--- a/src/FOL/ROOT Sat Dec 26 16:10:00 2015 +0100
+++ b/src/FOL/ROOT Sat Dec 26 19:27:46 2015 +0100
@@ -25,7 +25,6 @@
Examples for First-Order Logic.
*}
theories
- First_Order_Logic
Natural_Numbers
Intro
Nat
--- a/src/FOL/ex/First_Order_Logic.thy Sat Dec 26 16:10:00 2015 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,161 +0,0 @@
-(* Title: FOL/ex/First_Order_Logic.thy
- Author: Makarius
-*)
-
-section \<open>A simple formulation of First-Order Logic\<close>
-
-text \<open>
- The subsequent theory development illustrates single-sorted intuitionistic
- first-order logic with equality, formulated within the Pure framework. So
- this is strictly speaking an example of Isabelle/Pure, not Isabelle/FOL.
-\<close>
-
-theory First_Order_Logic
-imports Pure
-begin
-
-subsection \<open>Abstract syntax\<close>
-
-typedecl i
-typedecl o
-
-judgment Trueprop :: "o \<Rightarrow> prop" ("_" 5)
-
-
-subsection \<open>Propositional logic\<close>
-
-axiomatization false :: o ("\<bottom>")
- where falseE [elim]: "\<bottom> \<Longrightarrow> A"
-
-
-axiomatization imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25)
- where impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"
- and mp [dest]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B"
-
-
-axiomatization conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35)
- where conjI [intro]: "A \<Longrightarrow> B \<Longrightarrow> A \<and> B"
- and conjD1: "A \<and> B \<Longrightarrow> A"
- and conjD2: "A \<and> B \<Longrightarrow> B"
-
-theorem conjE [elim]:
- assumes "A \<and> B"
- obtains A and B
-proof
- from \<open>A \<and> B\<close> show A
- by (rule conjD1)
- from \<open>A \<and> B\<close> show B
- by (rule conjD2)
-qed
-
-
-axiomatization disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30)
- where disjE [elim]: "A \<or> B \<Longrightarrow> (A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C"
- and disjI1 [intro]: "A \<Longrightarrow> A \<or> B"
- and disjI2 [intro]: "B \<Longrightarrow> A \<or> B"
-
-
-definition true :: o ("\<top>")
- where "\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>"
-
-theorem trueI [intro]: \<top>
- unfolding true_def ..
-
-
-definition not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
- where "\<not> A \<equiv> A \<longrightarrow> \<bottom>"
-
-theorem notI [intro]: "(A \<Longrightarrow> \<bottom>) \<Longrightarrow> \<not> A"
- unfolding not_def ..
-
-theorem notE [elim]: "\<not> A \<Longrightarrow> A \<Longrightarrow> B"
- unfolding not_def
-proof -
- assume "A \<longrightarrow> \<bottom>" and A
- then have \<bottom> ..
- then show B ..
-qed
-
-
-definition iff :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longleftrightarrow>" 25)
- where "A \<longleftrightarrow> B \<equiv> (A \<longrightarrow> B) \<and> (B \<longrightarrow> A)"
-
-theorem iffI [intro]:
- assumes "A \<Longrightarrow> B"
- and "B \<Longrightarrow> A"
- shows "A \<longleftrightarrow> B"
- unfolding iff_def
-proof
- from \<open>A \<Longrightarrow> B\<close> show "A \<longrightarrow> B" ..
- from \<open>B \<Longrightarrow> A\<close> show "B \<longrightarrow> A" ..
-qed
-
-theorem iff1 [elim]:
- assumes "A \<longleftrightarrow> B" and A
- shows B
-proof -
- from \<open>A \<longleftrightarrow> B\<close> have "(A \<longrightarrow> B) \<and> (B \<longrightarrow> A)"
- unfolding iff_def .
- then have "A \<longrightarrow> B" ..
- from this and \<open>A\<close> show B ..
-qed
-
-theorem iff2 [elim]:
- assumes "A \<longleftrightarrow> B" and B
- shows A
-proof -
- from \<open>A \<longleftrightarrow> B\<close> have "(A \<longrightarrow> B) \<and> (B \<longrightarrow> A)"
- unfolding iff_def .
- then have "B \<longrightarrow> A" ..
- from this and \<open>B\<close> show A ..
