author | ballarin |
Wed, 15 Aug 2012 23:06:17 +0200 | |
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theory Generic |
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imports Base 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 \label{sec:config} *} |
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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"} like |
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this: |
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*} |
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declare [[show_main_goal = false]] |
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notepad |
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begin |
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note [[show_main_goal = true]] |
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end |
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text {* 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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@{rail " |
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@{syntax name} ('=' ('true' | 'false' | @{syntax int} | @{syntax float} | @{syntax name}))? |
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"} |
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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 intro} & : & @{text method} \\ |
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@{method_def elim} & : & @{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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@{rail " |
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(@@{method fold} | @@{method unfold} | @@{method insert}) @{syntax thmrefs} |
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; |
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(@@{method erule} | @@{method drule} | @@{method frule}) |
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('(' @{syntax nat} ')')? @{syntax thmrefs} |
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; |
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(@@{method intro} | @@{method elim}) @{syntax thmrefs}? |
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"} |
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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 intro} and @{method elim} repeatedly refine some goal |
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by intro- or elim-resolution, after having inserted any chained |
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facts. Exactly the rules given as arguments are taken into account; |
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this allows fine-tuned decomposition of a proof problem, in contrast |
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to common automated tools. |
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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 unfolded} & : & @{text attribute} \\ |
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@{attribute_def folded} & : & @{text attribute} \\ |
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@{attribute_def abs_def} & : & @{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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@{rail " |
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@@{attribute tagged} @{syntax name} @{syntax name} |
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@@{attribute untagged} @{syntax name} |
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@@{attribute THEN} ('[' @{syntax nat} ']')? @{syntax thmref} |
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(@@{attribute unfolded} | @@{attribute folded}) @{syntax thmrefs} |
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@@{attribute rotated} @{syntax int}? |
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"} |
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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} composes rules by resolution; it |
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resolves with the first premise of @{text a} (an alternative |
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position may be also specified). See also @{ML_op "RS"} in |
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\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 abs_def} turns an equation of the form @{prop "f x |
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y \<equiv> t"} into @{prop "f \<equiv> \<lambda>x y. t"}, which ensures that @{method |
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simp} or @{method unfold} steps always expand it. This also works |
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for object-logic equality. |
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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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@{rail " |
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@@{method subst} ('(' 'asm' ')')? \\ ('(' (@{syntax nat}+) ')')? @{syntax thmref} |
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@@{method split} @{syntax thmrefs} |
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"} |
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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} |
228 |
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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|
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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 |
27071 | 240 |
substitutions in the assumptions. The positions refer to the |
241 |
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 = |
248 |
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. Splitting is performed in the |
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conclusion or some assumption of the subgoal, depending of the |
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structure of the rule. |
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|
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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, using |
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@{attribute split}, for example. |
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\end{description} |
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*} |
261 |
||
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||
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subsection {* Further tactic emulations \label{sec:tactics} *} |
264 |
||
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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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||
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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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||
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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. |
|
285 |
||
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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}). |
|
292 |
||
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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} |
306 |
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@{rail " |
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(@@{method rule_tac} | @@{method erule_tac} | @@{method drule_tac} | |
42705 | 309 |
@@{method frule_tac} | @@{method cut_tac} | @@{method thin_tac}) @{syntax goal_spec}? \\ |
42617 | 310 |
( dynamic_insts @'in' @{syntax thmref} | @{syntax thmrefs} ) |
26782 | 311 |
; |
42705 | 312 |
@@{method subgoal_tac} @{syntax goal_spec}? (@{syntax prop} +) |
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; |
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@@{method rename_tac} @{syntax goal_spec}? (@{syntax name} +) |
26782 | 315 |
; |
42705 | 316 |
@@{method rotate_tac} @{syntax goal_spec}? @{syntax int}? |
26782 | 317 |
; |
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(@@{method tactic} | @@{method raw_tactic}) @{syntax text} |
26782 | 319 |
; |
320 |
||
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dynamic_insts: ((@{syntax name} '=' @{syntax term}) + @'and') |
322 |
"} |
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26782 | 323 |
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\begin{description} |
26782 | 325 |
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\item @{method rule_tac} etc. do resolution of rules with explicit |
26782 | 327 |
instantiation. This works the same way as the ML tactics @{ML |
30397 | 328 |
res_inst_tac} etc. (see \cite{isabelle-implementation}) |
26782 | 329 |
|
330 |
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 | 332 |
\cite{isabelle-implementation}). |
26782 | 333 |
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\item @{method cut_tac} inserts facts into the proof state as |
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assumption of a subgoal; instantiations may be given as well. Note |
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that the scope of schematic variables is spread over the main goal |
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statement and rule premises are turned into new subgoals. This is |
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in contrast to the regular method @{method insert} which inserts |
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closed rule statements. |
26782 | 340 |
|
46277 | 341 |
\item @{method thin_tac}~@{text \<phi>} deletes the specified premise |
342 |
from a subgoal. Note that @{text \<phi>} may contain schematic |
|
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variables, to abbreviate the intended proposition; the first |
|
344 |
matching subgoal premise will be deleted. Removing useless premises |
|
345 |
from a subgoal increases its readability and can make search tactics |
|
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run faster. |
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|
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\item @{method subgoal_tac}~@{text "\<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} adds the propositions |
349 |
