Theory Metamath_interface

(* 
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\_interface.thy}*}

theory Metamath_interface imports Complex_ZF MMI_prelude

begin

text{*This theory contains some lemmas that make it possible to use 
  the theorems translated from Metamath in a the @{text "complex0"} 
  context.*}

text{*The next lemma states that we can use 
  the theorems proven in the @{text "MMIsar0"} context in
  the @{text "complex0"} context. Unfortunately we have to 
  use low level Isabelle methods "rule" and "unfold" in the proof, simp
  and blast fail on the order axioms.
  *}

lemma (in complex0) MMIsar_valid: 
  shows "MMIsar0(\<real>,\<complex>,\<one>,\<zero>,\<i>,CplxAdd(R,A),CplxMul(R,A,M),
  StrictVersion(CplxROrder(R,A,r)))"
proof -;
  let ?real = "\<real>"
  let ?complex = "\<complex>"
  let ?zero = "\<zero>"
  let ?one = "\<one>"
  let ?iunit = "\<i>"
  let ?caddset = "CplxAdd(R,A)"
  let ?cmulset = "CplxMul(R,A,M)"
  let ?lessrrel = "StrictVersion(CplxROrder(R,A,r))"
  have "\<real> ⊆ \<complex>" using axresscn by simp;
  moreover have "\<one> ≠ \<zero>" using ax1ne0 by simp;
  moreover have "\<complex> isASet" by simp;
  moreover have " CplxAdd(R,A) : \<complex> × \<complex> -> \<complex>" 
    using axaddopr by simp;
  moreover have "CplxMul(R,A,M) : \<complex> × \<complex> -> \<complex>" 
    using axmulopr by simp;
  moreover have 
    "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> --> a· b = b · a"
    using axmulcom by simp;
  moreover have "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> --> a \<ca> b ∈ \<complex>"
    using axaddcl by simp;
  moreover have "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> --> a · b ∈ \<complex>"
    using axmulcl by simp;
  moreover have 
    "∀a b C. a ∈ \<complex> ∧ b ∈ \<complex> ∧ C ∈ \<complex> --> 
    a · (b \<ca> C) = a · b \<ca> a · C"
    using axdistr by simp;
  moreover have "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> -->
         a \<ca> b = b \<ca> a"
    using axaddcom by simp;
  moreover have "∀a b C. a ∈ \<complex> ∧ b ∈ \<complex> ∧ C ∈ \<complex> -->
      a \<ca> b \<ca> C = a \<ca> (b \<ca> C)"
    using axaddass by simp;
  moreover have 
    "∀a b c. a ∈ \<complex> ∧ b ∈ \<complex> ∧ c ∈ \<complex> --> a·b·c = a·(b·c)"
    using axmulass by simp;
  moreover have "\<one> ∈ \<real>" using ax1re by simp;
