diff -r 42671298f037 -r 313a24b66a8d doc-src/TutorialI/Inductive/document/Advanced.tex --- a/doc-src/TutorialI/Inductive/document/Advanced.tex Sun Nov 07 23:32:26 2010 +0100 +++ b/doc-src/TutorialI/Inductive/document/Advanced.tex Mon Nov 08 00:00:47 2010 +0100 @@ -42,7 +42,7 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{datatype}\isamarkupfalse% -\ {\isacharprime}f\ gterm\ {\isacharequal}\ Apply\ {\isacharprime}f\ {\isachardoublequoteopen}{\isacharprime}f\ gterm\ list{\isachardoublequoteclose}% +\ {\isaliteral{27}{\isacharprime}}f\ gterm\ {\isaliteral{3D}{\isacharequal}}\ Apply\ {\isaliteral{27}{\isacharprime}}f\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}f\ gterm\ list{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% To try it out, we declare a datatype of some integer operations: integer constants, the unary minus operator and the addition @@ -50,16 +50,16 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{datatype}\isamarkupfalse% -\ integer{\isacharunderscore}op\ {\isacharequal}\ Number\ int\ {\isacharbar}\ UnaryMinus\ {\isacharbar}\ Plus% +\ integer{\isaliteral{5F}{\isacharunderscore}}op\ {\isaliteral{3D}{\isacharequal}}\ Number\ int\ {\isaliteral{7C}{\isacharbar}}\ UnaryMinus\ {\isaliteral{7C}{\isacharbar}}\ Plus% \begin{isamarkuptext}% -Now the type \isa{integer{\isacharunderscore}op\ gterm} denotes the ground +Now the type \isa{integer{\isaliteral{5F}{\isacharunderscore}}op\ gterm} denotes the ground terms built over those symbols. The type constructor \isa{gterm} can be generalized to a function over sets. It returns the set of ground terms that can be formed over a set \isa{F} of function symbols. For example, we could consider the set of ground terms formed from the finite -set \isa{{\isacharbraceleft}Number\ {\isadigit{2}}{\isacharcomma}\ UnaryMinus{\isacharcomma}\ Plus{\isacharbraceright}}. +set \isa{{\isaliteral{7B}{\isacharbraceleft}}Number\ {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ UnaryMinus{\isaliteral{2C}{\isacharcomma}}\ Plus{\isaliteral{7D}{\isacharbraceright}}}. This concept is inductive. If we have a list \isa{args} of ground terms over~\isa{F} and a function symbol \isa{f} in \isa{F}, then we @@ -74,13 +74,13 @@ list.% \end{isamarkuptext}% \isamarkuptrue% -\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse% +\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse% \isanewline -\ \ gterms\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}f\ set\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{for}\ F\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}f\ set{\isachardoublequoteclose}\isanewline +\ \ gterms\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}f\ set\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}f\ gterm\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{for}\ F\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}f\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isakeyword{where}\isanewline -step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}{\isasymforall}t\ {\isasymin}\ set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ \ f\ {\isasymin}\ F{\isasymrbrakk}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ gterms\ F{\isachardoublequoteclose}% +step{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ args{\isaliteral{2E}{\isachardot}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F{\isaliteral{3B}{\isacharsemicolon}}\ \ f\ {\isaliteral{5C3C696E3E}{\isasymin}}\ F{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Apply\ f\ args{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% To demonstrate a proof from this definition, let us show that the function \isa{gterms} @@ -88,7 +88,7 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{lemma}\isamarkupfalse% -\ gterms{\isacharunderscore}mono{\isacharcolon}\ {\isachardoublequoteopen}F{\isasymsubseteq}G\ {\isasymLongrightarrow}\ gterms\ F\ {\isasymsubseteq}\ gterms\ G{\isachardoublequoteclose}\isanewline +\ gterms{\isaliteral{5F}{\isacharunderscore}}mono{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}F{\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}G\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ gterms\ F\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ gterms\ G{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -98,7 +98,7 @@ \isacommand{apply}\isamarkupfalse% \ clarify\isanewline \isacommand{apply}\isamarkupfalse% -\ {\isacharparenleft}erule\ gterms{\isachardot}induct{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}erule\ gterms{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline \isacommand{apply}\isamarkupfalse% \ blast\isanewline \isacommand{done}\isamarkupfalse% @@ -124,9 +124,9 @@ \isa{gterms\ F}. (We could have