--- a/src/HOL/IsaMakefile Tue Dec 04 14:26:22 2001 +0100
+++ b/src/HOL/IsaMakefile Tue Dec 04 17:59:36 2001 +0100
@@ -528,7 +528,7 @@
HOL-ex: HOL $(LOG)/HOL-ex.gz
$(LOG)/HOL-ex.gz: $(OUT)/HOL ex/AVL.ML ex/AVL.thy ex/Antiquote.thy \
- ex/BT.thy ex/BinEx.thy ex/Group.ML ex/Group.thy \
+ ex/BT.thy ex/BinEx.thy ex/Group.ML ex/Group.thy ex/Higher_Order_Logic.thy \
ex/Hilbert_Classical.thy ex/InSort.ML ex/InSort.thy ex/IntRing.ML \
ex/IntRing.thy ex/Lagrange.ML ex/Lagrange.thy ex/Locales.thy \
ex/MT.ML ex/MT.thy ex/MonoidGroup.thy ex/Multiquote.thy \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Higher_Order_Logic.thy Tue Dec 04 17:59:36 2001 +0100
@@ -0,0 +1,311 @@
+(* Title: HOL/ex/Higher_Order_Logic.thy
+ ID: $Id$
+ Author: Gertrud Bauer and Markus Wenzel, TU Muenchen
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Foundations of HOL *}
+
+theory Higher_Order_Logic = CPure:
+
+text {*
+ The following theory development demonstrates Higher-Order Logic
+ itself, represented directly within the Pure framework of Isabelle.
+ The ``HOL'' logic given here is essentially that of Gordon
+ \cite{Gordon:1985:HOL}, although we prefer to present basic concepts
+ in a slightly more conventional manner oriented towards plain
+ Natural Deduction.
+*}
+
+
+subsection {* Pure Logic *}
+
+classes type \<subseteq> logic
+defaultsort type
+
+typedecl o
+arities
+ o :: type
+ fun :: (type, type) type
+
+
+subsubsection {* Basic logical connectives *}
+
+judgment
+ Trueprop :: "o \<Rightarrow> prop" ("_" 5)
+
+consts
+ imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25)
+ All :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10)
+
+axioms
+ impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"
+ impE [dest, trans]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B"
+ allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x"
+ allE [dest]: "\<forall>x. P x \<Longrightarrow> P a"
+
+
+subsubsection {* Extensional equality *}
+
+consts
+ equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "=" 50)
+
+axioms
+ refl [intro]: "x = x"
+ subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y"
+ ext [intro]: "(\<And>x. f x = g x) \<Longrightarrow> f = g"
+ iff [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A = B"
+
+theorem sym [elim]: "x = y \<Longrightarrow> y = x"
+proof -
+ assume "x = y"
+ thus "y = x" by (rule subst) (rule refl)
+qed
+
+lemma [trans]: "x = y \<Longrightarrow> P y \<Longrightarrow> P x"
+ by (rule subst) (rule sym)
+
+lemma [trans]: "P x \<Longrightarrow> x = y \<Longrightarrow> P y"
+ by (rule subst)
+
+theorem trans [trans]: "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"
+ by (rule subst)
+
+theorem iff1 [elim]: "A = B \<Longrightarrow> A \<Longrightarrow> B"
+ by (rule subst)
+
+theorem iff2 [elim]: "A = B \<Longrightarrow> B \<Longrightarrow> A"
+ by (rule subst) (rule sym)
+
+
+subsubsection {* Derived connectives *}
+
+constdefs
+ false :: o ("\<bottom>")
+ "\<bottom> \<equiv> \<forall>A. A"
+ true :: o ("\<top>")
+ "\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>"
+ not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
+ "not \<equiv> \<lambda>A. A \<longrightarrow> \<bottom>"
+ conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35)
+ "conj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+ disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30)
+ "disj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ Ex :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10)
+ "Ex \<equiv> \<lambda>P. \<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
+
+syntax
+ "_not_equal" :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "\<noteq>" 50)
+translations
+ "x \<noteq> y" \<rightleftharpoons> "\<not> (x = y)"
+
+theorem falseE [elim]: "\<bottom> \<Longrightarrow> A"
+proof (unfold false_def)
+ assume "\<forall>A. A"
+ thus A ..
+qed
+
+theorem trueI [intro]: \<top>
+proof (unfold true_def)
+ show "\<bottom> \<longrightarrow> \<bottom>" ..
+qed
+
+theorem notI [intro]: "(A \<Longrightarrow> \<bottom>) \<Longrightarrow> \<not> A"
+proof (unfold not_def)
+ assume "A \<Longrightarrow> \<bottom>"
+ thus "A \<longrightarrow> \<bottom>" ..
+qed
+
+theorem notE [elim]: "\<not> A \<Longrightarrow> A \<Longrightarrow> B"
+proof (unfold not_def)
+ assume "A \<longrightarrow> \<bottom>"
+ also assume A
+ finally have \<bottom> ..
+ thus B ..
+qed
+
+lemma notE': "A \<Longrightarrow> \<not> A \<Longrightarrow> B"
+ by (rule notE)
+
+lemmas contradiction = notE notE' -- {* proof by contradiction in any order *}
+
+theorem conjI [intro]: "A \<Longrightarrow> B \<Longrightarrow> A \<and> B"
+proof (unfold conj_def)
+ assume A and B
+ show "\<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ fix C show "(A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "A \<longrightarrow> B \<longrightarrow> C"
+ also have A .
+ also have B .
+ finally show C .
+ qed
+ qed
+qed
+
+theorem conjE [elim]: "A \<and> B \<Longrightarrow> (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C"
+proof (unfold conj_def)
+ assume c: "\<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+ assume "A \<Longrightarrow> B \<Longrightarrow> C"
+ moreover {
+ from c have "(A \<longrightarrow> B \<longrightarrow> A) \<longrightarrow> A" ..
