ZU0006

dirichlet_signed_unit_multiply_involution

Multiplication by either actual signed unit is an involution on canonical signed codes.

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

∀ u. ∀ a. ∀ b. SignedUnit(u) → SignedMul(a,u,b) → SignedMul(b,u,a)

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

Definition DAG

SignedMul(a,b,c) · 3SignedUnit(u) · 1

Actual proof prerequisites

signed_mul_total · checked external prerequisitesigned_mul_functional · checked external prerequisitesigned_mul_associative · checked external prerequisitedirichlet_signed_unit_self_productsigned_mul_one_right · checked external prerequisite
Original expanded first-order statement 
forall u a b. (((u) = 2 \/ (u) = 1)) -> (exists sto_ap_involution_source sto_an_involution_source sto_bp_involution_source sto_bn_involution_source sto_cp_involution_source sto_cn_involution_source. (((((a) = 2 * (sto_ap_involution_source) /\ (sto_an_involution_source) = 0) \/ exists ge_signed_half_involution_sourceleft. (((a) = 2 * ge_signed_half_involution_sourceleft + 1 /\ (sto_ap_involution_source) = 0) /\ (sto_an_involution_source) = S ge_signed_half_involution_sourceleft))) /\ ((((((u) = 2 * (sto_bp_involution_source) /\ (sto_bn_involution_source) = 0) \/ exists ge_signed_half_involution_sourceright. (((u) = 2 * ge_signed_half_involution_sourceright + 1 /\ (sto_bp_involution_source) = 0) /\ (sto_bn_involution_source) = S ge_signed_half_involution_sourceright))) /\ ((((((b) = 2 * (sto_cp_involution_source) /\ (sto_cn_involution_source) = 0) \/ exists ge_signed_half_involution_sourceoutput. (((b) = 2 * ge_signed_half_involution_sourceoutput + 1 /\ (sto_cp_involution_source) = 0) /\ (sto_cn_involution_source) = S ge_signed_half_involution_sourceoutput))) /\ ((sto_ap_involution_source * sto_bp_involution_source + sto_an_involution_source * sto_bn_involution_source) + sto_cn_involution_source = (sto_ap_involution_source * sto_bn_involution_source + sto_an_involution_source * sto_bp_involution_source) + sto_cp_involution_source))))))) -> (exists sto_ap_involution_target sto_an_involution_target sto_bp_involution_target sto_bn_involution_target sto_cp_involution_target sto_cn_involution_target. (((((b) = 2 * (sto_ap_involution_target) /\ (sto_an_involution_target) = 0) \/ exists ge_signed_half_involution_targetleft. (((b) = 2 * ge_signed_half_involution_targetleft + 1 /\ (sto_ap_involution_target) = 0) /\ (sto_an_involution_target) = S ge_signed_half_involution_targetleft))) /\ ((((((u) = 2 * (sto_bp_involution_target) /\ (sto_bn_involution_target) = 0) \/ exists ge_signed_half_involution_targetright. (((u) = 2 * ge_signed_half_involution_targetright + 1 /\ (sto_bp_involution_target) = 0) /\ (sto_bn_involution_target) = S ge_signed_half_involution_targetright))) /\ ((((((a) = 2 * (sto_cp_involution_target) /\ (sto_cn_involution_target) = 0) \/ exists ge_signed_half_involution_targetoutput. (((a) = 2 * ge_signed_half_involution_targetoutput + 1 /\ (sto_cp_involution_target) = 0) /\ (sto_cn_involution_target) = S ge_signed_half_involution_targetoutput))) /\ ((sto_ap_involution_target * sto_bp_involution_target + sto_an_involution_target * sto_bn_involution_target) + sto_cn_involution_target = (sto_ap_involution_target * sto_bn_involution_target + sto_an_involution_target * sto_bp_involution_target) + sto_cp_involution_target)))))))

Complete tactic proof in conservative notation

All 33 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

33 script commands · 7 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.

Named ingredients (1)
dirichlet signed unit self product
01Fix variables and assumptionsL1–5

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

  1. L1
    intro u
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hu
  5. L5
    intro hab
02Establish hcL6–9

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

  1. L6
    have hc : ∃ c. SignedMul(b,u,c)Definitions: SignedMul(b,u,c)Original native command in the exact edition
  2. L7
    specialize signed_mul_total (b)
  3. L8
    specialize signed_mul_total (u)
  4. L9
    apply signed_mul_total
03Separate the logical casesL10–10

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

  1. L10
    cases hc
04Establish heqL11–20

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

  1. L11
    have heq : x=a
  2. L12
    specialize signed_mul_functional (a)
  3. L13
    specialize signed_mul_functional (2)
  4. L14
    specialize signed_mul_functional (x)
  5. L15
    specialize signed_mul_functional (a)
  6. L16
    apply signed_mul_functional
  7. L17
    specialize signed_mul_associative (a)
  8. L18
    specialize signed_mul_associative (u)
  9. L19
    specialize signed_mul_associative (u)
  10. L20
    specialize signed_mul_associative (b)
05Use earlier factsL21–30

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

  1. L21
    specialize signed_mul_associative (2)
  2. L22
    specialize signed_mul_associative (x)
  3. L23
    apply signed_mul_associative
  4. L24
    exact hab
  5. L25
    exact hc_witness
  6. L26
    specialize dirichlet_signed_unit_self_product (u)
  7. L27
    apply dirichlet_signed_unit_self_product
  8. L28
    exact hu
  9. L29
    specialize signed_mul_one_right (a)
  10. L30
    apply signed_mul_one_right
06Calculate and transport equalitiesL31–32

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

  1. L31
    rewrite heq at hc_witness
  2. L32
    rewrite heq at hc_witness
07Use earlier factsL33–33

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

  1. L33
    exact hc_witness

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro u
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hu
  5. 0005intro hab
  6. 0006have hc : ∃ c. SignedMul(b,u,c)
  7. 0007specialize signed_mul_total (b)
  8. 0008specialize signed_mul_total (u)
  9. 0009apply signed_mul_total
  10. 0010cases hc
  11. 0011have heq : x=a
  12. 0012specialize signed_mul_functional (a)
  13. 0013specialize signed_mul_functional (2)
  14. 0014specialize signed_mul_functional (x)
  15. 0015specialize signed_mul_functional (a)
  16. 0016apply signed_mul_functional
  17. 0017specialize signed_mul_associative (a)
  18. 0018specialize signed_mul_associative (u)
  19. 0019specialize signed_mul_associative (u)
  20. 0020specialize signed_mul_associative (b)
  21. 0021specialize signed_mul_associative (2)
  22. 0022specialize signed_mul_associative (x)
  23. 0023apply signed_mul_associative
  24. 0024exact hab
  25. 0025exact hc_witness
  26. 0026specialize dirichlet_signed_unit_self_product (u)
  27. 0027apply dirichlet_signed_unit_self_product
  28. 0028exact hu
  29. 0029specialize signed_mul_one_right (a)
  30. 0030apply signed_mul_one_right
  31. 0031rewrite heq at hc_witness
  32. 0032rewrite heq at hc_witness
  33. 0033exact hc_witness