author | wenzelm |
Sat, 30 Oct 2010 16:33:58 +0200 | |
changeset 40291 | 012ed4426fda |
parent 40255 | 9ffbc25e1606 |
child 42596 | 6c621a9d612a |
permissions | -rw-r--r-- |
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theory Generic |
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imports Main |
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begin |
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chapter {* Generic tools and packages \label{ch:gen-tools} *} |
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section {* Configuration options *} |
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text {* Isabelle/Pure maintains a record of named configuration |
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options within the theory or proof context, with values of type |
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@{ML_type bool}, @{ML_type int}, @{ML_type real}, or @{ML_type |
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string}. Tools may declare options in ML, and then refer to these |
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values (relative to the context). Thus global reference variables |
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are easily avoided. The user may change the value of a |
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configuration option by means of an associated attribute of the same |
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name. This form of context declaration works particularly well with |
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commands such as @{command "declare"} or @{command "using"}. |
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For historical reasons, some tools cannot take the full proof |
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context into account and merely refer to the background theory. |
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This is accommodated by configuration options being declared as |
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``global'', which may not be changed within a local context. |
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\begin{matharray}{rcll} |
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@{command_def "print_configs"} & : & @{text "context \<rightarrow>"} \\ |
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\end{matharray} |
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\begin{rail} |
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name ('=' ('true' | 'false' | int | float | name))? |
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\end{rail} |
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\begin{description} |
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\item @{command "print_configs"} prints the available configuration |
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options, with names, types, and current values. |
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\item @{text "name = value"} as an attribute expression modifies the |
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named option, with the syntax of the value depending on the option's |
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type. For @{ML_type bool} the default value is @{text true}. Any |
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attempt to change a global option in a local context is ignored. |
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\end{description} |
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*} |
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section {* Basic proof tools *} |
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subsection {* Miscellaneous methods and attributes \label{sec:misc-meth-att} *} |
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text {* |
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\begin{matharray}{rcl} |
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@{method_def unfold} & : & @{text method} \\ |
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@{method_def fold} & : & @{text method} \\ |
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@{method_def insert} & : & @{text method} \\[0.5ex] |
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@{method_def erule}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def drule}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def frule}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def succeed} & : & @{text method} \\ |
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@{method_def fail} & : & @{text method} \\ |
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\end{matharray} |
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\begin{rail} |
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('fold' | 'unfold' | 'insert') thmrefs |
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; |
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('erule' | 'drule' | 'frule') ('('nat')')? thmrefs |
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; |
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\end{rail} |
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\begin{description} |
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\item @{method unfold}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{method fold}~@{text |
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"a\<^sub>1 \<dots> a\<^sub>n"} expand (or fold back) the given definitions throughout |
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all goals; any chained facts provided are inserted into the goal and |
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subject to rewriting as well. |
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\item @{method insert}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} inserts theorems as facts |
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into all goals of the proof state. Note that current facts |
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indicated for forward chaining are ignored. |
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\item @{method erule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, @{method |
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drule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, and @{method frule}~@{text |
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"a\<^sub>1 \<dots> a\<^sub>n"} are similar to the basic @{method rule} |
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method (see \secref{sec:pure-meth-att}), but apply rules by |
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elim-resolution, destruct-resolution, and forward-resolution, |
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respectively \cite{isabelle-implementation}. The optional natural |
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number argument (default 0) specifies additional assumption steps to |
