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theory HOL_Specific |
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imports Base Main "~~/src/HOL/Library/Old_Recdef" |
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begin |
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chapter {* Isabelle/HOL \label{ch:hol} *} |
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section {* Higher-Order Logic *} |
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text {* Isabelle/HOL is based on Higher-Order Logic, a polymorphic |
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version of Church's Simple Theory of Types. HOL can be best |
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understood as a simply-typed version of classical set theory. The |
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logic was first implemented in Gordon's HOL system |
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\cite{mgordon-hol}. It extends Church's original logic |
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\cite{church40} by explicit type variables (naive polymorphism) and |
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a sound axiomatization scheme for new types based on subsets of |
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existing types. |
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Andrews's book \cite{andrews86} is a full description of the |
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original Church-style higher-order logic, with proofs of correctness |
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and completeness wrt.\ certain set-theoretic interpretations. The |
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particular extensions of Gordon-style HOL are explained semantically |
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in two chapters of the 1993 HOL book \cite{pitts93}. |
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Experience with HOL over decades has demonstrated that higher-order |
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logic is widely applicable in many areas of mathematics and computer |
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science. In a sense, Higher-Order Logic is simpler than First-Order |
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Logic, because there are fewer restrictions and special cases. Note |
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that HOL is \emph{weaker} than FOL with axioms for ZF set theory, |
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which is traditionally considered the standard foundation of regular |
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mathematics, but for most applications this does not matter. If you |
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prefer ML to Lisp, you will probably prefer HOL to ZF. |
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\medskip The syntax of HOL follows @{text "\<lambda>"}-calculus and |
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functional programming. Function application is curried. To apply |
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the function @{text f} of type @{text "\<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2 \<Rightarrow> \<tau>\<^sub>3"} to the |
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arguments @{text a} and @{text b} in HOL, you simply write @{text "f |
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a b"} (as in ML or Haskell). There is no ``apply'' operator; the |
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existing application of the Pure @{text "\<lambda>"}-calculus is re-used. |
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Note that in HOL @{text "f (a, b)"} means ``@{text "f"} applied to |
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the pair @{text "(a, b)"} (which is notation for @{text "Pair a |
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b"}). The latter typically introduces extra formal efforts that can |
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be avoided by currying functions by default. Explicit tuples are as |
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infrequent in HOL formalizations as in good ML or Haskell programs. |
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\medskip Isabelle/HOL has a distinct feel, compared to other |
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object-logics like Isabelle/ZF. It identifies object-level types |
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with meta-level types, taking advantage of the default |
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type-inference mechanism of Isabelle/Pure. HOL fully identifies |
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object-level functions with meta-level functions, with native |
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abstraction and application. |
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These identifications allow Isabelle to support HOL particularly |
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nicely, but they also mean that HOL requires some sophistication |
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from the user. In particular, an understanding of Hindley-Milner |
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type-inference with type-classes, which are both used extensively in |
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the standard libraries and applications. Beginners can set |
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@{attribute show_types} or even @{attribute show_sorts} to get more |
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explicit information about the result of type-inference. *} |
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section {* Inductive and coinductive definitions \label{sec:hol-inductive} *} |
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text {* |
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\begin{matharray}{rcl} |
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@{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{attribute_def (HOL) mono} & : & @{text attribute} \\ |
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\end{matharray} |
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An \emph{inductive definition} specifies the least predicate or set |
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@{text R} closed under given rules: applying a rule to elements of |
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@{text R} yields a result within @{text R}. For example, a |
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structural operational semantics is an inductive definition of an |
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evaluation relation. |
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Dually, a \emph{coinductive definition} specifies the greatest |
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predicate or set @{text R} that is consistent with given rules: |
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every element of @{text R} can be seen as arising by applying a rule |
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to elements of @{text R}. An important example is using |
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bisimulation relations to formalise equivalence of processes and |
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infinite data structures. |
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Both inductive and coinductive definitions are based on the |
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Knaster-Tarski fixed-point theorem for complete lattices. The |
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collection of introduction rules given by the user determines a |
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functor on subsets of set-theoretic relations. The required |
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monotonicity of the recursion scheme is proven as a prerequisite to |
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the fixed-point definition and the resulting consequences. This |
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works by pushing inclusion through logical connectives and any other |
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operator that might be wrapped around recursive occurrences of the |
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defined relation: there must be a monotonicity theorem of the form |
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@{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for each premise @{text "\<M> R t"} in an |
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introduction rule. The default rule declarations of Isabelle/HOL |
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already take care of most common situations. |
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@{rail " |
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(@@{command (HOL) inductive} | @@{command (HOL) inductive_set} | |
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@@{command (HOL) coinductive} | @@{command (HOL) coinductive_set}) |
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@{syntax target}? \\ |
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@{syntax \"fixes\"} (@'for' @{syntax \"fixes\"})? (@'where' clauses)? \\ |
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(@'monos' @{syntax thmrefs})? |
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clauses: (@{syntax thmdecl}? @{syntax prop} + '|') |
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; |
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@@{attribute (HOL) mono} (() | 'add' | 'del') |
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"} |
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\begin{description} |
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\item @{command (HOL) "inductive"} and @{command (HOL) |
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"coinductive"} define (co)inductive predicates from the introduction |
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rules. |
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The propositions given as @{text "clauses"} in the @{keyword |
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"where"} part are either rules of the usual @{text "\<And>/\<Longrightarrow>"} format |
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(with arbitrary nesting), or equalities using @{text "\<equiv>"}. The |
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latter specifies extra-logical abbreviations in the sense of |
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@{command_ref abbreviation}. Introducing abstract syntax |
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simultaneously with the actual introduction rules is occasionally |
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useful for complex specifications. |
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The optional @{keyword "for"} part contains a list of parameters of |
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the (co)inductive predicates that remain fixed throughout the |
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definition, in contrast to arguments of the relation that may vary |
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in each occurrence within the given @{text "clauses"}. |
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The optional @{keyword "monos"} declaration contains additional |
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\emph{monotonicity theorems}, which are required for each operator |
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applied to a recursive set in the introduction rules. |
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\item @{command (HOL) "inductive_set"} and @{command (HOL) |
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"coinductive_set"} are wrappers for to the previous commands for |
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native HOL predicates. This allows to define (co)inductive sets, |
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where multiple arguments are simulated via tuples. |
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\item @{attribute (HOL) mono} declares monotonicity rules in the |
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context. These rule are involved in the automated monotonicity |
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proof of the above inductive and coinductive definitions. |
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\end{description} |
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*} |
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subsection {* Derived rules *} |
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text {* A (co)inductive definition of @{text R} provides the following |
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main theorems: |
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\begin{description} |
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\item @{text R.intros} is the list of introduction rules as proven |
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theorems, for the recursive predicates (or sets). The rules are |
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also available individually, using the names given them in the |
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theory file; |
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\item @{text R.cases} is the case analysis (or elimination) rule; |
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\item @{text R.induct} or @{text R.coinduct} is the (co)induction |
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rule; |
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\item @{text R.simps} is the equation unrolling the fixpoint of the |
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predicate one step. |
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\end{description} |
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When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are |
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defined simultaneously, the list of introduction rules is called |
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@{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are |
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called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list |
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of mutual induction rules is called @{text |
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"R\<^sub>1_\<dots>_R\<^sub>n.inducts"}. |
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*} |
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subsection {* Monotonicity theorems *} |
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text {* The context maintains a default set of theorems that are used |
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in monotonicity proofs. New rules can be declared via the |
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@{attribute (HOL) mono} attribute. See the main Isabelle/HOL |
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sources for some examples. The general format of such monotonicity |
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theorems is as follows: |
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\begin{itemize} |
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\item Theorems of the form @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for proving |
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monotonicity of inductive definitions whose introduction rules have |
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premises involving terms such as @{text "\<M> R t"}. |
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\item Monotonicity theorems for logical operators, which are of the |
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general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in |
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the case of the operator @{text "\<or>"}, the corresponding theorem is |
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\[ |
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\infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}} |
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\] |
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\item De Morgan style equations for reasoning about the ``polarity'' |
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of expressions, e.g. |
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\[ |
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@{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad |
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@{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"} |
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\] |
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\item Equations for reducing complex operators to more primitive |
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ones whose monotonicity can easily be proved, e.g. |
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\[ |
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@{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad |
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@{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"} |
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\] |
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\end{itemize} |
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*} |
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subsubsection {* Examples *} |
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text {* The finite powerset operator can be defined inductively like this: *} |
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inductive_set Fin :: "'a set \<Rightarrow> 'a set set" for A :: "'a set" |
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where |
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empty: "{} \<in> Fin A" |
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| insert: "a \<in> A \<Longrightarrow> B \<in> Fin A \<Longrightarrow> insert a B \<in> Fin A" |
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text {* The accessible part of a relation is defined as follows: *} |
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|
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inductive acc :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" |
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for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<prec>" 50) |
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where acc: "(\<And>y. y \<prec> x \<Longrightarrow> acc r y) \<Longrightarrow> acc r x" |
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|
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text {* Common logical connectives can be easily characterized as |
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non-recursive inductive definitions with parameters, but without |
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arguments. *} |
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|
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inductive AND for A B :: bool |
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where "A \<Longrightarrow> B \<Longrightarrow> AND A B" |
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|
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inductive OR for A B :: bool |
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where "A \<Longrightarrow> OR A B" |
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| "B \<Longrightarrow> OR A B" |
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|
