EU001A

euler_unit_scaled_prefix_drop_last

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.

Restrict the independently specified unit-scaled action to its predecessor prefix.

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

The exact G014 theorem is proved in this research checkpoint for m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem also handles m=1 by congruence, not by asserting that one is a canonical remainder. No multiplicative-order or RSA theorem is claimed. The published atlas and Alpha membership are unchanged.

Exact theorem in conservative defined notation

∀ a. ∀ m. ∀ b. ∀ c. ∀ d. ∀ e. ∀ l. UnitScaledPrefix(a,m,b,c,d,e,S l) → UnitScaledPrefix(a,m,b,c,d,e,l)

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

Definition DAG

UnitScaledPrefix(a,m,b,c,d,e,l) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall a m b c d e l. (forall eu_scale_index_scale_drop_old eu_scale_source_scale_drop_old eu_scale_target_scale_drop_old. (exists eut_gap_eu_scale_drop_old_index. eut_gap_eu_scale_drop_old_index + S (eu_scale_index_scale_drop_old) = (S l)) -> (((exists fs_h_eu_scale_drop_old_source. fs_h_eu_scale_drop_old_source + S (eu_scale_source_scale_drop_old) = S ((S (eu_scale_index_scale_drop_old)) * c)) /\ exists fs_q_eu_scale_drop_old_source. b = fs_q_eu_scale_drop_old_source * S ((S (eu_scale_index_scale_drop_old)) * c) + (eu_scale_source_scale_drop_old))) -> (((exists fs_h_eu_scale_drop_old_target. fs_h_eu_scale_drop_old_target + S (eu_scale_target_scale_drop_old) = S ((S (eu_scale_index_scale_drop_old)) * e)) /\ exists fs_q_eu_scale_drop_old_target. d = fs_q_eu_scale_drop_old_target * S ((S (eu_scale_index_scale_drop_old)) * e) + (eu_scale_target_scale_drop_old))) -> (((forall eut_divisor_eu_scale_drop_old_unit. (exists eut_left_eu_scale_drop_old_unit. (eu_scale_index_scale_drop_old) = eut_divisor_eu_scale_drop_old_unit * eut_left_eu_scale_drop_old_unit) -> (exists eut_right_eu_scale_drop_old_unit. (m) = eut_divisor_eu_scale_drop_old_unit * eut_right_eu_scale_drop_old_unit) -> eut_divisor_eu_scale_drop_old_unit = 1) -> (exists eu_mod_left_scale_drop_old_scaled eu_mod_right_scale_drop_old_scaled. ((a)*eu_scale_source_scale_drop_old) + (m) * eu_mod_left_scale_drop_old_scaled = (eu_scale_target_scale_drop_old) + (m) * eu_mod_right_scale_drop_old_scaled)) /\ (~(forall eut_divisor_eu_scale_drop_old_unit. (exists eut_left_eu_scale_drop_old_unit. (eu_scale_index_scale_drop_old) = eut_divisor_eu_scale_drop_old_unit * eut_left_eu_scale_drop_old_unit) -> (exists eut_right_eu_scale_drop_old_unit. (m) = eut_divisor_eu_scale_drop_old_unit * eut_right_eu_scale_drop_old_unit) -> eut_divisor_eu_scale_drop_old_unit = 1) -> (exists eu_mod_left_scale_drop_old_unchanged eu_mod_right_scale_drop_old_unchanged. (eu_scale_source_scale_drop_old) + (m) * eu_mod_left_scale_drop_old_unchanged = (eu_scale_target_scale_drop_old) + (m) * eu_mod_right_scale_drop_old_unchanged)))) -> (forall eu_scale_index_scale_drop_new eu_scale_source_scale_drop_new eu_scale_target_scale_drop_new. (exists eut_gap_eu_scale_drop_new_index. eut_gap_eu_scale_drop_new_index + S (eu_scale_index_scale_drop_new) = (l)) -> (((exists fs_h_eu_scale_drop_new_source. fs_h_eu_scale_drop_new_source + S (eu_scale_source_scale_drop_new) = S ((S (eu_scale_index_scale_drop_new)) * c)) /\ exists fs_q_eu_scale_drop_new_source. b = fs_q_eu_scale_drop_new_source * S ((S (eu_scale_index_scale_drop_new)) * c) + (eu_scale_source_scale_drop_new))) -> (((exists fs_h_eu_scale_drop_new_target. fs_h_eu_scale_drop_new_target + S (eu_scale_target_scale_drop_new) = S ((S (eu_scale_index_scale_drop_new)) * e)) /\ exists fs_q_eu_scale_drop_new_target. d = fs_q_eu_scale_drop_new_target * S ((S (eu_scale_index_scale_drop_new)) * e) + (eu_scale_target_scale_drop_new))) -> (((forall eut_divisor_eu_scale_drop_new_unit. (exists eut_left_eu_scale_drop_new_unit. (eu_scale_index_scale_drop_new) = eut_divisor_eu_scale_drop_new_unit * eut_left_eu_scale_drop_new_unit) -> (exists eut_right_eu_scale_drop_new_unit. (m) = eut_divisor_eu_scale_drop_new_unit * eut_right_eu_scale_drop_new_unit) -> eut_divisor_eu_scale_drop_new_unit = 1) -> (exists eu_mod_left_scale_drop_new_scaled eu_mod_right_scale_drop_new_scaled. ((a)*eu_scale_source_scale_drop_new) + (m) * eu_mod_left_scale_drop_new_scaled = (eu_scale_target_scale_drop_new) + (m) * eu_mod_right_scale_drop_new_scaled)) /\ (~(forall eut_divisor_eu_scale_drop_new_unit. (exists eut_left_eu_scale_drop_new_unit. (eu_scale_index_scale_drop_new) = eut_divisor_eu_scale_drop_new_unit * eut_left_eu_scale_drop_new_unit) -> (exists eut_right_eu_scale_drop_new_unit. (m) = eut_divisor_eu_scale_drop_new_unit * eut_right_eu_scale_drop_new_unit) -> eut_divisor_eu_scale_drop_new_unit = 1) -> (exists eu_mod_left_scale_drop_new_unchanged eu_mod_right_scale_drop_new_unchanged. (eu_scale_source_scale_drop_new) + (m) * eu_mod_left_scale_drop_new_unchanged = (eu_scale_target_scale_drop_new) + (m) * eu_mod_right_scale_drop_new_unchanged))))

