FS002A · theorem body

four_square_euler_mixed_bd

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

The two mixed Hamilton products for the bd left-coordinate pair cancel exactly.

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 d e f g h. ((((b * f) * (d * h) = (b * h) * (d * f))) /\ (((b * g) * (d * e) = (b * e) * (d * g))))

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 d e f g h. ((((b * f) * (d * h) = (b * h) * (d * f))) /\ (((b * g) * (d * e) = (b * e) * (d * g))))

Proof neighborhood

Direct theorem prerequisites

FS0025 four_square_euler_cross_swap

Direct theorem dependents

FS002C four_square_euler_all_mixed_cancel

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

19 script commands · 3 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.

Named ingredients (1)
four square euler cross swap
01Fix variables and assumptionsL1–8

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
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
02Separate the logical casesL9–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    split
03Use earlier factsL10–19

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

  1. L10
    specialize four_square_euler_cross_swap b
  2. L11
    specialize four_square_euler_cross_swap f
  3. L12
    specialize four_square_euler_cross_swap d
  4. L13
    specialize four_square_euler_cross_swap h
  5. L14
    exact four_square_euler_cross_swap
  6. L15
    specialize four_square_euler_cross_swap b
  7. L16
    specialize four_square_euler_cross_swap g
  8. L17
    specialize four_square_euler_cross_swap d
  9. L18
    specialize four_square_euler_cross_swap e
  10. L19
    exact four_square_euler_cross_swap

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009split
  10. 0010specialize four_square_euler_cross_swap b
  11. 0011specialize four_square_euler_cross_swap f
  12. 0012specialize four_square_euler_cross_swap d
  13. 0013specialize four_square_euler_cross_swap h
  14. 0014exact four_square_euler_cross_swap
  15. 0015specialize four_square_euler_cross_swap b
  16. 0016specialize four_square_euler_cross_swap g
  17. 0017specialize four_square_euler_cross_swap d
  18. 0018specialize four_square_euler_cross_swap e
  19. 0019exact four_square_euler_cross_swap