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inductive EXISTS for B :: "'a \<Rightarrow> bool" |
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where "B a \<Longrightarrow> EXISTS B" |
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|
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text {* Here the @{text "cases"} or @{text "induct"} rules produced by |
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the @{command inductive} package coincide with the expected |
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elimination rules for Natural Deduction. Already in the original |
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article by Gerhard Gentzen \cite{Gentzen:1935} there is a hint that |
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each connective can be characterized by its introductions, and the |
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elimination can be constructed systematically. *} |
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252 |
section {* Recursive functions \label{sec:recursion} *} |
|
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254 |
text {* |
|
255 |
\begin{matharray}{rcl} |
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256 |
@{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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257 |
@{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
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@{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
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\end{matharray} |
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@{rail " |
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@@{command (HOL) primrec} @{syntax target}? @{syntax \"fixes\"} @'where' equations |
|
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; |
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(@@{command (HOL) fun} | @@{command (HOL) function}) @{syntax target}? functionopts? |
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@{syntax \"fixes\"} \\ @'where' equations |
|
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; |
268 |
||
42908 | 269 |
equations: (@{syntax thmdecl}? @{syntax prop} + '|') |
26849 | 270 |
; |
42908 | 271 |
functionopts: '(' (('sequential' | 'domintros') + ',') ')' |
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; |
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@@{command (HOL) termination} @{syntax term}? |
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"} |
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\begin{description} |
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|
42908 | 278 |
\item @{command (HOL) "primrec"} defines primitive recursive |
42912 | 279 |
functions over datatypes (see also @{command_ref (HOL) datatype} and |
280 |
@{command_ref (HOL) rep_datatype}). The given @{text equations} |
|
281 |
specify reduction rules that are produced by instantiating the |
|
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generic combinator for primitive recursion that is available for |
|
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each datatype. |
|
284 |
||
285 |
Each equation needs to be of the form: |
|
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||
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@{text [display] "f x\<^sub>1 \<dots> x\<^sub>m (C y\<^sub>1 \<dots> y\<^sub>k) z\<^sub>1 \<dots> z\<^sub>n = rhs"} |
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289 |
such that @{text C} is a datatype constructor, @{text rhs} contains |
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only the free variables on the left-hand side (or from the context), |
|
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and all recursive occurrences of @{text "f"} in @{text "rhs"} are of |
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the form @{text "f \<dots> y\<^sub>i \<dots>"} for some @{text i}. At most one |
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reduction rule for each constructor can be given. The order does |
|
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not matter. For missing constructors, the function is defined to |
|
295 |
return a default value, but this equation is made difficult to |
|
296 |
access for users. |
|
297 |
||
298 |
The reduction rules are declared as @{attribute simp} by default, |
|
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which enables standard proof methods like @{method simp} and |
|
300 |
@{method auto} to normalize expressions of @{text "f"} applied to |
|
301 |
datatype constructions, by simulating symbolic computation via |
|
302 |
rewriting. |
|
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|
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\item @{command (HOL) "function"} defines functions by general |
305 |
wellfounded recursion. A detailed description with examples can be |
|
306 |
found in \cite{isabelle-function}. The function is specified by a |
|
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set of (possibly conditional) recursive equations with arbitrary |
|
308 |
pattern matching. The command generates proof obligations for the |
|
309 |
completeness and the compatibility of patterns. |
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The defined function is considered partial, and the resulting |
312 |
simplification rules (named @{text "f.psimps"}) and induction rule |
|
313 |
(named @{text "f.pinduct"}) are guarded by a generated domain |
|
314 |
predicate @{text "f_dom"}. The @{command (HOL) "termination"} |
|
315 |
command can then be used to establish that the function is total. |
|
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|
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\item @{command (HOL) "fun"} is a shorthand notation for ``@{command |
318 |
(HOL) "function"}~@{text "(sequential)"}, followed by automated |
|
319 |
proof attempts regarding pattern matching and termination. See |
|
320 |
\cite{isabelle-function} for further details. |
|
42123 | 321 |
|
42908 | 322 |
\item @{command (HOL) "termination"}~@{text f} commences a |
323 |
termination proof for the previously defined function @{text f}. If |
|
324 |
this is omitted, the command refers to the most recent function |
|
325 |
definition. After the proof is closed, the recursive equations and |
|
326 |
the induction principle is established. |
|
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\end{description} |
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Recursive definitions introduced by the @{command (HOL) "function"} |
42912 | 331 |
command accommodate reasoning by induction (cf.\ @{method induct}): |
332 |
rule @{text "f.induct"} refers to a specific induction rule, with |
|
333 |
parameters named according to the user-specified equations. Cases |
|
334 |
are numbered starting from 1. For @{command (HOL) "primrec"}, the |
|
335 |
induction principle coincides with structural recursion on the |
|
336 |
datatype where the recursion is carried out. |
|
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|
338 |
The equations provided by these packages may be referred later as |
|
339 |
theorem list @{text "f.simps"}, where @{text f} is the (collective) |
|
340 |
name of the functions defined. Individual equations may be named |
|
341 |
explicitly as well. |
|
342 |
||
343 |
The @{command (HOL) "function"} command accepts the following |
|
344 |
options. |
|
345 |
||
346 |
\begin{description} |
|
347 |
||
348 |
\item @{text sequential} enables a preprocessor which disambiguates |
|
349 |
overlapping patterns by making them mutually disjoint. Earlier |
|
350 |
equations take precedence over later ones. This allows to give the |
|
351 |
specification in a format very similar to functional programming. |
|
352 |
Note that the resulting simplification and induction rules |
|
353 |
correspond to the transformed specification, not the one given |
|
354 |
originally. This usually means that each equation given by the user |
|
355 |
may result in several theorems. Also note that this automatic |
|
356 |
transformation only works for ML-style datatype patterns. |
|
357 |
||
358 |
\item @{text domintros} enables the automated generation of |
|
359 |
introduction rules for the domain predicate. While mostly not |
|
360 |
needed, they can be helpful in some proofs about partial functions. |
|
361 |
||
362 |
\end{description} |
|
26849 | 363 |
*} |
364 |
||
42912 | 365 |
subsubsection {* Example: evaluation of expressions *} |
366 |
||
367 |
text {* Subsequently, we define mutual datatypes for arithmetic and |
|
368 |
boolean expressions, and use @{command primrec} for evaluation |
|
369 |
functions that follow the same recursive structure. *} |
|
370 |
||
371 |
datatype 'a aexp = |
|
372 |
IF "'a bexp" "'a aexp" "'a aexp" |
|
373 |
| Sum "'a aexp" "'a aexp" |
|
374 |
| Diff "'a aexp" "'a aexp" |
|
375 |
| Var 'a |
|
376 |
| Num nat |
|
377 |
and 'a bexp = |
|
378 |
Less "'a aexp" "'a aexp" |
|
379 |
| And "'a bexp" "'a bexp" |
|
380 |
| Neg "'a bexp" |
|
381 |
||
382 |
||
383 |
text {* \medskip Evaluation of arithmetic and boolean expressions *} |
|
384 |
||
385 |
primrec evala :: "('a \<Rightarrow> nat) \<Rightarrow> 'a aexp \<Rightarrow> nat" |
|
386 |
and evalb :: "('a \<Rightarrow> nat) \<Rightarrow> 'a bexp \<Rightarrow> bool" |
|
387 |
where |
|
388 |
"evala env (IF b a1 a2) = (if evalb env b then evala env a1 else evala env a2)" |
|
389 |
| "evala env (Sum a1 a2) = evala env a1 + evala env a2" |
|
390 |
| "evala env (Diff a1 a2) = evala env a1 - evala env a2" |
|
391 |
| "evala env (Var v) = env v" |
|
392 |
| "evala env (Num n) = n" |
|
393 |
| "evalb env (Less a1 a2) = (evala env a1 < evala env a2)" |
|
394 |
| "evalb env (And b1 b2) = (evalb env b1 \<and> evalb env b2)" |
|
395 |
| "evalb env (Neg b) = (\<not> evalb env b)" |
|
396 |
||
397 |
text {* Since the value of an expression depends on the value of its |
|
398 |
variables, the functions @{const evala} and @{const evalb} take an |
|
399 |
additional parameter, an \emph{environment} that maps variables to |
|
400 |
their values. |
|
401 |
||
402 |
\medskip Substitution on expressions can be defined similarly. The |
|
403 |
mapping @{text f} of type @{typ "'a \<Rightarrow> 'a aexp"} given as a |
|
404 |
parameter is lifted canonically on the types @{typ "'a aexp"} and |
|
405 |
@{typ "'a bexp"}, respectively. |
|
406 |
*} |
|
407 |
||
408 |
primrec substa :: "('a \<Rightarrow> 'b aexp) \<Rightarrow> 'a aexp \<Rightarrow> 'b aexp" |
|
409 |
and substb :: "('a \<Rightarrow> 'b aexp) \<Rightarrow> 'a bexp \<Rightarrow> 'b bexp" |
|
410 |
where |
|
411 |
"substa f (IF b a1 a2) = IF (substb f b) (substa f a1) (substa f a2)" |
|
412 |
| "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)" |
|
413 |
| "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)" |
|
414 |
| "substa f (Var v) = f v" |
|
415 |
| "substa f (Num n) = Num n" |
|
416 |
| "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)" |
|
417 |
| "substb f (And b1 b2) = And (substb f b1) (substb f b2)" |
|
418 |
| "substb f (Neg b) = Neg (substb f b)" |
|
419 |
||
420 |
text {* In textbooks about semantics one often finds substitution |
|
421 |
theorems, which express the relationship between substitution and |
|
422 |
evaluation. For @{typ "'a aexp"} and @{typ "'a bexp"}, we can prove |
|
423 |
such a theorem by mutual induction, followed by simplification. |
|
424 |
*} |
|
425 |
||
426 |
lemma subst_one: |
|
427 |
"evala env (substa (Var (v := a')) a) = evala (env (v := evala env a')) a" |
|
428 |
"evalb env (substb (Var (v := a')) b) = evalb (env (v := evala env a')) b" |
|
429 |
by (induct a and b) simp_all |
|
430 |
||
431 |
lemma subst_all: |
|
432 |
"evala env (substa s a) = evala (\<lambda>x. evala env (s x)) a" |
|
433 |
"evalb env (substb s b) = evalb (\<lambda>x. evala env (s x)) b" |
|
434 |
by (induct a and b) simp_all |
|
435 |
||
436 |
||
437 |
subsubsection {* Example: a substitution function for terms *} |
|
438 |
||
439 |
text {* Functions on datatypes with nested recursion are also defined |
|
440 |
by mutual primitive recursion. *} |
|
441 |
||
442 |
datatype ('a, 'b) "term" = Var 'a | App 'b "('a, 'b) term list" |
|
443 |
||
444 |
text {* A substitution function on type @{typ "('a, 'b) term"} can be |
|
445 |
defined as follows, by working simultaneously on @{typ "('a, 'b) |
|
446 |
term list"}: *} |
|
447 |
||
448 |
primrec subst_term :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term \<Rightarrow> ('a, 'b) term" and |
|
449 |
subst_term_list :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term list \<Rightarrow> ('a, 'b) term list" |
|
450 |
where |
|
451 |
"subst_term f (Var a) = f a" |
|
452 |
| "subst_term f (App b ts) = App b (subst_term_list f ts)" |
|
453 |
| "subst_term_list f [] = []" |
|
454 |
| "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts" |
|
455 |
||
456 |
text {* The recursion scheme follows the structure of the unfolded |
|
457 |
definition of type @{typ "('a, 'b) term"}. To prove properties of this |
|
458 |
substitution function, mutual induction is needed: |
|
459 |
*} |
|
460 |
||
461 |
lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)" and |
|
462 |
"subst_term_list (subst_term f1 \<circ> f2) ts = subst_term_list f1 (subst_term_list f2 ts)" |
|
463 |
by (induct t and ts) simp_all |
|
464 |
||
465 |
||
466 |
subsubsection {* Example: a map function for infinitely branching trees *} |
|
467 |
||
468 |
text {* Defining functions on infinitely branching datatypes by |
|
469 |
primitive recursion is just as easy. |
|
470 |
*} |
|
471 |
||
472 |
datatype 'a tree = Atom 'a | Branch "nat \<Rightarrow> 'a tree" |
|
473 |
||
474 |
primrec map_tree :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a tree \<Rightarrow> 'b tree" |
|
475 |
where |
|
476 |
"map_tree f (Atom a) = Atom (f a)" |
|
477 |
| "map_tree f (Branch ts) = Branch (\<lambda>x. map_tree f (ts x))" |
|
478 |
||
479 |
text {* Note that all occurrences of functions such as @{text ts} |
|
480 |
above must be applied to an argument. In particular, @{term |
|
481 |
"map_tree f \<circ> ts"} is not allowed here. *} |
|
482 |
||
483 |
text {* Here is a simple composition lemma for @{term map_tree}: *} |
|
484 |
||
485 |
lemma "map_tree g (map_tree f t) = map_tree (g \<circ> f) t" |
|
486 |
by (induct t) simp_all |
|
487 |
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488 |
|
42908 | 489 |
subsection {* Proof methods related to recursive definitions *} |
26849 | 490 |
|
491 |
text {* |
|
492 |
\begin{matharray}{rcl} |
|
42908 | 493 |
@{method_def (HOL) pat_completeness} & : & @{text method} \\ |
494 |
@{method_def (HOL) relation} & : & @{text method} \\ |
|
495 |
@{method_def (HOL) lexicographic_order} & : & @{text method} \\ |
|
496 |
@{method_def (HOL) size_change} & : & @{text method} \\ |
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@{method_def (HOL) induction_schema} & : & @{text method} \\ |
26849 | 498 |
\end{matharray} |
499 |
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@{rail " |
42908 | 501 |
@@{method (HOL) relation} @{syntax term} |
502 |
; |
|
503 |
@@{method (HOL) lexicographic_order} (@{syntax clasimpmod} * ) |
|
504 |
; |
|
505 |
@@{method (HOL) size_change} ( orders (@{syntax clasimpmod} * ) ) |
|
506 |
; |
|
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@@{method (HOL) induction_schema} |
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|
508 |
; |
42908 | 509 |
orders: ( 'max' | 'min' | 'ms' ) * |
510 |
"} |
|
511 |
||
512 |
\begin{description} |
|
513 |
||
514 |
\item @{method (HOL) pat_completeness} is a specialized method to |
|
515 |
solve goals regarding the completeness of pattern matching, as |
|
516 |
required by the @{command (HOL) "function"} package (cf.\ |
|
517 |
\cite{isabelle-function}). |
|
518 |
||
519 |
\item @{method (HOL) relation}~@{text R} introduces a termination |
|
520 |
proof using the relation @{text R}. The resulting proof state will |
|
521 |
contain goals expressing that @{text R} is wellfounded, and that the |
|
522 |
arguments of recursive calls decrease with respect to @{text R}. |
|
523 |
Usually, this method is used as the initial proof step of manual |
|
524 |
termination proofs. |
|
525 |
||
526 |
\item @{method (HOL) "lexicographic_order"} attempts a fully |
|
527 |
automated termination proof by searching for a lexicographic |
|
528 |
combination of size measures on the arguments of the function. The |
|
529 |
method accepts the same arguments as the @{method auto} method, |
|
42930 | 530 |
which it uses internally to prove local descents. The @{syntax |
531 |
clasimpmod} modifiers are accepted (as for @{method auto}). |
|
42908 | 532 |
|
533 |
In case of failure, extensive information is printed, which can help |
|
534 |
to analyse the situation (cf.\ \cite{isabelle-function}). |
|
535 |
||
536 |
\item @{method (HOL) "size_change"} also works on termination goals, |
|
537 |
using a variation of the size-change principle, together with a |
|
538 |
graph decomposition technique (see \cite{krauss_phd} for details). |
|
539 |
Three kinds of orders are used internally: @{text max}, @{text min}, |
|
540 |
and @{text ms} (multiset), which is only available when the theory |
|
541 |
@{text Multiset} is loaded. When no order kinds are given, they are |
|
542 |
tried in order. The search for a termination proof uses SAT solving |
|
543 |
internally. |
|
544 |
||
42930 | 545 |
For local descent proofs, the @{syntax clasimpmod} modifiers are |
546 |
accepted (as for @{method auto}). |
|
42908 | 547 |
|
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|
548 |
\item @{method (HOL) induction_schema} derives user-specified |
46283 | 549 |
induction rules from well-founded induction and completeness of |
550 |
patterns. This factors out some operations that are done internally |
|
551 |
by the function package and makes them available separately. See |
|
552 |
@{file "~~/src/HOL/ex/Induction_Schema.thy"} for examples. |
|
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|
553 |
|
42908 | 554 |
\end{description} |
555 |
*} |
|
556 |
||
557 |
||
558 |
subsection {* Functions with explicit partiality *} |
|
559 |
||
560 |
text {* |
|
561 |
\begin{matharray}{rcl} |
|
562 |
@{command_def (HOL) "partial_function"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
|
563 |
@{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\ |
|
564 |
\end{matharray} |
|
565 |
||
566 |
@{rail " |
|
567 |
@@{command (HOL) partial_function} @{syntax target}? |
|
568 |
'(' @{syntax nameref} ')' @{syntax \"fixes\"} \\ |
|
569 |
@'where' @{syntax thmdecl}? @{syntax prop} |
|
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modernized rail diagrams using @{rail} antiquotation;
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|
570 |
"} |
26849 | 571 |
|
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|
572 |
\begin{description} |
42123 | 573 |
|
42908 | 574 |
\item @{command (HOL) "partial_function"}~@{text "(mode)"} defines |
575 |
recursive functions based on fixpoints in complete partial |
|
576 |
orders. No termination proof is required from the user or |
|
577 |
constructed internally. Instead, the possibility of non-termination |
|
578 |
is modelled explicitly in the result type, which contains an |
|
579 |
explicit bottom element. |
|
580 |
||
581 |
Pattern matching and mutual recursion are currently not supported. |
|
582 |
Thus, the specification consists of a single function described by a |
|
583 |
single recursive equation. |
|
584 |
||
585 |
There are no fixed syntactic restrictions on the body of the |
|
586 |
function, but the induced functional must be provably monotonic |
|
587 |
wrt.\ the underlying order. The monotonicitity proof is performed |
|
588 |
internally, and the definition is rejected when it fails. The proof |
|
589 |
can be influenced by declaring hints using the |
|
590 |
@{attribute (HOL) partial_function_mono} attribute. |
|
591 |
||
592 |
The mandatory @{text mode} argument specifies the mode of operation |
