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Step * of Lemma nim-sum-same

∀[x:ℕ]. (nim-sum(x;x) = 0 ∈ ℤ)
BY
{ ((CompleteInductionOnNat THEN Auto)
   THEN CaseNat 0 `x'
   THEN Reduce 0
   THEN Auto
   THEN (InstLemma `div_rem_sum` [⌜x⌝;⌜2⌝]⋅ THENA Auto)
   THEN (InstLemma `rem_bounds_1` [⌜x⌝;⌜2⌝]⋅ THENA Auto)
   THEN (InstHyp [⌜x ÷ 2⌝] 2⋅ THENA Auto)
   THEN RWO "nim-sum-rec" 0⋅
   THEN Auto) }

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Latex:

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Latex:
\mforall{}[x:\mBbbN{}].  (nim-sum(x;x)  =  0)


By

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Latex:
((CompleteInductionOnNat  THEN  Auto)
  THEN  CaseNat  0  `x'
  THEN  Reduce  0
  THEN  Auto
  THEN  (InstLemma  `div\_rem\_sum`  [\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}2\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (InstLemma  `rem\_bounds\_1`  [\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}2\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (InstHyp  [\mkleeneopen{}x  \mdiv{}  2\mkleeneclose{}]  2\mcdot{}  THENA  Auto)
  THEN  RWO  "nim-sum-rec"  0\mcdot{}
  THEN  Auto)

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