(* 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 following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Metamath interface› 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.*} subsection{*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(ℝ,ℂ,𝟭,𝟬,𝗂,CplxAdd(R,A),CplxMul(R,A,M), StrictVersion(CplxROrder(R,A,r)))" proof - let ?real = "ℝ" let ?complex = "ℂ" let ?zero = "𝟬" let ?one = "𝟭" 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 ∈ ℝ ∧ b ∈ ℝ ⟶ (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 ∈ ℝ ∧ b ∈ ℝ ∧ c ∈ ℝ ⟶ ((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 "ℝ ⊆ ℂ" using axresscn by simp moreover have "𝟭 ≠ 𝟬" using ax1ne0 by simp moreover have "ℂ isASet" by simp moreover have " CplxAdd(R,A) : ℂ × ℂ → ℂ" using axaddopr by simp moreover have "CplxMul(R,A,M) : ℂ × ℂ → ℂ" using axmulopr by simp moreover have "∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ a⋅ b = b ⋅ a" using axmulcom by simp hence "(∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?cmulset ` ⟨a, b⟩ = ?cmulset ` ⟨b, a⟩ )" by simp moreover have "∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ a \<ca> b ∈ ℂ" using axaddcl by simp hence "(∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?caddset ` ⟨a, b⟩ ∈ ℂ )" by simp moreover have "∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ a ⋅ b ∈ ℂ" using axmulcl by simp hence "(∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?cmulset ` ⟨a, b⟩ ∈ ℂ )" by simp moreover have "∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ a ⋅ (b \<ca> C) = a ⋅ b \<ca> a ⋅ C" using axdistr by simp hence "∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ ?cmulset ` ⟨a, ?caddset ` ⟨b, C⟩⟩ = ?caddset ` ⟨?cmulset ` ⟨a, b⟩, ?cmulset ` ⟨a, C⟩⟩" by simp moreover have "∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ a \<ca> b = b \<ca> a" using axaddcom by simp hence "∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?caddset ` ⟨a, b⟩ = ?caddset ` ⟨b, a⟩" by simp moreover have "∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ a \<ca> b \<ca> C = a \<ca> (b \<ca> C)" using axaddass by simp hence "∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ ?caddset ` ⟨?caddset ` ⟨a, b⟩, C⟩ = ?caddset ` ⟨a, ?caddset ` ⟨b, C⟩⟩" by simp moreover have "∀a b c. a ∈ ℂ ∧ b ∈ ℂ ∧ c ∈ ℂ ⟶ a⋅b⋅c = a⋅(b⋅c)" using axmulass by simp hence "∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ ?cmulset ` ⟨?cmulset ` ⟨a, b⟩, C⟩ = ?cmulset ` ⟨a, ?cmulset ` ⟨b, C⟩⟩" by simp moreover have "𝟭 ∈ ℝ" using ax1re by simp moreover have "𝗂⋅𝗂 \<ca> 𝟭 = 𝟬" using axi2m1 by simp hence "?caddset ` ⟨?cmulset ` ⟨𝗂, 𝗂⟩, 𝟭⟩ = 𝟬" by simp moreover have "∀a. a ∈ ℂ ⟶ a \<ca> 𝟬 = a" using ax0id by simp hence "∀a. a ∈ ℂ ⟶ ?caddset ` ⟨a, 𝟬⟩ = a" by simp moreover have "𝗂 ∈ ℂ" using axicn by simp moreover have "∀a. a ∈ ℂ ⟶ (∃x∈ℂ. a \<ca> x = 𝟬)" using axnegex by simp hence "∀a. a ∈ ℂ ⟶ (∃x∈ℂ. ?caddset ` ⟨a, x⟩ = 𝟬)" by simp moreover have "∀a. a ∈ ℂ ∧ a ≠ 𝟬 ⟶ (∃x∈ℂ. a ⋅ x = 𝟭)" using axrecex by simp hence "∀a. a ∈ ℂ ∧ a ≠ 𝟬 ⟶ ( ∃x∈ℂ. ?cmulset ` ⟨a, x⟩ = 𝟭 )" by simp moreover have "∀a. a ∈ ℂ ⟶ a⋅𝟭 = a" using ax1id by simp hence " ∀a. a ∈ ℂ ⟶ ?cmulset ` ⟨a, 𝟭⟩ = a" by simp moreover have "∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ a \<ca> b ∈ ℝ" using axaddrcl by simp hence "∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ ?caddset ` ⟨a, b⟩ ∈ ℝ" by simp moreover have "∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ a ⋅ b ∈ ℝ" using axmulrcl by simp hence "∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ ?cmulset ` ⟨a, b⟩ ∈ ℝ" by simp moreover have "∀a. a ∈ ℝ ⟶ (∃x∈ℝ. a \<ca> x = 𝟬)" using axrnegex by simp hence "∀a. a ∈ ℝ ⟶ ( ∃x∈ℝ. ?caddset ` ⟨a, x⟩ = 𝟬 )" by simp moreover have "∀a. a ∈ ℝ ∧ a≠𝟬 ⟶ (∃x∈ℝ. a ⋅ x = 𝟭)" using axrrecex by simp hence "∀a. a ∈ ℝ ∧ a ≠ 𝟬 ⟶ ( ∃x∈ℝ. ?cmulset ` ⟨a, x⟩ = 𝟭)" by simp ultimately have "( ( ( ( ∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ ⟨a, b⟩ ∈ ?lessrrel ⟷ ¬ (a = b ∨ ⟨b, a⟩ ∈ ?lessrrel) ) ∧ ( ∀a b C. a ∈ ℝ ∧ b ∈ ℝ ∧ C ∈ ℝ ⟶ ⟨a, b⟩ ∈ ?lessrrel ∧ ⟨b, C⟩ ∈ ?lessrrel ⟶ ⟨a, C⟩ ∈ ?lessrrel ) ∧ (∀a b C. a ∈ ℝ ∧ b ∈ ℝ ∧ C ∈ ℝ ⟶ ⟨a, b⟩ ∈ ?lessrrel ⟶ ⟨?caddset ` ⟨C, a⟩, ?caddset ` ⟨C, b⟩⟩ ∈ ?lessrrel ) ) ∧ ( ( ∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ ⟨𝟬, a⟩ ∈ ?lessrrel ∧ ⟨𝟬, b⟩ ∈ ?lessrrel ⟶ ⟨𝟬, ?cmulset ` ⟨a, b⟩⟩ ∈ ?lessrrel ) ∧ ( ∀S. S ⊆ ℝ ∧ S ≠ 0 ∧ ( ∃x∈ℝ. ∀y∈S. ⟨y, x⟩ ∈ ?lessrrel ) ⟶ ( ∃x∈ℝ. ( ∀y∈S. ⟨x, y⟩ ∉ ?lessrrel ) ∧ ( ∀y∈ℝ. ⟨y, x⟩ ∈ ?lessrrel ⟶ ( ∃z∈S. ⟨y, z⟩ ∈ ?lessrrel ) ) ) ) ) ∧ ℝ ⊆ ℂ ∧ 𝟭 ≠ 𝟬 ) ∧ ( ℂ isASet ∧ ?caddset ∈ ℂ × ℂ → ℂ ∧ ?cmulset ∈ ℂ × ℂ → ℂ ) ∧ ( (∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?cmulset ` ⟨a, b⟩ = ?cmulset ` ⟨b, a⟩ ) ∧ (∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?caddset ` ⟨a, b⟩ ∈ ℂ ) ) ∧ (∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?cmulset ` ⟨a, b⟩ ∈ ℂ ) ∧ (∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ ?cmulset ` ⟨a, ?caddset ` ⟨b, C⟩⟩ = ?caddset ` ⟨?cmulset ` ⟨a, b⟩, ?cmulset ` ⟨a, C⟩⟩ ) ) ∧ ( ( (∀a b. a ∈ ℂ ∧ b ∈ ℂ ⟶ ?caddset ` ⟨a, b⟩ = ?caddset ` ⟨b, a⟩ ) ∧ (∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ ?caddset ` ⟨?caddset ` ⟨a, b⟩, C⟩ = ?caddset ` ⟨a, ?caddset ` ⟨b, C⟩⟩ ) ∧ (∀a b C. a ∈ ℂ ∧ b ∈ ℂ ∧ C ∈ ℂ ⟶ ?cmulset ` ⟨?cmulset ` ⟨a, b⟩, C⟩ = ?cmulset ` ⟨a, ?cmulset ` ⟨b, C⟩⟩ ) ) ∧ (𝟭 ∈ ℝ ∧ ?caddset ` ⟨?cmulset ` ⟨𝗂, 𝗂⟩, 𝟭⟩ = 𝟬 ) ∧ (∀a. a ∈ ℂ ⟶ ?caddset ` ⟨a, 𝟬⟩ = a ) ∧ 𝗂 ∈ ℂ ) ∧ ( (∀a. a ∈ ℂ ⟶ (∃x∈ℂ. ?caddset ` ⟨a, x⟩ = 𝟬 ) ) ∧ ( ∀a. a ∈ ℂ ∧ a ≠ 𝟬 ⟶ ( ∃x∈ℂ. ?cmulset ` ⟨a, x⟩ = 𝟭 ) ) ∧ ( ∀a. a ∈ ℂ ⟶ ?cmulset ` ⟨a, 𝟭⟩ = a ) ) ∧ ( ( ∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ ?caddset ` ⟨a, b⟩ ∈ ℝ ) ∧ ( ∀a b. a ∈ ℝ ∧ b ∈ ℝ ⟶ ?cmulset ` ⟨a, b⟩ ∈ ℝ ) ) ∧ ( ∀a. a ∈ ℝ ⟶ ( ∃x∈ℝ. ?caddset ` ⟨a, x⟩ = 𝟬 ) ) ∧ ( ∀a. a ∈ ℝ ∧ a ≠ 𝟬 ⟶ ( ∃x∈ℝ. ?cmulset ` ⟨a, x⟩ = 𝟭 ) )" by blast then show "MMIsar0(ℝ,ℂ,𝟭,𝟬,𝗂,CplxAdd(R,A),CplxMul(R,A,M), StrictVersion(CplxROrder(R,A,r)))" unfolding MMIsar0_def by blast qed end