# HG changeset patch
# User wenzelm
# Date 1307015963 -7200
# Node ID ec270c6cb942249d734db9f66bd8250861ffb035
# Parent  8c0a49d7285739a9931e08286b0fa80f73f83a7a
some material on "Structured induction proofs";

diff -r 8c0a49d72857 -r ec270c6cb942 doc-src/IsarRef/Thy/Synopsis.thy
--- a/doc-src/IsarRef/Thy/Synopsis.thy	Wed Jun 01 13:06:45 2011 +0200
+++ b/doc-src/IsarRef/Thy/Synopsis.thy	Thu Jun 02 13:59:23 2011 +0200
@@ -214,7 +214,7 @@
 end
 
 
-section {* Calculational reasoning *}
+section {* Calculational reasoning \label{sec:calculations-synopsis} *}
 
 text {*
   For example, see @{file "~~/src/HOL/Isar_Examples/Group.thy"}.
@@ -363,7 +363,205 @@
 end
 
 
-section {* Structured Natural Deduction *}
+section {* Structured induction proofs *}
+
+subsection {* Induction as Natural Deduction *}
+
+text {* In principle, induction is just a special case of Natural
+  Deduction (see also \secref{sec:natural-deduction-synopsis}).  For
+  example: *}
+
+thm nat.induct
+print_statement nat.induct
+
+notepad
+begin
+  fix n :: nat
+  have "P n"
+  proof (rule nat.induct)  -- {* fragile rule application! *}
+    show "P 0" sorry
+  next
+    fix n :: nat
+    assume "P n"
+    show "P (Suc n)" sorry
+  qed
+end
+
+text {*
+  In practice, much more proof infrastructure is required.
+
+  The proof method @{method induct} provides:
+  \begin{itemize}
+
+  \item implicit rule selection and robust instantiation
+
+  \item context elements via symbolic case names
+
+  \item support for rule-structured induction statements, with local
+    parameters, premises, etc.
+
+  \end{itemize}
+*}
+
+notepad
+begin
+  fix n :: nat
+  have "P n"
+  proof (induct n)
+    case 0
+    show ?case sorry
+  next
+    case (Suc n)
+    from Suc.hyps show ?case sorry
+  qed
+end
+
+
+subsubsection {* Example *}
+
+text {*
+  The subsequent example combines the following proof patterns:
+  \begin{itemize}
+
+  \item outermost induction (over the datatype structure of natural
+  numbers), to decompose the proof problem in top-down manner
+
+  \item calculational reasoning (\secref{sec:calculations-synopsis})
+  to compose the result in each case
+
+  \item solving local claims within the calculation by simplification
+
+  \end{itemize}
+*}
+
+lemma
+  fixes n :: nat
+  shows "(\<Sum>i=0..n. i) = n * (n + 1) div 2"
+proof (induct n)
+  case 0
+  have "(\<Sum>i=0..0. i) = (0::nat)" by simp
+  also have "\<dots> = 0 * (0 + 1) div 2" by simp
+  finally show ?case .
+next
+  case (Suc n)
+  have "(\<Sum>i=0..Suc n. i) = (\<Sum>i=0..n. i) + (n + 1)" by simp
+  also have "\<dots> = n * (n + 1) div 2 + (n + 1)" by (simp add: Suc.hyps)
+  also have "\<dots> = (n * (n + 1) + 2 * (n + 1)) div 2" by simp
+  also have "\<dots> = (Suc n * (Suc n + 1)) div 2" by simp
+  finally show ?case .
+qed
+
+text {* This demonstrates how induction proofs can be done without
