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doc-src/IsarImplementation/Thy/logic.thy

changeset 20537 | b6b49903db7e |

parent 20521 | 189811b39869 |

child 20542 | a54ca4e90874 |

1.1 --- a/doc-src/IsarImplementation/Thy/logic.thy Thu Sep 14 15:27:08 2006 +0200 1.2 +++ b/doc-src/IsarImplementation/Thy/logic.thy Thu Sep 14 15:51:20 2006 +0200 1.3 @@ -20,12 +20,12 @@ 1.4 "\<And>"} for universal quantification (proofs depending on terms), and 1.5 @{text "\<Longrightarrow>"} for implication (proofs depending on proofs). 1.6 1.7 - Pure derivations are relative to a logical theory, which declares 1.8 - type constructors, term constants, and axioms. Theory declarations 1.9 - support schematic polymorphism, which is strictly speaking outside 1.10 - the logic.\footnote{Incidently, this is the main logical reason, why 1.11 - the theory context @{text "\<Theta>"} is separate from the context @{text 1.12 - "\<Gamma>"} of the core calculus.} 1.13 + Derivations are relative to a logical theory, which declares type 1.14 + constructors, constants, and axioms. Theory declarations support 1.15 + schematic polymorphism, which is strictly speaking outside the 1.16 + logic.\footnote{This is the deeper logical reason, why the theory 1.17 + context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"} 1.18 + of the core calculus.} 1.19 *} 1.20 1.21 1.22 @@ -42,8 +42,8 @@ 1.23 internally. The resulting relation is an ordering: reflexive, 1.24 transitive, and antisymmetric. 1.25 1.26 - A \emph{sort} is a list of type classes written as @{text 1.27 - "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic 1.28 + A \emph{sort} is a list of type classes written as @{text "s = 1.29 + {c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic 1.30 intersection. Notationally, the curly braces are omitted for 1.31 singleton intersections, i.e.\ any class @{text "c"} may be read as 1.32 a sort @{text "{c}"}. The ordering on type classes is extended to 1.33 @@ -56,11 +56,11 @@ 1.34 elements wrt.\ the sort order. 1.35 1.36 \medskip A \emph{fixed type variable} is a pair of a basic name 1.37 - (starting with a @{text "'"} character) and a sort constraint. For 1.38 - example, @{text "('a, s)"} which is usually printed as @{text 1.39 - "\<alpha>\<^isub>s"}. A \emph{schematic type variable} is a pair of an 1.40 - indexname and a sort constraint. For example, @{text "(('a, 0), 1.41 - s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}. 1.42 + (starting with a @{text "'"} character) and a sort constraint, e.g.\ 1.43 + @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}. 1.44 + A \emph{schematic type variable} is a pair of an indexname and a 1.45 + sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually 1.46 + printed as @{text "?\<alpha>\<^isub>s"}. 1.47 1.48 Note that \emph{all} syntactic components contribute to the identity 1.49 of type variables, including the sort constraint. The core logic 1.50 @@ -70,23 +70,23 @@ 1.51 1.52 A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator 1.53 on types declared in the theory. Type constructor application is 1.54 - usually written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}. 1.55 - For @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text 1.56 - "prop"} instead of @{text "()prop"}. For @{text "k = 1"} the 1.57 - parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text 1.58 - "(\<alpha>)list"}. Further notation is provided for specific constructors, 1.59 - notably the right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of 1.60 - @{text "(\<alpha>, \<beta>)fun"}. 1.61 + written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}. For 1.62 + @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"} 1.63 + instead of @{text "()prop"}. For @{text "k = 1"} the parentheses 1.64 + are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}. 1.65 + Further notation is provided for specific constructors, notably the 1.66 + right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>, 1.67 + \<beta>)fun"}. 1.68 1.69 - A \emph{type} @{text "\<tau>"} is defined inductively over type variables 1.70 - and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | 1.71 - ?\<alpha>\<^isub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}. 1.72 + A \emph{type} is defined inductively over type variables and type 1.73 + constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s | 1.74 + (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}. 