FS005Q · theorem body

four_square_signed_mod_zero_equivalent

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

Any two natural signed blocks that both vanish modulo the multiplier are congruent.

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. ModEq(k,a,0) → ModEq(k,b,0) → ModEq(k,a,b)

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. (exists ftcn_left_fssq_zero_equiv_left ftcn_right_fssq_zero_equiv_left. (a) + (k) * ftcn_left_fssq_zero_equiv_left = (0) + (k) * ftcn_right_fssq_zero_equiv_left) -> (exists ftcn_left_fssq_zero_equiv_right ftcn_right_fssq_zero_equiv_right. (b) + (k) * ftcn_left_fssq_zero_equiv_right = (0) + (k) * ftcn_right_fssq_zero_equiv_right) -> (exists ftcn_left_fssq_zero_equiv_result ftcn_right_fssq_zero_equiv_result. (a) + (k) * ftcn_left_fssq_zero_equiv_result = (b) + (k) * ftcn_right_fssq_zero_equiv_result)

Proof neighborhood

Direct theorem prerequisites

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

18 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.

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 b
  4. L4
    intro ha
  5. L5
    intro hb
02Establish hreverseL6–15

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

  1. L6
    have hreverse : ModEq(k,0,b)Definitions: ModEq(k,0,b)Original native command in the exact edition
  2. L7
    specialize mod_eq_symm k
  3. L8
    specialize mod_eq_symm b
  4. L9
    specialize mod_eq_symm 0
  5. L10
    apply mod_eq_symm
  6. L11
    exact hb
  7. L12
    specialize mod_eq_trans k
  8. L13
    specialize mod_eq_trans a
  9. L14
    specialize mod_eq_trans 0
  10. L15
    specialize mod_eq_trans b
03Use earlier factsL16–18

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

  1. L16
    apply mod_eq_trans
  2. L17
    exact ha
  3. L18
    exact hreverse

Library-wide reading audit

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