(* Title: HOL/Library/refute.ML
Author: Tjark Weber, TU Muenchen
Finite model generation for HOL formulas, using a SAT solver.
*)
signature REFUTE =
sig
exception REFUTE of string * string
(* ------------------------------------------------------------------------- *)
(* Model/interpretation related code (translation HOL -> propositional logic *)
(* ------------------------------------------------------------------------- *)
type params
type interpretation
type model
type arguments
exception MAXVARS_EXCEEDED
val add_interpreter : string -> (Proof.context -> model -> arguments -> term ->
(interpretation * model * arguments) option) -> theory -> theory
val add_printer : string -> (Proof.context -> model -> typ ->
interpretation -> (int -> bool) -> term option) -> theory -> theory
val interpret : Proof.context -> model -> arguments -> term ->
(interpretation * model * arguments)
val print : Proof.context -> model -> typ -> interpretation -> (int -> bool) -> term
val print_model : Proof.context -> model -> (int -> bool) -> string
(* ------------------------------------------------------------------------- *)
(* Interface *)
(* ------------------------------------------------------------------------- *)
val set_default_param : (string * string) -> theory -> theory
val get_default_param : Proof.context -> string -> string option
val get_default_params : Proof.context -> (string * string) list
val actual_params : Proof.context -> (string * string) list -> params
val find_model :
Proof.context -> params -> term list -> term -> bool -> string
(* tries to find a model for a formula: *)
val satisfy_term :
Proof.context -> (string * string) list -> term list -> term -> string
(* tries to find a model that refutes a formula: *)
val refute_term :
Proof.context -> (string * string) list -> term list -> term -> string
val refute_goal :
Proof.context -> (string * string) list -> thm -> int -> string
(* ------------------------------------------------------------------------- *)
(* Additional functions used by Nitpick (to be factored out) *)
(* ------------------------------------------------------------------------- *)
val get_classdef : theory -> string -> (string * term) option
val norm_rhs : term -> term
val get_def : theory -> string * typ -> (string * term) option
val get_typedef : theory -> typ -> (string * term) option
val is_IDT_constructor : theory -> string * typ -> bool
val is_IDT_recursor : theory -> string * typ -> bool
val is_const_of_class: theory -> string * typ -> bool
val string_of_typ : typ -> string
end;
structure Refute : REFUTE =
struct
open Prop_Logic;
(* We use 'REFUTE' only for internal error conditions that should *)
(* never occur in the first place (i.e. errors caused by bugs in our *)
(* code). Otherwise (e.g. to indicate invalid input data) we use *)
(* 'error'. *)
exception REFUTE of string * string; (* ("in function", "cause") *)
(* should be raised by an interpreter when more variables would be *)
(* required than allowed by 'maxvars' *)
exception MAXVARS_EXCEEDED;
(* ------------------------------------------------------------------------- *)
(* TREES *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* tree: implements an arbitrarily (but finitely) branching tree as a list *)
(* of (lists of ...) elements *)
(* ------------------------------------------------------------------------- *)
datatype 'a tree =
Leaf of 'a
| Node of ('a tree) list;
fun tree_map f tr =
case tr of
Leaf x => Leaf (f x)
| Node xs => Node (map (tree_map f) xs);
fun tree_pair (t1, t2) =
case t1 of
Leaf x =>
(case t2 of
Leaf y => Leaf (x,y)
| Node _ => raise REFUTE ("tree_pair",
"trees are of different height (second tree is higher)"))
| Node xs =>
(case t2 of
(* '~~' will raise an exception if the number of branches in *)
(* both trees is different at the current node *)
Node ys => Node (map tree_pair (xs ~~ ys))
| Leaf _ => raise REFUTE ("tree_pair",
"trees are of different height (first tree is higher)"));
(* ------------------------------------------------------------------------- *)
(* params: parameters that control the translation into a propositional *)
(* formula/model generation *)
(* *)
(* The following parameters are supported (and required (!), except for *)
(* "sizes" and "expect"): *)
(* *)
(* Name Type Description *)
(* *)
(* "sizes" (string * int) list *)
(* Size of ground types (e.g. 'a=2), or depth of IDTs. *)
(* "minsize" int If >0, minimal size of each ground type/IDT depth. *)
(* "maxsize" int If >0, maximal size of each ground type/IDT depth. *)
(* "maxvars" int If >0, use at most 'maxvars' Boolean variables *)
(* when transforming the term into a propositional *)
(* formula. *)
(* "maxtime" int If >0, terminate after at most 'maxtime' seconds. *)
(* "satsolver" string SAT solver to be used. *)
(* "no_assms" bool If "true", assumptions in structured proofs are *)
(* not considered. *)
(* "expect" string Expected result ("genuine", "potential", "none", or *)
(* "unknown"). *)
(* ------------------------------------------------------------------------- *)
type params =
{
sizes : (string * int) list,
minsize : int,
maxsize : int,
maxvars : int,
maxtime : int,
satsolver: string,
no_assms : bool,
expect : string
};
(* ------------------------------------------------------------------------- *)
(* interpretation: a term's interpretation is given by a variable of type *)
(* 'interpretation' *)
(* ------------------------------------------------------------------------- *)
type interpretation =
prop_formula list tree;
(* ------------------------------------------------------------------------- *)
(* model: a model specifies the size of types and the interpretation of *)
(* terms *)
(* ------------------------------------------------------------------------- *)
type model =
(typ * int) list * (term * interpretation) list;
(* ------------------------------------------------------------------------- *)
(* arguments: additional arguments required during interpretation of terms *)
(* ------------------------------------------------------------------------- *)
type arguments =
{
(* just passed unchanged from 'params': *)
maxvars : int,
(* whether to use 'make_equality' or 'make_def_equality': *)
def_eq : bool,
(* the following may change during the translation: *)
next_idx : int,
bounds : interpretation list,
wellformed: prop_formula
};
structure Data = Theory_Data
(
type T =
{interpreters: (string * (Proof.context -> model -> arguments -> term ->
(interpretation * model * arguments) option)) list,
printers: (string * (Proof.context -> model -> typ -> interpretation ->
(int -> bool) -> term option)) list,
parameters: string Symtab.table};