1.75 1.76 A \emph{type abbreviation} is a syntactic definition @{text 1.77 "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over 1.78 - variables @{text "\<^vec>\<alpha>"}. Type abbreviations looks like type 1.79 - constructors at the surface, but are fully expanded before entering 1.80 - the logical core. 1.81 + variables @{text "\<^vec>\<alpha>"}. Type abbreviations appear as type 1.82 + constructors in the syntax, but are expanded before entering the 1.83 + logical core. 1.84 1.85 A \emph{type arity} declares the image behavior of a type 1.86 constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>, 1.87 @@ -98,22 +98,22 @@ 1.88 1.89 \medskip The sort algebra is always maintained as \emph{coregular}, 1.90 which means that type arities are consistent with the subclass 1.91 - relation: for each type constructor @{text "\<kappa>"} and classes @{text 1.92 - "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> :: 1.93 - (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa> 1.94 - :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq> 1.95 - \<^vec>s\<^isub>2"} holds component-wise. 1.96 + relation: for any type constructor @{text "\<kappa>"}, and classes @{text 1.97 + "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> :: 1.98 + (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> :: 1.99 + (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq> 1.100 + \<^vec>s\<^isub>2"} component-wise. 1.101 1.102 The key property of a coregular order-sorted algebra is that sort 1.103 - constraints may be always solved in a most general fashion: for each 1.104 - type constructor @{text "\<kappa>"} and sort @{text "s"} there is a most 1.105 - general vector of argument sorts @{text "(s\<^isub>1, \<dots>, 1.106 - s\<^isub>k)"} such that a type scheme @{text 1.107 - "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is 1.108 - of sort @{text "s"}. Consequently, the unification problem on the 1.109 - algebra of types has most general solutions (modulo renaming and 1.110 - equivalence of sorts). Moreover, the usual type-inference algorithm 1.111 - will produce primary types as expected \cite{nipkow-prehofer}. 1.112 + constraints can be solved in a most general fashion: for each type 1.113 + constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general 1.114 + vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such 1.115 + that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, 1.116 + \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}. 1.117 + Consequently, unification on the algebra of types has most general 1.118 + solutions (modulo equivalence of sorts). This means that 1.119 + type-inference will produce primary types as expected 1.120 + \cite{nipkow-prehofer}. 1.121 *} 1.122 1.123 text %mlref {* 1.124 @@ -149,20 +149,21 @@ 1.125 \item @{ML_type typ} represents types; this is a datatype with 1.126 constructors @{ML TFree}, @{ML TVar}, @{ML Type}. 1.127 1.128 - \item @{ML map_atyps}~@{text "f \<tau>"} applies mapping @{text "f"} to 1.129 - all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text "\<tau>"}. 1.130 + \item @{ML map_atyps}~@{text "f \<tau>"} applies the mapping @{text "f"} 1.131 + to all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text 1.132 + "\<tau>"}. 1.133 1.134 - \item @{ML fold_atyps}~@{text "f \<tau>"} iterates operation @{text "f"} 1.135 - over all occurrences of atoms (@{ML TFree}, @{ML TVar}) in @{text 1.136 - "\<tau>"}; the type structure is traversed from left to right. 1.137 + \item @{ML fold_atyps}~@{text "f \<tau>"} iterates the operation @{text 1.138 + "f"} over all occurrences of atomic types (@{ML TFree}, @{ML TVar}) 1.139 + in @{text "\<tau>"}; the type structure is traversed from left to right. 1.140 1.141 \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"} 1.142 tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}. 1.143 1.144 - \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type 1.145 - is of a given sort. 1.146 + \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type 1.147 + @{text "\<tau>"} is of sort @{text "s"}. 1.148 1.149 - \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares new 1.150 + \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares a new 1.151 type constructors @{text "\<kappa>"} with @{text "k"} arguments and 1.152 optional mixfix syntax. 1.153 1.154 @@ -171,7 +172,7 @@ 1.155 optional mixfix syntax. 1.156 1.157 \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>, 1.158 - c\<^isub>n])"} declares new class @{text "c"}, together with class 1.159 + c\<^isub>n])"} declares a new class @{text "c"}, together with class 1.160 relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}. 1.161 1.162 \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1, 1.163 @@ -179,7 +180,7 @@ 1.164 c\<^isub>2"}. 1.165 1.166 \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares 1.167 - arity @{text "\<kappa> :: (\<^vec>s)s"}. 1.168 + the arity @{text "\<kappa> :: (\<^vec>s)s"}. 1.169 1.170 \end{description} 1.171 *} 1.172 @@ -193,62 +194,66 @@ 1.173 1.174 The language of terms is that of simply-typed @{text "\<lambda>"}-calculus 1.175 with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72} 1.176 - or \cite{paulson-ml2}), and named free variables and constants. 1.177 - Terms with loose bound variables are usually considered malformed. 