FP0016

prime_field_multiply_one_right

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

Natural one is the actual multiplicative identity, including at p=2.

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 p a. (~((p) = 1) /\ forall pfa_factor_left_multiply_one_domain pfa_factor_right_multiply_one_domain. (p) = pfa_factor_left_multiply_one_domain * pfa_factor_right_multiply_one_domain -> pfa_factor_left_multiply_one_domain = 1 \/ pfa_factor_right_multiply_one_domain = 1) -> (exists pfa_gap_multiply_one_bound. pfa_gap_multiply_one_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_oneleft. pfa_gap_multiply_oneleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_oneright. pfa_gap_multiply_oneright + S (1) = (p)) /\ ((((exists pfa_gap_multiply_oneresultbound. pfa_gap_multiply_oneresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_multiply_oneresultcongruence pfa_offset_right_multiply_oneresultcongruence. ((a) * (1)) + (p) * pfa_offset_left_multiply_oneresultcongruence = (a) + (p) * pfa_offset_right_multiply_oneresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Natural one is the actual multiplicative identity, including at p=2.

The unchanged tactic script uses 3 declared prerequisites and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

prime_two_le Alpha theorem; checked-use authorized FP0001 prime_field_mod_of_equal mul_one Stable theorem; checked-use authorized

Direct dependents

FP0017 prime_field_multiply_one_left FP002A prime_field_arithmetic_laws

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

18 script commands · 7 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.

Named ingredients (1)
prime field mod of equal
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro hp
  4. L4
    intro ha
02Separate the logical casesL5–5

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

  1. L5
    split
03Use earlier factsL6–6

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

  1. L6
    exact ha
04Separate the logical casesL7–7

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

  1. L7
    split
05Use earlier factsL8–10

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

  1. L8
    specialize prime_two_le (p)
  2. L9
    apply prime_two_le
  3. L10
    exact hp
06Separate the logical casesL11–11

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

  1. L11
    split
07Use earlier factsL12–18

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

  1. L12
    exact ha
  2. L13
    specialize prime_field_mod_of_equal (p)
  3. L14
    specialize prime_field_mod_of_equal (a * 1)
  4. L15
    specialize prime_field_mod_of_equal (a)
  5. L16
    apply prime_field_mod_of_equal
  6. L17
    specialize mul_one (a)
  7. L18
    apply mul_one

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005split
  6. 0006exact ha
  7. 0007split
  8. 0008specialize prime_two_le (p)
  9. 0009apply prime_two_le
  10. 0010exact hp
  11. 0011split
  12. 0012exact ha
  13. 0013specialize prime_field_mod_of_equal (p)
  14. 0014specialize prime_field_mod_of_equal (a * 1)
  15. 0015specialize prime_field_mod_of_equal (a)
  16. 0016apply prime_field_mod_of_equal
  17. 0017specialize mul_one (a)
  18. 0018apply mul_one