FP0006

prime_field_residue_bounded_value

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

No two distinct representatives below p denote the same residue class.

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. (exists pfa_gap_bounded_value. pfa_gap_bounded_value + S (a) = (p)) -> (((exists pfa_gap_bounded_resultbound. pfa_gap_bounded_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_bounded_resultcongruence pfa_offset_right_bounded_resultcongruence. (a) + (p) * pfa_offset_left_bounded_resultcongruence = (r) + (p) * pfa_offset_right_bounded_resultcongruence)))) -> r = a

Constructive proof overview

Generated structural guide

No two distinct representatives below p denote the same residue class.

The unchanged tactic script uses 2 declared prerequisites and contains 15 exact native proof lines.

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

Proof neighborhood

Direct dependencies

binary_canonical_residue_functional Alpha theorem; checked-use authorized FP0003 prime_field_residue_reflexive

Direct dependents

FP0029 prime_field_positive_below_modulus_not_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

15 script commands · 2 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 residue reflexive
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 ha
  5. L5
    intro hr
02Use earlier factsL6–15

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

  1. L6
    specialize binary_canonical_residue_functional (p)
  2. L7
    specialize binary_canonical_residue_functional (a)
  3. L8
    specialize binary_canonical_residue_functional (r)
  4. L9
    specialize binary_canonical_residue_functional (a)
  5. L10
    apply binary_canonical_residue_functional
  6. L11
    exact hr
  7. L12
    specialize prime_field_residue_reflexive (p)
  8. L13
    specialize prime_field_residue_reflexive (a)
  9. L14
    apply prime_field_residue_reflexive
  10. L15
    exact ha

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro r
  4. 0004intro ha
  5. 0005intro hr
  6. 0006specialize binary_canonical_residue_functional (p)
  7. 0007specialize binary_canonical_residue_functional (a)
  8. 0008specialize binary_canonical_residue_functional (r)
  9. 0009specialize binary_canonical_residue_functional (a)
  10. 0010apply binary_canonical_residue_functional
  11. 0011exact hr
  12. 0012specialize prime_field_residue_reflexive (p)
  13. 0013specialize prime_field_residue_reflexive (a)
  14. 0014apply prime_field_residue_reflexive
  15. 0015exact ha