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diff --git a/matcher.mli b/matcher.mli new file mode 100644 index 0000000..a6b6f46 --- /dev/null +++ b/matcher.mli @@ -0,0 +1,189 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Standalone module containing the algorithm for matching modulo + associativity and associativity and commutativity + (AAC). Additionnaly, some A or AC operators can have units (U). + + This module could be reused outside of the Coq plugin. + + Matching a pattern [p] against a term [t] modulo AACU boils down + to finding a substitution [env] such that the pattern [p] + instantiated with [env] is equal to [t] modulo AACU. + + We proceed by structural decomposition of the pattern, trying all + possible non-deterministic splittings of the subject, when needed. The + function {!matcher} is limited to top-level matching, that is, the + subject must make a perfect match against the pattern ([x+x] does + not match [a+a+b] ). + + We use a search monad {!Search_monad} to perform non-deterministic + choices in an almost transparent way. + + We also provide a function {!subterm} for finding a match that is + a subterm of the subject modulo AACU. In particular, this function + gives a solution to the aforementioned case ([x+x] against + [a+b+a]). + + On a slightly more involved level : + - it must be noted that we allow several AC/A operators to share + the same units, but that a given AC/A operator can have at most + one unit. + + - if the pattern does not contain "hard" symbols (like constants, + function symbols, AC or A symbols without units), there can be + infinitely many subterms such that the pattern matches: it is + possible to build "subterms" modulo AAC and U that make the size + of the term increase (by making neutral elements appear in a + layered fashion). Hence, in this case, a warning is issued, and + some solutions are omitted. + +*) + +(** {2 Utility functions} *) + +type symbol = int +type var = int + +(** Relationship between units and operators. This is a sparse + representation of a matrix of couples [(op,unit)] where [op] is + the index of the operation, and [unit] the index of the relevant + unit. We make the assumption that any operation has 0 or 1 unit, + and that operations can share a unit). *) + +type units =(symbol * symbol) list (* from AC/A symbols to the unit *) +type ext_units = + { + unit_for : units; (* from AC/A symbols to the unit *) + is_ac : (symbol * bool) list + } + + +(** The arguments of sums (or AC operators) are represented using finite multisets. + (Typically, [a+b+a] corresponds to [2.a+b], i.e. [Sum[a,2;b,1]]) *) +type 'a mset = ('a * int) list + +(** [linear] expands a multiset into a simple list *) +val linear : 'a mset -> 'a list + +(** Representations of expressions + + The module {!Terms} defines two different types for expressions. + - a public type {!Terms.t} that represents abstract syntax trees + of expressions with binary associative and commutative operators + - a private type {!Terms.nf_term}, corresponding to a canonical + representation for the above terms modulo AACU. The construction + functions on this type ensure that these terms are in normal form + (that is, no sum can appear as a subterm of the same sum, no + trailing units, lists corresponding to multisets are sorted, + etc...). + +*) +module Terms : +sig + + (** {2 Abstract syntax tree of terms and patterns} + + We represent both terms and patterns using the following datatype. + + Values of type [symbol] are used to index symbols. Typically, + given two associative operations [(^)] and [( * )], and two + morphisms [f] and [g], the term [f (a^b) (a*g b)] is represented + by the following value + [Sym(0,[| Dot(1, Sym(2,[||]), Sym(3,[||])); + Dot(4, Sym(2,[||]), Sym(5,[|Sym(3,[||])|])) |])] + where the implicit symbol environment associates + [f] to [0], [(^)] to [1], [a] to [2], [b] to [3], [( * )] to [4], and [g] to [5], + + Accordingly, the following value, that contains "variables" + [Sym(0,[| Dot(1, Var x, Unit (1); Dot(4, Var x, + Sym(5,[|Sym(3,[||])|])) |])] represents the pattern [forall x, f + (x^1) (x*g b)]. The relationship between [1] and [( * )] is only + mentionned in the units table. *) + + type t = + Dot of (symbol * t * t) + | Plus of (symbol * t * t) + | Sym of (symbol * t array) + | Var of var + | Unit of symbol + + (** Test for equality of terms modulo AACU (relies on the following + canonical representation of terms). Note than two different + units of a same operator are not considered equal. *) + val equal_aac : units -> t -> t -> bool + + + (** {2 Normalised terms (canonical representation) } + + A term in normal form is the canonical representative of the + equivalence class of all the terms that are equal modulo AACU. + This representation is only used internally; it is exported here + for the sake of completeness *) + + type nf_term + + (** {3 Comparisons} *) + + val nf_term_compare : nf_term -> nf_term -> int + val nf_equal : nf_term -> nf_term -> bool + + (** {3 Printing function} *) + + val sprint_nf_term : nf_term -> string + + (** {3 Conversion functions} *) + + (** we have the following property: [a] and [b] are equal modulo AACU + iif [nf_equal (term_of_t a) (term_of_t b) = true] *) + val term_of_t : units -> t -> nf_term + val t_of_term : nf_term -> t + +end + + +(** Substitutions (or environments) + + The module {!Subst} contains infrastructure to deal with + substitutions, i.e., functions from variables to terms. Only a + restricted subsets of these functions need to be exported. + + As expected, a particular substitution can be used to + instantiate a pattern. +*) +module Subst : +sig + type t + val sprint : t -> string + val instantiate : t -> Terms.t-> Terms.t + val to_list : t -> (var*Terms.t) list +end + + +(** {2 Main functions exported by this module} *) + +(** [matcher p t] computes the set of solutions to the given top-level + matching problem ([p] is the pattern, [t] is the term). If the + [strict] flag is set, solutions where units are used to + instantiate some variables are excluded, unless this unit appears + directly under a function symbol (e.g., f(x) still matches f(1), + while x+x+y does not match a+b+c, since this would require to + assign 1 to x). +*) +val matcher : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> Subst.t Search_monad.m + +(** [subterm p t] computes a set of solutions to the given + subterm-matching problem. + + Return a collection of possible solutions (each with the + associated depth, the context, and the solutions of the matching + problem). The context is actually a {!Terms.t} where the variables + are yet to be instantiated by one of the associated substitutions +*) +val subterm : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> (int * Terms.t * Subst.t Search_monad.m) Search_monad.m + |