val empty = {interpreters = [], printers = [], parameters = Symtab.empty};
val extend = I;
fun merge
({interpreters = in1, printers = pr1, parameters = pa1},
{interpreters = in2, printers = pr2, parameters = pa2}) : T =
{interpreters = AList.merge (op =) (K true) (in1, in2),
printers = AList.merge (op =) (K true) (pr1, pr2),
parameters = Symtab.merge (op =) (pa1, pa2)};
);
val get_data = Data.get o Proof_Context.theory_of;
(* ------------------------------------------------------------------------- *)
(* interpret: interprets the term 't' using a suitable interpreter; returns *)
(* the interpretation and a (possibly extended) model that keeps *)
(* track of the interpretation of subterms *)
(* ------------------------------------------------------------------------- *)
fun interpret ctxt model args t =
case get_first (fn (_, f) => f ctxt model args t)
(#interpreters (get_data ctxt)) of
NONE => raise REFUTE ("interpret",
"no interpreter for term " ^ quote (Syntax.string_of_term ctxt t))
| SOME x => x;
(* ------------------------------------------------------------------------- *)
(* print: converts the interpretation 'intr', which must denote a term of *)
(* type 'T', into a term using a suitable printer *)
(* ------------------------------------------------------------------------- *)
fun print ctxt model T intr assignment =
case get_first (fn (_, f) => f ctxt model T intr assignment)
(#printers (get_data ctxt)) of
NONE => raise REFUTE ("print",
"no printer for type " ^ quote (Syntax.string_of_typ ctxt T))
| SOME x => x;
(* ------------------------------------------------------------------------- *)
(* print_model: turns the model into a string, using a fixed interpretation *)
(* (given by an assignment for Boolean variables) and suitable *)
(* printers *)
(* ------------------------------------------------------------------------- *)
fun print_model ctxt model assignment =
let
val (typs, terms) = model
val typs_msg =
if null typs then
"empty universe (no type variables in term)\n"
else
"Size of types: " ^ commas (map (fn (T, i) =>
Syntax.string_of_typ ctxt T ^ ": " ^ string_of_int i) typs) ^ "\n"
val show_consts_msg =
if not (Config.get ctxt show_consts) andalso Library.exists (is_Const o fst) terms then
"enable \"show_consts\" to show the interpretation of constants\n"
else
""
val terms_msg =
if null terms then
"empty interpretation (no free variables in term)\n"
else
cat_lines (map_filter (fn (t, intr) =>
(* print constants only if 'show_consts' is true *)
if Config.get ctxt show_consts orelse not (is_Const t) then
SOME (Syntax.string_of_term ctxt t ^ ": " ^
Syntax.string_of_term ctxt
(print ctxt model (Term.type_of t) intr assignment))
else
NONE) terms) ^ "\n"
in
typs_msg ^ show_consts_msg ^ terms_msg
end;
(* ------------------------------------------------------------------------- *)
(* PARAMETER MANAGEMENT *)
(* ------------------------------------------------------------------------- *)
fun add_interpreter name f = Data.map (fn {interpreters, printers, parameters} =>
case AList.lookup (op =) interpreters name of
NONE => {interpreters = (name, f) :: interpreters,
printers = printers, parameters = parameters}
| SOME _ => error ("Interpreter " ^ name ^ " already declared"));
fun add_printer name f = Data.map (fn {interpreters, printers, parameters} =>
case AList.lookup (op =) printers name of
NONE => {interpreters = interpreters,
printers = (name, f) :: printers, parameters = parameters}
| SOME _ => error ("Printer " ^ name ^ " already declared"));
(* ------------------------------------------------------------------------- *)
(* set_default_param: stores the '(name, value)' pair in Data's *)
(* parameter table *)
(* ------------------------------------------------------------------------- *)
fun set_default_param (name, value) = Data.map
(fn {interpreters, printers, parameters} =>
{interpreters = interpreters, printers = printers,
parameters = Symtab.update (name, value) parameters});
(* ------------------------------------------------------------------------- *)
(* get_default_param: retrieves the value associated with 'name' from *)
(* Data's parameter table *)
(* ------------------------------------------------------------------------- *)
val get_default_param = Symtab.lookup o #parameters o get_data;
(* ------------------------------------------------------------------------- *)
(* get_default_params: returns a list of all '(name, value)' pairs that are *)
(* stored in Data's parameter table *)
(* ------------------------------------------------------------------------- *)
val get_default_params = Symtab.dest o #parameters o get_data;
(* ------------------------------------------------------------------------- *)
(* actual_params: takes a (possibly empty) list 'params' of parameters that *)
(* override the default parameters currently specified, and *)
(* returns a record that can be passed to 'find_model'. *)
(* ------------------------------------------------------------------------- *)
fun actual_params ctxt override =
let
fun read_bool (parms, name) =
case AList.lookup (op =) parms name of
SOME "true" => true
| SOME "false" => false
| SOME s => error ("parameter " ^ quote name ^
" (value is " ^ quote s ^ ") must be \"true\" or \"false\"")
| NONE => error ("parameter " ^ quote name ^
" must be assigned a value")
fun read_int (parms, name) =
case AList.lookup (op =) parms name of
SOME s =>
(case Int.fromString s of
SOME i => i
| NONE => error ("parameter " ^ quote name ^
" (value is " ^ quote s ^ ") must be an integer value"))
| NONE => error ("parameter " ^ quote name ^
" must be assigned a value")
fun read_string (parms, name) =
case AList.lookup (op =) parms name of
SOME s => s
| NONE => error ("parameter " ^ quote name ^
" must be assigned a value")
(* 'override' first, defaults last: *)
val allparams = override @ get_default_params ctxt
val minsize = read_int (allparams, "minsize")
val maxsize = read_int (allparams, "maxsize")
val maxvars = read_int (allparams, "maxvars")
val maxtime = read_int (allparams, "maxtime")
val satsolver = read_string (allparams, "satsolver")
val no_assms = read_bool (allparams, "no_assms")
val expect = the_default "" (AList.lookup (op =) allparams "expect")
(* all remaining parameters of the form "string=int" are collected in *)
(* 'sizes' *)
(* TODO: it is currently not possible to specify a size for a type *)
(* whose name is one of the other parameters (e.g. 'maxvars') *)
(* (string * int) list *)
val sizes = map_filter
(fn (name, value) => Option.map (pair name) (Int.fromString value))
(filter (fn (name, _) => name<>"minsize" andalso name<>"maxsize"
andalso name<>"maxvars" andalso name<>"maxtime"