  moreover have "\<i>·\<i> \<ca> \<one> = \<zero>"
    using axi2m1 by simp;
  moreover have "∀a. a ∈ \<complex> --> a \<ca> \<zero> = a"
    using ax0id by simp;
  moreover have "\<i> ∈ \<complex>" using axicn by simp;
  moreover have "∀a. a ∈ \<complex> --> (∃x∈\<complex>. a \<ca> x = \<zero>)"
    using axnegex by simp;
  moreover have "∀a. a ∈ \<complex> ∧ a ≠ \<zero> --> (∃x∈\<complex>. a · x = \<one>)"
    using axrecex by simp;
  moreover have "∀a. a ∈ \<complex> --> a·\<one> = a"
    using ax1id by simp;
  moreover have "∀a b. a ∈ \<real> ∧ b ∈ \<real> --> a \<ca> b ∈ \<real>"
    using axaddrcl by simp;
  moreover have "∀a b. a ∈ \<real> ∧ b ∈ \<real> --> a · b ∈ \<real>"
    using axmulrcl by simp;
  moreover have "∀a. a ∈ \<real> --> (∃x∈\<real>. a \<ca> x = \<zero>)"
    using axrnegex by simp;
  moreover have "∀a. a ∈ \<real> ∧ a≠\<zero> --> (∃x∈\<real>. a · x = \<one>)"
    using axrrecex by simp
  ultimately have "?real ⊆ ?complex ∧
    ?one ≠ ?zero ∧
    ?complex isASet ∧
    ?caddset ∈ ?complex × ?complex -> ?complex ∧
    ?cmulset ∈ ?complex × ?complex -> ?complex ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex -->
    ?cmulset ` ⟨A, B⟩ = ?cmulset ` ⟨B, A⟩) ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex --> ?caddset ` ⟨A, B⟩ ∈ ?complex) ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex --> ?cmulset ` ⟨A, B⟩ ∈ ?complex) ∧
    (∀A B C.
    A ∈ ?complex ∧ B ∈ ?complex ∧ C ∈ ?complex -->
    ?cmulset ` ⟨A, ?caddset ` ⟨B, C⟩⟩ =
    ?caddset ` ⟨?cmulset ` ⟨A, B⟩, ?cmulset ` ⟨A, C⟩⟩) ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex -->
    ?caddset ` ⟨A, B⟩ = ?caddset ` ⟨B, A⟩) ∧
    (∀A B C.
    A ∈ ?complex ∧ B ∈ ?complex ∧ C ∈ ?complex -->
    ?caddset ` ⟨?caddset ` ⟨A, B⟩, C⟩ =
    ?caddset ` ⟨A, ?caddset ` ⟨B, C⟩⟩) ∧
    (∀A B C.
    A ∈ ?complex ∧ B ∈ ?complex ∧ C ∈ ?complex -->
    ?cmulset ` ⟨?cmulset ` ⟨A, B⟩, C⟩ =
    ?cmulset ` ⟨A, ?cmulset ` ⟨B, C⟩⟩) ∧
    ?one ∈ ?real ∧
    ?caddset ` ⟨?cmulset ` ⟨?iunit, ?iunit⟩, ?one⟩ = ?zero ∧
    (∀A. A ∈ ?complex --> ?caddset ` ⟨A, ?zero⟩ = A) ∧
    ?iunit ∈ ?complex ∧
    (∀A. A ∈ ?complex --> (∃x∈?complex. ?caddset ` ⟨A, x⟩ = ?zero)) ∧
    (∀A. A ∈ ?complex ∧ A ≠ ?zero --> 
    (∃x∈?complex. ?cmulset ` ⟨A, x⟩ = ?one)) ∧
    (∀A. A ∈ ?complex --> ?cmulset ` ⟨A, ?one⟩ = A) ∧
    (∀A B. A ∈ ?real ∧ B ∈ ?real --> ?caddset ` ⟨A, B⟩ ∈ ?real) ∧
