FS005R · theorem body

four_square_signed_mod_zero_swap

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

Swapping the two summands of a modular zero balance preserves that 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. ∀ b. ModEq(k,a + b,0) → ModEq(k,b + a,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

none
Exact expanded first-order statement
forall k a b. (exists ftcn_left_fssq_zero_swap_source ftcn_right_fssq_zero_swap_source. (a + b) + (k) * ftcn_left_fssq_zero_swap_source = (0) + (k) * ftcn_right_fssq_zero_swap_source) -> (exists ftcn_left_fssq_zero_swap_result ftcn_right_fssq_zero_swap_result. (b + a) + (k) * ftcn_left_fssq_zero_swap_result = (0) + (k) * ftcn_right_fssq_zero_swap_result)

Proof neighborhood

Direct theorem prerequisites

add_comm · Stable closed

Direct theorem dependents

FS005T four_square_signed_cross_mixed_zero_reversed

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

8 script commands · 2 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 assumptionsL1–4

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 hsum
02Establish hswapL5–8

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

  1. L5
    have hswap : a + b = b + a
  2. L6
    apply add_comm
  3. L7
    rewrite hswap at hsum
  4. L8
    exact hsum

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hsum
  5. 0005have hswap : a + b = b + a
  6. 0006apply add_comm
  7. 0007rewrite hswap at hsum
  8. 0008exact hsum