|
593 |
of the command, which directly corresponds to a complete partial |
|
594 |
order on the result type. By default, the following modes are |
|
595 |
defined: |
|
26849 | 596 |
|
42908 | 597 |
\begin{description} |
46283 | 598 |
|
42908 | 599 |
\item @{text option} defines functions that map into the @{type |
600 |
option} type. Here, the value @{term None} is used to model a |
|
601 |
non-terminating computation. Monotonicity requires that if @{term |
|
46283 | 602 |
None} is returned by a recursive call, then the overall result must |
603 |
also be @{term None}. This is best achieved through the use of the |
|
604 |
monadic operator @{const "Option.bind"}. |
|
42908 | 605 |
|
606 |
\item @{text tailrec} defines functions with an arbitrary result |
|
607 |
type and uses the slightly degenerated partial order where @{term |
|
608 |
"undefined"} is the bottom element. Now, monotonicity requires that |
|
609 |
if @{term undefined} is returned by a recursive call, then the |
|
610 |
overall result must also be @{term undefined}. In practice, this is |
|
611 |
only satisfied when each recursive call is a tail call, whose result |
|
612 |
is directly returned. Thus, this mode of operation allows the |
|
613 |
definition of arbitrary tail-recursive functions. |
|
46283 | 614 |
|
42908 | 615 |
\end{description} |
616 |
||
617 |
Experienced users may define new modes by instantiating the locale |
|
618 |
@{const "partial_function_definitions"} appropriately. |
|
619 |
||
620 |
\item @{attribute (HOL) partial_function_mono} declares rules for |
|
621 |
use in the internal monononicity proofs of partial function |
|
622 |
definitions. |
|
26849 | 623 |
|
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|
624 |
\end{description} |
42908 | 625 |
|
626 |
*} |
|
627 |
||
628 |
||
629 |
subsection {* Old-style recursive function definitions (TFL) *} |
|
630 |
||
631 |
text {* |
|
632 |
\begin{matharray}{rcl} |
|
633 |
@{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\ |
|
634 |
@{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
|
635 |
\end{matharray} |
|
636 |
||
46280 | 637 |
The old TFL commands @{command (HOL) "recdef"} and @{command (HOL) |
638 |
"recdef_tc"} for defining recursive are mostly obsolete; @{command |
|
639 |
(HOL) "function"} or @{command (HOL) "fun"} should be used instead. |
|
640 |
||
42908 | 641 |
@{rail " |
642 |
@@{command (HOL) recdef} ('(' @'permissive' ')')? \\ |
|
643 |
@{syntax name} @{syntax term} (@{syntax prop} +) hints? |
|
644 |
; |
|
645 |
recdeftc @{syntax thmdecl}? tc |
|
646 |
; |
|
647 |
hints: '(' @'hints' ( recdefmod * ) ')' |
|
648 |
; |
|
649 |
recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf') |
|
650 |
(() | 'add' | 'del') ':' @{syntax thmrefs}) | @{syntax clasimpmod} |
|
651 |
; |
|
652 |
tc: @{syntax nameref} ('(' @{syntax nat} ')')? |
|
653 |
"} |
|
654 |
||
655 |
\begin{description} |
|
656 |
||
657 |
\item @{command (HOL) "recdef"} defines general well-founded |
|
658 |
recursive functions (using the TFL package), see also |
|
659 |
\cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells |
|
660 |
TFL to recover from failed proof attempts, returning unfinished |
|
661 |
results. The @{text recdef_simp}, @{text recdef_cong}, and @{text |
|
662 |
recdef_wf} hints refer to auxiliary rules to be used in the internal |
|
663 |
automated proof process of TFL. Additional @{syntax clasimpmod} |
|
42930 | 664 |
declarations may be given to tune the context of the Simplifier |
665 |
(cf.\ \secref{sec:simplifier}) and Classical reasoner (cf.\ |
|
666 |
\secref{sec:classical}). |
|
42908 | 667 |
|
668 |
\item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the |
|
669 |
proof for leftover termination condition number @{text i} (default |
|
670 |
1) as generated by a @{command (HOL) "recdef"} definition of |
|
671 |
constant @{text c}. |
|
672 |
||
673 |
Note that in most cases, @{command (HOL) "recdef"} is able to finish |
|
674 |
its internal proofs without manual intervention. |
|
675 |
||
676 |
\end{description} |
|
677 |
||
678 |
\medskip Hints for @{command (HOL) "recdef"} may be also declared |
|
679 |
globally, using the following attributes. |
|
680 |
||
681 |
\begin{matharray}{rcl} |
|
682 |
@{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\ |
|
683 |
@{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\ |
|
684 |
@{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\ |
|
685 |
\end{matharray} |
|
686 |
||
687 |
@{rail " |
|
688 |
(@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} | |
|
689 |
@@{attribute (HOL) recdef_wf}) (() | 'add' | 'del') |
|
690 |
"} |
|
691 |
*} |
|
692 |
||
693 |
||
694 |
section {* Datatypes \label{sec:hol-datatype} *} |
|
695 |
||
696 |
text {* |
|
697 |
\begin{matharray}{rcl} |
|
698 |
@{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
699 |
@{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
|
700 |
\end{matharray} |
|
701 |
||
702 |
@{rail " |
|
703 |
@@{command (HOL) datatype} (spec + @'and') |
|
704 |
; |
|
705 |
@@{command (HOL) rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +) |
|
706 |
; |
|
707 |
||
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|
708 |
spec: @{syntax typespec_sorts} @{syntax mixfix}? '=' (cons + '|') |
42908 | 709 |
; |
710 |
cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}? |
|
711 |
"} |
|
712 |
||
713 |
\begin{description} |
|
714 |
||
715 |
\item @{command (HOL) "datatype"} defines inductive datatypes in |
|
716 |
HOL. |
|
717 |
||
718 |
\item @{command (HOL) "rep_datatype"} represents existing types as |
|
42909 | 719 |
datatypes. |
720 |
||
721 |
For foundational reasons, some basic types such as @{typ nat}, @{typ |
|
722 |
"'a \<times> 'b"}, @{typ "'a + 'b"}, @{typ bool} and @{typ unit} are |
|
723 |
introduced by more primitive means using @{command_ref typedef}. To |
|
724 |
recover the rich infrastructure of @{command datatype} (e.g.\ rules |
|
725 |
for @{method cases} and @{method induct} and the primitive recursion |
|
726 |
combinators), such types may be represented as actual datatypes |
|
727 |
later. This is done by specifying the constructors of the desired |
|
728 |
type, and giving a proof of the induction rule, distinctness and |
|
729 |
injectivity of constructors. |
|
730 |
||
731 |
For example, see @{file "~~/src/HOL/Sum_Type.thy"} for the |
|
732 |
representation of the primitive sum type as fully-featured datatype. |
|
42908 | 733 |
|
734 |
\end{description} |
|
735 |
||
42909 | 736 |
The generated rules for @{method induct} and @{method cases} provide |
737 |
case names according to the given constructors, while parameters are |
|
738 |
named after the types (see also \secref{sec:cases-induct}). |
|
42908 | 739 |
|
740 |
See \cite{isabelle-HOL} for more details on datatypes, but beware of |
|
741 |
the old-style theory syntax being used there! Apart from proper |
|
742 |
proof methods for case-analysis and induction, there are also |
|
743 |
emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL) |
|
744 |
induct_tac} available, see \secref{sec:hol-induct-tac}; these admit |
|
745 |
to refer directly to the internal structure of subgoals (including |
|
746 |
internally bound parameters). |
|
26849 | 747 |
*} |
748 |
||
749 |
||
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|
750 |
subsubsection {* Examples *} |
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|
751 |
|
6834af822a8b
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changeset
|
752 |
text {* We define a type of finite sequences, with slightly different |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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diff
changeset
|
753 |
names than the existing @{typ "'a list"} that is already in @{theory |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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|
754 |
Main}: *} |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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changeset
|
755 |
|
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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changeset
|
756 |
datatype 'a seq = Empty | Seq 'a "'a seq" |
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changeset
|
757 |
|
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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changeset
|
758 |
text {* We can now prove some simple lemma by structural induction: *} |
6834af822a8b
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diff
changeset
|
759 |
|
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
760 |
lemma "Seq x xs \<noteq> xs" |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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|
761 |
proof (induct xs arbitrary: x) |
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wenzelm
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changeset
|
762 |
case Empty |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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changeset
|
763 |
txt {* This case can be proved using the simplifier: the freeness |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
764 |
properties of the datatype are already declared as @{attribute |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
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diff
changeset
|
765 |
simp} rules. *} |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
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42909
diff
changeset
|
766 |
show "Seq x Empty \<noteq> Empty" |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
767 |
by simp |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
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diff
changeset
|
768 |
next |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
769 |
case (Seq y ys) |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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changeset
|
770 |
txt {* The step case is proved similarly. *} |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
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parents:
42909
diff
changeset
|
771 |
show "Seq x (Seq y ys) \<noteq> Seq y ys" |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
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diff
changeset
|
772 |
using `Seq y ys \<noteq> ys` by simp |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
773 |
qed |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
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diff
changeset
|
774 |
|
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
775 |
text {* Here is a more succinct version of the same proof: *} |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
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diff
changeset
|
776 |
|
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
777 |
lemma "Seq x xs \<noteq> xs" |
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
diff
changeset
|
778 |
by (induct xs arbitrary: x) simp_all |
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updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
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diff
changeset
|
779 |
|
6834af822a8b
updated and simplified HOL datatype examples (NB: special treatment of distinctness has been discontinued in the vicinity of 542b34b178ec);
wenzelm
parents:
42909
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changeset
|
780 |
|
26849 | 781 |
section {* Records \label{sec:hol-record} *} |
782 |
||
783 |
text {* |
|
784 |
In principle, records merely generalize the concept of tuples, where |
|
785 |
components may be addressed by labels instead of just position. The |
|
786 |
logical infrastructure of records in Isabelle/HOL is slightly more |
|
787 |
advanced, though, supporting truly extensible record schemes. This |
|
788 |
admits operations that are polymorphic with respect to record |
|
789 |
extension, yielding ``object-oriented'' effects like (single) |
|
790 |
inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more |
|
791 |
details on object-oriented verification and record subtyping in HOL. |
|
792 |
*} |
|
793 |
||
794 |
||
795 |
subsection {* Basic concepts *} |
|
796 |
||
797 |
text {* |
|
798 |
Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records |
|
799 |
at the level of terms and types. The notation is as follows: |
|
800 |
||
801 |
\begin{center} |
|
802 |
\begin{tabular}{l|l|l} |
|
803 |
& record terms & record types \\ \hline |
|
804 |
fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\ |
|
805 |
schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} & |
|
806 |
@{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\ |
|
807 |
\end{tabular} |
|
808 |
\end{center} |
|
809 |
||
810 |
\noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text |
|
811 |
"(| x = a |)"}. |
|
812 |
||
813 |
A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value |
|
814 |
@{text a} and field @{text y} of value @{text b}. The corresponding |
|
815 |
type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"} |
|
816 |
and @{text "b :: B"}. |
|
817 |
||
818 |
A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields |
|
819 |
@{text x} and @{text y} as before, but also possibly further fields |
|
820 |
as indicated by the ``@{text "\<dots>"}'' notation (which is actually part |
|
821 |
of the syntax). The improper field ``@{text "\<dots>"}'' of a record |
|
822 |
scheme is called the \emph{more part}. Logically it is just a free |
|
823 |
variable, which is occasionally referred to as ``row variable'' in |
|
824 |
the literature. The more part of a record scheme may be |
|
825 |
instantiated by zero or more further components. For example, the |
|
826 |
previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z = |
|
26852 | 827 |
c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part. |
26849 | 828 |
Fixed records are special instances of record schemes, where |
829 |
``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"} |
|
830 |
element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation |
|
831 |
for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}. |
|
42123 | 832 |
|
26849 | 833 |
\medskip Two key observations make extensible records in a simply |
834 |
typed language like HOL work out: |
|
835 |
||
836 |
\begin{enumerate} |
|
837 |
||
838 |
\item the more part is internalized, as a free term or type |
|
839 |
variable, |
|
840 |
||
26852 | 841 |
\item field names are externalized, they cannot be accessed within |
842 |
the logic as first-class values. |
|
26849 | 843 |
|
844 |
\end{enumerate} |
|
845 |
||
846 |
\medskip In Isabelle/HOL record types have to be defined explicitly, |
|
847 |
fixing their field names and types, and their (optional) parent |
|
848 |
record. Afterwards, records may be formed using above syntax, while |
|
849 |
obeying the canonical order of fields as given by their declaration. |
|
850 |
The record package provides several standard operations like |
|
851 |
selectors and updates. The common setup for various generic proof |
|
852 |
tools enable succinct reasoning patterns. See also the Isabelle/HOL |
|
853 |
tutorial \cite{isabelle-hol-book} for further instructions on using |
|
854 |
records in practice. |
|
855 |
*} |
|
856 |
||
857 |
||
858 |
subsection {* Record specifications *} |
|
859 |
||
860 |
text {* |
|
861 |
\begin{matharray}{rcl} |
|
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|
862 |
@{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\ |
26849 | 863 |
\end{matharray} |
864 |
||
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|
865 |
@{rail " |
42705 | 866 |
@@{command (HOL) record} @{syntax typespec_sorts} '=' \\ |
46494
ea2ae63336f3
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|
867 |
(@{syntax type} '+')? (constdecl +) |
ea2ae63336f3
clarified outer syntax "constdecl", which is only local to some rail diagrams;
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diff
changeset
|
868 |
; |
ea2ae63336f3
clarified outer syntax "constdecl", which is only local to some rail diagrams;
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parents:
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diff
changeset
|
869 |
constdecl: @{syntax name} '::' @{syntax type} @{syntax mixfix}? |
42596
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|
870 |
"} |
26849 | 871 |
|
28760
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changeset
|
872 |
\begin{description} |
26849 | 873 |
|
28760
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|
874 |
\item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 |
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|
875 |
\<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"}, |
26849 | 876 |
derived from the optional parent record @{text "\<tau>"} by adding new |
877 |
field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc. |
|
878 |
||
879 |
The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be |
|
880 |
covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>, |
|
881 |
\<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text |
|
882 |
\<tau>} needs to specify an instance of an existing record type. At |
|
883 |
least one new field @{text "c\<^sub>i"} has to be specified. |
|
884 |
Basically, field names need to belong to a unique record. This is |
|
885 |
not a real restriction in practice, since fields are qualified by |
|
886 |
the record name internally. |
|
887 |
||
888 |
The parent record specification @{text \<tau>} is optional; if omitted |
|
889 |
@{text t} becomes a root record. The hierarchy of all records |
|
890 |
declared within a theory context forms a forest structure, i.e.\ a |
|
891 |
set of trees starting with a root record each. There is no way to |
|
892 |
merge multiple parent records! |
|
893 |
||
894 |
For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a |
|
895 |
type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 :: |
|
896 |
\<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text |
|
897 |
"(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for |
|
898 |
@{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> :: |
|
899 |
\<zeta>\<rparr>"}. |
|
900 |
||
28760
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unified use of declaration environment with IsarImplementation;
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diff
changeset
|
901 |
\end{description} |
26849 | 902 |
*} |
903 |
||
904 |
||
905 |
subsection {* Record operations *} |
|
906 |
||
907 |
text {* |
|
908 |
Any record definition of the form presented above produces certain |
|
909 |
standard operations. Selectors and updates are provided for any |
|
910 |
field, including the improper one ``@{text more}''. There are also |
|
911 |
cumulative record constructor functions. To simplify the |
|
912 |
presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>, |
|
913 |
\<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 :: |
|
914 |
\<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}. |
|
915 |
||
916 |
\medskip \textbf{Selectors} and \textbf{updates} are available for |
|
917 |
any field (including ``@{text more}''): |
|
918 |
||
919 |
\begin{matharray}{lll} |
|
26852 | 920 |
@{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ |
921 |
@{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
26849 | 922 |
\end{matharray} |
923 |
||
924 |
There is special syntax for application of updates: @{text "r\<lparr>x := |
|
925 |
a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for |
|
926 |
repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z := |
|
927 |
c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that |
|
928 |
because of postfix notation the order of fields shown here is |
|
929 |
reverse than in the actual term. Since repeated updates are just |
|
930 |
function applications, fields may be freely permuted in @{text "\<lparr>x |
|
931 |
:= a, y := b, z := c\<rparr>"}, as far as logical equality is concerned. |
|
932 |
Thus commutativity of independent updates can be proven within the |