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be performed here. |
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Note that these methods are improper ones, mainly serving for |
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experimentation and tactic script emulation. Different modes of |
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basic rule application are usually expressed in Isar at the proof |
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language level, rather than via implicit proof state manipulations. |
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For example, a proper single-step elimination would be done using |
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the plain @{method rule} method, with forward chaining of current |
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facts. |
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\item @{method succeed} yields a single (unchanged) result; it is |
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the identity of the ``@{text ","}'' method combinator (cf.\ |
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\secref{sec:proof-meth}). |
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\item @{method fail} yields an empty result sequence; it is the |
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identity of the ``@{text "|"}'' method combinator (cf.\ |
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\secref{sec:proof-meth}). |
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\end{description} |
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\begin{matharray}{rcl} |
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@{attribute_def tagged} & : & @{text attribute} \\ |
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@{attribute_def untagged} & : & @{text attribute} \\[0.5ex] |
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@{attribute_def THEN} & : & @{text attribute} \\ |
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@{attribute_def COMP} & : & @{text attribute} \\[0.5ex] |
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@{attribute_def unfolded} & : & @{text attribute} \\ |
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@{attribute_def folded} & : & @{text attribute} \\[0.5ex] |
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@{attribute_def rotated} & : & @{text attribute} \\ |
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@{attribute_def (Pure) elim_format} & : & @{text attribute} \\ |
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@{attribute_def standard}@{text "\<^sup>*"} & : & @{text attribute} \\ |
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@{attribute_def no_vars}@{text "\<^sup>*"} & : & @{text attribute} \\ |
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\end{matharray} |
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\begin{rail} |
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'tagged' name name |
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'untagged' name |
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('THEN' | 'COMP') ('[' nat ']')? thmref |
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; |
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('unfolded' | 'folded') thmrefs |
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'rotated' ( int )? |
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\end{rail} |
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\begin{description} |
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\item @{attribute tagged}~@{text "name value"} and @{attribute |
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untagged}~@{text name} add and remove \emph{tags} of some theorem. |
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Tags may be any list of string pairs that serve as formal comment. |
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The first string is considered the tag name, the second its value. |
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Note that @{attribute untagged} removes any tags of the same name. |
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\item @{attribute THEN}~@{text a} and @{attribute COMP}~@{text a} |
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compose rules by resolution. @{attribute THEN} resolves with the |
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first premise of @{text a} (an alternative position may be also |
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specified); the @{attribute COMP} version skips the automatic |
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lifting process that is normally intended (cf.\ @{ML "op RS"} and |
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@{ML "op COMP"} in \cite{isabelle-implementation}). |
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\item @{attribute unfolded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{attribute |
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folded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} expand and fold back again the given |
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definitions throughout a rule. |
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\item @{attribute rotated}~@{text n} rotate the premises of a |
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theorem by @{text n} (default 1). |
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\item @{attribute (Pure) elim_format} turns a destruction rule into |
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elimination rule format, by resolving with the rule @{prop "PROP A \<Longrightarrow> |
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(PROP A \<Longrightarrow> PROP B) \<Longrightarrow> PROP B"}. |
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Note that the Classical Reasoner (\secref{sec:classical}) provides |
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its own version of this operation. |
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\item @{attribute standard} puts a theorem into the standard form of |
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object-rules at the outermost theory level. Note that this |
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operation violates the local proof context (including active |
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locales). |
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\item @{attribute no_vars} replaces schematic variables by free |
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ones; this is mainly for tuning output of pretty printed theorems. |
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\end{description} |
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*} |
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subsection {* Low-level equational reasoning *} |
