(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2005 - 2008 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 ‹First Order Logic› theory Fol1 imports ZF.Trancl begin text‹Isabelle/ZF builds on the first order logic. Almost everything one would like to have in this area is covered in the standard Isabelle libraries. The material in this theory provides some lemmas that are missing or allow for a more readable proof style.› subsection‹Notions and lemmas in FOL› text‹This section contains mostly shortcuts and workarounds that allow to use more readable coding style.› text‹The next lemma serves as a workaround to problems with applying the definition of transitivity (of a relation) in our coding style (any attempt to do something like ‹using trans_def› puts Isabelle in an infinite loop).› lemma Fol1_L2: assumes A1: "∀ x y z. ⟨x, y⟩ ∈ r ∧ ⟨y, z⟩ ∈ r ⟶ ⟨x, z⟩ ∈ r" shows "trans(r)" proof - from A1 have "∀ x y z. ⟨x, y⟩ ∈ r ⟶ ⟨y, z⟩ ∈ r ⟶ ⟨x, z⟩ ∈ r" using imp_conj by blast then show ?thesis unfolding trans_def by blast qed text‹Another workaround for the problem of Isabelle simplifier looping when the transitivity definition is used.› lemma Fol1_L3: assumes A1: "trans(r)" and A2: "⟨ a,b⟩ ∈ r ∧ ⟨ b,c⟩ ∈ r" shows "⟨ a,c⟩ ∈ r" proof - from A1 have "∀x y z. ⟨x, y⟩ ∈ r ⟶ ⟨y, z⟩ ∈ r ⟶ ⟨x, z⟩ ∈ r" unfolding trans_def by blast with A2 show ?thesis using imp_conj by fast qed text‹There is a problem with application of the definition of asymetry for relations. The next lemma is a workaround.› lemma Fol1_L4: assumes A1: "antisym(r)" and A2: "⟨ a,b⟩ ∈ r" "⟨ b,a⟩ ∈ r" shows "a=b" proof - from A1 have "∀ x y. ⟨ x,y⟩ ∈ r ⟶ ⟨ y,x⟩ ∈ r ⟶ x=y" unfolding antisym_def by blast with A2 show "a=b" using imp_conj by fast qed text‹The definition below implements a common idiom that states that (perhaps under some assumptions) exactly one of given three statements is true.› definition "Exactly_1_of_3_holds(p,q,r) ≡ (p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" text‹The next lemma allows to prove statements of the form ‹Exactly_1_of_3_holds(p,q,r)›.› lemma Fol1_L5: assumes "p∨q∨r" and "p ⟶ ¬q ∧ ¬r" and "q ⟶ ¬p ∧ ¬r" and "r ⟶ ¬p ∧ ¬q" shows "Exactly_1_of_3_holds(p,q,r)" proof - from assms have "(p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" by blast then show "Exactly_1_of_3_holds (p,q,r)" unfolding Exactly_1_of_3_holds_def by fast qed text‹If exactly one of $p,q,r$ holds and $p$ is not true, then $q$ or $r$.› lemma Fol1_L6: assumes A1: "¬p" and A2: "Exactly_1_of_3_holds(p,q,r)" shows "q∨r" proof - from A2 have "(p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" unfolding Exactly_1_of_3_holds_def by fast hence "p ∨ q ∨ r" by blast with A1 show "q ∨ r" by simp qed text‹If exactly one of $p,q,r$ holds and $q$ is true, then $r$ can not be true.› lemma Fol1_L7: assumes A1: "q" and A2: "Exactly_1_of_3_holds(p,q,r)" shows "¬r" proof - from A2 have "(p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" unfolding Exactly_1_of_3_holds_def by fast with A1 show "¬r" by blast qed text‹The next lemma demonstrates an elegant form of the ‹Exactly_1_of_3_holds(p,q,r)› predicate.› lemma Fol1_L8: shows "Exactly_1_of_3_holds(p,q,r) ⟷ (p⟷q⟷r) ∧ ¬(p∧q∧r)" proof assume "Exactly_1_of_3_holds(p,q,r)" then have "(p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" unfolding Exactly_1_of_3_holds_def by fast thus "(p⟷q⟷r) ∧ ¬(p∧q∧r)" by blast next assume "(p⟷q⟷r) ∧ ¬(p∧q∧r)" hence "(p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" by auto then show "Exactly_1_of_3_holds(p,q,r)" unfolding Exactly_1_of_3_holds_def by fast qed text‹A property of the ‹Exactly_1_of_3_holds› predicate.› lemma Fol1_L8A: assumes A1: "Exactly_1_of_3_holds(p,q,r)" shows "p ⟷ ¬(q ∨ r)" proof - from A1 have "(p∨q∨r) ∧ (p ⟶ ¬q ∧ ¬r) ∧ (q ⟶ ¬p ∧ ¬r) ∧ (r ⟶ ¬p ∧ ¬q)" unfolding Exactly_1_of_3_holds_def by fast then show "p ⟷ ¬(q ∨ r)" by blast qed text‹Exclusive or definition. There is one also defined in the standard Isabelle, denoted ‹xor›, but it relates to boolean values, which are sets. Here we define a logical functor.› definition Xor (infixl "Xor" 66) where "p Xor q ≡ (p∨q) ∧ ¬(p ∧ q)" text‹The "exclusive or" is the same as negation of equivalence.› lemma Fol1_L9: shows "p Xor q ⟷ ¬(p⟷q)" using Xor_def by auto text‹Constructions from the same sets are the same. It is suprising but we do have to use this as a rule in rare cases.› lemma same_constr: assumes "x=y" shows "P(x) = P(y)" using assms by simp text‹Equivalence relations are symmetric.› lemma equiv_is_sym: assumes A1: "equiv(X,r)" and A2: "⟨x,y⟩ ∈ r" shows "⟨y,x⟩ ∈ r" proof - from A1 have "sym(r)" using equiv_def by simp then have "∀x y. ⟨x,y⟩ ∈ r ⟶ ⟨y,x⟩ ∈ r" unfolding sym_def by fast with A2 show "⟨y,x⟩ ∈ r" by blast qed (* In Isabelle/ZF conjunction associates to the right!. lemma test: assumes A1: "P" "Q∧R" shows "P∧Q∧R" proof - from A1 show "P∧Q∧R" by (rule one_more_conj); qed; *) end