|
933 |
logic for any two fields, but not as a general theorem. |
|
934 |
||
935 |
\medskip The \textbf{make} operation provides a cumulative record |
|
936 |
constructor function: |
|
937 |
||
938 |
\begin{matharray}{lll} |
|
26852 | 939 |
@{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
26849 | 940 |
\end{matharray} |
941 |
||
942 |
\medskip We now reconsider the case of non-root records, which are |
|
943 |
derived of some parent. In general, the latter may depend on |
|
944 |
another parent as well, resulting in a list of \emph{ancestor |
|
945 |
records}. Appending the lists of fields of all ancestors results in |
|
946 |
a certain field prefix. The record package automatically takes care |
|
947 |
of this by lifting operations over this context of ancestor fields. |
|
948 |
Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor |
|
949 |
fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"}, |
|
950 |
the above record operations will get the following types: |
|
951 |
||
26852 | 952 |
\medskip |
953 |
\begin{tabular}{lll} |
|
954 |
@{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ |
|
42123 | 955 |
@{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> |
26852 | 956 |
\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> |
957 |
\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
958 |
@{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> |
|
959 |
\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
960 |
\end{tabular} |
|
961 |
\medskip |
|
26849 | 962 |
|
26852 | 963 |
\noindent Some further operations address the extension aspect of a |
26849 | 964 |
derived record scheme specifically: @{text "t.fields"} produces a |
965 |
record fragment consisting of exactly the new fields introduced here |
|
966 |
(the result may serve as a more part elsewhere); @{text "t.extend"} |
|
967 |
takes a fixed record and adds a given more part; @{text |
|
968 |
"t.truncate"} restricts a record scheme to a fixed record. |
|
969 |
||
26852 | 970 |
\medskip |
971 |
\begin{tabular}{lll} |
|
972 |
@{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
973 |
@{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow> |
|
974 |
\<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
975 |
@{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
976 |
\end{tabular} |
|
977 |
\medskip |
|
26849 | 978 |
|
979 |
\noindent Note that @{text "t.make"} and @{text "t.fields"} coincide |
|
980 |
for root records. |
|
981 |
*} |
|
982 |
||
983 |
||
984 |
subsection {* Derived rules and proof tools *} |
|
985 |
||
986 |
text {* |
|
987 |
The record package proves several results internally, declaring |
|
988 |
these facts to appropriate proof tools. This enables users to |
|
989 |
reason about record structures quite conveniently. Assume that |
|
990 |
@{text t} is a record type as specified above. |
|
991 |
||
992 |
\begin{enumerate} |
|
42123 | 993 |
|
26849 | 994 |
\item Standard conversions for selectors or updates applied to |
995 |
record constructor terms are made part of the default Simplifier |
|
996 |
context; thus proofs by reduction of basic operations merely require |
|
997 |
the @{method simp} method without further arguments. These rules |
|
998 |
are available as @{text "t.simps"}, too. |
|
42123 | 999 |
|
26849 | 1000 |
\item Selectors applied to updated records are automatically reduced |
1001 |
by an internal simplification procedure, which is also part of the |
|
1002 |
standard Simplifier setup. |
|
1003 |
||
1004 |
\item Inject equations of a form analogous to @{prop "(x, y) = (x', |
|
1005 |
y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical |
|
1006 |
Reasoner as @{attribute iff} rules. These rules are available as |
|
1007 |
@{text "t.iffs"}. |
|
1008 |
||
1009 |
\item The introduction rule for record equality analogous to @{text |
|
1010 |
"x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier, |
|
1011 |
and as the basic rule context as ``@{attribute intro}@{text "?"}''. |
|
1012 |
The rule is called @{text "t.equality"}. |
|
1013 |
||
1014 |
\item Representations of arbitrary record expressions as canonical |
|
1015 |
constructor terms are provided both in @{method cases} and @{method |
|
1016 |
induct} format (cf.\ the generic proof methods of the same name, |
|
1017 |
\secref{sec:cases-induct}). Several variations are available, for |
|
1018 |
fixed records, record schemes, more parts etc. |
|
42123 | 1019 |
|
26849 | 1020 |
The generic proof methods are sufficiently smart to pick the most |
1021 |
sensible rule according to the type of the indicated record |
|
1022 |
expression: users just need to apply something like ``@{text "(cases |
|
1023 |
r)"}'' to a certain proof problem. |
|
1024 |
||
1025 |
\item The derived record operations @{text "t.make"}, @{text |
|
1026 |
"t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not} |
|
1027 |
treated automatically, but usually need to be expanded by hand, |
|
1028 |
using the collective fact @{text "t.defs"}. |
|
1029 |
||
1030 |
\end{enumerate} |
|
1031 |
*} |
|
1032 |
||
1033 |
||
42911 | 1034 |
subsubsection {* Examples *} |
1035 |
||
1036 |
text {* See @{file "~~/src/HOL/ex/Records.thy"}, for example. *} |
|
1037 |
||
1038 |
||
42908 | 1039 |
section {* Adhoc tuples *} |
26849 | 1040 |
|
1041 |
text {* |
|
1042 |
\begin{matharray}{rcl} |
|
42908 | 1043 |
@{attribute_def (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\ |
26849 | 1044 |
\end{matharray} |
1045 |
||
42596
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|
1046 |
@{rail " |
42908 | 1047 |
@@{attribute (HOL) split_format} ('(' 'complete' ')')? |
1048 |
"} |
|
1049 |
||
1050 |
\begin{description} |
|
1051 |
||
1052 |
\item @{attribute (HOL) split_format}\ @{text "(complete)"} causes |
|
1053 |
arguments in function applications to be represented canonically |
|
1054 |
according to their tuple type structure. |
|
1055 |
||
1056 |
Note that this operation tends to invent funny names for new local |
|
1057 |
parameters introduced. |
|
1058 |
||
1059 |
\end{description} |
|
1060 |
*} |
|
1061 |
||
1062 |
||
1063 |
section {* Typedef axiomatization \label{sec:hol-typedef} *} |
|
1064 |
||
46280 | 1065 |
text {* |
1066 |
\begin{matharray}{rcl} |
|
1067 |
@{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
|
1068 |
\end{matharray} |
|
1069 |
||
1070 |
A Gordon/HOL-style type definition is a certain axiom scheme that |
|
1071 |
identifies a new type with a subset of an existing type. More |
|
42908 | 1072 |
precisely, the new type is defined by exhibiting an existing type |
1073 |
@{text \<tau>}, a set @{text "A :: \<tau> set"}, and a theorem that proves |
|
1074 |
@{prop "\<exists>x. x \<in> A"}. Thus @{text A} is a non-empty subset of @{text |
|
1075 |
\<tau>}, and the new type denotes this subset. New functions are |
|
1076 |
postulated that establish an isomorphism between the new type and |
|
1077 |
the subset. In general, the type @{text \<tau>} may involve type |
|
1078 |
variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} which means that the type definition |
|
1079 |
produces a type constructor @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} depending on |
|
1080 |
those type arguments. |
|
1081 |
||
1082 |
The axiomatization can be considered a ``definition'' in the sense |
|
1083 |
of the particular set-theoretic interpretation of HOL |
|
1084 |
\cite{pitts93}, where the universe of types is required to be |
|
1085 |
downwards-closed wrt.\ arbitrary non-empty subsets. Thus genuinely |
|
1086 |
new types introduced by @{command "typedef"} stay within the range |
|
1087 |
of HOL models by construction. Note that @{command_ref |
|
1088 |
type_synonym} from Isabelle/Pure merely introduces syntactic |
|
1089 |
abbreviations, without any logical significance. |
|
47859 | 1090 |
|
42908 | 1091 |
@{rail " |
1092 |
@@{command (HOL) typedef} alt_name? abs_type '=' rep_set |
|
26849 | 1093 |
; |
1094 |
||
42908 | 1095 |
alt_name: '(' (@{syntax name} | @'open' | @'open' @{syntax name}) ')' |
26849 | 1096 |
; |
42908 | 1097 |
abs_type: @{syntax typespec_sorts} @{syntax mixfix}? |
1098 |
; |
|
1099 |
rep_set: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})? |
|
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
1100 |
"} |
26849 | 1101 |
|
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unified use of declaration environment with IsarImplementation;
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diff
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|
1102 |
\begin{description} |
26849 | 1103 |
|
42908 | 1104 |
\item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} |
1105 |
axiomatizes a type definition in the background theory of the |
|
1106 |
current context, depending on a non-emptiness result of the set |
|
1107 |
@{text A} that needs to be proven here. The set @{text A} may |
|
1108 |
contain type variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} as specified on the LHS, |
|
1109 |
but no term variables. |
|
1110 |
||
1111 |
Even though a local theory specification, the newly introduced type |
|
1112 |
constructor cannot depend on parameters or assumptions of the |
|
1113 |
context: this is structurally impossible in HOL. In contrast, the |
|
1114 |
non-emptiness proof may use local assumptions in unusual situations, |
|
1115 |
which could result in different interpretations in target contexts: |
|
1116 |
the meaning of the bijection between the representing set @{text A} |
|
1117 |
and the new type @{text t} may then change in different application |
|
1118 |
contexts. |
|
1119 |
||
1120 |
By default, @{command (HOL) "typedef"} defines both a type |
|
1121 |
constructor @{text t} for the new type, and a term constant @{text |
|
1122 |
t} for the representing set within the old type. Use the ``@{text |
|
1123 |
"(open)"}'' option to suppress a separate constant definition |
|
1124 |
altogether. The injection from type to set is called @{text Rep_t}, |
|
1125 |
its inverse @{text Abs_t}, unless explicit @{keyword (HOL) |
|
1126 |
"morphisms"} specification provides alternative names. |
|
26849 | 1127 |
|
42908 | 1128 |
The core axiomatization uses the locale predicate @{const |
1129 |
type_definition} as defined in Isabelle/HOL. Various basic |
|
1130 |
consequences of that are instantiated accordingly, re-using the |
|
1131 |
locale facts with names derived from the new type constructor. Thus |
|
1132 |
the generic @{thm type_definition.Rep} is turned into the specific |
|
1133 |
@{text "Rep_t"}, for example. |
|
1134 |
||
1135 |
Theorems @{thm type_definition.Rep}, @{thm |
|
1136 |
type_definition.Rep_inverse}, and @{thm type_definition.Abs_inverse} |
|
1137 |
provide the most basic characterization as a corresponding |
|
1138 |
injection/surjection pair (in both directions). The derived rules |
|
1139 |
@{thm type_definition.Rep_inject} and @{thm |
|
1140 |
type_definition.Abs_inject} provide a more convenient version of |
|
1141 |
injectivity, suitable for automated proof tools (e.g.\ in |
|
1142 |
declarations involving @{attribute simp} or @{attribute iff}). |
|
1143 |
Furthermore, the rules @{thm type_definition.Rep_cases}~/ @{thm |
|
1144 |
type_definition.Rep_induct}, and @{thm type_definition.Abs_cases}~/ |
|
1145 |
@{thm type_definition.Abs_induct} provide alternative views on |
|
1146 |
surjectivity. These rules are already declared as set or type rules |
|
1147 |
for the generic @{method cases} and @{method induct} methods, |
|
1148 |
respectively. |
|
1149 |
||
1150 |
An alternative name for the set definition (and other derived |
|
1151 |
entities) may be specified in parentheses; the default is to use |
|
1152 |
@{text t} directly. |
|
26849 | 1153 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1154 |
\end{description} |
26849 | 1155 |
|
42908 | 1156 |
\begin{warn} |
1157 |
If you introduce a new type axiomatically, i.e.\ via @{command_ref |
|
1158 |
typedecl} and @{command_ref axiomatization}, the minimum requirement |
|
1159 |
is that it has a non-empty model, to avoid immediate collapse of the |
|
1160 |
HOL logic. Moreover, one needs to demonstrate that the |
|
1161 |
interpretation of such free-form axiomatizations can coexist with |
|
1162 |
that of the regular @{command_def typedef} scheme, and any extension |
|
1163 |
that other people might have introduced elsewhere (e.g.\ in HOLCF |
|
1164 |
\cite{MuellerNvOS99}). |
|
1165 |
\end{warn} |
|
1166 |
*} |
|
1167 |
||
1168 |
subsubsection {* Examples *} |
|
1169 |
||
1170 |
text {* Type definitions permit the introduction of abstract data |
|
1171 |
types in a safe way, namely by providing models based on already |
|
1172 |
existing types. Given some abstract axiomatic description @{text P} |
|
1173 |
of a type, this involves two steps: |
|
1174 |
||
1175 |
\begin{enumerate} |
|
1176 |
||
1177 |
\item Find an appropriate type @{text \<tau>} and subset @{text A} which |
|
1178 |
has the desired properties @{text P}, and make a type definition |
|
1179 |
based on this representation. |
|
1180 |
||
1181 |
\item Prove that @{text P} holds for @{text \<tau>} by lifting @{text P} |
|
1182 |
from the representation. |
|
26849 | 1183 |
|
42908 | 1184 |
\end{enumerate} |
1185 |
||
1186 |
You can later forget about the representation and work solely in |
|
1187 |
terms of the abstract properties @{text P}. |
|
1188 |
||
1189 |
\medskip The following trivial example pulls a three-element type |
|
1190 |
into existence within the formal logical environment of HOL. *} |
|
1191 |
||
49812
e3945ddcb9aa
eliminated some remaining uses of typedef with implicit set definition;
wenzelm
parents:
48985
diff
changeset
|
1192 |
typedef (open) three = "{(True, True), (True, False), (False, True)}" |
42908 | 1193 |
by blast |
1194 |
||
1195 |
definition "One = Abs_three (True, True)" |
|
1196 |
definition "Two = Abs_three (True, False)" |
|
1197 |
definition "Three = Abs_three (False, True)" |
|
1198 |
||
1199 |
lemma three_distinct: "One \<noteq> Two" "One \<noteq> Three" "Two \<noteq> Three" |
|
49812
e3945ddcb9aa
eliminated some remaining uses of typedef with implicit set definition;
wenzelm
parents:
48985
diff
changeset
|
1200 |
by (simp_all add: One_def Two_def Three_def Abs_three_inject) |
42908 | 1201 |
|
1202 |
lemma three_cases: |
|
1203 |
fixes x :: three obtains "x = One" | "x = Two" | "x = Three" |
|
49812
e3945ddcb9aa
eliminated some remaining uses of typedef with implicit set definition;
wenzelm
parents:
48985
diff
changeset
|
1204 |
by (cases x) (auto simp: One_def Two_def Three_def Abs_three_inject) |
42908 | 1205 |
|
1206 |
text {* Note that such trivial constructions are better done with |
|
1207 |
derived specification mechanisms such as @{command datatype}: *} |
|
1208 |
||
1209 |
datatype three' = One' | Two' | Three' |
|
1210 |
||
1211 |
text {* This avoids re-doing basic definitions and proofs from the |
|
1212 |
primitive @{command typedef} above. *} |
|
26849 | 1213 |
|
1214 |
||
41396 | 1215 |
section {* Functorial structure of types *} |
1216 |
||
1217 |
text {* |
|
1218 |
\begin{matharray}{rcl} |
|
41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41396
diff
changeset
|
1219 |
@{command_def (HOL) "enriched_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
41396 | 1220 |
\end{matharray} |
1221 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
1222 |
@{rail " |
42617 | 1223 |
@@{command (HOL) enriched_type} (@{syntax name} ':')? @{syntax term} |
41396 | 1224 |
; |
42617 | 1225 |
"} |
41396 | 1226 |
|
1227 |
\begin{description} |
|
1228 |
||
42617 | 1229 |
\item @{command (HOL) "enriched_type"}~@{text "prefix: m"} allows to |
1230 |
prove and register properties about the functorial structure of type |
|
1231 |
constructors. These properties then can be used by other packages |
|
1232 |
to deal with those type constructors in certain type constructions. |
|
1233 |
Characteristic theorems are noted in the current local theory. By |
|
1234 |
default, they are prefixed with the base name of the type |
|
1235 |
constructor, an explicit prefix can be given alternatively. |
|
41396 | 1236 |
|
1237 |
The given term @{text "m"} is considered as \emph{mapper} for the |
|
1238 |
corresponding type constructor and must conform to the following |
|
1239 |
type pattern: |
|
1240 |
||
1241 |
\begin{matharray}{lll} |
|
1242 |
@{text "m"} & @{text "::"} & |
|
1243 |
@{text "\<sigma>\<^isub>1 \<Rightarrow> \<dots> \<sigma>\<^isub>k \<Rightarrow> (\<^vec>\<alpha>\<^isub>n) t \<Rightarrow> (\<^vec>\<beta>\<^isub>n) t"} \\ |
|
1244 |
\end{matharray} |
|
1245 |
||
1246 |
\noindent where @{text t} is the type constructor, @{text |
|
1247 |
"\<^vec>\<alpha>\<^isub>n"} and @{text "\<^vec>\<beta>\<^isub>n"} are distinct |
|
1248 |
type variables free in the local theory and @{text "\<sigma>\<^isub>1"}, |
|
1249 |
\ldots, @{text "\<sigma>\<^isub>k"} is a subsequence of @{text "\<alpha>\<^isub>1 \<Rightarrow> |
|
1250 |
\<beta>\<^isub>1"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<alpha>\<^isub>1"}, \ldots, |
|
1251 |
@{text "\<alpha>\<^isub>n \<Rightarrow> \<beta>\<^isub>n"}, @{text "\<beta>\<^isub>n \<Rightarrow> |
|
1252 |
\<alpha>\<^isub>n"}. |
|
1253 |
||
1254 |
\end{description} |
|
1255 |
*} |
|
1256 |
||
47859 | 1257 |
|
47821 | 1258 |
section {* Transfer package *} |
1259 |
||
1260 |
text {* |
|
1261 |
\begin{matharray}{rcl} |
|
1262 |
@{method_def (HOL) "transfer"} & : & @{text method} \\ |
|
1263 |
@{method_def (HOL) "transfer'"} & : & @{text method} \\ |
|
1264 |
@{method_def (HOL) "transfer_prover"} & : & @{text method} \\ |
|
1265 |
@{attribute_def (HOL) "transfer_rule"} & : & @{text attribute} \\ |
|
1266 |
@{attribute_def (HOL) "relator_eq"} & : & @{text attribute} \\ |
|
1267 |
\end{matharray} |
|
1268 |
||
1269 |
\begin{description} |
|
1270 |
||
1271 |
\item @{method (HOL) "transfer"} method replaces the current subgoal |
|
1272 |
with a logically equivalent one that uses different types and |
|
1273 |
constants. The replacement of types and constants is guided by the |
|
1274 |
database of transfer rules. Goals are generalized over all free |
|
1275 |
variables by default; this is necessary for variables whose types |
|
1276 |
change, but can be overridden for specific variables with e.g. |
|
1277 |
@{text "transfer fixing: x y z"}. |
|
1278 |
||
1279 |
\item @{method (HOL) "transfer'"} is a variant of @{method (HOL) |
|
1280 |
transfer} that allows replacing a subgoal with one that is |
|
1281 |
logically stronger (rather than equivalent). For example, a |
|
1282 |
subgoal involving equality on a quotient type could be replaced |
|
1283 |
with a subgoal involving equality (instead of the corresponding |
|
1284 |
equivalence relation) on the underlying raw type. |
|
1285 |
||
1286 |
\item @{method (HOL) "transfer_prover"} method assists with proving |
|
1287 |
a transfer rule for a new constant, provided the constant is |
|
1288 |
defined in terms of other constants that already have transfer |
|
1289 |
rules. It should be applied after unfolding the constant |
|
1290 |
definitions. |
|
1291 |
||
1292 |
\item @{attribute (HOL) "transfer_rule"} attribute maintains a |
|
1293 |
collection of transfer rules, which relate constants at two |
|
1294 |
different types. Typical transfer rules may relate different type |
|
1295 |
instances of the same polymorphic constant, or they may relate an |
|
1296 |
operation on a raw type to a corresponding operation on an |
|
1297 |
abstract type (quotient or subtype). For example: |
|
1298 |
||
1299 |
@{text "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"}\\ |
|
1300 |
@{text "(cr_int ===> cr_int ===> cr_int) (\<lambda>(x,y) (u,v). (x+u, y+v)) plus"} |
|
1301 |
||
1302 |
Lemmas involving predicates on relations can also be registered |
|
1303 |
using the same attribute. For example: |
|
1304 |
||
1305 |
@{text "bi_unique A \<Longrightarrow> (list_all2 A ===> op =) distinct distinct"}\\ |
|
1306 |
@{text "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (prod_rel A B)"} |
|
1307 |
||
1308 |
\item @{attribute (HOL) relator_eq} attribute collects identity laws |