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text {* |
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\begin{matharray}{rcl} |
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@{method_def subst} & : & @{text method} \\ |
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@{method_def hypsubst} & : & @{text method} \\ |
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@{method_def split} & : & @{text method} \\ |
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\end{matharray} |
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\begin{rail} |
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'subst' ('(' 'asm' ')')? ('(' (nat+) ')')? thmref |
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'split' ('(' 'asm' ')')? thmrefs |
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; |
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\end{rail} |
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These methods provide low-level facilities for equational reasoning |
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that are intended for specialized applications only. Normally, |
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single step calculations would be performed in a structured text |
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(see also \secref{sec:calculation}), while the Simplifier methods |
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provide the canonical way for automated normalization (see |
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\secref{sec:simplifier}). |
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\begin{description} |
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\item @{method subst}~@{text eq} performs a single substitution step |
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using rule @{text eq}, which may be either a meta or object |
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equality. |
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\item @{method subst}~@{text "(asm) eq"} substitutes in an |
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assumption. |
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\item @{method subst}~@{text "(i \<dots> j) eq"} performs several |
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substitutions in the conclusion. The numbers @{text i} to @{text j} |
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indicate the positions to substitute at. Positions are ordered from |
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the top of the term tree moving down from left to right. For |
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example, in @{text "(a + b) + (c + d)"} there are three positions |
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where commutativity of @{text "+"} is applicable: 1 refers to @{text |
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"a + b"}, 2 to the whole term, and 3 to @{text "c + d"}. |
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If the positions in the list @{text "(i \<dots> j)"} are non-overlapping |
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(e.g.\ @{text "(2 3)"} in @{text "(a + b) + (c + d)"}) you may |
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assume all substitutions are performed simultaneously. Otherwise |
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the behaviour of @{text subst} is not specified. |
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\item @{method subst}~@{text "(asm) (i \<dots> j) eq"} performs the |
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substitutions in the assumptions. The positions refer to the |
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assumptions in order from left to right. For example, given in a |
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goal of the form @{text "P (a + b) \<Longrightarrow> P (c + d) \<Longrightarrow> \<dots>"}, position 1 of |
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commutativity of @{text "+"} is the subterm @{text "a + b"} and |
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position 2 is the subterm @{text "c + d"}. |
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\item @{method hypsubst} performs substitution using some |
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assumption; this only works for equations of the form @{text "x = |
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t"} where @{text x} is a free or bound variable. |
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\item @{method split}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} performs single-step case |
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splitting using the given rules. By default, splitting is performed |
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in the conclusion of a goal; the @{text "(asm)"} option indicates to |
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operate on assumptions instead. |
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Note that the @{method simp} method already involves repeated |
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application of split rules as declared in the current context. |
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\end{description} |
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*} |
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||
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subsection {* Further tactic emulations \label{sec:tactics} *} |
242 |
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text {* |
|
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The following improper proof methods emulate traditional tactics. |
|
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These admit direct access to the goal state, which is normally |
|
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considered harmful! In particular, this may involve both numbered |
|
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goal addressing (default 1), and dynamic instantiation within the |
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scope of some subgoal. |
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\begin{warn} |
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Dynamic instantiations refer to universally quantified parameters |
|
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of a subgoal (the dynamic context) rather than fixed variables and |
|
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term abbreviations of a (static) Isar context. |
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\end{warn} |
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Tactic emulation methods, unlike their ML counterparts, admit |
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simultaneous instantiation from both dynamic and static contexts. |
