Theory Metamath_sampler

(* 
This file is a part of IsarMathLib - 
a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2006  Slawomir Kolodynski

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*)

header {*\isaheader{Metamath\_sampler.thy}*}

theory Metamath_sampler imports Metamath_interface MMI_Complex_ZF_1

begin

text{*This theory file contains some examples of theorems 
 translated fro Metamath and formulated in the @{text "complex0"} context.*}

text{*Metamath uses the set of real numbers extended with $+\infty$ and $-\infty$. 
  The $+\infty$ and $-\infty$ symbols are defined quite arbitrarily as $\mathbb{C}$
  and $\mathbb{\{ C\} }$, respectively. The next lemma that corresponds to 
  Metamath's @{text "renfdisj"} states that $+\infty$ and $-\infty$ are not 
  elements of $\mathbb{R}$.*}

lemma (in complex0) renfdisj: shows "\<real> ∩ {\<cpnf>,\<cmnf>} = 0"
proof -
  let ?real = "\<real>"
  let ?complex = "\<complex>"
  let ?one = "\<one>"
  let ?zero = "\<zero>"
  let ?iunit = "\<i>"
  let ?caddset = "CplxAdd(R,A)"
  let ?cmulset= "CplxMul(R,A,M)"
  let ?lessrrel = "StrictVersion(CplxROrder(R,A,r))"
  have "MMIsar0
    (?real, ?complex, ?one, ?zero, ?iunit, ?caddset, ?cmulset, ?lessrrel)"
    using MMIsar_valid by simp;
  then have "?real ∩ {?complex, {?complex}} = 0"
    by (rule MMIsar0.MMI_renfdisj);
  thus "\<real> ∩ {\<cpnf>,\<cmnf>} = 0" by simp;
qed;
  
text{*  The order relation used most often in Metamath is defined on 
  the set of complex reals extended with   $+\infty$ and $-\infty$. 
  The next lemma
  allows to use Metamath's @{text "xrltso"} that states that the @{text "\<ls>"}
  relations is a strict linear order on the extended set.*}

lemma (in complex0) xrltso: "\<cltrrset> Orders \<real>*"
proof -
  let ?real = "\<real>"
  let ?complex = "\<complex>"
  let ?one = "\<one>"
  let ?zero = "\<zero>"
  let ?iunit = "\<i>"
  let ?caddset = "CplxAdd(R,A)"
  let ?cmulset= "CplxMul(R,A,M)"
  let ?lessrrel = "StrictVersion(CplxROrder(R,A,r))"
  have "MMIsar0
    (?real, ?complex, ?one, ?zero, ?iunit, ?caddset, ?cmulset, ?lessrrel)"
    using MMIsar_valid by simp;
  then have 
    "(?lessrrel ∩ ?real × ?real ∪ 
    {⟨{?complex}, ?complex⟩} ∪ ?real × {?complex} ∪
      {{?complex}} × ?real) Orders (?real ∪ {?complex, {?complex}})"
    by (rule MMIsar0.MMI_xrltso);
  moreover have "?lessrrel ∩ ?real × ?real = ?lessrrel"
    using cplx_strict_ord_on_cplx_reals by auto
  ultimately show "\<cltrrset> Orders \<real>*" by simp;
qed;

end