EL0003

lte_first_correction_successor

The first correction term advances by b*C+Q without division or subtraction.

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

∀ a. ∀ b. ∀ d. ∀ n. ∀ R. ∀ Q. ∀ C. a = b + d → Q = n · R + d · C → a · Q + b · R = S n · (b · R) + d · (b · C + Q)

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

Definition DAG

none

Actual proof prerequisites

add_mul · checked external prerequisitemul_add · checked external prerequisitemul_assoc · checked external prerequisitemul_comm · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisitenatural_mul_swap_right_tail · checked external prerequisitemul_succ_left · checked external prerequisitemul_zero_left · checked external prerequisitezero_add · checked external prerequisiteone_mul · checked external prerequisitemul_one · checked external prerequisite
Original expanded first-order statement 
forall a b d n R Q C. a = b + d -> Q = n * R + d * C -> a * Q + b * R = S n * (b * R) + d * (b * C + Q)

Complete tactic proof in conservative notation

All 133 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

133 script commands · 30 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 assumptionsL1–9

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro d
  4. L4
    intro n
  5. L5
    intro R
  6. L6
    intro Q
  7. L7
    intro C
  8. L8
    intro ha
  9. L9
    intro hQ
02Calculate and transport equalitiesL10–19

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

  1. L10
    rewrite ha
  2. L11
    rewrite hQ
  3. L12
    rewrite hQ
  4. L13
    trans ((((b) * (((n) * (R))))) + ((((b) * (((d) * (C))))) + ((((d) * (((n) * (R))))) + ((((d) * (((d) * (C))))) + (((b) * (R)))))))
  5. L14
    simp [add_mul, mul_add, mul_assoc, add_assoc, mul_succ_left, mul_zero_left, zero_add, one_mul, mul_one]
  6. L15
    trans ((((R) * (((b) * (n))))) + ((((C) * (((b) * (d))))) + ((((R) * (((d) * (n))))) + ((((C) * (((d) * (d))))) + (((R) * (b)))))))
  7. L16
    congr
  8. L17
    trans ((R) * (((b) * (n))))
  9. L18
    trans ((b) * (((R) * (n))))
  10. L19
    congr
03Calculate and transport equalitiesL20–20

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

  1. L20
    refl
04Use earlier factsL21–22

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

  1. L21
    apply mul_comm
  2. L22
    apply natural_mul_swap_right_tail
05Calculate and transport equalitiesL23–30

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

  1. L23
    congr
  2. L24
    refl
  3. L25
    refl
  4. L26
    congr
  5. L27
    trans ((C) * (((b) * (d))))
  6. L28
    trans ((b) * (((C) * (d))))
  7. L29
    congr
  8. L30
    refl
06Use earlier factsL31–32

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

  1. L31
    apply mul_comm
  2. L32
    apply natural_mul_swap_right_tail
07Calculate and transport equalitiesL33–40

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

  1. L33
    congr
  2. L34
    refl
  3. L35
    refl
  4. L36
    congr
  5. L37
    trans ((R) * (((d) * (n))))
  6. L38
    trans ((d) * (((R) * (n))))
  7. L39
    congr
  8. L40
    refl
08Use earlier factsL41–42

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

  1. L41
    apply mul_comm
  2. L42
    apply natural_mul_swap_right_tail
09Calculate and transport equalitiesL43–50

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

  1. L43
    congr
  2. L44
    refl
  3. L45
    refl
  4. L46
    congr
  5. L47
    trans ((C) * (((d) * (d))))
  6. L48
    trans ((d) * (((C) * (d))))
  7. L49
    congr
  8. L50
    refl
10Use earlier factsL51–52

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

  1. L51
    apply mul_comm
  2. L52
    apply natural_mul_swap_right_tail
11Calculate and transport equalitiesL53–56

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

  1. L53
    congr
  2. L54
    refl
  3. L55
    refl
  4. L56
    trans ((R) * (b))
12Use earlier factsL57–57

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

  1. L57
    apply mul_comm
13Calculate and transport equalitiesL58–67

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

  1. L58
    congr
  2. L59
    refl
  3. L60
    refl
  4. L61
    trans ((((R) * (((b) * (n))))) + ((((R) * (b))) + ((((C) * (((b) * (d))))) + ((((R) * (((d) * (n))))) + (((C) * (((d) * (d)))))))))
  5. L62
    congr
  6. L63
    refl
  7. L64
    trans ((((R) * (b))) + ((((C) * (((b) * (d))))) + ((((R) * (((d) * (n))))) + (((C) * (((d) * (d))))))))
  8. L65
    trans ((((C) * (((b) * (d))))) + ((((R) * (b))) + ((((R) * (((d) * (n))))) + (((C) * (((d) * (d))))))))
  9. L66
    congr
  10. L67
    refl
14Calculate and transport equalitiesL68–70

