--- a/src/Doc/Implementation/Logic.thy Wed Dec 16 16:31:36 2015 +0100
+++ b/src/Doc/Implementation/Logic.thy Wed Dec 16 17:28:49 2015 +0100
@@ -7,110 +7,97 @@
chapter \<open>Primitive logic \label{ch:logic}\<close>
text \<open>
- The logical foundations of Isabelle/Isar are that of the Pure logic,
- which has been introduced as a Natural Deduction framework in
- @{cite paulson700}. This is essentially the same logic as ``\<open>\<lambda>HOL\<close>'' in the more abstract setting of Pure Type Systems (PTS)
- @{cite "Barendregt-Geuvers:2001"}, although there are some key
- differences in the specific treatment of simple types in
- Isabelle/Pure.
+ The logical foundations of Isabelle/Isar are that of the Pure logic, which
+ has been introduced as a Natural Deduction framework in @{cite paulson700}.
+ This is essentially the same logic as ``\<open>\<lambda>HOL\<close>'' in the more abstract
+ setting of Pure Type Systems (PTS) @{cite "Barendregt-Geuvers:2001"},
+ although there are some key differences in the specific treatment of simple
+ types in Isabelle/Pure.
- Following type-theoretic parlance, the Pure logic consists of three
- levels of \<open>\<lambda>\<close>-calculus with corresponding arrows, \<open>\<Rightarrow>\<close> for syntactic function space (terms depending on terms), \<open>\<And>\<close> for universal quantification (proofs depending on terms), and
- \<open>\<Longrightarrow>\<close> for implication (proofs depending on proofs).
+ Following type-theoretic parlance, the Pure logic consists of three levels
+ of \<open>\<lambda>\<close>-calculus with corresponding arrows, \<open>\<Rightarrow>\<close> for syntactic function space
+ (terms depending on terms), \<open>\<And>\<close> for universal quantification (proofs
+ depending on terms), and \<open>\<Longrightarrow>\<close> for implication (proofs depending on proofs).
Derivations are relative to a logical theory, which declares type
- constructors, constants, and axioms. Theory declarations support
- schematic polymorphism, which is strictly speaking outside the
- logic.\<^footnote>\<open>This is the deeper logical reason, why the theory
- context \<open>\<Theta>\<close> is separate from the proof context \<open>\<Gamma>\<close>
- of the core calculus: type constructors, term constants, and facts
- (proof constants) may involve arbitrary type schemes, but the type
- of a locally fixed term parameter is also fixed!\<close>
+ constructors, constants, and axioms. Theory declarations support schematic
+ polymorphism, which is strictly speaking outside the logic.\<^footnote>\<open>This is the
+ deeper logical reason, why the theory context \<open>\<Theta>\<close> is separate from the proof
+ context \<open>\<Gamma>\<close> of the core calculus: type constructors, term constants, and
+ facts (proof constants) may involve arbitrary type schemes, but the type of
+ a locally fixed term parameter is also fixed!\<close>
\<close>
section \<open>Types \label{sec:types}\<close>
text \<open>
- The language of types is an uninterpreted order-sorted first-order
- algebra; types are qualified by ordered type classes.
+ The language of types is an uninterpreted order-sorted first-order algebra;
+ types are qualified by ordered type classes.
\<^medskip>
- A \<^emph>\<open>type class\<close> is an abstract syntactic entity
- declared in the theory context. The \<^emph>\<open>subclass relation\<close> \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close> is specified by stating an acyclic
- generating relation; the transitive closure is maintained
- internally. The resulting relation is an ordering: reflexive,
- transitive, and antisymmetric.
+ A \<^emph>\<open>type class\<close> is an abstract syntactic entity declared in the theory
+ context. The \<^emph>\<open>subclass relation\<close> \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close> is specified by stating an
+ acyclic generating relation; the transitive closure is maintained
+ internally. The resulting relation is an ordering: reflexive, transitive,
+ and antisymmetric.
- A \<^emph>\<open>sort\<close> is a list of type classes written as \<open>s = {c\<^sub>1,
- \<dots>, c\<^sub>m}\<close>, it represents symbolic intersection. Notationally, the
- curly braces are omitted for singleton intersections, i.e.\ any
- class \<open>c\<close> may be read as a sort \<open>{c}\<close>. The ordering
- on type classes is extended to sorts according to the meaning of
- intersections: \<open>{c\<^sub>1, \<dots> c\<^sub>m} \<subseteq> {d\<^sub>1, \<dots>, d\<^sub>n}\<close> iff \<open>\<forall>j. \<exists>i. c\<^sub>i \<subseteq> d\<^sub>j\<close>. The empty intersection \<open>{}\<close> refers to
- the universal sort, which is the largest element wrt.\ the sort
- order. Thus \<open>{}\<close> represents the ``full sort'', not the
- empty one! The intersection of all (finitely many) classes declared
- in the current theory is the least element wrt.\ the sort ordering.
+ A \<^emph>\<open>sort\<close> is a list of type classes written as \<open>s = {c\<^sub>1, \<dots>, c\<^sub>m}\<close>, it
+ represents symbolic intersection. Notationally, the curly braces are omitted
+ for singleton intersections, i.e.\ any class \<open>c\<close> may be read as a sort
+ \<open>{c}\<close>. The ordering on type classes is extended to sorts according to the
+ meaning of intersections: \<open>{c\<^sub>1, \<dots> c\<^sub>m} \<subseteq> {d\<^sub>1, \<dots>, d\<^sub>n}\<close> iff \<open>\<forall>j. \<exists>i. c\<^sub>i \<subseteq>
+ d\<^sub>j\<close>. The empty intersection \<open>{}\<close> refers to the universal sort, which is the
+ largest element wrt.\ the sort order. Thus \<open>{}\<close> represents the ``full
+ sort'', not the empty one! The intersection of all (finitely many) classes
+ declared in the current theory is the least element wrt.\ the sort ordering.
\<^medskip>
- A \<^emph>\<open>fixed type variable\<close> is a pair of a basic name
- (starting with a \<open>'\<close> character) and a sort constraint, e.g.\
- \<open>('a, s)\<close> which is usually printed as \<open>\<alpha>\<^sub>s\<close>.
- A \<^emph>\<open>schematic type variable\<close> is a pair of an indexname and a
- sort constraint, e.g.\ \<open>(('a, 0), s)\<close> which is usually
- printed as \<open>?\<alpha>\<^sub>s\<close>.
+ A \<^emph>\<open>fixed type variable\<close> is a pair of a basic name (starting with a \<open>'\<close>
+ character) and a sort constraint, e.g.\ \<open>('a, s)\<close> which is usually printed
+ as \<open>\<alpha>\<^sub>s\<close>. A \<^emph>\<open>schematic type variable\<close> is a pair of an indexname and a sort
+ constraint, e.g.\ \<open>(('a, 0), s)\<close> which is usually printed as \<open>?\<alpha>\<^sub>s\<close>.
- Note that \<^emph>\<open>all\<close> syntactic components contribute to the identity
- of type variables: basic name, index, and sort constraint. The core
- logic handles type variables with the same name but different sorts
- as different, although the type-inference layer (which is outside
- the core) rejects anything like that.
+ Note that \<^emph>\<open>all\<close> syntactic components contribute to the identity of type
+ variables: basic name, index, and sort constraint. The core logic handles
+ type variables with the same name but different sorts as different, although
+ the type-inference layer (which is outside the core) rejects anything like
+ that.
- A \<^emph>\<open>type constructor\<close> \<open>\<kappa>\<close> is a \<open>k\<close>-ary operator
- on types declared in the theory. Type constructor application is
- written postfix as \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>k)\<kappa>\<close>. For
- \<open>k = 0\<close> the argument tuple is omitted, e.g.\ \<open>prop\<close>
- instead of \<open>()prop\<close>. For \<open>k = 1\<close> the parentheses
- are omitted, e.g.\ \<open>\<alpha> list\<close> instead of \<open>(\<alpha>)list\<close>.
- Further notation is provided for specific constructors, notably the
- right-associative infix \<open>\<alpha> \<Rightarrow> \<beta>\<close> instead of \<open>(\<alpha>,
- \<beta>)fun\<close>.
+ A \<^emph>\<open>type constructor\<close> \<open>\<kappa>\<close> is a \<open>k\<close>-ary operator on types declared in the
+ theory. Type constructor application is written postfix as \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>k)\<kappa>\<close>.
+ For \<open>k = 0\<close> the argument tuple is omitted, e.g.\ \<open>prop\<close> instead of \<open>()prop\<close>.
+ For \<open>k = 1\<close> the parentheses are omitted, e.g.\ \<open>\<alpha> list\<close> instead of
+ \<open>(\<alpha>)list\<close>. Further notation is provided for specific constructors, notably
+ the right-associative infix \<open>\<alpha> \<Rightarrow> \<beta>\<close> instead of \<open>(\<alpha>, \<beta>)fun\<close>.
- The logical category \<^emph>\<open>type\<close> is defined inductively over type
- variables and type constructors as follows: \<open>\<tau> = \<alpha>\<^sub>s | ?\<alpha>\<^sub>s |
- (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close>.
+ The logical category \<^emph>\<open>type\<close> is defined inductively over type variables and
+ type constructors as follows: \<open>\<tau> = \<alpha>\<^sub>s | ?\<alpha>\<^sub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close>.
- A \<^emph>\<open>type abbreviation\<close> is a syntactic definition \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close> of an arbitrary type expression \<open>\<tau>\<close> over
- variables \<open>\<^vec>\<alpha>\<close>. Type abbreviations appear as type
- constructors in the syntax, but are expanded before entering the
- logical core.
+ A \<^emph>\<open>type abbreviation\<close> is a syntactic definition \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close> of an
+ arbitrary type expression \<open>\<tau>\<close> over variables \<open>\<^vec>\<alpha>\<close>. Type abbreviations
+ appear as type constructors in the syntax, but are expanded before entering
+ the logical core.
- A \<^emph>\<open>type arity\<close> declares the image behavior of a type
- constructor wrt.\ the algebra of sorts: \<open>\<kappa> :: (s\<^sub>1, \<dots>,
- s\<^sub>k)s\<close> means that \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close> is
- of sort \<open>s\<close> if every argument type \<open>\<tau>\<^sub>i\<close> is
- of sort \<open>s\<^sub>i\<close>. Arity declarations are implicitly
- completed, i.e.\ \<open>\<kappa> :: (\<^vec>s)c\<close> entails \<open>\<kappa> ::
+ A \<^emph>\<open>type arity\<close> declares the image behavior of a type constructor wrt.\ the
+ algebra of sorts: \<open>\<kappa> :: (s\<^sub>1, \<dots>, s\<^sub>k)s\<close> means that \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close> is of
+ sort \<open>s\<close> if every argument type \<open>\<tau>\<^sub>i\<close> is of sort \<open>s\<^sub>i\<close>. Arity declarations
+ are implicitly completed, i.e.\ \<open>\<kappa> :: (\<^vec>s)c\<close> entails \<open>\<kappa> ::
(\<^vec>s)c'\<close> for any \<open>c' \<supseteq> c\<close>.
\<^medskip>
- The sort algebra is always maintained as \<^emph>\<open>coregular\<close>,
- which means that type arities are consistent with the subclass
- relation: for any type constructor \<open>\<kappa>\<close>, and classes \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>, and arities \<open>\<kappa> ::
- (\<^vec>s\<^sub>1)c\<^sub>1\<close> and \<open>\<kappa> ::
- (\<^vec>s\<^sub>2)c\<^sub>2\<close> holds \<open>\<^vec>s\<^sub>1 \<subseteq>
- \<^vec>s\<^sub>2\<close> component-wise.
+ The sort algebra is always maintained as \<^emph>\<open>coregular\<close>, which means that type
+ arities are consistent with the subclass relation: for any type constructor
+ \<open>\<kappa>\<close>, and classes \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>, and arities \<open>\<kappa> :: (\<^vec>s\<^sub>1)c\<^sub>1\<close> and \<open>\<kappa> ::
+ (\<^vec>s\<^sub>2)c\<^sub>2\<close> holds \<open>\<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2\<close> component-wise.
The key property of a coregular order-sorted algebra is that sort
constraints can be solved in a most general fashion: for each type
- constructor \<open>\<kappa>\<close> and sort \<open>s\<close> there is a most general
- vector of argument sorts \<open>(s\<^sub>1, \<dots>, s\<^sub>k)\<close> such
- that a type scheme \<open>(\<alpha>\<^bsub>s\<^sub>1\<^esub>, \<dots>,
- \<alpha>\<^bsub>s\<^sub>k\<^esub>)\<kappa>\<close> is of sort \<open>s\<close>.
- Consequently, type unification has most general solutions (modulo
- equivalence of sorts), so type-inference produces primary types as
- expected @{cite "nipkow-prehofer"}.
+ constructor \<open>\<kappa>\<close> and sort \<open>s\<close> there is a most general vector of argument
+ sorts \<open>(s\<^sub>1, \<dots>, s\<^sub>k)\<close> such that a type scheme \<open>(\<alpha>\<^bsub>s\<^sub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^sub>k\<^esub>)\<kappa>\<close> is of
+ sort \<open>s\<close>. Consequently, type unification has most general solutions (modulo
+ equivalence of sorts), so type-inference produces primary types as expected
+ @{cite "nipkow-prehofer"}.
\<close>
text %mlref \<open>
@@ -135,48 +122,42 @@
\<^descr> Type @{ML_type class} represents type classes.
- \<^descr> Type @{ML_type sort} represents sorts, i.e.\ finite
- intersections of classes. The empty list @{ML "[]: sort"} refers to
- the empty class intersection, i.e.\ the ``full sort''.
+ \<^descr> Type @{ML_type sort} represents sorts, i.e.\ finite intersections of
+ classes. The empty list @{ML "[]: sort"} refers to the empty class
+ intersection, i.e.\ the ``full sort''.
- \<^descr> Type @{ML_type arity} represents type arities. A triple
- \<open>(\<kappa>, \<^vec>s, s) : arity\<close> represents \<open>\<kappa> ::
- (\<^vec>s)s\<close> as described above.
+ \<^descr> Type @{ML_type arity} represents type arities. A triple \<open>(\<kappa>, \<^vec>s, s)
+ : arity\<close> represents \<open>\<kappa> :: (\<^vec>s)s\<close> as described above.
