(* Title: HOL/Tools/Nitpick/nitpick_mono.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2009, 2010
Monotonicity inference for higher-order logic.
*)
signature NITPICK_MONO =
sig
type hol_context = Nitpick_HOL.hol_context
val trace : bool Unsynchronized.ref
val formulas_monotonic :
hol_context -> bool -> typ -> term list * term list -> bool
val finitize_funs :
hol_context -> bool -> (typ option * bool option) list -> typ
-> term list * term list -> term list * term list
end;
structure Nitpick_Mono : NITPICK_MONO =
struct
open Nitpick_Util
open Nitpick_HOL
type var = int
datatype sign = Plus | Minus
datatype sign_atom = S of sign | V of var
type literal = var * sign
datatype mtyp =
MAlpha |
MFun of mtyp * sign_atom * mtyp |
MPair of mtyp * mtyp |
MType of string * mtyp list |
MRec of string * typ list
datatype mterm =
MRaw of term * mtyp |
MAbs of string * typ * mtyp * sign_atom * mterm |
MApp of mterm * mterm
type mdata =
{hol_ctxt: hol_context,
binarize: bool,
alpha_T: typ,
no_harmless: bool,
max_fresh: int Unsynchronized.ref,
datatype_mcache: ((string * typ list) * mtyp) list Unsynchronized.ref,
constr_mcache: (styp * mtyp) list Unsynchronized.ref}
exception UNSOLVABLE of unit
exception MTYPE of string * mtyp list * typ list
exception MTERM of string * mterm list
val trace = Unsynchronized.ref false
fun trace_msg msg = if !trace then tracing (msg ()) else ()
val string_for_var = signed_string_of_int
fun string_for_vars sep [] = "0\<^bsub>" ^ sep ^ "\<^esub>"
| string_for_vars sep xs = space_implode sep (map string_for_var xs)
fun subscript_string_for_vars sep xs =
if null xs then "" else "\<^bsub>" ^ string_for_vars sep xs ^ "\<^esub>"
fun string_for_sign Plus = "+"
| string_for_sign Minus = "-"
fun xor sn1 sn2 = if sn1 = sn2 then Plus else Minus
val negate = xor Minus
fun string_for_sign_atom (S sn) = string_for_sign sn
| string_for_sign_atom (V x) = string_for_var x
fun string_for_literal (x, sn) = string_for_var x ^ " = " ^ string_for_sign sn
val bool_M = MType (@{type_name bool}, [])
val dummy_M = MType (nitpick_prefix ^ "dummy", [])
fun is_MRec (MRec _) = true
| is_MRec _ = false
fun dest_MFun (MFun z) = z
| dest_MFun M = raise MTYPE ("Nitpick_Mono.dest_MFun", [M], [])
val no_prec = 100
fun precedence_of_mtype (MFun _) = 1
| precedence_of_mtype (MPair _) = 2
| precedence_of_mtype _ = no_prec
val string_for_mtype =
let
fun aux outer_prec M =
let
val prec = precedence_of_mtype M
val need_parens = (prec < outer_prec)
in
(if need_parens then "(" else "") ^
(if M = dummy_M then
"_"
else case M of
MAlpha => "\<alpha>"
| MFun (M1, a, M2) =>
aux (prec + 1) M1 ^ " \<Rightarrow>\<^bsup>" ^
string_for_sign_atom a ^ "\<^esup> " ^ aux prec M2
| MPair (M1, M2) => aux (prec + 1) M1 ^ " \<times> " ^ aux prec M2
| MType (s, []) =>
if s = @{type_name prop} orelse s = @{type_name bool} then "o"
else s
| MType (s, Ms) => "(" ^ commas (map (aux 0) Ms) ^ ") " ^ s
| MRec (s, _) => "[" ^ s ^ "]") ^
(if need_parens then ")" else "")
