FS005W · theorem body

four_square_signed_mod_zero_plus_congruent

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

A vanishing natural block can be prepended to any modular congruence.

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. ∀ c. ModEq(k,a,0) → ModEq(k,b,c) → ModEq(k,a + b,c)

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 c. (exists ftcn_left_fssq_zero_plus_first ftcn_right_fssq_zero_plus_first. (a) + (k) * ftcn_left_fssq_zero_plus_first = (0) + (k) * ftcn_right_fssq_zero_plus_first) -> (exists ftcn_left_fssq_zero_plus_second ftcn_right_fssq_zero_plus_second. (b) + (k) * ftcn_left_fssq_zero_plus_second = (c) + (k) * ftcn_right_fssq_zero_plus_second) -> (exists ftcn_left_fssq_zero_plus_result ftcn_right_fssq_zero_plus_result. (a + b) + (k) * ftcn_left_fssq_zero_plus_result = (c) + (k) * ftcn_right_fssq_zero_plus_result)

Proof neighborhood

Direct theorem prerequisites

mod_eq_add · Stable closed zero_add · Stable closed

Direct theorem dependents

FS004O four_square_signed_natural_positive_first_blocks 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

19 script commands · 3 reading checkpoints · 2 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 assumptionsL1–6

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 c
  5. L5
    intro ha
  6. L6
    intro hb
02Establish hsumL7–15

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

  1. L7
    have hsum : ModEq(k,a + b,0 + c)Definitions: ModEq(k,a + b,0 + c)Original native command in the exact edition
  2. L8
    specialize mod_eq_add k
  3. L9
    specialize mod_eq_add a
  4. L10
    specialize mod_eq_add 0
  5. L11
    specialize mod_eq_add b
  6. L12
    specialize mod_eq_add c
  7. L13
    apply mod_eq_add
  8. L14
    exact ha
  9. L15
    exact hb
03Establish hzeroL16–19

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

  1. L16
    have hzero : 0 + c = c
  2. L17
    apply zero_add
  3. L18
    rewrite hzero at hsum
  4. L19
    exact hsum

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro ha
  6. 0006intro hb
  7. 0007have hsum : ModEq(k,a + b,0 + c)
    Exact native replay linehave hsum : exists ftcn_left_fssq_zero_plus_sum ftcn_right_fssq_zero_plus_sum. (a + b) + (k) * ftcn_left_fssq_zero_plus_sum = (0 + c) + (k) * ftcn_right_fssq_zero_plus_sum
  8. 0008specialize mod_eq_add k
  9. 0009specialize mod_eq_add a
  10. 0010specialize mod_eq_add 0
  11. 0011specialize mod_eq_add b
  12. 0012specialize mod_eq_add c
  13. 0013apply mod_eq_add
  14. 0014exact ha
  15. 0015exact hb
  16. 0016have hzero : 0 + c = c
  17. 0017apply zero_add
  18. 0018rewrite hzero at hsum
  19. 0019exact hsum