+ also have "A \<longrightarrow> B \<longrightarrow> A"
+ proof
+ assume A
+ thus "B \<longrightarrow> A" ..
+ qed
+ finally have A .
+ } moreover {
+ from c have "(A \<longrightarrow> B \<longrightarrow> B) \<longrightarrow> B" ..
+ also have "A \<longrightarrow> B \<longrightarrow> B"
+ proof
+ show "B \<longrightarrow> B" ..
+ qed
+ finally have B .
+ } ultimately show C .
+qed
+
+theorem disjI1 [intro]: "A \<Longrightarrow> A \<or> B"
+proof (unfold disj_def)
+ assume A
+ show "\<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ fix C show "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "A \<longrightarrow> C"
+ also have A .
+ finally have C .
+ thus "(B \<longrightarrow> C) \<longrightarrow> C" ..
+ qed
+ qed
+qed
+
+theorem disjI2 [intro]: "B \<Longrightarrow> A \<or> B"
+proof (unfold disj_def)
+ assume B
+ show "\<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ fix C show "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ show "(B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "B \<longrightarrow> C"
+ also have B .
+ finally show C .
+ qed
+ qed
+ qed
+qed
+
+theorem disjE [elim]: "A \<or> B \<Longrightarrow> (A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C"
+proof (unfold disj_def)
+ assume c: "\<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ assume r1: "A \<Longrightarrow> C" and r2: "B \<Longrightarrow> C"
+ from c have "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" ..
+ also have "A \<longrightarrow> C"
+ proof
+ assume A thus C by (rule r1)
+ qed
+ also have "B \<longrightarrow> C"
+ proof
+ assume B thus C by (rule r2)
+ qed
+ finally show C .
+qed
+
+theorem exI [intro]: "P a \<Longrightarrow> \<exists>x. P x"
+proof (unfold Ex_def)
+ assume "P a"
+ show "\<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ fix C show "(\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "\<forall>x. P x \<longrightarrow> C"
+ hence "P a \<longrightarrow> C" ..
+ also have "P a" .
+ finally show C .
+ qed
+ qed
+qed
+
+theorem exE [elim]: "\<exists>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> C) \<Longrightarrow> C"
+proof (unfold Ex_def)
+ assume c: "\<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
+ assume r: "\<And>x. P x \<Longrightarrow> C"
+ from c have "(\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C" ..
+ also have "\<forall>x. P x \<longrightarrow> C"
+ proof
+ fix x show "P x \<longrightarrow> C"
+ proof
+ assume "P x"
+ thus C by (rule r)
+ qed
+ qed
+ finally show C .
+qed
+
+
+subsection {* Classical logic *}
+
+locale classical =
+ assumes classical: "(\<not> A \<Longrightarrow> A) \<Longrightarrow> A"
+
+theorem (in classical)
+ Peirce's_Law: "((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A"
+proof
+ assume a: "(A \<longrightarrow> B) \<longrightarrow> A"
+ show A
+ proof (rule classical)
+ assume "\<not> A"
+ have "A \<longrightarrow> B"
+ proof
+ assume A
+ thus B by (rule contradiction)
+ qed
+ with a show A ..
+ qed
+qed
+
+theorem (in classical)
+ double_negation: "\<not> \<not> A \<Longrightarrow> A"
+proof -
+ assume "\<not> \<not> A"
+ show A
+ proof (rule classical)
+ assume "\<not> A"
+ thus ?thesis by (rule contradiction)
+ qed
+qed
+
+theorem (in classical)
+ tertium_non_datur: "A \<or> \<not> A"
+proof (rule double_negation)
+ show "\<not> \<not> (A \<or> \<not> A)"
+ proof
+ assume "\<not> (A \<or> \<not> A)"
+ have "\<not> A"
+ proof
+ assume A hence "A \<or> \<not> A" ..
+ thus \<bottom> by (rule contradiction)
+ qed
+ hence "A \<or> \<not> A" ..
+ thus \<bottom> by (rule contradiction)
+ qed
+qed
+
+theorem (in classical)
+ classical_cases: "(A \<Longrightarrow> C) \<Longrightarrow> (\<not> A \<Longrightarrow> C) \<Longrightarrow> C"
+proof -
+ assume r1: "A \<Longrightarrow> C" and r2: "\<not> A \<Longrightarrow> C"
+ from tertium_non_datur show C
+ proof
+ assume A
+ thus ?thesis by (rule r1)
+ next
+ assume "\<not> A"
+ thus ?thesis by (rule r2)
+ qed
+qed
+
+end
--- a/src/HOL/ex/ROOT.ML Tue Dec 04 14:26:22 2001 +0100
+++ b/src/HOL/ex/ROOT.ML Tue Dec 04 17:59:36 2001 +0100
@@ -4,6 +4,8 @@
Miscellaneous examples for Higher-Order Logic.
*)
+time_use_thy "Higher_Order_Logic";
+
time_use_thy "Recdefs";
time_use_thy "Primrec";
time_use_thy "Locales";
--- a/src/HOL/ex/document/root.bib Tue Dec 04 14:26:22 2001 +0100
+++ b/src/HOL/ex/document/root.bib Tue Dec 04 17:59:36 2001 +0100
@@ -1,3 +1,11 @@
+
+@TechReport{Gordon:1985:HOL,
+ author = {M. J. C. Gordon},
+ title = {{HOL}: A machine oriented formulation of higher order logic},
+ institution = {University of Cambridge Computer Laboratory},
+ year = 1985,
+ number = 68
+}
@InProceedings{Kamm-et-al:1999,
author = {Florian Kamm{\"u}ller and Markus Wenzel and