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Step * 2 1 of Lemma rcos-seq-converges-to-half-pi


1. π/2(slower)
= (λn.eval m = 4 * n in
      (rcos-seq(exp-ratio(1;3164556962025316455;0;m;1000000000) + 1) m) ÷ 4)
∈ (ℕ+ ⟶ ℤ)
2. snd(TERMOF{rcos-seq-converges-ext:o, \\v:l}) ∈ lim n→∞.rcos-seq(n) = fst(TERMOF{rcos-seq-converges-ext:o, \\v:l})
⊢ lim n→∞.rcos-seq(n) = π/2(slower)
BY
{ ((Subst' TERMOF{rcos-seq-converges-ext:o, \\v:l} ~ <λn.eval m = 4 * n in
                                                         (rcos-seq(exp-ratio(1;3164556962025316455;0;m;1000000000) + 1) 
                                                          m) ÷ 4
                                                     , λk.((exp-ratio(1;3164556962025316455;0;4 * k;1000000000) + 1)
                                                          + 1)
                                                     > -1
    THENA (RW (AddrC [1] (TagC (mk_tag_term 1))) 0 THEN Auto)
    )
   THEN Reduce -1
   ) }

1
1. π/2(slower)
= (λn.eval m = 4 * n in
      (rcos-seq(exp-ratio(1;3164556962025316455;0;m;1000000000) + 1) m) ÷ 4)
∈ (ℕ+ ⟶ ℤ)
2. λk.((exp-ratio(1;3164556962025316455;0;4 * k;1000000000) + 1) + 1)
   ∈ lim n→∞.rcos-seq(n) = λn.eval m = 4 * n in
                              (rcos-seq(exp-ratio(1;3164556962025316455;0;m;1000000000) + 1) m) ÷ 4
⊢ lim n→∞.rcos-seq(n) = π/2(slower)

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Latex:

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Latex:

1.  \mpi{}/2(slower)
=  (\mlambda{}n.eval  m  =  4  *  n  in
            (rcos-seq(exp-ratio(1;3164556962025316455;0;m;1000000000)  +  1)  m)  \mdiv{}  4)
2.  snd(TERMOF\{rcos-seq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\})
      \mmember{}  lim  n\mrightarrow{}\minfty{}.rcos-seq(n)  =  fst(TERMOF\{rcos-seq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\})
\mvdash{}  lim  n\mrightarrow{}\minfty{}.rcos-seq(n)  =  \mpi{}/2(slower)


By

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Latex:
((Subst'  TERMOF\{rcos-seq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\} 
    \msim{}  <\mlambda{}n.eval  m  =  4  *  n  in
                (rcos-seq(exp-ratio(1;3164556962025316455;0;m;1000000000)  +  1)  m)  \mdiv{}  4
        ,  \mlambda{}k.((exp-ratio(1;3164556962025316455;0;4  *  k;1000000000)  +  1)  +  1)
        >  -1
    THENA  (RW  (AddrC  [1]  (TagC  (mk\_tag\_term  1)))  0  THEN  Auto)
    )
  THEN  Reduce  -1
  )

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