1.178 - The types of variables and constants is stored explicitly at each 1.179 - occurrence in the term. 1.180 + or \cite{paulson-ml2}), with the types being determined determined 1.181 + by the corresponding binders. In contrast, free variables and 1.182 + constants are have an explicit name and type in each occurrence. 1.183 1.184 \medskip A \emph{bound variable} is a natural number @{text "b"}, 1.185 - which refers to the next binder that is @{text "b"} steps upwards 1.186 - from the occurrence of @{text "b"} (counting from zero). Bindings 1.187 - may be introduced as abstractions within the term, or as a separate 1.188 - context (an inside-out list). This associates each bound variable 1.189 - with a type. A \emph{loose variables} is a bound variable that is 1.190 - outside the current scope of local binders or the context. For 1.191 + which accounts for the number of intermediate binders between the 1.192 + variable occurrence in the body and its binding position. For 1.193 example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"} 1.194 - corresponds to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a named 1.195 - representation. Also note that the very same bound variable may get 1.196 - different numbers at different occurrences. 1.197 + would correspond to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a 1.198 + named representation. Note that a bound variable may be represented 1.199 + by different de-Bruijn indices at different occurrences, depending 1.200 + on the nesting of abstractions. 1.201 + 1.202 + A \emph{loose variables} is a bound variable that is outside the 1.203 + scope of local binders. The types (and names) for loose variables 1.204 + can be managed as a separate context, that is maintained inside-out 1.205 + like a stack of hypothetical binders. The core logic only operates 1.206 + on closed terms, without any loose variables. 1.207 1.208 - A \emph{fixed variable} is a pair of a basic name and a type. For 1.209 - example, @{text "(x, \<tau>)"} which is usually printed @{text 1.210 - "x\<^isub>\<tau>"}. A \emph{schematic variable} is a pair of an 1.211 - indexname and a type. For example, @{text "((x, 0), \<tau>)"} which is 1.212 - usually printed as @{text "?x\<^isub>\<tau>"}. 1.213 + A \emph{fixed variable} is a pair of a basic name and a type, e.g.\ 1.214 + @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}. A 1.215 + \emph{schematic variable} is a pair of an indexname and a type, 1.216 + e.g.\ @{text "((x, 0), \<tau>)"} which is usually printed as @{text 1.217 + "?x\<^isub>\<tau>"}. 1.218 1.219 - \medskip A \emph{constant} is a atomic terms consisting of a basic 1.220 - name and a type. Constants are declared in the context as 1.221 - polymorphic families @{text "c :: \<sigma>"}, meaning that any @{text 1.222 - "c\<^isub>\<tau>"} is a valid constant for all substitution instances 1.223 - @{text "\<tau> \<le> \<sigma>"}. 1.224 + \medskip A \emph{constant} is a pair of a basic name and a type, 1.225 + e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text 1.226 + "c\<^isub>\<tau>"}. Constants are declared in the context as polymorphic 1.227 + families @{text "c :: \<sigma>"}, meaning that valid all substitution 1.228 + instances @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid. 1.229 1.230 - The list of \emph{type arguments} of @{text "c\<^isub>\<tau>"} wrt.\ the 1.231 - declaration @{text "c :: \<sigma>"} is the codomain of the type matcher 1.232 - presented in canonical order (according to the left-to-right 1.233 - occurrences of type variables in in @{text "\<sigma>"}). Thus @{text 1.234 - "c\<^isub>\<tau>"} can be represented more compactly as @{text 1.235 - "c(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. For example, the instance @{text 1.236 - "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} of some @{text "plus :: \<alpha> \<Rightarrow> \<alpha> 1.237 - \<Rightarrow> \<alpha>"} has the singleton list @{text "nat"} as type arguments, the 1.238 - constant may be represented as @{text "plus(nat)"}. 1.239 + The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"} 1.240 + wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of 1.241 + the matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>, 1.242 + ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in canonical order @{text 1.243 + "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. Within a given theory context, 1.244 + there is a one-to-one correspondence between any constant @{text 1.245 + "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1, \<dots>, 1.246 + \<tau>\<^isub>n)"} of its type arguments. For example, with @{text "plus 1.247 + :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> 1.248 + nat\<^esub>"} corresponds to @{text "plus(nat)"}. 1.249 1.250 Constant declarations @{text "c :: \<sigma>"} may contain sort constraints 1.251 for type variables in @{text "\<sigma>"}. These are observed by 1.252 type-inference as expected, but \emph{ignored} by the core logic. 