author | wenzelm |
Fri, 10 Jul 2020 21:58:49 +0200 | |
changeset 72007 | 13890356df78 |
parent 71924 | e5df9c8d9d4b |
permissions | -rw-r--r-- |
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clarified sessions: "Notable Examples in Isabelle/Pure";
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(* Title: Pure/Examples/First_Order_Logic.thy |
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Author: Makarius |
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*) |
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section \<open>A simple formulation of First-Order Logic\<close> |
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text \<open> |
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The subsequent theory development illustrates single-sorted intuitionistic |
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first-order logic with equality, formulated within the Pure framework. |
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\<close> |
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theory First_Order_Logic |
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imports Pure |
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begin |
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subsection \<open>Abstract syntax\<close> |
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typedecl i |
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typedecl o |
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judgment Trueprop :: "o \<Rightarrow> prop" ("_" 5) |
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subsection \<open>Propositional logic\<close> |
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axiomatization false :: o ("\<bottom>") |
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where falseE [elim]: "\<bottom> \<Longrightarrow> A" |
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axiomatization imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25) |
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where impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B" |
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and mp [dest]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B" |
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axiomatization conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35) |
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where conjI [intro]: "A \<Longrightarrow> B \<Longrightarrow> A \<and> B" |
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and conjD1: "A \<and> B \<Longrightarrow> A" |
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and conjD2: "A \<and> B \<Longrightarrow> B" |
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theorem conjE [elim]: |
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assumes "A \<and> B" |
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obtains A and B |
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proof |
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from \<open>A \<and> B\<close> show A |
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by (rule conjD1) |
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from \<open>A \<and> B\<close> show B |
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by (rule conjD2) |
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qed |
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axiomatization disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30) |
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where disjE [elim]: "A \<or> B \<Longrightarrow> (A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C" |
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and disjI1 [intro]: "A \<Longrightarrow> A \<or> B" |
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and disjI2 [intro]: "B \<Longrightarrow> A \<or> B" |
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definition true :: o ("\<top>") |
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where "\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>" |
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theorem trueI [intro]: \<top> |
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unfolding true_def .. |
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definition not :: "o \<Rightarrow> o" ("\<not> _" [40] 40) |
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where "\<not> A \<equiv> A \<longrightarrow> \<bottom>" |
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theorem notI [intro]: "(A \<Longrightarrow> \<bottom>) \<Longrightarrow> \<not> A" |
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unfolding not_def .. |
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theorem notE [elim]: "\<not> A \<Longrightarrow> A \<Longrightarrow> B" |
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unfolding not_def |
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proof - |
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assume "A \<longrightarrow> \<bottom>" and A |
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then have \<bottom> .. |
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then show B .. |
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qed |
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definition iff :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longleftrightarrow>" 25) |
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where "A \<longleftrightarrow> B \<equiv> (A \<longrightarrow> B) \<and> (B \<longrightarrow> A)" |
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theorem iffI [intro]: |
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assumes "A \<Longrightarrow> B" |
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and "B \<Longrightarrow> A" |
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shows "A \<longleftrightarrow> B" |
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unfolding iff_def |
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proof |
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from \<open>A \<Longrightarrow> B\<close> show "A \<longrightarrow> B" .. |
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from \<open>B \<Longrightarrow> A\<close> show "B \<longrightarrow> A" .. |
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qed |
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theorem iff1 [elim]: |
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assumes "A \<longleftrightarrow> B" and A |
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shows B |
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proof - |
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from \<open>A \<longleftrightarrow> B\<close> have "(A \<longrightarrow> B) \<and> (B \<longrightarrow> A)" |
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unfolding iff_def . |
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then have "A \<longrightarrow> B" .. |
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from this and \<open>A\<close> show B .. |
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qed |
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theorem iff2 [elim]: |
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assumes "A \<longleftrightarrow> B" and B |
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shows A |
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proof - |
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from \<open>A \<longleftrightarrow> B\<close> have "(A \<longrightarrow> B) \<and> (B \<longrightarrow> A)" |
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unfolding iff_def . |
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then have "B \<longrightarrow> A" .. |
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from this and \<open>B\<close> show A .. |
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qed |
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subsection \<open>Equality\<close> |
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axiomatization equal :: "i \<Rightarrow> i \<Rightarrow> o" (infixl "=" 50) |
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where refl [intro]: "x = x" |
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and subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y" |
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theorem trans [trans]: "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z" |
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by (rule subst) |
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theorem sym [sym]: "x = y \<Longrightarrow> y = x" |
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proof - |
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assume "x = y" |
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from this and refl show "y = x" |
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by (rule subst) |
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qed |
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subsection \<open>Quantifiers\<close> |
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axiomatization All :: "(i \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10) |
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where allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x" |
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and allD [dest]: "\<forall>x. P x \<Longrightarrow> P a" |
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axiomatization Ex :: "(i \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10) |
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where exI [intro]: "P a \<Longrightarrow> \<exists>x. P x" |
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and exE [elim]: "\<exists>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> C) \<Longrightarrow> C" |
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lemma "(\<exists>x. P (f x)) \<longrightarrow> (\<exists>y. P y)" |
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proof |
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assume "\<exists>x. P (f x)" |
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then obtain x where "P (f x)" .. |
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then show "\<exists>y. P y" .. |
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qed |
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lemma "(\<exists>x. \<forall>y. R x y) \<longrightarrow> (\<forall>y. \<exists>x. R x y)" |
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proof |
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assume "\<exists>x. \<forall>y. R x y" |
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then obtain x where "\<forall>y. R x y" .. |
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show "\<forall>y. \<exists>x. R x y" |
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proof |
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fix y |
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from \<open>\<forall>y. R x y\<close> have "R x y" .. |
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then show "\<exists>x. R x y" .. |
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qed |
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qed |
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end |