FP0025

prime_field_multiply_cancel_nonzero_left

Cancellation by any nonzero element is proved, not assumed from a field certificate.

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. Prime(p) → ¬a = 0 → FpMul(p,a,b,z) → FpMul(p,a,c,z) → b = c

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

Definition DAG

FpMul(p,a,b,c) · 2Prime(p) · 1

Actual proof prerequisites

mod_eq_bounded_unique · checked external prerequisitemod_eq_cancel_coprime · checked external prerequisiteprime_nonzero · checked external prerequisiteprime_field_nonzero_coprimemod_eq_trans · checked external prerequisitemod_eq_symm · checked external prerequisite
Original expanded first-order statement 
forall p a b c z. (~((p) = 1) /\ forall pfa_factor_left_cancel_mul_domain pfa_factor_right_cancel_mul_domain. (p) = pfa_factor_left_cancel_mul_domain * pfa_factor_right_cancel_mul_domain -> pfa_factor_left_cancel_mul_domain = 1 \/ pfa_factor_right_cancel_mul_domain = 1) -> ~(a = 0) -> (((exists pfa_gap_cancel_mul_firstleft. pfa_gap_cancel_mul_firstleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_mul_firstright. pfa_gap_cancel_mul_firstright + S (b) = (p)) /\ ((((exists pfa_gap_cancel_mul_firstresultbound. pfa_gap_cancel_mul_firstresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_mul_firstresultcongruence pfa_offset_right_cancel_mul_firstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_cancel_mul_firstresultcongruence = (z) + (p) * pfa_offset_right_cancel_mul_firstresultcongruence))))))))) -> (((exists pfa_gap_cancel_mul_secondleft. pfa_gap_cancel_mul_secondleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_mul_secondright. pfa_gap_cancel_mul_secondright + S (c) = (p)) /\ ((((exists pfa_gap_cancel_mul_secondresultbound. pfa_gap_cancel_mul_secondresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_mul_secondresultcongruence pfa_offset_right_cancel_mul_secondresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_cancel_mul_secondresultcongruence = (z) + (p) * pfa_offset_right_cancel_mul_secondresultcongruence))))))))) -> b = c

Complete tactic proof in conservative notation

All 48 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

48 script commands · 8 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.

Named ingredients (1)
prime field nonzero coprime
01Fix variables and assumptionsL1–9

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 hp
  7. L7
    intro hn
  8. L8
    intro hb
  9. L9
    intro hc
02Separate the logical casesL10–15

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

  1. L10
    cases hb
  2. L11
    cases hb_right
  3. L12
    cases hb_right_right
  4. L13
    cases hc
  5. L14
    cases hc_right
  6. L15
    cases hc_right_right
03Use earlier factsL16–25

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

  1. L16
    specialize mod_eq_bounded_unique (p)
  2. L17
    specialize mod_eq_bounded_unique (b)
  3. L18
    specialize mod_eq_bounded_unique (c)
  4. L19
    apply mod_eq_bounded_unique
  5. L20
    exact hb_right_left
  6. L21
    exact hc_right_left
  7. L22
    specialize mod_eq_cancel_coprime (p)
  8. L23
    specialize mod_eq_cancel_coprime (a)
  9. L24
    specialize mod_eq_cancel_coprime (b)
  10. L25
    specialize mod_eq_cancel_coprime (c)
04Use earlier factsL26–26

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

  1. L26
    apply mod_eq_cancel_coprime
05Fix variables and assumptionsL27–27

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

  1. L27
    intro hz
06Use earlier factsL28–37

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

  1. L28
    specialize prime_nonzero (p)
  2. L29
    apply prime_nonzero
  3. L30
    exact hp
  4. L31
    exact hz
  5. L32
    specialize prime_field_nonzero_coprime (p)
  6. L33
    specialize prime_field_nonzero_coprime (a)
  7. L34
    apply prime_field_nonzero_coprime
  8. L35
    exact hp
  9. L36
    exact hb_left
  10. L37
    exact hn
07Use earlier factsL38–47

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

  1. L38
    specialize mod_eq_trans (p)
  2. L39
    specialize mod_eq_trans (a * b)
  3. L40
    specialize mod_eq_trans (z)
  4. L41
    specialize mod_eq_trans (a * c)
  5. L42
    apply mod_eq_trans
  6. L43
    exact hb_right_right_right
  7. L44
    specialize mod_eq_symm (p)
  8. L45
    specialize mod_eq_symm (a * c)
  9. L46
    specialize mod_eq_symm (z)
  10. L47
    apply mod_eq_symm
08Use earlier factsL48–48

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

  1. L48
    exact hc_right_right_right

Library-wide reading audit

Original defined command ledger · 48 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro z
  6. 0006intro hp
  7. 0007intro hn
  8. 0008intro hb
  9. 0009intro hc
  10. 0010cases hb
  11. 0011cases hb_right
  12. 0012cases hb_right_right
  13. 0013cases hc
  14. 0014cases hc_right
  15. 0015cases hc_right_right
  16. 0016specialize mod_eq_bounded_unique (p)
  17. 0017specialize mod_eq_bounded_unique (b)
  18. 0018specialize mod_eq_bounded_unique (c)
  19. 0019apply mod_eq_bounded_unique
  20. 0020exact hb_right_left
  21. 0021exact hc_right_left
  22. 0022specialize mod_eq_cancel_coprime (p)
  23. 0023specialize mod_eq_cancel_coprime (a)
  24. 0024specialize mod_eq_cancel_coprime (b)
  25. 0025specialize mod_eq_cancel_coprime (c)
  26. 0026apply mod_eq_cancel_coprime
  27. 0027intro hz
  28. 0028specialize prime_nonzero (p)
  29. 0029apply prime_nonzero
  30. 0030exact hp
  31. 0031exact hz
  32. 0032specialize prime_field_nonzero_coprime (p)
  33. 0033specialize prime_field_nonzero_coprime (a)
  34. 0034apply prime_field_nonzero_coprime
  35. 0035exact hp
  36. 0036exact hb_left
  37. 0037exact hn
  38. 0038specialize mod_eq_trans (p)
  39. 0039specialize mod_eq_trans (a * b)
  40. 0040specialize mod_eq_trans (z)
  41. 0041specialize mod_eq_trans (a * c)
  42. 0042apply mod_eq_trans
  43. 0043exact hb_right_right_right
  44. 0044specialize mod_eq_symm (p)
  45. 0045specialize mod_eq_symm (a * c)
  46. 0046specialize mod_eq_symm (z)
  47. 0047apply mod_eq_symm
  48. 0048exact hc_right_right_right