andalso name<>"satsolver" andalso name<>"no_assms") allparams)
in
{sizes=sizes, minsize=minsize, maxsize=maxsize, maxvars=maxvars,
maxtime=maxtime, satsolver=satsolver, no_assms=no_assms, expect=expect}
end;
(* ------------------------------------------------------------------------- *)
(* TRANSLATION HOL -> PROPOSITIONAL LOGIC, BOOLEAN ASSIGNMENT -> MODEL *)
(* ------------------------------------------------------------------------- *)
fun typ_of_dtyp _ typ_assoc (Old_Datatype_Aux.DtTFree a) =
the (AList.lookup (op =) typ_assoc (Old_Datatype_Aux.DtTFree a))
| typ_of_dtyp descr typ_assoc (Old_Datatype_Aux.DtType (s, Us)) =
Type (s, map (typ_of_dtyp descr typ_assoc) Us)
| typ_of_dtyp descr typ_assoc (Old_Datatype_Aux.DtRec i) =
let val (s, ds, _) = the (AList.lookup (op =) descr i) in
Type (s, map (typ_of_dtyp descr typ_assoc) ds)
end
val close_form = ATP_Util.close_form
val specialize_type = ATP_Util.specialize_type
(* ------------------------------------------------------------------------- *)
(* is_const_of_class: returns 'true' iff 'Const (s, T)' is a constant that *)
(* denotes membership to an axiomatic type class *)
(* ------------------------------------------------------------------------- *)
fun is_const_of_class thy (s, _) =
let
val class_const_names = map Logic.const_of_class (Sign.all_classes thy)
in
(* I'm not quite sure if checking the name 's' is sufficient, *)
(* or if we should also check the type 'T'. *)
member (op =) class_const_names s
end;
(* ------------------------------------------------------------------------- *)
(* is_IDT_constructor: returns 'true' iff 'Const (s, T)' is the constructor *)
(* of an inductive datatype in 'thy' *)
(* ------------------------------------------------------------------------- *)
fun is_IDT_constructor thy (s, T) =
(case body_type T of
Type (s', _) =>
(case BNF_LFP_Compat.get_constrs thy s' of
SOME constrs =>
List.exists (fn (cname, cty) =>
cname = s andalso Sign.typ_instance thy (T, cty)) constrs
| NONE => false)
| _ => false);
(* ------------------------------------------------------------------------- *)
(* is_IDT_recursor: returns 'true' iff 'Const (s, T)' is the recursion *)
(* operator of an inductive datatype in 'thy' *)
(* ------------------------------------------------------------------------- *)
fun is_IDT_recursor thy (s, _) =
let
val rec_names = Symtab.fold (append o #rec_names o snd) (BNF_LFP_Compat.get_all thy []) []
in
(* I'm not quite sure if checking the name 's' is sufficient, *)
(* or if we should also check the type 'T'. *)
member (op =) rec_names s
end;
(* ------------------------------------------------------------------------- *)
(* norm_rhs: maps f ?t1 ... ?tn == rhs to %t1...tn. rhs *)
(* ------------------------------------------------------------------------- *)
fun norm_rhs eqn =
let
fun lambda (v as Var ((x, _), T)) t = Abs (x, T, abstract_over (v, t))
| lambda v t = raise TERM ("lambda", [v, t])
val (lhs, rhs) = Logic.dest_equals eqn
val (_, args) = Term.strip_comb lhs
in
fold lambda (rev args) rhs
end
(* ------------------------------------------------------------------------- *)
(* get_def: looks up the definition of a constant *)
(* ------------------------------------------------------------------------- *)
fun get_def thy (s, T) =
let
fun get_def_ax [] = NONE
| get_def_ax ((axname, ax) :: axioms) =
(let
val (lhs, _) = Logic.dest_equals ax (* equations only *)
val c = Term.head_of lhs
val (s', T') = Term.dest_Const c
in
if s=s' then
let
val typeSubs = Sign.typ_match thy (T', T) Vartab.empty
val ax' = Envir.subst_term_types typeSubs ax
val rhs = norm_rhs ax'
in
SOME (axname, rhs)
end
else
get_def_ax axioms
end handle ERROR _ => get_def_ax axioms
| TERM _ => get_def_ax axioms
| Type.TYPE_MATCH => get_def_ax axioms)
in
get_def_ax (Theory.all_axioms_of thy)
end;
(* ------------------------------------------------------------------------- *)
(* get_typedef: looks up the definition of a type, as created by "typedef" *)
(* ------------------------------------------------------------------------- *)
fun get_typedef thy T =
let
fun get_typedef_ax [] = NONE
| get_typedef_ax ((axname, ax) :: axioms) =
(let
fun type_of_type_definition (Const (s', T')) =
if s'= \<^const_name>\type_definition\ then
SOME T'
else
NONE
| type_of_type_definition (Free _) = NONE
| type_of_type_definition (Var _) = NONE
| type_of_type_definition (Bound _) = NONE
| type_of_type_definition (Abs (_, _, body)) =
type_of_type_definition body
| type_of_type_definition (t1 $ t2) =
(case type_of_type_definition t1 of
SOME x => SOME x
| NONE => type_of_type_definition t2)
in
case type_of_type_definition ax of
SOME T' =>
let
val T'' = domain_type (domain_type T')
val typeSubs = Sign.typ_match thy (T'', T) Vartab.empty
in
SOME (axname, Envir.subst_term_types typeSubs ax)
end
| NONE => get_typedef_ax axioms
end handle ERROR _ => get_typedef_ax axioms
| TERM _ => get_typedef_ax axioms
| Type.TYPE_MATCH => get_typedef_ax axioms)
in
get_typedef_ax (Theory.all_axioms_of thy)
end;
(* ------------------------------------------------------------------------- *)
(* get_classdef: looks up the defining axiom for an axiomatic type class, as *)
(* created by the "axclass" command *)
(* ------------------------------------------------------------------------- *)
fun get_classdef thy class =
let
val axname = class ^ "_class_def"
in
Option.map (pair axname)
(AList.lookup (op =) (Theory.all_axioms_of thy) axname)
end;
(* ------------------------------------------------------------------------- *)
(* unfold_defs: unfolds all defined constants in a term 't', beta-eta *)
(* normalizes the result term; certain constants are not *)
(* unfolded (cf. 'collect_axioms' and the various interpreters *)
(* below): if the interpretation respects a definition anyway, *)
(* that definition does not need to be unfolded *)
(* ------------------------------------------------------------------------- *)
(* Note: we could intertwine unfolding of constants and beta-(eta-) *)
(* normalization; this would save some unfolding for terms where *)
(* constants are eliminated by beta-reduction (e.g. 'K c1 c2'). On *)
(* the other hand, this would cause additional work for terms where *)
(* constants are duplicated by beta-reduction (e.g. 'S c1 c2 c3'). *)
fun unfold_defs thy t =
let
fun unfold_loop t =
case t of
(* Pure *)
Const (\<^const_name>\<open>Pure.all\<close>, _) => t
| Const (\<^const_name>\<open>Pure.eq\<close>, _) => t
| Const (\<^const_name>\<open>Pure.imp\<close>, _) => t
| Const (\<^const_name>\<open>Pure.type\<close>, _) => t (* axiomatic type classes *)
(* HOL *)
| Const (\<^const_name>\<open>Trueprop\<close>, _) => t