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If names occur in both contexts goal parameters hide locally fixed |
|
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variables. Likewise, schematic variables refer to term |
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abbreviations, if present in the static context. Otherwise the |
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schematic variable is interpreted as a schematic variable and left |
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to be solved by unification with certain parts of the subgoal. |
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Note that the tactic emulation proof methods in Isabelle/Isar are |
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consistently named @{text foo_tac}. Note also that variable names |
|
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occurring on left hand sides of instantiations must be preceded by a |
|
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question mark if they coincide with a keyword or contain dots. This |
|
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is consistent with the attribute @{attribute "where"} (see |
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\secref{sec:pure-meth-att}). |
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\begin{matharray}{rcl} |
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@{method_def rule_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def erule_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def drule_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def frule_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def cut_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def thin_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def subgoal_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def rename_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def rotate_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def tactic}@{text "\<^sup>*"} & : & @{text method} \\ |
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@{method_def raw_tactic}@{text "\<^sup>*"} & : & @{text method} \\ |
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\end{matharray} |
284 |
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\begin{rail} |
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( 'rule_tac' | 'erule_tac' | 'drule_tac' | 'frule_tac' | 'cut_tac' | 'thin_tac' ) goalspec? |
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( insts thmref | thmrefs ) |
288 |
; |
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'subgoal_tac' goalspec? (prop +) |
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; |
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'rename_tac' goalspec? (name +) |
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; |
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'rotate_tac' goalspec? int? |
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; |
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('tactic' | 'raw_tactic') text |
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; |
297 |
||
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insts: ((name '=' term) + 'and') 'in' |
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; |
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\end{rail} |
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\begin{description} |
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\item @{method rule_tac} etc. do resolution of rules with explicit |
26782 | 305 |
instantiation. This works the same way as the ML tactics @{ML |
30397 | 306 |
res_inst_tac} etc. (see \cite{isabelle-implementation}) |
26782 | 307 |
|
308 |
Multiple rules may be only given if there is no instantiation; then |
|
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@{method rule_tac} is the same as @{ML resolve_tac} in ML (see |
|
30397 | 310 |
\cite{isabelle-implementation}). |
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\item @{method cut_tac} inserts facts into the proof state as |
27209 | 313 |
assumption of a subgoal, see also @{ML Tactic.cut_facts_tac} in |
30397 | 314 |
\cite{isabelle-implementation}. Note that the scope of schematic |
26782 | 315 |
variables is spread over the main goal statement. Instantiations |
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may be given as well, see also ML tactic @{ML cut_inst_tac} in |
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\cite{isabelle-implementation}. |
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\item @{method thin_tac}~@{text \<phi>} deletes the specified assumption |
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from a subgoal; note that @{text \<phi>} may contain schematic variables. |
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See also @{ML thin_tac} in \cite{isabelle-implementation}. |
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\item @{method subgoal_tac}~@{text \<phi>} adds @{text \<phi>} as an |
27239 | 324 |
assumption to a subgoal. See also @{ML subgoal_tac} and @{ML |
30397 | 325 |
subgoals_tac} in \cite{isabelle-implementation}. |
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\item @{method rename_tac}~@{text "x\<^sub>1 \<dots> x\<^sub>n"} renames parameters of a |
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goal according to the list @{text "x\<^sub>1, \<dots>, x\<^sub>n"}, which refers to the |
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\emph{suffix} of variables. |
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\item @{method rotate_tac}~@{text n} rotates the assumptions of a |
26782 | 332 |
goal by @{text n} positions: from right to left if @{text n} is |
333 |
positive, and from left to right if @{text n} is negative; the |
|
334 |
default value is 1. See also @{ML rotate_tac} in |
|
30397 | 335 |
\cite{isabelle-implementation}. |
26782 | 336 |
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\item @{method tactic}~@{text "text"} produces a proof method from |
26782 | 338 |
any ML text of type @{ML_type tactic}. Apart from the usual ML |
27223 | 339 |
environment and the current proof context, the ML code may refer to |
340 |
the locally bound values @{ML_text facts}, which indicates any |
|
341 |
current facts used for forward-chaining. |
|
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\item @{method raw_tactic} is similar to @{method tactic}, but |
27223 | 344 |
presents the goal state in its raw internal form, where simultaneous |
345 |
subgoals appear as conjunction of the logical framework instead of |
|
346 |
the usual split into several subgoals. While feature this is useful |
|
347 |
for debugging of complex method definitions, it should not never |
|
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appear in production theories. |
|
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\end{description} |
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*} |
