EL0012

lte_odd_prime_quotient_unit

The prime-step quotient is exactly p times a genuine p-nondivisible cofactor; no valuation conclusion is assumed.

Alpha v34 checked-use · first admitted v29 · 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. Exact original first-admission records.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ p. ∀ d. ∀ r. ∀ R. ∀ T. ∀ Q. ∀ C. ∀ H. ¬p = 1 ∧ (∀ x. ∀ y. p = x · y → x = 1 ∨ y = 1) → ¬p = 2 → Dvd(p,d) → ¬Dvd(p,R) → Q = p · R + d · C → 2 · C = p · r · T + d · H → ∃ x. Q = p · x ∧ ¬Dvd(p,x)

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

Definition DAG

Dvd(d,n) · 4

Actual proof prerequisites

lte_prime_divides_correctionlte_nondivisor_add_multipleadd_mul · checked external prerequisitemul_add · checked external prerequisitemul_assoc · checked external prerequisitemul_comm · checked external prerequisiteadd_assoc · checked external prerequisitenatural_mul_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall p d r R T Q C H. (~((p) = 1) /\ forall pvs_left_quotient_prime pvs_right_quotient_prime. (p) = pvs_left_quotient_prime * pvs_right_quotient_prime -> pvs_left_quotient_prime = 1 \/ pvs_right_quotient_prime = 1) -> ~(p = 2) -> (exists olte_factor_quotient_difference. (d) = (p) * olte_factor_quotient_difference) -> ~(exists olte_factor_quotient_base. (R) = (p) * olte_factor_quotient_base) -> Q = p * R + d * C -> 2 * C = (p * r) * T + d * H -> exists u. Q = p * u /\ ~(exists olte_factor_quotient_unit. (u) = (p) * olte_factor_quotient_unit)

Complete tactic proof in conservative notation

All 67 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

67 script commands · 16 reading checkpoints · 1 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 (2)
lte prime divides correctionlte nondivisor add multiple
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro d
  3. L3
    intro r
  4. L4
    intro R
  5. L5
    intro T
  6. L6
    intro Q
  7. L7
    intro C
  8. L8
    intro H
  9. L9
    intro hp
  10. L10
    intro hne
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hd
  2. L12
    intro hR
  3. L13
    intro hQ
  4. L14
    intro hC
03Establish hdivL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply lte prime divides correction.

  1. L15
  2. L16
    specialize lte_prime_divides_correction (p)
  3. L17
    specialize lte_prime_divides_correction (d)
  4. L18
    specialize lte_prime_divides_correction (r)
  5. L19
    specialize lte_prime_divides_correction (T)
  6. L20
    specialize lte_prime_divides_correction (C)
  7. L21
    specialize lte_prime_divides_correction (H)
  8. L22
    apply lte_prime_divides_correction
  9. L23
    exact hp
  10. L24
    exact hne
04Use earlier factsL25–26

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

  1. L25
    exact hd
  2. L26
    exact hC
05Separate the logical casesL27–27

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

  1. L27
    cases hdiv
06Construct an explicit witnessL28–28

Supply the displayed value, then prove that it has the required property.

  1. L28
    exists R + d * x
07Separate the logical casesL29–29

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

  1. L29
    split
08Calculate and transport equalitiesL30–36

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L30
    rewrite hQ
  2. L31
    rewrite hdiv_witness
  3. L32
    trans ((((p) * (R))) + (((d) * (((p) * (x))))))
  4. L33
    simp [add_mul, mul_add, mul_assoc, add_assoc]
  5. L34
    trans ((((R) * (p))) + (((d) * (((p) * (x))))))
  6. L35
    congr
  7. L36
    trans ((R) * (p))
09Use earlier factsL37–37

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

  1. L37
    apply mul_comm
10Calculate and transport equalitiesL38–47

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L38
    congr
  2. L39
    refl
  3. L40
    refl
  4. L41
    refl
  5. L42
    trans ((((R) * (p))) + (((d) * (((p) * (x))))))
  6. L43
    refl
  7. L44
    trans ((((p) * (R))) + (((p) * (((d) * (x))))))
  8. L45
    symm
  9. L46
    congr
  10. L47
    trans ((R) * (p))
11Use earlier factsL48–48