used \isa{intro\ subsetI}.) We then apply rule induction. Here is the resulting subgoal: \begin{isabelle}% -\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}F\ {\isasymsubseteq}\ G{\isacharsemicolon}\ {\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F\ {\isasymand}\ t\ {\isasymin}\ gterms\ G{\isacharsemicolon}\ f\ {\isasymin}\ F{\isasymrbrakk}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ G% +\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ args\ f{\isaliteral{2E}{\isachardot}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}F\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ G{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t{\isaliteral{5C3C696E3E}{\isasymin}}set\ args{\isaliteral{2E}{\isachardot}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F\ {\isaliteral{5C3C616E643E}{\isasymand}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G{\isaliteral{3B}{\isacharsemicolon}}\ f\ {\isaliteral{5C3C696E3E}{\isasymin}}\ F{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G% \end{isabelle} The assumptions state that \isa{f} belongs to~\isa{F}, which is included in~\isa{G}, and that every element of the list \isa{args} is @@ -146,8 +146,8 @@ Why do we call this function \isa{gterms} instead of \isa{gterm}? A constant may have the same name as a type. However, name clashes could arise in the theorems that Isabelle generates. -Our choice of names keeps \isa{gterms{\isachardot}induct} separate from -\isa{gterm{\isachardot}induct}. +Our choice of names keeps \isa{gterms{\isaliteral{2E}{\isachardot}}induct} separate from +\isa{gterm{\isaliteral{2E}{\isachardot}}induct}. \end{warn} Call a term \textbf{well-formed} if each symbol occurring in it is applied @@ -162,14 +162,14 @@ term.% \end{isamarkuptext}% \isamarkuptrue% -\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse% +\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse% \isanewline -\ \ well{\isacharunderscore}formed{\isacharunderscore}gterm\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{for}\ arity\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline +\ \ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}f\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}f\ gterm\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{for}\ arity\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}f\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isakeyword{where}\isanewline -step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}{\isasymforall}t\ {\isasymin}\ set\ args{\isachardot}\ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharsemicolon}\ \ \isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequoteclose}% +step{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ args{\isaliteral{2E}{\isachardot}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity{\isaliteral{3B}{\isacharsemicolon}}\ \ \isanewline +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isaliteral{3D}{\isacharequal}}\ arity\ f{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Apply\ f\ args{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% The inductive definition neatly captures the reasoning above. The universal quantification over the @@ -200,22 +200,22 @@ direct, if more obscure, than using a universal quantifier.% \end{isamarkuptext}% \isamarkuptrue% -\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse% +\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse% \isanewline -\ \ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{for}\ arity\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline +\ \ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}f\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}f\ gterm\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{for}\ arity\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}f\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isakeyword{where}\isanewline -step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}args\ {\isasymin}\ lists\ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharparenright}{\isacharsemicolon}\ \ \isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequoteclose}\isanewline -\isakeyword{monos}\ lists{\isacharunderscore}mono% +step{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ {\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ \ \isanewline +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isaliteral{3D}{\isacharequal}}\ arity\ f{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Apply\ f\ args{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\isakeyword{monos}\ lists{\isaliteral{5F}{\isacharunderscore}}mono% \begin{isamarkuptext}% -We cite the theorem \isa{lists{\isacharunderscore}mono} to justify +We cite the theorem \isa{lists{\isaliteral{5F}{\isacharunderscore}}mono} to justify using the function \isa{lists}.