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 exchange positions without disturbing their shared right tail.
The unchanged tactic script uses 2 declared prerequisites and contains 11 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_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorizedDirect 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_blocksFormal 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
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
02Calculate and transport equalitiesL4–5
03Use earlier factsL6–6
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L6
apply add_assoc
04Calculate and transport equalitiesL7–8
05Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- 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.
- L10
refl
07Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
apply add_assoc