- \<^descr> Type @{ML_type typ} represents types; this is a datatype with
- constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
+ \<^descr> Type @{ML_type typ} represents types; this is a datatype with constructors
+ @{ML TFree}, @{ML TVar}, @{ML Type}.
- \<^descr> @{ML Term.map_atyps}~\<open>f \<tau>\<close> applies the mapping \<open>f\<close> to all atomic types (@{ML TFree}, @{ML TVar}) occurring in
- \<open>\<tau>\<close>.
+ \<^descr> @{ML Term.map_atyps}~\<open>f \<tau>\<close> applies the mapping \<open>f\<close> to all atomic types
+ (@{ML TFree}, @{ML TVar}) occurring in \<open>\<tau>\<close>.
- \<^descr> @{ML Term.fold_atyps}~\<open>f \<tau>\<close> iterates the operation
- \<open>f\<close> over all occurrences of atomic types (@{ML TFree}, @{ML
- TVar}) in \<open>\<tau>\<close>; the type structure is traversed from left to
- right.
+ \<^descr> @{ML Term.fold_atyps}~\<open>f \<tau>\<close> iterates the operation \<open>f\<close> over all
+ occurrences of atomic types (@{ML TFree}, @{ML TVar}) in \<open>\<tau>\<close>; the type
+ structure is traversed from left to right.
- \<^descr> @{ML Sign.subsort}~\<open>thy (s\<^sub>1, s\<^sub>2)\<close>
- tests the subsort relation \<open>s\<^sub>1 \<subseteq> s\<^sub>2\<close>.
+ \<^descr> @{ML Sign.subsort}~\<open>thy (s\<^sub>1, s\<^sub>2)\<close> tests the subsort relation \<open>s\<^sub>1 \<subseteq>
+ s\<^sub>2\<close>.
- \<^descr> @{ML Sign.of_sort}~\<open>thy (\<tau>, s)\<close> tests whether type
- \<open>\<tau>\<close> is of sort \<open>s\<close>.
+ \<^descr> @{ML Sign.of_sort}~\<open>thy (\<tau>, s)\<close> tests whether type \<open>\<tau>\<close> is of sort \<open>s\<close>.
- \<^descr> @{ML Sign.add_type}~\<open>ctxt (\<kappa>, k, mx)\<close> declares a
- new type constructors \<open>\<kappa>\<close> with \<open>k\<close> arguments and
- optional mixfix syntax.
+ \<^descr> @{ML Sign.add_type}~\<open>ctxt (\<kappa>, k, mx)\<close> declares a new type constructors \<open>\<kappa>\<close>
+ with \<open>k\<close> arguments and optional mixfix syntax.
- \<^descr> @{ML Sign.add_type_abbrev}~\<open>ctxt (\<kappa>, \<^vec>\<alpha>, \<tau>)\<close>
- defines a new type abbreviation \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close>.
+ \<^descr> @{ML Sign.add_type_abbrev}~\<open>ctxt (\<kappa>, \<^vec>\<alpha>, \<tau>)\<close> defines a new type
+ abbreviation \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close>.
- \<^descr> @{ML Sign.primitive_class}~\<open>(c, [c\<^sub>1, \<dots>,
- c\<^sub>n])\<close> declares a new class \<open>c\<close>, together with class
- relations \<open>c \<subseteq> c\<^sub>i\<close>, for \<open>i = 1, \<dots>, n\<close>.
+ \<^descr> @{ML Sign.primitive_class}~\<open>(c, [c\<^sub>1, \<dots>, c\<^sub>n])\<close> declares a new class \<open>c\<close>,
+ together with class relations \<open>c \<subseteq> c\<^sub>i\<close>, for \<open>i = 1, \<dots>, n\<close>.
- \<^descr> @{ML Sign.primitive_classrel}~\<open>(c\<^sub>1,
- c\<^sub>2)\<close> declares the class relation \<open>c\<^sub>1 \<subseteq>
- c\<^sub>2\<close>.
+ \<^descr> @{ML Sign.primitive_classrel}~\<open>(c\<^sub>1, c\<^sub>2)\<close> declares the class relation
+ \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>.
- \<^descr> @{ML Sign.primitive_arity}~\<open>(\<kappa>, \<^vec>s, s)\<close> declares
- the arity \<open>\<kappa> :: (\<^vec>s)s\<close>.
+ \<^descr> @{ML Sign.primitive_arity}~\<open>(\<kappa>, \<^vec>s, s)\<close> declares the arity \<open>\<kappa> ::
+ (\<^vec>s)s\<close>.
\<close>
text %mlantiq \<open>
@@ -201,92 +182,84 @@
@@{ML_antiquotation typ} type
\<close>}
- \<^descr> \<open>@{class c}\<close> inlines the internalized class \<open>c\<close> --- as @{ML_type string} literal.
-
- \<^descr> \<open>@{sort s}\<close> inlines the internalized sort \<open>s\<close>
- --- as @{ML_type "string list"} literal.
-
- \<^descr> \<open>@{type_name c}\<close> inlines the internalized type
- constructor \<open>c\<close> --- as @{ML_type string} literal.
-
- \<^descr> \<open>@{type_abbrev c}\<close> inlines the internalized type
- abbreviation \<open>c\<close> --- as @{ML_type string} literal.
-
- \<^descr> \<open>@{nonterminal c}\<close> inlines the internalized syntactic
- type~/ grammar nonterminal \<open>c\<close> --- as @{ML_type string}
+ \<^descr> \<open>@{class c}\<close> inlines the internalized class \<open>c\<close> --- as @{ML_type string}
literal.
- \<^descr> \<open>@{typ \<tau>}\<close> inlines the internalized type \<open>\<tau>\<close>
- --- as constructor term for datatype @{ML_type typ}.
+ \<^descr> \<open>@{sort s}\<close> inlines the internalized sort \<open>s\<close> --- as @{ML_type "string
+ list"} literal.
+
+ \<^descr> \<open>@{type_name c}\<close> inlines the internalized type constructor \<open>c\<close> --- as
+ @{ML_type string} literal.
+
+ \<^descr> \<open>@{type_abbrev c}\<close> inlines the internalized type abbreviation \<open>c\<close> --- as
+ @{ML_type string} literal.
+
+ \<^descr> \<open>@{nonterminal c}\<close> inlines the internalized syntactic type~/ grammar
+ nonterminal \<open>c\<close> --- as @{ML_type string} literal.
+
+ \<^descr> \<open>@{typ \<tau>}\<close> inlines the internalized type \<open>\<tau>\<close> --- as constructor term for
+ datatype @{ML_type typ}.
\<close>
section \<open>Terms \label{sec:terms}\<close>
text \<open>
- The language of terms is that of simply-typed \<open>\<lambda>\<close>-calculus
- with de-Bruijn indices for bound variables (cf.\ @{cite debruijn72}
- or @{cite "paulson-ml2"}), with the types being determined by the
- corresponding binders. In contrast, free variables and constants
- have an explicit name and type in each occurrence.
+ The language of terms is that of simply-typed \<open>\<lambda>\<close>-calculus with de-Bruijn
+ indices for bound variables (cf.\ @{cite debruijn72} or @{cite
+ "paulson-ml2"}), with the types being determined by the corresponding
+ binders. In contrast, free variables and constants have an explicit name and
+ type in each occurrence.
\<^medskip>
- A \<^emph>\<open>bound variable\<close> is a natural number \<open>b\<close>,
- which accounts for the number of intermediate binders between the
- variable occurrence in the body and its binding position. For
- example, the de-Bruijn term \<open>\<lambda>\<^bsub>bool\<^esub>. \<lambda>\<^bsub>bool\<^esub>. 1 \<and> 0\<close> would
- correspond to \<open>\<lambda>x\<^bsub>bool\<^esub>. \<lambda>y\<^bsub>bool\<^esub>. x \<and> y\<close> in a named
- representation. Note that a bound variable may be represented by
- different de-Bruijn indices at different occurrences, depending on
- the nesting of abstractions.
+ A \<^emph>\<open>bound variable\<close> is a natural number \<open>b\<close>, which accounts for the number
+ of intermediate binders between the variable occurrence in the body and its
+ binding position. For example, the de-Bruijn term \<open>\<lambda>\<^bsub>bool\<^esub>. \<lambda>\<^bsub>bool\<^esub>. 1 \<and> 0\<close>
+ would correspond to \<open>\<lambda>x\<^bsub>bool\<^esub>. \<lambda>y\<^bsub>bool\<^esub>. x \<and> y\<close> in a named representation.
+ Note that a bound variable may be represented by different de-Bruijn indices
+ at different occurrences, depending on the nesting of abstractions.
- A \<^emph>\<open>loose variable\<close> is a bound variable that is outside the
- scope of local binders. The types (and names) for loose variables
- can be managed as a separate context, that is maintained as a stack
- of hypothetical binders. The core logic operates on closed terms,
- without any loose variables.
+ A \<^emph>\<open>loose variable\<close> is a bound variable that is outside the scope of local
+ binders. The types (and names) for loose variables can be managed as a
+ separate context, that is maintained as a stack of hypothetical binders. The
+ core logic operates on closed terms, without any loose variables.
- A \<^emph>\<open>fixed variable\<close> is a pair of a basic name and a type, e.g.\
- \<open>(x, \<tau>)\<close> which is usually printed \<open>x\<^sub>\<tau>\<close> here. A
- \<^emph>\<open>schematic variable\<close> is a pair of an indexname and a type,
- e.g.\ \<open>((x, 0), \<tau>)\<close> which is likewise printed as \<open>?x\<^sub>\<tau>\<close>.
+ A \<^emph>\<open>fixed variable\<close> is a pair of a basic name and a type, e.g.\ \<open>(x, \<tau>)\<close>
+ which is usually printed \<open>x\<^sub>\<tau>\<close> here. A \<^emph>\<open>schematic variable\<close> is a pair of an
+ indexname and a type, e.g.\ \<open>((x, 0), \<tau>)\<close> which is likewise printed as
+ \<open>?x\<^sub>\<tau>\<close>.
\<^medskip>
- A \<^emph>\<open>constant\<close> is a pair of a basic name and a type,
- e.g.\ \<open>(c, \<tau>)\<close> which is usually printed as \<open>c\<^sub>\<tau>\<close>
- here. Constants are declared in the context as polymorphic families
- \<open>c :: \<sigma>\<close>, meaning that all substitution instances \<open>c\<^sub>\<tau>\<close> for \<open>\<tau> = \<sigma>\<vartheta>\<close> are valid.
+ A \<^emph>\<open>constant\<close> is a pair of a basic name and a type, e.g.\ \<open>(c, \<tau>)\<close> which is
+ usually printed as \<open>c\<^sub>\<tau>\<close> here. Constants are declared in the context as
+ polymorphic families \<open>c :: \<sigma>\<close>, meaning that all substitution instances \<open>c\<^sub>\<tau>\<close>
+ for \<open>\<tau> = \<sigma>\<vartheta>\<close> are valid.
- The vector of \<^emph>\<open>type arguments\<close> of constant \<open>c\<^sub>\<tau>\<close> wrt.\
- the declaration \<open>c :: \<sigma>\<close> is defined as the codomain of the
- matcher \<open>\<vartheta> = {?\<alpha>\<^sub>1 \<mapsto> \<tau>\<^sub>1, \<dots>, ?\<alpha>\<^sub>n \<mapsto> \<tau>\<^sub>n}\<close> presented in
- canonical order \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close>, corresponding to the
- left-to-right occurrences of the \<open>\<alpha>\<^sub>i\<close> in \<open>\<sigma>\<close>.
- Within a given theory context, there is a one-to-one correspondence
- between any constant \<open>c\<^sub>\<tau>\<close> and the application \<open>c(\<tau>\<^sub>1,
- \<dots>, \<tau>\<^sub>n)\<close> of its type arguments. For example, with \<open>plus :: \<alpha>
- \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>\<close>, the instance \<open>plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>\<close> corresponds to
+ The vector of \<^emph>\<open>type arguments\<close> of constant \<open>c\<^sub>\<tau>\<close> wrt.\ the declaration \<open>c
+ :: \<sigma>\<close> is defined as the codomain of the matcher \<open>\<vartheta> = {?\<alpha>\<^sub>1 \<mapsto> \<tau>\<^sub>1,
+ \<dots>, ?\<alpha>\<^sub>n \<mapsto> \<tau>\<^sub>n}\<close> presented in canonical order \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close>, corresponding
+ to the left-to-right occurrences of the \<open>\<alpha>\<^sub>i\<close> in \<open>\<sigma>\<close>. Within a given theory
+ context, there is a one-to-one correspondence between any constant \<open>c\<^sub>\<tau>\<close> and
+ the application \<open>c(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close> of its type arguments. For example, with
+ \<open>plus :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>\<close>, the instance \<open>plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>\<close> corresponds to
\<open>plus(nat)\<close>.
- Constant declarations \<open>c :: \<sigma>\<close> may contain sort constraints
- for type variables in \<open>\<sigma>\<close>. These are observed by
- type-inference as expected, but \<^emph>\<open>ignored\<close> by the core logic.
- This means the primitive logic is able to reason with instances of
- polymorphic constants that the user-level type-checker would reject
- due to violation of type class restrictions.
+ Constant declarations \<open>c :: \<sigma>\<close> may contain sort constraints for type
+ variables in \<open>\<sigma>\<close>. These are observed by type-inference as expected, but
+ \<^emph>\<open>ignored\<close> by the core logic. This means the primitive logic is able to
+ reason with instances of polymorphic constants that the user-level
+ type-checker would reject due to violation of type class restrictions.
\<^medskip>
- An \<^emph>\<open>atomic term\<close> is either a variable or constant.
- The logical category \<^emph>\<open>term\<close> is defined inductively over atomic
- terms, with abstraction and application as follows: \<open>t = b |
- x\<^sub>\<tau> | ?x\<^sub>\<tau> | c\<^sub>\<tau> | \<lambda>\<^sub>\<tau>. t | t\<^sub>1 t\<^sub>2\<close>. Parsing and printing takes care of
- converting between an external representation with named bound
- variables. Subsequently, we shall use the latter notation instead
- of internal de-Bruijn representation.