| Const (\<^const_name>\<open>Not\<close>, _) => t
| (* redundant, since 'True' is also an IDT constructor *)
Const (\<^const_name>\<open>True\<close>, _) => t
| (* redundant, since 'False' is also an IDT constructor *)
Const (\<^const_name>\<open>False\<close>, _) => t
| Const (\<^const_name>\<open>undefined\<close>, _) => t
| Const (\<^const_name>\<open>The\<close>, _) => t
| Const (\<^const_name>\<open>Hilbert_Choice.Eps\<close>, _) => t
| Const (\<^const_name>\<open>All\<close>, _) => t
| Const (\<^const_name>\<open>Ex\<close>, _) => t
| Const (\<^const_name>\<open>HOL.eq\<close>, _) => t
| Const (\<^const_name>\<open>HOL.conj\<close>, _) => t
| Const (\<^const_name>\<open>HOL.disj\<close>, _) => t
| Const (\<^const_name>\<open>HOL.implies\<close>, _) => t
(* sets *)
| Const (\<^const_name>\<open>Collect\<close>, _) => t
| Const (\<^const_name>\<open>Set.member\<close>, _) => t
(* other optimizations *)
| Const (\<^const_name>\<open>Finite_Set.card\<close>, _) => t
| Const (\<^const_name>\<open>Finite_Set.finite\<close>, _) => t
| Const (\<^const_name>\<open>Orderings.less\<close>, Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>bool\<close>])])) => t
| Const (\<^const_name>\<open>Groups.plus\<close>, Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>nat\<close>])])) => t
| Const (\<^const_name>\<open>Groups.minus\<close>, Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>nat\<close>])])) => t
| Const (\<^const_name>\<open>Groups.times\<close>, Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>nat\<close>])])) => t
| Const (\<^const_name>\<open>append\<close>, _) => t
| Const (\<^const_name>\<open>fst\<close>, _) => t
| Const (\<^const_name>\<open>snd\<close>, _) => t
(* simply-typed lambda calculus *)
| Const (s, T) =>
(if is_IDT_constructor thy (s, T)
orelse is_IDT_recursor thy (s, T) then
t (* do not unfold IDT constructors/recursors *)
(* unfold the constant if there is a defining equation *)
else
case get_def thy (s, T) of
SOME ((*axname*) _, rhs) =>
(* Note: if the term to be unfolded (i.e. 'Const (s, T)') *)
(* occurs on the right-hand side of the equation, i.e. in *)
(* 'rhs', we must not use this equation to unfold, because *)
(* that would loop. Here would be the right place to *)
(* check this. However, getting this really right seems *)
(* difficult because the user may state arbitrary axioms, *)
(* which could interact with overloading to create loops. *)
((*tracing (" unfolding: " ^ axname);*)
unfold_loop rhs)
| NONE => t)
| Free _ => t
| Var _ => t
| Bound _ => t
| Abs (s, T, body) => Abs (s, T, unfold_loop body)
| t1 $ t2 => (unfold_loop t1) $ (unfold_loop t2)
val result = Envir.beta_eta_contract (unfold_loop t)
in
result
end;
(* ------------------------------------------------------------------------- *)
(* collect_axioms: collects (monomorphic, universally quantified, unfolded *)
(* versions of) all HOL axioms that are relevant w.r.t 't' *)
(* ------------------------------------------------------------------------- *)
(* Note: to make the collection of axioms more easily extensible, this *)
(* function could be based on user-supplied "axiom collectors", *)
(* similar to 'interpret'/interpreters or 'print'/printers *)
(* Note: currently we use "inverse" functions to the definitional *)
(* mechanisms provided by Isabelle/HOL, e.g. for "axclass", *)
(* "typedef", "definition". A more general approach could consider *)
(* *every* axiom of the theory and collect it if it has a constant/ *)
(* type/typeclass in common with the term 't'. *)
(* Which axioms are "relevant" for a particular term/type goes hand in *)
(* hand with the interpretation of that term/type by its interpreter (see *)
(* way below): if the interpretation respects an axiom anyway, the axiom *)
(* does not need to be added as a constraint here. *)
(* To avoid collecting the same axiom multiple times, we use an *)
(* accumulator 'axs' which contains all axioms collected so far. *)
local
fun get_axiom thy xname =
Name_Space.check (Context.Theory thy) (Theory.axiom_table thy) (xname, Position.none);
val the_eq_trivial = get_axiom \<^theory>\<open>HOL\<close> "the_eq_trivial";
val someI = get_axiom \<^theory>\<open>Hilbert_Choice\<close> "someI";
in
fun collect_axioms ctxt t =
let
val thy = Proof_Context.theory_of ctxt
val _ = tracing "Adding axioms..."
fun collect_this_axiom (axname, ax) axs =
let
val ax' = unfold_defs thy ax
in
if member (op aconv) axs ax' then axs
else (tracing axname; collect_term_axioms ax' (ax' :: axs))
end
and collect_sort_axioms T axs =
let
val sort =
(case T of
TFree (_, sort) => sort
| TVar (_, sort) => sort
| _ => raise REFUTE ("collect_axioms",
"type " ^ Syntax.string_of_typ ctxt T ^ " is not a variable"))
(* obtain axioms for all superclasses *)
val superclasses = sort @ maps (Sign.super_classes thy) sort
(* merely an optimization, because 'collect_this_axiom' disallows *)
(* duplicate axioms anyway: *)
val superclasses = distinct (op =) superclasses
val class_axioms = maps (fn class => map (fn ax =>
("<" ^ class ^ ">", Thm.prop_of ax))
(#axioms (Axclass.get_info thy class) handle ERROR _ => []))
superclasses
(* replace the (at most one) schematic type variable in each axiom *)
(* by the actual type 'T' *)
val monomorphic_class_axioms = map (fn (axname, ax) =>
(case Term.add_tvars ax [] of
[] => (axname, ax)
| [(idx, S)] => (axname, Envir.subst_term_types (Vartab.make [(idx, (S, T))]) ax)
| _ =>
raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^
Syntax.string_of_term ctxt ax ^
") contains more than one type variable")))
class_axioms
in
fold collect_this_axiom monomorphic_class_axioms axs
end
and collect_type_axioms T axs =
case T of
(* simple types *)
Type (\<^type_name>\<open>prop\<close>, []) => axs
| Type (\<^type_name>\<open>fun\<close>, [T1, T2]) => collect_type_axioms T2 (collect_type_axioms T1 axs)
| Type (\<^type_name>\<open>set\<close>, [T1]) => collect_type_axioms T1 axs
(* axiomatic type classes *)
| Type (\<^type_name>\<open>itself\<close>, [T1]) => collect_type_axioms T1 axs
| Type (s, Ts) =>
(case BNF_LFP_Compat.get_info thy [] s of
SOME _ => (* inductive datatype *)
(* only collect relevant type axioms for the argument types *)
fold collect_type_axioms Ts axs
| NONE =>
(case get_typedef thy T of
SOME (axname, ax) =>
collect_this_axiom (axname, ax) axs
| NONE =>
(* unspecified type, perhaps introduced with "typedecl" *)
(* at least collect relevant type axioms for the argument types *)
fold collect_type_axioms Ts axs))
(* axiomatic type classes *)
| TFree _ => collect_sort_axioms T axs
(* axiomatic type classes *)
| TVar _ => collect_sort_axioms T axs
and collect_term_axioms t axs =
case t of