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

  1. L68
    trans ((((R) * (((d) * (n))))) + ((((R) * (b))) + (((C) * (((d) * (d)))))))
  2. L69
    congr
  3. L70
    refl
15Use earlier factsL71–73

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

  1. L71
    apply add_comm
  2. L72
    apply four_square_add_swap_right_tail
  3. L73
    apply four_square_add_swap_right_tail
16Calculate and transport equalitiesL74–83

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

  1. L74
    congr
  2. L75
    refl
  3. L76
    refl
  4. L77
    trans ((((n) * (((b) * (R))))) + ((((b) * (R))) + ((((d) * (((b) * (C))))) + ((((d) * (((n) * (R))))) + (((d) * (((d) * (C)))))))))
  5. L78
    symm
  6. L79
    congr
  7. L80
    trans ((R) * (((n) * (b))))
  8. L81
    trans ((n) * (((R) * (b))))
  9. L82
    congr
  10. L83
    refl
17Use earlier factsL84–85

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

  1. L84
    apply mul_comm
  2. L85
    apply natural_mul_swap_right_tail
18Calculate and transport equalitiesL86–88

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

  1. L86
    congr
  2. L87
    refl
  3. L88
    trans ((b) * (n))
19Use earlier factsL89–89

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

  1. L89
    apply mul_comm
20Calculate and transport equalitiesL90–94

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

  1. L90
    congr
  2. L91
    refl
  3. L92
    refl
  4. L93
    congr
  5. L94
    trans ((R) * (b))
21Use earlier factsL95–95

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

  1. L95
    apply mul_comm
22Calculate and transport equalitiesL96–103

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

  1. L96
    congr
  2. L97
    refl
  3. L98
    refl
  4. L99
    congr
  5. L100
    trans ((C) * (((d) * (b))))
  6. L101
    trans ((d) * (((C) * (b))))
  7. L102
    congr
  8. L103
    refl
23Use earlier factsL104–105

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

  1. L104
    apply mul_comm
  2. L105
    apply natural_mul_swap_right_tail
24Calculate and transport equalitiesL106–108

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

  1. L106
    congr
  2. L107
    refl
  3. L108
    trans ((b) * (d))
25Use earlier factsL109–109

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

  1. L109
    apply mul_comm
26Calculate and transport equalitiesL110–117

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

  1. L110
    congr
  2. L111
    refl
  3. L112
    refl
  4. L113
    congr
  5. L114
    trans ((R) * (((d) * (n))))
  6. L115
    trans ((d) * (((R) * (n))))
  7. L116
    congr
  8. L117
    refl
27Use earlier factsL118–119

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

  1. L118
    apply mul_comm
  2. L119
    apply natural_mul_swap_right_tail
28Calculate and transport equalitiesL120–126

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

  1. L120
    congr
  2. L121
    refl
  3. L122
    refl
  4. L123
    trans ((C) * (((d) * (d))))
  5. L124
    trans ((d) * (((C) * (d))))
  6. L125
    congr
  7. L126
    refl
29Use earlier factsL127–128

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

  1. L127
    apply mul_comm
  2. L128
    apply natural_mul_swap_right_tail
30Calculate and transport equalitiesL129–133

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

  1. L129
    congr
  2. L130
    refl
  3. L131
    refl
  4. L132
    symm
  5. L133
    simp [add_mul, mul_add, mul_assoc, add_assoc, mul_succ_left, mul_zero_left, zero_add, one_mul, mul_one]

