square_conjugate_diagonal_regroup FS000Z four_square_conjugate_mixed_decomposition FS001M four_square_descent_centered_signed_remainder_exists FS002J four_square_euler_add_permute_six FS002K four_square_euler_add_permute_nine FS002L four_square_euler_add_permute_twelve FS002M four_square_euler_add_permute_sixteen FS004O four_square_signed_natural_positive_first_blocks FS004Q four_square_signed_orientation_mask_00 FS004R four_square_signed_orientation_mask_01 FS004S four_square_signed_orientation_mask_02 FS004T four_square_signed_orientation_mask_03 FS004U four_square_signed_orientation_mask_04 FS004V four_square_signed_orientation_mask_05 FS004W four_square_signed_orientation_mask_06 FS004X four_square_signed_orientation_mask_07 FS004Y four_square_signed_orientation_mask_08 FS004Z four_square_signed_orientation_mask_09 FS0050 four_square_signed_orientation_mask_10 FS0051 four_square_signed_orientation_mask_11 FS0052 four_square_signed_orientation_mask_12 FS0053 four_square_signed_orientation_mask_13 FS0054 four_square_signed_orientation_mask_14 FS0055 four_square_signed_orientation_mask_15 FS005F four_square_signed_sum_two_decomposition FS005G four_square_signed_sum_four_decomposition FS005I four_square_signed_pair_cross_decomposition FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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