% \footnote{This particular theorem is installed by default already, but we include the \isakeyword{monos} declaration in order to illustrate its syntax.} \begin{isabelle}% -A\ {\isasymsubseteq}\ B\ {\isasymLongrightarrow}\ lists\ A\ {\isasymsubseteq}\ lists\ B\rulename{lists{\isacharunderscore}mono}% +A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ lists\ A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ lists\ B\rulename{lists{\isaliteral{5F}{\isacharunderscore}}mono}% \end{isabelle} Why must the function be monotone? An inductive definition describes an iterative construction: each element of the set is constructed by a @@ -223,8 +223,8 @@ elements of \isa{even} are constructed by finitely many applications of the rules \begin{isabelle}% -{\isadigit{0}}\ {\isasymin}\ even\isasep\isanewline% -n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even% +{\isadigit{0}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\isasep\isanewline% +n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even% \end{isabelle} All references to a set in its inductive definition must be positive. Applications of an @@ -232,8 +232,8 @@ construction process to converge. The following pair of rules do not constitute an inductive definition: \begin{trivlist} -\item \isa{{\isadigit{0}}\ {\isasymin}\ even} -\item \isa{n\ {\isasymnotin}\ even\ {\isasymLongrightarrow}\ Suc\ n\ {\isasymin}\ even} +\item \isa{{\isadigit{0}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even} +\item \isa{n\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ even\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Suc\ n\ {\isaliteral{5C3C696E3E}{\isasymin}}\ even} \end{trivlist} Showing that 4 is even using these rules requires showing that 3 is not even. It is far from trivial to show that this set of rules @@ -242,7 +242,7 @@ Even with its use of the function \isa{lists}, the premise of our introduction rule is positive: \begin{isabelle}% -args\ {\isasymin}\ lists\ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharparenright}% +args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ {\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{29}{\isacharparenright}}% \end{isabelle} To apply the rule we construct a list \isa{args} of previously constructed well-formed terms. We obtain a @@ -265,7 +265,7 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{lemma}\isamarkupfalse% -\ {\isachardoublequoteopen}well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -275,7 +275,7 @@ \isacommand{apply}\isamarkupfalse% \ clarify\isanewline \isacommand{apply}\isamarkupfalse% -\ {\isacharparenleft}erule\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isachardot}induct{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}erule\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline \isacommand{apply}\isamarkupfalse% \ auto\isanewline \isacommand{done}\isamarkupfalse% @@ -295,14 +295,14 @@ % \begin{isamarkuptxt}% The \isa{clarify} method gives -us an element of \isa{well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity} on which to perform +us an element of \isa{well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity} on which to perform induction. The resulting subgoal can be proved automatically: \begin{isabelle}% -\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}\ \ \ }t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymand}\ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharsemicolon}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity% +\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ args\ f{\isaliteral{2E}{\isachardot}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t{\isaliteral{5C3C696E3E}{\isasymin}}set\ args{\isaliteral{2E}{\isachardot}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ \ \ }t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity\ {\isaliteral{5C3C616E643E}{\isasymand}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{3B}{\isacharsemicolon}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ \ }length\ args\ {\isaliteral{3D}{\isacharequal}}\ arity\ f{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity% \end{isabelle} This proof resembles the one given in {\S}\ref{sec:gterm-datatype} above, especially in the form of the @@ -317,7 +317,7 @@ % \endisadelimproof \isacommand{lemma}\isamarkupfalse% -\ {\isachardoublequoteopen}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -327,7 +327,7 @@ \isacommand{apply}\isamarkupfalse% \ clarify\isanewline \isacommand{apply}\isamarkupfalse% -\ {\isacharparenleft}erule\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}{\isachardot}induct{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}erule\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline \isacommand{apply}\isamarkupfalse% \ auto\isanewline \isacommand{done}\isamarkupfalse% @@ -349,28 +349,28 @@ The proof script is virtually identical, but the subgoal after applying induction may be surprising: \begin{isabelle}% -\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}args\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}}{\isasymin}\ lists\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}{\isasymin}\ \ }{\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasyminter}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}{\isasymin}\ \ {\isacharparenleft}}{\isacharbraceleft}a{\isachardot}\ a\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharbraceright}{\isacharparenright}{\isacharsemicolon}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity% +\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ args\ f{\isaliteral{2E}{\isachardot}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}args\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}}{\isaliteral{5C3C696E3E}{\isasymin}}\ lists\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C696E3E}{\isasymin}}\ \ }{\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C696E3E}{\isasymin}}\ \ {\isaliteral{28}{\isacharparenleft}}}{\isaliteral{7B}{\isacharbraceleft}}a{\isaliteral{2E}{\isachardot}}\ a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ \ }length\ args\ {\isaliteral{3D}{\isacharequal}}\ arity\ f{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity% \end{isabelle} The induction hypothesis contains an application of \isa{lists}. Using a monotone function in the inductive definition always has this effect. The subgoal may look uninviting, but fortunately \isa{lists} distributes over intersection: \begin{isabelle}% -listsp\ {\isacharparenleft}{\isacharparenleft}{\isasymlambda}x{\isachardot}\ x\ {\isasymin}\ A{\isacharparenright}\ {\isasyminter}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ x\ {\isasymin}\ B{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ x\ {\isasymin}\ lists\ A{\isacharparenright}\ {\isasyminter}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ x\ {\isasymin}\ lists\ B{\isacharparenright}\rulename{lists{\isacharunderscore}Int{\isacharunderscore}eq}% +listsp\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ B{\isaliteral{29}{\isacharparenright}}\rulename{lists{\isaliteral{5F}{\isacharunderscore}}Int{\isaliteral{5F}{\isacharunderscore}}eq}% \end{isabelle} Thanks to this default simplification rule, the induction hypothesis is quickly replaced by its two parts: \begin{trivlist} -\item \isa{args\ {\isasymin}\ lists\ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharparenright}} -\item \isa{args\ {\isasymin}\ lists\ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharparenright}} +\item \isa{args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ {\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{29}{\isacharparenright}}} +\item \isa{args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ {\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm\ arity{\isaliteral{29}{\isacharparenright}}} \end{trivlist} -Invoking the rule \isa{well{\isacharunderscore}formed{\isacharunderscore}gterm{\isachardot}step} completes the proof. The +Invoking the rule \isa{well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{2E}{\isachardot}}step} completes the proof. The call to \isa{auto} does all this work. This example is typical of how monotone functions @@ -378,7 +378,7 @@ distribute over intersection. Monotonicity implies one direction of this set equality; we have this theorem: \begin{isabelle}% -mono\ f\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}A\ {\isasyminter}\ B{\isacharparenright}\ {\isasymsubseteq}\ f\ A\ {\isasyminter}\ f\ B\rulename{mono{\isacharunderscore}Int}% +mono\ f\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ f\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ f\ B\rulename{mono{\isaliteral{5F}{\isacharunderscore}}Int}% \end{isabelle}% \end{isamarkuptxt}% \isamarkuptrue% @@ -397,16 +397,16 @@ \begin{isamarkuptext}% \index{rule inversion|(}% Does \isa{gterms} distribute over intersection? We have proved that this -function is monotone, so \isa{mono{\isacharunderscore}Int} gives one of the inclusions. The +function is monotone, so \isa{mono{\isaliteral{5F}{\isacharunderscore}}Int} gives one of the inclusions. The opposite inclusion asserts that if \isa{t} is a ground term over both of the sets \isa{F} and~\isa{G} then it is also a ground term over their intersection, -\isa{F\ {\isasyminter}\ G}.% +\isa{F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ G}.