@{text "\<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} as local premises to a subgoal, and poses the same |
|
350 |
as new subgoals (in the original context). |
|
26782 | 351 |
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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. |
26782 | 355 |
|
46274 | 356 |
\item @{method rotate_tac}~@{text n} rotates the premises of a |
357 |
subgoal by @{text n} positions: from right to left if @{text n} is |
|
26782 | 358 |
positive, and from left to right if @{text n} is negative; the |
46274 | 359 |
default value is 1. |
26782 | 360 |
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\item @{method tactic}~@{text "text"} produces a proof method from |
26782 | 362 |
any ML text of type @{ML_type tactic}. Apart from the usual ML |
27223 | 363 |
environment and the current proof context, the ML code may refer to |
364 |
the locally bound values @{ML_text facts}, which indicates any |
|
365 |
current facts used for forward-chaining. |
|
26782 | 366 |
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\item @{method raw_tactic} is similar to @{method tactic}, but |
27223 | 368 |
presents the goal state in its raw internal form, where simultaneous |
369 |
subgoals appear as conjunction of the logical framework instead of |
|
370 |
the usual split into several subgoals. While feature this is useful |
|
371 |
for debugging of complex method definitions, it should not never |
|
372 |
appear in production theories. |
|
26782 | 373 |
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\end{description} |
26782 | 375 |
*} |
376 |
||
377 |
||
27040 | 378 |
section {* The Simplifier \label{sec:simplifier} *} |
26782 | 379 |
|
27040 | 380 |
subsection {* Simplification methods *} |
26782 | 381 |
|
382 |
text {* |
|
383 |
\begin{matharray}{rcl} |
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@{method_def simp} & : & @{text method} \\ |
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@{method_def simp_all} & : & @{text method} \\ |
26782 | 386 |
\end{matharray} |
387 |
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@{rail " |
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(@@{method simp} | @@{method simp_all}) opt? (@{syntax simpmod} * ) |
26782 | 390 |
; |
391 |
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opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use' | 'asm_lr' ) ')' |
26782 | 393 |
; |
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@{syntax_def simpmod}: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') | |
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'split' (() | 'add' | 'del')) ':' @{syntax thmrefs} |
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"} |
26782 | 397 |
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398 |
\begin{description} |
26782 | 399 |
|
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\item @{method simp} invokes the Simplifier, after declaring |
26782 | 401 |
additional rules according to the arguments given. Note that the |
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402 |
@{text only} modifier first removes all other rewrite rules, |
26782 | 403 |
congruences, and looper tactics (including splits), and then behaves |
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like @{text add}. |
26782 | 405 |
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406 |
\medskip The @{text cong} modifiers add or delete Simplifier |
45645 | 407 |
congruence rules (see also \secref{sec:simp-cong}), the default is |
408 |
to add. |
|
26782 | 409 |
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\medskip The @{text split} modifiers add or delete rules for the |
26782 | 411 |
Splitter (see also \cite{isabelle-ref}), the default is to add. |
412 |
This works only if the Simplifier method has been properly setup to |
|
413 |
include the Splitter (all major object logics such HOL, HOLCF, FOL, |
|
414 |
ZF do this already). |
|
415 |
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\item @{method simp_all} is similar to @{method simp}, but acts on |
26782 | 417 |
all goals (backwards from the last to the first one). |
418 |
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\end{description} |
26782 | 420 |
|
421 |
By default the Simplifier methods take local assumptions fully into |
|
422 |
account, using equational assumptions in the subsequent |
|
423 |
normalization process, or simplifying assumptions themselves (cf.\ |
|
30397 | 424 |
@{ML asm_full_simp_tac} in \cite{isabelle-ref}). In structured |
425 |
proofs this is usually quite well behaved in practice: just the |
|
426 |
local premises of the actual goal are involved, additional facts may |
|
427 |
be inserted via explicit forward-chaining (via @{command "then"}, |
|
35613 | 428 |
@{command "from"}, @{command "using"} etc.). |
26782 | 429 |
|
430 |
Additional Simplifier options may be specified to tune the behavior |
|
431 |
further (mostly for unstructured scripts with many accidental local |
|
432 |
facts): ``@{text "(no_asm)"}'' means assumptions are ignored |
|
433 |
completely (cf.\ @{ML simp_tac}), ``@{text "(no_asm_simp)"}'' means |
|
434 |
assumptions are used in the simplification of the conclusion but are |
|
435 |
not themselves simplified (cf.\ @{ML asm_simp_tac}), and ``@{text |
|
436 |
"(no_asm_use)"}'' means assumptions are simplified but are not used |
|
437 |
in the simplification of each other or the conclusion (cf.\ @{ML |
|
438 |
full_simp_tac}). For compatibility reasons, there is also an option |
|
439 |
``@{text "(asm_lr)"}'', which means that an assumption is only used |
|
440 |
for simplifying assumptions which are to the right of it (cf.\ @{ML |
|
441 |
asm_lr_simp_tac}). |
|
442 |
||
27092 | 443 |
The configuration option @{text "depth_limit"} limits the number of |
26782 | 444 |
recursive invocations of the simplifier during conditional |
445 |
rewriting. |
|
446 |
||
447 |
\medskip The Splitter package is usually configured to work as part |
|
448 |
of the Simplifier. The effect of repeatedly applying @{ML |
|
449 |
split_tac} can be simulated by ``@{text "(simp only: split: |
|
450 |
a\<^sub>1 \<dots> a\<^sub>n)"}''. There is also a separate @{text split} |
|
451 |
method available for single-step case splitting. |
|
452 |
*} |
|
453 |
||
454 |
||
27040 | 455 |
subsection {* Declaring rules *} |
26782 | 456 |
|
457 |
text {* |
|
458 |
\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 split} & : & @{text attribute} \\ |
26782 | 462 |
\end{matharray} |
463 |
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@{rail " |
45645 | 465 |
(@@{attribute simp} | @@{attribute split}) (() | 'add' | 'del') |
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|
466 |
"} |
26782 | 467 |
|
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468 |
\begin{description} |
26782 | 469 |
|
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470 |
\item @{command "print_simpset"} prints the collection of rules |
26782 | 471 |
declared to the Simplifier, which is also known as ``simpset'' |
472 |
internally \cite{isabelle-ref}. |
|
473 |
||
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474 |
\item @{attribute simp} declares simplification rules. |
26782 | 475 |
|
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476 |
\item @{attribute split} declares case split rules. |
26782 | 477 |
|
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478 |
\end{description} |
26782 | 479 |
*} |
480 |
||
481 |
||
45645 | 482 |
subsection {* Congruence rules\label{sec:simp-cong} *} |
483 |
||
484 |
text {* |
|
485 |
\begin{matharray}{rcl} |
|
486 |
@{attribute_def cong} & : & @{text attribute} \\ |
|
487 |
\end{matharray} |
|
488 |
||
489 |
@{rail " |
|
490 |
@@{attribute cong} (() | 'add' | 'del') |
|
491 |
"} |
|
492 |
||
493 |
\begin{description} |
|
494 |
||
495 |
\item @{attribute cong} declares congruence rules to the Simplifier |
|
496 |
context. |
|
497 |
||
498 |
\end{description} |
|
499 |
||
500 |
Congruence rules are equalities of the form @{text [display] |
|
501 |
"\<dots> \<Longrightarrow> f ?x\<^sub>1 \<dots> ?x\<^sub>n = f ?y\<^sub>1 \<dots> ?y\<^sub>n"} |
|
502 |
||
503 |
This controls the simplification of the arguments of @{text f}. For |
|
504 |
example, some arguments can be simplified under additional |
|
505 |
assumptions: @{text [display] "?P\<^sub>1 \<longleftrightarrow> ?Q\<^sub>1 \<Longrightarrow> (?Q\<^sub>1 \<Longrightarrow> ?P\<^sub>2 \<longleftrightarrow> ?Q\<^sub>2) \<Longrightarrow> |
|
506 |
(?P\<^sub>1 \<longrightarrow> ?P\<^sub>2) \<longleftrightarrow> (?Q\<^sub>1 \<longrightarrow> ?Q\<^sub>2)"} |
|
507 |
||
508 |
Given this rule, the simplifier assumes @{text "?Q\<^sub>1"} and extracts |
|
509 |
rewrite rules from it when simplifying @{text "?P\<^sub>2"}. Such local |
|
510 |
assumptions are effective for rewriting formulae such as @{text "x = |
|
511 |
0 \<longrightarrow> y + x = y"}. |
|
512 |
||
513 |
%FIXME |
|
514 |
%The local assumptions are also provided as theorems to the solver; |
|
515 |
%see \secref{sec:simp-solver} below. |
|
516 |
||
517 |
\medskip The following congruence rule for bounded quantifiers also |
|
518 |
supplies contextual information --- about the bound variable: |
|
519 |
@{text [display] "(?A = ?B) \<Longrightarrow> (\<And>x. x \<in> ?B \<Longrightarrow> ?P x \<longleftrightarrow> ?Q x) \<Longrightarrow> |
|
520 |
(\<forall>x \<in> ?A. ?P x) \<longleftrightarrow> (\<forall>x \<in> ?B. ?Q x)"} |
|
521 |
||
522 |
\medskip This congruence rule for conditional expressions can |
|
523 |
supply contextual information for simplifying the arms: |
|
524 |
@{text [display] "?p = ?q \<Longrightarrow> (?q \<Longrightarrow> ?a = ?c) \<Longrightarrow> (\<not> ?q \<Longrightarrow> ?b = ?d) \<Longrightarrow> |
|
525 |
(if ?p then ?a else ?b) = (if ?q then ?c else ?d)"} |
|
526 |
||
527 |
A congruence rule can also \emph{prevent} simplification of some |
|
528 |
arguments. Here is an alternative congruence rule for conditional |
|
529 |
expressions that conforms to non-strict functional evaluation: |
|
530 |
@{text [display] "?p = ?q \<Longrightarrow> (if ?p then ?a else ?b) = (if ?q then ?a else ?b)"} |
|
531 |
||
532 |
Only the first argument is simplified; the others remain unchanged. |