|
1309 |
for relators of various type constructors, e.g. @{text "list_all2 |
|
1310 |
(op =) = (op =)"}. The @{method (HOL) transfer} method uses these |
|
1311 |
lemmas to infer transfer rules for non-polymorphic constants on |
|
1312 |
the fly. |
|
1313 |
||
1314 |
\end{description} |
|
1315 |
||
1316 |
*} |
|
1317 |
||
47859 | 1318 |
|
47802 | 1319 |
section {* Lifting package *} |
1320 |
||
1321 |
text {* |
|
1322 |
\begin{matharray}{rcl} |
|
1323 |
@{command_def (HOL) "setup_lifting"} & : & @{text "local_theory \<rightarrow> local_theory"}\\ |
|
1324 |
@{command_def (HOL) "lift_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\ |
|
1325 |
@{command_def (HOL) "print_quotmaps"} & : & @{text "context \<rightarrow>"}\\ |
|
1326 |
@{command_def (HOL) "print_quotients"} & : & @{text "context \<rightarrow>"}\\ |
|
1327 |
@{attribute_def (HOL) "quot_map"} & : & @{text attribute} \\ |
|
1328 |
@{attribute_def (HOL) "invariant_commute"} & : & @{text attribute} \\ |
|
1329 |
\end{matharray} |
|
1330 |
||
1331 |
@{rail " |
|
1332 |
@@{command (HOL) setup_lifting} ('(' 'no_abs_code' ')')? \\ |
|
1333 |
@{syntax thmref} @{syntax thmref}?; |
|
1334 |
"} |
|
1335 |
||
1336 |
@{rail " |
|
1337 |
@@{command (HOL) lift_definition} @{syntax name} '::' @{syntax type} @{syntax mixfix}? \\ |
|
1338 |
'is' @{syntax term}; |
|
1339 |
"} |
|
1340 |
||
1341 |
\begin{description} |
|
1342 |
||
47859 | 1343 |
\item @{command (HOL) "setup_lifting"} Sets up the Lifting package |
1344 |
to work with a user-defined type. The user must provide either a |
|
1345 |
quotient theorem @{text "Quotient R Abs Rep T"} or a |
|
1346 |
type_definition theorem @{text "type_definition Rep Abs A"}. The |
|
1347 |
package configures transfer rules for equality and quantifiers on |
|
1348 |
the type, and sets up the @{command_def (HOL) "lift_definition"} |
|
1349 |
command to work with the type. In the case of a quotient theorem, |
|
1350 |
an optional theorem @{text "reflp R"} can be provided as a second |
|
1351 |
argument. This allows the package to generate stronger transfer |
|
1352 |
rules. |
|
1353 |
||
1354 |
@{command (HOL) "setup_lifting"} is called automatically if a |
|
1355 |
quotient type is defined by the command @{command (HOL) |
|
1356 |
"quotient_type"} from the Quotient package. |
|
1357 |
||
1358 |
If @{command (HOL) "setup_lifting"} is called with a |
|
1359 |
type_definition theorem, the abstract type implicitly defined by |
|
1360 |
the theorem is declared as an abstract type in the code |
|
1361 |
generator. This allows @{command (HOL) "lift_definition"} to |
|
1362 |
register (generated) code certificate theorems as abstract code |
|
1363 |
equations in the code generator. The option @{text "no_abs_code"} |
|
1364 |
of the command @{command (HOL) "setup_lifting"} can turn off that |
|
1365 |
behavior and causes that code certificate theorems generated by |
|
1366 |
@{command (HOL) "lift_definition"} are not registred as abstract |
|
1367 |
code equations. |
|
1368 |
||
1369 |
\item @{command (HOL) "lift_definition"} @{text "f :: \<tau> is t"} |
|
1370 |
Defines a new function @{text f} with an abstract type @{text \<tau>} |
|
1371 |
in terms of a corresponding operation @{text t} on a |
|
1372 |
representation type. The term @{text t} doesn't have to be |
|
1373 |
necessarily a constant but it can be any term. |
|
47802 | 1374 |
|
1375 |
Users must discharge a respectfulness proof obligation when each |
|
47859 | 1376 |
constant is defined. For a type copy, i.e. a typedef with @{text |
1377 |
UNIV}, the proof is discharged automatically. The obligation is |
|
47802 | 1378 |
presented in a user-friendly, readable form. A respectfulness |
47859 | 1379 |
theorem in the standard format @{text f.rsp} and a transfer rule |
1380 |
@{text f.tranfer} for the Transfer package are generated by the |
|
1381 |
package. |
|
47802 | 1382 |
|
1383 |
Integration with code_abstype: For typedefs (e.g. subtypes |
|
47859 | 1384 |
corresponding to a datatype invariant, such as dlist), @{command |
1385 |
(HOL) "lift_definition"} generates a code certificate theorem |
|
1386 |
@{text f.rep_eq} and sets up code generation for each constant. |
|
47802 | 1387 |
|
1388 |
\item @{command (HOL) "print_quotmaps"} prints stored quotient map |
|
47859 | 1389 |
theorems. |
1390 |
||
1391 |
\item @{command (HOL) "print_quotients"} prints stored quotient |
|
1392 |
theorems. |
|
1393 |
||
1394 |
\item @{attribute (HOL) quot_map} registers a quotient map |
|
1395 |
theorem. For examples see @{file |
|
1396 |
"~~/src/HOL/Library/Quotient_List.thy"} or other Quotient_*.thy |
|
1397 |
files. |
|
1398 |
||
1399 |
\item @{attribute (HOL) invariant_commute} registers a theorem which |
|
1400 |
shows a relationship between the constant @{text |
|
1401 |
Lifting.invariant} (used for internal encoding of proper subtypes) |
|
1402 |
and a relator. Such theorems allows the package to hide @{text |
|
1403 |
Lifting.invariant} from a user in a user-readable form of a |
|
1404 |
respectfulness theorem. For examples see @{file |
|
1405 |
"~~/src/HOL/Library/Quotient_List.thy"} or other Quotient_*.thy |
|
1406 |
files. |
|
47802 | 1407 |
|
1408 |
\end{description} |
|
1409 |
*} |
|
1410 |
||
47859 | 1411 |
|
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1412 |
section {* Quotient types *} |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1413 |
|
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1414 |
text {* |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1415 |
\begin{matharray}{rcl} |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1416 |
@{command_def (HOL) "quotient_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\ |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1417 |
@{command_def (HOL) "quotient_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\ |
47802 | 1418 |
@{command_def (HOL) "print_quotmapsQ3"} & : & @{text "context \<rightarrow>"}\\ |
1419 |
@{command_def (HOL) "print_quotientsQ3"} & : & @{text "context \<rightarrow>"}\\ |
|
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1420 |
@{command_def (HOL) "print_quotconsts"} & : & @{text "context \<rightarrow>"}\\ |
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1421 |
@{method_def (HOL) "lifting"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1422 |
@{method_def (HOL) "lifting_setup"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1423 |
@{method_def (HOL) "descending"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1424 |
@{method_def (HOL) "descending_setup"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1425 |
@{method_def (HOL) "partiality_descending"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1426 |
@{method_def (HOL) "partiality_descending_setup"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1427 |
@{method_def (HOL) "regularize"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1428 |
@{method_def (HOL) "injection"} & : & @{text method} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1429 |
@{method_def (HOL) "cleaning"} & : & @{text method} \\ |
46448
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1430 |
@{attribute_def (HOL) "quot_thm"} & : & @{text attribute} \\ |
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1431 |
@{attribute_def (HOL) "quot_lifted"} & : & @{text attribute} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1432 |
@{attribute_def (HOL) "quot_respect"} & : & @{text attribute} \\ |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1433 |
@{attribute_def (HOL) "quot_preserve"} & : & @{text attribute} \\ |
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1434 |
\end{matharray} |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1435 |
|
46280 | 1436 |
The quotient package defines a new quotient type given a raw type |
1437 |
and a partial equivalence relation. It also includes automation for |
|
1438 |
transporting definitions and theorems. It can automatically produce |
|
1439 |
definitions and theorems on the quotient type, given the |
|
1440 |
corresponding constants and facts on the raw type. |
|
1441 |
||
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1442 |
@{rail " |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1443 |
@@{command (HOL) quotient_type} (spec + @'and'); |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1444 |
|
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1445 |
spec: @{syntax typespec} @{syntax mixfix}? '=' \\ |
45678 | 1446 |
@{syntax type} '/' ('partial' ':')? @{syntax term} \\ |
47859 | 1447 |
(@'morphisms' @{syntax name} @{syntax name})?; |
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1448 |
"} |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1449 |
|
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1450 |
@{rail " |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1451 |
@@{command (HOL) quotient_definition} constdecl? @{syntax thmdecl}? \\ |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1452 |
@{syntax term} 'is' @{syntax term}; |
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1453 |
|
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1454 |
constdecl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}? |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1455 |
"} |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1456 |
|
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1457 |
@{rail " |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1458 |
@@{method (HOL) lifting} @{syntax thmrefs}? |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1459 |
; |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1460 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1461 |
@@{method (HOL) lifting_setup} @{syntax thmrefs}? |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1462 |
; |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1463 |
"} |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1464 |
|
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1465 |
\begin{description} |
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1466 |
|
46457 | 1467 |
\item @{command (HOL) "quotient_type"} defines quotient types. The |
46280 | 1468 |
injection from a quotient type to a raw type is called @{text |
1469 |
rep_t}, its inverse @{text abs_t} unless explicit @{keyword (HOL) |
|
46457 | 1470 |
"morphisms"} specification provides alternative names. @{command |
1471 |
(HOL) "quotient_type"} requires the user to prove that the relation |
|
1472 |
is an equivalence relation (predicate @{text equivp}), unless the |
|
1473 |
user specifies explicitely @{text partial} in which case the |
|
1474 |
obligation is @{text part_equivp}. A quotient defined with @{text |
|
1475 |
partial} is weaker in the sense that less things can be proved |
|
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1476 |
automatically. |
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1477 |
|
46280 | 1478 |
\item @{command (HOL) "quotient_definition"} defines a constant on |
1479 |
the quotient type. |
|
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1480 |
|
47802 | 1481 |
\item @{command (HOL) "print_quotmapsQ3"} prints quotient map |
46280 | 1482 |
functions. |
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1483 |
|
47802 | 1484 |
\item @{command (HOL) "print_quotientsQ3"} prints quotients. |
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1485 |
|
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1486 |
\item @{command (HOL) "print_quotconsts"} prints quotient constants. |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1487 |
|
46447
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1488 |
\item @{method (HOL) "lifting"} and @{method (HOL) "lifting_setup"} |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1489 |
methods match the current goal with the given raw theorem to be |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1490 |
lifted producing three new subgoals: regularization, injection and |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1491 |
cleaning subgoals. @{method (HOL) "lifting"} tries to apply the |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1492 |
heuristics for automatically solving these three subgoals and |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1493 |
leaves only the subgoals unsolved by the heuristics to the user as |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1494 |
opposed to @{method (HOL) "lifting_setup"} which leaves the three |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1495 |
subgoals unsolved. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1496 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1497 |
\item @{method (HOL) "descending"} and @{method (HOL) |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1498 |
"descending_setup"} try to guess a raw statement that would lift |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1499 |
to the current subgoal. Such statement is assumed as a new subgoal |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1500 |
and @{method (HOL) "descending"} continues in the same way as |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1501 |
@{method (HOL) "lifting"} does. @{method (HOL) "descending"} tries |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1502 |
to solve the arising regularization, injection and cleaning |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1503 |
subgoals with the analoguous method @{method (HOL) |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1504 |
"descending_setup"} which leaves the four unsolved subgoals. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1505 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1506 |
\item @{method (HOL) "partiality_descending"} finds the regularized |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1507 |
theorem that would lift to the current subgoal, lifts it and |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1508 |
leaves as a subgoal. This method can be used with partial |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1509 |
equivalence quotients where the non regularized statements would |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1510 |
not be true. @{method (HOL) "partiality_descending_setup"} leaves |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1511 |
the injection and cleaning subgoals unchanged. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1512 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1513 |
\item @{method (HOL) "regularize"} applies the regularization |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1514 |
heuristics to the current subgoal. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1515 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1516 |
\item @{method (HOL) "injection"} applies the injection heuristics |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1517 |
to the current goal using the stored quotient respectfulness |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1518 |
theorems. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1519 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1520 |
\item @{method (HOL) "cleaning"} applies the injection cleaning |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1521 |
heuristics to the current subgoal using the stored quotient |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1522 |
preservation theorems. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1523 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1524 |
\item @{attribute (HOL) quot_lifted} attribute tries to |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1525 |
automatically transport the theorem to the quotient type. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1526 |
The attribute uses all the defined quotients types and quotient |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1527 |
constants often producing undesired results or theorems that |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1528 |
cannot be lifted. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1529 |
|
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1530 |
\item @{attribute (HOL) quot_respect} and @{attribute (HOL) |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1531 |
quot_preserve} attributes declare a theorem as a respectfulness |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1532 |
and preservation theorem respectively. These are stored in the |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1533 |
local theory store and used by the @{method (HOL) "injection"} |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1534 |
and @{method (HOL) "cleaning"} methods respectively. |
f37da60a8cc6
specification of the quotient package
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46028
diff
changeset
|
1535 |
|
46448
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1536 |
\item @{attribute (HOL) quot_thm} declares that a certain theorem |
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1537 |
is a quotient extension theorem. Quotient extension theorems |
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1538 |
allow for quotienting inside container types. Given a polymorphic |
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1539 |
type that serves as a container, a map function defined for this |
47349
803729c9fd4d
documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
bulwahn
parents:
46641
diff
changeset
|
1540 |
container using @{command (HOL) "enriched_type"} and a relation |
46448
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1541 |
map defined for for the container type, the quotient extension |
47349
803729c9fd4d
documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
bulwahn
parents:
46641
diff
changeset
|
1542 |
theorem should be @{term "Quotient3 R Abs Rep \<Longrightarrow> Quotient3 |
46448
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1543 |
(rel_map R) (map Abs) (map Rep)"}. Quotient extension theorems |
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1544 |
are stored in a database and are used all the steps of lifting |
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1545 |
theorems. |
f1201fac7398
more specification of the quotient package in IsarRef
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
46447
diff
changeset
|
1546 |
|
43993
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1547 |
\end{description} |
b141d7a3d4e3
rudimentary documentation of the quotient package in the isar reference manual
bulwahn
parents:
43914
diff
changeset
|
1548 |
*} |
41396 | 1549 |
|
1550 |
||
43994 | 1551 |
section {* Coercive subtyping *} |
1552 |
||
1553 |
text {* |
|
1554 |
\begin{matharray}{rcl} |
|
1555 |
@{attribute_def (HOL) coercion} & : & @{text attribute} \\ |
|
1556 |
@{attribute_def (HOL) coercion_enabled} & : & @{text attribute} \\ |
|
1557 |
@{attribute_def (HOL) coercion_map} & : & @{text attribute} \\ |
|
1558 |
\end{matharray} |
|
1559 |
||
46280 | 1560 |
Coercive subtyping allows the user to omit explicit type |
1561 |
conversions, also called \emph{coercions}. Type inference will add |
|
1562 |
them as necessary when parsing a term. See |
|
1563 |
\cite{traytel-berghofer-nipkow-2011} for details. |
|
1564 |
||
43994 | 1565 |
@{rail " |
1566 |
@@{attribute (HOL) coercion} (@{syntax term})? |
|
1567 |
; |
|
1568 |
"} |
|
1569 |
@{rail " |
|
1570 |
@@{attribute (HOL) coercion_map} (@{syntax term})? |
|
1571 |
; |
|
1572 |
"} |
|
1573 |
||
1574 |
\begin{description} |
|
1575 |
||
1576 |
\item @{attribute (HOL) "coercion"}~@{text "f"} registers a new |
|
46280 | 1577 |
coercion function @{text "f :: \<sigma>\<^isub>1 \<Rightarrow> \<sigma>\<^isub>2"} where @{text "\<sigma>\<^isub>1"} and |
1578 |
@{text "\<sigma>\<^isub>2"} are type constructors without arguments. Coercions are |
|
1579 |
composed by the inference algorithm if needed. Note that the type |
|
1580 |
inference algorithm is complete only if the registered coercions |
|
1581 |
form a lattice. |
|
43994 | 1582 |
|
46280 | 1583 |