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

  1. L48
    apply mul_comm
12Calculate and transport equalitiesL49–52

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L49
    congr
  2. L50
    refl
  3. L51
    refl
  4. L52
    trans ((d) * (((p) * (x))))
13Use earlier factsL53–53

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

  1. L53
    apply natural_mul_swap_right_tail
14Calculate and transport equalitiesL54–58

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L54
    congr
  2. L55
    refl
  3. L56
    refl
  4. L57
    symm
  5. L58
    simp [add_mul, mul_add, mul_assoc, add_assoc]
15Fix variables and assumptionsL59–59

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

  1. L59
    intro hunit
16Use earlier factsL60–67

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

  1. L60
    specialize lte_nondivisor_add_multiple (p)
  2. L61
    specialize lte_nondivisor_add_multiple (R)
  3. L62
    specialize lte_nondivisor_add_multiple (d)
  4. L63
    specialize lte_nondivisor_add_multiple (x)
  5. L64
    apply lte_nondivisor_add_multiple
  6. L65
    exact hR
  7. L66
    exact hd
  8. L67
    exact hunit

Library-wide reading audit

Original defined command ledger · 67 lines
  1. 0001intro p
  2. 0002intro d
  3. 0003intro r
  4. 0004intro R
  5. 0005intro T
  6. 0006intro Q
  7. 0007intro C
  8. 0008intro H
  9. 0009intro hp
  10. 0010intro hne
  11. 0011intro hd
  12. 0012intro hR
  13. 0013intro hQ
  14. 0014intro hC
  15. 0015have hdiv : Dvd(p,C)
  16. 0016specialize lte_prime_divides_correction (p)
  17. 0017specialize lte_prime_divides_correction (d)
  18. 0018specialize lte_prime_divides_correction (r)
  19. 0019specialize lte_prime_divides_correction (T)
  20. 0020specialize lte_prime_divides_correction (C)
  21. 0021specialize lte_prime_divides_correction (H)
  22. 0022apply lte_prime_divides_correction
  23. 0023exact hp
  24. 0024exact hne
  25. 0025exact hd
  26. 0026exact hC
  27. 0027cases hdiv
  28. 0028exists R + d * x
  29. 0029split
  30. 0030rewrite hQ
  31. 0031rewrite hdiv_witness
  32. 0032trans ((((p) * (R))) + (((d) * (((p) * (x))))))
  33. 0033simp [add_mul, mul_add, mul_assoc, add_assoc]
  34. 0034trans ((((R) * (p))) + (((d) * (((p) * (x))))))
  35. 0035congr
  36. 0036trans ((R) * (p))
  37. 0037apply mul_comm
  38. 0038congr
  39. 0039refl
  40. 0040refl
  41. 0041refl
  42. 0042trans ((((R) * (p))) + (((d) * (((p) * (x))))))
  43. 0043refl
  44. 0044trans ((((p) * (R))) + (((p) * (((d) * (x))))))
  45. 0045symm
  46. 0046congr
  47. 0047trans ((R) * (p))
  48. 0048apply mul_comm
  49. 0049congr
  50. 0050refl
  51. 0051refl
  52. 0052trans ((d) * (((p) * (x))))
  53. 0053apply natural_mul_swap_right_tail
  54. 0054congr
  55. 0055refl
  56. 0056refl
  57. 0057symm
  58. 0058simp [add_mul, mul_add, mul_assoc, add_assoc]
  59. 0059intro hunit
  60. 0060specialize lte_nondivisor_add_multiple (p)
  61. 0061specialize lte_nondivisor_add_multiple (R)
  62. 0062specialize lte_nondivisor_add_multiple (d)
  63. 0063specialize lte_nondivisor_add_multiple (x)
  64. 0064apply lte_nondivisor_add_multiple
  65. 0065exact hR
  66. 0066exact hd
  67. 0067exact hunit