|
533 |
This can make simplification much faster, but may require an extra |
|
534 |
case split over the condition @{text "?q"} to prove the goal. |
|
535 |
*} |
|
536 |
||
537 |
||
27040 | 538 |
subsection {* Simplification procedures *} |
26782 | 539 |
|
42925 | 540 |
text {* Simplification procedures are ML functions that produce proven |
541 |
rewrite rules on demand. They are associated with higher-order |
|
542 |
patterns that approximate the left-hand sides of equations. The |
|
543 |
Simplifier first matches the current redex against one of the LHS |
|
544 |
patterns; if this succeeds, the corresponding ML function is |
|
545 |
invoked, passing the Simplifier context and redex term. Thus rules |
|
546 |
may be specifically fashioned for particular situations, resulting |
|
547 |
in a more powerful mechanism than term rewriting by a fixed set of |
|
548 |
rules. |
|
549 |
||
550 |
Any successful result needs to be a (possibly conditional) rewrite |
|
551 |
rule @{text "t \<equiv> u"} that is applicable to the current redex. The |
|
552 |
rule will be applied just as any ordinary rewrite rule. It is |
|
553 |
expected to be already in \emph{internal form}, bypassing the |
|
554 |
automatic preprocessing of object-level equivalences. |
|
555 |
||
26782 | 556 |
\begin{matharray}{rcl} |
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557 |
@{command_def "simproc_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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558 |
simproc & : & @{text attribute} \\ |
26782 | 559 |
\end{matharray} |
560 |
||
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561 |
@{rail " |
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|
562 |
@@{command simproc_setup} @{syntax name} '(' (@{syntax term} + '|') ')' '=' |
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563 |
@{syntax text} \\ (@'identifier' (@{syntax nameref}+))? |
26782 | 564 |
; |
565 |
||
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|
566 |
@@{attribute simproc} (('add' ':')? | 'del' ':') (@{syntax name}+) |
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|
567 |
"} |
26782 | 568 |
|
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|
569 |
\begin{description} |
26782 | 570 |
|
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571 |
\item @{command "simproc_setup"} defines a named simplification |
26782 | 572 |
procedure that is invoked by the Simplifier whenever any of the |
573 |
given term patterns match the current redex. The implementation, |
|
574 |
which is provided as ML source text, needs to be of type @{ML_type |
|
575 |
"morphism -> simpset -> cterm -> thm option"}, where the @{ML_type |
|
576 |
cterm} represents the current redex @{text r} and the result is |
|
577 |
supposed to be some proven rewrite rule @{text "r \<equiv> r'"} (or a |
|
578 |
generalized version), or @{ML NONE} to indicate failure. The |
|
579 |
@{ML_type simpset} argument holds the full context of the current |
|
580 |
Simplifier invocation, including the actual Isar proof context. The |
|
581 |
@{ML_type morphism} informs about the difference of the original |
|
582 |
compilation context wrt.\ the one of the actual application later |
|
583 |
on. The optional @{keyword "identifier"} specifies theorems that |
|
584 |
represent the logical content of the abstract theory of this |
|
585 |
simproc. |
|
586 |
||
587 |
Morphisms and identifiers are only relevant for simprocs that are |
|
588 |
defined within a local target context, e.g.\ in a locale. |
|
589 |
||
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|
590 |
\item @{text "simproc add: name"} and @{text "simproc del: name"} |
26782 | 591 |
add or delete named simprocs to the current Simplifier context. The |
592 |
default is to add a simproc. Note that @{command "simproc_setup"} |
|
593 |
already adds the new simproc to the subsequent context. |
|
594 |
||
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|
595 |
\end{description} |
26782 | 596 |
*} |
597 |
||
598 |
||
42925 | 599 |
subsubsection {* Example *} |
600 |
||
601 |
text {* The following simplification procedure for @{thm |
|
602 |
[source=false, show_types] unit_eq} in HOL performs fine-grained |
|
603 |
control over rule application, beyond higher-order pattern matching. |
|
604 |
Declaring @{thm unit_eq} as @{attribute simp} directly would make |
|
605 |
the simplifier loop! Note that a version of this simplification |
|
606 |
procedure is already active in Isabelle/HOL. *} |
|
607 |
||
608 |
simproc_setup unit ("x::unit") = {* |
|
609 |
fn _ => fn _ => fn ct => |
|
610 |
if HOLogic.is_unit (term_of ct) then NONE |
|
611 |
else SOME (mk_meta_eq @{thm unit_eq}) |
|
612 |
*} |
|
613 |
||
614 |
text {* Since the Simplifier applies simplification procedures |
|
615 |
frequently, it is important to make the failure check in ML |
|
616 |
reasonably fast. *} |
|
617 |
||
618 |
||
27040 | 619 |
subsection {* Forward simplification *} |
26782 | 620 |
|
621 |
text {* |
|
622 |
\begin{matharray}{rcl} |
|
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|
623 |
@{attribute_def simplified} & : & @{text attribute} \\ |
26782 | 624 |
\end{matharray} |
625 |
||
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|
626 |
@{rail " |
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|
627 |
@@{attribute simplified} opt? @{syntax thmrefs}? |
26782 | 628 |
; |
629 |
||
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|
630 |
opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use') ')' |
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|
631 |
"} |
26782 | 632 |
|
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|
633 |
\begin{description} |
26782 | 634 |
|
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|
635 |
\item @{attribute simplified}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} causes a theorem to |
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|
636 |
be simplified, either by exactly the specified rules @{text "a\<^sub>1, \<dots>, |
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|
637 |
a\<^sub>n"}, or the implicit Simplifier context if no arguments are given. |
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|
638 |
The result is fully simplified by default, including assumptions and |
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|
639 |
conclusion; the options @{text no_asm} etc.\ tune the Simplifier in |
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|
640 |
the same way as the for the @{text simp} method. |
26782 | 641 |
|
642 |
Note that forward simplification restricts the simplifier to its |
|
643 |
most basic operation of term rewriting; solver and looper tactics |
|
644 |
\cite{isabelle-ref} are \emph{not} involved here. The @{text |
|
645 |
simplified} attribute should be only rarely required under normal |
|
646 |
circumstances. |
|
647 |
||
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|
648 |
\end{description} |
26782 | 649 |
*} |
650 |
||
651 |
||
27040 | 652 |
section {* The Classical Reasoner \label{sec:classical} *} |
26782 | 653 |
|
42930 | 654 |
subsection {* Basic concepts *} |
42927
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|
655 |
|
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|
656 |
text {* Although Isabelle is generic, many users will be working in |
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|
657 |
some extension of classical first-order logic. Isabelle/ZF is built |
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|
658 |
upon theory FOL, while Isabelle/HOL conceptually contains |
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|
659 |
first-order logic as a fragment. Theorem-proving in predicate logic |
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|
660 |
is undecidable, but many automated strategies have been developed to |
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|
661 |
assist in this task. |
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|
662 |
|
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|
663 |
Isabelle's classical reasoner is a generic package that accepts |
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|
664 |
certain information about a logic and delivers a suite of automatic |
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|
665 |
proof tools, based on rules that are classified and declared in the |
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|
666 |
context. These proof procedures are slow and simplistic compared |
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|
667 |
with high-end automated theorem provers, but they can save |
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|
668 |
considerable time and effort in practice. They can prove theorems |
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|
669 |
such as Pelletier's \cite{pelletier86} problems 40 and 41 in a few |
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|
670 |
milliseconds (including full proof reconstruction): *} |
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|
671 |
|
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|
672 |
lemma "(\<exists>y. \<forall>x. F x y \<longleftrightarrow> F x x) \<longrightarrow> \<not> (\<forall>x. \<exists>y. \<forall>z. F z y \<longleftrightarrow> \<not> F z x)" |
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|
673 |
by blast |
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|
674 |
|
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|
675 |
lemma "(\<forall>z. \<exists>y. \<forall>x. f x y \<longleftrightarrow> f x z \<and> \<not> f x x) \<longrightarrow> \<not> (\<exists>z. \<forall>x. f x z)" |
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|
676 |
by blast |
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|
677 |
|
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|
678 |
text {* The proof tools are generic. They are not restricted to |
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|
679 |
first-order logic, and have been heavily used in the development of |
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|
680 |
the Isabelle/HOL library and applications. The tactics can be |
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|