\item @{attribute (HOL) "coercion_map"}~@{text "map"} registers a |
1584 |
new map function to lift coercions through type constructors. The |
|
1585 |
function @{text "map"} must conform to the following type pattern |
|
43994 | 1586 |
|
1587 |
\begin{matharray}{lll} |
|
1588 |
@{text "map"} & @{text "::"} & |
|
1589 |
@{text "f\<^isub>1 \<Rightarrow> \<dots> \<Rightarrow> f\<^isub>n \<Rightarrow> (\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n) t \<Rightarrow> (\<beta>\<^isub>1, \<dots>, \<beta>\<^isub>n) t"} \\ |
|
1590 |
\end{matharray} |
|
1591 |
||
46280 | 1592 |
where @{text "t"} is a type constructor and @{text "f\<^isub>i"} is of type |
1593 |
@{text "\<alpha>\<^isub>i \<Rightarrow> \<beta>\<^isub>i"} or @{text "\<beta>\<^isub>i \<Rightarrow> \<alpha>\<^isub>i"}. Registering a map function |
|
1594 |
overwrites any existing map function for this particular type |
|
1595 |
constructor. |
|
43994 | 1596 |
|
1597 |
\item @{attribute (HOL) "coercion_enabled"} enables the coercion |
|
1598 |
inference algorithm. |
|
1599 |
||
1600 |
\end{description} |
|
1601 |
*} |
|
1602 |
||
1603 |
||
26849 | 1604 |
section {* Arithmetic proof support *} |
1605 |
||
1606 |
text {* |
|
1607 |
\begin{matharray}{rcl} |
|
28761
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1608 |
@{method_def (HOL) arith} & : & @{text method} \\ |
30863 | 1609 |
@{attribute_def (HOL) arith} & : & @{text attribute} \\ |
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|
1610 |
@{attribute_def (HOL) arith_split} & : & @{text attribute} \\ |
26849 | 1611 |
\end{matharray} |
1612 |
||
46280 | 1613 |
\begin{description} |
26849 | 1614 |
|
46280 | 1615 |
\item @{method (HOL) arith} decides linear arithmetic problems (on |
1616 |
types @{text nat}, @{text int}, @{text real}). Any current facts |
|
1617 |
are inserted into the goal before running the procedure. |
|
26849 | 1618 |
|
46280 | 1619 |
\item @{attribute (HOL) arith} declares facts that are supplied to |
1620 |
the arithmetic provers implicitly. |
|
1621 |
||
1622 |
\item @{attribute (HOL) arith_split} attribute declares case split |
|
30865 | 1623 |
rules to be expanded before @{method (HOL) arith} is invoked. |
30863 | 1624 |
|
46280 | 1625 |
\end{description} |
1626 |
||
1627 |
Note that a simpler (but faster) arithmetic prover is already |
|
1628 |
invoked by the Simplifier. |
|
26849 | 1629 |
*} |
1630 |
||
1631 |
||
30169 | 1632 |
section {* Intuitionistic proof search *} |
1633 |
||
1634 |
text {* |
|
1635 |
\begin{matharray}{rcl} |
|
30171 | 1636 |
@{method_def (HOL) iprover} & : & @{text method} \\ |
30169 | 1637 |
\end{matharray} |
1638 |
||
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1639 |
@{rail " |
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|
1640 |
@@{method (HOL) iprover} ( @{syntax rulemod} * ) |
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|
1641 |
"} |
30169 | 1642 |
|
46280 | 1643 |
\begin{description} |
1644 |
||
1645 |
\item @{method (HOL) iprover} performs intuitionistic proof search, |
|
1646 |
depending on specifically declared rules from the context, or given |
|
1647 |
as explicit arguments. Chained facts are inserted into the goal |
|
1648 |
before commencing proof search. |
|
35613 | 1649 |
|
30169 | 1650 |
Rules need to be classified as @{attribute (Pure) intro}, |
1651 |
@{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the |
|
1652 |
``@{text "!"}'' indicator refers to ``safe'' rules, which may be |
|
1653 |
applied aggressively (without considering back-tracking later). |
|
1654 |
Rules declared with ``@{text "?"}'' are ignored in proof search (the |
|
42626 | 1655 |
single-step @{method (Pure) rule} method still observes these). An |
30169 | 1656 |
explicit weight annotation may be given as well; otherwise the |
1657 |
number of rule premises will be taken into account here. |
|
46280 | 1658 |
|
1659 |
\end{description} |
|
30169 | 1660 |
*} |
1661 |
||
46280 | 1662 |
|
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|
1663 |
section {* Model Elimination and Resolution *} |
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|
1664 |
|
36ba44fe0781
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|
1665 |
text {* |
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|
1666 |
\begin{matharray}{rcl} |
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|
1667 |
@{method_def (HOL) "meson"} & : & @{text method} \\ |
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|
1668 |
@{method_def (HOL) "metis"} & : & @{text method} \\ |
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|
1669 |
\end{matharray} |
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|
1670 |
|
36ba44fe0781
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|
1671 |
@{rail " |
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|
1672 |
@@{method (HOL) meson} @{syntax thmrefs}? |
36ba44fe0781
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changeset
|
1673 |
; |
36ba44fe0781
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changeset
|
1674 |
|
46280 | 1675 |
@@{method (HOL) metis} |
1676 |
('(' ('partial_types' | 'full_types' | 'no_types' | @{syntax name}) ')')? |
|
1677 |
@{syntax thmrefs}? |
|
43578
36ba44fe0781
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|
1678 |
"} |
36ba44fe0781
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parents:
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changeset
|
1679 |
|
46280 | 1680 |
\begin{description} |
1681 |
||
1682 |
\item @{method (HOL) meson} implements Loveland's model elimination |
|
1683 |
procedure \cite{loveland-78}. See @{file |
|
1684 |
"~~/src/HOL/ex/Meson_Test.thy"} for examples. |
|
43578
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|
1685 |
|
46280 | 1686 |
\item @{method (HOL) metis} combines ordered resolution and ordered |
1687 |
paramodulation to find first-order (or mildly higher-order) proofs. |
|
1688 |
The first optional argument specifies a type encoding; see the |
|
1689 |
Sledgehammer manual \cite{isabelle-sledgehammer} for details. The |
|
1690 |
directory @{file "~~/src/HOL/Metis_Examples"} contains several small |
|
1691 |
theories developed to a large extent using @{method (HOL) metis}. |
|
1692 |
||
1693 |
\end{description} |
|
43578
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|
1694 |
*} |
30169 | 1695 |
|
46280 | 1696 |
|
30171 | 1697 |
section {* Coherent Logic *} |
1698 |
||
1699 |
text {* |
|
1700 |
\begin{matharray}{rcl} |
|
1701 |
@{method_def (HOL) "coherent"} & : & @{text method} \\ |
|
1702 |
\end{matharray} |
|
1703 |
||
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|
1704 |
@{rail " |
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|
1705 |
@@{method (HOL) coherent} @{syntax thmrefs}? |
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|
1706 |
"} |
30171 | 1707 |
|
46280 | 1708 |
\begin{description} |
1709 |
||
1710 |
\item @{method (HOL) coherent} solves problems of \emph{Coherent |
|
1711 |
Logic} \cite{Bezem-Coquand:2005}, which covers applications in |
|
1712 |
confluence theory, lattice theory and projective geometry. See |
|
1713 |
@{file "~~/src/HOL/ex/Coherent.thy"} for some examples. |
|
1714 |
||
1715 |
\end{description} |
|
30171 | 1716 |
*} |
1717 |
||
1718 |
||
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|
1719 |
section {* Proving propositions *} |
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|
1720 |
|
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|
1721 |
text {* |
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|
1722 |
In addition to the standard proof methods, a number of diagnosis |
de9d43c427ae
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changeset
|
1723 |
tools search for proofs and provide an Isar proof snippet on success. |
de9d43c427ae
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|
1724 |
These tools are available via the following commands. |
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|
1725 |
|
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|
1726 |
\begin{matharray}{rcl} |
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|
1727 |
@{command_def (HOL) "solve_direct"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
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|
1728 |
@{command_def (HOL) "try"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
46641 | 1729 |
@{command_def (HOL) "try0"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
42215
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|
1730 |
@{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
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|
1731 |
@{command_def (HOL) "sledgehammer_params"} & : & @{text "theory \<rightarrow> theory"} |
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|
1732 |
\end{matharray} |
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|
1733 |
|
42596
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modernized rail diagrams using @{rail} antiquotation;
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|
1734 |
@{rail " |
43040 | 1735 |
@@{command (HOL) try} |
1736 |
; |
|
1737 |
||
46641 | 1738 |
@@{command (HOL) try0} ( ( ( 'simp' | 'intro' | 'elim' | 'dest' ) ':' @{syntax thmrefs} ) + ) ? |
42596
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|
1739 |
@{syntax nat}? |
42215
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|
1740 |
; |
43040 | 1741 |
|
42596
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|
1742 |
@@{command (HOL) sledgehammer} ( '[' args ']' )? facts? @{syntax nat}? |
42215
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|
1743 |
; |
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|
1744 |
|
42596
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|
1745 |
@@{command (HOL) sledgehammer_params} ( ( '[' args ']' ) ? ) |
42215
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|
1746 |
; |
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|
1747 |
|
42596
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changeset
|
1748 |
args: ( @{syntax name} '=' value + ',' ) |
42215
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|
1749 |
; |
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|
1750 |
|
42596
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|
1751 |
facts: '(' ( ( ( ( 'add' | 'del' ) ':' ) ? @{syntax thmrefs} ) + ) ? ')' |
42215
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changeset
|
1752 |
; |
43019
619f16bf2150
removed FIXME: "try_methods" doesn't take a clasimpmod, because it needs to generate Isar proof text in case of success and hence needs total control over its arguments
blanchet
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43016
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|
1753 |
"} % FIXME check args "value" |
42215
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|
1754 |
|
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|
1755 |
\begin{description} |
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|
1756 |
|
46283 | 1757 |
\item @{command (HOL) "solve_direct"} checks whether the current |
1758 |
subgoals can be solved directly by an existing theorem. Duplicate |
|
1759 |
lemmas can be detected in this way. |
|
42215
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|
1760 |
|
46641 | 1761 |
\item @{command (HOL) "try0"} attempts to prove a subgoal |
46283 | 1762 |
using a combination of standard proof methods (@{method auto}, |
1763 |
@{method simp}, @{method blast}, etc.). Additional facts supplied |
|
1764 |
via @{text "simp:"}, @{text "intro:"}, @{text "elim:"}, and @{text |
|
1765 |
"dest:"} are passed to the appropriate proof methods. |
|
42215
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|
1766 |
|
43914
64819f353c53
updating documentation about quickcheck; adding information about try
bulwahn
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diff
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|
1767 |
\item @{command (HOL) "try"} attempts to prove or disprove a subgoal |
46283 | 1768 |
using a combination of provers and disprovers (@{command (HOL) |
1769 |
"solve_direct"}, @{command (HOL) "quickcheck"}, @{command (HOL) |
|
46641 | 1770 |
"try0"}, @{command (HOL) "sledgehammer"}, @{command (HOL) |
46283 | 1771 |
"nitpick"}). |
43914
64819f353c53
updating documentation about quickcheck; adding information about try
bulwahn
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|
1772 |
|
46283 | 1773 |
\item @{command (HOL) "sledgehammer"} attempts to prove a subgoal |
1774 |
using external automatic provers (resolution provers and SMT |
|
1775 |
solvers). See the Sledgehammer manual \cite{isabelle-sledgehammer} |
|
1776 |
for details. |
|
42215
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|
1777 |
|
46283 | 1778 |
\item @{command (HOL) "sledgehammer_params"} changes @{command (HOL) |
1779 |
"sledgehammer"} configuration options persistently. |
|
42215
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|
1780 |
|
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|
1781 |
\end{description} |
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|
1782 |
*} |
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|
1783 |
|
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|
1784 |
|
31912
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|
1785 |
section {* Checking and refuting propositions *} |
f5bd306f5e9d
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diff
changeset
|
1786 |
|
f5bd306f5e9d
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|
1787 |
text {* |
f5bd306f5e9d
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haftmann
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changeset
|
1788 |
Identifying incorrect propositions usually involves evaluation of |
42215
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|
1789 |
particular assignments and systematic counterexample search. This |
31912
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haftmann
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diff
changeset
|
1790 |
is supported by the following commands. |
f5bd306f5e9d
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haftmann
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diff
changeset
|
1791 |
|
f5bd306f5e9d
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haftmann
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changeset
|
1792 |
\begin{matharray}{rcl} |
f5bd306f5e9d
more friendly wrt. PGs interpretation of compound *); added dedicated section on value and quickcheck
haftmann
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diff
changeset
|
1793 |
@{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
45409
5abb0e738b00
adding some documentation about the values command to the isar reference
bulwahn
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45408
diff
changeset
|
1794 |
@{command_def (HOL) "values"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
31912
f5bd306f5e9d
more friendly wrt. PGs interpretation of compound *); added dedicated section on value and quickcheck
haftmann
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changeset
|
1795 |
@{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
42215
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|
1796 |
@{command_def (HOL) "refute"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
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changeset
|
1797 |
@{command_def (HOL) "nitpick"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
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changeset
|
1798 |
@{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"} \\ |
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changeset
|
1799 |
@{command_def (HOL) "refute_params"} & : & @{text "theory \<rightarrow> theory"} \\ |
45943
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|
1800 |
@{command_def (HOL) "nitpick_params"} & : & @{text "theory \<rightarrow> theory"} \\ |
46592
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|
1801 |
@{command_def (HOL) "quickcheck_generator"} & : & @{text "theory \<rightarrow> theory"} \\ |
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|
1802 |
@{command_def (HOL) "find_unused_assms"} & : & @{text "context \<rightarrow>"} |
31912
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|
1803 |
\end{matharray} |
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|
1804 |
|
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|
1805 |
@{rail " |
46628 | 1806 |
@@{command (HOL) value} ( '[' @{syntax name} ']' )? modes? @{syntax term} |
31912
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|
1807 |
; |
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|
1808 |
|
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|
1809 |
@@{command (HOL) values} modes? @{syntax nat}? @{syntax term} |
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|
1810 |
; |
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changeset
|
1811 |
|
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|
1812 |
(@@{command (HOL) quickcheck} | @@{command (HOL) refute} | @@{command (HOL) nitpick}) |
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|
1813 |
( '[' args ']' )? @{syntax nat}? |
31912
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|
1814 |
; |
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|
1815 |
|
42596
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|
1816 |
(@@{command (HOL) quickcheck_params} | @@{command (HOL) refute_params} | |
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|
1817 |
@@{command (HOL) nitpick_params}) ( '[' args ']' )? |
31912
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|
1818 |
; |
46592
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|
1819 |
|
46628 | 1820 |
@@{command (HOL) quickcheck_generator} @{syntax nameref} \\ |
45943
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|
1821 |
'operations:' ( @{syntax term} +) |
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|
1822 |
; |
31912
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|
1823 |
|
46628 | 1824 |
@@{command (HOL) find_unused_assms} @{syntax name}? |
46592
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|
1825 |
; |
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adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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|
1826 |
|
42596
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|
1827 |
modes: '(' (@{syntax name} +) ')' |
31912
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|
1828 |
; |
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changeset
|
1829 |
|
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|
1830 |
args: ( @{syntax name} '=' value + ',' ) |
31912
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|
1831 |
; |
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|
1832 |
"} % FIXME check "value" |
31912
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|
1833 |
|
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|
1834 |
\begin{description} |
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|
1835 |
|
f5bd306f5e9d
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|
1836 |
\item @{command (HOL) "value"}~@{text t} evaluates and prints a |
46283 | 1837 |
term; optionally @{text modes} can be specified, which are appended |
1838 |
to the current print mode; see \secref{sec:print-modes}. |
|
1839 |
Internally, the evaluation is performed by registered evaluators, |
|
1840 |
which are invoked sequentially until a result is returned. |
|
1841 |
Alternatively a specific evaluator can be selected using square |
|
1842 |
brackets; typical evaluators use the current set of code equations |
|
1843 |
to normalize and include @{text simp} for fully symbolic evaluation |
|
1844 |
using the simplifier, @{text nbe} for \emph{normalization by |
|
1845 |
evaluation} and \emph{code} for code generation in SML. |
|
31912
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|
1846 |
|
46283 | 1847 |
\item @{command (HOL) "values"}~@{text t} enumerates a set |
1848 |
comprehension by evaluation and prints its values up to the given |
