PQ0002

prime_field_subtract_equal_zero

The genuine bounded difference of a canonical coefficient from itself is natural zero.

Alpha v34 checked-use · first admitted v32 · 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.

All coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ p. ∀ a. ∀ r. Prime(p) → FpAdd(p,a,r,a) → r = 0

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

Definition DAG

FpAdd(p,a,b,c) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_add_cancel_left · checked external prerequisiteprime_field_add_zero_right · checked external prerequisite
Original expanded first-order statement 
forall p a r. (~((p) = 1) /\ forall pfa_factor_left_scalar_equal_prime pfa_factor_right_scalar_equal_prime. (p) = pfa_factor_left_scalar_equal_prime * pfa_factor_right_scalar_equal_prime -> pfa_factor_left_scalar_equal_prime = 1 \/ pfa_factor_right_scalar_equal_prime = 1) -> (((exists pfa_gap_scalar_equal_sumleft. pfa_gap_scalar_equal_sumleft + S (a) = (p)) /\ (((exists pfa_gap_scalar_equal_sumright. pfa_gap_scalar_equal_sumright + S (r) = (p)) /\ ((((exists pfa_gap_scalar_equal_sumresultbound. pfa_gap_scalar_equal_sumresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_scalar_equal_sumresultcongruence pfa_offset_right_scalar_equal_sumresultcongruence. ((a) + (r)) + (p) * pfa_offset_left_scalar_equal_sumresultcongruence = (a) + (p) * pfa_offset_right_scalar_equal_sumresultcongruence))))))))) -> r=0

Complete tactic proof in conservative notation

All 18 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

18 script commands · 5 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 assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro r
  4. L4
    intro hp
  5. L5
    intro h
02Use earlier factsL6–15

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

  1. L6
    specialize prime_field_add_cancel_left (p)
  2. L7
    specialize prime_field_add_cancel_left (a)
  3. L8
    specialize prime_field_add_cancel_left (r)
  4. L9
    specialize prime_field_add_cancel_left (0)
  5. L10
    specialize prime_field_add_cancel_left (a)
  6. L11
    apply prime_field_add_cancel_left
  7. L12
    exact h
  8. L13
    specialize prime_field_add_zero_right (p)
  9. L14
    specialize prime_field_add_zero_right (a)
  10. L15
    apply prime_field_add_zero_right
03Use earlier factsL16–16

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

  1. L16
    exact hp
04Separate the logical casesL17–17

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

  1. L17
    cases h
05Use earlier factsL18–18

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

  1. L18
    exact h_left

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro r
  4. 0004intro hp
  5. 0005intro h
  6. 0006specialize prime_field_add_cancel_left (p)
  7. 0007specialize prime_field_add_cancel_left (a)
  8. 0008specialize prime_field_add_cancel_left (r)
  9. 0009specialize prime_field_add_cancel_left (0)
  10. 0010specialize prime_field_add_cancel_left (a)
  11. 0011apply prime_field_add_cancel_left
  12. 0012exact h
  13. 0013specialize prime_field_add_zero_right (p)
  14. 0014specialize prime_field_add_zero_right (a)
  15. 0015apply prime_field_add_zero_right
  16. 0016exact hp
  17. 0017cases h
  18. 0018exact h_left