-qed
-
-
-subsection \<open>Equality\<close>
-
-axiomatization equal :: "i \<Rightarrow> i \<Rightarrow> o" (infixl "=" 50)
- where refl [intro]: "x = x"
- and subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y"
-
-theorem trans [trans]: "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"
- by (rule subst)
-
-theorem sym [sym]: "x = y \<Longrightarrow> y = x"
-proof -
- assume "x = y"
- from this and refl show "y = x"
- by (rule subst)
-qed
-
-
-subsection \<open>Quantifiers\<close>
-
-axiomatization All :: "(i \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10)
- where allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x"
- and allD [dest]: "\<forall>x. P x \<Longrightarrow> P a"
-
-axiomatization Ex :: "(i \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10)
- where exI [intro]: "P a \<Longrightarrow> \<exists>x. P x"
- and exE [elim]: "\<exists>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> C) \<Longrightarrow> C"
-
-
-lemma "(\<exists>x. P (f x)) \<longrightarrow> (\<exists>y. P y)"
-proof
- assume "\<exists>x. P (f x)"
- then obtain x where "P (f x)" ..
- then show "\<exists>y. P y" ..
-qed
-
-lemma "(\<exists>x. \<forall>y. R x y) \<longrightarrow> (\<forall>y. \<exists>x. R x y)"
-proof
- assume "\<exists>x. \<forall>y. R x y"
- then obtain x where "\<forall>y. R x y" ..
- show "\<forall>y. \<exists>x. R x y"
- proof
- fix y
- from \<open>\<forall>y. R x y\<close> have "R x y" ..
- then show "\<exists>x. R x y" ..
- qed
-qed
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Induct/Nested_Datatype.thy Sat Dec 26 19:27:46 2015 +0100
@@ -0,0 +1,61 @@
+section \<open>Nested datatypes\<close>
+
+theory Nested_Datatype
+imports Main
+begin
+
+subsection \<open>Terms and substitution\<close>
+
+datatype ('a, 'b) "term" =
+ Var 'a
+| App 'b "('a, 'b) term list"
+
+primrec subst_term :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term \<Rightarrow> ('a, 'b) term"
+ and subst_term_list :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term list \<Rightarrow> ('a, 'b) term list"
+where
+ "subst_term f (Var a) = f a"
+| "subst_term f (App b ts) = App b (subst_term_list f ts)"
+| "subst_term_list f [] = []"
+| "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts"
+
+lemmas subst_simps = subst_term.simps subst_term_list.simps
+
+text \<open>\<^medskip> A simple lemma about composition of substitutions.\<close>
+
+lemma
+ "subst_term (subst_term f1 \<circ> f2) t =
+ subst_term f1 (subst_term f2 t)"
+ and
+ "subst_term_list (subst_term f1 \<circ> f2) ts =
+ subst_term_list f1 (subst_term_list f2 ts)"
+ by (induct t and ts rule: subst_term.induct subst_term_list.induct) simp_all
+
+lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)"
+proof -
+ let "?P t" = ?thesis
+ let ?Q = "\<lambda>ts. subst_term_list (subst_term f1 \<circ> f2) ts =
+ subst_term_list f1 (subst_term_list f2 ts)"
+ show ?thesis
+ proof (induct t rule: subst_term.induct)
+ show "?P (Var a)" for a by simp
+ show "?P (App b ts)" if "?Q ts" for b ts
+ using that by (simp only: subst_simps)
+ show "?Q []" by simp
+ show "?Q (t # ts)" if "?P t" "?Q ts" for t ts
+ using that by (simp only: subst_simps)
+ qed
+qed
+
+
+subsection \<open>Alternative induction\<close>
+
+lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)"
+proof (induct t rule: term.induct)
+ case (Var a)
+ show ?case by (simp add: o_def)
+next
+ case (App b ts)
+ then show ?case by (induct ts) simp_all
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Isar_Examples/First_Order_Logic.thy Sat Dec 26 19:27:46 2015 +0100
@@ -0,0 +1,160 @@
+(* Title: HOL/Isar_Examples/First_Order_Logic.thy
+ Author: Makarius
+*)
+
+section \<open>A simple formulation of First-Order Logic\<close>
+
+text \<open>
+ The subsequent theory development illustrates single-sorted intuitionistic
+ first-order logic with equality, formulated within the Pure framework.