681 |
traced, and their components can be called directly; in this manner, |
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|
682 |
any proof can be viewed interactively. *} |
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|
683 |
|
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|
684 |
|
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|
685 |
subsubsection {* The sequent calculus *} |
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|
686 |
|
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|
687 |
text {* Isabelle supports natural deduction, which is easy to use for |
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|
688 |
interactive proof. But natural deduction does not easily lend |
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|
689 |
itself to automation, and has a bias towards intuitionism. For |
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|
690 |
certain proofs in classical logic, it can not be called natural. |
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|
691 |
The \emph{sequent calculus}, a generalization of natural deduction, |
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|
692 |
is easier to automate. |
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|
693 |
|
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|
694 |
A \textbf{sequent} has the form @{text "\<Gamma> \<turnstile> \<Delta>"}, where @{text "\<Gamma>"} |
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|
695 |
and @{text "\<Delta>"} are sets of formulae.\footnote{For first-order |
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|
696 |
logic, sequents can equivalently be made from lists or multisets of |
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|
697 |
formulae.} The sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} is |
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|
698 |
\textbf{valid} if @{text "P\<^sub>1 \<and> \<dots> \<and> P\<^sub>m"} implies @{text "Q\<^sub>1 \<or> \<dots> \<or> |
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|
699 |
Q\<^sub>n"}. Thus @{text "P\<^sub>1, \<dots>, P\<^sub>m"} represent assumptions, each of which |
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|
700 |
is true, while @{text "Q\<^sub>1, \<dots>, Q\<^sub>n"} represent alternative goals. A |
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|
701 |
sequent is \textbf{basic} if its left and right sides have a common |
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|
702 |
formula, as in @{text "P, Q \<turnstile> Q, R"}; basic sequents are trivially |
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|
703 |
valid. |
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|
704 |
|
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|
705 |
Sequent rules are classified as \textbf{right} or \textbf{left}, |
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|
706 |
indicating which side of the @{text "\<turnstile>"} symbol they operate on. |
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|
707 |
Rules that operate on the right side are analogous to natural |
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|
708 |
deduction's introduction rules, and left rules are analogous to |
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|
709 |
elimination rules. The sequent calculus analogue of @{text "(\<longrightarrow>I)"} |
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|
710 |
is the rule |
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|
711 |
\[ |
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|
712 |
\infer[@{text "(\<longrightarrow>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<longrightarrow> Q"}}{@{text "P, \<Gamma> \<turnstile> \<Delta>, Q"}} |
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|
713 |
\] |
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|
714 |
Applying the rule backwards, this breaks down some implication on |
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|
715 |
the right side of a sequent; @{text "\<Gamma>"} and @{text "\<Delta>"} stand for |
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|
716 |
the sets of formulae that are unaffected by the inference. The |
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|
717 |
analogue of the pair @{text "(\<or>I1)"} and @{text "(\<or>I2)"} is the |
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|
718 |
single rule |
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|
719 |
\[ |
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|
720 |
\infer[@{text "(\<or>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<or> Q"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P, Q"}} |
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|
721 |
\] |
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|
722 |
This breaks down some disjunction on the right side, replacing it by |
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|
723 |
both disjuncts. Thus, the sequent calculus is a kind of |
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|
724 |
multiple-conclusion logic. |
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|
725 |
|
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|
726 |
To illustrate the use of multiple formulae on the right, let us |
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|
727 |
prove the classical theorem @{text "(P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}. Working |
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|
728 |
backwards, we reduce this formula to a basic sequent: |
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|
729 |
\[ |
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|
730 |
\infer[@{text "(\<or>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}} |
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|
731 |
{\infer[@{text "(\<longrightarrow>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q), (Q \<longrightarrow> P)"}} |
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|
732 |
{\infer[@{text "(\<longrightarrow>R)"}]{@{text "P \<turnstile> Q, (Q \<longrightarrow> P)"}} |
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|
733 |
{@{text "P, Q \<turnstile> Q, P"}}}} |
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|
734 |
\] |
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|
735 |
|
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|
736 |
This example is typical of the sequent calculus: start with the |
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|
737 |
desired theorem and apply rules backwards in a fairly arbitrary |
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|
738 |
manner. This yields a surprisingly effective proof procedure. |
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|
739 |
Quantifiers add only few complications, since Isabelle handles |
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|
740 |
parameters and schematic variables. See \cite[Chapter |
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|
741 |
10]{paulson-ml2} for further discussion. *} |
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|
742 |
|
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|
743 |
|
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|
744 |
subsubsection {* Simulating sequents by natural deduction *} |
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|
745 |
|
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|
746 |
text {* Isabelle can represent sequents directly, as in the |
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|
747 |
object-logic LK. But natural deduction is easier to work with, and |
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|
748 |
most object-logics employ it. Fortunately, we can simulate the |
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|
749 |
sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} by the Isabelle formula |
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|
750 |
@{text "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>2 \<Longrightarrow> ... \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> Q\<^sub>1"} where the order of |
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|
751 |
the assumptions and the choice of @{text "Q\<^sub>1"} are arbitrary. |
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|
752 |
Elim-resolution plays a key role in simulating sequent proofs. |
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|
753 |
|
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|
754 |
We can easily handle reasoning on the left. Elim-resolution with |
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|
755 |
the rules @{text "(\<or>E)"}, @{text "(\<bottom>E)"} and @{text "(\<exists>E)"} achieves |
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|
756 |
a similar effect as the corresponding sequent rules. For the other |
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|
757 |
connectives, we use sequent-style elimination rules instead of |
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|
758 |
destruction rules such as @{text "(\<and>E1, 2)"} and @{text "(\<forall>E)"}. |
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|
759 |
But note that the rule @{text "(\<not>L)"} has no effect under our |
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|
760 |
representation of sequents! |
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|
761 |
\[ |
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|
762 |
\infer[@{text "(\<not>L)"}]{@{text "\<not> P, \<Gamma> \<turnstile> \<Delta>"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P"}} |
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|
763 |
\] |
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|
764 |
|
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|
765 |
What about reasoning on the right? Introduction rules can only |
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|
766 |
affect the formula in the conclusion, namely @{text "Q\<^sub>1"}. The |
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|
767 |
other right-side formulae are represented as negated assumptions, |
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|
768 |