+ An \<^emph>\<open>atomic term\<close> is either a variable or constant. The logical category
+ \<^emph>\<open>term\<close> is defined inductively over atomic terms, with abstraction and
+ application as follows: \<open>t = b | x\<^sub>\<tau> | ?x\<^sub>\<tau> | c\<^sub>\<tau> | \<lambda>\<^sub>\<tau>. t | t\<^sub>1 t\<^sub>2\<close>.
+ Parsing and printing takes care of converting between an external
+ representation with named bound variables. Subsequently, we shall use the
+ latter notation instead of internal de-Bruijn representation.
- The inductive relation \<open>t :: \<tau>\<close> assigns a (unique) type to a
- term according to the structure of atomic terms, abstractions, and
- applications:
+ The inductive relation \<open>t :: \<tau>\<close> assigns a (unique) type to a term according
+ to the structure of atomic terms, abstractions, and applications:
\[
\infer{\<open>a\<^sub>\<tau> :: \<tau>\<close>}{}
\qquad
@@ -296,47 +269,46 @@
\]
A \<^emph>\<open>well-typed term\<close> is a term that can be typed according to these rules.
- Typing information can be omitted: type-inference is able to
- reconstruct the most general type of a raw term, while assigning
- most general types to all of its variables and constants.
- Type-inference depends on a context of type constraints for fixed
- variables, and declarations for polymorphic constants.
+ Typing information can be omitted: type-inference is able to reconstruct the
+ most general type of a raw term, while assigning most general types to all
+ of its variables and constants. Type-inference depends on a context of type
+ constraints for fixed variables, and declarations for polymorphic constants.
The identity of atomic terms consists both of the name and the type
- component. This means that different variables \<open>x\<^bsub>\<tau>\<^sub>1\<^esub>\<close> and \<open>x\<^bsub>\<tau>\<^sub>2\<^esub>\<close> may become the same after
- type instantiation. Type-inference rejects variables of the same
- name, but different types. In contrast, mixed instances of
+ component. This means that different variables \<open>x\<^bsub>\<tau>\<^sub>1\<^esub>\<close> and \<open>x\<^bsub>\<tau>\<^sub>2\<^esub>\<close> may
+ become the same after type instantiation. Type-inference rejects variables
+ of the same name, but different types. In contrast, mixed instances of
polymorphic constants occur routinely.
\<^medskip>
- The \<^emph>\<open>hidden polymorphism\<close> of a term \<open>t :: \<sigma>\<close>
- is the set of type variables occurring in \<open>t\<close>, but not in
- its type \<open>\<sigma>\<close>. This means that the term implicitly depends
- on type arguments that are not accounted in the result type, i.e.\
- there are different type instances \<open>t\<vartheta> :: \<sigma>\<close> and
- \<open>t\<vartheta>' :: \<sigma>\<close> with the same type. This slightly
- pathological situation notoriously demands additional care.
+ The \<^emph>\<open>hidden polymorphism\<close> of a term \<open>t :: \<sigma>\<close> is the set of type variables
+ occurring in \<open>t\<close>, but not in its type \<open>\<sigma>\<close>. This means that the term
+ implicitly depends on type arguments that are not accounted in the result
+ type, i.e.\ there are different type instances \<open>t\<vartheta> :: \<sigma>\<close> and
+ \<open>t\<vartheta>' :: \<sigma>\<close> with the same type. This slightly pathological
+ situation notoriously demands additional care.
\<^medskip>
- A \<^emph>\<open>term abbreviation\<close> is a syntactic definition \<open>c\<^sub>\<sigma> \<equiv> t\<close> of a closed term \<open>t\<close> of type \<open>\<sigma>\<close>,
- without any hidden polymorphism. A term abbreviation looks like a
- constant in the syntax, but is expanded before entering the logical
- core. Abbreviations are usually reverted when printing terms, using
- \<open>t \<rightarrow> c\<^sub>\<sigma>\<close> as rules for higher-order rewriting.
+ A \<^emph>\<open>term abbreviation\<close> is a syntactic definition \<open>c\<^sub>\<sigma> \<equiv> t\<close> of a closed term
+ \<open>t\<close> of type \<open>\<sigma>\<close>, without any hidden polymorphism. A term abbreviation looks
+ like a constant in the syntax, but is expanded before entering the logical
+ core. Abbreviations are usually reverted when printing terms, using \<open>t \<rightarrow>
+ c\<^sub>\<sigma>\<close> as rules for higher-order rewriting.
\<^medskip>
- Canonical operations on \<open>\<lambda>\<close>-terms include \<open>\<alpha>\<beta>\<eta>\<close>-conversion: \<open>\<alpha>\<close>-conversion refers to capture-free
- renaming of bound variables; \<open>\<beta>\<close>-conversion contracts an
- abstraction applied to an argument term, substituting the argument
- in the body: \<open>(\<lambda>x. b)a\<close> becomes \<open>b[a/x]\<close>; \<open>\<eta>\<close>-conversion contracts vacuous application-abstraction: \<open>\<lambda>x. f x\<close> becomes \<open>f\<close>, provided that the bound variable
- does not occur in \<open>f\<close>.
+ Canonical operations on \<open>\<lambda>\<close>-terms include \<open>\<alpha>\<beta>\<eta>\<close>-conversion: \<open>\<alpha>\<close>-conversion
+ refers to capture-free renaming of bound variables; \<open>\<beta>\<close>-conversion contracts
+ an abstraction applied to an argument term, substituting the argument in the
+ body: \<open>(\<lambda>x. b)a\<close> becomes \<open>b[a/x]\<close>; \<open>\<eta>\<close>-conversion contracts vacuous
+ application-abstraction: \<open>\<lambda>x. f x\<close> becomes \<open>f\<close>, provided that the bound
+ variable does not occur in \<open>f\<close>.
- Terms are normally treated modulo \<open>\<alpha>\<close>-conversion, which is
- implicit in the de-Bruijn representation. Names for bound variables
- in abstractions are maintained separately as (meaningless) comments,
- mostly for parsing and printing. Full \<open>\<alpha>\<beta>\<eta>\<close>-conversion is
- commonplace in various standard operations (\secref{sec:obj-rules})
- that are based on higher-order unification and matching.
+ Terms are normally treated modulo \<open>\<alpha>\<close>-conversion, which is implicit in the
+ de-Bruijn representation. Names for bound variables in abstractions are
+ maintained separately as (meaningless) comments, mostly for parsing and
+ printing. Full \<open>\<alpha>\<beta>\<eta>\<close>-conversion is commonplace in various standard
+ operations (\secref{sec:obj-rules}) that are based on higher-order
+ unification and matching.
\<close>
text %mlref \<open>
@@ -361,56 +333,52 @@
@{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
\end{mldecls}
- \<^descr> Type @{ML_type term} represents de-Bruijn terms, with comments
- in abstractions, and explicitly named free variables and constants;
- this is a datatype with constructors @{index_ML Bound}, @{index_ML
- Free}, @{index_ML Var}, @{index_ML Const}, @{index_ML Abs},
- @{index_ML_op "$"}.
+ \<^descr> Type @{ML_type term} represents de-Bruijn terms, with comments in
+ abstractions, and explicitly named free variables and constants; this is a
+ datatype with constructors @{index_ML Bound}, @{index_ML Free}, @{index_ML
+ Var}, @{index_ML Const}, @{index_ML Abs}, @{index_ML_op "$"}.
- \<^descr> \<open>t\<close>~@{ML_text aconv}~\<open>u\<close> checks \<open>\<alpha>\<close>-equivalence of two terms. This is the basic equality relation
- on type @{ML_type term}; raw datatype equality should only be used
- for operations related to parsing or printing!
-
- \<^descr> @{ML Term.map_types}~\<open>f t\<close> applies the mapping \<open>f\<close> to all types occurring in \<open>t\<close>.
+ \<^descr> \<open>t\<close>~@{ML_text aconv}~\<open>u\<close> checks \<open>\<alpha>\<close>-equivalence of two terms. This is the
+ basic equality relation on type @{ML_type term}; raw datatype equality
+ should only be used for operations related to parsing or printing!
- \<^descr> @{ML Term.fold_types}~\<open>f t\<close> iterates the operation
- \<open>f\<close> over all occurrences of types in \<open>t\<close>; the term
- structure is traversed from left to right.
+ \<^descr> @{ML Term.map_types}~\<open>f t\<close> applies the mapping \<open>f\<close> to all types occurring
+ in \<open>t\<close>.
+
+ \<^descr> @{ML Term.fold_types}~\<open>f t\<close> iterates the operation \<open>f\<close> over all
+ occurrences of types in \<open>t\<close>; the term structure is traversed from left to
+ right.
- \<^descr> @{ML Term.map_aterms}~\<open>f t\<close> applies the mapping \<open>f\<close> to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
- Const}) occurring in \<open>t\<close>.
+ \<^descr> @{ML Term.map_aterms}~\<open>f t\<close> applies the mapping \<open>f\<close> to all atomic terms
+ (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}) occurring in \<open>t\<close>.
- \<^descr> @{ML Term.fold_aterms}~\<open>f t\<close> iterates the operation
- \<open>f\<close> over all occurrences of atomic terms (@{ML Bound}, @{ML
- Free}, @{ML Var}, @{ML Const}) in \<open>t\<close>; the term structure is
- traversed from left to right.
+ \<^descr> @{ML Term.fold_aterms}~\<open>f t\<close> iterates the operation \<open>f\<close> over all
+ occurrences of atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
+ Const}) in \<open>t\<close>; the term structure is traversed from left to right.
- \<^descr> @{ML fastype_of}~\<open>t\<close> determines the type of a
- well-typed term. This operation is relatively slow, despite the
- omission of any sanity checks.
+ \<^descr> @{ML fastype_of}~\<open>t\<close> determines the type of a well-typed term. This
+ operation is relatively slow, despite the omission of any sanity checks.
- \<^descr> @{ML lambda}~\<open>a b\<close> produces an abstraction \<open>\<lambda>a. b\<close>, where occurrences of the atomic term \<open>a\<close> in the
- body \<open>b\<close> are replaced by bound variables.
+ \<^descr> @{ML lambda}~\<open>a b\<close> produces an abstraction \<open>\<lambda>a. b\<close>, where occurrences of
+ the atomic term \<open>a\<close> in the body \<open>b\<close> are replaced by bound variables.
- \<^descr> @{ML betapply}~\<open>(t, u)\<close> produces an application \<open>t u\<close>, with topmost \<open>\<beta>\<close>-conversion if \<open>t\<close> is an
- abstraction.
+ \<^descr> @{ML betapply}~\<open>(t, u)\<close> produces an application \<open>t u\<close>, with topmost
+ \<open>\<beta>\<close>-conversion if \<open>t\<close> is an abstraction.
- \<^descr> @{ML incr_boundvars}~\<open>j\<close> increments a term's dangling
- bound variables by the offset \<open>j\<close>. This is required when
- moving a subterm into a context where it is enclosed by a different
- number of abstractions. Bound variables with a matching abstraction
- are unaffected.
+ \<^descr> @{ML incr_boundvars}~\<open>j\<close> increments a term's dangling bound variables by
+ the offset \<open>j\<close>. This is required when moving a subterm into a context where
+ it is enclosed by a different number of abstractions. Bound variables with a
+ matching abstraction are unaffected.
- \<^descr> @{ML Sign.declare_const}~\<open>ctxt ((c, \<sigma>), mx)\<close> declares
- a new constant \<open>c :: \<sigma>\<close> with optional mixfix syntax.
+ \<^descr> @{ML Sign.declare_const}~\<open>ctxt ((c, \<sigma>), mx)\<close> declares a new constant \<open>c ::
+ \<sigma>\<close> with optional mixfix syntax.
- \<^descr> @{ML Sign.add_abbrev}~\<open>print_mode (c, t)\<close>
- introduces a new term abbreviation \<open>c \<equiv> t\<close>.
+ \<^descr> @{ML Sign.add_abbrev}~\<open>print_mode (c, t)\<close> introduces a new term
+ abbreviation \<open>c \<equiv> t\<close>.
- \<^descr> @{ML Sign.const_typargs}~\<open>thy (c, \<tau>)\<close> and @{ML
- Sign.const_instance}~\<open>thy (c, [\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n])\<close>
- convert between two representations of polymorphic constants: full
- type instance vs.\ compact type arguments form.
+ \<^descr> @{ML Sign.const_typargs}~\<open>thy (c, \<tau>)\<close> and @{ML Sign.const_instance}~\<open>thy
+ (c, [\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n])\<close> convert between two representations of polymorphic
+ constants: full type instance vs.\ compact type arguments form.
\<close>
text %mlantiq \<open>
@@ -433,33 +401,31 @@
@@{ML_antiquotation prop} prop
\<close>}
- \<^descr> \<open>@{const_name c}\<close> inlines the internalized logical
- constant name \<open>c\<close> --- as @{ML_type string} literal.
+ \<^descr> \<open>@{const_name c}\<close> inlines the internalized logical constant name \<open>c\<close> ---
+ as @{ML_type string} literal.
+
+ \<^descr> \<open>@{const_abbrev c}\<close> inlines the internalized abbreviated constant name \<open>c\<close>
+ --- as @{ML_type string} literal.
- \<^descr> \<open>@{const_abbrev c}\<close> inlines the internalized
- abbreviated constant name \<open>c\<close> --- as @{ML_type string}
- literal.
+ \<^descr> \<open>@{const c(\<^vec>\<tau>)}\<close> inlines the internalized constant \<open>c\<close> with precise
+ type instantiation in the sense of @{ML Sign.const_instance} --- as @{ML
+ Const} constructor term for datatype @{ML_type term}.
- \<^descr> \<open>@{const c(\<^vec>\<tau>)}\<close> inlines the internalized
- constant \<open>c\<close> with precise type instantiation in the sense of
- @{ML Sign.const_instance} --- as @{ML Const} constructor term for
+ \<^descr> \<open>@{term t}\<close> inlines the internalized term \<open>t\<close> --- as constructor term for
datatype @{ML_type term}.
- \<^descr> \<open>@{term t}\<close> inlines the internalized term \<open>t\<close>
- --- as constructor term for datatype @{ML_type term}.
-
- \<^descr> \<open>@{prop \<phi>}\<close> inlines the internalized proposition
- \<open>\<phi>\<close> --- as constructor term for datatype @{ML_type term}.