1.253 This means the primitive logic is able to reason with instances of 1.254 - polymorphic constants that the user-level type-checker would reject. 1.255 + polymorphic constants that the user-level type-checker would reject 1.256 + due to violation of type class restrictions. 1.257 1.258 - \medskip A \emph{term} @{text "t"} is defined inductively over 1.259 - variables and constants, with abstraction and application as 1.260 - follows: @{text "t = b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | 1.261 - \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}. Parsing and printing takes 1.262 - care of converting between an external representation with named 1.263 - bound variables. Subsequently, we shall use the latter notation 1.264 - instead of internal de-Bruijn representation. 1.265 + \medskip A \emph{term} is defined inductively over variables and 1.266 + constants, with abstraction and application as follows: @{text "t = 1.267 + b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | 1.268 + t\<^isub>1 t\<^isub>2"}. Parsing and printing takes care of 1.269 + converting between an external representation with named bound 1.270 + variables. Subsequently, we shall use the latter notation instead 1.271 + of internal de-Bruijn representation. 1.272 1.273 - The subsequent inductive relation @{text "t :: \<tau>"} assigns a 1.274 - (unique) type to a term, using the special type constructor @{text 1.275 - "(\<alpha>, \<beta>)fun"}, which is written @{text "\<alpha> \<Rightarrow> \<beta>"}. 1.276 + The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a 1.277 + term according to the structure of atomic terms, abstractions, and 1.278 + applicatins: 1.279 \[ 1.280 \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{} 1.281 \qquad 1.282 @@ -264,46 +269,47 @@ 1.283 Type-inference depends on a context of type constraints for fixed 1.284 variables, and declarations for polymorphic constants. 1.285 1.286 - The identity of atomic terms consists both of the name and the type. 1.287 - Thus different entities @{text "c\<^bsub>\<tau>\<^isub>1\<^esub>"} and 1.288 - @{text "c\<^bsub>\<tau>\<^isub>2\<^esub>"} may well identified by type 1.289 - instantiation, by mapping @{text "\<tau>\<^isub>1"} and @{text 1.290 - "\<tau>\<^isub>2"} to the same @{text "\<tau>"}. Although, 1.291 - different type instances of constants of the same basic name are 1.292 - commonplace, this rarely happens for variables: type-inference 1.293 - always demands ``consistent'' type constraints. 1.294 + The identity of atomic terms consists both of the name and the type 1.295 + component. This means that different variables @{text 1.296 + "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text 1.297 + "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type 1.298 + instantiation. Some outer layers of the system make it hard to 1.299 + produce variables of the same name, but different types. In 1.300 + particular, type-inference always demands ``consistent'' type 1.301 + constraints for free variables. In contrast, mixed instances of 1.302 + polymorphic constants occur frequently. 1.303 1.304 \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"} 1.305 is the set of type variables occurring in @{text "t"}, but not in 1.306 - @{text "\<sigma>"}. This means that the term implicitly depends on the 1.307 - values of various type variables that are not visible in the overall 1.308 - type, i.e.\ there are different type instances @{text "t\<vartheta> 1.309 - :: \<sigma>"} and @{text "t\<vartheta>' :: \<sigma>"} with the same type. This 1.310 - slightly pathological situation is apt to cause strange effects. 1.311 + @{text "\<sigma>"}. This means that the term implicitly depends on type 1.312 + arguments that are not accounted in result type, i.e.\ there are 1.313 + different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text 1.314 + "t\<vartheta>' :: \<sigma>"} with the same type. This slightly 1.315 + pathological situation demands special care. 1.316 1.317 \medskip A \emph{term abbreviation} is a syntactic definition @{text 1.318 - "c\<^isub>\<sigma> \<equiv> t"} of an arbitrary closed term @{text "t"} of type 1.319 - @{text "\<sigma>"} without any hidden polymorphism. A term abbreviation 1.320 - looks like a constant at the surface, but is fully expanded before 1.321 - entering the logical core. Abbreviations are usually reverted when 1.322 - printing terms, using rules @{text "t \<rightarrow> c\<^isub>\<sigma>"} has a 1.323 - higher-order term rewrite system. 1.324 + "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"}, 1.325 + without any hidden polymorphism. A term abbreviation looks like a 1.326 + constant in the syntax, but is fully expanded before entering the 1.327 + logical core. Abbreviations are usually reverted when printing 1.328 + terms, using the collective @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for 1.329 + higher-order rewriting. 