FS000C · theorem body

signed_balance_square_transport

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

Every canonical signed balance supplies a natural magnitude, constructive sign branch, and exact squared cross-term transport.

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

forall code left right. (exists sb_pos_fs_abs sb_neg_fs_abs. (((code = 2 * sb_pos_fs_abs /\ sb_neg_fs_abs = 0) \/ exists sd_half_fs_abs. ((code = 2 * sd_half_fs_abs + 1 /\ sb_pos_fs_abs = 0) /\ sb_neg_fs_abs = S sd_half_fs_abs)) /\ left + sb_neg_fs_abs = right + sb_pos_fs_abs)) -> exists magnitude. ((left = right + magnitude \/ right = left + magnitude) /\ left * left + right * right = magnitude * magnitude + (left * right + right * left))

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

none

In local proof propositions

none
Exact expanded first-order statement
forall code left right. (exists sb_pos_fs_abs sb_neg_fs_abs. (((code = 2 * sb_pos_fs_abs /\ sb_neg_fs_abs = 0) \/ exists sd_half_fs_abs. ((code = 2 * sd_half_fs_abs + 1 /\ sb_pos_fs_abs = 0) /\ sb_neg_fs_abs = S sd_half_fs_abs)) /\ left + sb_neg_fs_abs = right + sb_pos_fs_abs)) -> exists magnitude. ((left = right + magnitude \/ right = left + magnitude) /\ left * left + right * right = magnitude * magnitude + (left * right + right * left))

Proof neighborhood

Direct theorem prerequisites

FS0002 signed_balance_absolute_exists FS000B four_square_absolute_square_balance

Direct theorem dependents

none

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 · 6 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.

Named ingredients (2)
signed balance absolute existsfour square absolute square balance
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro code
  2. L2
    intro left
  3. L3
    intro right
  4. L4
    intro hbalance
02Establish habsoluteL5–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed balance absolute exists.

  1. L5
    have habsolute : exists magnitude. (left = right + magnitude \/ right = left + magnitude)
  2. L6
    specialize signed_balance_absolute_exists code
  3. L7
    specialize signed_balance_absolute_exists left
  4. L8
    specialize signed_balance_absolute_exists right
  5. L9
    apply signed_balance_absolute_exists
  6. L10
    exact hbalance
03Separate the logical casesL11–11

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases habsolute
04Construct an explicit witnessL12–12

Supply the displayed value, then prove that it has the required property.

  1. L12
    exists x
05Separate the logical casesL13–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L13
    split
06Use earlier factsL14–19

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

  1. L14
    exact habsolute_witness
  2. L15
    specialize four_square_absolute_square_balance left
  3. L16
    specialize four_square_absolute_square_balance right
  4. L17
    specialize four_square_absolute_square_balance x
  5. L18
    apply four_square_absolute_square_balance
  6. L19
    exact habsolute_witness

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro code
  2. 0002intro left
  3. 0003intro right
  4. 0004intro hbalance
  5. 0005have habsolute : exists magnitude. (left = right + magnitude \/ right = left + magnitude)
  6. 0006specialize signed_balance_absolute_exists code
  7. 0007specialize signed_balance_absolute_exists left
  8. 0008specialize signed_balance_absolute_exists right
  9. 0009apply signed_balance_absolute_exists
  10. 0010exact hbalance
  11. 0011cases habsolute
  12. 0012exists x
  13. 0013split
  14. 0014exact habsolute_witness
  15. 0015specialize four_square_absolute_square_balance left
  16. 0016specialize four_square_absolute_square_balance right
  17. 0017specialize four_square_absolute_square_balance x
  18. 0018apply four_square_absolute_square_balance
  19. 0019exact habsolute_witness