GI003B

gaussian_signed_balance_integer_transport

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The unchanged canonical signed code continues to represent every equal signed difference.

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 code p n q m. (((p) + (m)) = ((q) + (n))) -> (exists ge_balance_positive_balance_source ge_balance_negative_balance_source. (((((code) = 2 * (ge_balance_positive_balance_source) /\ (ge_balance_negative_balance_source) = 0) \/ exists ge_signed_half_balance_sourcedecode. (((code) = 2 * ge_signed_half_balance_sourcedecode + 1 /\ (ge_balance_positive_balance_source) = 0) /\ (ge_balance_negative_balance_source) = S ge_signed_half_balance_sourcedecode))) /\ ((p) + ge_balance_negative_balance_source = (n) + ge_balance_positive_balance_source))) -> (exists ge_balance_positive_balance_target ge_balance_negative_balance_target. (((((code) = 2 * (ge_balance_positive_balance_target) /\ (ge_balance_negative_balance_target) = 0) \/ exists ge_signed_half_balance_targetdecode. (((code) = 2 * ge_signed_half_balance_targetdecode + 1 /\ (ge_balance_positive_balance_target) = 0) /\ (ge_balance_negative_balance_target) = S ge_signed_half_balance_targetdecode))) /\ ((q) + ge_balance_negative_balance_target = (m) + ge_balance_positive_balance_target)))

Constructive proof overview

Generated structural guide

The unchanged canonical signed code continues to represent every equal signed difference.

The unchanged tactic script uses 2 declared prerequisites and contains 30 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

integer_span_pair_equal_transitive Alpha theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

GI0041 gaussian_representation_integer_transport GI0049 gaussian_signed_add_of_balances GI004B gaussian_signed_mul_of_balances

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

30 script commands · 10 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro code
  2. L2
    intro p
  3. L3
    intro n
  4. L4
    intro q
  5. L5
    intro m
  6. L6
    intro hequal
  7. L7
    intro hbalance
02Separate the logical casesL8–10

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

  1. L8
    cases hbalance
  2. L9
    cases hbalance_witness
  3. L10
    cases hbalance_witness_witness
03Establish htransportL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply integer span pair equal transitive.

  1. L11
    have htransport : q + x1 = x + m
  2. L12
    specialize integer_span_pair_equal_transitive q
  3. L13
    specialize integer_span_pair_equal_transitive m
  4. L14
    specialize integer_span_pair_equal_transitive p
  5. L15
    specialize integer_span_pair_equal_transitive n
  6. L16
    specialize integer_span_pair_equal_transitive x
  7. L17
    specialize integer_span_pair_equal_transitive x1
  8. L18
    apply integer_span_pair_equal_transitive
  9. L19
    symm
  10. L20
    exact hequal
04Calculate and transport equalitiesL21–21

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

  1. L21
    trans n + x
05Use earlier factsL22–23

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

  1. L22
    exact hbalance_witness_witness_right
  2. L23
    apply add_comm
06Construct an explicit witnessL24–25

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

  1. L24
    exists x
  2. L25
    exists x1
07Separate the logical casesL26–26

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

  1. L26
    split
08Use earlier factsL27–27

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

  1. L27
    exact hbalance_witness_witness_left
09Calculate and transport equalitiesL28–28

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

  1. L28
    trans x + m
10Use earlier factsL29–30

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

  1. L29
    exact htransport
  2. L30
    apply add_comm

Library-wide reading audit

Original exact command ledger · 30 lines
  1. 0001intro code
  2. 0002intro p
  3. 0003intro n
  4. 0004intro q
  5. 0005intro m
  6. 0006intro hequal
  7. 0007intro hbalance
  8. 0008cases hbalance
  9. 0009cases hbalance_witness
  10. 0010cases hbalance_witness_witness
  11. 0011have htransport : q + x1 = x + m
  12. 0012specialize integer_span_pair_equal_transitive q
  13. 0013specialize integer_span_pair_equal_transitive m
  14. 0014specialize integer_span_pair_equal_transitive p
  15. 0015specialize integer_span_pair_equal_transitive n
  16. 0016specialize integer_span_pair_equal_transitive x
  17. 0017specialize integer_span_pair_equal_transitive x1
  18. 0018apply integer_span_pair_equal_transitive
  19. 0019symm
  20. 0020exact hequal
  21. 0021trans n + x
  22. 0022exact hbalance_witness_witness_right
  23. 0023apply add_comm
  24. 0024exists x
  25. 0025exists x1
  26. 0026split
  27. 0027exact hbalance_witness_witness_left
  28. 0028trans x + m
  29. 0029exact htransport
  30. 0030apply add_comm