(* Pure *)
Const (\<^const_name>\<open>Pure.all\<close>, _) => axs
| Const (\<^const_name>\<open>Pure.eq\<close>, _) => axs
| Const (\<^const_name>\<open>Pure.imp\<close>, _) => axs
(* axiomatic type classes *)
| Const (\<^const_name>\<open>Pure.type\<close>, T) => collect_type_axioms T axs
(* HOL *)
| Const (\<^const_name>\<open>Trueprop\<close>, _) => axs
| Const (\<^const_name>\<open>Not\<close>, _) => axs
(* redundant, since 'True' is also an IDT constructor *)
| Const (\<^const_name>\<open>True\<close>, _) => axs
(* redundant, since 'False' is also an IDT constructor *)
| Const (\<^const_name>\<open>False\<close>, _) => axs
| Const (\<^const_name>\<open>undefined\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>The\<close>, T) =>
let
val ax = specialize_type thy (\<^const_name>\<open>The\<close>, T) (#2 the_eq_trivial)
in
collect_this_axiom (#1 the_eq_trivial, ax) axs
end
| Const (\<^const_name>\<open>Hilbert_Choice.Eps\<close>, T) =>
let
val ax = specialize_type thy (\<^const_name>\<open>Hilbert_Choice.Eps\<close>, T) (#2 someI)
in
collect_this_axiom (#1 someI, ax) axs
end
| Const (\<^const_name>\<open>All\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>Ex\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>HOL.eq\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>HOL.conj\<close>, _) => axs
| Const (\<^const_name>\<open>HOL.disj\<close>, _) => axs
| Const (\<^const_name>\<open>HOL.implies\<close>, _) => axs
(* sets *)
| Const (\<^const_name>\<open>Collect\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>Set.member\<close>, T) => collect_type_axioms T axs
(* other optimizations *)
| Const (\<^const_name>\<open>Finite_Set.card\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>Finite_Set.finite\<close>, T) =>
collect_type_axioms T axs
| Const (\<^const_name>\<open>Orderings.less\<close>, T as Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>bool\<close>])])) =>
collect_type_axioms T axs
| Const (\<^const_name>\<open>Groups.plus\<close>, T as Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>nat\<close>])])) =>
collect_type_axioms T axs
| Const (\<^const_name>\<open>Groups.minus\<close>, T as Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>nat\<close>])])) =>
collect_type_axioms T axs
| Const (\<^const_name>\<open>Groups.times\<close>, T as Type ("fun", [\<^typ>\<open>nat\<close>,
Type ("fun", [\<^typ>\<open>nat\<close>, \<^typ>\<open>nat\<close>])])) =>
collect_type_axioms T axs
| Const (\<^const_name>\<open>append\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>fst\<close>, T) => collect_type_axioms T axs
| Const (\<^const_name>\<open>snd\<close>, T) => collect_type_axioms T axs
(* simply-typed lambda calculus *)
| Const (s, T) =>
if is_const_of_class thy (s, T) then
(* axiomatic type classes: add "OFCLASS(?'a::c, c_class)" *)
(* and the class definition *)
let
val class = Logic.class_of_const s
val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]), class)
val ax_in = SOME (specialize_type thy (s, T) of_class)
(* type match may fail due to sort constraints *)
handle Type.TYPE_MATCH => NONE
val ax_1 = Option.map (fn ax => (Syntax.string_of_term ctxt ax, ax)) ax_in
val ax_2 = Option.map (apsnd (specialize_type thy (s, T))) (get_classdef thy class)
in
collect_type_axioms T (fold collect_this_axiom (map_filter I [ax_1, ax_2]) axs)
end
else if is_IDT_constructor thy (s, T)
orelse is_IDT_recursor thy (s, T)
then
(* only collect relevant type axioms *)
collect_type_axioms T axs
else
(* other constants should have been unfolded, with some *)
(* exceptions: e.g. Abs_xxx/Rep_xxx functions for *)
(* typedefs, or type-class related constants *)
(* only collect relevant type axioms *)
collect_type_axioms T axs
| Free (_, T) => collect_type_axioms T axs
| Var (_, T) => collect_type_axioms T axs
| Bound _ => axs
| Abs (_, T, body) => collect_term_axioms body (collect_type_axioms T axs)
| t1 $ t2 => collect_term_axioms t2 (collect_term_axioms t1 axs)
val result = map close_form (collect_term_axioms t [])
val _ = tracing " ...done."
in
result
end;
end;
(* ------------------------------------------------------------------------- *)
(* ground_types: collects all ground types in a term (including argument *)
(* types of other types), suppressing duplicates. Does not *)
(* return function types, set types, non-recursive IDTs, or *)
(* 'propT'. For IDTs, also the argument types of constructors *)
(* and all mutually recursive IDTs are considered. *)
(* ------------------------------------------------------------------------- *)
fun ground_types ctxt t =
let
val thy = Proof_Context.theory_of ctxt
fun collect_types T acc =
(case T of
Type (\<^type_name>\<open>fun\<close>, [T1, T2]) => collect_types T1 (collect_types T2 acc)
| Type (\<^type_name>\<open>prop\<close>, []) => acc
| Type (\<^type_name>\<open>set\<close>, [T1]) => collect_types T1 acc
| Type (s, Ts) =>
(case BNF_LFP_Compat.get_info thy [] s of
SOME info => (* inductive datatype *)
let
val index = #index info
val descr = #descr info
val (_, typs, _) = the (AList.lookup (op =) descr index)
val typ_assoc = typs ~~ Ts
(* sanity check: every element in 'dtyps' must be a *)
(* 'DtTFree' *)
val _ = if Library.exists (fn d =>
case d of Old_Datatype_Aux.DtTFree _ => false | _ => true) typs then
raise REFUTE ("ground_types", "datatype argument (for type "
^ Syntax.string_of_typ ctxt T ^ ") is not a variable")
else ()
(* required for mutually recursive datatypes; those need to *)
(* be added even if they are an instance of an otherwise non- *)
(* recursive datatype *)
fun collect_dtyp d acc =
let
val dT = typ_of_dtyp descr typ_assoc d
in
case d of
Old_Datatype_Aux.DtTFree _ =>
collect_types dT acc
| Old_Datatype_Aux.DtType (_, ds) =>
collect_types dT (fold_rev collect_dtyp ds acc)
| Old_Datatype_Aux.DtRec i =>
if member (op =) acc dT then
acc (* prevent infinite recursion *)
else
let
val (_, dtyps, dconstrs) = the (AList.lookup (op =) descr i)
(* if the current type is a recursive IDT (i.e. a depth *)
(* is required), add it to 'acc' *)
val acc_dT = if Library.exists (fn (_, ds) =>
Library.exists Old_Datatype_Aux.is_rec_type ds) dconstrs then
insert (op =) dT acc
else acc
(* collect argument types *)
val acc_dtyps = fold_rev collect_dtyp dtyps acc_dT
(* collect constructor types *)
val acc_dconstrs = fold_rev collect_dtyp (maps snd dconstrs) acc_dtyps
in
acc_dconstrs
end
end
in
(* argument types 'Ts' could be added here, but they are also *)
(* added by 'collect_dtyp' automatically *)
collect_dtyp (Old_Datatype_Aux.DtRec index) acc
end
| NONE =>
(* not an inductive datatype, e.g. defined via "typedef" or *)
(* "typedecl" *)
insert (op =) T (fold collect_types Ts acc))