+\<close>
+
+theory First_Order_Logic
+imports Pure
+begin
+
+subsection \<open>Abstract syntax\<close>
+
+typedecl i
+typedecl o
+
+judgment Trueprop :: "o \<Rightarrow> prop" ("_" 5)
+
+
+subsection \<open>Propositional logic\<close>
+
+axiomatization false :: o ("\<bottom>")
+ where falseE [elim]: "\<bottom> \<Longrightarrow> A"
+
+
+axiomatization imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25)
+ where impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"
+ and mp [dest]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B"
+
+
+axiomatization conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35)
+ where conjI [intro]: "A \<Longrightarrow> B \<Longrightarrow> A \<and> B"
+ and conjD1: "A \<and> B \<Longrightarrow> A"
+ and conjD2: "A \<and> B \<Longrightarrow> B"
+
+theorem conjE [elim]:
+ assumes "A \<and> B"
+ obtains A and B
+proof
+ from \<open>A \<and> B\<close> show A
+ by (rule conjD1)
+ from \<open>A \<and> B\<close> show B
+ by (rule conjD2)
+qed
+
+
+axiomatization disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30)
+ where disjE [elim]: "A \<or> B \<Longrightarrow> (A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C"
+ and disjI1 [intro]: "A \<Longrightarrow> A \<or> B"
+ and disjI2 [intro]: "B \<Longrightarrow> A \<or> B"
+
+
+definition true :: o ("\<top>")
+ where "\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>"
+
+theorem trueI [intro]: \<top>
+ unfolding true_def ..
+
+
+definition not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
+ where "\<not> A \<equiv> A \<longrightarrow> \<bottom>"
+
+theorem notI [intro]: "(A \<Longrightarrow> \<bottom>) \<Longrightarrow> \<not> A"
+ unfolding not_def ..
+
+theorem notE [elim]: "\<not> A \<Longrightarrow> A \<Longrightarrow> B"
+ unfolding not_def
+proof -
+ assume "A \<longrightarrow> \<bottom>" and A
+ then have \<bottom> ..
+ then show B ..
+qed
+
+
+definition iff :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longleftrightarrow>" 25)
+ where "A \<longleftrightarrow> B \<equiv> (A \<longrightarrow> B) \<and> (B \<longrightarrow> A)"
+
+theorem iffI [intro]:
+ assumes "A \<Longrightarrow> B"
+ and "B \<Longrightarrow> A"
+ shows "A \<longleftrightarrow> B"
+ unfolding iff_def
+proof
+ from \<open>A \<Longrightarrow> B\<close> show "A \<longrightarrow> B" ..
+ from \<open>B \<Longrightarrow> A\<close> show "B \<longrightarrow> A" ..
+qed
+
+theorem iff1 [elim]:
+ assumes "A \<longleftrightarrow> B" and A
+ shows B
+proof -
+ from \<open>A \<longleftrightarrow> B\<close> have "(A \<longrightarrow> B) \<and> (B \<longrightarrow> A)"
+ unfolding iff_def .
+ then have "A \<longrightarrow> B" ..
+ from this and \<open>A\<close> show B ..
+qed
+
+theorem iff2 [elim]:
+ assumes "A \<longleftrightarrow> B" and B
+ shows A
+proof -
+ from \<open>A \<longleftrightarrow> B\<close> have "(A \<longrightarrow> B) \<and> (B \<longrightarrow> A)"
+ unfolding iff_def .
+ then have "B \<longrightarrow> A" ..
+ from this and \<open>B\<close> show A ..
+qed
+
+
+subsection \<open>Equality\<close>
+
+axiomatization equal :: "i \<Rightarrow> i \<Rightarrow> o" (infixl "=" 50)
+ where refl [intro]: "x = x"
+ and subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y"
+
+theorem trans [trans]: "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"
+ by (rule subst)
+
+theorem sym [sym]: "x = y \<Longrightarrow> y = x"
+proof -
+ assume "x = y"
+ from this and refl show "y = x"
+ by (rule subst)
+qed
+
+
+subsection \<open>Quantifiers\<close>
+
+axiomatization All :: "(i \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10)
+ where allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x"
+ and allD [dest]: "\<forall>x. P x \<Longrightarrow> P a"
+
+axiomatization Ex :: "(i \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10)
+ where exI [intro]: "P a \<Longrightarrow> \<exists>x. P x"
+ and exE [elim]: "\<exists>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> C) \<Longrightarrow> C"
+
+
+lemma "(\<exists>x. P (f x)) \<longrightarrow> (\<exists>y. P y)"
+proof
+ assume "\<exists>x. P (f x)"
+ then obtain x where "P (f x)" ..