352 |
||
353 |
||
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section {* The Simplifier \label{sec:simplifier} *} |
26782 | 355 |
|
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subsection {* Simplification methods *} |
26782 | 357 |
|
358 |
text {* |
|
359 |
\begin{matharray}{rcl} |
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@{method_def simp} & : & @{text method} \\ |
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@{method_def simp_all} & : & @{text method} \\ |
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\end{matharray} |
363 |
||
364 |
\indexouternonterm{simpmod} |
|
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\begin{rail} |
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('simp' | 'simp_all') opt? (simpmod *) |
26782 | 367 |
; |
368 |
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opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use' | 'asm_lr' ) ')' |
26782 | 370 |
; |
371 |
simpmod: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') | |
|
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'split' (() | 'add' | 'del')) ':' thmrefs |
|
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; |
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\end{rail} |
|
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\begin{description} |
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\item @{method simp} invokes the Simplifier, after declaring |
26782 | 379 |
additional rules according to the arguments given. Note that the |
380 |
\railtterm{only} modifier first removes all other rewrite rules, |
|
381 |
congruences, and looper tactics (including splits), and then behaves |
|
382 |
like \railtterm{add}. |
|
383 |
||
384 |
\medskip The \railtterm{cong} modifiers add or delete Simplifier |
|
385 |
congruence rules (see also \cite{isabelle-ref}), the default is to |
|
386 |
add. |
|
387 |
||
388 |
\medskip The \railtterm{split} modifiers add or delete rules for the |
|
389 |
Splitter (see also \cite{isabelle-ref}), the default is to add. |
|
390 |
This works only if the Simplifier method has been properly setup to |
|
391 |
include the Splitter (all major object logics such HOL, HOLCF, FOL, |
|
392 |
ZF do this already). |
|
393 |
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\item @{method simp_all} is similar to @{method simp}, but acts on |
26782 | 395 |
all goals (backwards from the last to the first one). |
396 |
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\end{description} |
26782 | 398 |
|
399 |
By default the Simplifier methods take local assumptions fully into |
|
400 |
account, using equational assumptions in the subsequent |
|
401 |
normalization process, or simplifying assumptions themselves (cf.\ |
|
30397 | 402 |
@{ML asm_full_simp_tac} in \cite{isabelle-ref}). In structured |
403 |
proofs this is usually quite well behaved in practice: just the |
|
404 |
local premises of the actual goal are involved, additional facts may |
|
405 |
be inserted via explicit forward-chaining (via @{command "then"}, |
|
35613 | 406 |
@{command "from"}, @{command "using"} etc.). |
26782 | 407 |
|
408 |
Additional Simplifier options may be specified to tune the behavior |
|
409 |
further (mostly for unstructured scripts with many accidental local |
|
410 |
facts): ``@{text "(no_asm)"}'' means assumptions are ignored |
|
411 |
completely (cf.\ @{ML simp_tac}), ``@{text "(no_asm_simp)"}'' means |
|
412 |
assumptions are used in the simplification of the conclusion but are |
|
413 |
not themselves simplified (cf.\ @{ML asm_simp_tac}), and ``@{text |
|
414 |
"(no_asm_use)"}'' means assumptions are simplified but are not used |
|
415 |
in the simplification of each other or the conclusion (cf.\ @{ML |
|
416 |
full_simp_tac}). For compatibility reasons, there is also an option |
|
417 |
``@{text "(asm_lr)"}'', which means that an assumption is only used |
|
418 |
for simplifying assumptions which are to the right of it (cf.\ @{ML |
|
419 |
asm_lr_simp_tac}). |
|
420 |
||
27092 | 421 |
The configuration option @{text "depth_limit"} limits the number of |
26782 | 422 |
recursive invocations of the simplifier during conditional |
423 |
rewriting. |
|
424 |
||
425 |
\medskip The Splitter package is usually configured to work as part |
|
426 |
of the Simplifier. The effect of repeatedly applying @{ML |
|
427 |
split_tac} can be simulated by ``@{text "(simp only: split: |
|
428 |
a\<^sub>1 \<dots> a\<^sub>n)"}''. There is also a separate @{text split} |
|
429 |
method available for single-step case splitting. |
|
430 |
*} |
|
431 |
||
432 |
||
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subsection {* Declaring rules *} |
26782 | 434 |
|
435 |
text {* |
|
436 |
\begin{matharray}{rcl} |
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@{command_def "print_simpset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
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@{attribute_def simp} & : & @{text attribute} \\ |
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@{attribute_def cong} & : & @{text attribute} \\ |
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@{attribute_def split} & : & @{text attribute} \\ |
26782 | 441 |
\end{matharray} |
442 |
||
443 |
\begin{rail} |
|
444 |
('simp' | 'cong' | 'split') (() | 'add' | 'del') |
|
445 |
; |
|
446 |
\end{rail} |
|
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\begin{description} |
26782 | 449 |
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\item @{command "print_simpset"} prints the collection of rules |
26782 | 451 |
declared to the Simplifier, which is also known as ``simpset'' |
452 |
internally \cite{isabelle-ref}. |
|
453 |
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\item @{attribute simp} declares simplification rules. |
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\item @{attribute cong} declares congruence rules. |
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\item @{attribute split} declares case split rules. |
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\end{description} |
26782 | 461 |
*} |
462 |
||
463 |
||
27040 | 464 |
subsection {* Simplification procedures *} |
26782 | 465 |
|
466 |
text {* |
|
467 |
\begin{matharray}{rcl} |
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@{command_def "simproc_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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simproc & : & @{text attribute} \\ |
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\end{matharray} |
471 |
||
472 |
\begin{rail} |
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'simproc_setup' name '(' (term + '|') ')' '=' text \\ ('identifier' (nameref+))? |
26782 | 474 |
; |
475 |
||
476 |
'simproc' (('add' ':')? | 'del' ':') (name+) |
|
477 |
; |
|
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\end{rail} |
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479 |
