EU000D

euler_multiplier_permutation_exists

Every coprime multiplier at every positive modulus constructs a genuine canonical finite permutation, including modulus one.

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.

The exact G014 theorem covers m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem handles m=1 by congruence, not by asserting that one is a canonical remainder. Multiplicative-order and RSA statements are not claimed.

Exact theorem in conservative defined notation

∀ a. ∀ m. ¬m = 0 → Coprime(a,m) → ∃ x. ∃ y. UnitMultiplierPrefix(a,m,x,y,m) ∧ PermutationPrefix(x,y,m)

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

Definition DAG

PermutationPrefix(b,c,l) · 1UnitMultiplierPrefix(a,m,b,c,l) · 2Coprime(a,b) · 1

Actual proof prerequisites

euler_multiplier_prefix_existseuler_multiplier_prefix_permutation
Original expanded first-order statement 
forall a m. ~(m=0) -> (forall eut_divisor_eu_exists_unit. (exists eut_left_eu_exists_unit. (a) = eut_divisor_eu_exists_unit * eut_left_eu_exists_unit) -> (exists eut_right_eu_exists_unit. (m) = eut_divisor_eu_exists_unit * eut_right_eu_exists_unit) -> eut_divisor_eu_exists_unit = 1) -> exists b c. (forall eu_index_exists_map. (exists eut_gap_eu_exists_map_index. eut_gap_eu_exists_map_index + S (eu_index_exists_map) = (m)) -> exists eu_residue_exists_map. (((exists fs_h_eu_exists_map_at. fs_h_eu_exists_map_at + S (eu_residue_exists_map) = S ((S (eu_index_exists_map)) * c)) /\ exists fs_q_eu_exists_map_at. b = fs_q_eu_exists_map_at * S ((S (eu_index_exists_map)) * c) + (eu_residue_exists_map))) /\ ((exists eut_gap_eu_exists_map_bound. eut_gap_eu_exists_map_bound + S (eu_residue_exists_map) = (m)) /\ (exists eu_mod_left_exists_map_mod eu_mod_right_exists_map_mod. ((a)*eu_index_exists_map) + (m) * eu_mod_left_exists_map_mod = (eu_residue_exists_map) + (m) * eu_mod_right_exists_map_mod))) /\ (((forall fp_i_eu_exists_permutation_bounded. (exists fp_gap_eu_exists_permutation_bounded_index. fp_gap_eu_exists_permutation_bounded_index + S fp_i_eu_exists_permutation_bounded = m) -> exists fp_value_eu_exists_permutation_bounded. ((((exists ff_h_eu_exists_permutation_bounded_entry. ff_h_eu_exists_permutation_bounded_entry + S (fp_value_eu_exists_permutation_bounded) = S ((S (fp_i_eu_exists_permutation_bounded)) * c)) /\ exists ff_q_eu_exists_permutation_bounded_entry. b = ff_q_eu_exists_permutation_bounded_entry * S ((S (fp_i_eu_exists_permutation_bounded)) * c) + (fp_value_eu_exists_permutation_bounded))) /\ (exists fp_gap_eu_exists_permutation_bounded_value. fp_gap_eu_exists_permutation_bounded_value + S fp_value_eu_exists_permutation_bounded = m))) /\ ((forall fp_i_eu_exists_permutation_injective fp_j_eu_exists_permutation_injective fp_value_eu_exists_permutation_injective. (exists fp_gap_eu_exists_permutation_injective_i. fp_gap_eu_exists_permutation_injective_i + S fp_i_eu_exists_permutation_injective = m) -> (exists fp_gap_eu_exists_permutation_injective_j. fp_gap_eu_exists_permutation_injective_j + S fp_j_eu_exists_permutation_injective = m) -> (((exists ff_h_eu_exists_permutation_injective_left. ff_h_eu_exists_permutation_injective_left + S (fp_value_eu_exists_permutation_injective) = S ((S (fp_i_eu_exists_permutation_injective)) * c)) /\ exists ff_q_eu_exists_permutation_injective_left. b = ff_q_eu_exists_permutation_injective_left * S ((S (fp_i_eu_exists_permutation_injective)) * c) + (fp_value_eu_exists_permutation_injective))) -> (((exists ff_h_eu_exists_permutation_injective_right. ff_h_eu_exists_permutation_injective_right + S (fp_value_eu_exists_permutation_injective) = S ((S (fp_j_eu_exists_permutation_injective)) * c)) /\ exists ff_q_eu_exists_permutation_injective_right. b = ff_q_eu_exists_permutation_injective_right * S ((S (fp_j_eu_exists_permutation_injective)) * c) + (fp_value_eu_exists_permutation_injective))) -> fp_i_eu_exists_permutation_injective = fp_j_eu_exists_permutation_injective) /\ (forall fp_value_eu_exists_permutation_surjective. (exists fp_gap_eu_exists_permutation_surjective_value. fp_gap_eu_exists_permutation_surjective_value + S fp_value_eu_exists_permutation_surjective = m) -> exists fp_i_eu_exists_permutation_surjective. ((exists fp_gap_eu_exists_permutation_surjective_index. fp_gap_eu_exists_permutation_surjective_index + S fp_i_eu_exists_permutation_surjective = m) /\ (((exists ff_h_eu_exists_permutation_surjective_entry. ff_h_eu_exists_permutation_surjective_entry + S (fp_value_eu_exists_permutation_surjective) = S ((S (fp_i_eu_exists_permutation_surjective)) * c)) /\ exists ff_q_eu_exists_permutation_surjective_entry. b = ff_q_eu_exists_permutation_surjective_entry * S ((S (fp_i_eu_exists_permutation_surjective)) * c) + (fp_value_eu_exists_permutation_surjective))))))))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 6 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.