+ then show "\<exists>y. P y" ..
+qed
+
+lemma "(\<exists>x. \<forall>y. R x y) \<longrightarrow> (\<forall>y. \<exists>x. R x y)"
+proof
+ assume "\<exists>x. \<forall>y. R x y"
+ then obtain x where "\<forall>y. R x y" ..
+ show "\<forall>y. \<exists>x. R x y"
+ proof
+ fix y
+ from \<open>\<forall>y. R x y\<close> have "R x y" ..
+ then show "\<exists>x. R x y" ..
+ qed
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Isar_Examples/Higher_Order_Logic.thy Sat Dec 26 19:27:46 2015 +0100
@@ -0,0 +1,327 @@
+(* Title: HOL/Isar_Examples/Higher_Order_Logic.thy
+ Author: Makarius
+*)
+
+section \<open>Foundations of HOL\<close>
+
+theory Higher_Order_Logic
+imports Pure
+begin
+
+text \<open>
+ 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.
+\<close>
+
+
+subsection \<open>Pure Logic\<close>
+
+class type
+default_sort type
+
+typedecl o
+instance o :: type ..
+instance "fun" :: (type, type) type ..
+
+
+subsubsection \<open>Basic logical connectives\<close>
+
+judgment Trueprop :: "o \<Rightarrow> prop" ("_" 5)
+
+axiomatization imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25)
+ where impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"
+ and impE [dest, trans]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B"
+
+axiomatization All :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10)
+ where allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x"
+ and allE [dest]: "\<forall>x. P x \<Longrightarrow> P a"
+
+
+subsubsection \<open>Extensional equality\<close>
+
+axiomatization equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "=" 50)
+ where refl [intro]: "x = x"
+ and subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y"
+
+axiomatization
+ where ext [intro]: "(\<And>x. f x = g x) \<Longrightarrow> f = g"
+ and iff [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A = B"
+
+theorem sym [sym]:
+ assumes "x = y"
+ shows "y = x"
+ using assms by (rule subst) (rule refl)
+
+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 \<open>Derived connectives\<close>
+
+definition false :: o ("\<bottom>") where "\<bottom> \<equiv> \<forall>A. A"
+
+theorem falseE [elim]:
+ assumes "\<bottom>"
+ shows A
+proof -
+ from \<open>\<bottom>\<close> have "\<forall>A. A" unfolding false_def .
+ then show A ..
+qed
+
+
+definition true :: o ("\<top>") where "\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>"
+
+theorem trueI [intro]: \<top>
+ unfolding true_def ..
+
+
+definition not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
+ where "not \<equiv> \<lambda>A. A \<longrightarrow> \<bottom>"
+
+abbreviation not_equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "\<noteq>" 50)
+ where "x \<noteq> y \<equiv> \<not> (x = y)"
+
+theorem notI [intro]:
+ assumes "A \<Longrightarrow> \<bottom>"
+ shows "\<not> A"
+ using assms unfolding not_def ..
+
+theorem notE [elim]:
+ assumes "\<not> A" and A
+ shows B
+proof -
+ from \<open>\<not> A\<close> have "A \<longrightarrow> \<bottom>" unfolding not_def .
+ from this and \<open>A\<close> have "\<bottom>" ..
+ then show B ..
+qed
+
+lemma notE': "A \<Longrightarrow> \<not> A \<Longrightarrow> B"
+ by (rule notE)
+
+lemmas contradiction = notE notE' \<comment> \<open>proof by contradiction in any order\<close>
+
+
+definition conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35)
+ where "conj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+
+theorem conjI [intro]:
+ assumes A and B
+ shows "A \<and> B"
+ unfolding conj_def
+proof
+ fix C
+ show "(A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "A \<longrightarrow> B \<longrightarrow> C"
+ also note \<open>A\<close>
+ also note \<open>B\<close>
+ finally show C .