end
in aux 0 end
fun flatten_mtype (MPair (M1, M2)) = maps flatten_mtype [M1, M2]
| flatten_mtype (MType (_, Ms)) = maps flatten_mtype Ms
| flatten_mtype M = [M]
fun precedence_of_mterm (MRaw _) = no_prec
| precedence_of_mterm (MAbs _) = 1
| precedence_of_mterm (MApp _) = 2
fun string_for_mterm ctxt =
let
fun mtype_annotation M = "\<^bsup>" ^ string_for_mtype M ^ "\<^esup>"
fun aux outer_prec m =
let
val prec = precedence_of_mterm m
val need_parens = (prec < outer_prec)
in
(if need_parens then "(" else "") ^
(case m of
MRaw (t, M) => Syntax.string_of_term ctxt t ^ mtype_annotation M
| MAbs (s, _, M, a, m) =>
"\<lambda>" ^ s ^ mtype_annotation M ^ ".\<^bsup>" ^
string_for_sign_atom a ^ "\<^esup> " ^ aux prec m
| MApp (m1, m2) => aux prec m1 ^ " " ^ aux (prec + 1) m2) ^
(if need_parens then ")" else "")
end
in aux 0 end
fun mtype_of_mterm (MRaw (_, M)) = M
| mtype_of_mterm (MAbs (_, _, M, a, m)) = MFun (M, a, mtype_of_mterm m)
| mtype_of_mterm (MApp (m1, _)) =
case mtype_of_mterm m1 of
MFun (_, _, M12) => M12
| M1 => raise MTYPE ("Nitpick_Mono.mtype_of_mterm", [M1], [])
fun strip_mcomb (MApp (m1, m2)) = strip_mcomb m1 ||> (fn ms => append ms [m2])
| strip_mcomb m = (m, [])
fun initial_mdata hol_ctxt binarize no_harmless alpha_T =
({hol_ctxt = hol_ctxt, binarize = binarize, alpha_T = alpha_T,
no_harmless = no_harmless, max_fresh = Unsynchronized.ref 0,
datatype_mcache = Unsynchronized.ref [],
constr_mcache = Unsynchronized.ref []} : mdata)
fun could_exist_alpha_subtype alpha_T (T as Type (_, Ts)) =
T = alpha_T orelse (not (is_fp_iterator_type T) andalso
exists (could_exist_alpha_subtype alpha_T) Ts)
| could_exist_alpha_subtype alpha_T T = (T = alpha_T)
fun could_exist_alpha_sub_mtype _ (alpha_T as TFree _) T =
could_exist_alpha_subtype alpha_T T
| could_exist_alpha_sub_mtype ctxt alpha_T T =
(T = alpha_T orelse is_datatype ctxt [(NONE, true)] T)
fun exists_alpha_sub_mtype MAlpha = true
| exists_alpha_sub_mtype (MFun (M1, _, M2)) =
exists exists_alpha_sub_mtype [M1, M2]
| exists_alpha_sub_mtype (MPair (M1, M2)) =
exists exists_alpha_sub_mtype [M1, M2]
| exists_alpha_sub_mtype (MType (_, Ms)) = exists exists_alpha_sub_mtype Ms
| exists_alpha_sub_mtype (MRec _) = true
fun exists_alpha_sub_mtype_fresh MAlpha = true
| exists_alpha_sub_mtype_fresh (MFun (_, V _, _)) = true
| exists_alpha_sub_mtype_fresh (MFun (_, _, M2)) =
exists_alpha_sub_mtype_fresh M2
| exists_alpha_sub_mtype_fresh (MPair (M1, M2)) =
exists exists_alpha_sub_mtype_fresh [M1, M2]
| exists_alpha_sub_mtype_fresh (MType (_, Ms)) =
exists exists_alpha_sub_mtype_fresh Ms
| exists_alpha_sub_mtype_fresh (MRec _) = true
fun constr_mtype_for_binders z Ms =
fold_rev (fn M => curry3 MFun M (S Minus)) Ms (MRec z)
fun repair_mtype _ _ MAlpha = MAlpha
| repair_mtype cache seen (MFun (M1, a, M2)) =
MFun (repair_mtype cache seen M1, a, repair_mtype cache seen M2)
| repair_mtype cache seen (MPair Mp) =
MPair (pairself (repair_mtype cache seen) Mp)
| repair_mtype cache seen (MType (s, Ms)) =
