EU0006

euler_multiplier_residue_exists

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Actual Euclidean division constructs a canonical residue of every multiplied index, including index zero.

Exact expanded first-order arithmetic statement

forall a m i. ~(m=0) -> exists r. (exists eut_gap_eu_residue_bound. eut_gap_eu_residue_bound + S (r) = (m)) /\ (exists eu_mod_left_residue_congruence eu_mod_right_residue_congruence. (a*i) + (m) * eu_mod_left_residue_congruence = (r) + (m) * eu_mod_right_residue_congruence)

Constructive proof overview

Generated structural guide

Actual Euclidean division constructs a canonical residue of every multiplied index, including index zero.

The unchanged tactic script uses 3 declared prerequisites and contains 27 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

division_remainder_exists Alpha theorem; checked-use authorized remainder_decomposition_to_mod_eq Alpha theorem; checked-use authorized mul_comm Alpha theorem; checked-use authorized

Direct dependents

EU0009 euler_multiplier_prefix_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

Read the argument

Proof checkpoints

27 script commands · 12 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro i
  4. L4
    intro hm
02Use earlier factsL5–6

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

  1. L5
    specialize division_remainder_exists m
  2. L6
    specialize division_remainder_exists (a*i)
03Establish hdL7–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder exists.

  1. L7
    have hd : exists q r. a*i=m*q+r /\ (exists eut_gap_eu_actual_division. eut_gap_eu_actual_division + S (r) = (m))
  2. L8
    apply division_remainder_exists
  3. L9
    exact hm
04Separate the logical casesL10–12

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

  1. L10
    cases hd
  2. L11
    cases hd_witness
  3. L12
    cases hd_witness_witness
05Construct an explicit witnessL13–13

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

  1. L13
    exists x1
06Separate the logical casesL14–14

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

  1. L14
    split
07Use earlier factsL15–20

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

  1. L15
    exact hd_witness_witness_right
  2. L16
    specialize remainder_decomposition_to_mod_eq (m)
  3. L17
    specialize remainder_decomposition_to_mod_eq (a*i)
  4. L18
    specialize remainder_decomposition_to_mod_eq (x)
  5. L19
    specialize remainder_decomposition_to_mod_eq (x1)
  6. L20
    apply remainder_decomposition_to_mod_eq
08Calculate and transport equalitiesL21–21

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

  1. L21
    trans m*x+x1
09Use earlier factsL22–22

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

  1. L22
    exact hd_witness_witness_left
10Calculate and transport equalitiesL23–23

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

  1. L23
    congr
11Use earlier factsL24–26

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

  1. L24
    specialize mul_comm (m)
  2. L25
    specialize mul_comm (x)
  3. L26
    apply mul_comm
12Calculate and transport equalitiesL27–27

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

  1. L27
    refl

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro i
  4. 0004intro hm
  5. 0005specialize division_remainder_exists m
  6. 0006specialize division_remainder_exists (a*i)
  7. 0007have hd : exists q r. a*i=m*q+r /\ (exists eut_gap_eu_actual_division. eut_gap_eu_actual_division + S (r) = (m))
  8. 0008apply division_remainder_exists
  9. 0009exact hm
  10. 0010cases hd
  11. 0011cases hd_witness
  12. 0012cases hd_witness_witness
  13. 0013exists x1
  14. 0014split
  15. 0015exact hd_witness_witness_right
  16. 0016specialize remainder_decomposition_to_mod_eq (m)
  17. 0017specialize remainder_decomposition_to_mod_eq (a*i)
  18. 0018specialize remainder_decomposition_to_mod_eq (x)
  19. 0019specialize remainder_decomposition_to_mod_eq (x1)
  20. 0020apply remainder_decomposition_to_mod_eq
  21. 0021trans m*x+x1
  22. 0022exact hd_witness_witness_left
  23. 0023congr
  24. 0024specialize mul_comm (m)
  25. 0025specialize mul_comm (x)
  26. 0026apply mul_comm
  27. 0027refl