CD0026

finite_modular_inverse_shift

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

Opposite additive shifts invert one another in balanced modular arithmetic by explicit witnesses.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall p t d a b. t+d=p -> (exists fms_u_mod fms_v_mod. (a+t) + (p) * fms_u_mod = (b) + (p) * fms_v_mod) -> (exists fms_u_mod fms_v_mod. (b+d) + (p) * fms_u_mod = (a) + (p) * fms_v_mod)

Constructive proof overview

Generated structural guide

Opposite additive shifts invert one another in balanced modular arithmetic by explicit witnesses.

The unchanged tactic script uses 2 declared prerequisites and contains 22 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

CD0028 finite_modular_pushforward_membership_witness CD0037 finite_modular_dyson_lower_from_pullback

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

22 script commands · 6 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro t
  3. L3
    intro d
  4. L4
    intro a
  5. L5
    intro b
  6. L6
    intro htd
  7. L7
    intro hmod
02Separate the logical casesL8–9

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

  1. L8
    cases hmod
  2. L9
    cases hmod_witness
03Construct an explicit witnessL10–11

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

  1. L10
    exists x1
  2. L11
    exists S x
04Calculate and transport equalitiesL12–16

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

  1. L12
    trans (b+p*x1)+d
  2. L13
    simp [add_assoc, add_comm]
  3. L14
    trans ((a+t)+p*x)+d
  4. L15
    congr
  5. L16
    symm
05Use earlier factsL17–17

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

  1. L17
    exact hmod_witness_witness
06Calculate and transport equalitiesL18–22

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

  1. L18
    refl
  2. L19
    trans (a+(t+d))+p*x
  3. L20
    simp [add_assoc, add_comm]
  4. L21
    rewrite htd
  5. L22
    simp [add_assoc, add_comm]

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro p
  2. 0002intro t
  3. 0003intro d
  4. 0004intro a
  5. 0005intro b
  6. 0006intro htd
  7. 0007intro hmod
  8. 0008cases hmod
  9. 0009cases hmod_witness
  10. 0010exists x1
  11. 0011exists S x
  12. 0012trans (b+p*x1)+d
  13. 0013simp [add_assoc, add_comm]
  14. 0014trans ((a+t)+p*x)+d
  15. 0015congr
  16. 0016symm
  17. 0017exact hmod_witness_witness
  18. 0018refl
  19. 0019trans (a+(t+d))+p*x
  20. 0020simp [add_assoc, add_comm]
  21. 0021rewrite htd
  22. 0022simp [add_assoc, add_comm]