author | wenzelm |
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(* Title: ZF/QPair.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1993 University of Cambridge |
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Many proofs are borrowed from pair.thy and sum.thy |
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Do we EVER have rank(a) < rank(<a;b>) ? Perhaps if the latter rank |
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is not a limit ordinal? |
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*) |
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section\<open>Quine-Inspired Ordered Pairs and Disjoint Sums\<close> |
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theory QPair imports Sum func begin |
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text\<open>For non-well-founded data |
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structures in ZF. Does not precisely follow Quine's construction. Thanks |
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to Thomas Forster for suggesting this approach! |
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W. V. Quine, On Ordered Pairs and Relations, in Selected Logic Papers, |
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1966. |
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\<close> |
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definition |
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QPair :: "[i, i] \<Rightarrow> i" (\<open><(_;/ _)>\<close>) where |
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"<a;b> \<equiv> a+b" |
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definition |
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qfst :: "i \<Rightarrow> i" where |
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"qfst(p) \<equiv> THE a. \<exists>b. p=<a;b>" |
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definition |
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qsnd :: "i \<Rightarrow> i" where |
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"qsnd(p) \<equiv> THE b. \<exists>a. p=<a;b>" |
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definition |
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qsplit :: "[[i, i] \<Rightarrow> 'a, i] \<Rightarrow> 'a::{}" (*for pattern-matching*) where |
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"qsplit(c,p) \<equiv> c(qfst(p), qsnd(p))" |
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definition |
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qconverse :: "i \<Rightarrow> i" where |
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"qconverse(r) \<equiv> {z. w \<in> r, \<exists>x y. w=<x;y> \<and> z=<y;x>}" |
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definition |
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QSigma :: "[i, i \<Rightarrow> i] \<Rightarrow> i" where |
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"QSigma(A,B) \<equiv> \<Union>x\<in>A. \<Union>y\<in>B(x). {<x;y>}" |
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syntax |
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"_QSUM" :: "[idt, i, i] \<Rightarrow> i" (\<open>(3QSUM _ \<in> _./ _)\<close> 10) |
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translations |
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"QSUM x \<in> A. B" => "CONST QSigma(A, \<lambda>x. B)" |
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abbreviation |
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qprod (infixr \<open><*>\<close> 80) where |
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"A <*> B \<equiv> QSigma(A, \<lambda>_. B)" |
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definition |
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qsum :: "[i,i]\<Rightarrow>i" (infixr \<open><+>\<close> 65) where |
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"A <+> B \<equiv> ({0} <*> A) \<union> ({1} <*> B)" |
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definition |
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QInl :: "i\<Rightarrow>i" where |
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"QInl(a) \<equiv> <0;a>" |
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definition |
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QInr :: "i\<Rightarrow>i" where |
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"QInr(b) \<equiv> <1;b>" |
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definition |
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qcase :: "[i\<Rightarrow>i, i\<Rightarrow>i, i]\<Rightarrow>i" where |
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"qcase(c,d) \<equiv> qsplit(\<lambda>y z. cond(y, d(z), c(z)))" |
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subsection\<open>Quine ordered pairing\<close> |
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(** Lemmas for showing that <a;b> uniquely determines a and b **) |
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lemma QPair_empty [simp]: "<0;0> = 0" |
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by (simp add: QPair_def) |
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lemma QPair_iff [simp]: "<a;b> = <c;d> \<longleftrightarrow> a=c \<and> b=d" |
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apply (simp add: QPair_def) |
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apply (rule sum_equal_iff) |
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done |
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lemmas QPair_inject = QPair_iff [THEN iffD1, THEN conjE, elim!] |
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lemma QPair_inject1: "<a;b> = <c;d> \<Longrightarrow> a=c" |
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by blast |
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lemma QPair_inject2: "<a;b> = <c;d> \<Longrightarrow> b=d" |
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by blast |
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subsubsection\<open>QSigma: Disjoint union of a family of sets |
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Generalizes Cartesian product\<close> |
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lemma QSigmaI [intro!]: "\<lbrakk>a \<in> A; b \<in> B(a)\<rbrakk> \<Longrightarrow> <a;b> \<in> QSigma(A,B)" |
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by (simp add: QSigma_def) |
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(** Elimination rules for <a;b>:A*B -- introducing no eigenvariables **) |
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lemma QSigmaE [elim!]: |
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"\<lbrakk>c \<in> QSigma(A,B); |
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\<And>x y.\<lbrakk>x \<in> A; y \<in> B(x); c=<x;y>\<rbrakk> \<Longrightarrow> P |
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\<rbrakk> \<Longrightarrow> P" |
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by (simp add: QSigma_def, blast) |
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lemma QSigmaE2 [elim!]: |