+  having to consider the raw Natural Deduction structure. *}
+
+
+subsection {* Induction with local parameters and premises *}
+
+text {* Idea: Pure rule statements are passed through the induction
+  rule.  This achieves convenient proof patterns, thanks to some
+  internal trickery in the @{method induct} method.
+
+  Important: Using compact HOL formulae with @{text "\<forall>/\<longrightarrow>"} is a
+  well-known anti-pattern! It would produce useless formal noise.
+*}
+
+notepad
+begin
+  fix n :: nat
+  fix P :: "nat \<Rightarrow> bool"
+  fix Q :: "'a \<Rightarrow> nat \<Rightarrow> bool"
+
+  have "P n"
+  proof (induct n)
+    case 0
+    show "P 0" sorry
+  next
+    case (Suc n)
+    from `P n` show "P (Suc n)" sorry
+  qed
+
+  have "A n \<Longrightarrow> P n"
+  proof (induct n)
+    case 0
+    from `A 0` show "P 0" sorry
+  next
+    case (Suc n)
+    from `A n \<Longrightarrow> P n`
+      and `A (Suc n)` show "P (Suc n)" sorry
+  qed
+
+  have "\<And>x. Q x n"
+  proof (induct n)
+    case 0
+    show "Q x 0" sorry
+  next
+    case (Suc n)
+    from `\<And>x. Q x n` show "Q x (Suc n)" sorry
+    txt {* Local quantification admits arbitrary instances: *}
+    note `Q a n` and `Q b n`
+  qed
+end
+
+
+subsection {* Implicit induction context *}
+
+text {* The @{method induct} method can isolate local parameters and
+  premises directly from the given statement.  This is convenient in
+  practical applications, but requires some understanding of what is
+  going on internally (as explained above).  *}
+
+notepad
+begin
+  fix n :: nat
+  fix Q :: "'a \<Rightarrow> nat \<Rightarrow> bool"
+
+  fix x :: 'a
+  assume "A x n"
+  then have "Q x n"
+  proof (induct n arbitrary: x)
+    case 0
+    from `A x 0` show "Q x 0" sorry
+  next
+    case (Suc n)
+    from `\<And>x. A x n \<Longrightarrow> Q x n`  -- {* arbitrary instances can be produced here *}
+      and `A x (Suc n)` show "Q x (Suc n)" sorry
+  qed
+end
+
+
+subsection {* Advanced induction with term definitions *}
+
+text {* Induction over subexpressions of a certain shape are delicate
+  to formalize.  The Isar @{method induct} method provides
+  infrastructure for this.
+
+  Idea: sub-expressions of the problem are turned into a defined
+  induction variable; often accompanied with fixing of auxiliary
+  parameters in the original expression.  *}
+
+notepad
+begin
+  fix a :: "'a \<Rightarrow> nat"
+  fix A :: "nat \<Rightarrow> bool"
+
+  assume "A (a x)"
+  then have "P (a x)"
+  proof (induct "a x" arbitrary: x)
+    case 0
+    note prem = `A (a x)`
+      and defn = `0 = a x`
+    show "P (a x)" sorry
+  next
+    case (Suc n)
+    note hyp = `\<And>x. n = a x \<Longrightarrow> A (a x) \<Longrightarrow> P (a x)`
+      and prem = `A (a x)`
+      and defn = `Suc n = a x`
+    show "P (a x)" sorry
+  qed
+end
+
+
+section {* Structured Natural Deduction \label{sec:natural-deduction-synopsis} *}
 
