Theory Metamath_interface

theory Metamath_interface
imports Complex_ZF MMI_prelude
(* 
This file is a part of IsarMathLib -
a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2006 Slawomir Kolodynski

This program is free software; Redistribution and use in source and binary forms,
with or without modification, are permitted provided that the
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derived from this software without specific prior written permission.

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WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
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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.*}


section{*MMisar0 and complex0 contexts.*}

text{*In the section we show a lemma that the assumptions in
@{text "complex0"} context imply the assumptions of the @{text "MMIsar0"}
context. The @{text "Metamath_sampler"} theory provides examples
how this lemma can be used.*}


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 "(∀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
"(∀S. S ⊆ ?real ∧ S ≠ 0 ∧ (∃x∈?real. ∀y∈S. ⟨y, x⟩ ∈ ?lessrrel) -->
(∃x∈?real.
(∀y∈S. ⟨x, y⟩ ∉ ?lessrrel) ∧
(∀y∈?real. ⟨y, x⟩ ∈ ?lessrrel --> (∃z∈S. ⟨y, z⟩ ∈ ?lessrrel))))"

using pre_axsup by simp;
moreover 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;
hence "(∀a b. a ∈ \<complex> ∧ b ∈ \<complex> -->
?cmulset ` ⟨a, b⟩ = ?cmulset ` ⟨b, a⟩
)"
by simp;
moreover have "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> --> a \<ca> b ∈ \<complex>"
using axaddcl by simp;
hence "(∀a b. a ∈ \<complex> ∧ b ∈ \<complex> -->
?caddset ` ⟨a, b⟩ ∈ \<complex>
)"
by simp;
moreover have "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> --> a · b ∈ \<complex>"
using axmulcl by simp
hence "(∀a b. a ∈ \<complex> ∧ b ∈ \<complex> -->
?cmulset ` ⟨a, b⟩ ∈ \<complex> )"
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
hence "∀a b C.
a ∈ \<complex> ∧ b ∈ \<complex> ∧ C ∈ \<complex> -->
?cmulset ` ⟨a, ?caddset ` ⟨b, C⟩⟩ =
?caddset `
⟨?cmulset ` ⟨a, b⟩, ?cmulset ` ⟨a, C⟩⟩"

by simp;
moreover have "∀a b. a ∈ \<complex> ∧ b ∈ \<complex> -->
a \<ca> b = b \<ca> a"

using axaddcom by simp;
hence "∀a b.
a ∈ \<complex> ∧ b ∈ \<complex> -->
?caddset ` ⟨a, b⟩ = ?caddset ` ⟨b, a⟩"
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
hence "∀a b C.
a ∈ \<complex> ∧ b ∈ \<complex> ∧ C ∈ \<complex> -->
?caddset ` ⟨?caddset ` ⟨a, b⟩, C⟩ =
?caddset ` ⟨a, ?caddset ` ⟨b, C⟩⟩"
by simp;
moreover have
"∀a b c. a ∈ \<complex> ∧ b ∈ \<complex> ∧ c ∈ \<complex> --> a·b·c = a·(b·c)"
using axmulass by simp
hence "∀a b C.
a ∈ \<complex> ∧ b ∈ \<complex> ∧ C ∈ \<complex> -->
?cmulset ` ⟨?cmulset ` ⟨a, b⟩, C⟩ =
?cmulset ` ⟨a, ?cmulset ` ⟨b, C⟩⟩"
by simp;
moreover have "\<one> ∈ \<real>" using ax1re by simp;
moreover have "\<i>·\<i> \<ca> \<one> = \<zero>"
using axi2m1 by simp
hence "?caddset ` ⟨?cmulset ` ⟨\<i>, \<i>⟩, \<one>⟩ = \<zero>" by simp;
moreover have "∀a. a ∈ \<complex> --> a \<ca> \<zero> = a"
using ax0id by simp;
hence "∀a. a ∈ \<complex> --> ?caddset ` ⟨a, \<zero>⟩ = a" 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;
hence "∀a. a ∈ \<complex> -->
(∃x∈\<complex>. ?caddset ` ⟨a, x⟩ = \<zero>)"
by simp;
moreover have "∀a. a ∈ \<complex> ∧ a ≠ \<zero> --> (∃x∈\<complex>. a · x = \<one>)"
using axrecex by simp
hence "∀a. a ∈ \<complex> ∧ a ≠ \<zero> -->
( ∃x∈\<complex>. ?cmulset ` ⟨a, x⟩ = \<one> )"
by simp;
moreover have "∀a. a ∈ \<complex> --> a·\<one> = a"
using ax1id by simp
hence " ∀a. a ∈ \<complex> -->
?cmulset ` ⟨a, \<one>⟩ = a"
by simp;
moreover have "∀a b. a ∈ \<real> ∧ b ∈ \<real> --> a \<ca> b ∈ \<real>"
using axaddrcl by simp;
hence "∀a b. a ∈ \<real> ∧ b ∈ \<real> -->
?caddset ` ⟨a, b⟩ ∈ \<real>"
by simp;
moreover have "∀a b. a ∈ \<real> ∧ b ∈ \<real> --> a · b ∈ \<real>"
using axmulrcl by simp;
hence "∀a b. a ∈ \<real> ∧ b ∈ \<real> -->
?cmulset ` ⟨a, b⟩ ∈ \<real>"
by simp;
moreover have "∀a. a ∈ \<real> --> (∃x∈\<real>. a \<ca> x = \<zero>)"
using axrnegex by simp
hence "∀a. a ∈ \<real> -->
( ∃x∈\<real>. ?caddset ` ⟨a, x⟩ = \<zero> )"
by simp
moreover have "∀a. a ∈ \<real> ∧ a≠\<zero> --> (∃x∈\<real>. a · x = \<one>)"
using axrrecex by simp
hence "∀a. a ∈ \<real> ∧ a ≠ \<zero> -->
( ∃x∈\<real>. ?cmulset ` ⟨a, x⟩ = \<one>)"
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
) ∧

( ∀S. S ⊆ \<real> ∧ S ≠ 0 ∧
( ∃x∈\<real>. ∀y∈S. ⟨y, x⟩ ∈ ?lessrrel
) -->
( ∃x∈\<real>.
( ∀y∈S. ⟨x, y⟩ ∉ ?lessrrel
) ∧
( ∀y∈\<real>. ⟨y, x⟩ ∈ ?lessrrel -->
( ∃z∈S. ⟨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 ` ⟨\<i>, \<i>⟩, \<one>⟩ = \<zero>
) ∧

(∀a. a ∈ \<complex> --> ?caddset ` ⟨a, \<zero>⟩ = a
) ∧

\<i> ∈ \<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 blast;
then show "MMIsar0(\<real>,\<complex>,\<one>,\<zero>,\<i>,CplxAdd(R,A),CplxMul(R,A,M),
StrictVersion(CplxROrder(R,A,r)))"
unfolding MMIsar0_def by blast;
qed;


end