FS005N — four_square_signed_cross_negative

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 k a b e f. (exists ftcn_left_fssq_cross_negative_a ftcn_right_fssq_cross_negative_a. (a + e) + (k) * ftcn_left_fssq_cross_negative_a = (0) + (k) * ftcn_right_fssq_cross_negative_a) -> (exists ftcn_left_fssq_cross_negative_b ftcn_right_fssq_cross_negative_b. (b + f) + (k) * ftcn_left_fssq_cross_negative_b = (0) + (k) * ftcn_right_fssq_cross_negative_b) -> (exists ftcn_left_fssq_cross_negative_result ftcn_right_fssq_cross_negative_result. (a * f) + (k) * ftcn_left_fssq_cross_negative_result = (b * e) + (k) * ftcn_right_fssq_cross_negative_result)

Constructive proof overview

Generated structural guide

Two negative signed coordinate orientations also have congruent crossed bilinear products.

The unchanged tactic script uses 3 declared prerequisites and contains 31 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

FS005K four_square_signed_negative_scale_zero mul_comm Stable theorem; checked-use authorized FS005L four_square_signed_common_zero_cancel

Direct dependents

FS004N four_square_signed_conjugate_negative_blocks FS004O four_square_signed_natural_positive_first_blocks FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

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

31 script commands · 4 reading checkpoints · 3 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 (2)

four square signed negative scale zero four square signed common zero cancel

01Fix variables and assumptionsL1–7

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

L1
intro k
L2
intro a
L3
intro b
L4
intro e
L5
intro f
L6
intro ha
L7
intro hb

02Establish hfirstL8–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square signed negative scale zero.

L8
have hfirst : exists ftcn_left_fssq_cross_negative_first ftcn_right_fssq_cross_negative_first. (a * f + e * f) + (k) * ftcn_left_fssq_cross_negative_first = (0) + (k) * ftcn_right_fssq_cross_negative_first
L9
specialize four_square_signed_negative_scale_zero k
L10
specialize four_square_signed_negative_scale_zero a
L11
specialize four_square_signed_negative_scale_zero e
L12
specialize four_square_signed_negative_scale_zero f
L13
apply four_square_signed_negative_scale_zero
L14
exact ha

03Establish hsecondL15–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square signed negative scale zero.

L15
have hsecond : exists ftcn_left_fssq_cross_negative_second ftcn_right_fssq_cross_negative_second. (b * e + f * e) + (k) * ftcn_left_fssq_cross_negative_second = (0) + (k) * ftcn_right_fssq_cross_negative_second
L16
specialize four_square_signed_negative_scale_zero k
L17
specialize four_square_signed_negative_scale_zero b
L18
specialize four_square_signed_negative_scale_zero f
L19
specialize four_square_signed_negative_scale_zero e
L20
apply four_square_signed_negative_scale_zero
L21
exact hb

04Establish hswapL22–31

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul comm.

L22
have hswap : f * e = e * f
L23
apply mul_comm
L24
rewrite hswap at hsecond
L25
specialize four_square_signed_common_zero_cancel k
L26
specialize four_square_signed_common_zero_cancel (a * f)
L27
specialize four_square_signed_common_zero_cancel (b * e)
L28
specialize four_square_signed_common_zero_cancel (e * f)
L29
apply four_square_signed_common_zero_cancel
L30
exact hfirst
L31
exact hsecond

Library-wide reading audit

Original exact command ledger · 31 lines

0001intro k
0002intro a
0003intro b
0004intro e
0005intro f
0006intro ha
0007intro hb
0008have hfirst : exists ftcn_left_fssq_cross_negative_first ftcn_right_fssq_cross_negative_first. (a * f + e * f) + (k) * ftcn_left_fssq_cross_negative_first = (0) + (k) * ftcn_right_fssq_cross_negative_first
0009specialize four_square_signed_negative_scale_zero k
0010specialize four_square_signed_negative_scale_zero a
0011specialize four_square_signed_negative_scale_zero e
0012specialize four_square_signed_negative_scale_zero f
0013apply four_square_signed_negative_scale_zero
0014exact ha
0015have hsecond : exists ftcn_left_fssq_cross_negative_second ftcn_right_fssq_cross_negative_second. (b * e + f * e) + (k) * ftcn_left_fssq_cross_negative_second = (0) + (k) * ftcn_right_fssq_cross_negative_second
0016specialize four_square_signed_negative_scale_zero k
0017specialize four_square_signed_negative_scale_zero b
0018specialize four_square_signed_negative_scale_zero f
0019specialize four_square_signed_negative_scale_zero e
0020apply four_square_signed_negative_scale_zero
0021exact hb
0022have hswap : f * e = e * f
0023apply mul_comm
0024rewrite hswap at hsecond
0025specialize four_square_signed_common_zero_cancel k
0026specialize four_square_signed_common_zero_cancel (a * f)
0027specialize four_square_signed_common_zero_cancel (b * e)
0028specialize four_square_signed_common_zero_cancel (e * f)
0029apply four_square_signed_common_zero_cancel
0030exact hfirst
0031exact hsecond