MType (s, maps (flatten_mtype o repair_mtype cache seen) Ms)
| repair_mtype cache seen (MRec (z as (s, _))) =
case AList.lookup (op =) cache z |> the of
MRec _ => MType (s, [])
| M => if member (op =) seen M then MType (s, [])
else repair_mtype cache (M :: seen) M
fun repair_datatype_mcache cache =
let
fun repair_one (z, M) =
Unsynchronized.change cache
(AList.update (op =) (z, repair_mtype (!cache) [] M))
in List.app repair_one (rev (!cache)) end
fun repair_constr_mcache dtype_cache constr_mcache =
let
fun repair_one (x, M) =
Unsynchronized.change constr_mcache
(AList.update (op =) (x, repair_mtype dtype_cache [] M))
in List.app repair_one (!constr_mcache) end
fun is_fin_fun_supported_type @{typ prop} = true
| is_fin_fun_supported_type @{typ bool} = true
| is_fin_fun_supported_type (Type (@{type_name option}, _)) = true
| is_fin_fun_supported_type _ = false
fun fin_fun_body _ _ (t as @{term False}) = SOME t
| fin_fun_body _ _ (t as Const (@{const_name None}, _)) = SOME t
| fin_fun_body dom_T ran_T
((t0 as Const (@{const_name If}, _))
$ (t1 as Const (@{const_name HOL.eq}, _) $ Bound 0 $ t1')
$ t2 $ t3) =
(if loose_bvar1 (t1', 0) then
NONE
else case fin_fun_body dom_T ran_T t3 of
NONE => NONE
| SOME t3 =>
SOME (t0 $ (Const (@{const_name is_unknown}, dom_T --> bool_T) $ t1')
$ (Const (@{const_name unknown}, ran_T)) $ (t0 $ t1 $ t2 $ t3)))
| fin_fun_body _ _ _ = NONE
fun fresh_mfun_for_fun_type (mdata as {max_fresh, ...} : mdata) all_minus
T1 T2 =
let
val M1 = fresh_mtype_for_type mdata all_minus T1
val M2 = fresh_mtype_for_type mdata all_minus T2
val a = if not all_minus andalso exists_alpha_sub_mtype_fresh M1 andalso
is_fin_fun_supported_type (body_type T2) then
V (Unsynchronized.inc max_fresh)
else
S Minus
in (M1, a, M2) end
and fresh_mtype_for_type (mdata as {hol_ctxt as {ctxt, ...}, binarize, alpha_T,
datatype_mcache, constr_mcache, ...})
all_minus =
let
fun do_type T =
if T = alpha_T then
MAlpha
else case T of
Type (@{type_name fun}, [T1, T2]) =>
MFun (fresh_mfun_for_fun_type mdata all_minus T1 T2)
| Type (@{type_name prod}, [T1, T2]) => MPair (pairself do_type (T1, T2))
| Type (z as (s, _)) =>
if could_exist_alpha_sub_mtype ctxt alpha_T T then
case AList.lookup (op =) (!datatype_mcache) z of
SOME M => M
| NONE =>
let
val _ = Unsynchronized.change datatype_mcache (cons (z, MRec z))
val xs = binarized_and_boxed_datatype_constrs hol_ctxt binarize T
val (all_Ms, constr_Ms) =
fold_rev (fn (_, T') => fn (all_Ms, constr_Ms) =>
let
val binder_Ms = map do_type (binder_types T')
val new_Ms = filter exists_alpha_sub_mtype_fresh
binder_Ms
val constr_M = constr_mtype_for_binders z
binder_Ms
in
(union (op =) new_Ms all_Ms,
constr_M :: constr_Ms)
end)
xs ([], [])
val M = MType (s, all_Ms)
val _ = Unsynchronized.change datatype_mcache
(AList.update (op =) (z, M))
val _ = Unsynchronized.change constr_mcache
(append (xs ~~ constr_Ms))
in
if forall (not o is_MRec o snd) (!datatype_mcache) then
(repair_datatype_mcache datatype_mcache;