@{text "\<not> Q\<^sub>2, \<dots>, \<not> Q\<^sub>n"}. In order to operate on one of these, it |
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|
769 |
must first be exchanged with @{text "Q\<^sub>1"}. Elim-resolution with the |
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|
770 |
@{text swap} rule has this effect: @{text "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"} |
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|
771 |
|
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|
772 |
To ensure that swaps occur only when necessary, each introduction |
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|
773 |
rule is converted into a swapped form: it is resolved with the |
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|
774 |
second premise of @{text "(swap)"}. The swapped form of @{text |
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|
775 |
"(\<and>I)"}, which might be called @{text "(\<not>\<and>E)"}, is |
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|
776 |
@{text [display] "\<not> (P \<and> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (\<not> R \<Longrightarrow> Q) \<Longrightarrow> R"} |
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|
777 |
|
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|
778 |
Similarly, the swapped form of @{text "(\<longrightarrow>I)"} is |
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|
779 |
@{text [display] "\<not> (P \<longrightarrow> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P \<Longrightarrow> Q) \<Longrightarrow> R"} |
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|
780 |
|
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|
781 |
Swapped introduction rules are applied using elim-resolution, which |
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|
782 |
deletes the negated formula. Our representation of sequents also |
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|
783 |
requires the use of ordinary introduction rules. If we had no |
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|
784 |
regard for readability of intermediate goal states, we could treat |
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|
785 |
the right side more uniformly by representing sequents as @{text |
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|
786 |
[display] "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> \<bottom>"} |
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|
787 |
*} |
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|
788 |
|
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|
789 |
|
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|
790 |
subsubsection {* Extra rules for the sequent calculus *} |
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|
791 |
|
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|
792 |
text {* As mentioned, destruction rules such as @{text "(\<and>E1, 2)"} and |
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|
793 |
@{text "(\<forall>E)"} must be replaced by sequent-style elimination rules. |
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|
794 |
In addition, we need rules to embody the classical equivalence |
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|
795 |
between @{text "P \<longrightarrow> Q"} and @{text "\<not> P \<or> Q"}. The introduction |
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|
796 |
rules @{text "(\<or>I1, 2)"} are replaced by a rule that simulates |
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|
797 |
@{text "(\<or>R)"}: @{text [display] "(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q"} |
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|
798 |
|
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|
799 |
The destruction rule @{text "(\<longrightarrow>E)"} is replaced by @{text [display] |
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|
800 |
"(P \<longrightarrow> Q) \<Longrightarrow> (\<not> P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"} |
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|
801 |
|
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|
802 |
Quantifier replication also requires special rules. In classical |
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|
803 |
logic, @{text "\<exists>x. P x"} is equivalent to @{text "\<not> (\<forall>x. \<not> P x)"}; |
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|
804 |
the rules @{text "(\<exists>R)"} and @{text "(\<forall>L)"} are dual: |
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|
805 |
\[ |
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|
806 |
\infer[@{text "(\<exists>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x"}}{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x, P t"}} |
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|
807 |
\qquad |
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|
808 |
\infer[@{text "(\<forall>L)"}]{@{text "\<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}{@{text "P t, \<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}} |
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|
809 |
\] |
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|
810 |
Thus both kinds of quantifier may be replicated. Theorems requiring |
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|
811 |
multiple uses of a universal formula are easy to invent; consider |
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|
812 |
@{text [display] "(\<forall>x. P x \<longrightarrow> P (f x)) \<and> P a \<longrightarrow> P (f\<^sup>n a)"} for any |
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|
813 |
@{text "n > 1"}. Natural examples of the multiple use of an |
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|
814 |
existential formula are rare; a standard one is @{text "\<exists>x. \<forall>y. P x |
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|
815 |
\<longrightarrow> P y"}. |
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|
816 |
|
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|
817 |
Forgoing quantifier replication loses completeness, but gains |
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|
818 |
decidability, since the search space becomes finite. Many useful |
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|
819 |
theorems can be proved without replication, and the search generally |
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|
820 |
delivers its verdict in a reasonable time. To adopt this approach, |
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moved/updated introduction to Classical Reasoner;
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|
821 |
represent the sequent rules @{text "(\<exists>R)"}, @{text "(\<exists>L)"} and |
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|
822 |
@{text "(\<forall>R)"} by @{text "(\<exists>I)"}, @{text "(\<exists>E)"} and @{text "(\<forall>I)"}, |
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|
823 |
respectively, and put @{text "(\<forall>E)"} into elimination form: @{text |
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|
824 |
[display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"} |
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|
825 |
|
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|
826 |
Elim-resolution with this rule will delete the universal formula |
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|
827 |
after a single use. To replicate universal quantifiers, replace the |
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|
828 |
rule by @{text [display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"} |
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|
829 |
|
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|
830 |
To replicate existential quantifiers, replace @{text "(\<exists>I)"} by |
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|
831 |
@{text [display] "(\<not> (\<exists>x. P x) \<Longrightarrow> P t) \<Longrightarrow> \<exists>x. P x"} |
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|
832 |
|
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|
833 |
All introduction rules mentioned above are also useful in swapped |
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|
834 |
form. |
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|
835 |
|
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|
836 |
Replication makes the search space infinite; we must apply the rules |
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|
837 |
with care. The classical reasoner distinguishes between safe and |
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|
838 |
unsafe rules, applying the latter only when there is no alternative. |
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|
839 |
Depth-first search may well go down a blind alley; best-first search |
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|
840 |
is better behaved in an infinite search space. However, quantifier |
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|
841 |
replication is too expensive to prove any but the simplest theorems. |
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|
842 |
*} |
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|
843 |
|
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|
844 |
|
42928
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|
845 |
subsection {* Rule declarations *} |
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|
846 |
|
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|
847 |
text {* The proof tools of the Classical Reasoner depend on |
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|
848 |
collections of rules declared in the context, which are classified |
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|
849 |
as introduction, elimination or destruction and as \emph{safe} or |
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|
850 |
\emph{unsafe}. In general, safe rules can be attempted blindly, |
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|
851 |
while unsafe rules must be used with care. A safe rule must never |
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|
852 |
reduce a provable goal to an unprovable set of subgoals. |
