FS0007

four_square_additive_gap_reorder

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

The five additive square-gap contributions reorder without subtraction or a polynomial normalizer.

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

Constructive proof overview

Generated structural guide

The five additive square-gap contributions reorder without subtraction or a polynomial normalizer.

The unchanged tactic script uses 4 declared prerequisites and contains 25 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

add_shuffle_middle Alpha theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized FS0006 four_square_add_swap_right_tail

Direct dependents

FS0009 four_square_gap_balance_right

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

25 script commands · 11 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.

Named ingredients (1)
four square add swap right tail
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
02Calculate and transport equalitiesL5–6

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

  1. L5
    trans ((a + c) + (b + d)) + a
  2. L6
    congr
03Use earlier factsL7–7

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

  1. L7
    apply add_shuffle_middle
04Calculate and transport equalitiesL8–9

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

  1. L8
    refl
  2. L9
    trans (a + c) + ((b + d) + a)
05Use earlier factsL10–10

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

  1. L10
    apply add_assoc
06Calculate and transport equalitiesL11–13

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

  1. L11
    trans (a + c) + (b + (d + a))
  2. L12
    congr
  3. L13
    refl
07Use earlier factsL14–14

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

  1. L14
    apply add_assoc
08Calculate and transport equalitiesL15–17

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

  1. L15
    trans (a + c) + (d + (b + a))
  2. L16
    congr
  3. L17
    refl
09Use earlier factsL18–18

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

  1. L18
    apply four_square_add_swap_right_tail
10Calculate and transport equalitiesL19–23

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

  1. L19
    trans (a + c) + (d + (a + b))
  2. L20
    congr
  3. L21
    refl
  4. L22
    congr
  5. L23
    refl
11Use earlier factsL24–25

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

  1. L24
    apply add_comm
  2. L25
    apply four_square_add_swap_right_tail

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005trans ((a + c) + (b + d)) + a
  6. 0006congr
  7. 0007apply add_shuffle_middle
  8. 0008refl
  9. 0009trans (a + c) + ((b + d) + a)
  10. 0010apply add_assoc
  11. 0011trans (a + c) + (b + (d + a))
  12. 0012congr
  13. 0013refl
  14. 0014apply add_assoc
  15. 0015trans (a + c) + (d + (b + a))
  16. 0016congr
  17. 0017refl
  18. 0018apply four_square_add_swap_right_tail
  19. 0019trans (a + c) + (d + (a + b))
  20. 0020congr
  21. 0021refl
  22. 0022congr
  23. 0023refl
  24. 0024apply add_comm
  25. 0025apply four_square_add_swap_right_tail