Step
*
2
of Lemma
Hofstadter_wf
1. ∀n:ℕ. ((0 < n
⇒ (HofstadterF(n) ∈ ℕn + 1)) ∧ (HofstadterM(n) ∈ ℕn + 1))
⊢ ∀n:ℤ. ((HofstadterF(n) ∈ if 0 <z n then ℕn + 1 else ℕ+2 fi ) ∧ (HofstadterM(n) ∈ if 0 ≤z n then ℕn + 1 else ℕ1 fi ))
BY
{ ((D 0 THENA Auto) THEN (Decide ⌜0 ≤ n⌝⋅ THENA Auto)) }
1
1. ∀n:ℕ. ((0 < n
⇒ (HofstadterF(n) ∈ ℕn + 1)) ∧ (HofstadterM(n) ∈ ℕn + 1))
2. n : ℤ@i
3. 0 ≤ n
⊢ (HofstadterF(n) ∈ if 0 <z n then ℕn + 1 else ℕ+2 fi ) ∧ (HofstadterM(n) ∈ if 0 ≤z n then ℕn + 1 else ℕ1 fi )
2
1. ∀n:ℕ. ((0 < n
⇒ (HofstadterF(n) ∈ ℕn + 1)) ∧ (HofstadterM(n) ∈ ℕn + 1))
2. n : ℤ@i
3. ¬(0 ≤ n)
⊢ (HofstadterF(n) ∈ if 0 <z n then ℕn + 1 else ℕ+2 fi ) ∧ (HofstadterM(n) ∈ if 0 ≤z n then ℕn + 1 else ℕ1 fi )
Latex:
Latex:
1. \mforall{}n:\mBbbN{}. ((0 < n {}\mRightarrow{} (HofstadterF(n) \mmember{} \mBbbN{}n + 1)) \mwedge{} (HofstadterM(n) \mmember{} \mBbbN{}n + 1))
\mvdash{} \mforall{}n:\mBbbZ{}
((HofstadterF(n) \mmember{} if 0 <z n then \mBbbN{}n + 1 else \mBbbN{}\msupplus{}2 fi )
\mwedge{} (HofstadterM(n) \mmember{} if 0 \mleq{}z n then \mBbbN{}n + 1 else \mBbbN{}1 fi ))
By
Latex:
((D 0 THENA Auto) THEN (Decide \mkleeneopen{}0 \mleq{} n\mkleeneclose{}\mcdot{} THENA Auto))
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