FS005M · theorem body

four_square_signed_cross_positive

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

Two positive signed coordinate orientations have congruent crossed bilinear products.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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) → ModEq(k,b,f) → ModEq(k,a · f,b · e)

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_positive_a ftcn_right_fssq_cross_positive_a. (a) + (k) * ftcn_left_fssq_cross_positive_a = (e) + (k) * ftcn_right_fssq_cross_positive_a) -> (exists ftcn_left_fssq_cross_positive_b ftcn_right_fssq_cross_positive_b. (b) + (k) * ftcn_left_fssq_cross_positive_b = (f) + (k) * ftcn_right_fssq_cross_positive_b) -> (exists ftcn_left_fssq_cross_positive_result ftcn_right_fssq_cross_positive_result. (a * f) + (k) * ftcn_left_fssq_cross_positive_result = (b * e) + (k) * ftcn_right_fssq_cross_positive_result)

Proof neighborhood

Direct theorem prerequisites

mod_eq_mul_right · Stable closed mul_comm · Stable closed mod_eq_symm · Stable closed mod_eq_trans · Stable closed

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

37 script commands · 6 reading checkpoints · 4 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.

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 hfirstL8–14

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

  1. L8
    have hfirst : ModEq(k,a · f,e · f)Definitions: ModEq(k,a · f,e · f)Original native command in the exact edition
  2. L9
    specialize mod_eq_mul_right k
  3. L10
    specialize mod_eq_mul_right a
  4. L11
    specialize mod_eq_mul_right e
  5. L12
    specialize mod_eq_mul_right f
  6. L13
    apply mod_eq_mul_right
  7. L14
    exact ha
03Establish hsecondL15–21

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

  1. L15
    have hsecond : ModEq(k,b · e,f · e)Definitions: ModEq(k,b · e,f · e)Original native command in the exact edition
  2. L16
    specialize mod_eq_mul_right k
  3. L17
    specialize mod_eq_mul_right b
  4. L18
    specialize mod_eq_mul_right f
  5. L19
    specialize mod_eq_mul_right e
  6. L20
    apply mod_eq_mul_right
  7. L21
    exact hb
04Establish hswapL22–24

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

  1. L22
    have hswap : f * e = e * f
  2. L23
    apply mul_comm
  3. L24
    rewrite hswap at hsecond
05Establish hreverseL25–34

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq symm.

  1. L25
    have hreverse : ModEq(k,e · f,b · e)Definitions: ModEq(k,e · f,b · e)Original native command in the exact edition
  2. L26
    specialize mod_eq_symm k
  3. L27
    specialize mod_eq_symm (b * e)
  4. L28
    specialize mod_eq_symm (e * f)
  5. L29
    apply mod_eq_symm
  6. L30
    exact hsecond
  7. L31
    specialize mod_eq_trans k
  8. L32
    specialize mod_eq_trans (a * f)
  9. L33
    specialize mod_eq_trans (e * f)
  10. L34
    specialize mod_eq_trans (b * e)
06Use earlier factsL35–37

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

  1. L35
    apply mod_eq_trans
  2. L36
    exact hfirst
  3. L37
    exact hreverse

Library-wide reading audit

Original defined command ledger · 37 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro e
  5. 0005intro f
  6. 0006intro ha
  7. 0007intro hb
  8. 0008have hfirst : ModEq(k,a · f,e · f)
    Exact native replay linehave hfirst : exists ftcn_left_fssq_cross_positive_first ftcn_right_fssq_cross_positive_first. (a * f) + (k) * ftcn_left_fssq_cross_positive_first = (e * f) + (k) * ftcn_right_fssq_cross_positive_first
  9. 0009specialize mod_eq_mul_right k
  10. 0010specialize mod_eq_mul_right a
  11. 0011specialize mod_eq_mul_right e
  12. 0012specialize mod_eq_mul_right f
  13. 0013apply mod_eq_mul_right
  14. 0014exact ha
  15. 0015have hsecond : ModEq(k,b · e,f · e)
    Exact native replay linehave hsecond : exists ftcn_left_fssq_cross_positive_second ftcn_right_fssq_cross_positive_second. (b * e) + (k) * ftcn_left_fssq_cross_positive_second = (f * e) + (k) * ftcn_right_fssq_cross_positive_second
  16. 0016specialize mod_eq_mul_right k
  17. 0017specialize mod_eq_mul_right b
  18. 0018specialize mod_eq_mul_right f
  19. 0019specialize mod_eq_mul_right e
  20. 0020apply mod_eq_mul_right
  21. 0021exact hb
  22. 0022have hswap : f * e = e * f
  23. 0023apply mul_comm
  24. 0024rewrite hswap at hsecond
  25. 0025have hreverse : ModEq(k,e · f,b · e)
    Exact native replay linehave hreverse : exists ftcn_left_fssq_cross_positive_reverse ftcn_right_fssq_cross_positive_reverse. (e * f) + (k) * ftcn_left_fssq_cross_positive_reverse = (b * e) + (k) * ftcn_right_fssq_cross_positive_reverse
  26. 0026specialize mod_eq_symm k
  27. 0027specialize mod_eq_symm (b * e)
  28. 0028specialize mod_eq_symm (e * f)
  29. 0029apply mod_eq_symm
  30. 0030exact hsecond
  31. 0031specialize mod_eq_trans k
  32. 0032specialize mod_eq_trans (a * f)
  33. 0033specialize mod_eq_trans (e * f)
  34. 0034specialize mod_eq_trans (b * e)
  35. 0035apply mod_eq_trans
  36. 0036exact hfirst
  37. 0037exact hreverse