+ qed
+qed
+
+theorem conjE [elim]:
+ assumes "A \<and> B"
+ obtains A and B
+proof
+ from \<open>A \<and> B\<close> have *: "(A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C" for C
+ unfolding conj_def ..
+ show A
+ proof -
+ note * [of A]
+ also have "A \<longrightarrow> B \<longrightarrow> A"
+ proof
+ assume A
+ then show "B \<longrightarrow> A" ..
+ qed
+ finally show ?thesis .
+ qed
+ show B
+ proof -
+ note * [of B]
+ also have "A \<longrightarrow> B \<longrightarrow> B"
+ proof
+ show "B \<longrightarrow> B" ..
+ qed
+ finally show ?thesis .
+ qed
+qed
+
+
+definition disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30)
+ where "disj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+
+theorem disjI1 [intro]:
+ assumes A
+ shows "A \<or> B"
+ unfolding disj_def
+proof
+ fix C
+ show "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "A \<longrightarrow> C"
+ from this and \<open>A\<close> have C ..
+ then show "(B \<longrightarrow> C) \<longrightarrow> C" ..
+ qed
+qed
+
+theorem disjI2 [intro]:
+ assumes B
+ shows "A \<or> B"
+ unfolding disj_def
+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"
+ from this and \<open>B\<close> show C ..
+ qed
+ qed
+qed
+
+theorem disjE [elim]:
+ assumes "A \<or> B"
+ obtains (a) A | (b) B
+proof -
+ from \<open>A \<or> B\<close> have "(A \<longrightarrow> thesis) \<longrightarrow> (B \<longrightarrow> thesis) \<longrightarrow> thesis"
+ unfolding disj_def ..
+ also have "A \<longrightarrow> thesis"
+ proof
+ assume A
+ then show thesis by (rule a)
+ qed
+ also have "B \<longrightarrow> thesis"
+ proof
+ assume B
+ then show thesis by (rule b)
+ qed
+ finally show thesis .
+qed
+
+
+definition Ex :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10)
+ where "\<exists>x. P x \<equiv> \<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
+
+theorem exI [intro]: "P a \<Longrightarrow> \<exists>x. P x"
+ unfolding Ex_def
+proof
+ fix C
+ assume "P a"
+ show "(\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
+ proof
+ assume "\<forall>x. P x \<longrightarrow> C"
+ then have "P a \<longrightarrow> C" ..
+ from this and \<open>P a\<close> show C ..
+ qed
+qed
+
+theorem exE [elim]:
+ assumes "\<exists>x. P x"
+ obtains (that) x where "P x"
+proof -
+ from \<open>\<exists>x. P x\<close> have "(\<forall>x. P x \<longrightarrow> thesis) \<longrightarrow> thesis"
+ unfolding Ex_def ..
+ also have "\<forall>x. P x \<longrightarrow> thesis"
+ proof
+ fix x
+ show "P x \<longrightarrow> thesis"
+ proof
+ assume "P x"
+ then show thesis by (rule that)
+ qed
+ qed
+ finally show thesis .
+qed
+
+
+subsection \<open>Classical logic\<close>
+
+text \<open>
+ The subsequent rules of classical reasoning are all equivalent.
+\<close>
+
+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
+ with \<open>\<not> A\<close> show B by (rule contradiction)
+ qed
+ with a show A ..
+ qed
+qed
+
+theorem (in classical) double_negation:
+ assumes "\<not> \<not> A"
+ shows A
+proof (rule classical)
+ assume "\<not> A"
+ with \<open>\<not> \<not> A\<close> show ?thesis by (rule contradiction)
+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 then have "A \<or> \<not> A" ..
+ with \<open>\<not> (A \<or> \<not> A)\<close> show \<bottom> by (rule contradiction)
+ qed
+ then have "A \<or> \<not> A" ..
+ with \<open>\<not> (A \<or> \<not> A)\<close> show \<bottom> by (rule contradiction)
+ qed
+qed
+
+theorem (in classical) classical_cases:
+ obtains A | "\<not> A"
+ using tertium_non_datur
+proof
+ assume A
+ then show thesis ..
+next
+ assume "\<not> A"
+ then show thesis ..
+qed
+
+lemma
+ assumes classical_cases: "\<And>A C. (A \<Longrightarrow> C) \<Longrightarrow> (\<not> A \<Longrightarrow> C) \<Longrightarrow> C"
+ shows "PROP classical"
+proof
+ fix A
+ assume *: "\<not> A \<Longrightarrow> A"
+ show A
+ proof (rule classical_cases)
+ assume A
+ then show A .
