FP000B

prime_field_add_commutative

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

Commutativity of actual canonical add, not an assumed table axiom.

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 b c. (((exists pfa_gap_addcomm_sourceleft. pfa_gap_addcomm_sourceleft + S (a) = (p)) /\ (((exists pfa_gap_addcomm_sourceright. pfa_gap_addcomm_sourceright + S (b) = (p)) /\ ((((exists pfa_gap_addcomm_sourceresultbound. pfa_gap_addcomm_sourceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_sourceresultcongruence pfa_offset_right_addcomm_sourceresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addcomm_sourceresultcongruence = (c) + (p) * pfa_offset_right_addcomm_sourceresultcongruence))))))))) -> (((exists pfa_gap_addcomm_targetleft. pfa_gap_addcomm_targetleft + S (b) = (p)) /\ (((exists pfa_gap_addcomm_targetright. pfa_gap_addcomm_targetright + S (a) = (p)) /\ ((((exists pfa_gap_addcomm_targetresultbound. pfa_gap_addcomm_targetresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_targetresultcongruence pfa_offset_right_addcomm_targetresultcongruence. ((b) + (a)) + (p) * pfa_offset_left_addcomm_targetresultcongruence = (c) + (p) * pfa_offset_right_addcomm_targetresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Commutativity of actual canonical add, not an assumed table axiom.

The unchanged tactic script uses 2 declared prerequisites and contains 20 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_comm Stable theorem; checked-use authorized FP0004 prime_field_residue_input_equal

Direct dependents

FP0015 prime_field_add_zero_left FP002A prime_field_arithmetic_laws FP0042 prime_field_add_table_commutative

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

20 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.

Named ingredients (1)
prime field residue input equal
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 b
  4. L4
    intro c
  5. L5
    intro h
02Separate the logical casesL6–8

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

  1. L6
    cases h
  2. L7
    cases h_right
  3. L8
    split
03Use earlier factsL9–9

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

  1. L9
    exact h_right_left
04Separate the logical casesL10–10

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

  1. L10
    split
05Use earlier factsL11–20

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

  1. L11
    exact h_left
  2. L12
    specialize prime_field_residue_input_equal (p)
  3. L13
    specialize prime_field_residue_input_equal (b + a)
  4. L14
    specialize prime_field_residue_input_equal (a + b)
  5. L15
    specialize prime_field_residue_input_equal (c)
  6. L16
    apply prime_field_residue_input_equal
  7. L17
    specialize add_comm (b)
  8. L18
    specialize add_comm (a)
  9. L19
    apply add_comm
  10. L20
    exact h_right_right

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro h
  6. 0006cases h
  7. 0007cases h_right
  8. 0008split
  9. 0009exact h_right_left
  10. 0010split
  11. 0011exact h_left
  12. 0012specialize prime_field_residue_input_equal (p)
  13. 0013specialize prime_field_residue_input_equal (b + a)
  14. 0014specialize prime_field_residue_input_equal (a + b)
  15. 0015specialize prime_field_residue_input_equal (c)
  16. 0016apply prime_field_residue_input_equal
  17. 0017specialize add_comm (b)
  18. 0018specialize add_comm (a)
  19. 0019apply add_comm
  20. 0020exact h_right_right