FS005T · theorem body

four_square_signed_cross_mixed_zero_reversed

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

The negative-positive orientation also makes the crossed bilinear sum vanish.

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

∀ k. ∀ a. ∀ b. ∀ e. ∀ f. ModEq(k,a + e,0) → ModEq(k,b,f) → ModEq(k,a · f + b · e,0)

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

PD0008 ModEq

In local proof propositions

PD0008 ModEq
Exact expanded first-order statement
forall k a b e f. (exists ftcn_left_fssq_cross_reverse_a ftcn_right_fssq_cross_reverse_a. (a + e) + (k) * ftcn_left_fssq_cross_reverse_a = (0) + (k) * ftcn_right_fssq_cross_reverse_a) -> (exists ftcn_left_fssq_cross_reverse_b ftcn_right_fssq_cross_reverse_b. (b) + (k) * ftcn_left_fssq_cross_reverse_b = (f) + (k) * ftcn_right_fssq_cross_reverse_b) -> (exists ftcn_left_fssq_cross_reverse_result ftcn_right_fssq_cross_reverse_result. (a * f + b * e) + (k) * ftcn_left_fssq_cross_reverse_result = (0) + (k) * ftcn_right_fssq_cross_reverse_result)

Proof neighborhood

Direct theorem prerequisites

FS005O four_square_signed_cross_mixed_zero FS005R four_square_signed_mod_zero_swap

Direct theorem dependents

FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

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

21 script commands · 3 reading checkpoints · 1 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 (2)
four square signed cross mixed zerofour square signed mod zero swap
01Fix variables and assumptionsL1–7

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro e
  5. L5
    intro f
  6. L6
    intro ha
  7. L7
    intro hb
02Establish hreverseL8–17

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

  1. L8
    have hreverse : ModEq(k,b · e + a · f,0)Definitions: ModEq(k,b · e + a · f,0)Original native command in the exact edition
  2. L9
    specialize four_square_signed_cross_mixed_zero k
  3. L10
    specialize four_square_signed_cross_mixed_zero b
  4. L11
    specialize four_square_signed_cross_mixed_zero a
  5. L12
    specialize four_square_signed_cross_mixed_zero f
  6. L13
    specialize four_square_signed_cross_mixed_zero e
  7. L14
    apply four_square_signed_cross_mixed_zero
  8. L15
    exact hb
  9. L16
    exact ha
  10. L17
    specialize four_square_signed_mod_zero_swap k
03Use earlier factsL18–21

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

  1. L18
    specialize four_square_signed_mod_zero_swap (b * e)
  2. L19
    specialize four_square_signed_mod_zero_swap (a * f)
  3. L20
    apply four_square_signed_mod_zero_swap
  4. L21
    exact hreverse

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro e
  5. 0005intro f
  6. 0006intro ha
  7. 0007intro hb
  8. 0008have hreverse : ModEq(k,b · e + a · f,0)
    Exact native replay linehave hreverse : exists ftcn_left_fssq_cross_reverse_order ftcn_right_fssq_cross_reverse_order. (b * e + a * f) + (k) * ftcn_left_fssq_cross_reverse_order = (0) + (k) * ftcn_right_fssq_cross_reverse_order
  9. 0009specialize four_square_signed_cross_mixed_zero k
  10. 0010specialize four_square_signed_cross_mixed_zero b
  11. 0011specialize four_square_signed_cross_mixed_zero a
  12. 0012specialize four_square_signed_cross_mixed_zero f
  13. 0013specialize four_square_signed_cross_mixed_zero e
  14. 0014apply four_square_signed_cross_mixed_zero
  15. 0015exact hb
  16. 0016exact ha
  17. 0017specialize four_square_signed_mod_zero_swap k
  18. 0018specialize four_square_signed_mod_zero_swap (b * e)
  19. 0019specialize four_square_signed_mod_zero_swap (a * f)
  20. 0020apply four_square_signed_mod_zero_swap
  21. 0021exact hreverse