+ next
+ assume "\<not> A"
+ then show A by (rule *)
+ qed
+qed
+
+end
--- a/src/HOL/Isar_Examples/Nested_Datatype.thy Sat Dec 26 16:10:00 2015 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,61 +0,0 @@
-section \<open>Nested datatypes\<close>
-
-theory Nested_Datatype
-imports Main
-begin
-
-subsection \<open>Terms and substitution\<close>
-
-datatype ('a, 'b) "term" =
- Var 'a
-| App 'b "('a, 'b) term list"
-
-primrec subst_term :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term \<Rightarrow> ('a, 'b) term"
- and subst_term_list :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term list \<Rightarrow> ('a, 'b) term list"
-where
- "subst_term f (Var a) = f a"
-| "subst_term f (App b ts) = App b (subst_term_list f ts)"
-| "subst_term_list f [] = []"
-| "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts"
-
-lemmas subst_simps = subst_term.simps subst_term_list.simps
-
-text \<open>\<^medskip> A simple lemma about composition of substitutions.\<close>
-
-lemma
- "subst_term (subst_term f1 \<circ> f2) t =
- subst_term f1 (subst_term f2 t)"
- and
- "subst_term_list (subst_term f1 \<circ> f2) ts =
- subst_term_list f1 (subst_term_list f2 ts)"
- by (induct t and ts rule: subst_term.induct subst_term_list.induct) simp_all
-
-lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)"
-proof -
- let "?P t" = ?thesis
- let ?Q = "\<lambda>ts. subst_term_list (subst_term f1 \<circ> f2) ts =
- subst_term_list f1 (subst_term_list f2 ts)"
- show ?thesis
- proof (induct t rule: subst_term.induct)
- show "?P (Var a)" for a by simp
- show "?P (App b ts)" if "?Q ts" for b ts
- using that by (simp only: subst_simps)
- show "?Q []" by simp
- show "?Q (t # ts)" if "?P t" "?Q ts" for t ts
- using that by (simp only: subst_simps)
- qed
-qed
-
-
-subsection \<open>Alternative induction\<close>
-
-lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)"
-proof (induct t rule: term.induct)
- case (Var a)
- show ?case by (simp add: o_def)
-next
- case (App b ts)
- then show ?case by (induct ts) simp_all
-qed
-
-end
--- a/src/HOL/Isar_Examples/document/root.bib Sat Dec 26 16:10:00 2015 +0100
+++ b/src/HOL/Isar_Examples/document/root.bib Sat Dec 26 19:27:46 2015 +0100
@@ -44,6 +44,14 @@
publisher = CUP,
year = 1990}
+@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
+}
+
@manual{isabelle-HOL,
author = {Tobias Nipkow and Lawrence C. Paulson and Markus Wenzel},
title = {{Isabelle}'s Logics: {HOL}},
--- a/src/HOL/Isar_Examples/document/root.tex Sat Dec 26 16:10:00 2015 +0100
+++ b/src/HOL/Isar_Examples/document/root.tex Sat Dec 26 19:27:46 2015 +0100
@@ -15,7 +15,7 @@
\begin{document}
-\title{Miscellaneous Isabelle/Isar examples for Higher-Order Logic}
+\title{Miscellaneous Isabelle/Isar examples}
\author{Makarius Wenzel \\[2ex]
With contributions by Gertrud Bauer and Tobias Nipkow}
\maketitle
--- a/src/HOL/ROOT Sat Dec 26 16:10:00 2015 +0100
+++ b/src/HOL/ROOT Sat Dec 26 19:27:46 2015 +0100
@@ -99,6 +99,7 @@
theories [quick_and_dirty]
Common_Patterns
theories
+ Nested_Datatype
QuoDataType
QuoNestedDataType
Term
@@ -548,7 +549,6 @@
Adhoc_Overloading_Examples
Iff_Oracle
Coercion_Examples
- Higher_Order_Logic
Abstract_NAT
Guess
Fundefs
@@ -622,11 +622,13 @@
session "HOL-Isar_Examples" in Isar_Examples = HOL +
description {*
- Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
+ Miscellaneous Isabelle/Isar examples.
