FP001A

prime_field_add_cancel_left

Additive cancellation follows from genuine balanced congruence cancellation and canonical bounds.

Alpha v34 checked-use · first admitted v31 · 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.

This checkpoint constructs prime-order fields (k=1), with genuine finite arithmetic tables, cardinality, and characteristic. Inversion is proved only for nonzero elements; the table's zero entry is a zero-to-zero convention. G091 for every prime power p^k, with an irreducible polynomial of degree k, remains open. No extension-field construction or G091 closure is claimed.

Exact theorem in conservative defined notation

∀ p. ∀ a. ∀ b. ∀ c. ∀ z. FpAdd(p,a,b,z) → FpAdd(p,a,c,z) → b = c

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

Definition DAG

FpAdd(p,a,b,c) · 2

Actual proof prerequisites

mod_eq_bounded_unique · checked external prerequisitemod_eq_add_cancel_left · checked external prerequisitemod_eq_trans · checked external prerequisitemod_eq_symm · checked external prerequisite
Original expanded first-order statement 
forall p a b c z. (((exists pfa_gap_cancel_add_firstleft. pfa_gap_cancel_add_firstleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_add_firstright. pfa_gap_cancel_add_firstright + S (b) = (p)) /\ ((((exists pfa_gap_cancel_add_firstresultbound. pfa_gap_cancel_add_firstresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_add_firstresultcongruence pfa_offset_right_cancel_add_firstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_cancel_add_firstresultcongruence = (z) + (p) * pfa_offset_right_cancel_add_firstresultcongruence))))))))) -> (((exists pfa_gap_cancel_add_secondleft. pfa_gap_cancel_add_secondleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_add_secondright. pfa_gap_cancel_add_secondright + S (c) = (p)) /\ ((((exists pfa_gap_cancel_add_secondresultbound. pfa_gap_cancel_add_secondresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_add_secondresultcongruence pfa_offset_right_cancel_add_secondresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_cancel_add_secondresultcongruence = (z) + (p) * pfa_offset_right_cancel_add_secondresultcongruence))))))))) -> b = c

Complete tactic proof in conservative notation

All 35 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

35 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 assumptionsL1–7

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 z
  6. L6
    intro hb
  7. L7
    intro hc
02Separate the logical casesL8–13

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

  1. L8
    cases hb
  2. L9
    cases hb_right
  3. L10
    cases hb_right_right
  4. L11
    cases hc
  5. L12
    cases hc_right
  6. L13
    cases hc_right_right
03Use earlier factsL14–23

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

  1. L14
    specialize mod_eq_bounded_unique (p)
  2. L15
    specialize mod_eq_bounded_unique (b)
  3. L16
    specialize mod_eq_bounded_unique (c)
  4. L17
    apply mod_eq_bounded_unique
  5. L18
    exact hb_right_left
  6. L19
    exact hc_right_left
  7. L20
    specialize mod_eq_add_cancel_left (p)
  8. L21
    specialize mod_eq_add_cancel_left (a)
  9. L22
    specialize mod_eq_add_cancel_left (b)
  10. L23
    specialize mod_eq_add_cancel_left (c)
04Use earlier factsL24–33

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

  1. L24
    apply mod_eq_add_cancel_left
  2. L25
    specialize mod_eq_trans (p)
  3. L26
    specialize mod_eq_trans (a + b)
  4. L27
    specialize mod_eq_trans (z)
  5. L28
    specialize mod_eq_trans (a + c)
  6. L29
    apply mod_eq_trans
  7. L30
    exact hb_right_right_right
  8. L31
    specialize mod_eq_symm (p)
  9. L32
    specialize mod_eq_symm (a + c)
  10. L33
    specialize mod_eq_symm (z)
05Use earlier factsL34–35

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

  1. L34
    apply mod_eq_symm
  2. L35
    exact hc_right_right_right

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro z
  6. 0006intro hb
  7. 0007intro hc
  8. 0008cases hb
  9. 0009cases hb_right
  10. 0010cases hb_right_right
  11. 0011cases hc
  12. 0012cases hc_right
  13. 0013cases hc_right_right
  14. 0014specialize mod_eq_bounded_unique (p)
  15. 0015specialize mod_eq_bounded_unique (b)
  16. 0016specialize mod_eq_bounded_unique (c)
  17. 0017apply mod_eq_bounded_unique
  18. 0018exact hb_right_left
  19. 0019exact hc_right_left
  20. 0020specialize mod_eq_add_cancel_left (p)
  21. 0021specialize mod_eq_add_cancel_left (a)
  22. 0022specialize mod_eq_add_cancel_left (b)
  23. 0023specialize mod_eq_add_cancel_left (c)
  24. 0024apply mod_eq_add_cancel_left
  25. 0025specialize mod_eq_trans (p)
  26. 0026specialize mod_eq_trans (a + b)
  27. 0027specialize mod_eq_trans (z)
  28. 0028specialize mod_eq_trans (a + c)
  29. 0029apply mod_eq_trans
  30. 0030exact hb_right_right_right
  31. 0031specialize mod_eq_symm (p)
  32. 0032specialize mod_eq_symm (a + c)
  33. 0033specialize mod_eq_symm (z)
  34. 0034apply mod_eq_symm
  35. 0035exact hc_right_right_right