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"\<lbrakk><a;b>: QSigma(A,B); \<lbrakk>a \<in> A; b \<in> B(a)\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
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by (simp add: QSigma_def) |
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lemma QSigmaD1: "<a;b> \<in> QSigma(A,B) \<Longrightarrow> a \<in> A" |
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by blast |
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lemma QSigmaD2: "<a;b> \<in> QSigma(A,B) \<Longrightarrow> b \<in> B(a)" |
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by blast |
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lemma QSigma_cong: |
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"\<lbrakk>A=A'; \<And>x. x \<in> A' \<Longrightarrow> B(x)=B'(x)\<rbrakk> \<Longrightarrow> |
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QSigma(A,B) = QSigma(A',B')" |
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by (simp add: QSigma_def) |
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lemma QSigma_empty1 [simp]: "QSigma(0,B) = 0" |
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by blast |
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lemma QSigma_empty2 [simp]: "A <*> 0 = 0" |
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by blast |
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subsubsection\<open>Projections: qfst, qsnd\<close> |
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lemma qfst_conv [simp]: "qfst(<a;b>) = a" |
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by (simp add: qfst_def) |
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lemma qsnd_conv [simp]: "qsnd(<a;b>) = b" |
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by (simp add: qsnd_def) |
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lemma qfst_type [TC]: "p \<in> QSigma(A,B) \<Longrightarrow> qfst(p) \<in> A" |
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by auto |
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lemma qsnd_type [TC]: "p \<in> QSigma(A,B) \<Longrightarrow> qsnd(p) \<in> B(qfst(p))" |
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by auto |
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lemma QPair_qfst_qsnd_eq: "a \<in> QSigma(A,B) \<Longrightarrow> <qfst(a); qsnd(a)> = a" |
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by auto |
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subsubsection\<open>Eliminator: qsplit\<close> |
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(*A META-equality, so that it applies to higher types as well...*) |
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lemma qsplit [simp]: "qsplit(\<lambda>x y. c(x,y), <a;b>) \<equiv> c(a,b)" |
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by (simp add: qsplit_def) |
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lemma qsplit_type [elim!]: |
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"\<lbrakk>p \<in> QSigma(A,B); |
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\<And>x y.\<lbrakk>x \<in> A; y \<in> B(x)\<rbrakk> \<Longrightarrow> c(x,y):C(<x;y>) |
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\<rbrakk> \<Longrightarrow> qsplit(\<lambda>x y. c(x,y), p) \<in> C(p)" |
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by auto |
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lemma expand_qsplit: |
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"u \<in> A<*>B \<Longrightarrow> R(qsplit(c,u)) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B. u = <x;y> \<longrightarrow> R(c(x,y)))" |
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apply (simp add: qsplit_def, auto) |
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done |
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subsubsection\<open>qsplit for predicates: result type o\<close> |
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lemma qsplitI: "R(a,b) \<Longrightarrow> qsplit(R, <a;b>)" |
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by (simp add: qsplit_def) |
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lemma qsplitE: |
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"\<lbrakk>qsplit(R,z); z \<in> QSigma(A,B); |
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\<And>x y. \<lbrakk>z = <x;y>; R(x,y)\<rbrakk> \<Longrightarrow> P |
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\<rbrakk> \<Longrightarrow> P" |
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by (simp add: qsplit_def, auto) |
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lemma qsplitD: "qsplit(R,<a;b>) \<Longrightarrow> R(a,b)" |
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by (simp add: qsplit_def) |
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subsubsection\<open>qconverse\<close> |
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lemma qconverseI [intro!]: "<a;b>:r \<Longrightarrow> <b;a>:qconverse(r)" |
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by (simp add: qconverse_def, blast) |
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lemma qconverseD [elim!]: "<a;b> \<in> qconverse(r) \<Longrightarrow> <b;a> \<in> r" |
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by (simp add: qconverse_def, blast) |
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lemma qconverseE [elim!]: |
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"\<lbrakk>yx \<in> qconverse(r); |
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\<And>x y. \<lbrakk>yx=<y;x>; <x;y>:r\<rbrakk> \<Longrightarrow> P |
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\<rbrakk> \<Longrightarrow> P" |
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by (simp add: qconverse_def, blast) |
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lemma qconverse_qconverse: "r<=QSigma(A,B) \<Longrightarrow> qconverse(qconverse(r)) = r" |
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by blast |
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lemma qconverse_type: "r \<subseteq> A <*> B \<Longrightarrow> qconverse(r) \<subseteq> B <*> A" |
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by blast |
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lemma qconverse_prod: "qconverse(A <*> B) = B <*> A" |
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by blast |
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lemma qconverse_empty: "qconverse(0) = 0" |
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by blast |
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subsection\<open>The Quine-inspired notion of disjoint sum\<close> |
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lemmas qsum_defs = qsum_def QInl_def QInr_def qcase_def |
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(** Introduction rules for the injections **) |
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lemma QInlI [intro!]: "a \<in> A \<Longrightarrow> QInl(a) \<in> A <+> B" |
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by (simp add: qsum_defs, blast) |
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lemma QInrI [intro!]: "b \<in> B \<Longrightarrow> QInr(b) \<in> A <+> B" |