*}
theories [document = false]
"~~/src/HOL/Library/Lattice_Syntax"
"../Number_Theory/Primes"
+ theories [quick_and_dirty]
+ Structured_Statements
theories
Basic_Logic
Cantor
@@ -639,12 +641,11 @@
Hoare_Ex
Knaster_Tarski
Mutilated_Checkerboard
- Nested_Datatype
Peirce
Puzzle
Summation
- theories [quick_and_dirty]
- Structured_Statements
+ First_Order_Logic
+ Higher_Order_Logic
document_files
"root.bib"
"root.tex"
--- a/src/HOL/ex/Higher_Order_Logic.thy Sat Dec 26 16:10:00 2015 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,327 +0,0 @@
-(* Title: HOL/ex/Higher_Order_Logic.thy
- Author: Makarius
-*)
-
-section \<open>Foundations of HOL\<close>
-
-theory Higher_Order_Logic
-imports Pure
-begin
-
-text \<open>
- 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.
-\<close>
-
-
-subsection \<open>Pure Logic\<close>
-
-class type
-default_sort type
-
-typedecl o
-instance o :: type ..
-instance "fun" :: (type, type) type ..
-
-
-subsubsection \<open>Basic logical connectives\<close>
-
-judgment Trueprop :: "o \<Rightarrow> prop" ("_" 5)
-
-axiomatization imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25)
- where impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"
- and impE [dest, trans]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B"
-
-axiomatization All :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10)
- where allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x"
- and allE [dest]: "\<forall>x. P x \<Longrightarrow> P a"
-
-
-subsubsection \<open>Extensional equality\<close>
-
-axiomatization equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "=" 50)
- where refl [intro]: "x = x"
- and subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y"
-
-axiomatization
- where ext [intro]: "(\<And>x. f x = g x) \<Longrightarrow> f = g"
- and iff [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A = B"
-
-theorem sym [sym]:
- assumes "x = y"
- shows "y = x"
- using assms by (rule subst) (rule refl)
-
-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 \<open>Derived connectives\<close>
-
-definition false :: o ("\<bottom>") where "\<bottom> \<equiv> \<forall>A. A"
-
-theorem falseE [elim]:
- assumes "\<bottom>"
- shows A
-proof -
- from \<open>\<bottom>\<close> have "\<forall>A. A" unfolding false_def .
- then show A ..
-qed
-
-
-definition true :: o ("\<top>") where "\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>"
-
-theorem trueI [intro]: \<top>
- unfolding true_def ..
-
-
-definition not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
- where "not \<equiv> \<lambda>A. A \<longrightarrow> \<bottom>"
-
-abbreviation not_equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "\<noteq>" 50)
- where "x \<noteq> y \<equiv> \<not> (x = y)"
-
-theorem notI [intro]:
- assumes "A \<Longrightarrow> \<bottom>"
- shows "\<not> A"
- using assms unfolding not_def ..
-
-theorem notE [elim]:
- assumes "\<not> A" and A
- shows B
-proof -
- from \<open>\<not> A\<close> have "A \<longrightarrow> \<bottom>" unfolding not_def .
- from this and \<open>A\<close> have "\<bottom>" ..
- then show B ..
-qed
-
-lemma notE': "A \<Longrightarrow> \<not> A \<Longrightarrow> B"
- by (rule notE)
-
-lemmas contradiction = notE notE' \<comment> \<open>proof by contradiction in any order\<close>
-
-
-definition conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35)
- where "conj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
-
-theorem conjI [intro]:
- assumes A and B
- shows "A \<and> B"
- unfolding conj_def
-proof
- fix C
- show "(A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
- proof
- assume "A \<longrightarrow> B \<longrightarrow> C"
- also note \<open>A\<close>
- also note \<open>B\<close>
- finally show C .
- qed
-qed
-
-theorem conjE [elim]:
- assumes "A \<and> B"
- obtains A and B
-proof
- from \<open>A \<and> B\<close> have *: "(A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C" for C
- unfolding conj_def ..
- show A
- proof -
- note * [of A]
- also have "A \<longrightarrow> B \<longrightarrow> A"
- proof
- assume A
- then show "B \<longrightarrow> A" ..
- qed
- finally show ?thesis .
- qed
- show B
- proof -
- note * [of B]
- also have "A \<longrightarrow> B \<longrightarrow> B"
- proof
- show "B \<longrightarrow> B" ..
