FS0010 — four_square_conjugate_global_compensation

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 e f g h. (((a * a + b * b + c * c + d * d) * (e * e + f * f + g * g + h * h)) + (((((a * e + b * f + c * g + d * h) * (0) + (0) * (a * e + b * f + c * g + d * h)) + ((a * f + c * h) * (b * e + d * g) + (b * e + d * g) * (a * f + c * h))) + (((a * g + d * f) * (c * e + b * h) + (c * e + b * h) * (a * g + d * f)) + ((a * h + b * g) * (d * e + c * f) + (d * e + c * f) * (a * h + b * g)))))) = (((((a * e + b * f + c * g + d * h) * (a * e + b * f + c * g + d * h) + (0) * (0)) + ((a * f + c * h) * (a * f + c * h) + (b * e + d * g) * (b * e + d * g))) + (((a * g + d * f) * (a * g + d * f) + (c * e + b * h) * (c * e + b * h)) + ((a * h + b * g) * (a * h + b * g) + (d * e + c * f) * (d * e + c * f)))))

Constructive proof overview

Generated structural guide

The complete subtraction-free conjugate quaternion compensation equation follows from sixteen diagonal and twelve explicitly crossed mixed blocks.

The unchanged tactic script uses 5 declared prerequisites and contains 28 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

Proof neighborhood

Direct dependencies

FS002H four_square_euler_diagonal_expansion FS000V four_square_conjugate_diagonal_regroup FS000Y four_square_conjugate_cross_decomposition FS000Z four_square_conjugate_mixed_decomposition FS000X four_square_conjugate_left_decomposition

Direct dependents

FS0012 four_square_signed_conjugate_quaternion

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

28 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 (5)

four square euler diagonal expansion four square conjugate diagonal regroup four square conjugate cross decomposition four square conjugate mixed decomposition four square conjugate left decomposition

01Fix variables and assumptionsL1–8

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

L1
intro a
L2
intro b
L3
intro c
L4
intro d
L5
intro e
L6
intro f
L7
intro g
L8
intro h

02Calculate and transport equalitiesL9–10

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

L9
trans ((((((((((a * e) * (a * e)) + ((a * f) * (a * f))) + ((a * g) * (a * g))) + ((a * h) * (a * h)))) + · expand full local formula (707 characters)
trans ((((((((((a * e) * (a * e)) + ((a * f) * (a * f))) + ((a * g) * (a * g))) + ((a * h) * (a * h)))) + ((((((b * e) * (b * e)) + ((b * f) * (b * f))) + ((b * g) * (b * g))) + ((b * h) * (b * h))))) + ((((((c * e) * (c * e)) + ((c * f) * (c * f))) + ((c * g) * (c * g))) + ((c * h) * (c * h))))) + ((((((d * e) * (d * e)) + ((d * f) * (d * f))) + ((d * g) * (d * g))) + ((d * h) * (d * h)))))) + (((((a * e + b * f + c * g + d * h) * (0) + (0) * (a * e + b * f + c * g + d * h)) + ((a * f + c * h) * (b * e + d * g) + (b * e + d * g) * (a * f + c * h))) + (((a * g + d * f) * (c * e + b * h) + (c * e + b * h) * (a * g + d * f)) + ((a * h + b * g) * (d * e + c * f) + (d * e + c * f) * (a * h + b * g)))))
L10
congr

03Use earlier factsL11–11