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\begin{description} |
26782 | 481 |
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\item @{command "simproc_setup"} defines a named simplification |
26782 | 483 |
procedure that is invoked by the Simplifier whenever any of the |
484 |
given term patterns match the current redex. The implementation, |
|
485 |
which is provided as ML source text, needs to be of type @{ML_type |
|
486 |
"morphism -> simpset -> cterm -> thm option"}, where the @{ML_type |
|
487 |
cterm} represents the current redex @{text r} and the result is |
|
488 |
supposed to be some proven rewrite rule @{text "r \<equiv> r'"} (or a |
|
489 |
generalized version), or @{ML NONE} to indicate failure. The |
|
490 |
@{ML_type simpset} argument holds the full context of the current |
|
491 |
Simplifier invocation, including the actual Isar proof context. The |
|
492 |
@{ML_type morphism} informs about the difference of the original |
|
493 |
compilation context wrt.\ the one of the actual application later |
|
494 |
on. The optional @{keyword "identifier"} specifies theorems that |
|
495 |
represent the logical content of the abstract theory of this |
|
496 |
simproc. |
|
497 |
||
498 |
Morphisms and identifiers are only relevant for simprocs that are |
|
499 |
defined within a local target context, e.g.\ in a locale. |
|
500 |
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\item @{text "simproc add: name"} and @{text "simproc del: name"} |
26782 | 502 |
add or delete named simprocs to the current Simplifier context. The |
503 |
default is to add a simproc. Note that @{command "simproc_setup"} |
|
504 |
already adds the new simproc to the subsequent context. |
|
505 |
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\end{description} |
26782 | 507 |
*} |
508 |
||
509 |
||
27040 | 510 |
subsection {* Forward simplification *} |
26782 | 511 |
|
512 |
text {* |
|
513 |
\begin{matharray}{rcl} |
|
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514 |
@{attribute_def simplified} & : & @{text attribute} \\ |
26782 | 515 |
\end{matharray} |
516 |
||
517 |
\begin{rail} |
|
518 |
'simplified' opt? thmrefs? |
|
519 |
; |
|
520 |
||
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|
521 |
opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use') ')' |
26782 | 522 |
; |
523 |
\end{rail} |
|
524 |
||
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525 |
\begin{description} |
26782 | 526 |
|
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|
527 |
\item @{attribute simplified}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} causes a theorem to |
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|
528 |
be simplified, either by exactly the specified rules @{text "a\<^sub>1, \<dots>, |
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|
529 |
a\<^sub>n"}, or the implicit Simplifier context if no arguments are given. |
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|
530 |
The result is fully simplified by default, including assumptions and |
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531 |
conclusion; the options @{text no_asm} etc.\ tune the Simplifier in |
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|
532 |
the same way as the for the @{text simp} method. |
26782 | 533 |
|
534 |
Note that forward simplification restricts the simplifier to its |
|
535 |
most basic operation of term rewriting; solver and looper tactics |
|
536 |
\cite{isabelle-ref} are \emph{not} involved here. The @{text |
|
537 |
simplified} attribute should be only rarely required under normal |
|
538 |
circumstances. |
|
539 |
||
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540 |
\end{description} |
26782 | 541 |
*} |
542 |
||
543 |
||
27040 | 544 |
section {* The Classical Reasoner \label{sec:classical} *} |
26782 | 545 |
|
27040 | 546 |
subsection {* Basic methods *} |
26782 | 547 |
|
548 |
text {* |
|
549 |
\begin{matharray}{rcl} |
|
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550 |
@{method_def rule} & : & @{text method} \\ |
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551 |
@{method_def contradiction} & : & @{text method} \\ |
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|
552 |
@{method_def intro} & : & @{text method} \\ |
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553 |
@{method_def elim} & : & @{text method} \\ |
26782 | 554 |
\end{matharray} |
555 |
||
556 |
\begin{rail} |
|
557 |
('rule' | 'intro' | 'elim') thmrefs? |
|
558 |
; |
|
559 |
\end{rail} |
|
560 |
||
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561 |
\begin{description} |
26782 | 562 |
|
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|
563 |
\item @{method rule} as offered by the Classical Reasoner is a |
26782 | 564 |
refinement over the primitive one (see \secref{sec:pure-meth-att}). |
565 |
Both versions essentially work the same, but the classical version |
|
566 |
observes the classical rule context in addition to that of |
|
567 |
Isabelle/Pure. |
|
568 |
||
569 |
Common object logics (HOL, ZF, etc.) declare a rich collection of |
|
570 |
classical rules (even if these would qualify as intuitionistic |
|
571 |
ones), but only few declarations to the rule context of |
|
572 |
Isabelle/Pure (\secref{sec:pure-meth-att}). |
|
573 |
||
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574 |
\item @{method contradiction} solves some goal by contradiction, |
26782 | 575 |
deriving any result from both @{text "\<not> A"} and @{text A}. Chained |
576 |
facts, which are guaranteed to participate, may appear in either |
|
577 |
order. |
|
578 |
||
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579 |
\item @{method intro} and @{method elim} repeatedly refine some goal |
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|
580 |
by intro- or elim-resolution, after having inserted any chained |
26901 | 581 |
facts. Exactly the rules given as arguments are taken into account; |
582 |
this allows fine-tuned decomposition of a proof problem, in contrast |
|
583 |
to common automated tools. |
|
26782 | 584 |
|
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|
585 |
\end{description} |
26782 | 586 |
*} |
587 |
||
588 |
||
27040 | 589 |
subsection {* Automated methods *} |
26782 | 590 |
|
591 |
text {* |
|
592 |
\begin{matharray}{rcl} |
|
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593 |
@{method_def blast} & : & @{text method} \\ |
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594 |
@{method_def fast} & : & @{text method} \\ |
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595 |
@{method_def slow} & : & @{text method} \\ |
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596 |
@{method_def best} & : & @{text method} \\ |
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|
597 |
@{method_def safe} & : & @{text method} \\ |
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598 |
@{method_def clarify} & : & @{text method} \\ |
