PA0071 · theorem

gauss_pointwise_signed_half_choice

Alpha v34 checked-use theorem · independently closed; not Stable

A canonical nonzero remainder yields one explicit zero/one signed-half choice at its decoded source index.

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.

Statement with defined notation

∀ p. ∀ h. ∀ a. ∀ b. ∀ c. ∀ i. ∀ x. ∀ q. ∀ r. p = 2 · h + 1 → BetaAt(b,c,i,x) → a · x = q · p + r → Lt(r,p) → ¬r = 0 → ∃ y. ∃ z. ∃ n. BetaAt(b,c,i,y) ∧ (Lt(0,z) ∧ (Le(z,h) ∧ ((n = 0 ∨ n = 1) ∧ (n = 0 ∧ ModEq(p,a · y,z) ∨ n = 1 ∧ ModEq(p,a · y,2 · h · z)))))

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0001 Le PD0002 Lt PD0008 ModEq PD0013 BetaAt

7 occurrences

In local proof propositions

PD0001 Le PD0002 Lt PD0008 ModEq

4 occurrences

Exact expanded native-PA statement
forall p h a b c i x q r. p = 2 * h + 1 -> (((exists ff_h_gsp_point_source. ff_h_gsp_point_source + S (x) = S ((S (i)) * c)) /\ exists ff_q_gsp_point_source. b = ff_q_gsp_point_source * S ((S (i)) * c) + (x))) -> a * x = q * p + r -> (exists gsp_lt_gap_point_remainder_bound. gsp_lt_gap_point_remainder_bound + S r = p) -> ~(r = 0) -> (exists gsp_value_point_result gsp_magnitude_point_result gsp_sign_point_result. (((exists ff_h_gsp_point_result_source. ff_h_gsp_point_result_source + S (gsp_value_point_result) = S ((S (i)) * c)) /\ exists ff_q_gsp_point_result_source. b = ff_q_gsp_point_result_source * S ((S (i)) * c) + (gsp_value_point_result))) /\ ((exists gsp_lt_gap_point_result_positive. gsp_lt_gap_point_result_positive + S 0 = gsp_magnitude_point_result) /\ ((exists gsp_le_gap_point_result_bounded. gsp_le_gap_point_result_bounded + gsp_magnitude_point_result = h) /\ ((gsp_sign_point_result = 0 \/ gsp_sign_point_result = 1) /\ (((gsp_sign_point_result = 0 /\ (exists gsp_mod_left_point_result_lower gsp_mod_right_point_result_lower. (a * gsp_value_point_result) + p * gsp_mod_left_point_result_lower = (gsp_magnitude_point_result) + p * gsp_mod_right_point_result_lower)) \/ (gsp_sign_point_result = 1 /\ (exists gsp_mod_left_point_result_reflected gsp_mod_right_point_result_reflected. (a * gsp_value_point_result) + p * gsp_mod_left_point_result_reflected = ((2 * h) * gsp_magnitude_point_result) + p * gsp_mod_right_point_result_reflected))))))))

Proof neighborhood

Direct theorem prerequisites

PA0070 gauss_pointwise_signed_half_representative

Direct theorem dependents

PA0072 gauss_half_range_signed_choices

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

62 script commands · 29 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 (1)
gauss pointwise signed half representative
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro a
  4. L4
    intro b
  5. L5
    intro c
  6. L6
    intro i
  7. L7
    intro x
  8. L8
    intro q
  9. L9
    intro r
  10. L10
    intro hp
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hsource
  2. L12
    intro hdecomp
  3. L13
    intro hrp
  4. L14
    intro hr0
03Establish hrepresentativeL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gauss pointwise signed half representative.

  1. L15
    have hrepresentative : ∃ m. Lt(0,m) ∧ (Le(m,h) ∧ (ModEq(p,a · x,m) ∨ ModEq(p,a · x,2 · h · m)))Definitions: Lt(0,m)Le(m,h)ModEq(p,a · x,m)ModEq(p,a · x,2 · h · m)Original native command in the exact edition
  2. L16
    specialize gauss_pointwise_signed_half_representative p
  3. L17
    specialize gauss_pointwise_signed_half_representative h
  4. L18
    specialize gauss_pointwise_signed_half_representative a
  5. L19
    specialize gauss_pointwise_signed_half_representative x
  6. L20
    specialize gauss_pointwise_signed_half_representative q
  7. L21
    specialize gauss_pointwise_signed_half_representative r
  8. L22
    apply gauss_pointwise_signed_half_representative
  9. L23
    exact hp
  10. L24
    exact hdecomp
04Use earlier factsL25–26

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

  1. L25
    exact hrp
  2. L26
    exact hr0
05Separate the logical casesL27–30

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

  1. L27
    cases hrepresentative
  2. L28
    cases hrepresentative_witness
  3. L29
    cases hrepresentative_witness_right
  4. L30
    cases hrepresentative_witness_right_right
06Construct an explicit witnessL31–33