+ \<^descr> \<open>@{prop \<phi>}\<close> inlines the internalized proposition \<open>\<phi>\<close> --- as constructor
+ term for datatype @{ML_type term}.
\<close>
section \<open>Theorems \label{sec:thms}\<close>
text \<open>
- A \<^emph>\<open>proposition\<close> is a well-typed term of type \<open>prop\<close>, a
- \<^emph>\<open>theorem\<close> is a proven proposition (depending on a context of
- hypotheses and the background theory). Primitive inferences include
- plain Natural Deduction rules for the primary connectives \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> of the framework. There is also a builtin
+ A \<^emph>\<open>proposition\<close> is a well-typed term of type \<open>prop\<close>, a \<^emph>\<open>theorem\<close> is a
+ proven proposition (depending on a context of hypotheses and the background
+ theory). Primitive inferences include plain Natural Deduction rules for the
+ primary connectives \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> of the framework. There is also a builtin
notion of equality/equivalence \<open>\<equiv>\<close>.
\<close>
@@ -467,16 +433,14 @@
subsection \<open>Primitive connectives and rules \label{sec:prim-rules}\<close>
text \<open>
- The theory \<open>Pure\<close> contains constant declarations for the
- primitive connectives \<open>\<And>\<close>, \<open>\<Longrightarrow>\<close>, and \<open>\<equiv>\<close> of
- the logical framework, see \figref{fig:pure-connectives}. The
- derivability judgment \<open>A\<^sub>1, \<dots>, A\<^sub>n \<turnstile> B\<close> is
- defined inductively by the primitive inferences given in
- \figref{fig:prim-rules}, with the global restriction that the
- hypotheses must \<^emph>\<open>not\<close> contain any schematic variables. The
- builtin equality is conceptually axiomatized as shown in
- \figref{fig:pure-equality}, although the implementation works
- directly with derived inferences.
+ The theory \<open>Pure\<close> contains constant declarations for the primitive
+ connectives \<open>\<And>\<close>, \<open>\<Longrightarrow>\<close>, and \<open>\<equiv>\<close> of the logical framework, see
+ \figref{fig:pure-connectives}. The derivability judgment \<open>A\<^sub>1, \<dots>, A\<^sub>n \<turnstile> B\<close>
+ is defined inductively by the primitive inferences given in
+ \figref{fig:prim-rules}, with the global restriction that the hypotheses
+ must \<^emph>\<open>not\<close> contain any schematic variables. The builtin equality is
+ conceptually axiomatized as shown in \figref{fig:pure-equality}, although
+ the implementation works directly with derived inferences.
\begin{figure}[htb]
\begin{center}
@@ -523,26 +487,29 @@
\end{center}
\end{figure}
- The introduction and elimination rules for \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> are analogous to formation of dependently typed \<open>\<lambda>\<close>-terms representing the underlying proof objects. Proof terms
- are irrelevant in the Pure logic, though; they cannot occur within
- propositions. The system provides a runtime option to record
+ The introduction and elimination rules for \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> are analogous to
+ formation of dependently typed \<open>\<lambda>\<close>-terms representing the underlying proof
+ objects. Proof terms are irrelevant in the Pure logic, though; they cannot
+ occur within propositions. The system provides a runtime option to record
explicit proof terms for primitive inferences, see also
- \secref{sec:proof-terms}. Thus all three levels of \<open>\<lambda>\<close>-calculus become explicit: \<open>\<Rightarrow>\<close> for terms, and \<open>\<And>/\<Longrightarrow>\<close> for proofs (cf.\ @{cite "Berghofer-Nipkow:2000:TPHOL"}).
+ \secref{sec:proof-terms}. Thus all three levels of \<open>\<lambda>\<close>-calculus become
+ explicit: \<open>\<Rightarrow>\<close> for terms, and \<open>\<And>/\<Longrightarrow>\<close> for proofs (cf.\ @{cite
+ "Berghofer-Nipkow:2000:TPHOL"}).
- Observe that locally fixed parameters (as in \<open>\<And>\<hyphen>intro\<close>) need not be recorded in the hypotheses, because
- the simple syntactic types of Pure are always inhabitable.
- ``Assumptions'' \<open>x :: \<tau>\<close> for type-membership are only
- present as long as some \<open>x\<^sub>\<tau>\<close> occurs in the statement
- body.\<^footnote>\<open>This is the key difference to ``\<open>\<lambda>HOL\<close>'' in
- the PTS framework @{cite "Barendregt-Geuvers:2001"}, where hypotheses
- \<open>x : A\<close> are treated uniformly for propositions and types.\<close>
+ Observe that locally fixed parameters (as in \<open>\<And>\<hyphen>intro\<close>) need not be recorded
+ in the hypotheses, because the simple syntactic types of Pure are always
+ inhabitable. ``Assumptions'' \<open>x :: \<tau>\<close> for type-membership are only present
+ as long as some \<open>x\<^sub>\<tau>\<close> occurs in the statement body.\<^footnote>\<open>This is the key
+ difference to ``\<open>\<lambda>HOL\<close>'' in the PTS framework @{cite
+ "Barendregt-Geuvers:2001"}, where hypotheses \<open>x : A\<close> are treated uniformly
+ for propositions and types.\<close>
\<^medskip>
- The axiomatization of a theory is implicitly closed by
- forming all instances of type and term variables: \<open>\<turnstile>
- A\<vartheta>\<close> holds for any substitution instance of an axiom
- \<open>\<turnstile> A\<close>. By pushing substitutions through derivations
- inductively, we also get admissible \<open>generalize\<close> and \<open>instantiate\<close> rules as shown in \figref{fig:subst-rules}.
+ The axiomatization of a theory is implicitly closed by forming all instances
+ of type and term variables: \<open>\<turnstile> A\<vartheta>\<close> holds for any substitution
+ instance of an axiom \<open>\<turnstile> A\<close>. By pushing substitutions through derivations
+ inductively, we also get admissible \<open>generalize\<close> and \<open>instantiate\<close> rules as
+ shown in \figref{fig:subst-rules}.
\begin{figure}[htb]
\begin{center}
@@ -560,40 +527,39 @@
\end{center}
\end{figure}
- Note that \<open>instantiate\<close> does not require an explicit
- side-condition, because \<open>\<Gamma>\<close> may never contain schematic
- variables.
+ Note that \<open>instantiate\<close> does not require an explicit side-condition, because
+ \<open>\<Gamma>\<close> may never contain schematic variables.
- In principle, variables could be substituted in hypotheses as well,
- but this would disrupt the monotonicity of reasoning: deriving
- \<open>\<Gamma>\<vartheta> \<turnstile> B\<vartheta>\<close> from \<open>\<Gamma> \<turnstile> B\<close> is
- correct, but \<open>\<Gamma>\<vartheta> \<supseteq> \<Gamma>\<close> does not necessarily hold:
- the result belongs to a different proof context.
+ In principle, variables could be substituted in hypotheses as well, but this
+ would disrupt the monotonicity of reasoning: deriving \<open>\<Gamma>\<vartheta> \<turnstile>
+ B\<vartheta>\<close> from \<open>\<Gamma> \<turnstile> B\<close> is correct, but \<open>\<Gamma>\<vartheta> \<supseteq> \<Gamma>\<close> does not
+ necessarily hold: the result belongs to a different proof context.
\<^medskip>
- An \<^emph>\<open>oracle\<close> is a function that produces axioms on the
- fly. Logically, this is an instance of the \<open>axiom\<close> rule
- (\figref{fig:prim-rules}), but there is an operational difference.
- The system always records oracle invocations within derivations of
- theorems by a unique tag.
+ An \<^emph>\<open>oracle\<close> is a function that produces axioms on the fly. Logically, this
+ is an instance of the \<open>axiom\<close> rule (\figref{fig:prim-rules}), but there is
+ an operational difference. The system always records oracle invocations
+ within derivations of theorems by a unique tag.
- Axiomatizations should be limited to the bare minimum, typically as
- part of the initial logical basis of an object-logic formalization.
- Later on, theories are usually developed in a strictly definitional
- fashion, by stating only certain equalities over new constants.
+ Axiomatizations should be limited to the bare minimum, typically as part of
+ the initial logical basis of an object-logic formalization. Later on,
+ theories are usually developed in a strictly definitional fashion, by
+ stating only certain equalities over new constants.
- A \<^emph>\<open>simple definition\<close> consists of a constant declaration \<open>c :: \<sigma>\<close> together with an axiom \<open>\<turnstile> c \<equiv> t\<close>, where \<open>t
- :: \<sigma>\<close> is a closed term without any hidden polymorphism. The RHS
- may depend on further defined constants, but not \<open>c\<close> itself.
- Definitions of functions may be presented as \<open>c \<^vec>x \<equiv>
- t\<close> instead of the puristic \<open>c \<equiv> \<lambda>\<^vec>x. t\<close>.
+ A \<^emph>\<open>simple definition\<close> consists of a constant declaration \<open>c :: \<sigma>\<close> together
+ with an axiom \<open>\<turnstile> c \<equiv> t\<close>, where \<open>t :: \<sigma>\<close> is a closed term without any hidden
+ polymorphism. The RHS may depend on further defined constants, but not \<open>c\<close>
+ itself. Definitions of functions may be presented as \<open>c \<^vec>x \<equiv> t\<close>
+ instead of the puristic \<open>c \<equiv> \<lambda>\<^vec>x. t\<close>.
- An \<^emph>\<open>overloaded definition\<close> consists of a collection of axioms
- for the same constant, with zero or one equations \<open>c((\<^vec>\<alpha>)\<kappa>) \<equiv> t\<close> for each type constructor \<open>\<kappa>\<close> (for
- distinct variables \<open>\<^vec>\<alpha>\<close>). The RHS may mention
- previously defined constants as above, or arbitrary constants \<open>d(\<alpha>\<^sub>i)\<close> for some \<open>\<alpha>\<^sub>i\<close> projected from \<open>\<^vec>\<alpha>\<close>. Thus overloaded definitions essentially work by
- primitive recursion over the syntactic structure of a single type
- argument. See also @{cite \<open>\S4.3\<close> "Haftmann-Wenzel:2006:classes"}.
+ An \<^emph>\<open>overloaded definition\<close> consists of a collection of axioms for the same
+ constant, with zero or one equations \<open>c((\<^vec>\<alpha>)\<kappa>) \<equiv> t\<close> for each type
+ constructor \<open>\<kappa>\<close> (for distinct variables \<open>\<^vec>\<alpha>\<close>). The RHS may mention
+ previously defined constants as above, or arbitrary constants \<open>d(\<alpha>\<^sub>i)\<close> for
+ some \<open>\<alpha>\<^sub>i\<close> projected from \<open>\<^vec>\<alpha>\<close>. Thus overloaded definitions
+ essentially work by primitive recursion over the syntactic structure of a
+ single type argument. See also @{cite \<open>\S4.3\<close>
+ "Haftmann-Wenzel:2006:classes"}.
\<close>
text %mlref \<open>
@@ -635,101 +601,89 @@
Defs.entry -> Defs.entry list -> theory -> theory"} \\
\end{mldecls}
- \<^descr> @{ML Thm.peek_status}~\<open>thm\<close> informs about the current
- status of the derivation object behind the given theorem. This is a
- snapshot of a potentially ongoing (parallel) evaluation of proofs.
- The three Boolean values indicate the following: \<^verbatim>\<open>oracle\<close>
- if the finished part contains some oracle invocation; \<^verbatim>\<open>unfinished\<close>
- if some future proofs are still pending; \<^verbatim>\<open>failed\<close> if some future
- proof has failed, rendering the theorem invalid!
+ \<^descr> @{ML Thm.peek_status}~\<open>thm\<close> informs about the current status of the
+ derivation object behind the given theorem. This is a snapshot of a
+ potentially ongoing (parallel) evaluation of proofs. The three Boolean
+ values indicate the following: \<^verbatim>\<open>oracle\<close> if the finished part contains some
+ oracle invocation; \<^verbatim>\<open>unfinished\<close> if some future proofs are still pending;
+ \<^verbatim>\<open>failed\<close> if some future proof has failed, rendering the theorem invalid!
- \<^descr> @{ML Logic.all}~\<open>a B\<close> produces a Pure quantification
- \<open>\<And>a. B\<close>, where occurrences of the atomic term \<open>a\<close> in
- the body proposition \<open>B\<close> are replaced by bound variables.
- (See also @{ML lambda} on terms.)
+ \<^descr> @{ML Logic.all}~\<open>a B\<close> produces a Pure quantification \<open>\<And>a. B\<close>, where
+ occurrences of the atomic term \<open>a\<close> in the body proposition \<open>B\<close> are replaced
+ by bound variables. (See also @{ML lambda} on terms.)
- \<^descr> @{ML Logic.mk_implies}~\<open>(A, B)\<close> produces a Pure
- implication \<open>A \<Longrightarrow> B\<close>.
+ \<^descr> @{ML Logic.mk_implies}~\<open>(A, B)\<close> produces a Pure implication \<open>A \<Longrightarrow> B\<close>.
- \<^descr> Types @{ML_type ctyp} and @{ML_type cterm} represent certified
- types and terms, respectively. These are abstract datatypes that
- guarantee that its values have passed the full well-formedness (and
- well-typedness) checks, relative to the declarations of type
- constructors, constants etc.\ in the background theory. The
- abstract types @{ML_type ctyp} and @{ML_type cterm} are part of the
- same inference kernel that is mainly responsible for @{ML_type thm}.
- Thus syntactic operations on @{ML_type ctyp} and @{ML_type cterm}
- are located in the @{ML_structure Thm} module, even though theorems are
- not yet involved at that stage.
+ \<^descr> Types @{ML_type ctyp} and @{ML_type cterm} represent certified types and
+ terms, respectively. These are abstract datatypes that guarantee that its
+ values have passed the full well-formedness (and well-typedness) checks,
+ relative to the declarations of type constructors, constants etc.\ in the
+ background theory. The abstract types @{ML_type ctyp} and @{ML_type cterm}
+ are part of the same inference kernel that is mainly responsible for
+ @{ML_type thm}. Thus syntactic operations on @{ML_type ctyp} and @{ML_type
+ cterm} are located in the @{ML_structure Thm} module, even though theorems
+ are not yet involved at that stage.