 subsection {* Rule statements *}
 
diff -r 8c0a49d72857 -r ec270c6cb942 doc-src/IsarRef/Thy/document/Synopsis.tex
--- a/doc-src/IsarRef/Thy/document/Synopsis.tex	Wed Jun 01 13:06:45 2011 +0200
+++ b/doc-src/IsarRef/Thy/document/Synopsis.tex	Thu Jun 02 13:59:23 2011 +0200
@@ -481,7 +481,7 @@
 \isanewline
 \isacommand{end}\isamarkupfalse%
 %
-\isamarkupsection{Calculational reasoning%
+\isamarkupsection{Calculational reasoning \label{sec:calculations-synopsis}%
 }
 \isamarkuptrue%
 %
@@ -843,7 +843,437 @@
 \isanewline
 \isacommand{end}\isamarkupfalse%
 %
-\isamarkupsection{Structured Natural Deduction%
+\isamarkupsection{Structured induction proofs%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Induction as Natural Deduction%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In principle, induction is just a special case of Natural
+  Deduction (see also \secref{sec:natural-deduction-synopsis}).  For
+  example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ nat{\isaliteral{2E}{\isachardot}}induct\isanewline
+\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}\isamarkupfalse%
+\ nat{\isaliteral{2E}{\isachardot}}induct\isanewline
+\isanewline
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}rule\ nat{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\ \ %
+\isamarkupcmt{fragile rule application!%
+}
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\begin{isamarkuptext}%
+In practice, much more proof infrastructure is required.
+
+  The proof method \hyperlink{method.induct}{\mbox{\isa{induct}}} provides:
+  \begin{itemize}
+
+  \item implicit rule selection and robust instantiation
+
+  \item context elements via symbolic case names
+
+  \item support for rule-structured induction statements, with local
+    parameters, premises, etc.
+
+  \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ Suc{\isaliteral{2E}{\isachardot}}hyps\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\isamarkupsubsubsection{Example%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The subsequent example combines the following proof patterns:
+  \begin{itemize}
+
+  \item outermost induction (over the datatype structure of natural
+  numbers), to decompose the proof problem in top-down manner
+
+  \item calculational reasoning (\secref{sec:calculations-synopsis})
+  to compose the result in each case
+
+  \item solving local claims within the calculation by simplification
+
+  \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\isanewline
+\ \ \isakeyword{fixes}\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
+\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C53756D3E}{\isasymSum}}i{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{2E}{\isachardot}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C53756D3E}{\isasymSum}}i{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
+\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C53756D3E}{\isasymSum}}i{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}Suc\ n{\isaliteral{2E}{\isachardot}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C53756D3E}{\isasymSum}}i{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{2E}{\isachardot}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ Suc{\isaliteral{2E}{\isachardot}}hyps{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+This demonstrates how induction proofs can be done without
+  having to consider the raw Natural Deduction structure.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Induction with local parameters and premises%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Idea: Pure rule statements are passed through the induction
+  rule.  This achieves convenient proof patterns, thanks to some
+  internal trickery in the \hyperlink{method.induct}{\mbox{\isa{induct}}} method.
+
+  Important: Using compact HOL formulae with \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} is a
+  well-known anti-pattern! It would produce useless formal noise.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ P\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ Q\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}P\ n{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isadigit{0}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}A\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ n{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ Q\ x\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q\ x\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ Q\ x\ n{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q\ x\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+Local quantification admits arbitrary instances:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{note}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}Q\ a\ n{\isaliteral{60}{\isacharbackquoteclose}}\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}Q\ b\ n{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\isamarkupsubsection{Implicit induction context%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The \hyperlink{method.induct}{\mbox{\isa{induct}}} method can isolate local parameters and
+  premises directly from the given statement.  This is convenient in
+  practical applications, but requires some understanding of what is
+  going on internally (as explained above).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ Q\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q\ x\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ n\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ x{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}A\ x\ {\isadigit{0}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q\ x\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ A\ x\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q\ x\ n{\isaliteral{60}{\isacharbackquoteclose}}\ \ %
+\isamarkupcmt{arbitrary instances can be produced here%
+}
+\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}A\ x\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q\ x\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\isamarkupsubsection{Advanced induction with term definitions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Induction over subexpressions of a certain shape are delicate
+  to formalize.  The Isar \hyperlink{method.induct}{\mbox{\isa{induct}}} method provides
+  infrastructure for this.
+
+  Idea: sub-expressions of the problem are turned into a defined
+  induction variable; often accompanied with fixing of auxiliary
+  parameters in the original expression.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ a\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ A\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induct\ {\isaliteral{22}{\isachardoublequoteopen}}a\ x{\isaliteral{22}{\isachardoublequoteclose}}\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ x{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \ \ \isacommand{note}\isamarkupfalse%
+\ prem\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ defn\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{60}{\isacharbackquoteopen}}{\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ a\ x{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{note}\isamarkupfalse%
+\ hyp\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{3D}{\isacharequal}}\ a\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ prem\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ defn\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{60}{\isacharbackquoteopen}}Suc\ n\ {\isaliteral{3D}{\isacharequal}}\ a\ x{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}a\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\isamarkupsection{Structured Natural Deduction \label{sec:natural-deduction-synopsis}%
 }
 \isamarkuptrue%
 %