FP0014 — prime_field_add_zero_right

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_add_zero_domain pfa_factor_right_add_zero_domain. (p) = pfa_factor_left_add_zero_domain * pfa_factor_right_add_zero_domain -> pfa_factor_left_add_zero_domain = 1 \/ pfa_factor_right_add_zero_domain = 1) -> (exists pfa_gap_add_zero_bound. pfa_gap_add_zero_bound + S (a) = (p)) -> (((exists pfa_gap_add_zeroleft. pfa_gap_add_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_add_zeroright. pfa_gap_add_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_add_zeroresultbound. pfa_gap_add_zeroresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_add_zeroresultcongruence pfa_offset_right_add_zeroresultcongruence. ((a) + (0)) + (p) * pfa_offset_left_add_zeroresultcongruence = (a) + (p) * pfa_offset_right_add_zeroresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Natural zero is the actual additive identity, not an arbitrary chosen code.

The unchanged tactic script uses 2 declared prerequisites and contains 17 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FP0002 prime_field_zero_below_prime FP0001 prime_field_mod_of_equal

Direct dependents

FP0015 prime_field_add_zero_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

17 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 (2)

prime field zero below prime prime field mod of equal

01Fix variables and assumptionsL1–4

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

L1
intro p
L2
intro a
L3
intro hp
L4
intro ha

02Separate the logical casesL5–5

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

L5
split

03Use earlier factsL6–6

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

L6
exact ha

04Separate the logical casesL7–7

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

L7
split

05Use earlier factsL8–10

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

L8
specialize prime_field_zero_below_prime (p)
L9
apply prime_field_zero_below_prime
L10
exact hp

06Separate the logical casesL11–11

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

L11
split

07Use earlier factsL12–17

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

L12
exact ha
L13
specialize prime_field_mod_of_equal (p)
L14
specialize prime_field_mod_of_equal (a + 0)
L15
specialize prime_field_mod_of_equal (a)
L16
apply prime_field_mod_of_equal
L17
apply PA3

Library-wide reading audit

Original exact command ledger · 17 lines

0001intro p
0002intro a
0003intro hp
0004intro ha
0005split
0006exact ha
0007split
0008specialize prime_field_zero_below_prime (p)
0009apply prime_field_zero_below_prime
0010exact hp
0011split
0012exact ha
0013specialize prime_field_mod_of_equal (p)
0014specialize prime_field_mod_of_equal (a + 0)
0015specialize prime_field_mod_of_equal (a)
0016apply prime_field_mod_of_equal
0017apply PA3