|
1849 |
number of solutions; optionally @{text modes} can be specified, |
|
1850 |
which are appended to the current print mode; see |
|
1851 |
\secref{sec:print-modes}. |
|
45409
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|
1852 |
|
31912
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|
1853 |
\item @{command (HOL) "quickcheck"} tests the current goal for |
46283 | 1854 |
counterexamples using a series of assignments for its free |
1855 |
variables; by default the first subgoal is tested, an other can be |
|
1856 |
selected explicitly using an optional goal index. Assignments can |
|
1857 |
be chosen exhausting the search space upto a given size, or using a |
|
1858 |
fixed number of random assignments in the search space, or exploring |
|
1859 |
the search space symbolically using narrowing. By default, |
|
1860 |
quickcheck uses exhaustive testing. A number of configuration |
|
1861 |
options are supported for @{command (HOL) "quickcheck"}, notably: |
|
31912
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|
1862 |
|
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|
1863 |
\begin{description} |
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|
1864 |
|
43914
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|
1865 |
\item[@{text tester}] specifies which testing approach to apply. |
46283 | 1866 |
There are three testers, @{text exhaustive}, @{text random}, and |
1867 |
@{text narrowing}. An unknown configuration option is treated as |
|
1868 |
an argument to tester, making @{text "tester ="} optional. When |
|
1869 |
multiple testers are given, these are applied in parallel. If no |
|
1870 |
tester is specified, quickcheck uses the testers that are set |
|
1871 |
active, i.e., configurations @{attribute |
|
1872 |
quickcheck_exhaustive_active}, @{attribute |
|
1873 |
quickcheck_random_active}, @{attribute |
|
1874 |
quickcheck_narrowing_active} are set to true. |
|
1875 |
||
40254 | 1876 |
\item[@{text size}] specifies the maximum size of the search space |
1877 |
for assignment values. |
|
31912
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|
1878 |
|
45758
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documenting the genuine_only option in quickcheck;
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|
1879 |
\item[@{text genuine_only}] sets quickcheck only to return genuine |
46283 | 1880 |
counterexample, but not potentially spurious counterexamples due |
1881 |
to underspecified functions. |
|
46498
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adding documentation for abort_potential option in quickcheck
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|
1882 |
|
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|
1883 |
\item[@{text abort_potential}] sets quickcheck to abort once it |
2754784e9153
adding documentation for abort_potential option in quickcheck
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|
1884 |
found a potentially spurious counterexample and to not continue |
2754784e9153
adding documentation for abort_potential option in quickcheck
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|
1885 |
to search for a further genuine counterexample. |
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|
1886 |
For this option to be effective, the @{text genuine_only} option |
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|
1887 |
must be set to false. |
47859 | 1888 |
|
42092
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|
1889 |
\item[@{text eval}] takes a term or a list of terms and evaluates |
46283 | 1890 |
these terms under the variable assignment found by quickcheck. |
48159 | 1891 |
This option is currently only supported by the default |
1892 |
(exhaustive) tester. |
|
42123 | 1893 |
|
40254 | 1894 |
\item[@{text iterations}] sets how many sets of assignments are |
1895 |
generated for each particular size. |
|
31912
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|
1896 |
|
40254 | 1897 |
\item[@{text no_assms}] specifies whether assumptions in |
1898 |
structured proofs should be ignored. |
|
35331 | 1899 |
|
47349
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|
1900 |
\item[@{text locale}] specifies how to process conjectures in |
803729c9fd4d
documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
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|
1901 |
a locale context, i.e., they can be interpreted or expanded. |
803729c9fd4d
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|
1902 |
The option is a whitespace-separated list of the two words |
803729c9fd4d
documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
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|
1903 |
@{text interpret} and @{text expand}. The list determines the |
47859 | 1904 |
order they are employed. The default setting is to first use |
47349
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documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
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|
1905 |
interpretations and then test the expanded conjecture. |
803729c9fd4d
documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
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|
1906 |
The option is only provided as attribute declaration, but not |
47859 | 1907 |
as parameter to the command. |
47349
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documenting options quickcheck_locale; adjusting IsarRef documentation of Quotient predicate; NEWS
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|
1908 |
|
40254 | 1909 |
\item[@{text timeout}] sets the time limit in seconds. |
40245
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|
1910 |
|
40254 | 1911 |
\item[@{text default_type}] sets the type(s) generally used to |
1912 |
instantiate type variables. |
|
40245
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updating documentation on quickcheck in the Isar reference
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|
1913 |
|
40254 | 1914 |
\item[@{text report}] if set quickcheck reports how many tests |
1915 |
fulfilled the preconditions. |
|
40245
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updating documentation on quickcheck in the Isar reference
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changeset
|
1916 |
|
46592
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|
1917 |
\item[@{text use_subtype}] if set quickcheck automatically lifts |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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|
1918 |
conjectures to registered subtypes if possible, and tests the |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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parents:
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changeset
|
1919 |
lifted conjecture. |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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changeset
|
1920 |
|
45766
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updating documentation about quiet and verbose options in quickcheck
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|
1921 |
\item[@{text quiet}] if set quickcheck does not output anything |
46046d8e9659
updating documentation about quiet and verbose options in quickcheck
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|
1922 |
while testing. |
47859 | 1923 |
|
46283 | 1924 |
\item[@{text verbose}] if set quickcheck informs about the current |
1925 |
size and cardinality while testing. |
|
40245
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|
1926 |
|
40254 | 1927 |
\item[@{text expect}] can be used to check if the user's |
1928 |
expectation was met (@{text no_expectation}, @{text |
|
1929 |
no_counterexample}, or @{text counterexample}). |
|
40245
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updating documentation on quickcheck in the Isar reference
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|
1930 |
|
31912
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|
1931 |
\end{description} |
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|
1932 |
|
46283 | 1933 |
These option can be given within square brackets. |
31912
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|
1934 |
|
46283 | 1935 |
\item @{command (HOL) "quickcheck_params"} changes @{command (HOL) |
1936 |
"quickcheck"} configuration options persistently. |
|
42215
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|
1937 |
|
45943
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adding documentation about the quickcheck_generator command in the IsarRef
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changeset
|
1938 |
\item @{command (HOL) "quickcheck_generator"} creates random and |
46283 | 1939 |
exhaustive value generators for a given type and operations. It |
1940 |
generates values by using the operations as if they were |
|
1941 |
constructors of that type. |
|
45943
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adding documentation about the quickcheck_generator command in the IsarRef
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changeset
|
1942 |
|
42215
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diff
changeset
|
1943 |
\item @{command (HOL) "refute"} tests the current goal for |
46283 | 1944 |
counterexamples using a reduction to SAT. The following |
1945 |
configuration options are supported: |
|
42215
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changeset
|
1946 |
|
de9d43c427ae
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changeset
|
1947 |
\begin{description} |
de9d43c427ae
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diff
changeset
|
1948 |
|
46283 | 1949 |
\item[@{text minsize}] specifies the minimum size (cardinality) of |
1950 |
the models to search for. |
|
42215
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diff
changeset
|
1951 |
|
46283 | 1952 |
\item[@{text maxsize}] specifies the maximum size (cardinality) of |
1953 |
the models to search for. Nonpositive values mean @{text "\<infinity>"}. |
|
42215
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changeset
|
1954 |
|
46283 | 1955 |
\item[@{text maxvars}] specifies the maximum number of Boolean |
1956 |
variables to use when transforming the term into a propositional |
|
1957 |
formula. Nonpositive values mean @{text "\<infinity>"}. |
|
42215
de9d43c427ae
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blanchet
parents:
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diff
changeset
|
1958 |
|
de9d43c427ae
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parents:
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diff
changeset
|
1959 |
\item[@{text satsolver}] specifies the SAT solver to use. |
de9d43c427ae
document "nitpick(_params)", "refute(_params)", "try", "sledgehammer(_params)", and "solve_direct"
blanchet
parents:
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diff
changeset
|
1960 |
|
de9d43c427ae
document "nitpick(_params)", "refute(_params)", "try", "sledgehammer(_params)", and "solve_direct"
blanchet
parents:
42123
diff
changeset
|
1961 |
\item[@{text no_assms}] specifies whether assumptions in |
de9d43c427ae
document "nitpick(_params)", "refute(_params)", "try", "sledgehammer(_params)", and "solve_direct"
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|
1962 |
structured proofs should be ignored. |
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|
1963 |
|
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|
1964 |
\item[@{text maxtime}] sets the time limit in seconds. |
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|
1965 |
|
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|
1966 |
\item[@{text expect}] can be used to check if the user's |
46283 | 1967 |
expectation was met (@{text genuine}, @{text potential}, @{text |
1968 |
none}, or @{text unknown}). |
|
42215
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|
1969 |
|
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|
1970 |
\end{description} |
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|
1971 |
|
46283 | 1972 |
These option can be given within square brackets. |
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|
1973 |
|
46283 | 1974 |
\item @{command (HOL) "refute_params"} changes @{command (HOL) |
1975 |
"refute"} configuration options persistently. |
|
42215
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|
1976 |
|
46283 | 1977 |
\item @{command (HOL) "nitpick"} tests the current goal for |
1978 |
counterexamples using a reduction to first-order relational |
|
1979 |
logic. See the Nitpick manual \cite{isabelle-nitpick} for details. |
|
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changeset
|
1980 |
|
46283 | 1981 |
\item @{command (HOL) "nitpick_params"} changes @{command (HOL) |
1982 |
"nitpick"} configuration options persistently. |
|
31912
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changeset
|
1983 |
|
46592
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changeset
|
1984 |
\item @{command (HOL) "find_unused_assms"} finds potentially superfluous |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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changeset
|
1985 |
assumptions in theorems using quickcheck. |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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changeset
|
1986 |
It takes the theory name to be checked for superfluous assumptions as |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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|
1987 |
optional argument. If not provided, it checks the current theory. |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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changeset
|
1988 |
Options to the internal quickcheck invocations can be changed with |
d5d49bd4a7b4
adding documentation about find_unused_assms command and use_subtype option in the IsarRef
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|
1989 |
common configuration declarations. |
d5d49bd4a7b4
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changeset
|
1990 |
|
31912
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|
1991 |
\end{description} |
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|
1992 |
*} |
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changeset
|
1993 |
|
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|
1994 |
|
28752 | 1995 |
section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *} |
26849 | 1996 |
|
1997 |
text {* |
|
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|
1998 |
The following tools of Isabelle/HOL support cases analysis and |
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|
1999 |
induction in unstructured tactic scripts; see also |
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|
2000 |
\secref{sec:cases-induct} for proper Isar versions of similar ideas. |
26849 | 2001 |
|
2002 |
\begin{matharray}{rcl} |
|
28761
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|
2003 |
@{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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|
2004 |
@{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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|
2005 |
@{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\ |
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|
2006 |
@{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
26849 | 2007 |
\end{matharray} |
2008 |
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|
2009 |
@{rail " |
42705 | 2010 |
@@{method (HOL) case_tac} @{syntax goal_spec}? @{syntax term} rule? |
26849 | 2011 |
; |
42705 | 2012 |
@@{method (HOL) induct_tac} @{syntax goal_spec}? (@{syntax insts} * @'and') rule? |
26849 | 2013 |
; |
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|
2014 |
@@{method (HOL) ind_cases} (@{syntax prop}+) (@'for' (@{syntax name}+))? |
26849 | 2015 |
; |
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|
2016 |
@@{command (HOL) inductive_cases} (@{syntax thmdecl}? (@{syntax prop}+) + @'and') |
26849 | 2017 |
; |
2018 |
||
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|
2019 |
rule: 'rule' ':' @{syntax thmref} |
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|
2020 |
"} |
26849 | 2021 |
|
28760
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|
2022 |
\begin{description} |
26849 | 2023 |
|
28760
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changeset
|
2024 |
\item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit |
cbc435f7b16b
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changeset
|
2025 |
to reason about inductive types. Rules are selected according to |
cbc435f7b16b
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changeset
|
2026 |
the declarations by the @{attribute cases} and @{attribute induct} |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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changeset
|
2027 |
attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL) |
cbc435f7b16b
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changeset
|
2028 |
datatype} package already takes care of this. |
27123
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changeset
|
2029 |
|
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changeset
|
2030 |
These unstructured tactics feature both goal addressing and dynamic |
26849 | 2031 |
instantiation. Note that named rule cases are \emph{not} provided |
27123
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|
2032 |
as would be by the proper @{method cases} and @{method induct} proof |
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|
2033 |
methods (see \secref{sec:cases-induct}). Unlike the @{method |
11fcdd5897dd
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|
2034 |
induct} method, @{method induct_tac} does not handle structured rule |
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|
2035 |
statements, only the compact object-logic conclusion of the subgoal |
11fcdd5897dd
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changeset
|
2036 |
being addressed. |
42123 | 2037 |
|
28760
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|
2038 |
\item @{method (HOL) ind_cases} and @{command (HOL) |
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changeset
|
2039 |
"inductive_cases"} provide an interface to the internal @{ML_text |
26860 | 2040 |
mk_cases} operation. Rules are simplified in an unrestricted |
2041 |
forward manner. |
|
26849 | 2042 |
|
2043 |
While @{method (HOL) ind_cases} is a proof method to apply the |
|
2044 |
result immediately as elimination rules, @{command (HOL) |
|
2045 |
"inductive_cases"} provides case split theorems at the theory level |
|
2046 |
for later use. The @{keyword "for"} argument of the @{method (HOL) |
|
2047 |
ind_cases} method allows to specify a list of variables that should |
|
2048 |
be generalized before applying the resulting rule. |
|
2049 |
||
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|
2050 |
\end{description} |
26849 | 2051 |
*} |
2052 |
||
2053 |
||
2054 |
section {* Executable code *} |
|
2055 |
||
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|
2056 |
text {* For validation purposes, it is often useful to \emph{execute} |
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|
2057 |
specifications. In principle, execution could be simulated by |
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changeset
|
2058 |
Isabelle's inference kernel, i.e. by a combination of resolution and |
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changeset
|
2059 |
simplification. Unfortunately, this approach is rather inefficient. |
8749742785b8
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changeset
|
2060 |
A more efficient way of executing specifications is to translate |
8749742785b8
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changeset
|
2061 |
them into a functional programming language such as ML. |
26849 | 2062 |
|
45192 | 2063 |
Isabelle provides a generic framework to support code generation |