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by (simp add: qsum_defs, blast) |
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(** Elimination rules **) |
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lemma qsumE [elim!]: |
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"\<lbrakk>u \<in> A <+> B; |
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\<And>x. \<lbrakk>x \<in> A; u=QInl(x)\<rbrakk> \<Longrightarrow> P; |
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\<And>y. \<lbrakk>y \<in> B; u=QInr(y)\<rbrakk> \<Longrightarrow> P |
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\<rbrakk> \<Longrightarrow> P" |
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by (simp add: qsum_defs, blast) |
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(** Injection and freeness equivalences, for rewriting **) |
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lemma QInl_iff [iff]: "QInl(a)=QInl(b) \<longleftrightarrow> a=b" |
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by (simp add: qsum_defs ) |
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lemma QInr_iff [iff]: "QInr(a)=QInr(b) \<longleftrightarrow> a=b" |
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by (simp add: qsum_defs ) |
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lemma QInl_QInr_iff [simp]: "QInl(a)=QInr(b) \<longleftrightarrow> False" |
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by (simp add: qsum_defs ) |
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lemma QInr_QInl_iff [simp]: "QInr(b)=QInl(a) \<longleftrightarrow> False" |
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by (simp add: qsum_defs ) |
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lemma qsum_empty [simp]: "0<+>0 = 0" |
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by (simp add: qsum_defs ) |
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(*Injection and freeness rules*) |
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lemmas QInl_inject = QInl_iff [THEN iffD1] |
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lemmas QInr_inject = QInr_iff [THEN iffD1] |
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lemmas QInl_neq_QInr = QInl_QInr_iff [THEN iffD1, THEN FalseE, elim!] |
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lemmas QInr_neq_QInl = QInr_QInl_iff [THEN iffD1, THEN FalseE, elim!] |
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lemma QInlD: "QInl(a): A<+>B \<Longrightarrow> a \<in> A" |
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by blast |
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lemma QInrD: "QInr(b): A<+>B \<Longrightarrow> b \<in> B" |
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by blast |
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(** <+> is itself injective... who cares?? **) |
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lemma qsum_iff: |
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"u \<in> A <+> B \<longleftrightarrow> (\<exists>x. x \<in> A \<and> u=QInl(x)) | (\<exists>y. y \<in> B \<and> u=QInr(y))" |
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by blast |
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lemma qsum_subset_iff: "A <+> B \<subseteq> C <+> D \<longleftrightarrow> A<=C \<and> B<=D" |
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by blast |
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lemma qsum_equal_iff: "A <+> B = C <+> D \<longleftrightarrow> A=C \<and> B=D" |
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apply (simp (no_asm) add: extension qsum_subset_iff) |
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apply blast |
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done |
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subsubsection\<open>Eliminator -- qcase\<close> |
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lemma qcase_QInl [simp]: "qcase(c, d, QInl(a)) = c(a)" |
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by (simp add: qsum_defs ) |
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lemma qcase_QInr [simp]: "qcase(c, d, QInr(b)) = d(b)" |
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by (simp add: qsum_defs ) |
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lemma qcase_type: |
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"\<lbrakk>u \<in> A <+> B; |
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\<And>x. x \<in> A \<Longrightarrow> c(x): C(QInl(x)); |
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\<And>y. y \<in> B \<Longrightarrow> d(y): C(QInr(y)) |
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\<rbrakk> \<Longrightarrow> qcase(c,d,u) \<in> C(u)" |
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by (simp add: qsum_defs, auto) |
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(** Rules for the Part primitive **) |
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lemma Part_QInl: "Part(A <+> B,QInl) = {QInl(x). x \<in> A}" |
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by blast |
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lemma Part_QInr: "Part(A <+> B,QInr) = {QInr(y). y \<in> B}" |
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by blast |
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lemma Part_QInr2: "Part(A <+> B, \<lambda>x. QInr(h(x))) = {QInr(y). y \<in> Part(B,h)}" |
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by blast |
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lemma Part_qsum_equality: "C \<subseteq> A <+> B \<Longrightarrow> Part(C,QInl) \<union> Part(C,QInr) = C" |
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by blast |
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subsubsection\<open>Monotonicity\<close> |
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lemma QPair_mono: "\<lbrakk>a<=c; b<=d\<rbrakk> \<Longrightarrow> <a;b> \<subseteq> <c;d>" |
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by (simp add: QPair_def sum_mono) |
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lemma QSigma_mono [rule_format]: |
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"\<lbrakk>A<=C; \<forall>x\<in>A. B(x) \<subseteq> D(x)\<rbrakk> \<Longrightarrow> QSigma(A,B) \<subseteq> QSigma(C,D)" |
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by blast |
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lemma QInl_mono: "a<=b \<Longrightarrow> QInl(a) \<subseteq> QInl(b)" |
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by (simp add: QInl_def subset_refl [THEN QPair_mono]) |
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lemma QInr_mono: "a<=b \<Longrightarrow> QInr(a) \<subseteq> QInr(b)" |
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by (simp add: QInr_def subset_refl [THEN QPair_mono]) |
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lemma qsum_mono: "\<lbrakk>A<=C; B<=D\<rbrakk> \<Longrightarrow> A <+> B \<subseteq> C <+> D" |
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by blast |
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0 | 326 |
end |