1.330 1.331 \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text 1.332 - "\<alpha>\<beta>\<eta>"}-conversion. @{text "\<alpha>"}-conversion refers to capture-free 1.333 + "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free 1.334 renaming of bound variables; @{text "\<beta>"}-conversion contracts an 1.335 - abstraction applied to some argument term, substituting the argument 1.336 + abstraction applied to an argument term, substituting the argument 1.337 in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text 1.338 "\<eta>"}-conversion contracts vacuous application-abstraction: @{text 1.339 "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable 1.340 - @{text "0"} does not occur in @{text "f"}. 1.341 + does not occur in @{text "f"}. 1.342 1.343 - Terms are almost always treated module @{text "\<alpha>"}-conversion, which 1.344 - is implicit in the de-Bruijn representation. The names in 1.345 - abstractions of bound variables are maintained only as a comment for 1.346 - parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-equivalence is usually 1.347 - taken for granted higher rules (\secref{sec:rules}), anything 1.348 - depending on higher-order unification or rewriting. 1.349 + Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is 1.350 + implicit in the de-Bruijn representation. Names for bound variables 1.351 + in abstractions are maintained separately as (meaningless) comments, 1.352 + mostly for parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-conversion is 1.353 + commonplace in various higher operations (\secref{sec:rules}) that 1.354 + are based on higher-order unification and matching. 1.355 *} 1.356 1.357 text %mlref {* 1.358 @@ -326,43 +332,43 @@ 1.359 1.360 \begin{description} 1.361 1.362 - \item @{ML_type term} represents de-Bruijn terms with comments in 1.363 - abstractions for bound variable names. This is a datatype with 1.364 - constructors @{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}, @{ML 1.365 - Abs}, @{ML "op $"}. 1.366 + \item @{ML_type term} represents de-Bruijn terms, with comments in 1.367 + abstractions, and explicitly named free variables and constants; 1.368 + this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML 1.369 + Var}, @{ML Const}, @{ML Abs}, @{ML "op $"}. 1.370 1.371 \item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text 1.372 "\<alpha>"}-equivalence of two terms. This is the basic equality relation 1.373 on type @{ML_type term}; raw datatype equality should only be used 1.374 for operations related to parsing or printing! 1.375 1.376 - \item @{ML map_term_types}~@{text "f t"} applies mapping @{text "f"} 1.377 - to all types occurring in @{text "t"}. 1.378 + \item @{ML map_term_types}~@{text "f t"} applies the mapping @{text 1.379 + "f"} to all types occurring in @{text "t"}. 1.380 + 1.381 + \item @{ML fold_types}~@{text "f t"} iterates the operation @{text 1.382 + "f"} over all occurrences of types in @{text "t"}; the term 1.383 + structure is traversed from left to right. 1.384 1.385 - \item @{ML fold_types}~@{text "f t"} iterates operation @{text "f"} 1.386 - over all occurrences of types in @{text "t"}; the term structure is 1.387 + \item @{ML map_aterms}~@{text "f t"} applies the mapping @{text "f"} 1.388 + to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML 1.389 + Const}) occurring in @{text "t"}. 1.390 + 1.391 + \item @{ML fold_aterms}~@{text "f t"} iterates the operation @{text 1.392 + "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML Free}, 1.393 + @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is 1.394 traversed from left to right. 1.395 1.396 - \item @{ML map_aterms}~@{text "f t"} applies mapping @{text "f"} to 1.397 - all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}) 1.398 - occurring in @{text "t"}. 1.399 - 1.400 - \item @{ML fold_aterms}~@{text "f t"} iterates operation @{text "f"} 1.401 - over all occurrences of atomic terms in (@{ML Bound}, @{ML Free}, 1.402 - @{ML Var}, @{ML Const}) @{text "t"}; the term structure is traversed 1.403 - from left to right. 1.404 - 1.405 - \item @{ML fastype_of}~@{text "t"} recomputes the type of a 1.406 - well-formed term, while omitting any sanity checks. This operation 1.407 - is relatively slow. 1.408 + \item @{ML fastype_of}~@{text "t"} determines the type of a 1.409 + well-typed term. This operation is relatively slow, despite the 1.410 + omission of any sanity checks. 1.411 1.412 \item @{ML lambda}~@{text "a b"} produces an abstraction @{text 1.413 - "\<lambda>a. b"}, where occurrences of the original (atomic) term @{text 1.414 - "a"} in the body @{text "b"} are replaced by bound variables. 1.415 + "\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the 1.416 + body @{text "b"} are replaced by bound variables. 