FS0002 · theorem body

signed_balance_absolute_exists

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

Every canonical balanced signed coordinate has an explicit natural absolute magnitude with a constructive sign choice.

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

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))

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))

Proof neighborhood

Direct theorem prerequisites

signed_decode_normal · Alpha closed

Direct theorem dependents

FS0005 quaternion_coordinate_absolute_total FS000C signed_balance_square_transport FS000H four_square_absolute_difference_total

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

25 script commands · 13 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 code
  2. L2
    intro left
  3. L3
    intro right
  4. L4
    intro hbalance
02Separate the logical casesL5–7

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

  1. L5
    cases hbalance
  2. L6
    cases hbalance_witness
  3. L7
    cases hbalance_witness_witness
03Use earlier factsL8–10

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

  1. L8
    specialize signed_decode_normal code
  2. L9
    specialize signed_decode_normal x
  3. L10
    specialize signed_decode_normal x1
04Establish hnormalL11–13

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

  1. L11
    have hnormal : x = 0 \/ x1 = 0
  2. L12
    apply signed_decode_normal
  3. L13
    exact hbalance_witness_witness_left
05Separate the logical casesL14–14

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

  1. L14
    cases hnormal
06Construct an explicit witnessL15–15

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

  1. L15
    exists x1
07Separate the logical casesL16–16

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

  1. L16
    right
08Calculate and transport equalitiesL17–19

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L17
    symm
  2. L18
    rewrite hnormal_left at hbalance_witness_witness_right
  3. L19
    rewrite PA3 at hbalance_witness_witness_right
09Use earlier factsL20–20

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

  1. L20
    exact hbalance_witness_witness_right
10Construct an explicit witnessL21–21

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

  1. L21
    exists x
11Separate the logical casesL22–22

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

  1. L22
    left
12Calculate and transport equalitiesL23–24

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L23
    rewrite hnormal_right at hbalance_witness_witness_right
  2. L24
    rewrite PA3 at hbalance_witness_witness_right
13Use earlier factsL25–25

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

  1. L25
    exact hbalance_witness_witness_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro code
  2. 0002intro left
  3. 0003intro right
  4. 0004intro hbalance
  5. 0005cases hbalance
  6. 0006cases hbalance_witness
  7. 0007cases hbalance_witness_witness
  8. 0008specialize signed_decode_normal code
  9. 0009specialize signed_decode_normal x
  10. 0010specialize signed_decode_normal x1
  11. 0011have hnormal : x = 0 \/ x1 = 0
  12. 0012apply signed_decode_normal
  13. 0013exact hbalance_witness_witness_left
  14. 0014cases hnormal
  15. 0015exists x1
  16. 0016right
  17. 0017symm
  18. 0018rewrite hnormal_left at hbalance_witness_witness_right
  19. 0019rewrite PA3 at hbalance_witness_witness_right
  20. 0020exact hbalance_witness_witness_right
  21. 0021exists x
  22. 0022left
  23. 0023rewrite hnormal_right at hbalance_witness_witness_right
  24. 0024rewrite PA3 at hbalance_witness_witness_right
  25. 0025exact hbalance_witness_witness_right