42627
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changeset
|
2064 |
from executable specifications. Isabelle/HOL instantiates these |
45192 | 2065 |
mechanisms in a way that is amenable to end-user applications. Code |
2066 |
can be generated for functional programs (including overloading |
|
2067 |
using type classes) targeting SML \cite{SML}, OCaml \cite{OCaml}, |
|
2068 |
Haskell \cite{haskell-revised-report} and Scala |
|
42627
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changeset
|
2069 |
\cite{scala-overview-tech-report}. Conceptually, code generation is |
8749742785b8
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changeset
|
2070 |
split up in three steps: \emph{selection} of code theorems, |
8749742785b8
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changeset
|
2071 |
\emph{translation} into an abstract executable view and |
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changeset
|
2072 |
\emph{serialization} to a specific \emph{target language}. |
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changeset
|
2073 |
Inductive specifications can be executed using the predicate |
8749742785b8
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|
2074 |
compiler which operates within HOL. See \cite{isabelle-codegen} for |
8749742785b8
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|
2075 |
an introduction. |
37422 | 2076 |
|
2077 |
\begin{matharray}{rcl} |
|
2078 |
@{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2079 |
@{attribute_def (HOL) code} & : & @{text attribute} \\ |
|
2080 |
@{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2081 |
@{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2082 |
@{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
45232
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|
2083 |
@{attribute_def (HOL) code_unfold} & : & @{text attribute} \\ |
37422 | 2084 |
@{attribute_def (HOL) code_post} & : & @{text attribute} \\ |
46028
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|
2085 |
@{attribute_def (HOL) code_abbrev} & : & @{text attribute} \\ |
37422 | 2086 |
@{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
2087 |
@{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2088 |
@{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2089 |
@{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2090 |
@{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2091 |
@{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2092 |
@{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2093 |
@{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2094 |
@{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2095 |
@{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2096 |
@{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
45408
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|
2097 |
@{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"} \\ |
7156f63ce3c2
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|
2098 |
@{command_def (HOL) "code_pred"} & : & @{text "theory \<rightarrow> proof(prove)"} |
37422 | 2099 |
\end{matharray} |
2100 |
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|
2101 |
@{rail " |
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|
2102 |
@@{command (HOL) export_code} ( constexpr + ) \\ |
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|
2103 |
( ( @'in' target ( @'module_name' @{syntax string} ) ? \\ |
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|
2104 |
( @'file' ( @{syntax string} | '-' ) ) ? ( '(' args ')' ) ?) + ) ? |
37422 | 2105 |
; |
2106 |
||
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|
2107 |
const: @{syntax term} |
37422 | 2108 |
; |
2109 |
||
40711
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|
2110 |
constexpr: ( const | 'name._' | '_' ) |
37422 | 2111 |
; |
2112 |
||
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|
2113 |
typeconstructor: @{syntax nameref} |
37422 | 2114 |
; |
2115 |
||
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|
2116 |
class: @{syntax nameref} |
37422 | 2117 |
; |
2118 |
||
38814 | 2119 |
target: 'SML' | 'OCaml' | 'Haskell' | 'Scala' |
37422 | 2120 |
; |
2121 |
||
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|
2122 |
@@{attribute (HOL) code} ( 'del' | 'abstype' | 'abstract' )? |
37422 | 2123 |
; |
2124 |
||
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|
2125 |
@@{command (HOL) code_abort} ( const + ) |
37422 | 2126 |
; |
2127 |
||
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|
2128 |
@@{command (HOL) code_datatype} ( const + ) |
37422 | 2129 |
; |
2130 |
||
45232
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|
2131 |
@@{attribute (HOL) code_unfold} ( 'del' ) ? |
37422 | 2132 |
; |
2133 |
||
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|
2134 |
@@{attribute (HOL) code_post} ( 'del' ) ? |
37422 | 2135 |
; |
2136 |
||
46028
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|
2137 |
@@{attribute (HOL) code_abbrev} |
45232
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|
2138 |
; |
eb56e1774c26
updating documentation: code_inline -> code_unfold; added documentation about attribute code_unfold_post
bulwahn
parents:
45192
diff
changeset
|
2139 |
|
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2140 |
@@{command (HOL) code_thms} ( constexpr + ) ? |
37422 | 2141 |
; |
2142 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
2143 |
@@{command (HOL) code_deps} ( constexpr + ) ? |
37422 | 2144 |
; |
2145 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2146 |
@@{command (HOL) code_const} (const + @'and') \\ |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2147 |
( ( '(' target ( syntax ? + @'and' ) ')' ) + ) |
37422 | 2148 |
; |
2149 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2150 |
@@{command (HOL) code_type} (typeconstructor + @'and') \\ |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2151 |
( ( '(' target ( syntax ? + @'and' ) ')' ) + ) |
37422 | 2152 |
; |
2153 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2154 |
@@{command (HOL) code_class} (class + @'and') \\ |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2155 |
( ( '(' target \\ ( @{syntax string} ? + @'and' ) ')' ) + ) |
37422 | 2156 |
; |
2157 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2158 |
@@{command (HOL) code_instance} (( typeconstructor '::' class ) + @'and') \\ |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2159 |
( ( '(' target ( '-' ? + @'and' ) ')' ) + ) |
37422 | 2160 |
; |
2161 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2162 |
@@{command (HOL) code_reserved} target ( @{syntax string} + ) |
37422 | 2163 |
; |
2164 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2165 |
@@{command (HOL) code_monad} const const target |
37422 | 2166 |
; |
2167 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2168 |
@@{command (HOL) code_include} target ( @{syntax string} ( @{syntax string} | '-') ) |
37422 | 2169 |
; |
2170 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2171 |
@@{command (HOL) code_modulename} target ( ( @{syntax string} @{syntax string} ) + ) |
39608 | 2172 |
; |
2173 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2174 |
@@{command (HOL) code_reflect} @{syntax string} \\ |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2175 |
( @'datatypes' ( @{syntax string} '=' ( '_' | ( @{syntax string} + '|' ) + @'and' ) ) ) ? \\ |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2176 |
( @'functions' ( @{syntax string} + ) ) ? ( @'file' @{syntax string} ) ? |
37422 | 2177 |
; |
2178 |
||
45408
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2179 |
@@{command (HOL) code_pred} \\( '(' @'modes' ':' modedecl ')')? \\ const |
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2180 |
; |
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2181 |
|
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2182 |
syntax: @{syntax string} | ( @'infix' | @'infixl' | @'infixr' ) @{syntax nat} @{syntax string} |
45408
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2183 |
; |
47859 | 2184 |
|
45408
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2185 |
modedecl: (modes | ((const ':' modes) \\ |
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2186 |
(@'and' ((const ':' modes @'and') +))?)) |
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2187 |
; |
47859 | 2188 |
|
45408
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2189 |
modes: mode @'as' const |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2190 |
"} |
37422 | 2191 |
|
2192 |
\begin{description} |
|
2193 |
||
2194 |
\item @{command (HOL) "export_code"} generates code for a given list |
|
39608 | 2195 |
of constants in the specified target language(s). If no |
2196 |
serialization instruction is given, only abstract code is generated |
|
2197 |
internally. |
|
37422 | 2198 |
|
2199 |
Constants may be specified by giving them literally, referring to |
|
2200 |
all executable contants within a certain theory by giving @{text |
|
2201 |
"name.*"}, or referring to \emph{all} executable constants currently |
|
2202 |
available by giving @{text "*"}. |
|
2203 |
||
2204 |
By default, for each involved theory one corresponding name space |
|
2205 |
module is generated. Alternativly, a module name may be specified |
|
2206 |
after the @{keyword "module_name"} keyword; then \emph{all} code is |
|
2207 |
placed in this module. |
|
2208 |
||
39608 | 2209 |
For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification |
2210 |
refers to a single file; for \emph{Haskell}, it refers to a whole |
|
2211 |
directory, where code is generated in multiple files reflecting the |
|
2212 |
module hierarchy. Omitting the file specification denotes standard |
|
37749 | 2213 |
output. |
37422 | 2214 |
|
2215 |
Serializers take an optional list of arguments in parentheses. For |
|
2216 |
\emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits |
|
2217 |
explicit module signatures. |
|
42123 | 2218 |
|
39608 | 2219 |
For \emph{Haskell} a module name prefix may be given using the |
2220 |
``@{text "root:"}'' argument; ``@{text string_classes}'' adds a |
|
2221 |
``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate |
|
2222 |
datatype declaration. |
|
37422 | 2223 |
|
2224 |
\item @{attribute (HOL) code} explicitly selects (or with option |
|
38462
34d3de1254cd
formally document `code abstype` and `code abstract` attributes
haftmann
parents:
37820
diff
changeset
|
2225 |
``@{text "del"}'' deselects) a code equation for code generation. |
34d3de1254cd
formally document `code abstype` and `code abstract` attributes
haftmann
parents:
37820
diff
changeset
|
2226 |
Usually packages introducing code equations provide a reasonable |
34d3de1254cd
formally document `code abstype` and `code abstract` attributes
haftmann
parents:
37820
diff
changeset
|
2227 |
default setup for selection. Variants @{text "code abstype"} and |
34d3de1254cd
formally document `code abstype` and `code abstract` attributes
haftmann
parents:
37820
diff
changeset
|
2228 |
@{text "code abstract"} declare abstract datatype certificates or |
34d3de1254cd
formally document `code abstype` and `code abstract` attributes
haftmann
parents:
37820
diff
changeset
|
2229 |
code equations on abstract datatype representations respectively. |
37422 | 2230 |
|
2231 |
\item @{command (HOL) "code_abort"} declares constants which are not |
|
39608 | 2232 |
required to have a definition by means of code equations; if needed |
2233 |
these are implemented by program abort instead. |
|
37422 | 2234 |
|
2235 |
\item @{command (HOL) "code_datatype"} specifies a constructor set |
|
2236 |
for a logical type. |
|
2237 |
||
2238 |
\item @{command (HOL) "print_codesetup"} gives an overview on |
|
2239 |
selected code equations and code generator datatypes. |
|
2240 |
||
45232
eb56e1774c26
updating documentation: code_inline -> code_unfold; added documentation about attribute code_unfold_post
bulwahn
parents:
45192
diff
changeset
|
2241 |
\item @{attribute (HOL) code_unfold} declares (or with option |
46028
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2242 |
``@{text "del"}'' removes) theorems which during preprocessing |
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2243 |
are applied as rewrite rules to any code equation or evaluation |
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2244 |
input. |
37422 | 2245 |
|
39608 | 2246 |
\item @{attribute (HOL) code_post} declares (or with option ``@{text |
2247 |
"del"}'' removes) theorems which are applied as rewrite rules to any |
|
2248 |
result of an evaluation. |
|
37422 | 2249 |
|
46028
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2250 |
\item @{attribute (HOL) code_abbrev} declares equations which are |
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2251 |
applied as rewrite rules to any result of an evaluation and |
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2252 |
symmetrically during preprocessing to any code equation or evaluation |
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2253 |
input. |
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45944
diff
changeset
|
2254 |
|
39608 | 2255 |
\item @{command (HOL) "print_codeproc"} prints the setup of the code |
2256 |
generator preprocessor. |
|
37422 | 2257 |
|
2258 |
\item @{command (HOL) "code_thms"} prints a list of theorems |
|
2259 |
representing the corresponding program containing all given |
|
2260 |
constants after preprocessing. |
|
2261 |
||
2262 |
\item @{command (HOL) "code_deps"} visualizes dependencies of |
|
2263 |
theorems representing the corresponding program containing all given |
|
2264 |
constants after preprocessing. |
|
2265 |
||
2266 |
\item @{command (HOL) "code_const"} associates a list of constants |
|
2267 |
with target-specific serializations; omitting a serialization |
|
2268 |
deletes an existing serialization. |
|
2269 |
||
2270 |
\item @{command (HOL) "code_type"} associates a list of type |
|
2271 |
constructors with target-specific serializations; omitting a |
|
2272 |
serialization deletes an existing serialization. |
|
2273 |
||
2274 |
\item @{command (HOL) "code_class"} associates a list of classes |
|
2275 |
with target-specific class names; omitting a serialization deletes |
|
2276 |
an existing serialization. This applies only to \emph{Haskell}. |
|
2277 |
||
2278 |
\item @{command (HOL) "code_instance"} declares a list of type |
|
2279 |
constructor / class instance relations as ``already present'' for a |
|
2280 |
given target. Omitting a ``@{text "-"}'' deletes an existing |
|
2281 |
``already present'' declaration. This applies only to |
|
2282 |
\emph{Haskell}. |
|
2283 |
||
2284 |
\item @{command (HOL) "code_reserved"} declares a list of names as |
|
2285 |
reserved for a given target, preventing it to be shadowed by any |
|
2286 |
generated code. |
|
2287 |
||
2288 |
\item @{command (HOL) "code_monad"} provides an auxiliary mechanism |
|
2289 |
to generate monadic code for Haskell. |
|
2290 |
||
2291 |
\item @{command (HOL) "code_include"} adds arbitrary named content |
|
2292 |
(``include'') to generated code. A ``@{text "-"}'' as last argument |
|
2293 |
will remove an already added ``include''. |
|
2294 |
||
2295 |
\item @{command (HOL) "code_modulename"} declares aliasings from one |
|
2296 |
module name onto another. |
|
2297 |
||
39608 | 2298 |
\item @{command (HOL) "code_reflect"} without a ``@{text "file"}'' |
2299 |
argument compiles code into the system runtime environment and |
|
2300 |
modifies the code generator setup that future invocations of system |
|
2301 |
runtime code generation referring to one of the ``@{text |
|
47859 | 2302 |
"datatypes"}'' or ``@{text "functions"}'' entities use these |
2303 |
precompiled entities. With a ``@{text "file"}'' argument, the |
|
2304 |
corresponding code is generated into that specified file without |
|
2305 |
modifying the code generator setup. |
|
2306 |
||
2307 |
\item @{command (HOL) "code_pred"} creates code equations for a |
|
2308 |
predicate given a set of introduction rules. Optional mode |
|
2309 |
annotations determine which arguments are supposed to be input or |
|
2310 |
output. If alternative introduction rules are declared, one must |
|
2311 |
prove a corresponding elimination rule. |
|
45408
7156f63ce3c2
adding a minimal documentation about the code_pred command to the isar reference
bulwahn
parents:
45232
diff
changeset
|
2312 |
|
37422 | 2313 |
\end{description} |
42627
8749742785b8
moved material about old codegen to isar-ref manual;
wenzelm
parents:
42626
diff
changeset
|
2314 |
*} |
37422 | 2315 |
|
42627
8749742785b8
moved material about old codegen to isar-ref manual;
wenzelm
parents:
42626
diff
changeset
|
2316 |
|
27045 | 2317 |
section {* Definition by specification \label{sec:hol-specification} *} |
2318 |
||
2319 |
text {* |
|
2320 |
\begin{matharray}{rcl} |
|
28761
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
2321 |
@{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
2322 |
@{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
27045 | 2323 |
\end{matharray} |
2324 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2325 |
@{rail " |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2326 |
(@@{command (HOL) specification} | @@{command (HOL) ax_specification}) |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2327 |
'(' (decl +) ')' \\ (@{syntax thmdecl}? @{syntax prop} +) |
27045 | 2328 |
; |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2329 |
decl: ((@{syntax name} ':')? @{syntax term} '(' @'overloaded' ')'?) |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
42215
diff
changeset
|
2330 |
"} |
27045 | 2331 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2332 |
\begin{description} |
27045 | 2333 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2334 |
\item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a |
27045 | 2335 |
goal stating the existence of terms with the properties specified to |
2336 |
hold for the constants given in @{text decls}. After finishing the |
|
2337 |
proof, the theory will be augmented with definitions for the given |
|
2338 |
constants, as well as with theorems stating the properties for these |
|
2339 |
constants. |
|
2340 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2341 |
\item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2342 |
a goal stating the existence of terms with the properties specified |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2343 |
to hold for the constants given in @{text decls}. After finishing |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2344 |
the proof, the theory will be augmented with axioms expressing the |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2345 |
properties given in the first place. |
27045 | 2346 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2347 |
\item @{text decl} declares a constant to be defined by the |
27045 | 2348 |
specification given. The definition for the constant @{text c} is |
2349 |
bound to the name @{text c_def} unless a theorem name is given in |
|
2350 |
the declaration. Overloaded constants should be declared as such. |
|
2351 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
2352 |
\end{description} |
27045 | 2353 |
|
2354 |
Whether to use @{command (HOL) "specification"} or @{command (HOL) |
|
2355 |
"ax_specification"} is to some extent a matter of style. @{command |
|
2356 |
(HOL) "specification"} introduces no new axioms, and so by |
|
2357 |
construction cannot introduce inconsistencies, whereas @{command |
|
2358 |
(HOL) "ax_specification"} does introduce axioms, but only after the |
|
2359 |
user has explicitly proven it to be safe. A practical issue must be |
|
2360 |
considered, though: After introducing two constants with the same |
|
2361 |
properties using @{command (HOL) "specification"}, one can prove |
|
2362 |
that the two constants are, in fact, equal. If this might be a |
|
2363 |
problem, one should use @{command (HOL) "ax_specification"}. |
|
2364 |
*} |
|
2365 |
||
26840 | 2366 |
end |