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|
853 |
|
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|
854 |
The rule @{text "P \<Longrightarrow> P \<or> Q"} is unsafe because it reduces @{text "P |
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|
855 |
\<or> Q"} to @{text "P"}, which might turn out as premature choice of an |
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|
856 |
unprovable subgoal. Any rule is unsafe whose premises contain new |
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|
857 |
unknowns. The elimination rule @{text "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"} is |
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|
858 |
unsafe, since it is applied via elim-resolution, which discards the |
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|
859 |
assumption @{text "\<forall>x. P x"} and replaces it by the weaker |
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|
860 |
assumption @{text "P t"}. The rule @{text "P t \<Longrightarrow> \<exists>x. P x"} is |
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|
861 |
unsafe for similar reasons. The quantifier duplication rule @{text |
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parents:
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|
862 |
"\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"} is unsafe in a different sense: |
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|
863 |
since it keeps the assumption @{text "\<forall>x. P x"}, it is prone to |
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|
864 |
looping. In classical first-order logic, all rules are safe except |
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|
865 |
those mentioned above. |
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|
866 |
|
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|
867 |
The safe~/ unsafe distinction is vague, and may be regarded merely |
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|
868 |
as a way of giving some rules priority over others. One could argue |
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|
869 |
that @{text "(\<or>E)"} is unsafe, because repeated application of it |
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|
870 |
could generate exponentially many subgoals. Induction rules are |
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|
871 |
unsafe because inductive proofs are difficult to set up |
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|
872 |
automatically. Any inference is unsafe that instantiates an unknown |
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|
873 |
in the proof state --- thus matching must be used, rather than |
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|
874 |
unification. Even proof by assumption is unsafe if it instantiates |
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|
875 |
unknowns shared with other subgoals. |
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|
876 |
|
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|
877 |
\begin{matharray}{rcl} |
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|
878 |
@{command_def "print_claset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
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|
879 |
@{attribute_def intro} & : & @{text attribute} \\ |
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|
880 |
@{attribute_def elim} & : & @{text attribute} \\ |
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|
881 |
@{attribute_def dest} & : & @{text attribute} \\ |
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|
882 |
@{attribute_def rule} & : & @{text attribute} \\ |
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|
883 |
@{attribute_def iff} & : & @{text attribute} \\ |
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|
884 |
@{attribute_def swapped} & : & @{text attribute} \\ |
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|
885 |
\end{matharray} |
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|
886 |
|
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|
887 |
@{rail " |
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|
888 |
(@@{attribute intro} | @@{attribute elim} | @@{attribute dest}) ('!' | () | '?') @{syntax nat}? |
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|
889 |
; |
9d946de41120
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|
890 |
@@{attribute rule} 'del' |
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|
891 |
; |
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|
892 |
@@{attribute iff} (((() | 'add') '?'?) | 'del') |
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|
893 |
"} |
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|
894 |
|
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|
895 |
\begin{description} |
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|
896 |
|
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|
897 |
\item @{command "print_claset"} prints the collection of rules |
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|
898 |
declared to the Classical Reasoner, i.e.\ the @{ML_type claset} |
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|
899 |
within the context. |
9d946de41120
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|
900 |
|
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|
901 |
\item @{attribute intro}, @{attribute elim}, and @{attribute dest} |
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|
902 |
declare introduction, elimination, and destruction rules, |
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|
903 |
respectively. By default, rules are considered as \emph{unsafe} |
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|
904 |
(i.e.\ not applied blindly without backtracking), while ``@{text |
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|
905 |
"!"}'' classifies as \emph{safe}. Rule declarations marked by |
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|
906 |
``@{text "?"}'' coincide with those of Isabelle/Pure, cf.\ |
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|
907 |
\secref{sec:pure-meth-att} (i.e.\ are only applied in single steps |
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|
908 |
of the @{method rule} method). The optional natural number |
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|
909 |
specifies an explicit weight argument, which is ignored by the |
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|
910 |
automated reasoning tools, but determines the search order of single |
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|
911 |
rule steps. |
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|
912 |
|
9d946de41120
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|
913 |
Introduction rules are those that can be applied using ordinary |
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|
914 |
resolution. Their swapped forms are generated internally, which |
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|
915 |
will be applied using elim-resolution. Elimination rules are |
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|
916 |
applied using elim-resolution. Rules are sorted by the number of |
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|
917 |
new subgoals they will yield; rules that generate the fewest |
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|
918 |
subgoals will be tried first. Otherwise, later declarations take |
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|
919 |
precedence over earlier ones. |
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|
920 |
|
9d946de41120
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|
921 |
Rules already present in the context with the same classification |
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|
922 |
are ignored. A warning is printed if the rule has already been |
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|
923 |
added with some other classification, but the rule is added anyway |
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|
924 |
as requested. |
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|
925 |
|
9d946de41120
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|
926 |
\item @{attribute rule}~@{text del} deletes all occurrences of a |
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|
927 |
rule from the classical context, regardless of its classification as |
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|
928 |
introduction~/ elimination~/ destruction and safe~/ unsafe. |
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|
929 |
|
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|
930 |
\item @{attribute iff} declares logical equivalences to the |
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|
931 |
Simplifier and the Classical reasoner at the same time. |
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|
932 |
Non-conditional rules result in a safe introduction and elimination |
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|
933 |
pair; conditional ones are considered unsafe. Rules with negative |
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|
934 |
conclusion are automatically inverted (using @{text "\<not>"}-elimination |
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|
935 |
internally). |
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|
936 |
|
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|
937 |
The ``@{text "?"}'' version of @{attribute iff} declares rules to |
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|
938 |
the Isabelle/Pure context only, and omits the Simplifier |
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|
939 |
declaration. |
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|
940 |
|
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|
941 |
\item @{attribute swapped} turns an introduction rule into an |
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|
942 |
elimination, by resolving with the classical swap principle @{text |
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|
943 |
"\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"} in the second position. This is mainly for |
