ZU0005

dirichlet_signed_add_solve

Construct the signed addend taking any canonical signed r to any canonical signed e, including zero and negative values.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Canonical signed +1 is code 2 and -1 is code 1. The two-case unit graph does not assume an inverse or cancellation law: its actual product characterization and affine existence and uniqueness are proved. These scalar lemmas support the separately checked finite inverse criterion; full G009 remains broader.

Exact theorem in conservative defined notation

∀ r. ∀ e. ∃ y. SignedAdd(r,y,e)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

SignedDecode(z,p,n) · 2SignedAdd(a,b,c) · 2

Actual proof prerequisites

signed_negate_total · checked external prerequisitesigned_add_total · checked external prerequisitesigned_add_associative · checked external prerequisitesigned_add_negate_right_zero · checked external prerequisitesigned_add_zero_left · checked external prerequisite
Original expanded first-order statement 
forall r e. exists y. (exists dsa_ap_add_solve dsa_an_add_solve dsa_bp_add_solve dsa_bn_add_solve dsa_cp_add_solve dsa_cn_add_solve. (((((r) = 2 * (dsa_ap_add_solve) /\ (dsa_an_add_solve) = 0) \/ exists ge_signed_half_add_solveleft. (((r) = 2 * ge_signed_half_add_solveleft + 1 /\ (dsa_ap_add_solve) = 0) /\ (dsa_an_add_solve) = S ge_signed_half_add_solveleft))) /\ ((((((y) = 2 * (dsa_bp_add_solve) /\ (dsa_bn_add_solve) = 0) \/ exists ge_signed_half_add_solveright. (((y) = 2 * ge_signed_half_add_solveright + 1 /\ (dsa_bp_add_solve) = 0) /\ (dsa_bn_add_solve) = S ge_signed_half_add_solveright))) /\ ((((((e) = 2 * (dsa_cp_add_solve) /\ (dsa_cn_add_solve) = 0) \/ exists ge_signed_half_add_solveoutput. (((e) = 2 * ge_signed_half_add_solveoutput + 1 /\ (dsa_cp_add_solve) = 0) /\ (dsa_cn_add_solve) = S ge_signed_half_add_solveoutput))) /\ ((dsa_ap_add_solve + dsa_bp_add_solve) + dsa_cn_add_solve = (dsa_an_add_solve + dsa_bn_add_solve) + dsa_cp_add_solve)))))))

Complete tactic proof in conservative notation

All 26 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

26 script commands · 8 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro r
  2. L2
    intro e
02Establish hnL3–5

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

  1. L3
    have hn : ∃ n. ∃ x. ∃ y. SignedDecode(r,x,y) ∧ SignedDecode(n,y,x)Definitions: SignedDecode(r,x,y)SignedDecode(n,y,x)Original native command in the exact edition
  2. L4
    specialize signed_negate_total (r)
  3. L5
    apply signed_negate_total
03Separate the logical casesL6–6

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

  1. L6
    cases hn
04Establish hyL7–10

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

  1. L7
    have hy : ∃ y. SignedAdd(x,e,y)Definitions: SignedAdd(x,e,y)Original native command in the exact edition
  2. L8
    specialize signed_add_total (x)
  3. L9
    specialize signed_add_total (e)
  4. L10
    apply signed_add_total
05Separate the logical casesL11–11

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

  1. L11
    cases hy
06Construct an explicit witnessL12–12

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

  1. L12
    exists x1
07Use earlier factsL13–22

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

  1. L13
    specialize signed_add_associative (r)
  2. L14
    specialize signed_add_associative (x)
  3. L15
    specialize signed_add_associative (e)
  4. L16
    specialize signed_add_associative (0)
  5. L17
    specialize signed_add_associative (x1)
  6. L18
    specialize signed_add_associative (e)
  7. L19
    apply signed_add_associative
  8. L20
    specialize signed_add_negate_right_zero (r)
  9. L21
    specialize signed_add_negate_right_zero (x)
  10. L22
    apply signed_add_negate_right_zero
08Use earlier factsL23–26

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

  1. L23
    exact hn_witness
  2. L24
    specialize signed_add_zero_left (e)
  3. L25
    apply signed_add_zero_left
  4. L26
    exact hy_witness

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro r
  2. 0002intro e
  3. 0003have hn : ∃ n. ∃ x. ∃ y. SignedDecode(r,x,y) ∧ SignedDecode(n,y,x)
  4. 0004specialize signed_negate_total (r)
  5. 0005apply signed_negate_total
  6. 0006cases hn
  7. 0007have hy : ∃ y. SignedAdd(x,e,y)
  8. 0008specialize signed_add_total (x)
  9. 0009specialize signed_add_total (e)
  10. 0010apply signed_add_total
  11. 0011cases hy
  12. 0012exists x1
  13. 0013specialize signed_add_associative (r)
  14. 0014specialize signed_add_associative (x)
  15. 0015specialize signed_add_associative (e)
  16. 0016specialize signed_add_associative (0)
  17. 0017specialize signed_add_associative (x1)
  18. 0018specialize signed_add_associative (e)
  19. 0019apply signed_add_associative
  20. 0020specialize signed_add_negate_right_zero (r)
  21. 0021specialize signed_add_negate_right_zero (x)
  22. 0022apply signed_add_negate_right_zero
  23. 0023exact hn_witness
  24. 0024specialize signed_add_zero_left (e)
  25. 0025apply signed_add_zero_left
  26. 0026exact hy_witness