    (∀A B. A ∈ ?real ∧ B ∈ ?real --> ?cmulset ` ⟨A, B⟩ ∈ ?real) ∧
    (∀A. A ∈ ?real --> (∃x∈?real. ?caddset ` ⟨A, x⟩ = ?zero)) ∧
    (∀A. A ∈ ?real ∧ A ≠ ?zero --> (∃x∈?real. ?cmulset ` ⟨A, x⟩ = ?one))"
    by simp;
  moreover have "(∀a b. a ∈ ?real ∧ b ∈ ?real -->
    ⟨a, b⟩ ∈ ?lessrrel <-> ¬ (a = b ∨ ⟨b, a⟩ ∈ ?lessrrel))"
  proof -
    have I:
      "∀a b. a ∈ \<real> ∧ b ∈ \<real> --> (a \<lsr> b <-> ¬(a=b ∨ b \<lsr> a))"
      using pre_axlttri by blast;
    { fix a b assume "a ∈ ?real ∧ b ∈ ?real"
      with I have "(a \<lsr> b <-> ¬(a=b ∨ b \<lsr> a))"
        by blast;
      hence
        "⟨a, b⟩ ∈ ?lessrrel <-> ¬ (a = b ∨ ⟨b, a⟩ ∈ ?lessrrel)"
        by simp;
    } thus "(∀a b. a ∈ ?real ∧ b ∈ ?real -->
        (⟨a, b⟩ ∈ ?lessrrel <-> ¬ (a = b ∨ ⟨b, a⟩ ∈ ?lessrrel)))"
      by blast;
  qed;
  moreover have "(∀a b c.
    a ∈ ?real ∧ b ∈ ?real ∧ c ∈ ?real -->
    ⟨a, b⟩ ∈ ?lessrrel ∧ ⟨b, c⟩ ∈ ?lessrrel --> ⟨a, c⟩ ∈ ?lessrrel)"
  proof -
    have II: "∀a b c.  a ∈ \<real> ∧ b ∈ \<real> ∧ c ∈ \<real> --> 
      ((a \<lsr> b ∧ b \<lsr> c) --> a \<lsr> c)"
      using pre_axlttrn by blast;
    { fix a b c assume "a ∈ ?real ∧ b ∈ ?real ∧ c ∈ ?real"
      with II have "(a \<lsr> b ∧ b \<lsr> c) --> a \<lsr> c"
        by blast;
      hence     
        "⟨a, b⟩ ∈ ?lessrrel ∧ ⟨b, c⟩ ∈ ?lessrrel --> ⟨a, c⟩ ∈ ?lessrrel"
        by simp;
    } thus  "(∀a b c.
        a ∈ ?real ∧ b ∈ ?real ∧ c ∈ ?real -->
        ⟨a, b⟩ ∈ ?lessrrel ∧ ⟨b, c⟩ ∈ ?lessrrel --> ⟨a, c⟩ ∈ ?lessrrel)"
      by blast;
  qed;
  moreover have "(∀A B C.
    A ∈ ?real ∧ B ∈ ?real ∧ C ∈ ?real -->
    ⟨A, B⟩ ∈ ?lessrrel -->
    ⟨?caddset ` ⟨C, A⟩, ?caddset ` ⟨C, B⟩⟩ ∈ ?lessrrel)"
    using pre_axltadd by simp;
  moreover have "(∀A B. A ∈ ?real ∧ B ∈ ?real -->
    ⟨?zero, A⟩ ∈ ?lessrrel ∧ ⟨?zero, B⟩ ∈ ?lessrrel -->
    ⟨?zero, ?cmulset ` ⟨A, B⟩⟩ ∈ ?lessrrel)"
    using pre_axmulgt0 by simp;
  moreover have 
    "(∀A. A ⊆ ?real ∧ A ≠ 0 ∧ (∃x∈?real. ∀y∈A. ⟨y, x⟩ ∈ ?lessrrel) -->
    (∃x∈?real.
    (∀y∈A. ⟨x, y⟩ ∉ ?lessrrel) ∧
    (∀y∈?real. ⟨y, x⟩ ∈ ?lessrrel --> (∃z∈A. ⟨y, z⟩ ∈ ?lessrrel))))"
    using pre_axsup by simp;
  ultimately have 
    "(∀A B. A ∈ ?real ∧ B ∈ ?real -->
    ⟨A, B⟩ ∈ ?lessrrel <-> ¬ (A = B ∨ ⟨B, A⟩ ∈ ?lessrrel)) ∧
    (∀A B C.
    A ∈ ?real ∧ B ∈ ?real ∧ C ∈ ?real -->
    ⟨A, B⟩ ∈ ?lessrrel ∧ ⟨B, C⟩ ∈ ?lessrrel --> ⟨A, C⟩ ∈ ?lessrrel) ∧
    (∀A B C.