26782 | 599 |
\end{matharray} |
600 |
||
601 |
\indexouternonterm{clamod} |
|
602 |
\begin{rail} |
|
35613 | 603 |
'blast' nat? (clamod *) |
26782 | 604 |
; |
35613 | 605 |
('fast' | 'slow' | 'best' | 'safe' | 'clarify') (clamod *) |
26782 | 606 |
; |
607 |
||
608 |
clamod: (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' thmrefs |
|
609 |
; |
|
610 |
\end{rail} |
|
611 |
||
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612 |
\begin{description} |
26782 | 613 |
|
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|
614 |
\item @{method blast} refers to the classical tableau prover (see |
30397 | 615 |
@{ML blast_tac} in \cite{isabelle-ref}). The optional argument |
616 |
specifies a user-supplied search bound (default 20). |
|
26782 | 617 |
|
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|
618 |
\item @{method fast}, @{method slow}, @{method best}, @{method |
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|
619 |
safe}, and @{method clarify} refer to the generic classical |
26782 | 620 |
reasoner. See @{ML fast_tac}, @{ML slow_tac}, @{ML best_tac}, @{ML |
30397 | 621 |
safe_tac}, and @{ML clarify_tac} in \cite{isabelle-ref} for more |
622 |
information. |
|
26782 | 623 |
|
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624 |
\end{description} |
26782 | 625 |
|
626 |
Any of the above methods support additional modifiers of the context |
|
627 |
of classical rules. Their semantics is analogous to the attributes |
|
628 |
given before. Facts provided by forward chaining are inserted into |
|
35613 | 629 |
the goal before commencing proof search. |
26782 | 630 |
*} |
631 |
||
632 |
||
27040 | 633 |
subsection {* Combined automated methods \label{sec:clasimp} *} |
26782 | 634 |
|
635 |
text {* |
|
636 |
\begin{matharray}{rcl} |
|
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637 |
@{method_def auto} & : & @{text method} \\ |
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638 |
@{method_def force} & : & @{text method} \\ |
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639 |
@{method_def clarsimp} & : & @{text method} \\ |
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640 |
@{method_def fastsimp} & : & @{text method} \\ |
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|
641 |
@{method_def slowsimp} & : & @{text method} \\ |
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|
642 |
@{method_def bestsimp} & : & @{text method} \\ |
26782 | 643 |
\end{matharray} |
644 |
||
645 |
\indexouternonterm{clasimpmod} |
|
646 |
\begin{rail} |
|
35613 | 647 |
'auto' (nat nat)? (clasimpmod *) |
26782 | 648 |
; |
35613 | 649 |
('force' | 'clarsimp' | 'fastsimp' | 'slowsimp' | 'bestsimp') (clasimpmod *) |
26782 | 650 |
; |
651 |
||
652 |
clasimpmod: ('simp' (() | 'add' | 'del' | 'only') | |
|
653 |
('cong' | 'split') (() | 'add' | 'del') | |
|
654 |
'iff' (((() | 'add') '?'?) | 'del') | |
|
655 |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' thmrefs |
|
656 |
\end{rail} |
|
657 |
||
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658 |
\begin{description} |
26782 | 659 |
|
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|
660 |
\item @{method auto}, @{method force}, @{method clarsimp}, @{method |
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|
661 |
fastsimp}, @{method slowsimp}, and @{method bestsimp} provide access |
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|
662 |
to Isabelle's combined simplification and classical reasoning |
26782 | 663 |
tactics. These correspond to @{ML auto_tac}, @{ML force_tac}, @{ML |
664 |
clarsimp_tac}, and Classical Reasoner tactics with the Simplifier |
|
30397 | 665 |
added as wrapper, see \cite{isabelle-ref} for more information. The |
666 |
modifier arguments correspond to those given in |
|
26782 | 667 |
\secref{sec:simplifier} and \secref{sec:classical}. Just note that |
668 |
the ones related to the Simplifier are prefixed by \railtterm{simp} |
|
669 |
here. |
|
670 |
||
671 |
Facts provided by forward chaining are inserted into the goal before |
|
35613 | 672 |
doing the search. |
26782 | 673 |
|
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|
674 |
\end{description} |
26782 | 675 |
*} |
676 |
||
677 |
||
27040 | 678 |
subsection {* Declaring rules *} |
26782 | 679 |
|
680 |
text {* |
|
681 |
\begin{matharray}{rcl} |
|
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|
682 |
@{command_def "print_claset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
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|
683 |
@{attribute_def intro} & : & @{text attribute} \\ |
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|
684 |
@{attribute_def elim} & : & @{text attribute} \\ |
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|
685 |
@{attribute_def dest} & : & @{text attribute} \\ |
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|
686 |
@{attribute_def rule} & : & @{text attribute} \\ |
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|
687 |
@{attribute_def iff} & : & @{text attribute} \\ |
26782 | 688 |
\end{matharray} |
689 |
||
690 |
\begin{rail} |
|
691 |
('intro' | 'elim' | 'dest') ('!' | () | '?') nat? |
|
692 |
; |
|
693 |
'rule' 'del' |
|
694 |
; |
|
695 |
'iff' (((() | 'add') '?'?) | 'del') |
|
696 |
; |
|
697 |
\end{rail} |
|
698 |
||
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|
699 |
\begin{description} |
26782 | 700 |
|
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|
701 |
\item @{command "print_claset"} prints the collection of rules |
26782 | 702 |
declared to the Classical Reasoner, which is also known as |
703 |
``claset'' internally \cite{isabelle-ref}. |
|
704 |
||
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|
705 |
\item @{attribute intro}, @{attribute elim}, and @{attribute dest} |
26782 | 706 |
declare introduction, elimination, and destruction rules, |
707 |
respectively. By default, rules are considered as \emph{unsafe} |
|
708 |
(i.e.\ not applied blindly without backtracking), while ``@{text |
|
709 |
"!"}'' classifies as \emph{safe}. Rule declarations marked by |
|
710 |
``@{text "?"}'' coincide with those of Isabelle/Pure, cf.\ |
|
711 |
\secref{sec:pure-meth-att} (i.e.\ are only applied in single steps |
|
712 |
of the @{method rule} method). The optional natural number |
|
713 |
specifies an explicit weight argument, which is ignored by automated |
|
714 |
tools, but determines the search order of single rule steps. |
|
715 |
||
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|
716 |
\item @{attribute rule}~@{text del} deletes introduction, |
26782 | 717 |
elimination, or destruction rules from the context. |
718 |
||
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|
719 |
\item @{attribute iff} declares logical equivalences to the |
26782 | 720 |
Simplifier and the Classical reasoner at the same time. |
721 |
Non-conditional rules result in a ``safe'' introduction and |
|
722 |
