--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/Examples/Higher_Order_Logic.thy Mon Jun 08 15:09:57 2020 +0200
@@ -0,0 +1,520 @@
+(* Title: Pure/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 illustrates the foundations of Higher-Order
+ Logic. The ``HOL'' logic that is given here resembles @{cite
+ "Gordon:1985:HOL"} and its predecessor @{cite "church40"}, but the order of
+ axiomatizations and defined connectives has be adapted to modern
+ presentations of \<open>\<lambda>\<close>-calculus and Constructive Type Theory. Thus it fits
+ nicely to the underlying Natural Deduction framework of Isabelle/Pure and
+ Isabelle/Isar.
+\<close>
+
+
+section \<open>HOL syntax within Pure\<close>
+
+class type
+default_sort type
+
+typedecl o
+instance o :: type ..
+instance "fun" :: (type, type) type ..
+
+judgment Trueprop :: "o \<Rightarrow> prop" ("_" 5)
+
+
+section \<open>Minimal logic (axiomatization)\<close>
+
+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"
+
+lemma atomize_imp [atomize]: "(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)"
+ by standard (fact impI, fact impE)
+
+lemma atomize_all [atomize]: "(\<And>x. P x) \<equiv> Trueprop (\<forall>x. P x)"
+ by standard (fact allI, fact allE)
+
+
+subsubsection \<open>Derived connectives\<close>
+
+definition False :: o
+ where "False \<equiv> \<forall>A. A"
+
+lemma FalseE [elim]:
+ assumes "False"
+ shows A
+proof -
+ from \<open>False\<close> have "\<forall>A. A" by (simp only: False_def)
+ then show A ..
+qed
+
+
+definition True :: o
+ where "True \<equiv> False \<longrightarrow> False"
+
+lemma TrueI [intro]: True
+ unfolding True_def ..
+
+
+definition not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
+ where "not \<equiv> \<lambda>A. A \<longrightarrow> False"
+
+lemma notI [intro]:
+ assumes "A \<Longrightarrow> False"
+ shows "\<not> A"
+ using assms unfolding not_def ..
+
+lemma notE [elim]:
+ assumes "\<not> A" and A
+ shows B
+proof -
+ from \<open>\<not> A\<close> have "A \<longrightarrow> False" by (simp only: not_def)
+ from this and \<open>A\<close> have "False" ..
+ 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 "A \<and> B \<equiv> \<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C"
+
+lemma 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
+
+lemma 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 "A \<or> B \<equiv> \<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C"
+
+lemma 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
+
+lemma 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
+
+lemma 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"
+
+lemma 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
+
+lemma 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
+
+
+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"
+
+abbreviation not_equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "\<noteq>" 50)
+ where "x \<noteq> y \<equiv> \<not> (x = y)"
+
+abbreviation iff :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longleftrightarrow>" 25)
+ where "A \<longleftrightarrow> B \<equiv> A = B"
+
+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 \<longleftrightarrow> B"
+ for f g :: "'a \<Rightarrow> 'b"
+
+lemma sym [sym]: "y = x" if "x = y"
+ using that 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)
+
+lemma arg_cong: "f x = f y" if "x = y"
+ using that by (rule subst) (rule refl)
+
+lemma fun_cong: "f x = g x" if "f = g"
+ using that by (rule subst) (rule refl)
+
+lemma trans [trans]: "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"
+ by (rule subst)
+
+lemma iff1 [elim]: "A \<longleftrightarrow> B \<Longrightarrow> A \<Longrightarrow> B"
+ by (rule subst)
+
+lemma iff2 [elim]: "A \<longleftrightarrow> B \<Longrightarrow> B \<Longrightarrow> A"
+ by (rule subst) (rule sym)
+
+
+subsection \<open>Cantor's Theorem\<close>
+
+text \<open>
+ Cantor's Theorem states that there is no surjection from a set to its
+ powerset. The subsequent formulation uses elementary \<open>\<lambda>\<close>-calculus and
+ predicate logic, with standard introduction and elimination rules.
+\<close>
+
+lemma iff_contradiction:
+ assumes *: "\<not> A \<longleftrightarrow> A"
+ shows C
+proof (rule notE)
+ show "\<not> A"
+ proof
+ assume A
+ with * have "\<not> A" ..
+ from this and \<open>A\<close> show False ..
+ qed
+ with * show A ..
+qed
+
+theorem Cantor: "\<not> (\<exists>f :: 'a \<Rightarrow> 'a \<Rightarrow> o. \<forall>A. \<exists>x. A = f x)"
+proof
+ assume "\<exists>f :: 'a \<Rightarrow> 'a \<Rightarrow> o. \<forall>A. \<exists>x. A = f x"
+ then obtain f :: "'a \<Rightarrow> 'a \<Rightarrow> o" where *: "\<forall>A. \<exists>x. A = f x" ..
+ let ?D = "\<lambda>x. \<not> f x x"
+ from * have "\<exists>x. ?D = f x" ..
+ then obtain a where "?D = f a" ..
+ then have "?D a \<longleftrightarrow> f a a" using refl by (rule subst)
+ then have "\<not> f a a \<longleftrightarrow> f a a" .
+ then show False by (rule iff_contradiction)
+qed
+
+
+subsection \<open>Characterization of 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"
+ \<comment> \<open>predicate definition and hypothetical context\<close>
+begin
+
+lemma classical_contradiction:
+ assumes "\<not> A \<Longrightarrow> False"
+ shows A
+proof (rule classical)
+ assume "\<not> A"
+ then have False by (rule assms)
+ then show A ..