| TFree _ => insert (op =) T acc
| TVar _ => insert (op =) T acc)
in
fold_types collect_types t []
end;
(* ------------------------------------------------------------------------- *)
(* string_of_typ: (rather naive) conversion from types to strings, used to *)
(* look up the size of a type in 'sizes'. Parameterized *)
(* types with different parameters (e.g. "'a list" vs. "bool *)
(* list") are identified. *)
(* ------------------------------------------------------------------------- *)
fun string_of_typ (Type (s, _)) = s
| string_of_typ (TFree (s, _)) = s
| string_of_typ (TVar ((s,_), _)) = s;
(* ------------------------------------------------------------------------- *)
(* first_universe: returns the "first" (i.e. smallest) universe by assigning *)
(* 'minsize' to every type for which no size is specified in *)
(* 'sizes' *)
(* ------------------------------------------------------------------------- *)
fun first_universe xs sizes minsize =
let
fun size_of_typ T =
case AList.lookup (op =) sizes (string_of_typ T) of
SOME n => n
| NONE => minsize
in
map (fn T => (T, size_of_typ T)) xs
end;
(* ------------------------------------------------------------------------- *)
(* next_universe: enumerates all universes (i.e. assignments of sizes to *)
(* types), where the minimal size of a type is given by *)
(* 'minsize', the maximal size is given by 'maxsize', and a *)
(* type may have a fixed size given in 'sizes' *)
(* ------------------------------------------------------------------------- *)
fun next_universe xs sizes minsize maxsize =
let
(* creates the "first" list of length 'len', where the sum of all list *)
(* elements is 'sum', and the length of the list is 'len' *)
fun make_first _ 0 sum =
if sum = 0 then
SOME []
else
NONE
| make_first max len sum =
if sum <= max orelse max < 0 then
Option.map (fn xs' => sum :: xs') (make_first max (len-1) 0)
else
Option.map (fn xs' => max :: xs') (make_first max (len-1) (sum-max))
(* enumerates all int lists with a fixed length, where 0<=x<='max' for *)
(* all list elements x (unless 'max'<0) *)
fun next _ _ _ [] =
NONE
| next max len sum [x] =
(* we've reached the last list element, so there's no shift possible *)
make_first max (len+1) (sum+x+1) (* increment 'sum' by 1 *)
| next max len sum (x1::x2::xs) =
if x1>0 andalso (x2<max orelse max<0) then
(* we can shift *)
SOME (the (make_first max (len+1) (sum+x1-1)) @ (x2+1) :: xs)
else
(* continue search *)
next max (len+1) (sum+x1) (x2::xs)
(* only consider those types for which the size is not fixed *)
val mutables = filter_out (AList.defined (op =) sizes o string_of_typ o fst) xs
(* subtract 'minsize' from every size (will be added again at the end) *)
val diffs = map (fn (_, n) => n-minsize) mutables
in
case next (maxsize-minsize) 0 0 diffs of
SOME diffs' =>
(* merge with those types for which the size is fixed *)
SOME (fst (fold_map (fn (T, _) => fn ds =>
case AList.lookup (op =) sizes (string_of_typ T) of
(* return the fixed size *)
SOME n => ((T, n), ds)
(* consume the head of 'ds', add 'minsize' *)
| NONE => ((T, minsize + hd ds), tl ds))
xs diffs'))
| NONE => NONE
end;
(* ------------------------------------------------------------------------- *)
(* toTrue: converts the interpretation of a Boolean value to a propositional *)
(* formula that is true iff the interpretation denotes "true" *)
(* ------------------------------------------------------------------------- *)
fun toTrue (Leaf [fm, _]) = fm
| toTrue _ = raise REFUTE ("toTrue", "interpretation does not denote a Boolean value");
(* ------------------------------------------------------------------------- *)
(* toFalse: converts the interpretation of a Boolean value to a *)
(* propositional formula that is true iff the interpretation *)
(* denotes "false" *)
(* ------------------------------------------------------------------------- *)
fun toFalse (Leaf [_, fm]) = fm
| toFalse _ = raise REFUTE ("toFalse", "interpretation does not denote a Boolean value");
(* ------------------------------------------------------------------------- *)
(* find_model: repeatedly calls 'interpret' with appropriate parameters, *)
(* applies a SAT solver, and (in case a model is found) displays *)
(* the model to the user by calling 'print_model' *)
(* {...} : parameters that control the translation/model generation *)
(* assm_ts : assumptions to be considered unless "no_assms" is specified *)
(* t : term to be translated into a propositional formula *)
(* negate : if true, find a model that makes 't' false (rather than true) *)
(* ------------------------------------------------------------------------- *)
fun find_model ctxt
{sizes, minsize, maxsize, maxvars, maxtime, satsolver, no_assms, expect}
assm_ts t negate =
let
val thy = Proof_Context.theory_of ctxt
fun check_expect outcome_code =
if expect = "" orelse outcome_code = expect then outcome_code
else error ("Unexpected outcome: " ^ quote outcome_code ^ ".")
fun wrapper () =
let
val timer = Timer.startRealTimer ()
val t =
if no_assms then t
else if negate then Logic.list_implies (assm_ts, t)
else Logic.mk_conjunction_list (t :: assm_ts)
val u = unfold_defs thy t
val _ = tracing ("Unfolded term: " ^ Syntax.string_of_term ctxt u)
val axioms = collect_axioms ctxt u
val types = fold (union (op =) o ground_types ctxt) (u :: axioms) []
val _ = tracing ("Ground types: "
^ (if null types then "none."
else commas (map (Syntax.string_of_typ ctxt) types)))
(* we can only consider fragments of recursive IDTs, so we issue a *)
(* warning if the formula contains a recursive IDT *)
(* TODO: no warning needed for /positive/ occurrences of IDTs *)
val maybe_spurious = Library.exists (fn
Type (s, _) =>
(case BNF_LFP_Compat.get_info thy [] s of
SOME info => (* inductive datatype *)
let
val index = #index info
val descr = #descr info
val (_, _, constrs) = the (AList.lookup (op =) descr index)
in
(* recursive datatype? *)
Library.exists (fn (_, ds) =>
Library.exists Old_Datatype_Aux.is_rec_type ds) constrs
end
| NONE => false)
| _ => false) types
val _ =
if maybe_spurious andalso Context_Position.is_visible ctxt then
warning "Term contains a recursive datatype; countermodel(s) may be spurious!"
else ()
fun find_model_loop universe =
let
val msecs_spent = Time.toMilliseconds (Timer.checkRealTimer timer)
val _ = maxtime = 0 orelse msecs_spent < 1000 * maxtime
orelse raise Timeout.TIMEOUT (Time.fromMilliseconds msecs_spent)
val init_model = (universe, [])
val init_args = {maxvars = maxvars, def_eq = false, next_idx = 1,
bounds = [], wellformed = True}
val _ = tracing ("Translating term (sizes: "
^ commas (map (fn (_, n) => string_of_int n) universe) ^ ") ...")