1.417 1.418 - \item @{ML betapply}~@{text "t u"} produces an application @{text "t 1.419 - u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} happens to 1.420 - be an abstraction. 1.421 + \item @{ML betapply}~@{text "(t, u)"} produces an application @{text 1.422 + "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an 1.423 + abstraction. 1.424 1.425 \item @{ML Sign.add_consts_i}~@{text "[(c, \<sigma>, mx), \<dots>]"} declares a 1.426 new constant @{text "c :: \<sigma>"} with optional mixfix syntax. 1.427 @@ -373,9 +379,9 @@ 1.428 1.429 \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML 1.430 Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"} 1.431 - convert between the two representations of constants, namely full 1.432 - type instance vs.\ compact type arguments form (depending on the 1.433 - most general declaration given in the context). 1.434 + convert between the representations of polymorphic constants: the 1.435 + full type instance vs.\ the compact type arguments form (depending 1.436 + on the most general declaration given in the context). 1.437 1.438 \end{description} 1.439 *} 1.440 @@ -424,24 +430,23 @@ 1.441 \emph{theorem} is a proven proposition (depending on a context of 1.442 hypotheses and the background theory). Primitive inferences include 1.443 plain natural deduction rules for the primary connectives @{text 1.444 - "\<And>"} and @{text "\<Longrightarrow>"} of the framework. There are separate (derived) 1.445 - rules for equality/equivalence @{text "\<equiv>"} and internal conjunction 1.446 - @{text "&"}. 1.447 + "\<And>"} and @{text "\<Longrightarrow>"} of the framework. There is also a builtin 1.448 + notion of equality/equivalence @{text "\<equiv>"}. 1.449 *} 1.450 1.451 -subsection {* Standard connectives and rules *} 1.452 +subsection {* Primitive connectives and rules *} 1.453 1.454 text {* 1.455 - The basic theory is called @{text "Pure"}, it contains declarations 1.456 - for the standard logical connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and 1.457 - @{text "\<equiv>"} of the framework, see \figref{fig:pure-connectives}. 1.458 - The derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is 1.459 - defined inductively by the primitive inferences given in 1.460 - \figref{fig:prim-rules}, with the global syntactic restriction that 1.461 - hypotheses may never contain schematic variables. The builtin 1.462 - equality is conceptually axiomatized shown in 1.463 + The theory @{text "Pure"} contains declarations for the standard 1.464 + connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of the logical 1.465 + framework, see \figref{fig:pure-connectives}. The derivability 1.466 + judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined 1.467 + inductively by the primitive inferences given in 1.468 + \figref{fig:prim-rules}, with the global restriction that hypotheses 1.469 + @{text "\<Gamma>"} may \emph{not} contain schematic variables. The builtin 1.470 + equality is conceptually axiomatized as shown in 1.471 \figref{fig:pure-equality}, although the implementation works 1.472 - directly with (derived) inference rules. 1.473 + directly with derived inference rules. 1.474 1.475 \begin{figure}[htb] 1.476 \begin{center} 1.477 @@ -450,7 +455,7 @@ 1.478 @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\ 1.479 @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\ 1.480 \end{tabular} 1.481 - \caption{Standard connectives of Pure}\label{fig:pure-connectives} 1.482 + \caption{Primitive connectives of Pure}\label{fig:pure-connectives} 1.483 \end{center} 1.484 \end{figure} 1.485 1.486 @@ -462,9 +467,9 @@ 1.487 \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{} 1.488 \] 1.489 \[ 1.490 - \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}{@{text "\<Gamma> \<turnstile> b x"} & @{text "x \<notin> \<Gamma>"}} 1.491 + \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}} 1.492 \qquad 1.493 - \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b a"}}{@{text "\<Gamma> \<turnstile> \<And>x. b x"}} 1.494 + \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}} 1.495 \] 1.496 \[ 1.497 \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}} 1.498 @@ -478,34 +483,34 @@ 1.499 \begin{figure}[htb] 1.500 \begin{center} 1.501 \begin{tabular}{ll} 1.502 - @{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\ 1.503 + @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\ 1.504 @{text "\<turnstile> x \<equiv> x"} & reflexivity \\ 1.505 @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\ 1.506 @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\ 1.507 - @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & coincidence with equivalence \\ 1.508 + @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\ 1.509 \end{tabular} 1.510 - \caption{Conceptual axiomatization of builtin equality}\label{fig:pure-equality} 1.511 + \caption{Conceptual axiomatization of @{text "\<equiv>"}}\label{fig:pure-equality} 1.512 \end{center} 1.513 \end{figure} 1.514 1.515 The introduction and elimination rules for @{text "\<And>"} and @{text 1.516 - "\<Longrightarrow>"} are analogous to formation of (dependently typed) @{text 1.517 + "\<Longrightarrow>"} are analogous to formation of dependently typed @{text 1.518 "\<lambda>"}-terms representing the underlying proof objects. Proof terms 1.519 - are \emph{irrelevant} in the Pure logic, they may never occur within 1.520 - propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow is non-dependent. The 1.521 - system provides a runtime option to record explicit proof terms for 1.522 - primitive inferences, cf.