FS005K · theorem body

four_square_signed_negative_scale_zero

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

Multiplying an opposite signed congruence preserves its subtraction-free zero balance.

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. ∀ e. ∀ t. ModEq(k,a + e,0) → ModEq(k,a · t + e · t,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 e t. (exists ftcn_left_fssq_negative_scale_source ftcn_right_fssq_negative_scale_source. (a + e) + (k) * ftcn_left_fssq_negative_scale_source = (0) + (k) * ftcn_right_fssq_negative_scale_source) -> (exists ftcn_left_fssq_negative_scale_result ftcn_right_fssq_negative_scale_result. (a * t + e * t) + (k) * ftcn_left_fssq_negative_scale_result = (0) + (k) * ftcn_right_fssq_negative_scale_result)

Proof neighborhood

Direct theorem prerequisites

mod_eq_mul_right · Stable closed add_mul · Stable closed mul_zero_left · Stable closed

Direct theorem dependents

FS005N four_square_signed_cross_negative FS005O four_square_signed_cross_mixed_zero FS005V four_square_signed_dot_negative_zero

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 · 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.

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 assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro e
  4. L4
    intro t
  5. L5
    intro hsource
02Establish hscaledL6–12

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

  1. L6
    have hscaled : ModEq(k,(a + e) · t,0 · t)Definitions: ModEq(k,(a + e) · t,0 · t)Original native command in the exact edition
  2. L7
    specialize mod_eq_mul_right k
  3. L8
    specialize mod_eq_mul_right (a + e)
  4. L9
    specialize mod_eq_mul_right 0
  5. L10
    specialize mod_eq_mul_right t
  6. L11
    apply mod_eq_mul_right
  7. L12
    exact hsource
03Establish hleftL13–14

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

  1. L13
    have hleft : (a + e) * t = a * t + e * t
  2. L14
    apply add_mul
04Establish hrightL15–19

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

  1. L15
    have hright : 0 * t = 0
  2. L16
    apply mul_zero_left
  3. L17
    rewrite hleft at hscaled
  4. L18
    rewrite hright at hscaled
  5. L19
    exact hscaled

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro e
  4. 0004intro t
  5. 0005intro hsource
  6. 0006have hscaled : ModEq(k,(a + e) · t,0 · t)
    Exact native replay linehave hscaled : exists ftcn_left_fssq_negative_scaled ftcn_right_fssq_negative_scaled. ((a + e) * t) + (k) * ftcn_left_fssq_negative_scaled = (0 * t) + (k) * ftcn_right_fssq_negative_scaled
  7. 0007specialize mod_eq_mul_right k
  8. 0008specialize mod_eq_mul_right (a + e)
  9. 0009specialize mod_eq_mul_right 0
  10. 0010specialize mod_eq_mul_right t
  11. 0011apply mod_eq_mul_right
  12. 0012exact hsource
  13. 0013have hleft : (a + e) * t = a * t + e * t
  14. 0014apply add_mul
  15. 0015have hright : 0 * t = 0
  16. 0016apply mul_zero_left
  17. 0017rewrite hleft at hscaled
  18. 0018rewrite hright at hscaled
  19. 0019exact hscaled