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|
944 |
illustrative purposes: the Classical Reasoner already swaps rules |
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|
945 |
internally as explained above. |
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|
946 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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|
947 |
\end{description} |
26782 | 948 |
*} |
949 |
||
950 |
||
43365 | 951 |
subsection {* Structured methods *} |
952 |
||
953 |
text {* |
|
954 |
\begin{matharray}{rcl} |
|
955 |
@{method_def rule} & : & @{text method} \\ |
|
956 |
@{method_def contradiction} & : & @{text method} \\ |
|
957 |
\end{matharray} |
|
958 |
||
959 |
@{rail " |
|
960 |
@@{method rule} @{syntax thmrefs}? |
|
961 |
"} |
|
962 |
||
963 |
\begin{description} |
|
964 |
||
965 |
\item @{method rule} as offered by the Classical Reasoner is a |
|
966 |
refinement over the Pure one (see \secref{sec:pure-meth-att}). Both |
|
967 |
versions work the same, but the classical version observes the |
|
968 |
classical rule context in addition to that of Isabelle/Pure. |
|
969 |
||
970 |
Common object logics (HOL, ZF, etc.) declare a rich collection of |
|
971 |
classical rules (even if these would qualify as intuitionistic |
|
972 |
ones), but only few declarations to the rule context of |
|
973 |
Isabelle/Pure (\secref{sec:pure-meth-att}). |
|
974 |
||
975 |
\item @{method contradiction} solves some goal by contradiction, |
|
976 |
deriving any result from both @{text "\<not> A"} and @{text A}. Chained |
|
977 |
facts, which are guaranteed to participate, may appear in either |
|
978 |
order. |
|
979 |
||
980 |
\end{description} |
|
981 |
*} |
|
982 |
||
983 |
||
27040 | 984 |
subsection {* Automated methods *} |
26782 | 985 |
|
986 |
text {* |
|
987 |
\begin{matharray}{rcl} |
|
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|
988 |
@{method_def blast} & : & @{text method} \\ |
42930 | 989 |
@{method_def auto} & : & @{text method} \\ |
990 |
@{method_def force} & : & @{text method} \\ |
|
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|
991 |
@{method_def fast} & : & @{text method} \\ |
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|
992 |
@{method_def slow} & : & @{text method} \\ |
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|
993 |
@{method_def best} & : & @{text method} \\ |
44911 | 994 |
@{method_def fastforce} & : & @{text method} \\ |
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|
995 |
@{method_def slowsimp} & : & @{text method} \\ |
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|
996 |
@{method_def bestsimp} & : & @{text method} \\ |
43367 | 997 |
@{method_def deepen} & : & @{text method} \\ |
26782 | 998 |
\end{matharray} |
999 |
||
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|
1000 |
@{rail " |
42930 | 1001 |
@@{method blast} @{syntax nat}? (@{syntax clamod} * ) |
1002 |
; |
|
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|
1003 |
@@{method auto} (@{syntax nat} @{syntax nat})? (@{syntax clasimpmod} * ) |
26782 | 1004 |
; |
42930 | 1005 |
@@{method force} (@{syntax clasimpmod} * ) |
1006 |
; |
|
1007 |
(@@{method fast} | @@{method slow} | @@{method best}) (@{syntax clamod} * ) |
|
26782 | 1008 |
; |
44911 | 1009 |
(@@{method fastforce} | @@{method slowsimp} | @@{method bestsimp}) |
42930 | 1010 |
(@{syntax clasimpmod} * ) |
1011 |
; |
|
43367 | 1012 |
@@{method deepen} (@{syntax nat} ?) (@{syntax clamod} * ) |
1013 |
; |
|
42930 | 1014 |
@{syntax_def clamod}: |
1015 |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' @{syntax thmrefs} |
|
1016 |
; |
|
42596
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|
1017 |
@{syntax_def clasimpmod}: ('simp' (() | 'add' | 'del' | 'only') | |
26782 | 1018 |
('cong' | 'split') (() | 'add' | 'del') | |
1019 |
'iff' (((() | 'add') '?'?) | 'del') | |
|
42596
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|
1020 |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' @{syntax thmrefs} |
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|
1021 |
"} |
26782 | 1022 |
|
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|
1023 |
\begin{description} |
26782 | 1024 |
|
42930 | 1025 |
\item @{method blast} is a separate classical tableau prover that |
1026 |
uses the same classical rule declarations as explained before. |
|
1027 |
||
1028 |
Proof search is coded directly in ML using special data structures. |
|
1029 |
A successful proof is then reconstructed using regular Isabelle |
|
1030 |
inferences. It is faster and more powerful than the other classical |
|
1031 |
reasoning tools, but has major limitations too. |
|
1032 |
||
1033 |
\begin{itemize} |
|
1034 |
||
1035 |
\item It does not use the classical wrapper tacticals, such as the |
|
44911 | 1036 |
integration with the Simplifier of @{method fastforce}. |
42930 | 1037 |
|
1038 |
\item It does not perform higher-order unification, as needed by the |
|
1039 |
rule @{thm [source=false] rangeI} in HOL. There are often |
|
1040 |
alternatives to such rules, for example @{thm [source=false] |
|
1041 |
range_eqI}. |
|
1042 |
||
1043 |
\item Function variables may only be applied to parameters of the |
|
1044 |
subgoal. (This restriction arises because the prover does not use |
|
1045 |
higher-order unification.) If other function variables are present |
|
1046 |
then the prover will fail with the message \texttt{Function Var's |
|
1047 |
argument not a bound variable}. |
|
1048 |
||
1049 |
\item Its proof strategy is more general than @{method fast} but can |
|
1050 |
be slower. If @{method blast} fails or seems to be running forever, |
|
1051 |
try @{method fast} and the other proof tools described below. |
|
1052 |
||
1053 |
\end{itemize} |
|
1054 |
||
1055 |
The optional integer argument specifies a bound for the number of |
|
1056 |
unsafe steps used in a proof. By default, @{method blast} starts |
|
1057 |
with a bound of 0 and increases it successively to 20. In contrast, |
|
1058 |
@{text "(blast lim)"} tries to prove the goal using a search bound |
|
1059 |
of @{text "lim"}. Sometimes a slow proof using @{method blast} can |
|
1060 |
be made much faster by supplying the successful search bound to this |
|
1061 |
proof method instead. |
|
1062 |
||
1063 |
\item @{method auto} combines classical reasoning with |
|
1064 |
simplification. It is intended for situations where there are a lot |
|
1065 |
of mostly trivial subgoals; it proves all the easy ones, leaving the |
|
1066 |
ones it cannot prove. Occasionally, attempting to prove the hard |
|
1067 |
ones may take a long time. |
|
1068 |
||
43332 | 1069 |
The optional depth arguments in @{text "(auto m n)"} refer to its |
1070 |
builtin classical reasoning procedures: @{text m} (default 4) is for |
|
1071 |
@{method blast}, which is tried first, and @{text n} (default 2) is |
|
1072 |
for a slower but more general alternative that also takes wrappers |
|
1073 |
into account. |
|
42930 | 1074 |
|
1075 |
\item @{method force} is intended to prove the first subgoal |
|
1076 |
completely, using many fancy proof tools and performing a rather |
|
1077 |
exhaustive search. As a result, proof attempts may take rather long |
|
1078 |
or diverge easily. |
|
1079 |
||
1080 |
\item @{method fast}, @{method best}, @{method slow} attempt to |
|
1081 |
prove the first subgoal using sequent-style reasoning as explained |
|
1082 |
before. Unlike @{method blast}, they construct proofs directly in |
|
1083 |
Isabelle. |
|
26782 | 1084 |
|
42930 | 1085 |
There is a difference in search strategy and back-tracking: @{method |
1086 |
fast} uses depth-first search and @{method best} uses best-first |
|
1087 |
search (guided by a heuristic function: normally the total size of |
|
1088 |
the proof state). |
|
1089 |
||
1090 |
Method @{method slow} is like @{method fast}, but conducts a broader |
|
1091 |
search: it may, when backtracking from a failed proof attempt, undo |
|
1092 |
even the step of proving a subgoal by assumption. |
|
1093 |
||
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|
1094 |
\item @{method fastforce}, @{method slowsimp}, @{method bestsimp} |
c422128d3889
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|
1095 |
are like @{method fast}, @{method slow}, @{method best}, |
c422128d3889
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diff
changeset
|
1096 |
respectively, but use the Simplifier as additional wrapper. The name |
c422128d3889
discontinued obsolete method fastsimp / tactic fast_simp_tac;
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changeset
|
1097 |
@{method fastforce}, reflects the behaviour of this popular method |
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|
1098 |
better without requiring an understanding of its implementation. |
42930 | 1099 |
|
43367 | 1100 |
\item @{method deepen} works by exhaustive search up to a certain |
1101 |
depth. The start depth is 4 (unless specified explicitly), and the |
|
1102 |
depth is increased iteratively up to 10. Unsafe rules are modified |
|
1103 |
to preserve the formula they act on, so that it be used repeatedly. |
|
1104 |
This method can prove more goals than @{method fast}, but is much |
|
1105 |
slower, for example if the assumptions have many universal |
|
1106 |
quantifiers. |
|
1107 |
||
42930 | 1108 |
\end{description} |
1109 |
||
1110 |
Any of the above methods support additional modifiers of the context |
|
1111 |
of classical (and simplifier) rules, but the ones related to the |
|
1112 |
Simplifier are explicitly prefixed by @{text simp} here. The |
|
1113 |
semantics of these ad-hoc rule declarations is analogous to the |
|
1114 |
attributes given before. Facts provided by forward chaining are |
|
1115 |
inserted into the goal before commencing proof search. |
|
1116 |
*} |
|
1117 |
||
1118 |
||
1119 |
subsection {* Semi-automated methods *} |
|
1120 |
||
1121 |
text {* These proof methods may help in situations when the |
|
1122 |
fully-automated tools fail. The result is a simpler subgoal that |
|
1123 |
can be tackled by other means, such as by manual instantiation of |
|
1124 |
quantifiers. |
|
1125 |
||
1126 |
\begin{matharray}{rcl} |
|
1127 |
@{method_def safe} & : & @{text method} \\ |
|
1128 |