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

  1. L31
    exists x
  2. L32
    exists x1
  3. L33
    exists 0
07Separate the logical casesL34–34

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

  1. L34
    split
08Use earlier factsL35–35

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

  1. L35
    exact hsource
09Separate the logical casesL36–36

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

  1. L36
    split
10Use earlier factsL37–37

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

  1. L37
    exact hrepresentative_witness_left
11Separate the logical casesL38–38

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

  1. L38
    split
12Use earlier factsL39–39

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

  1. L39
    exact hrepresentative_witness_right_left
13Separate the logical casesL40–41

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

  1. L40
    split
  2. L41
    left
14Calculate and transport equalitiesL42–42

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

  1. L42
    refl
15Separate the logical casesL43–44

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

  1. L43
    left
  2. L44
    split
16Calculate and transport equalitiesL45–45

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

  1. L45
    refl
17Use earlier factsL46–46

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

  1. L46
    exact hrepresentative_witness_right_right_left
18Construct an explicit witnessL47–49

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

  1. L47
    exists x
  2. L48
    exists x1
  3. L49
    exists 1
19Separate the logical casesL50–50

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

  1. L50
    split
20Use earlier factsL51–51

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

  1. L51
    exact hsource
21Separate the logical casesL52–52

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

  1. L52
    split
22Use earlier factsL53–53

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

  1. L53
    exact hrepresentative_witness_left
23Separate the logical casesL54–54

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

  1. L54
    split
24Use earlier factsL55–55

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

  1. L55
    exact hrepresentative_witness_right_left
25Separate the logical casesL56–57

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

  1. L56
    split
  2. L57
    right
26Calculate and transport equalitiesL58–58

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

  1. L58
    refl
27Separate the logical casesL59–60

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

  1. L59
    right
  2. L60
    split
28Calculate and transport equalitiesL61–61

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

  1. L61
    refl
29Use earlier factsL62–62

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

  1. L62
    exact hrepresentative_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 62 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro a
  4. 0004intro b
  5. 0005intro c
  6. 0006intro i
  7. 0007intro x
  8. 0008intro q
  9. 0009intro r
  10. 0010intro hp
  11. 0011intro hsource
  12. 0012intro hdecomp
  13. 0013intro hrp
  14. 0014intro hr0
  15. 0015have hrepresentative : ∃ m. Lt(0,m) ∧ (Le(m,h) ∧ (ModEq(p,a · x,m) ∨ ModEq(p,a · x,2 · h · m)))
    Exact native replay linehave hrepresentative : exists m. (exists gsp_lt_gap_point_result_positive. gsp_lt_gap_point_result_positive + S 0 = m) /\ ((exists gsp_le_gap_point_result_bounded. gsp_le_gap_point_result_bounded + m = h) /\ ((exists gsp_mod_left_point_result_lower gsp_mod_right_point_result_lower. (a * x) + p * gsp_mod_left_point_result_lower = (m) + p * gsp_mod_right_point_result_lower) \/ (exists gsp_mod_left_point_result_reflected gsp_mod_right_point_result_reflected. (a * x) + p * gsp_mod_left_point_result_reflected = ((2 * h) * m) + p * gsp_mod_right_point_result_reflected)))
  16. 0016specialize gauss_pointwise_signed_half_representative p
  17. 0017specialize gauss_pointwise_signed_half_representative h
  18. 0018specialize gauss_pointwise_signed_half_representative a
  19. 0019specialize gauss_pointwise_signed_half_representative x
  20. 0020specialize gauss_pointwise_signed_half_representative q
  21. 0021specialize gauss_pointwise_signed_half_representative r
  22. 0022apply gauss_pointwise_signed_half_representative
  23. 0023exact hp
  24. 0024exact hdecomp
  25. 0025exact hrp
  26. 0026exact hr0
  27. 0027cases hrepresentative
  28. 0028cases hrepresentative_witness
  29. 0029cases hrepresentative_witness_right
  30. 0030cases hrepresentative_witness_right_right
  31. 0031exists x
  32. 0032exists x1
  33. 0033exists 0
  34. 0034split
  35. 0035exact hsource
  36. 0036split
  37. 0037exact hrepresentative_witness_left
  38. 0038split
  39. 0039exact hrepresentative_witness_right_left
  40. 0040split
  41. 0041left
  42. 0042refl
  43. 0043left
  44. 0044split
  45. 0045refl
  46. 0046exact hrepresentative_witness_right_right_left
  47. 0047exists x
  48. 0048exists x1
  49. 0049exists 1
  50. 0050split
  51. 0051exact hsource
  52. 0052split
  53. 0053exact hrepresentative_witness_left
  54. 0054split
  55. 0055exact hrepresentative_witness_right_left
  56. 0056split
  57. 0057right
  58. 0058refl
  59. 0059right
  60. 0060split
  61. 0061refl
  62. 0062exact hrepresentative_witness_right_right_right