FS005S

four_square_signed_zero_cancel_right

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 17 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

zero_add Stable theorem; checked-use authorized FS005L four_square_signed_common_zero_cancel

Direct dependents

FS004N four_square_signed_conjugate_negative_blocks

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 : 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
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 exact command ledger · 17 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hsum
  5. 0005intro htail
  6. 0006have 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