- qed
- finally show ?thesis .
- qed
-qed
-
-
-definition disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30)
- where "disj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
-
-theorem disjI1 [intro]:
- assumes A
- shows "A \<or> B"
- unfolding disj_def
-proof
- fix C
- show "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
- proof
- assume "A \<longrightarrow> C"
- from this and \<open>A\<close> have C ..
- then show "(B \<longrightarrow> C) \<longrightarrow> C" ..
- qed
-qed
-
-theorem disjI2 [intro]:
- assumes B
- shows "A \<or> B"
- unfolding disj_def
-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"
- from this and \<open>B\<close> show C ..
- qed
- qed
-qed
-
-theorem disjE [elim]:
- assumes "A \<or> B"
- obtains (a) A | (b) B
-proof -
- from \<open>A \<or> B\<close> have "(A \<longrightarrow> thesis) \<longrightarrow> (B \<longrightarrow> thesis) \<longrightarrow> thesis"
- unfolding disj_def ..
- also have "A \<longrightarrow> thesis"
- proof
- assume A
- then show thesis by (rule a)
- qed
- also have "B \<longrightarrow> thesis"
- proof
- assume B
- then show thesis by (rule b)
- qed
- finally show thesis .
-qed
-
-
-definition Ex :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10)
- where "\<exists>x. P x \<equiv> \<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
-
-theorem exI [intro]: "P a \<Longrightarrow> \<exists>x. P x"
- unfolding Ex_def
-proof
- fix C
- assume "P a"
- show "(\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C"
- proof
- assume "\<forall>x. P x \<longrightarrow> C"
- then have "P a \<longrightarrow> C" ..
- from this and \<open>P a\<close> show C ..
- qed
-qed
-
-theorem exE [elim]:
- assumes "\<exists>x. P x"
- obtains (that) x where "P x"
-proof -
- from \<open>\<exists>x. P x\<close> have "(\<forall>x. P x \<longrightarrow> thesis) \<longrightarrow> thesis"
- unfolding Ex_def ..
- also have "\<forall>x. P x \<longrightarrow> thesis"
- proof
- fix x
- show "P x \<longrightarrow> thesis"
- proof
- assume "P x"
- then show thesis by (rule that)
- qed
- qed
- finally show thesis .
-qed
-
-
-subsection \<open>Classical logic\<close>
-
-text \<open>
- The subsequent rules of classical reasoning are all equivalent.
-\<close>
-
-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
- with \<open>\<not> A\<close> show B by (rule contradiction)
- qed
- with a show A ..
- qed
-qed
-
-theorem (in classical) double_negation:
- assumes "\<not> \<not> A"
- shows A
-proof (rule classical)
- assume "\<not> A"
- with \<open>\<not> \<not> A\<close> show ?thesis by (rule contradiction)
-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 then have "A \<or> \<not> A" ..
- with \<open>\<not> (A \<or> \<not> A)\<close> show \<bottom> by (rule contradiction)
- qed
- then have "A \<or> \<not> A" ..
- with \<open>\<not> (A \<or> \<not> A)\<close> show \<bottom> by (rule contradiction)
- qed
-qed
-
-theorem (in classical) classical_cases:
- obtains A | "\<not> A"
- using tertium_non_datur
-proof
- assume A
- then show thesis ..
-next
- assume "\<not> A"
- then show thesis ..
-qed
-
-lemma
- assumes classical_cases: "\<And>A C. (A \<Longrightarrow> C) \<Longrightarrow> (\<not> A \<Longrightarrow> C) \<Longrightarrow> C"
- shows "PROP classical"
-proof
- fix A
- assume *: "\<not> A \<Longrightarrow> A"
- show A
- proof (rule classical_cases)
- assume A
- then show A .
- next
- assume "\<not> A"
- then show A by (rule *)
- qed
-qed
-
-end
--- a/src/HOL/ex/document/root.bib Sat Dec 26 16:10:00 2015 +0100
+++ b/src/HOL/ex/document/root.bib Sat Dec 26 19:27:46 2015 +0100
@@ -1,11 +1,3 @@
-@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{HuttonW04,author={Graham Hutton and Joel Wright},
title={Compiling Exceptions Correctly},
booktitle={Proc.\ Conf.\ Mathematics of Program Construction},