@{method_def clarify} & : & @{text method} \\ |
|
1129 |
@{method_def clarsimp} & : & @{text method} \\ |
|
1130 |
\end{matharray} |
|
1131 |
||
1132 |
@{rail " |
|
1133 |
(@@{method safe} | @@{method clarify}) (@{syntax clamod} * ) |
|
1134 |
; |
|
1135 |
@@{method clarsimp} (@{syntax clasimpmod} * ) |
|
1136 |
"} |
|
1137 |
||
1138 |
\begin{description} |
|
1139 |
||
1140 |
\item @{method safe} repeatedly performs safe steps on all subgoals. |
|
1141 |
It is deterministic, with at most one outcome. |
|
1142 |
||
43366 | 1143 |
\item @{method clarify} performs a series of safe steps without |
1144 |
splitting subgoals; see also @{ML clarify_step_tac}. |
|
42930 | 1145 |
|
1146 |
\item @{method clarsimp} acts like @{method clarify}, but also does |
|
1147 |
simplification. Note that if the Simplifier context includes a |
|
1148 |
splitter for the premises, the subgoal may still be split. |
|
26782 | 1149 |
|
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|
1150 |
\end{description} |
26782 | 1151 |
*} |
1152 |
||
1153 |
||
43366 | 1154 |
subsection {* Single-step tactics *} |
1155 |
||
1156 |
text {* |
|
1157 |
\begin{matharray}{rcl} |
|
1158 |
@{index_ML safe_step_tac: "Proof.context -> int -> tactic"} \\ |
|
1159 |
@{index_ML inst_step_tac: "Proof.context -> int -> tactic"} \\ |
|
1160 |
@{index_ML step_tac: "Proof.context -> int -> tactic"} \\ |
|
1161 |
@{index_ML slow_step_tac: "Proof.context -> int -> tactic"} \\ |
|
1162 |
@{index_ML clarify_step_tac: "Proof.context -> int -> tactic"} \\ |
|
1163 |
\end{matharray} |
|
1164 |
||
1165 |
These are the primitive tactics behind the (semi)automated proof |
|
1166 |
methods of the Classical Reasoner. By calling them yourself, you |
|
1167 |
can execute these procedures one step at a time. |
|
1168 |
||
1169 |
\begin{description} |
|
1170 |
||
1171 |
\item @{ML safe_step_tac}~@{text "ctxt i"} performs a safe step on |
|
1172 |
subgoal @{text i}. The safe wrapper tacticals are applied to a |
|
1173 |
tactic that may include proof by assumption or Modus Ponens (taking |
|
1174 |
care not to instantiate unknowns), or substitution. |
|
1175 |
||
1176 |
\item @{ML inst_step_tac} is like @{ML safe_step_tac}, but allows |
|
1177 |
unknowns to be instantiated. |
|
1178 |
||
1179 |
\item @{ML step_tac}~@{text "ctxt i"} is the basic step of the proof |
|
1180 |
procedure. The unsafe wrapper tacticals are applied to a tactic |
|
1181 |
that tries @{ML safe_tac}, @{ML inst_step_tac}, or applies an unsafe |
|
1182 |
rule from the context. |
|
1183 |
||
1184 |
\item @{ML slow_step_tac} resembles @{ML step_tac}, but allows |
|
1185 |
backtracking between using safe rules with instantiation (@{ML |
|
1186 |
inst_step_tac}) and using unsafe rules. The resulting search space |
|
1187 |
is larger. |
|
1188 |
||
1189 |
\item @{ML clarify_step_tac}~@{text "ctxt i"} performs a safe step |
|
1190 |
on subgoal @{text i}. No splitting step is applied; for example, |
|
1191 |
the subgoal @{text "A \<and> B"} is left as a conjunction. Proof by |
|
1192 |
assumption, Modus Ponens, etc., may be performed provided they do |
|
1193 |
not instantiate unknowns. Assumptions of the form @{text "x = t"} |
|
1194 |
may be eliminated. The safe wrapper tactical is applied. |
|
1195 |
||
1196 |
\end{description} |
|
1197 |
*} |
|
1198 |
||
1199 |
||
27044 | 1200 |
section {* Object-logic setup \label{sec:object-logic} *} |
26790 | 1201 |
|
1202 |
text {* |
|
1203 |
\begin{matharray}{rcl} |
|
28761
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|
1204 |
@{command_def "judgment"} & : & @{text "theory \<rightarrow> theory"} \\ |
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|
1205 |
@{method_def atomize} & : & @{text method} \\ |
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changeset
|
1206 |
@{attribute_def atomize} & : & @{text attribute} \\ |
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diff
changeset
|
1207 |
@{attribute_def rule_format} & : & @{text attribute} \\ |
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diff
changeset
|
1208 |
@{attribute_def rulify} & : & @{text attribute} \\ |
26790 | 1209 |
\end{matharray} |
1210 |
||
1211 |
The very starting point for any Isabelle object-logic is a ``truth |
|
1212 |
judgment'' that links object-level statements to the meta-logic |
|
1213 |
(with its minimal language of @{text prop} that covers universal |
|
1214 |
quantification @{text "\<And>"} and implication @{text "\<Longrightarrow>"}). |
|
1215 |
||
1216 |
Common object-logics are sufficiently expressive to internalize rule |
|
1217 |
statements over @{text "\<And>"} and @{text "\<Longrightarrow>"} within their own |
|
1218 |
language. This is useful in certain situations where a rule needs |
|
1219 |
to be viewed as an atomic statement from the meta-level perspective, |
|
1220 |
e.g.\ @{text "\<And>x. x \<in> A \<Longrightarrow> P x"} versus @{text "\<forall>x \<in> A. P x"}. |
|
1221 |
||
1222 |
From the following language elements, only the @{method atomize} |
|
1223 |
method and @{attribute rule_format} attribute are occasionally |
|
1224 |
required by end-users, the rest is for those who need to setup their |
|
1225 |
own object-logic. In the latter case existing formulations of |
|
1226 |
Isabelle/FOL or Isabelle/HOL may be taken as realistic examples. |
|
1227 |
||
1228 |
Generic tools may refer to the information provided by object-logic |
|
1229 |
declarations internally. |
|
1230 |
||
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|
1231 |
@{rail " |
46494
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diff
changeset
|
1232 |
@@{command judgment} @{syntax name} '::' @{syntax type} @{syntax mixfix}? |
26790 | 1233 |
; |
42596
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modernized rail diagrams using @{rail} antiquotation;
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diff
changeset
|
1234 |
@@{attribute atomize} ('(' 'full' ')')? |
26790 | 1235 |
; |
42596
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modernized rail diagrams using @{rail} antiquotation;
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diff
changeset
|
1236 |
@@{attribute rule_format} ('(' 'noasm' ')')? |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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diff
changeset
|
1237 |
"} |
26790 | 1238 |
|
28760
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unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1239 |
\begin{description} |
26790 | 1240 |
|
28760
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diff
changeset
|
1241 |
\item @{command "judgment"}~@{text "c :: \<sigma> (mx)"} declares constant |
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unified use of declaration environment with IsarImplementation;
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|
1242 |
@{text c} as the truth judgment of the current object-logic. Its |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1243 |
type @{text \<sigma>} should specify a coercion of the category of |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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28754
diff
changeset
|
1244 |
object-level propositions to @{text prop} of the Pure meta-logic; |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1245 |
the mixfix annotation @{text "(mx)"} would typically just link the |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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parents:
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diff
changeset
|
1246 |
object language (internally of syntactic category @{text logic}) |
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unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1247 |
with that of @{text prop}. Only one @{command "judgment"} |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
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diff
changeset
|
1248 |
declaration may be given in any theory development. |
26790 | 1249 |
|
28760
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unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1250 |
\item @{method atomize} (as a method) rewrites any non-atomic |
26790 | 1251 |
premises of a sub-goal, using the meta-level equations declared via |
1252 |
@{attribute atomize} (as an attribute) beforehand. As a result, |
|
1253 |
heavily nested goals become amenable to fundamental operations such |
|
42626 | 1254 |
as resolution (cf.\ the @{method (Pure) rule} method). Giving the ``@{text |
26790 | 1255 |
"(full)"}'' option here means to turn the whole subgoal into an |
1256 |
object-statement (if possible), including the outermost parameters |
|
1257 |
and assumptions as well. |
|
1258 |
||
1259 |
A typical collection of @{attribute atomize} rules for a particular |
|
1260 |
object-logic would provide an internalization for each of the |
|
1261 |
connectives of @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"}. |
|
1262 |
Meta-level conjunction should be covered as well (this is |
|
1263 |
particularly important for locales, see \secref{sec:locale}). |
|
1264 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
1265 |
\item @{attribute rule_format} rewrites a theorem by the equalities |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
1266 |
declared as @{attribute rulify} rules in the current object-logic. |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
1267 |
By default, the result is fully normalized, including assumptions |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
1268 |
and conclusions at any depth. The @{text "(no_asm)"} option |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
1269 |
restricts the transformation to the conclusion of a rule. |
26790 | 1270 |
|
1271 |
In common object-logics (HOL, FOL, ZF), the effect of @{attribute |
|
1272 |
rule_format} is to replace (bounded) universal quantification |
|
1273 |
(@{text "\<forall>"}) and implication (@{text "\<longrightarrow>"}) by the corresponding |
|
1274 |
rule statements over @{text "\<And>"} and @{text "\<Longrightarrow>"}. |
|
1275 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28754
diff
changeset
|
1276 |
\end{description} |
26790 | 1277 |
*} |
1278 |
||
26782 | 1279 |
end |