PQ0002

prime_field_subtract_equal_zero

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

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

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 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 18 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_field_add_cancel_left Alpha theorem; checked-use authorized prime_field_add_zero_right Alpha theorem; checked-use authorized

Direct dependents

PQ0017 prime_field_polynomial_subtract_equal_entry_zero

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 · 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.

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 exact 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