elimination pair; conditional ones are considered ``unsafe''. Rules |
|
723 |
with negative conclusion are automatically inverted (using @{text |
|
26789 | 724 |
"\<not>"}-elimination internally). |
26782 | 725 |
|
726 |
The ``@{text "?"}'' version of @{attribute iff} declares rules to |
|
727 |
the Isabelle/Pure context only, and omits the Simplifier |
|
728 |
declaration. |
|
729 |
||
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|
730 |
\end{description} |
26782 | 731 |
*} |
732 |
||
733 |
||
27040 | 734 |
subsection {* Classical operations *} |
26782 | 735 |
|
736 |
text {* |
|
737 |
\begin{matharray}{rcl} |
|
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|
738 |
@{attribute_def swapped} & : & @{text attribute} \\ |
26782 | 739 |
\end{matharray} |
740 |
||
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|
741 |
\begin{description} |
26782 | 742 |
|
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|
743 |
\item @{attribute swapped} turns an introduction rule into an |
26782 | 744 |
elimination, by resolving with the classical swap principle @{text |
745 |
"(\<not> B \<Longrightarrow> A) \<Longrightarrow> (\<not> A \<Longrightarrow> B)"}. |
|
746 |
||
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|
747 |
\end{description} |
26782 | 748 |
*} |
749 |
||
750 |
||
27044 | 751 |
section {* Object-logic setup \label{sec:object-logic} *} |
26790 | 752 |
|
753 |
text {* |
|
754 |
\begin{matharray}{rcl} |
|
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|
755 |
@{command_def "judgment"} & : & @{text "theory \<rightarrow> theory"} \\ |
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|
756 |
@{method_def atomize} & : & @{text method} \\ |
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|
757 |
@{attribute_def atomize} & : & @{text attribute} \\ |
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|
758 |
@{attribute_def rule_format} & : & @{text attribute} \\ |
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|
759 |
@{attribute_def rulify} & : & @{text attribute} \\ |
26790 | 760 |
\end{matharray} |
761 |
||
762 |
The very starting point for any Isabelle object-logic is a ``truth |
|
763 |
judgment'' that links object-level statements to the meta-logic |
|
764 |
(with its minimal language of @{text prop} that covers universal |
|
765 |
quantification @{text "\<And>"} and implication @{text "\<Longrightarrow>"}). |
|
766 |
||
767 |
Common object-logics are sufficiently expressive to internalize rule |
|
768 |
statements over @{text "\<And>"} and @{text "\<Longrightarrow>"} within their own |
|
769 |
language. This is useful in certain situations where a rule needs |
|
770 |
to be viewed as an atomic statement from the meta-level perspective, |
|
771 |
e.g.\ @{text "\<And>x. x \<in> A \<Longrightarrow> P x"} versus @{text "\<forall>x \<in> A. P x"}. |
|
772 |
||
773 |
From the following language elements, only the @{method atomize} |
|
774 |
method and @{attribute rule_format} attribute are occasionally |
|
775 |
required by end-users, the rest is for those who need to setup their |
|
776 |
own object-logic. In the latter case existing formulations of |
|
777 |
Isabelle/FOL or Isabelle/HOL may be taken as realistic examples. |
|
778 |
||
779 |
Generic tools may refer to the information provided by object-logic |
|
780 |
declarations internally. |
|
781 |
||
782 |
\begin{rail} |
|
783 |
'judgment' constdecl |
|
784 |
; |
|
785 |
'atomize' ('(' 'full' ')')? |
|
786 |
; |
|
40255
9ffbc25e1606
eliminated obsolete \_ escapes in rail environments;
wenzelm
parents:
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diff
changeset
|
787 |
'rule_format' ('(' 'noasm' ')')? |
26790 | 788 |
; |
789 |
\end{rail} |
|
790 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
791 |
\begin{description} |
26790 | 792 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
793 |
\item @{command "judgment"}~@{text "c :: \<sigma> (mx)"} declares constant |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
794 |
@{text c} as the truth judgment of the current object-logic. Its |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
795 |
type @{text \<sigma>} should specify a coercion of the category of |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
796 |
object-level propositions to @{text prop} of the Pure meta-logic; |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
797 |
the mixfix annotation @{text "(mx)"} would typically just link the |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
798 |
object language (internally of syntactic category @{text logic}) |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
799 |
with that of @{text prop}. Only one @{command "judgment"} |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
800 |
declaration may be given in any theory development. |
26790 | 801 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
802 |
\item @{method atomize} (as a method) rewrites any non-atomic |
26790 | 803 |
premises of a sub-goal, using the meta-level equations declared via |
804 |
@{attribute atomize} (as an attribute) beforehand. As a result, |
|
805 |
heavily nested goals become amenable to fundamental operations such |
|
806 |
as resolution (cf.\ the @{method rule} method). Giving the ``@{text |
|
807 |
"(full)"}'' option here means to turn the whole subgoal into an |
|
808 |
object-statement (if possible), including the outermost parameters |
|
809 |
and assumptions as well. |
|
810 |
||
811 |
A typical collection of @{attribute atomize} rules for a particular |
|
812 |
object-logic would provide an internalization for each of the |
|
813 |
connectives of @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"}. |
|
814 |
Meta-level conjunction should be covered as well (this is |
|
815 |
particularly important for locales, see \secref{sec:locale}). |
|
816 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
817 |
\item @{attribute rule_format} rewrites a theorem by the equalities |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
818 |
declared as @{attribute rulify} rules in the current object-logic. |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
819 |
By default, the result is fully normalized, including assumptions |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
820 |
and conclusions at any depth. The @{text "(no_asm)"} option |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
821 |
restricts the transformation to the conclusion of a rule. |
26790 | 822 |
|
823 |
In common object-logics (HOL, FOL, ZF), the effect of @{attribute |
|
824 |
rule_format} is to replace (bounded) universal quantification |
|
825 |
(@{text "\<forall>"}) and implication (@{text "\<longrightarrow>"}) by the corresponding |
|
826 |
rule statements over @{text "\<And>"} and @{text "\<Longrightarrow>"}. |
|
827 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
828 |
\end{description} |
26790 | 829 |
*} |
830 |
||
26782 | 831 |
end |