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{lemma}\isamarkupfalse% -\ gterms{\isacharunderscore}IntI{\isacharcolon}\isanewline -\ \ \ \ \ {\isachardoublequoteopen}t\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F{\isasyminter}G{\isacharparenright}{\isachardoublequoteclose}% +\ gterms{\isaliteral{5F}{\isacharunderscore}}IntI{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{5C3C696E7465723E}{\isasyminter}}G{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% \isadelimproof % \endisadelimproof @@ -422,32 +422,32 @@ % \begin{isamarkuptext}% Attempting this proof, we get the assumption -\isa{Apply\ f\ args\ {\isasymin}\ gterms\ G}, which cannot be broken down. +\isa{Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G}, which cannot be broken down. It looks like a job for rule inversion:\cmmdx{inductive\protect\_cases}% \end{isamarkuptext}% \isamarkuptrue% -\isacommand{inductive{\isacharunderscore}cases}\isamarkupfalse% -\ gterm{\isacharunderscore}Apply{\isacharunderscore}elim\ {\isacharbrackleft}elim{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}Apply\ f\ args\ {\isasymin}\ gterms\ F{\isachardoublequoteclose}% +\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}\isamarkupfalse% +\ gterm{\isaliteral{5F}{\isacharunderscore}}Apply{\isaliteral{5F}{\isacharunderscore}}elim\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% Here is the result. \begin{isabelle}% -{\isasymlbrakk}Apply\ f\ args\ {\isasymin}\ gterms\ F{\isacharsemicolon}\isanewline -\isaindent{\ }{\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ f\ {\isasymin}\ F{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\isanewline -{\isasymLongrightarrow}\ P\rulename{gterm{\isacharunderscore}Apply{\isacharunderscore}elim}% +{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F{\isaliteral{3B}{\isacharsemicolon}}\isanewline +\isaindent{\ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t{\isaliteral{5C3C696E3E}{\isasymin}}set\ args{\isaliteral{2E}{\isachardot}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F{\isaliteral{3B}{\isacharsemicolon}}\ f\ {\isaliteral{5C3C696E3E}{\isasymin}}\ F{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\rulename{gterm{\isaliteral{5F}{\isacharunderscore}}Apply{\isaliteral{5F}{\isacharunderscore}}elim}% \end{isabelle} This rule replaces an assumption about \isa{Apply\ f\ args} by assumptions about \isa{f} and~\isa{args}. No cases are discarded (there was only one to begin with) but the rule applies specifically to the pattern \isa{Apply\ f\ args}. It can be applied repeatedly as an elimination rule without looping, so we -have given the \isa{elim{\isacharbang}} attribute. +have given the \isa{elim{\isaliteral{21}{\isacharbang}}} attribute. Now we can prove the other half of that distributive law.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{lemma}\isamarkupfalse% -\ gterms{\isacharunderscore}IntI\ {\isacharbrackleft}rule{\isacharunderscore}format{\isacharcomma}\ intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \ \ \ {\isachardoublequoteopen}t\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F{\isasyminter}G{\isacharparenright}{\isachardoublequoteclose}\isanewline +\ gterms{\isaliteral{5F}{\isacharunderscore}}IntI\ {\isaliteral{5B}{\isacharbrackleft}}rule{\isaliteral{5F}{\isacharunderscore}}format{\isaliteral{2C}{\isacharcomma}}\ intro{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{5C3C696E7465723E}{\isasyminter}}G{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -455,7 +455,7 @@ % \isatagproof \isacommand{apply}\isamarkupfalse% -\ {\isacharparenleft}erule\ gterms{\isachardot}induct{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}erule\ gterms{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline \isacommand{apply}\isamarkupfalse% \ blast\isanewline \isacommand{done}\isamarkupfalse% @@ -477,19 +477,19 @@ The proof begins with rule induction over the definition of \isa{gterms}, which leaves a single subgoal: \begin{isabelle}% -\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}args\ f{\isachardot}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}\ \ \ }t\ {\isasymin}\ gterms\ F\ {\isasymand}\ {\isacharparenleft}t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}{\isacharparenright}{\isacharsemicolon}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }f\ {\isasymin}\ F{\isasymrbrakk}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\isanewline -\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymLongrightarrow}\ }Apply\ f\ args\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}% +\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}args\ f{\isaliteral{2E}{\isachardot}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t{\isaliteral{5C3C696E3E}{\isasymin}}set\ args{\isaliteral{2E}{\isachardot}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ \ \ }t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ F\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ {\isaliteral{28}{\isacharparenleft}}F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ G{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ \ }f\ {\isaliteral{5C3C696E3E}{\isasymin}}\ F{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\isanewline +\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ }Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ {\isaliteral{28}{\isacharparenleft}}F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ G{\isaliteral{29}{\isacharparenright}}% \end{isabelle} -To prove this, we assume \isa{Apply\ f\ args\ {\isasymin}\ gterms\ G}. Rule inversion, -in the form of \isa{gterm{\isacharunderscore}Apply{\isacharunderscore}elim}, infers +To prove this, we assume \isa{Apply\ f\ args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ gterms\ G}. Rule inversion, +in the form of \isa{gterm{\isaliteral{5F}{\isacharunderscore}}Apply{\isaliteral{5F}{\isacharunderscore}}elim}, infers that every element of \isa{args} belongs to \isa{gterms\ G}; hence (by the induction hypothesis) it belongs -to \isa{gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}}. Rule inversion also yields -\isa{f\ {\isasymin}\ G} and hence \isa{f\ {\isasymin}\ F\ {\isasyminter}\ G}. +to \isa{gterms\ {\isaliteral{28}{\isacharparenleft}}F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ G{\isaliteral{29}{\isacharparenright}}}. Rule inversion also yields +\isa{f\ {\isaliteral{5C3C696E3E}{\isasymin}}\ G} and hence \isa{f\ {\isaliteral{5C3C696E3E}{\isasymin}}\ F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ G}. All of this reasoning is done by \isa{blast}. \smallskip @@ -504,8 +504,8 @@ % \endisadelimproof \isacommand{lemma}\isamarkupfalse% -\ gterms{\isacharunderscore}Int{\isacharunderscore}eq\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \ \ \ {\isachardoublequoteopen}gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}\ {\isacharequal}\ gterms\ F\ {\isasyminter}\ gterms\ G{\isachardoublequoteclose}\isanewline +\ gterms{\isaliteral{5F}{\isacharunderscore}}Int{\isaliteral{5F}{\isacharunderscore}}eq\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}gterms\ {\isaliteral{28}{\isacharparenleft}}F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ G{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ gterms\ F\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ gterms\ G{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -513,7 +513,7 @@ % \isatagproof \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}blast\ intro{\isacharbang}{\isacharcolon}\ mono{\isacharunderscore}Int\ monoI\ gterms{\isacharunderscore}mono{\isacharparenright}% +\ {\isaliteral{28}{\isacharparenleft}}blast\ intro{\isaliteral{21}{\isacharbang}}{\isaliteral{3A}{\isacharcolon}}\ mono{\isaliteral{5F}{\isacharunderscore}}Int\ monoI\ gterms{\isaliteral{5F}{\isacharunderscore}}mono{\isaliteral{29}{\isacharparenright}}% \endisatagproof {\isafoldproof}% % @@ -533,10 +533,10 @@ list of argument types paired with the result type. Complete this inductive definition: \begin{isabelle} -\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse% +\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse% \isanewline -\ \ well{\isacharunderscore}typed{\isacharunderscore}gterm\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ {\isacharprime}t\ list\ {\isacharasterisk}\ {\isacharprime}t{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}f\ gterm\ {\isacharasterisk}\ {\isacharprime}t{\isacharparenright}set{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{for}\ sig\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}f\ {\isasymRightarrow}\ {\isacharprime}t\ list\ {\isacharasterisk}\ {\isacharprime}t{\isachardoublequoteclose}% +\ \ well{\isaliteral{5F}{\isacharunderscore}}typed{\isaliteral{5F}{\isacharunderscore}}gterm\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}f\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}t\ list\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{27}{\isacharprime}}t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}f\ gterm\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{27}{\isacharprime}}t{\isaliteral{29}{\isacharparenright}}set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{for}\ sig\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}f\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}t\ list\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{27}{\isacharprime}}t{\isaliteral{22}{\isachardoublequoteclose}}% \end{isabelle} \end{exercise} \end{isamarkuptext}