ZU0006

dirichlet_signed_unit_multiply_involution

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

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

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 expanded first-order arithmetic 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)))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 5 declared prerequisites and contains 33 exact native proof lines.

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

Proof neighborhood

Direct dependencies

signed_mul_total Alpha theorem; checked-use authorized signed_mul_functional Alpha theorem; checked-use authorized signed_mul_associative Alpha theorem; checked-use authorized ZU0001 dirichlet_signed_unit_self_product signed_mul_one_right Alpha theorem; checked-use authorized

Direct dependents

ZU0007 dirichlet_signed_unit_multiply_cancel_right ZU0008 dirichlet_signed_unit_affine_solve

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

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.

Named ingredients (1)
dirichlet signed unit self product

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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
  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 exact command ledger · 33 lines
  1. 0001intro u
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hu
  5. 0005intro hab
  6. 0006have hc : exists c. (exists sto_ap_involution_construct sto_an_involution_construct sto_bp_involution_construct sto_bn_involution_construct sto_cp_involution_construct sto_cn_involution_construct. (((((b) = 2 * (sto_ap_involution_construct) /\ (sto_an_involution_construct) = 0) \/ exists ge_signed_half_involution_constructleft. (((b) = 2 * ge_signed_half_involution_constructleft + 1 /\ (sto_ap_involution_construct) = 0) /\ (sto_an_involution_construct) = S ge_signed_half_involution_constructleft))) /\ ((((((u) = 2 * (sto_bp_involution_construct) /\ (sto_bn_involution_construct) = 0) \/ exists ge_signed_half_involution_constructright. (((u) = 2 * ge_signed_half_involution_constructright + 1 /\ (sto_bp_involution_construct) = 0) /\ (sto_bn_involution_construct) = S ge_signed_half_involution_constructright))) /\ ((((((c) = 2 * (sto_cp_involution_construct) /\ (sto_cn_involution_construct) = 0) \/ exists ge_signed_half_involution_constructoutput. (((c) = 2 * ge_signed_half_involution_constructoutput + 1 /\ (sto_cp_involution_construct) = 0) /\ (sto_cn_involution_construct) = S ge_signed_half_involution_constructoutput))) /\ ((sto_ap_involution_construct * sto_bp_involution_construct + sto_an_involution_construct * sto_bn_involution_construct) + sto_cn_involution_construct = (sto_ap_involution_construct * sto_bn_involution_construct + sto_an_involution_construct * sto_bp_involution_construct) + sto_cp_involution_construct)))))))
  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