- \<^descr> @{ML Thm.ctyp_of}~\<open>ctxt \<tau>\<close> and @{ML
- Thm.cterm_of}~\<open>ctxt t\<close> explicitly check types and terms,
- respectively. This also involves some basic normalizations, such
- expansion of type and term abbreviations from the underlying
- theory context.
- Full re-certification is relatively slow and should be avoided in
- tight reasoning loops.
+ \<^descr> @{ML Thm.ctyp_of}~\<open>ctxt \<tau>\<close> and @{ML Thm.cterm_of}~\<open>ctxt t\<close> explicitly
+ check types and terms, respectively. This also involves some basic
+ normalizations, such expansion of type and term abbreviations from the
+ underlying theory context. Full re-certification is relatively slow and
+ should be avoided in tight reasoning loops.
- \<^descr> @{ML Thm.apply}, @{ML Thm.lambda}, @{ML Thm.all}, @{ML
- Drule.mk_implies} etc.\ compose certified terms (or propositions)
- incrementally. This is equivalent to @{ML Thm.cterm_of} after
- unchecked @{ML_op "$"}, @{ML lambda}, @{ML Logic.all}, @{ML
- Logic.mk_implies} etc., but there can be a big difference in
- performance when large existing entities are composed by a few extra
- constructions on top. There are separate operations to decompose
+ \<^descr> @{ML Thm.apply}, @{ML Thm.lambda}, @{ML Thm.all}, @{ML Drule.mk_implies}
+ etc.\ compose certified terms (or propositions) incrementally. This is
+ equivalent to @{ML Thm.cterm_of} after unchecked @{ML_op "$"}, @{ML lambda},
+ @{ML Logic.all}, @{ML Logic.mk_implies} etc., but there can be a big
+ difference in performance when large existing entities are composed by a few
+ extra constructions on top. There are separate operations to decompose
certified terms and theorems to produce certified terms again.
- \<^descr> Type @{ML_type thm} represents proven propositions. This is
- an abstract datatype that guarantees that its values have been
- constructed by basic principles of the @{ML_structure Thm} module.
- Every @{ML_type thm} value refers its background theory,
- cf.\ \secref{sec:context-theory}.
+ \<^descr> Type @{ML_type thm} represents proven propositions. This is an abstract
+ datatype that guarantees that its values have been constructed by basic
+ principles of the @{ML_structure Thm} module. Every @{ML_type thm} value
+ refers its background theory, cf.\ \secref{sec:context-theory}.
- \<^descr> @{ML Thm.transfer}~\<open>thy thm\<close> transfers the given
- theorem to a \<^emph>\<open>larger\<close> theory, see also \secref{sec:context}.
- This formal adjustment of the background context has no logical
- significance, but is occasionally required for formal reasons, e.g.\
- when theorems that are imported from more basic theories are used in
- the current situation.
+ \<^descr> @{ML Thm.transfer}~\<open>thy thm\<close> transfers the given theorem to a \<^emph>\<open>larger\<close>
+ theory, see also \secref{sec:context}. This formal adjustment of the
+ background context has no logical significance, but is occasionally required
+ for formal reasons, e.g.\ when theorems that are imported from more basic
+ theories are used in the current situation.
- \<^descr> @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML
- Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim}
- correspond to the primitive inferences of \figref{fig:prim-rules}.
+ \<^descr> @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML Thm.forall_elim}, @{ML
+ Thm.implies_intr}, and @{ML Thm.implies_elim} correspond to the primitive
+ inferences of \figref{fig:prim-rules}.
- \<^descr> @{ML Thm.generalize}~\<open>(\<^vec>\<alpha>, \<^vec>x)\<close>
- corresponds to the \<open>generalize\<close> rules of
- \figref{fig:subst-rules}. Here collections of type and term
- variables are generalized simultaneously, specified by the given
- basic names.
+ \<^descr> @{ML Thm.generalize}~\<open>(\<^vec>\<alpha>, \<^vec>x)\<close> corresponds to the
+ \<open>generalize\<close> rules of \figref{fig:subst-rules}. Here collections of type and
+ term variables are generalized simultaneously, specified by the given basic
+ names.
- \<^descr> @{ML Thm.instantiate}~\<open>(\<^vec>\<alpha>\<^sub>s,
- \<^vec>x\<^sub>\<tau>)\<close> corresponds to the \<open>instantiate\<close> rules
- of \figref{fig:subst-rules}. Type variables are substituted before
- term variables. Note that the types in \<open>\<^vec>x\<^sub>\<tau>\<close>
- refer to the instantiated versions.
+ \<^descr> @{ML Thm.instantiate}~\<open>(\<^vec>\<alpha>\<^sub>s, \<^vec>x\<^sub>\<tau>)\<close> corresponds to the
+ \<open>instantiate\<close> rules of \figref{fig:subst-rules}. Type variables are
+ substituted before term variables. Note that the types in \<open>\<^vec>x\<^sub>\<tau>\<close> refer
+ to the instantiated versions.
- \<^descr> @{ML Thm.add_axiom}~\<open>ctxt (name, A)\<close> declares an
- arbitrary proposition as axiom, and retrieves it as a theorem from
- the resulting theory, cf.\ \<open>axiom\<close> in
- \figref{fig:prim-rules}. Note that the low-level representation in
- the axiom table may differ slightly from the returned theorem.
+ \<^descr> @{ML Thm.add_axiom}~\<open>ctxt (name, A)\<close> declares an arbitrary proposition as
+ axiom, and retrieves it as a theorem from the resulting theory, cf.\ \<open>axiom\<close>
+ in \figref{fig:prim-rules}. Note that the low-level representation in the
+ axiom table may differ slightly from the returned theorem.
- \<^descr> @{ML Thm.add_oracle}~\<open>(binding, oracle)\<close> produces a named
- oracle rule, essentially generating arbitrary axioms on the fly,
- cf.\ \<open>axiom\<close> in \figref{fig:prim-rules}.
+ \<^descr> @{ML Thm.add_oracle}~\<open>(binding, oracle)\<close> produces a named oracle rule,
+ essentially generating arbitrary axioms on the fly, cf.\ \<open>axiom\<close> in
+ \figref{fig:prim-rules}.
- \<^descr> @{ML Thm.add_def}~\<open>ctxt unchecked overloaded (name, c
- \<^vec>x \<equiv> t)\<close> states a definitional axiom for an existing constant
- \<open>c\<close>. Dependencies are recorded via @{ML Theory.add_deps},
- unless the \<open>unchecked\<close> option is set. Note that the
- low-level representation in the axiom table may differ slightly from
- the returned theorem.
+ \<^descr> @{ML Thm.add_def}~\<open>ctxt unchecked overloaded (name, c \<^vec>x \<equiv> t)\<close>
+ states a definitional axiom for an existing constant \<open>c\<close>. Dependencies are
+ recorded via @{ML Theory.add_deps}, unless the \<open>unchecked\<close> option is set.
+ Note that the low-level representation in the axiom table may differ
+ slightly from the returned theorem.
- \<^descr> @{ML Theory.add_deps}~\<open>ctxt name c\<^sub>\<tau> \<^vec>d\<^sub>\<sigma>\<close>
- declares dependencies of a named specification for constant \<open>c\<^sub>\<tau>\<close>, relative to existing specifications for constants \<open>\<^vec>d\<^sub>\<sigma>\<close>. This also works for type constructors.
+ \<^descr> @{ML Theory.add_deps}~\<open>ctxt name c\<^sub>\<tau> \<^vec>d\<^sub>\<sigma>\<close> declares dependencies of
+ a named specification for constant \<open>c\<^sub>\<tau>\<close>, relative to existing
+ specifications for constants \<open>\<^vec>d\<^sub>\<sigma>\<close>. This also works for type
+ constructors.
\<close>
-
text %mlantiq \<open>
\begin{matharray}{rcl}
@{ML_antiquotation_def "ctyp"} & : & \<open>ML_antiquotation\<close> \\
@@ -755,46 +709,42 @@
@'by' method method?
\<close>}
- \<^descr> \<open>@{ctyp \<tau>}\<close> produces a certified type wrt.\ the
- current background theory --- as abstract value of type @{ML_type
- ctyp}.
+ \<^descr> \<open>@{ctyp \<tau>}\<close> produces a certified type wrt.\ the current background theory
+ --- as abstract value of type @{ML_type ctyp}.
+
+ \<^descr> \<open>@{cterm t}\<close> and \<open>@{cprop \<phi>}\<close> produce a certified term wrt.\ the current
+ background theory --- as abstract value of type @{ML_type cterm}.
- \<^descr> \<open>@{cterm t}\<close> and \<open>@{cprop \<phi>}\<close> produce a
- certified term wrt.\ the current background theory --- as abstract
- value of type @{ML_type cterm}.
+ \<^descr> \<open>@{thm a}\<close> produces a singleton fact --- as abstract value of type
+ @{ML_type thm}.
- \<^descr> \<open>@{thm a}\<close> produces a singleton fact --- as abstract
- value of type @{ML_type thm}.
-
- \<^descr> \<open>@{thms a}\<close> produces a general fact --- as abstract
- value of type @{ML_type "thm list"}.
+ \<^descr> \<open>@{thms a}\<close> produces a general fact --- as abstract value of type
+ @{ML_type "thm list"}.
- \<^descr> \<open>@{lemma \<phi> by meth}\<close> produces a fact that is proven on
- the spot according to the minimal proof, which imitates a terminal
- Isar proof. The result is an abstract value of type @{ML_type thm}
- or @{ML_type "thm list"}, depending on the number of propositions
- given here.
+ \<^descr> \<open>@{lemma \<phi> by meth}\<close> produces a fact that is proven on the spot according
+ to the minimal proof, which imitates a terminal Isar proof. The result is an
+ abstract value of type @{ML_type thm} or @{ML_type "thm list"}, depending on
+ the number of propositions given here.
- The internal derivation object lacks a proper theorem name, but it
- is formally closed, unless the \<open>(open)\<close> option is specified
- (this may impact performance of applications with proof terms).
+ The internal derivation object lacks a proper theorem name, but it is
+ formally closed, unless the \<open>(open)\<close> option is specified (this may impact
+ performance of applications with proof terms).
- Since ML antiquotations are always evaluated at compile-time, there
- is no run-time overhead even for non-trivial proofs. Nonetheless,
- the justification is syntactically limited to a single @{command
- "by"} step. More complex Isar proofs should be done in regular
- theory source, before compiling the corresponding ML text that uses
- the result.
+ Since ML antiquotations are always evaluated at compile-time, there is no
+ run-time overhead even for non-trivial proofs. Nonetheless, the
+ justification is syntactically limited to a single @{command "by"} step.
+ More complex Isar proofs should be done in regular theory source, before
+ compiling the corresponding ML text that uses the result.
\<close>
subsection \<open>Auxiliary connectives \label{sec:logic-aux}\<close>
-text \<open>Theory \<open>Pure\<close> provides a few auxiliary connectives
- that are defined on top of the primitive ones, see
- \figref{fig:pure-aux}. These special constants are useful in
- certain internal encodings, and are normally not directly exposed to
- the user.
+text \<open>
+ Theory \<open>Pure\<close> provides a few auxiliary connectives that are defined on top
+ of the primitive ones, see \figref{fig:pure-aux}. These special constants
+ are useful in certain internal encodings, and are normally not directly
+ exposed to the user.
\begin{figure}[htb]
\begin{center}
@@ -812,37 +762,32 @@
\end{center}
\end{figure}
- The introduction \<open>A \<Longrightarrow> B \<Longrightarrow> A &&& B\<close>, and eliminations
- (projections) \<open>A &&& B \<Longrightarrow> A\<close> and \<open>A &&& B \<Longrightarrow> B\<close> are
- available as derived rules. Conjunction allows to treat
- simultaneous assumptions and conclusions uniformly, e.g.\ consider
- \<open>A \<Longrightarrow> B \<Longrightarrow> C &&& D\<close>. In particular, the goal mechanism
- represents multiple claims as explicit conjunction internally, but
- this is refined (via backwards introduction) into separate sub-goals
- before the user commences the proof; the final result is projected
- into a list of theorems using eliminations (cf.\
- \secref{sec:tactical-goals}).
+ The introduction \<open>A \<Longrightarrow> B \<Longrightarrow> A &&& B\<close>, and eliminations (projections) \<open>A &&& B
+ \<Longrightarrow> A\<close> and \<open>A &&& B \<Longrightarrow> B\<close> are available as derived rules. Conjunction allows to
+ treat simultaneous assumptions and conclusions uniformly, e.g.\ consider \<open>A
+ \<Longrightarrow> B \<Longrightarrow> C &&& D\<close>. In particular, the goal mechanism represents multiple claims
+ as explicit conjunction internally, but this is refined (via backwards
+ introduction) into separate sub-goals before the user commences the proof;
+ the final result is projected into a list of theorems using eliminations
+ (cf.\ \secref{sec:tactical-goals}).
- The \<open>prop\<close> marker (\<open>#\<close>) makes arbitrarily complex
- propositions appear as atomic, without changing the meaning: \<open>\<Gamma> \<turnstile> A\<close> and \<open>\<Gamma> \<turnstile> #A\<close> are interchangeable. See
- \secref{sec:tactical-goals} for specific operations.
+ The \<open>prop\<close> marker (\<open>#\<close>) makes arbitrarily complex propositions appear as
+ atomic, without changing the meaning: \<open>\<Gamma> \<turnstile> A\<close> and \<open>\<Gamma> \<turnstile> #A\<close> are
+ interchangeable. See \secref{sec:tactical-goals} for specific operations.
- The \<open>term\<close> marker turns any well-typed term into a derivable
- proposition: \<open>\<turnstile> TERM t\<close> holds unconditionally. Although
- this is logically vacuous, it allows to treat terms and proofs
- uniformly, similar to a type-theoretic framework.
+ The \<open>term\<close> marker turns any well-typed term into a derivable proposition: \<open>\<turnstile>
+ TERM t\<close> holds unconditionally. Although this is logically vacuous, it allows
+ to treat terms and proofs uniformly, similar to a type-theoretic framework.
- The \<open>TYPE\<close> constructor is the canonical representative of
- the unspecified type \<open>\<alpha> itself\<close>; it essentially injects the
- language of types into that of terms. There is specific notation
- \<open>TYPE(\<tau>)\<close> for \<open>TYPE\<^bsub>\<tau> itself\<^esub>\<close>.
