FS005S · theorem body

four_square_signed_zero_cancel_right

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

Cancel a separately vanishing natural tail from a subtraction-free modular zero balance.

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,0) → ModEq(k,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

PD0008 ModEq
Exact expanded first-order statement
forall k a b. (exists ftcn_left_fssq_zero_cancel_sum ftcn_right_fssq_zero_cancel_sum. (a + b) + (k) * ftcn_left_fssq_zero_cancel_sum = (0) + (k) * ftcn_right_fssq_zero_cancel_sum) -> (exists ftcn_left_fssq_zero_cancel_tail ftcn_right_fssq_zero_cancel_tail. (b) + (k) * ftcn_left_fssq_zero_cancel_tail = (0) + (k) * ftcn_right_fssq_zero_cancel_tail) -> (exists ftcn_left_fssq_zero_cancel_result ftcn_right_fssq_zero_cancel_result. (a) + (k) * ftcn_left_fssq_zero_cancel_result = (0) + (k) * ftcn_right_fssq_zero_cancel_result)

Proof neighborhood

Direct theorem prerequisites

zero_add · Stable closed FS005L four_square_signed_common_zero_cancel

Direct theorem dependents

FS004N four_square_signed_conjugate_negative_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

17 script commands · 4 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.

Named ingredients (1)
four square signed common zero cancel
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 hsum
  5. L5
    intro htail
02Establish hpaddedL6–6

Establish this local claim before using it. It is not an additional assumption.

  1. L6
    have hpadded : ModEq(k,0 + b,0)Definitions: ModEq(k,0 + b,0)Original native command in the exact edition
03Establish hpadL7–16

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

  1. L7
    have hpad : 0 + b = b
  2. L8
    apply zero_add
  3. L9
    rewrite hpad
  4. L10
    exact htail
  5. L11
    specialize four_square_signed_common_zero_cancel k
  6. L12
    specialize four_square_signed_common_zero_cancel a
  7. L13
    specialize four_square_signed_common_zero_cancel 0
  8. L14
    specialize four_square_signed_common_zero_cancel b
  9. L15
    apply four_square_signed_common_zero_cancel
  10. L16
    exact hsum
04Use earlier factsL17–17

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

  1. L17
    exact hpadded

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hsum
  5. 0005intro htail
  6. 0006have hpadded : ModEq(k,0 + b,0)
    Exact native replay linehave hpadded : exists ftcn_left_fssq_zero_cancel_padded ftcn_right_fssq_zero_cancel_padded. (0 + b) + (k) * ftcn_left_fssq_zero_cancel_padded = (0) + (k) * ftcn_right_fssq_zero_cancel_padded
  7. 0007have hpad : 0 + b = b
  8. 0008apply zero_add
  9. 0009rewrite hpad
  10. 0010exact htail
  11. 0011specialize four_square_signed_common_zero_cancel k
  12. 0012specialize four_square_signed_common_zero_cancel a
  13. 0013specialize four_square_signed_common_zero_cancel 0
  14. 0014specialize four_square_signed_common_zero_cancel b
  15. 0015apply four_square_signed_common_zero_cancel
  16. 0016exact hsum
  17. 0017exact hpadded