\ \cite{Berghofer-Nipkow:2000:TPHOL}. Thus 1.523 - the three-fold @{text "\<lambda>"}-structure can be made explicit. 1.524 + are irrelevant in the Pure logic, though, they may never occur 1.525 + within propositions. The system provides a runtime option to record 1.526 + explicit proof terms for primitive inferences. Thus all three 1.527 + levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for 1.528 + terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\ 1.529 + \cite{Berghofer-Nipkow:2000:TPHOL}). 1.530 1.531 - Observe that locally fixed parameters (as used in rule @{text 1.532 - "\<And>_intro"}) need not be recorded in the hypotheses, because the 1.533 - simple syntactic types of Pure are always inhabitable. The typing 1.534 - ``assumption'' @{text "x :: \<tau>"} is logically vacuous, it disappears 1.535 - automatically whenever the statement body ceases to mention variable 1.536 - @{text "x\<^isub>\<tau>"}.\footnote{This greatly simplifies many basic 1.537 - reasoning steps, and is the key difference to the formulation of 1.538 - this logic as ``@{text "\<lambda>HOL"}'' in the PTS framework 1.539 - \cite{Barendregt-Geuvers:2001}.} 1.540 + Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need 1.541 + not be recorded in the hypotheses, because the simple syntactic 1.542 + types of Pure are always inhabitable. Typing ``assumptions'' @{text 1.543 + "x :: \<tau>"} are (implicitly) present only with occurrences of @{text 1.544 + "x\<^isub>\<tau>"} in the statement body.\footnote{This is the key 1.545 + difference ``@{text "\<lambda>HOL"}'' in the PTS framework 1.546 + \cite{Barendregt-Geuvers:2001}, where @{text "x : A"} hypotheses are 1.547 + treated explicitly for types, in the same way as propositions.} 1.548 1.549 \medskip FIXME @{text "\<alpha>\<beta>\<eta>"}-equivalence and primitive definitions 1.550 1.551 @@ -514,13 +519,11 @@ 1.552 "\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication. 1.553 1.554 \medskip The axiomatization of a theory is implicitly closed by 1.555 - forming all instances of type and term variables: @{text "\<turnstile> A\<vartheta>"} for 1.556 - any substitution instance of axiom @{text "\<turnstile> A"}. By pushing 1.557 - substitution through derivations inductively, we get admissible 1.558 - substitution rules for theorems shown in \figref{fig:subst-rules}. 1.559 - Alternatively, the term substitution rules could be derived from 1.560 - @{text "\<And>_intro/elim"}. The versions for types are genuine 1.561 - admissible rules, due to the lack of true polymorphism in the logic. 1.562 + forming all instances of type and term variables: @{text "\<turnstile> 1.563 + A\<vartheta>"} holds for any substitution instance of an axiom 1.564 + @{text "\<turnstile> A"}. By pushing substitution through derivations 1.565 + inductively, we get admissible @{text "generalize"} and @{text 1.566 + "instance"} rules shown in \figref{fig:subst-rules}. 1.567 1.568 \begin{figure}[htb] 1.569 \begin{center} 1.570 @@ -538,11 +541,15 @@ 1.571 \end{center} 1.572 \end{figure} 1.573 1.574 - Since @{text "\<Gamma>"} may never contain any schematic variables, the 1.575 - @{text "instantiate"} do not require an explicit side-condition. In 1.576 - principle, variables could be substituted in hypotheses as well, but 1.577 - this could disrupt monotonicity of the basic calculus: derivations 1.578 - could leave the current proof context. 1.579 + Note that @{text "instantiate"} does not require an explicit 1.580 + side-condition, because @{text "\<Gamma>"} may never contain schematic 1.581 + variables. 1.582 + 1.583 + In principle, variables could be substituted in hypotheses as well, 1.584 + but this would disrupt monotonicity reasoning: deriving @{text 1.585 + "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is correct, but 1.586 + @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold --- the result 1.587 + belongs to a different proof context. 