- Although being devoid of any particular meaning, the term \<open>TYPE(\<tau>)\<close> accounts for the type \<open>\<tau>\<close> within the term
- language. In particular, \<open>TYPE(\<alpha>)\<close> may be used as formal
- argument in primitive definitions, in order to circumvent hidden
- polymorphism (cf.\ \secref{sec:terms}). For example, \<open>c
- TYPE(\<alpha>) \<equiv> A[\<alpha>]\<close> defines \<open>c :: \<alpha> itself \<Rightarrow> prop\<close> in terms of
- a proposition \<open>A\<close> that depends on an additional type
- argument, which is essentially a predicate on types.
+ The \<open>TYPE\<close> constructor is the canonical representative of the unspecified
+ type \<open>\<alpha> itself\<close>; it essentially injects the language of types into that of
+ terms. There is specific notation \<open>TYPE(\<tau>)\<close> for \<open>TYPE\<^bsub>\<tau> itself\<^esub>\<close>. Although
+ being devoid of any particular meaning, the term \<open>TYPE(\<tau>)\<close> accounts for the
+ type \<open>\<tau>\<close> within the term language. In particular, \<open>TYPE(\<alpha>)\<close> may be used as
+ formal argument in primitive definitions, in order to circumvent hidden
+ polymorphism (cf.\ \secref{sec:terms}). For example, \<open>c TYPE(\<alpha>) \<equiv> A[\<alpha>]\<close>
+ defines \<open>c :: \<alpha> itself \<Rightarrow> prop\<close> in terms of a proposition \<open>A\<close> that depends on
+ an additional type argument, which is essentially a predicate on types.
\<close>
text %mlref \<open>
@@ -857,8 +802,7 @@
\<^descr> @{ML Conjunction.intr} derives \<open>A &&& B\<close> from \<open>A\<close> and \<open>B\<close>.
- \<^descr> @{ML Conjunction.elim} derives \<open>A\<close> and \<open>B\<close>
- from \<open>A &&& B\<close>.
+ \<^descr> @{ML Conjunction.elim} derives \<open>A\<close> and \<open>B\<close> from \<open>A &&& B\<close>.
\<^descr> @{ML Drule.mk_term} derives \<open>TERM t\<close>.
@@ -866,35 +810,35 @@
\<^descr> @{ML Logic.mk_type}~\<open>\<tau>\<close> produces the term \<open>TYPE(\<tau>)\<close>.
- \<^descr> @{ML Logic.dest_type}~\<open>TYPE(\<tau>)\<close> recovers the type
- \<open>\<tau>\<close>.
+ \<^descr> @{ML Logic.dest_type}~\<open>TYPE(\<tau>)\<close> recovers the type \<open>\<tau>\<close>.
\<close>
subsection \<open>Sort hypotheses\<close>
-text \<open>Type variables are decorated with sorts, as explained in
- \secref{sec:types}. This constrains type instantiation to certain
- ranges of types: variable \<open>\<alpha>\<^sub>s\<close> may only be assigned to types
- \<open>\<tau>\<close> that belong to sort \<open>s\<close>. Within the logic, sort
- constraints act like implicit preconditions on the result \<open>\<lparr>\<alpha>\<^sub>1 : s\<^sub>1\<rparr>, \<dots>, \<lparr>\<alpha>\<^sub>n : s\<^sub>n\<rparr>, \<Gamma> \<turnstile> \<phi>\<close> where the type variables \<open>\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n\<close> cover the propositions \<open>\<Gamma>\<close>, \<open>\<phi>\<close>, as
- well as the proof of \<open>\<Gamma> \<turnstile> \<phi>\<close>.
+text \<open>
+ Type variables are decorated with sorts, as explained in \secref{sec:types}.
+ This constrains type instantiation to certain ranges of types: variable
+ \<open>\<alpha>\<^sub>s\<close> may only be assigned to types \<open>\<tau>\<close> that belong to sort \<open>s\<close>. Within the
+ logic, sort constraints act like implicit preconditions on the result \<open>\<lparr>\<alpha>\<^sub>1
+ : s\<^sub>1\<rparr>, \<dots>, \<lparr>\<alpha>\<^sub>n : s\<^sub>n\<rparr>, \<Gamma> \<turnstile> \<phi>\<close> where the type variables \<open>\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n\<close> cover
+ the propositions \<open>\<Gamma>\<close>, \<open>\<phi>\<close>, as well as the proof of \<open>\<Gamma> \<turnstile> \<phi>\<close>.
- These \<^emph>\<open>sort hypothesis\<close> of a theorem are passed monotonically
- through further derivations. They are redundant, as long as the
- statement of a theorem still contains the type variables that are
- accounted here. The logical significance of sort hypotheses is
- limited to the boundary case where type variables disappear from the
- proposition, e.g.\ \<open>\<lparr>\<alpha>\<^sub>s : s\<rparr> \<turnstile> \<phi>\<close>. Since such dangling type
- variables can be renamed arbitrarily without changing the
- proposition \<open>\<phi>\<close>, the inference kernel maintains sort
- hypotheses in anonymous form \<open>s \<turnstile> \<phi>\<close>.
+ These \<^emph>\<open>sort hypothesis\<close> of a theorem are passed monotonically through
+ further derivations. They are redundant, as long as the statement of a
+ theorem still contains the type variables that are accounted here. The
+ logical significance of sort hypotheses is limited to the boundary case
+ where type variables disappear from the proposition, e.g.\ \<open>\<lparr>\<alpha>\<^sub>s : s\<rparr> \<turnstile> \<phi>\<close>.
+ Since such dangling type variables can be renamed arbitrarily without
+ changing the proposition \<open>\<phi>\<close>, the inference kernel maintains sort hypotheses
+ in anonymous form \<open>s \<turnstile> \<phi>\<close>.
- In most practical situations, such extra sort hypotheses may be
- stripped in a final bookkeeping step, e.g.\ at the end of a proof:
- they are typically left over from intermediate reasoning with type
- classes that can be satisfied by some concrete type \<open>\<tau>\<close> of
- sort \<open>s\<close> to replace the hypothetical type variable \<open>\<alpha>\<^sub>s\<close>.\<close>
+ In most practical situations, such extra sort hypotheses may be stripped in
+ a final bookkeeping step, e.g.\ at the end of a proof: they are typically
+ left over from intermediate reasoning with type classes that can be
+ satisfied by some concrete type \<open>\<tau>\<close> of sort \<open>s\<close> to replace the hypothetical
+ type variable \<open>\<alpha>\<^sub>s\<close>.
+\<close>
text %mlref \<open>
\begin{mldecls}
@@ -902,17 +846,18 @@
@{index_ML Thm.strip_shyps: "thm -> thm"} \\
\end{mldecls}
- \<^descr> @{ML Thm.extra_shyps}~\<open>thm\<close> determines the extraneous
- sort hypotheses of the given theorem, i.e.\ the sorts that are not
- present within type variables of the statement.
+ \<^descr> @{ML Thm.extra_shyps}~\<open>thm\<close> determines the extraneous sort hypotheses of
+ the given theorem, i.e.\ the sorts that are not present within type
+ variables of the statement.
- \<^descr> @{ML Thm.strip_shyps}~\<open>thm\<close> removes any extraneous
- sort hypotheses that can be witnessed from the type signature.
+ \<^descr> @{ML Thm.strip_shyps}~\<open>thm\<close> removes any extraneous sort hypotheses that
+ can be witnessed from the type signature.
\<close>
-text %mlex \<open>The following artificial example demonstrates the
- derivation of @{prop False} with a pending sort hypothesis involving
- a logically empty sort.\<close>
+text %mlex \<open>
+ The following artificial example demonstrates the derivation of @{prop
+ False} with a pending sort hypothesis involving a logically empty sort.
+\<close>
class empty =
assumes bad: "\<And>(x::'a) y. x \<noteq> y"
@@ -922,55 +867,54 @@
ML_val \<open>@{assert} (Thm.extra_shyps @{thm false} = [@{sort empty}])\<close>
-text \<open>Thanks to the inference kernel managing sort hypothesis
- according to their logical significance, this example is merely an
- instance of \<^emph>\<open>ex falso quodlibet consequitur\<close> --- not a collapse
- of the logical framework!\<close>
+text \<open>
+ Thanks to the inference kernel managing sort hypothesis according to their
+ logical significance, this example is merely an instance of \<^emph>\<open>ex falso
+ quodlibet consequitur\<close> --- not a collapse of the logical framework!
+\<close>
section \<open>Object-level rules \label{sec:obj-rules}\<close>
text \<open>
- The primitive inferences covered so far mostly serve foundational
- purposes. User-level reasoning usually works via object-level rules
- that are represented as theorems of Pure. Composition of rules
- involves \<^emph>\<open>backchaining\<close>, \<^emph>\<open>higher-order unification\<close> modulo
- \<open>\<alpha>\<beta>\<eta>\<close>-conversion of \<open>\<lambda>\<close>-terms, and so-called
- \<^emph>\<open>lifting\<close> of rules into a context of \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> connectives. Thus the full power of higher-order Natural
- Deduction in Isabelle/Pure becomes readily available.
+ The primitive inferences covered so far mostly serve foundational purposes.
+ User-level reasoning usually works via object-level rules that are
+ represented as theorems of Pure. Composition of rules involves
+ \<^emph>\<open>backchaining\<close>, \<^emph>\<open>higher-order unification\<close> modulo \<open>\<alpha>\<beta>\<eta>\<close>-conversion of
+ \<open>\<lambda>\<close>-terms, and so-called \<^emph>\<open>lifting\<close> of rules into a context of \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close>
+ connectives. Thus the full power of higher-order Natural Deduction in
+ Isabelle/Pure becomes readily available.
\<close>
subsection \<open>Hereditary Harrop Formulae\<close>
text \<open>
- The idea of object-level rules is to model Natural Deduction
- inferences in the style of Gentzen @{cite "Gentzen:1935"}, but we allow
- arbitrary nesting similar to @{cite extensions91}. The most basic
- rule format is that of a \<^emph>\<open>Horn Clause\<close>:
+ The idea of object-level rules is to model Natural Deduction inferences in
+ the style of Gentzen @{cite "Gentzen:1935"}, but we allow arbitrary nesting
+ similar to @{cite extensions91}. The most basic rule format is that of a
+ \<^emph>\<open>Horn Clause\<close>:
\[
\infer{\<open>A\<close>}{\<open>A\<^sub>1\<close> & \<open>\<dots>\<close> & \<open>A\<^sub>n\<close>}
\]
- where \<open>A, A\<^sub>1, \<dots>, A\<^sub>n\<close> are atomic propositions
- of the framework, usually of the form \<open>Trueprop B\<close>, where
- \<open>B\<close> is a (compound) object-level statement. This
- object-level inference corresponds to an iterated implication in
- Pure like this:
+ where \<open>A, A\<^sub>1, \<dots>, A\<^sub>n\<close> are atomic propositions of the framework, usually of
+ the form \<open>Trueprop B\<close>, where \<open>B\<close> is a (compound) object-level statement.
+ This object-level inference corresponds to an iterated implication in Pure
+ like this:
\[
\<open>A\<^sub>1 \<Longrightarrow> \<dots> A\<^sub>n \<Longrightarrow> A\<close>
\]
- As an example consider conjunction introduction: \<open>A \<Longrightarrow> B \<Longrightarrow> A \<and>
- B\<close>. Any parameters occurring in such rule statements are
- conceptionally treated as arbitrary:
+ As an example consider conjunction introduction: \<open>A \<Longrightarrow> B \<Longrightarrow> A \<and> B\<close>. Any
+ parameters occurring in such rule statements are conceptionally treated as
+ arbitrary:
\[
\<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. A\<^sub>1 x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> \<dots> A\<^sub>n x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> A x\<^sub>1 \<dots> x\<^sub>m\<close>
\]
- Nesting of rules means that the positions of \<open>A\<^sub>i\<close> may
- again hold compound rules, not just atomic propositions.
- Propositions of this format are called \<^emph>\<open>Hereditary Harrop
- Formulae\<close> in the literature @{cite "Miller:1991"}. Here we give an
- inductive characterization as follows:
+ Nesting of rules means that the positions of \<open>A\<^sub>i\<close> may again hold compound
+ rules, not just atomic propositions. Propositions of this format are called
+ \<^emph>\<open>Hereditary Harrop Formulae\<close> in the literature @{cite "Miller:1991"}. Here
+ we give an inductive characterization as follows:
\<^medskip>
\begin{tabular}{ll}
@@ -980,29 +924,26 @@
\end{tabular}
\<^medskip>
- Thus we essentially impose nesting levels on propositions formed
- from \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close>. At each level there is a prefix
- of parameters and compound premises, concluding an atomic
- proposition. Typical examples are \<open>\<longrightarrow>\<close>-introduction \<open>(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B\<close> or mathematical induction \<open>P 0 \<Longrightarrow> (\<And>n. P n
- \<Longrightarrow> P (Suc n)) \<Longrightarrow> P n\<close>. Even deeper nesting occurs in well-founded
- induction \<open>(\<And>x. (\<And>y. y \<prec> x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x\<close>, but this
- already marks the limit of rule complexity that is usually seen in
- practice.
+ Thus we essentially impose nesting levels on propositions formed from \<open>\<And>\<close>
+ and \<open>\<Longrightarrow>\<close>. At each level there is a prefix of parameters and compound
+ premises, concluding an atomic proposition. Typical examples are
+ \<open>\<longrightarrow>\<close>-introduction \<open>(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B\<close> or mathematical induction \<open>P 0 \<Longrightarrow> (\<And>n. P n
+ \<Longrightarrow> P (Suc n)) \<Longrightarrow> P n\<close>. Even deeper nesting occurs in well-founded induction
+ \<open>(\<And>x. (\<And>y. y \<prec> x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x\<close>, but this already marks the limit of
+ rule complexity that is usually seen in practice.
\<^medskip>
- Regular user-level inferences in Isabelle/Pure always
- maintain the following canonical form of results:
+ Regular user-level inferences in Isabelle/Pure always maintain the following
+ canonical form of results:
- \<^item> Normalization by \<open>(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)\<close>,
- which is a theorem of Pure, means that quantifiers are pushed in
- front of implication at each level of nesting. The normal form is a
- Hereditary Harrop Formula.