    A ∈ ?real ∧ B ∈ ?real ∧ C ∈ ?real -->
    ⟨A, B⟩ ∈ ?lessrrel -->
    ⟨?caddset ` ⟨C, A⟩, ?caddset ` ⟨C, B⟩⟩ ∈ ?lessrrel) ∧
    (∀A B. A ∈ ?real ∧ B ∈ ?real -->
    ⟨?zero, A⟩ ∈ ?lessrrel ∧ ⟨?zero, B⟩ ∈ ?lessrrel -->
    ⟨?zero, ?cmulset ` ⟨A, B⟩⟩ ∈ ?lessrrel) ∧
    (∀A. A ⊆ ?real ∧ A ≠ 0 ∧ (∃x∈?real. ∀y∈A. ⟨y, x⟩ ∈ ?lessrrel) -->
    (∃x∈?real.
    (∀y∈A. ⟨x, y⟩ ∉ ?lessrrel) ∧
    (∀y∈?real. ⟨y, x⟩ ∈ ?lessrrel --> (∃z∈A. ⟨y, z⟩ ∈ ?lessrrel)))) ∧
    ?real ⊆ ?complex ∧
    ?one ≠ ?zero ∧
    ?complex isASet ∧
    ?caddset ∈ ?complex × ?complex -> ?complex ∧
    ?cmulset ∈ ?complex × ?complex -> ?complex ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex -->
    ?cmulset ` ⟨A, B⟩ = ?cmulset ` ⟨B, A⟩) ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex --> ?caddset ` ⟨A, B⟩ ∈ ?complex) ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex --> ?cmulset ` ⟨A, B⟩ ∈ ?complex) ∧
    (∀A B C.
    A ∈ ?complex ∧ B ∈ ?complex ∧ C ∈ ?complex -->
    ?cmulset ` ⟨A, ?caddset ` ⟨B, C⟩⟩ =
    ?caddset ` ⟨?cmulset ` ⟨A, B⟩, ?cmulset ` ⟨A, C⟩⟩) ∧
    (∀A B. A ∈ ?complex ∧ B ∈ ?complex -->
    ?caddset ` ⟨A, B⟩ = ?caddset ` ⟨B, A⟩) ∧
    (∀A B C. A ∈ ?complex ∧ B ∈ ?complex ∧ C ∈ ?complex -->
    ?caddset ` ⟨?caddset ` ⟨A, B⟩, C⟩ =
    ?caddset ` ⟨A, ?caddset ` ⟨B, C⟩⟩) ∧
    (∀A B C. A ∈ ?complex ∧ B ∈ ?complex ∧ C ∈ ?complex -->
    ?cmulset ` ⟨?cmulset ` ⟨A, B⟩, C⟩ = ?cmulset ` ⟨A, ?cmulset ` ⟨B, C⟩⟩) ∧
    ?one ∈ ?real ∧
    ?caddset ` ⟨?cmulset ` ⟨?iunit, ?iunit⟩, ?one⟩ = ?zero ∧
    (∀A. A ∈ ?complex --> ?caddset ` ⟨A, ?zero⟩ = A) ∧
    ?iunit ∈ ?complex ∧
    (∀A. A ∈ ?complex --> (∃x∈?complex. ?caddset ` ⟨A, x⟩ = ?zero)) ∧
    (∀A. A ∈ ?complex ∧ A ≠ ?zero -->
    (∃x∈?complex. ?cmulset ` ⟨A, x⟩ = ?one)) ∧
    (∀A. A ∈ ?complex --> ?cmulset ` ⟨A, ?one⟩ = A) ∧
    (∀A B. A ∈ ?real ∧ B ∈ ?real --> ?caddset ` ⟨A, B⟩ ∈ ?real) ∧
    (∀A B. A ∈ ?real ∧ B ∈ ?real --> ?cmulset ` ⟨A, B⟩ ∈ ?real) ∧
    (∀A. A ∈ ?real --> (∃x∈?real. ?caddset ` ⟨A, x⟩ = ?zero)) ∧
    (∀A. A ∈ ?real ∧ A ≠ ?zero --> (∃x∈?real. ?cmulset ` ⟨A, x⟩ = ?one))"
    by (rule five_more_conj);
  thus  "MMIsar0(\<real>,\<complex>,\<one>,\<zero>,\<i>,CplxAdd(R,A),CplxMul(R,A,M),
    StrictVersion(CplxROrder(R,A,r)))" by (unfold MMIsar0_def);
qed;

text{*In @{text "complex0"} context the strict version of the order
  relation on complex reals is a relation on complex reals.*}

end