(* translate 'u' and all axioms *)
val (intrs, (model, args)) = fold_map (fn t' => fn (m, a) =>
let
val (i, m', a') = interpret ctxt m a t'
in
(* set 'def_eq' to 'true' *)
(i, (m', {maxvars = #maxvars a', def_eq = true,
next_idx = #next_idx a', bounds = #bounds a',
wellformed = #wellformed a'}))
end) (u :: axioms) (init_model, init_args)
(* make 'u' either true or false, and make all axioms true, and *)
(* add the well-formedness side condition *)
val fm_u = (if negate then toFalse else toTrue) (hd intrs)
val fm_ax = Prop_Logic.all (map toTrue (tl intrs))
val fm = Prop_Logic.all [#wellformed args, fm_ax, fm_u]
val _ =
(if member (op =) ["cdclite"] satsolver andalso Context_Position.is_visible ctxt then
warning ("Using SAT solver " ^ quote satsolver ^
"; for better performance, consider installing an \
\external solver.")
else ());
val solver =
SAT_Solver.invoke_solver satsolver
handle Option.Option =>
error ("Unknown SAT solver: " ^ quote satsolver ^
". Available solvers: " ^
commas (map (quote o fst) (SAT_Solver.get_solvers ())) ^ ".")
in
writeln "Invoking SAT solver...";
(case solver fm of
SAT_Solver.SATISFIABLE assignment =>
(writeln ("Model found:\n" ^ print_model ctxt model
(fn i => case assignment i of SOME b => b | NONE => true));
if maybe_spurious then "potential" else "genuine")
| SAT_Solver.UNSATISFIABLE _ =>
(writeln "No model exists.";
case next_universe universe sizes minsize maxsize of
SOME universe' => find_model_loop universe'
| NONE => (writeln
"Search terminated, no larger universe within the given limits.";
"none"))
| SAT_Solver.UNKNOWN =>
(writeln "No model found.";
case next_universe universe sizes minsize maxsize of
SOME universe' => find_model_loop universe'
| NONE => (writeln
"Search terminated, no larger universe within the given limits.";
"unknown"))) handle SAT_Solver.NOT_CONFIGURED =>
(error ("SAT solver " ^ quote satsolver ^ " is not configured.");
"unknown")
end
handle MAXVARS_EXCEEDED =>
(writeln ("Search terminated, number of Boolean variables ("
^ string_of_int maxvars ^ " allowed) exceeded.");
"unknown")
val outcome_code = find_model_loop (first_universe types sizes minsize)
in
check_expect outcome_code
end
in
(* some parameter sanity checks *)
minsize>=1 orelse
error ("\"minsize\" is " ^ string_of_int minsize ^ ", must be at least 1");
maxsize>=1 orelse
error ("\"maxsize\" is " ^ string_of_int maxsize ^ ", must be at least 1");
maxsize>=minsize orelse
error ("\"maxsize\" (=" ^ string_of_int maxsize ^
") is less than \"minsize\" (=" ^ string_of_int minsize ^ ").");
maxvars>=0 orelse
error ("\"maxvars\" is " ^ string_of_int maxvars ^ ", must be at least 0");
maxtime>=0 orelse
error ("\"maxtime\" is " ^ string_of_int maxtime ^ ", must be at least 0");
(* enter loop with or without time limit *)
writeln ("Trying to find a model that "
^ (if negate then "refutes" else "satisfies") ^ ": "
^ Syntax.string_of_term ctxt t);
if maxtime > 0 then (
Timeout.apply (Time.fromSeconds maxtime)
wrapper ()
handle Timeout.TIMEOUT _ =>
(writeln ("Search terminated, time limit (" ^
string_of_int maxtime
^ (if maxtime=1 then " second" else " seconds") ^ ") exceeded.");
check_expect "unknown")
) else wrapper ()
end;
(* ------------------------------------------------------------------------- *)
(* INTERFACE, PART 2: FINDING A MODEL *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* satisfy_term: calls 'find_model' to find a model that satisfies 't' *)
(* params : list of '(name, value)' pairs used to override default *)
(* parameters *)
(* ------------------------------------------------------------------------- *)
fun satisfy_term ctxt params assm_ts t =
find_model ctxt (actual_params ctxt params) assm_ts t false;
(* ------------------------------------------------------------------------- *)
(* refute_term: calls 'find_model' to find a model that refutes 't' *)
(* params : list of '(name, value)' pairs used to override default *)
(* parameters *)
(* ------------------------------------------------------------------------- *)
fun refute_term ctxt params assm_ts t =
let
(* disallow schematic type variables, since we cannot properly negate *)
(* terms containing them (their logical meaning is that there EXISTS a *)
(* type s.t. ...; to refute such a formula, we would have to show that *)
(* for ALL types, not ...) *)
val _ = null (Term.add_tvars t []) orelse
error "Term to be refuted contains schematic type variables"
(* existential closure over schematic variables *)
val vars = sort_by (fst o fst) (Term.add_vars t [])
(* Term.term *)
val ex_closure = fold (fn ((x, i), T) => fn t' =>
HOLogic.exists_const T $
Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t
(* Note: If 't' is of type 'propT' (rather than 'boolT'), applying *)
(* 'HOLogic.exists_const' is not type-correct. However, this is not *)
(* really a problem as long as 'find_model' still interprets the *)
(* resulting term correctly, without checking its type. *)
(* replace outermost universally quantified variables by Free's: *)
(* refuting a term with Free's is generally faster than refuting a *)
(* term with (nested) quantifiers, because quantifiers are expanded, *)
(* while the SAT solver searches for an interpretation for Free's. *)
(* Also we get more information back that way, namely an *)
(* interpretation which includes values for the (formerly) *)
(* quantified variables. *)
(* maps !!x1...xn. !xk...xm. t to t *)
fun strip_all_body (Const (\<^const_name>\<open>Pure.all\<close>, _) $ Abs (_, _, t)) =
strip_all_body t
| strip_all_body (Const (\<^const_name>\<open>Trueprop\<close>, _) $ t) =
strip_all_body t
| strip_all_body (Const (\<^const_name>\<open>All\<close>, _) $ Abs (_, _, t)) =
strip_all_body t
| strip_all_body t = t
(* maps !!x1...xn. !xk...xm. t to [x1, ..., xn, xk, ..., xm] *)
fun strip_all_vars (Const (\<^const_name>\<open>Pure.all\<close>, _) $ Abs (a, T, t)) =
(a, T) :: strip_all_vars t
| strip_all_vars (Const (\<^const_name>\<open>Trueprop\<close>, _) $ t) =
strip_all_vars t
| strip_all_vars (Const (\<^const_name>\<open>All\<close>, _) $ Abs (a, T, t)) =
(a, T) :: strip_all_vars t
| strip_all_vars _ = [] : (string * typ) list
val strip_t = strip_all_body ex_closure
val frees = Term.rename_wrt_term strip_t (strip_all_vars ex_closure)
val subst_t = Term.subst_bounds (map Free frees, strip_t)
in
find_model ctxt (actual_params ctxt params) assm_ts subst_t true
end;