+ \<^item> Normalization by \<open>(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)\<close>, which is a theorem of
+ Pure, means that quantifiers are pushed in front of implication at each
+ level of nesting. The normal form is a Hereditary Harrop Formula.
- \<^item> The outermost prefix of parameters is represented via
- schematic variables: instead of \<open>\<And>\<^vec>x. \<^vec>H \<^vec>x
- \<Longrightarrow> A \<^vec>x\<close> we have \<open>\<^vec>H ?\<^vec>x \<Longrightarrow> A ?\<^vec>x\<close>.
- Note that this representation looses information about the order of
- parameters, and vacuous quantifiers vanish automatically.
+ \<^item> The outermost prefix of parameters is represented via schematic variables:
+ instead of \<open>\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x\<close> we have \<open>\<^vec>H
+ ?\<^vec>x \<Longrightarrow> A ?\<^vec>x\<close>. Note that this representation looses information
+ about the order of parameters, and vacuous quantifiers vanish automatically.
\<close>
text %mlref \<open>
@@ -1010,43 +951,42 @@
@{index_ML Simplifier.norm_hhf: "Proof.context -> thm -> thm"} \\
\end{mldecls}
- \<^descr> @{ML Simplifier.norm_hhf}~\<open>ctxt thm\<close> normalizes the given
- theorem according to the canonical form specified above. This is
- occasionally helpful to repair some low-level tools that do not
- handle Hereditary Harrop Formulae properly.
+ \<^descr> @{ML Simplifier.norm_hhf}~\<open>ctxt thm\<close> normalizes the given theorem
+ according to the canonical form specified above. This is occasionally
+ helpful to repair some low-level tools that do not handle Hereditary Harrop
+ Formulae properly.
\<close>
subsection \<open>Rule composition\<close>
text \<open>
- The rule calculus of Isabelle/Pure provides two main inferences:
- @{inference resolution} (i.e.\ back-chaining of rules) and
- @{inference assumption} (i.e.\ closing a branch), both modulo
- higher-order unification. There are also combined variants, notably
- @{inference elim_resolution} and @{inference dest_resolution}.
+ The rule calculus of Isabelle/Pure provides two main inferences: @{inference
+ resolution} (i.e.\ back-chaining of rules) and @{inference assumption}
+ (i.e.\ closing a branch), both modulo higher-order unification. There are
+ also combined variants, notably @{inference elim_resolution} and @{inference
+ dest_resolution}.
- To understand the all-important @{inference resolution} principle,
- we first consider raw @{inference_def composition} (modulo
- higher-order unification with substitution \<open>\<vartheta>\<close>):
+ To understand the all-important @{inference resolution} principle, we first
+ consider raw @{inference_def composition} (modulo higher-order unification
+ with substitution \<open>\<vartheta>\<close>):
\[
\infer[(@{inference_def composition})]{\<open>\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
{\<open>\<^vec>A \<Longrightarrow> B\<close> & \<open>B' \<Longrightarrow> C\<close> & \<open>B\<vartheta> = B'\<vartheta>\<close>}
\]
- Here the conclusion of the first rule is unified with the premise of
- the second; the resulting rule instance inherits the premises of the
- first and conclusion of the second. Note that \<open>C\<close> can again
- consist of iterated implications. We can also permute the premises
- of the second rule back-and-forth in order to compose with \<open>B'\<close> in any position (subsequently we shall always refer to
- position 1 w.l.o.g.).
+ Here the conclusion of the first rule is unified with the premise of the
+ second; the resulting rule instance inherits the premises of the first and
+ conclusion of the second. Note that \<open>C\<close> can again consist of iterated
+ implications. We can also permute the premises of the second rule
+ back-and-forth in order to compose with \<open>B'\<close> in any position (subsequently
+ we shall always refer to position 1 w.l.o.g.).
- In @{inference composition} the internal structure of the common
- part \<open>B\<close> and \<open>B'\<close> is not taken into account. For
- proper @{inference resolution} we require \<open>B\<close> to be atomic,
- and explicitly observe the structure \<open>\<And>\<^vec>x. \<^vec>H
- \<^vec>x \<Longrightarrow> B' \<^vec>x\<close> of the premise of the second rule. The
- idea is to adapt the first rule by ``lifting'' it into this context,
- by means of iterated application of the following inferences:
+ In @{inference composition} the internal structure of the common part \<open>B\<close>
+ and \<open>B'\<close> is not taken into account. For proper @{inference resolution} we
+ require \<open>B\<close> to be atomic, and explicitly observe the structure \<open>\<And>\<^vec>x.
+ \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x\<close> of the premise of the second rule. The idea
+ is to adapt the first rule by ``lifting'' it into this context, by means of
+ iterated application of the following inferences:
\[
\infer[(@{inference_def imp_lift})]{\<open>(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)\<close>}{\<open>\<^vec>A \<Longrightarrow> B\<close>}
\]
@@ -1065,10 +1005,10 @@
\end{tabular}}
\]
- Continued resolution of rules allows to back-chain a problem towards
- more and sub-problems. Branches are closed either by resolving with
- a rule of 0 premises, or by producing a ``short-circuit'' within a
- solved situation (again modulo unification):
+ Continued resolution of rules allows to back-chain a problem towards more
+ and sub-problems. Branches are closed either by resolving with a rule of 0
+ premises, or by producing a ``short-circuit'' within a solved situation
+ (again modulo unification):
\[
\infer[(@{inference_def assumption})]{\<open>C\<vartheta>\<close>}
{\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C\<close> & \<open>A\<vartheta> = H\<^sub>i\<vartheta>\<close>~~\text{(for some~\<open>i\<close>)}}
@@ -1089,133 +1029,125 @@
@{index_ML_op "OF": "thm * thm list -> thm"} \\
\end{mldecls}
- \<^descr> \<open>rule\<^sub>1 RSN (i, rule\<^sub>2)\<close> resolves the conclusion of
- \<open>rule\<^sub>1\<close> with the \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>,
- according to the @{inference resolution} principle explained above.
- Unless there is precisely one resolvent it raises exception @{ML
- THM}.
+ \<^descr> \<open>rule\<^sub>1 RSN (i, rule\<^sub>2)\<close> resolves the conclusion of \<open>rule\<^sub>1\<close> with the
+ \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, according to the @{inference resolution}
+ principle explained above. Unless there is precisely one resolvent it raises
+ exception @{ML THM}.
- This corresponds to the rule attribute @{attribute THEN} in Isar
- source language.
+ This corresponds to the rule attribute @{attribute THEN} in Isar source
+ language.
- \<^descr> \<open>rule\<^sub>1 RS rule\<^sub>2\<close> abbreviates \<open>rule\<^sub>1 RSN (1,
- rule\<^sub>2)\<close>.
+ \<^descr> \<open>rule\<^sub>1 RS rule\<^sub>2\<close> abbreviates \<open>rule\<^sub>1 RSN (1, rule\<^sub>2)\<close>.
- \<^descr> \<open>rules\<^sub>1 RLN (i, rules\<^sub>2)\<close> joins lists of rules. For
- every \<open>rule\<^sub>1\<close> in \<open>rules\<^sub>1\<close> and \<open>rule\<^sub>2\<close> in
- \<open>rules\<^sub>2\<close>, it resolves the conclusion of \<open>rule\<^sub>1\<close> with
- the \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, accumulating multiple
- results in one big list. Note that such strict enumerations of
- higher-order unifications can be inefficient compared to the lazy
- variant seen in elementary tactics like @{ML resolve_tac}.
+ \<^descr> \<open>rules\<^sub>1 RLN (i, rules\<^sub>2)\<close> joins lists of rules. For every \<open>rule\<^sub>1\<close> in
+ \<open>rules\<^sub>1\<close> and \<open>rule\<^sub>2\<close> in \<open>rules\<^sub>2\<close>, it resolves the conclusion of \<open>rule\<^sub>1\<close>
+ with the \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, accumulating multiple results in one
+ big list. Note that such strict enumerations of higher-order unifications
+ can be inefficient compared to the lazy variant seen in elementary tactics
+ like @{ML resolve_tac}.
- \<^descr> \<open>rules\<^sub>1 RL rules\<^sub>2\<close> abbreviates \<open>rules\<^sub>1 RLN (1,
- rules\<^sub>2)\<close>.
+ \<^descr> \<open>rules\<^sub>1 RL rules\<^sub>2\<close> abbreviates \<open>rules\<^sub>1 RLN (1, rules\<^sub>2)\<close>.
- \<^descr> \<open>[rule\<^sub>1, \<dots>, rule\<^sub>n] MRS rule\<close> resolves \<open>rule\<^sub>i\<close>
- against premise \<open>i\<close> of \<open>rule\<close>, for \<open>i = n, \<dots>,
- 1\<close>. By working from right to left, newly emerging premises are
- concatenated in the result, without interfering.
+ \<^descr> \<open>[rule\<^sub>1, \<dots>, rule\<^sub>n] MRS rule\<close> resolves \<open>rule\<^sub>i\<close> against premise \<open>i\<close> of
+ \<open>rule\<close>, for \<open>i = n, \<dots>, 1\<close>. By working from right to left, newly emerging
+ premises are concatenated in the result, without interfering.
- \<^descr> \<open>rule OF rules\<close> is an alternative notation for \<open>rules MRS rule\<close>, which makes rule composition look more like
- function application. Note that the argument \<open>rules\<close> need
- not be atomic.
+ \<^descr> \<open>rule OF rules\<close> is an alternative notation for \<open>rules MRS rule\<close>, which
+ makes rule composition look more like function application. Note that the
+ argument \<open>rules\<close> need not be atomic.
- This corresponds to the rule attribute @{attribute OF} in Isar
- source language.
+ This corresponds to the rule attribute @{attribute OF} in Isar source
+ language.
\<close>
section \<open>Proof terms \label{sec:proof-terms}\<close>
-text \<open>The Isabelle/Pure inference kernel can record the proof of
- each theorem as a proof term that contains all logical inferences in
- detail. Rule composition by resolution (\secref{sec:obj-rules}) and
- type-class reasoning is broken down to primitive rules of the
- logical framework. The proof term can be inspected by a separate
- proof-checker, for example.
+text \<open>
+ The Isabelle/Pure inference kernel can record the proof of each theorem as a
+ proof term that contains all logical inferences in detail. Rule composition
+ by resolution (\secref{sec:obj-rules}) and type-class reasoning is broken
+ down to primitive rules of the logical framework. The proof term can be
+ inspected by a separate proof-checker, for example.
- According to the well-known \<^emph>\<open>Curry-Howard isomorphism\<close>, a proof
- can be viewed as a \<open>\<lambda>\<close>-term. Following this idea, proofs in
- Isabelle are internally represented by a datatype similar to the one
- for terms described in \secref{sec:terms}. On top of these
- syntactic terms, two more layers of \<open>\<lambda>\<close>-calculus are added,
- which correspond to \<open>\<And>x :: \<alpha>. B x\<close> and \<open>A \<Longrightarrow> B\<close>
- according to the propositions-as-types principle. The resulting
- 3-level \<open>\<lambda>\<close>-calculus resembles ``\<open>\<lambda>HOL\<close>'' in the
- more abstract setting of Pure Type Systems (PTS)
- @{cite "Barendregt-Geuvers:2001"}, if some fine points like schematic
- polymorphism and type classes are ignored.
+ According to the well-known \<^emph>\<open>Curry-Howard isomorphism\<close>, a proof can be
+ viewed as a \<open>\<lambda>\<close>-term. Following this idea, proofs in Isabelle are internally
+ represented by a datatype similar to the one for terms described in
+ \secref{sec:terms}. On top of these syntactic terms, two more layers of
+ \<open>\<lambda>\<close>-calculus are added, which correspond to \<open>\<And>x :: \<alpha>. B x\<close> and \<open>A \<Longrightarrow> B\<close>
+ according to the propositions-as-types principle. The resulting 3-level
+ \<open>\<lambda>\<close>-calculus resembles ``\<open>\<lambda>HOL\<close>'' in the more abstract setting of Pure Type
+ Systems (PTS) @{cite "Barendregt-Geuvers:2001"}, if some fine points like
+ schematic polymorphism and type classes are ignored.
\<^medskip>
- \<^emph>\<open>Proof abstractions\<close> of the form \<open>\<^bold>\<lambda>x :: \<alpha>. prf\<close>
- or \<open>\<^bold>\<lambda>p : A. prf\<close> correspond to introduction of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>, and \<^emph>\<open>proof applications\<close> of the form \<open>p \<cdot> t\<close> or \<open>p \<bullet> q\<close> correspond to elimination of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>. Actual types \<open>\<alpha>\<close>, propositions \<open>A\<close>, and terms \<open>t\<close> might be suppressed and reconstructed
- from the overall proof term.
+ \<^emph>\<open>Proof abstractions\<close> of the form \<open>\<^bold>\<lambda>x :: \<alpha>. prf\<close> or \<open>\<^bold>\<lambda>p : A. prf\<close>
+ correspond to introduction of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>, and \<^emph>\<open>proof applications\<close> of the form
+ \<open>p \<cdot> t\<close> or \<open>p \<bullet> q\<close> correspond to elimination of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>. Actual types \<open>\<alpha>\<close>,
+ propositions \<open>A\<close>, and terms \<open>t\<close> might be suppressed and reconstructed from
+ the overall proof term.
\<^medskip>
- Various atomic proofs indicate special situations within
- the proof construction as follows.
+ Various atomic proofs indicate special situations within the proof
+ construction as follows.
- A \<^emph>\<open>bound proof variable\<close> is a natural number \<open>b\<close> that
- acts as de-Bruijn index for proof term abstractions.
+ A \<^emph>\<open>bound proof variable\<close> is a natural number \<open>b\<close> that acts as de-Bruijn
+ index for proof term abstractions.
- A \<^emph>\<open>minimal proof\<close> ``\<open>?\<close>'' is a dummy proof term. This
- indicates some unrecorded part of the proof.
+ A \<^emph>\<open>minimal proof\<close> ``\<open>?\<close>'' is a dummy proof term. This indicates some
+ unrecorded part of the proof.