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 · 3 reading checkpoints · 0 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 assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro d
  6. L6
    intro e
  7. L7
    intro l
  8. L8
    intro h
  9. L9
    intro i
  10. L10
    intro u
02Fix variables and assumptionsL11–14

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

  1. L11
    intro v
  2. L12
    intro hi
  3. L13
    intro hu
  4. L14
    intro hv
03Use earlier factsL15–24

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

  1. L15
    specialize h (i)
  2. L16
    specialize h (u)
  3. L17
    specialize h (v)
  4. L18
    apply h
  5. L19
    specialize le_succ (S i)
  6. L20
    specialize le_succ (l)
  7. L21
    apply le_succ
  8. L22
    exact hi
  9. L23
    exact hu
  10. L24
    exact hv

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro e
  7. 0007intro l
  8. 0008intro h
  9. 0009intro i
  10. 0010intro u
  11. 0011intro v
  12. 0012intro hi
  13. 0013intro hu
  14. 0014intro hv
  15. 0015specialize h (i)
  16. 0016specialize h (u)
  17. 0017specialize h (v)
  18. 0018apply h
  19. 0019specialize le_succ (S i)
  20. 0020specialize le_succ (l)
  21. 0021apply le_succ
  22. 0022exact hi
  23. 0023exact hu
  24. 0024exact hv