Library-wide reading audit

Original defined command ledger · 133 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro d
  4. 0004intro n
  5. 0005intro R
  6. 0006intro Q
  7. 0007intro C
  8. 0008intro ha
  9. 0009intro hQ
  10. 0010rewrite ha
  11. 0011rewrite hQ
  12. 0012rewrite hQ
  13. 0013trans ((((b) * (((n) * (R))))) + ((((b) * (((d) * (C))))) + ((((d) * (((n) * (R))))) + ((((d) * (((d) * (C))))) + (((b) * (R)))))))
  14. 0014simp [add_mul, mul_add, mul_assoc, add_assoc, mul_succ_left, mul_zero_left, zero_add, one_mul, mul_one]
  15. 0015trans ((((R) * (((b) * (n))))) + ((((C) * (((b) * (d))))) + ((((R) * (((d) * (n))))) + ((((C) * (((d) * (d))))) + (((R) * (b)))))))
  16. 0016congr
  17. 0017trans ((R) * (((b) * (n))))
  18. 0018trans ((b) * (((R) * (n))))
  19. 0019congr
  20. 0020refl
  21. 0021apply mul_comm
  22. 0022apply natural_mul_swap_right_tail
  23. 0023congr
  24. 0024refl
  25. 0025refl
  26. 0026congr
  27. 0027trans ((C) * (((b) * (d))))
  28. 0028trans ((b) * (((C) * (d))))
  29. 0029congr
  30. 0030refl
  31. 0031apply mul_comm
  32. 0032apply natural_mul_swap_right_tail
  33. 0033congr
  34. 0034refl
  35. 0035refl
  36. 0036congr
  37. 0037trans ((R) * (((d) * (n))))
  38. 0038trans ((d) * (((R) * (n))))
  39. 0039congr
  40. 0040refl
  41. 0041apply mul_comm
  42. 0042apply natural_mul_swap_right_tail
  43. 0043congr
  44. 0044refl
  45. 0045refl
  46. 0046congr
  47. 0047trans ((C) * (((d) * (d))))
  48. 0048trans ((d) * (((C) * (d))))
  49. 0049congr
  50. 0050refl
  51. 0051apply mul_comm
  52. 0052apply natural_mul_swap_right_tail
  53. 0053congr
  54. 0054refl
  55. 0055refl
  56. 0056trans ((R) * (b))
  57. 0057apply mul_comm
  58. 0058congr
  59. 0059refl
  60. 0060refl
  61. 0061trans ((((R) * (((b) * (n))))) + ((((R) * (b))) + ((((C) * (((b) * (d))))) + ((((R) * (((d) * (n))))) + (((C) * (((d) * (d)))))))))
  62. 0062congr
  63. 0063refl
  64. 0064trans ((((R) * (b))) + ((((C) * (((b) * (d))))) + ((((R) * (((d) * (n))))) + (((C) * (((d) * (d))))))))
  65. 0065trans ((((C) * (((b) * (d))))) + ((((R) * (b))) + ((((R) * (((d) * (n))))) + (((C) * (((d) * (d))))))))
  66. 0066congr
  67. 0067refl
  68. 0068trans ((((R) * (((d) * (n))))) + ((((R) * (b))) + (((C) * (((d) * (d)))))))
  69. 0069congr
  70. 0070refl
  71. 0071apply add_comm
  72. 0072apply four_square_add_swap_right_tail
  73. 0073apply four_square_add_swap_right_tail
  74. 0074congr
  75. 0075refl
  76. 0076refl
  77. 0077trans ((((n) * (((b) * (R))))) + ((((b) * (R))) + ((((d) * (((b) * (C))))) + ((((d) * (((n) * (R))))) + (((d) * (((d) * (C)))))))))
  78. 0078symm
  79. 0079congr
  80. 0080trans ((R) * (((n) * (b))))
  81. 0081trans ((n) * (((R) * (b))))
  82. 0082congr
  83. 0083refl
  84. 0084apply mul_comm
  85. 0085apply natural_mul_swap_right_tail
  86. 0086congr
  87. 0087refl
  88. 0088trans ((b) * (n))
  89. 0089apply mul_comm
  90. 0090congr
  91. 0091refl
  92. 0092refl
  93. 0093congr
  94. 0094trans ((R) * (b))
  95. 0095apply mul_comm
  96. 0096congr
  97. 0097refl
  98. 0098refl
  99. 0099congr
  100. 0100trans ((C) * (((d) * (b))))
  101. 0101trans ((d) * (((C) * (b))))
  102. 0102congr
  103. 0103refl
  104. 0104apply mul_comm
  105. 0105apply natural_mul_swap_right_tail
  106. 0106congr
  107. 0107refl
  108. 0108trans ((b) * (d))
  109. 0109apply mul_comm
  110. 0110congr
  111. 0111refl
  112. 0112refl
  113. 0113congr
  114. 0114trans ((R) * (((d) * (n))))
  115. 0115trans ((d) * (((R) * (n))))
  116. 0116congr
  117. 0117refl
  118. 0118apply mul_comm
  119. 0119apply natural_mul_swap_right_tail
  120. 0120congr
  121. 0121refl
  122. 0122refl
  123. 0123trans ((C) * (((d) * (d))))
  124. 0124trans ((d) * (((C) * (d))))
  125. 0125congr
  126. 0126refl
  127. 0127apply mul_comm
  128. 0128apply natural_mul_swap_right_tail
  129. 0129congr
  130. 0130refl
  131. 0131refl
  132. 0132symm
  133. 0133simp [add_mul, mul_add, mul_assoc, add_assoc, mul_succ_left, mul_zero_left, zero_add, one_mul, mul_one]