- \<open>Hyp A\<close> refers to some pending hypothesis by giving its
- proposition. This indicates an open context of implicit hypotheses,
- similar to loose bound variables or free variables within a term
- (\secref{sec:terms}).
+ \<open>Hyp A\<close> refers to some pending hypothesis by giving its proposition. This
+ indicates an open context of implicit hypotheses, similar to loose bound
+ variables or free variables within a term (\secref{sec:terms}).
- An \<^emph>\<open>axiom\<close> or \<^emph>\<open>oracle\<close> \<open>a : A[\<^vec>\<tau>]\<close> refers
- some postulated \<open>proof constant\<close>, which is subject to
- schematic polymorphism of theory content, and the particular type
- instantiation may be given explicitly. The vector of types \<open>\<^vec>\<tau>\<close> refers to the schematic type variables in the generic
+ An \<^emph>\<open>axiom\<close> or \<^emph>\<open>oracle\<close> \<open>a : A[\<^vec>\<tau>]\<close> refers some postulated \<open>proof
+ constant\<close>, which is subject to schematic polymorphism of theory content, and
+ the particular type instantiation may be given explicitly. The vector of
+ types \<open>\<^vec>\<tau>\<close> refers to the schematic type variables in the generic
proposition \<open>A\<close> in canonical order.
- A \<^emph>\<open>proof promise\<close> \<open>a : A[\<^vec>\<tau>]\<close> is a placeholder
- for some proof of polymorphic proposition \<open>A\<close>, with explicit
- type instantiation as given by the vector \<open>\<^vec>\<tau>\<close>, as
- above. Unlike axioms or oracles, proof promises may be
- \<^emph>\<open>fulfilled\<close> eventually, by substituting \<open>a\<close> by some
- particular proof \<open>q\<close> at the corresponding type instance.
- This acts like Hindley-Milner \<open>let\<close>-polymorphism: a generic
- local proof definition may get used at different type instances, and
- is replaced by the concrete instance eventually.
+ A \<^emph>\<open>proof promise\<close> \<open>a : A[\<^vec>\<tau>]\<close> is a placeholder for some proof of
+ polymorphic proposition \<open>A\<close>, with explicit type instantiation as given by
+ the vector \<open>\<^vec>\<tau>\<close>, as above. Unlike axioms or oracles, proof promises
+ may be \<^emph>\<open>fulfilled\<close> eventually, by substituting \<open>a\<close> by some particular proof
+ \<open>q\<close> at the corresponding type instance. This acts like Hindley-Milner
+ \<open>let\<close>-polymorphism: a generic local proof definition may get used at
+ different type instances, and is replaced by the concrete instance
+ eventually.
- A \<^emph>\<open>named theorem\<close> wraps up some concrete proof as a closed
- formal entity, in the manner of constant definitions for proof
- terms. The \<^emph>\<open>proof body\<close> of such boxed theorems involves some
- digest about oracles and promises occurring in the original proof.
- This allows the inference kernel to manage this critical information
- without the full overhead of explicit proof terms.
+ A \<^emph>\<open>named theorem\<close> wraps up some concrete proof as a closed formal entity,
+ in the manner of constant definitions for proof terms. The \<^emph>\<open>proof body\<close> of
+ such boxed theorems involves some digest about oracles and promises
+ occurring in the original proof. This allows the inference kernel to manage
+ this critical information without the full overhead of explicit proof terms.
\<close>
subsection \<open>Reconstructing and checking proof terms\<close>
-text \<open>Fully explicit proof terms can be large, but most of this
- information is redundant and can be reconstructed from the context.
- Therefore, the Isabelle/Pure inference kernel records only
- \<^emph>\<open>implicit\<close> proof terms, by omitting all typing information in
- terms, all term and type labels of proof abstractions, and some
- argument terms of applications \<open>p \<cdot> t\<close> (if possible).
+text \<open>
+ Fully explicit proof terms can be large, but most of this information is
+ redundant and can be reconstructed from the context. Therefore, the
+ Isabelle/Pure inference kernel records only \<^emph>\<open>implicit\<close> proof terms, by
+ omitting all typing information in terms, all term and type labels of proof
+ abstractions, and some argument terms of applications \<open>p \<cdot> t\<close> (if possible).
- There are separate operations to reconstruct the full proof term
- later on, using \<^emph>\<open>higher-order pattern unification\<close>
- @{cite "nipkow-patterns" and "Berghofer-Nipkow:2000:TPHOL"}.
+ There are separate operations to reconstruct the full proof term later on,
+ using \<^emph>\<open>higher-order pattern unification\<close> @{cite "nipkow-patterns" and
+ "Berghofer-Nipkow:2000:TPHOL"}.
- The \<^emph>\<open>proof checker\<close> expects a fully reconstructed proof term,
- and can turn it into a theorem by replaying its primitive inferences
- within the kernel.\<close>
+ The \<^emph>\<open>proof checker\<close> expects a fully reconstructed proof term, and can turn
+ it into a theorem by replaying its primitive inferences within the kernel.
+\<close>
subsection \<open>Concrete syntax of proof terms\<close>
-text \<open>The concrete syntax of proof terms is a slight extension of
- the regular inner syntax of Isabelle/Pure @{cite "isabelle-isar-ref"}.
- Its main syntactic category @{syntax (inner) proof} is defined as
- follows:
+text \<open>
+ The concrete syntax of proof terms is a slight extension of the regular
+ inner syntax of Isabelle/Pure @{cite "isabelle-isar-ref"}. Its main
+ syntactic category @{syntax (inner) proof} is defined as follows:
\begin{center}
\begin{supertabular}{rclr}
@@ -1240,13 +1172,14 @@
\end{supertabular}
\end{center}
- Implicit term arguments in partial proofs are indicated by ``\<open>_\<close>''. Type arguments for theorems and axioms may be specified
- using \<open>p \<cdot> TYPE(type)\<close> (they must appear before any other
- term argument of a theorem or axiom, but may be omitted altogether).
+ Implicit term arguments in partial proofs are indicated by ``\<open>_\<close>''. Type
+ arguments for theorems and axioms may be specified using \<open>p \<cdot> TYPE(type)\<close>
+ (they must appear before any other term argument of a theorem or axiom, but
+ may be omitted altogether).
\<^medskip>
- There are separate read and print operations for proof
- terms, in order to avoid conflicts with the regular term language.
+ There are separate read and print operations for proof terms, in order to
+ avoid conflicts with the regular term language.
\<close>
text %mlref \<open>
@@ -1263,65 +1196,60 @@
@{index_ML Proof_Syntax.pretty_proof: "Proof.context -> proof -> Pretty.T"} \\
\end{mldecls}
- \<^descr> Type @{ML_type proof} represents proof terms; this is a
- datatype with constructors @{index_ML Abst}, @{index_ML AbsP},
- @{index_ML_op "%"}, @{index_ML_op "%%"}, @{index_ML PBound},
- @{index_ML MinProof}, @{index_ML Hyp}, @{index_ML PAxm}, @{index_ML
- Oracle}, @{index_ML Promise}, @{index_ML PThm} as explained above.
- %FIXME OfClass (!?)
+ \<^descr> Type @{ML_type proof} represents proof terms; this is a datatype with
+ constructors @{index_ML Abst}, @{index_ML AbsP}, @{index_ML_op "%"},
+ @{index_ML_op "%%"}, @{index_ML PBound}, @{index_ML MinProof}, @{index_ML
+ Hyp}, @{index_ML PAxm}, @{index_ML Oracle}, @{index_ML Promise}, @{index_ML
+ PThm} as explained above. %FIXME OfClass (!?)
+
+ \<^descr> Type @{ML_type proof_body} represents the nested proof information of a
+ named theorem, consisting of a digest of oracles and named theorem over some
+ proof term. The digest only covers the directly visible part of the proof:
+ in order to get the full information, the implicit graph of nested theorems
+ needs to be traversed (e.g.\ using @{ML Proofterm.fold_body_thms}).
- \<^descr> Type @{ML_type proof_body} represents the nested proof
- information of a named theorem, consisting of a digest of oracles
- and named theorem over some proof term. The digest only covers the
- directly visible part of the proof: in order to get the full
- information, the implicit graph of nested theorems needs to be
- traversed (e.g.\ using @{ML Proofterm.fold_body_thms}).
+ \<^descr> @{ML Thm.proof_of}~\<open>thm\<close> and @{ML Thm.proof_body_of}~\<open>thm\<close> produce the
+ proof term or proof body (with digest of oracles and theorems) from a given
+ theorem. Note that this involves a full join of internal futures that
+ fulfill pending proof promises, and thus disrupts the natural bottom-up
+ construction of proofs by introducing dynamic ad-hoc dependencies. Parallel
+ performance may suffer by inspecting proof terms at run-time.
- \<^descr> @{ML Thm.proof_of}~\<open>thm\<close> and @{ML
- Thm.proof_body_of}~\<open>thm\<close> produce the proof term or proof
- body (with digest of oracles and theorems) from a given theorem.
- Note that this involves a full join of internal futures that fulfill
- pending proof promises, and thus disrupts the natural bottom-up
- construction of proofs by introducing dynamic ad-hoc dependencies.
- Parallel performance may suffer by inspecting proof terms at
- run-time.
+ \<^descr> @{ML proofs} specifies the detail of proof recording within @{ML_type thm}
+ values produced by the inference kernel: @{ML 0} records only the names of
+ oracles, @{ML 1} records oracle names and propositions, @{ML 2} additionally
+ records full proof terms. Officially named theorems that contribute to a
+ result are recorded in any case.
- \<^descr> @{ML proofs} specifies the detail of proof recording within
- @{ML_type thm} values produced by the inference kernel: @{ML 0}
- records only the names of oracles, @{ML 1} records oracle names and
- propositions, @{ML 2} additionally records full proof terms.
- Officially named theorems that contribute to a result are recorded
- in any case.
+ \<^descr> @{ML Reconstruct.reconstruct_proof}~\<open>thy prop prf\<close> turns the implicit
+ proof term \<open>prf\<close> into a full proof of the given proposition.
- \<^descr> @{ML Reconstruct.reconstruct_proof}~\<open>thy prop prf\<close>
- turns the implicit proof term \<open>prf\<close> into a full proof of the
- given proposition.
-
- Reconstruction may fail if \<open>prf\<close> is not a proof of \<open>prop\<close>, or if it does not contain sufficient information for
- reconstruction. Failure may only happen for proofs that are
- constructed manually, but not for those produced automatically by
- the inference kernel.
+ Reconstruction may fail if \<open>prf\<close> is not a proof of \<open>prop\<close>, or if it does not
+ contain sufficient information for reconstruction. Failure may only happen
+ for proofs that are constructed manually, but not for those produced
+ automatically by the inference kernel.
- \<^descr> @{ML Reconstruct.expand_proof}~\<open>thy [thm\<^sub>1, \<dots>, thm\<^sub>n]
- prf\<close> expands and reconstructs the proofs of all specified theorems,
- with the given (full) proof. Theorems that are not unique specified
- via their name may be disambiguated by giving their proposition.
+ \<^descr> @{ML Reconstruct.expand_proof}~\<open>thy [thm\<^sub>1, \<dots>, thm\<^sub>n] prf\<close> expands and
+ reconstructs the proofs of all specified theorems, with the given (full)
+ proof. Theorems that are not unique specified via their name may be
+ disambiguated by giving their proposition.
- \<^descr> @{ML Proof_Checker.thm_of_proof}~\<open>thy prf\<close> turns the
- given (full) proof into a theorem, by replaying it using only
- primitive rules of the inference kernel.
+ \<^descr> @{ML Proof_Checker.thm_of_proof}~\<open>thy prf\<close> turns the given (full) proof
+ into a theorem, by replaying it using only primitive rules of the inference
+ kernel.
- \<^descr> @{ML Proof_Syntax.read_proof}~\<open>thy b\<^sub>1 b\<^sub>2 s\<close> reads in a
- proof term. The Boolean flags indicate the use of sort and type
- information. Usually, typing information is left implicit and is
- inferred during proof reconstruction. %FIXME eliminate flags!?
+ \<^descr> @{ML Proof_Syntax.read_proof}~\<open>thy b\<^sub>1 b\<^sub>2 s\<close> reads in a proof term. The
+ Boolean flags indicate the use of sort and type information. Usually, typing
+ information is left implicit and is inferred during proof reconstruction.
+ %FIXME eliminate flags!?
- \<^descr> @{ML Proof_Syntax.pretty_proof}~\<open>ctxt prf\<close>
- pretty-prints the given proof term.
+ \<^descr> @{ML Proof_Syntax.pretty_proof}~\<open>ctxt prf\<close> pretty-prints the given proof
+ term.
\<close>
-text %mlex \<open>Detailed proof information of a theorem may be retrieved
- as follows:\<close>
+text %mlex \<open>
+ Detailed proof information of a theorem may be retrieved as follows:
+\<close>
lemma ex: "A \<and> B \<longrightarrow> B \<and> A"
proof
@@ -1344,15 +1272,16 @@
(fn (name, _, _) => insert (op =) name) [body] [];
\<close>
-text \<open>The result refers to various basic facts of Isabelle/HOL:
- @{thm [source] HOL.impI}, @{thm [source] HOL.conjE}, @{thm [source]
- HOL.conjI} etc. The combinator @{ML Proofterm.fold_body_thms}
- recursively explores the graph of the proofs of all theorems being
- used here.
+text \<open>
+ The result refers to various basic facts of Isabelle/HOL: @{thm [source]
+ HOL.impI}, @{thm [source] HOL.conjE}, @{thm [source] HOL.conjI} etc. The
+ combinator @{ML Proofterm.fold_body_thms} recursively explores the graph of
+ the proofs of all theorems being used here.
\<^medskip>
- Alternatively, we may produce a proof term manually, and
- turn it into a theorem as follows:\<close>
+ Alternatively, we may produce a proof term manually, and turn it into a
+ theorem as follows:
+\<close>
ML_val \<open>
val thy = @{theory};
@@ -1371,9 +1300,8 @@
text \<open>
\<^medskip>
- See also @{file "~~/src/HOL/Proofs/ex/XML_Data.thy"}
- for further examples, with export and import of proof terms via
- XML/ML data representation.
+ See also @{file "~~/src/HOL/Proofs/ex/XML_Data.thy"} for further examples,
+ with export and import of proof terms via XML/ML data representation.
\<close>
end