(* ------------------------------------------------------------------------- *)
(* refute_goal *)
(* ------------------------------------------------------------------------- *)
fun refute_goal ctxt params th i =
let
val t = th |> Thm.prop_of
in
if Logic.count_prems t = 0 then
(writeln "No subgoal!"; "none")
else
let
val assms = map Thm.term_of (Assumption.all_assms_of ctxt)
val (t, frees) = Logic.goal_params t i
in
refute_term ctxt params assms (subst_bounds (frees, t))
end
end
(* ------------------------------------------------------------------------- *)
(* INTERPRETERS: Auxiliary Functions *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* make_constants: returns all interpretations for type 'T' that consist of *)
(* unit vectors with 'True'/'False' only (no Boolean *)
(* variables) *)
(* ------------------------------------------------------------------------- *)
fun make_constants ctxt model T =
let
(* returns a list with all unit vectors of length n *)
fun unit_vectors n =
let
(* returns the k-th unit vector of length n *)
fun unit_vector (k, n) =
Leaf ((replicate (k-1) False) @ (True :: (replicate (n-k) False)))
fun unit_vectors_loop k =
if k>n then [] else unit_vector (k,n) :: unit_vectors_loop (k+1)
in
unit_vectors_loop 1
end
(* returns a list of lists, each one consisting of n (possibly *)
(* identical) elements from 'xs' *)
fun pick_all 1 xs = map single xs
| pick_all n xs =
let val rec_pick = pick_all (n - 1) xs in
maps (fn x => map (cons x) rec_pick) xs
end
(* returns all constant interpretations that have the same tree *)
(* structure as the interpretation argument *)
fun make_constants_intr (Leaf xs) = unit_vectors (length xs)
| make_constants_intr (Node xs) = map Node (pick_all (length xs)
(make_constants_intr (hd xs)))
(* obtain the interpretation for a variable of type 'T' *)
val (i, _, _) = interpret ctxt model {maxvars=0, def_eq=false, next_idx=1,
bounds=[], wellformed=True} (Free ("dummy", T))
in
make_constants_intr i
end;
(* ------------------------------------------------------------------------- *)
(* size_of_type: returns the number of elements in a type 'T' (i.e. 'length *)
(* (make_constants T)', but implemented more efficiently) *)
(* ------------------------------------------------------------------------- *)
(* returns 0 for an empty ground type or a function type with empty *)
(* codomain, but fails for a function type with empty domain -- *)
(* admissibility of datatype constructor argument types (see "Inductive *)
(* datatypes in HOL - lessons learned ...", S. Berghofer, M. Wenzel, *)
(* TPHOLs 99) ensures that recursive, possibly empty, datatype fragments *)
(* never occur as the domain of a function type that is the type of a *)
(* constructor argument *)
fun size_of_type ctxt model T =
let
(* returns the number of elements that have the same tree structure as a *)
(* given interpretation *)
fun size_of_intr (Leaf xs) = length xs
| size_of_intr (Node xs) = Integer.pow (length xs) (size_of_intr (hd xs))
(* obtain the interpretation for a variable of type 'T' *)
val (i, _, _) = interpret ctxt model {maxvars=0, def_eq=false, next_idx=1,
bounds=[], wellformed=True} (Free ("dummy", T))
in
size_of_intr i
end;
(* ------------------------------------------------------------------------- *)
(* TT/FF: interpretations that denote "true" or "false", respectively *)
(* ------------------------------------------------------------------------- *)
val TT = Leaf [True, False];
val FF = Leaf [False, True];
(* ------------------------------------------------------------------------- *)
(* make_equality: returns an interpretation that denotes (extensional) *)
(* equality of two interpretations *)
(* - two interpretations are 'equal' iff they are both defined and denote *)
(* the same value *)
(* - two interpretations are 'not_equal' iff they are both defined at least *)
(* partially, and a defined part denotes different values *)
(* - a completely undefined interpretation is neither 'equal' nor *)
(* 'not_equal' to another interpretation *)
(* ------------------------------------------------------------------------- *)
(* We could in principle represent '=' on a type T by a particular *)
(* interpretation. However, the size of that interpretation is quadratic *)
(* in the size of T. Therefore comparing the interpretations 'i1' and *)
(* 'i2' directly is more efficient than constructing the interpretation *)
(* for equality on T first, and "applying" this interpretation to 'i1' *)
(* and 'i2' in the usual way (cf. 'interpretation_apply') then. *)
fun make_equality (i1, i2) =
let
fun equal (i1, i2) =
(case i1 of
Leaf xs =>
(case i2 of
Leaf ys => Prop_Logic.dot_product (xs, ys) (* defined and equal *)
| Node _ => raise REFUTE ("make_equality",
"second interpretation is higher"))
| Node xs =>
(case i2 of
Leaf _ => raise REFUTE ("make_equality",
"first interpretation is higher")
| Node ys => Prop_Logic.all (map equal (xs ~~ ys))))
fun not_equal (i1, i2) =
(case i1 of
Leaf xs =>
(case i2 of
(* defined and not equal *)
Leaf ys => Prop_Logic.all ((Prop_Logic.exists xs)
:: (Prop_Logic.exists ys)
:: (map (fn (x,y) => SOr (SNot x, SNot y)) (xs ~~ ys)))
| Node _ => raise REFUTE ("make_equality",
"second interpretation is higher"))
| Node xs =>
(case i2 of
Leaf _ => raise REFUTE ("make_equality",
"first interpretation is higher")
| Node ys => Prop_Logic.exists (map not_equal (xs ~~ ys))))
in
(* a value may be undefined; therefore 'not_equal' is not just the *)
(* negation of 'equal' *)
Leaf [equal (i1, i2), not_equal (i1, i2)]
end;
(* ------------------------------------------------------------------------- *)
(* make_def_equality: returns an interpretation that denotes (extensional) *)
(* equality of two interpretations *)
(* This function treats undefined/partially defined interpretations *)
(* different from 'make_equality': two undefined interpretations are *)
(* considered equal, while a defined interpretation is considered not equal *)
(* to an undefined interpretation. *)
(* ------------------------------------------------------------------------- *)
fun make_def_equality (i1, i2) =
let
fun equal (i1, i2) =
(case i1 of
Leaf xs =>
(case i2 of
(* defined and equal, or both undefined *)
Leaf ys => SOr (Prop_Logic.dot_product (xs, ys),
--> --------------------
--> maximum size reached
--> --------------------
¤ Dauer der Verarbeitung: 0.106 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|