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 (2)
euler multiplier prefix existseuler multiplier prefix permutation
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 hm
  4. L4
    intro hc
02Establish hL5–10

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

  1. L5
    have h : ∃ b. ∃ c. UnitMultiplierPrefix(a,m,b,c,m)Definitions: UnitMultiplierPrefix(a,m,b,c,m)Original native command in the exact edition
  2. L6
    specialize euler_multiplier_prefix_exists (a)
  3. L7
    specialize euler_multiplier_prefix_exists (m)
  4. L8
    specialize euler_multiplier_prefix_exists (m)
  5. L9
    apply euler_multiplier_prefix_exists
  6. L10
    exact hm
03Separate the logical casesL11–12

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

  1. L11
    cases h
  2. L12
    cases h_witness
04Construct an explicit witnessL13–14

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

  1. L13
    exists x
  2. L14
    exists x1
05Separate the logical casesL15–15

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

  1. L15
    split
06Use earlier factsL16–24

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

  1. L16
    exact h_witness_witness
  2. L17
    specialize euler_multiplier_prefix_permutation (a)
  3. L18
    specialize euler_multiplier_prefix_permutation (m)
  4. L19
    specialize euler_multiplier_prefix_permutation (x)
  5. L20
    specialize euler_multiplier_prefix_permutation (x1)
  6. L21
    apply euler_multiplier_prefix_permutation
  7. L22
    exact hm
  8. L23
    exact hc
  9. L24
    exact h_witness_witness

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro hm
  4. 0004intro hc
  5. 0005have h : ∃ b. ∃ c. UnitMultiplierPrefix(a,m,b,c,m)
  6. 0006specialize euler_multiplier_prefix_exists (a)
  7. 0007specialize euler_multiplier_prefix_exists (m)
  8. 0008specialize euler_multiplier_prefix_exists (m)
  9. 0009apply euler_multiplier_prefix_exists
  10. 0010exact hm
  11. 0011cases h
  12. 0012cases h_witness
  13. 0013exists x
  14. 0014exists x1
  15. 0015split
  16. 0016exact h_witness_witness
  17. 0017specialize euler_multiplier_prefix_permutation (a)
  18. 0018specialize euler_multiplier_prefix_permutation (m)
  19. 0019specialize euler_multiplier_prefix_permutation (x)
  20. 0020specialize euler_multiplier_prefix_permutation (x1)
  21. 0021apply euler_multiplier_prefix_permutation
  22. 0022exact hm
  23. 0023exact hc
  24. 0024exact h_witness_witness