+qed
+
+lemma double_negation:
+ assumes "\<not> \<not> A"
+ shows A
+proof (rule classical_contradiction)
+ assume "\<not> A"
+ with \<open>\<not> \<not> A\<close> show False by (rule contradiction)
+qed
+
+lemma 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 False by (rule contradiction)
+ qed
+ then have "A \<or> \<not> A" ..
+ with \<open>\<not> (A \<or> \<not> A)\<close> show False by (rule contradiction)
+ qed
+qed
+
+lemma classical_cases:
+ obtains A | "\<not> A"
+ using tertium_non_datur
+proof
+ assume A
+ then show thesis ..
+next
+ assume "\<not> A"
+ then show thesis ..
+qed
+
+end
+
+lemma classical_if_cases: classical
+ if cases: "\<And>A C. (A \<Longrightarrow> C) \<Longrightarrow> (\<not> A \<Longrightarrow> C) \<Longrightarrow> C"
+proof
+ fix A
+ assume *: "\<not> A \<Longrightarrow> A"
+ show A
+ proof (rule cases)
+ assume A
+ then show A .
+ next
+ assume "\<not> A"
+ then show A by (rule *)
+ qed
+qed
+
+
+section \<open>Peirce's Law\<close>
+
+text \<open>
+ Peirce's Law is another characterization of classical reasoning. Its
+ statement only requires implication.
+\<close>
+
+theorem (in classical) Peirce's_Law: "((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A"
+proof
+ assume *: "(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 * show A ..
+ qed
+qed
+
+
+section \<open>Hilbert's choice operator (axiomatization)\<close>
+
+axiomatization Eps :: "('a \<Rightarrow> o) \<Rightarrow> 'a"
+ where someI: "P x \<Longrightarrow> P (Eps P)"
+
+syntax "_Eps" :: "pttrn \<Rightarrow> o \<Rightarrow> 'a" ("(3SOME _./ _)" [0, 10] 10)
+translations "SOME x. P" \<rightleftharpoons> "CONST Eps (\<lambda>x. P)"
+
+text \<open>
+ \<^medskip>
+ It follows a derivation of the classical law of tertium-non-datur by
+ means of Hilbert's choice operator (due to Berghofer, Beeson, Harrison,
+ based on a proof by Diaconescu).
+ \<^medskip>
+\<close>
+
+theorem Diaconescu: "A \<or> \<not> A"
+proof -
+ let ?P = "\<lambda>x. (A \<and> x) \<or> \<not> x"
+ let ?Q = "\<lambda>x. (A \<and> \<not> x) \<or> x"
+
+ have a: "?P (Eps ?P)"
+ proof (rule someI)
+ have "\<not> False" ..
+ then show "?P False" ..
+ qed
+ have b: "?Q (Eps ?Q)"
+ proof (rule someI)
+ have True ..
+ then show "?Q True" ..
+ qed
+
+ from a show ?thesis
+ proof
+ assume "A \<and> Eps ?P"
+ then have A ..
+ then show ?thesis ..
+ next
+ assume "\<not> Eps ?P"
+ from b show ?thesis
+ proof
+ assume "A \<and> \<not> Eps ?Q"
+ then have A ..
+ then show ?thesis ..
+ next
+ assume "Eps ?Q"
+ have neq: "?P \<noteq> ?Q"
+ proof
+ assume "?P = ?Q"
+ then have "Eps ?P \<longleftrightarrow> Eps ?Q" by (rule arg_cong)
+ also note \<open>Eps ?Q\<close>
+ finally have "Eps ?P" .
+ with \<open>\<not> Eps ?P\<close> show False by (rule contradiction)
+ qed
+ have "\<not> A"
+ proof
+ assume A
+ have "?P = ?Q"
+ proof (rule ext)
+ show "?P x \<longleftrightarrow> ?Q x" for x
+ proof
+ assume "?P x"
+ then show "?Q x"
+ proof
+ assume "\<not> x"
+ with \<open>A\<close> have "A \<and> \<not> x" ..
+ then show ?thesis ..
+ next
+ assume "A \<and> x"
+ then have x ..
+ then show ?thesis ..
+ qed
+ next
+ assume "?Q x"
+ then show "?P x"
+ proof
+ assume "A \<and> \<not> x"
+ then have "\<not> x" ..
+ then show ?thesis ..
+ next
+ assume x
+ with \<open>A\<close> have "A \<and> x" ..
+ then show ?thesis ..
+ qed
+ qed
+ qed
+ with neq show False by (rule contradiction)
+ qed
+ then show ?thesis ..
+ qed
+ qed
+qed
+
+text \<open>
+ This means, the hypothetical predicate \<^const>\<open>classical\<close> always holds
+ unconditionally (with all consequences).
+\<close>
+
+interpretation classical
+proof (rule classical_if_cases)
+ fix A C
+ assume *: "A \<Longrightarrow> C"
+ and **: "\<not> A \<Longrightarrow> C"
+ from Diaconescu [of A] show C
+ proof
+ assume A
+ then show C by (rule *)
+ next
+ assume "\<not> A"
+ then show C by (rule **)
+ qed
+qed
+
+thm classical
+ classical_contradiction
+ double_negation
+ tertium_non_datur
+ classical_cases
+ Peirce's_Law
+
+end