1.588 *} 1.589 1.590 text %mlref {* 1.591 @@ -567,16 +574,16 @@ 1.592 subsection {* Auxiliary connectives *} 1.593 1.594 text {* 1.595 - Pure also provides various auxiliary connectives based on primitive 1.596 - definitions, see \figref{fig:pure-aux}. These are normally not 1.597 - exposed to the user, but appear in internal encodings only. 1.598 + Theory @{text "Pure"} also defines a few auxiliary connectives, see 1.599 + \figref{fig:pure-aux}. These are normally not exposed to the user, 1.600 + but appear in internal encodings only. 1.601 1.602 \begin{figure}[htb] 1.603 \begin{center} 1.604 \begin{tabular}{ll} 1.605 @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\ 1.606 @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex] 1.607 - @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}) \\ 1.608 + @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, hidden) \\ 1.609 @{text "#A \<equiv> A"} \\[1ex] 1.610 @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\ 1.611 @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex] 1.612 @@ -587,39 +594,38 @@ 1.613 \end{center} 1.614 \end{figure} 1.615 1.616 - Conjunction as an explicit connective allows to treat both 1.617 - simultaneous assumptions and conclusions uniformly. The definition 1.618 - allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow> B \<Longrightarrow> A & B"}, 1.619 - and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}. For 1.620 - example, several claims may be stated at the same time, which is 1.621 - intermediately represented as an assumption, but the user only 1.622 - encounters several sub-goals, and several resulting facts in the 1.623 - very end (cf.\ \secref{sec:tactical-goals}). 1.624 + Derived conjunction rules include introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A & 1.625 + B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}. 1.626 + Conjunction allows to treat simultaneous assumptions and conclusions 1.627 + uniformly. For example, multiple claims are intermediately 1.628 + represented as explicit conjunction, but this is usually refined 1.629 + into separate sub-goals before the user continues the proof; the 1.630 + final result is projected into a list of theorems (cf.\ 1.631 + \secref{sec:tactical-goals}). 1.632 1.633 - The @{text "#"} marker allows complex propositions (nested @{text 1.634 - "\<And>"} and @{text "\<Longrightarrow>"}) to appear formally as atomic, without changing 1.635 - the meaning: @{text "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are 1.636 - interchangeable. See \secref{sec:tactical-goals} for specific 1.637 - operations. 1.638 + The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex 1.639 + propositions appear as atomic, without changing the meaning: @{text 1.640 + "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable. See 1.641 + \secref{sec:tactical-goals} for specific operations. 1.642 1.643 - The @{text "TERM"} marker turns any well-formed term into a 1.644 - derivable proposition: @{text "\<turnstile> TERM t"} holds 1.645 - unconditionally. Despite its logically vacous meaning, this is 1.646 - occasionally useful to treat syntactic terms and proven propositions 1.647 - uniformly, as in a type-theoretic framework. 1.648 + The @{text "term"} marker turns any well-formed term into a 1.649 + derivable proposition: @{text "\<turnstile> TERM t"} holds unconditionally. 1.650 + Although this is logically vacuous, it allows to treat terms and 1.651 + proofs uniformly, similar to a type-theoretic framework. 1.652 1.653 - The @{text "TYPE"} constructor (which is the canonical 1.654 - representative of the unspecified type @{text "\<alpha> itself"}) injects 1.655 - the language of types into that of terms. There is specific 1.656 - notation @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau> 1.657 + The @{text "TYPE"} constructor is the canonical representative of 1.658 + the unspecified type @{text "\<alpha> itself"}; it essentially injects the 1.659 + language of types into that of terms. There is specific notation 1.660 + @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau> 1.661 itself\<^esub>"}. 1.662 - Although being devoid of any particular meaning, the term @{text 1.663 - "TYPE(\<tau>)"} is able to carry the type @{text "\<tau>"} formally. @{text 1.664 - "TYPE(\<alpha>)"} may be used as an additional formal argument in primitive 1.665 - definitions, in order to avoid hidden polymorphism (cf.\ 1.666 - \secref{sec:terms}). For example, @{text "c TYPE(\<alpha>) \<equiv> A[\<alpha>]"} turns 1.667 - out as a formally correct definition of some proposition @{text "A"} 1.668 - that depends on an additional type argument. 1.669 + Although being devoid of any particular meaning, the @{text 1.670 + "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term 1.671 + language. In particular, @{text "TYPE(\<alpha>)"} may be used as formal 1.672 + argument in primitive definitions, in order to circumvent hidden 1.673 + polymorphism (cf.\ \secref{sec:terms}). For example, @{text "c 1.674 + TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of 1.675 + a proposition @{text "A"} that depends on an additional type 1.676 + argument, which is essentially a predicate on types. 1.677 *} 1.678 1.679 text %mlref {*