repair_constr_mcache (!datatype_mcache) constr_mcache;
AList.lookup (op =) (!datatype_mcache) z |> the)
else
M
end
else
MType (s, [])
| _ => MType (simple_string_of_typ T, [])
in do_type end
fun prodM_factors (MPair (M1, M2)) = maps prodM_factors [M1, M2]
| prodM_factors M = [M]
fun curried_strip_mtype (MFun (M1, _, M2)) =
curried_strip_mtype M2 |>> append (prodM_factors M1)
| curried_strip_mtype M = ([], M)
fun sel_mtype_from_constr_mtype s M =
let val (arg_Ms, dataM) = curried_strip_mtype M in
MFun (dataM, S Minus,
case sel_no_from_name s of ~1 => bool_M | n => nth arg_Ms n)
end
fun mtype_for_constr (mdata as {hol_ctxt = {ctxt, ...}, alpha_T, constr_mcache,
...}) (x as (_, T)) =
if could_exist_alpha_sub_mtype ctxt alpha_T T then
case AList.lookup (op =) (!constr_mcache) x of
SOME M => M
| NONE => if T = alpha_T then
let val M = fresh_mtype_for_type mdata false T in
(Unsynchronized.change constr_mcache (cons (x, M)); M)
end
else
(fresh_mtype_for_type mdata false (body_type T);
AList.lookup (op =) (!constr_mcache) x |> the)
else
fresh_mtype_for_type mdata false T
fun mtype_for_sel (mdata as {hol_ctxt, binarize, ...}) (x as (s, _)) =
x |> binarized_and_boxed_constr_for_sel hol_ctxt binarize
|> mtype_for_constr mdata |> sel_mtype_from_constr_mtype s
fun resolve_sign_atom lits (V x) =
x |> AList.lookup (op =) lits |> Option.map S |> the_default (V x)
| resolve_sign_atom _ a = a
fun resolve_mtype lits =
let
fun aux MAlpha = MAlpha
| aux (MFun (M1, a, M2)) = MFun (aux M1, resolve_sign_atom lits a, aux M2)
| aux (MPair Mp) = MPair (pairself aux Mp)
| aux (MType (s, Ms)) = MType (s, map aux Ms)
| aux (MRec z) = MRec z
in aux end
datatype comp_op = Eq | Leq
type comp = sign_atom * sign_atom * comp_op * var list
type sign_expr = literal list
type constraint_set = literal list * comp list * sign_expr list
fun string_for_comp_op Eq = "="
| string_for_comp_op Leq = "\<le>"
fun string_for_sign_expr [] = "\<bot>"
| string_for_sign_expr lits =
space_implode " \<or> " (map string_for_literal lits)
fun do_literal _ NONE = NONE
| do_literal (x, sn) (SOME lits) =
case AList.lookup (op =) lits x of
SOME sn' => if sn = sn' then SOME lits else NONE
| NONE => SOME ((x, sn) :: lits)
fun do_sign_atom_comp Eq [] a1 a2 (accum as (lits, comps)) =
(case (a1, a2) of
(S sn1, S sn2) => if sn1 = sn2 then SOME accum else NONE
| (V x1, S sn2) =>
Option.map (rpair comps) (do_literal (x1, sn2) (SOME lits))
| (V _, V _) => SOME (lits, insert (op =) (a1, a2, Eq, []) comps)
| _ => do_sign_atom_comp Eq [] a2 a1 accum)
| do_sign_atom_comp Leq [] a1 a2 (accum as (lits, comps)) =
(case (a1, a2) of
(_, S Minus) => SOME accum
| (S Plus, _) => SOME accum
| (S Minus, S Plus) => NONE
| (V _, V _) => SOME (lits, insert (op =) (a1, a2, Leq, []) comps)
| _ => do_sign_atom_comp Eq [] a1 a2 accum)
| do_sign_atom_comp cmp xs a1 a2 (lits, comps) =
SOME (lits, insert (op =) (a1, a2, cmp, xs) comps)
fun do_mtype_comp _ _ _ _ NONE = NONE
| do_mtype_comp _ _ MAlpha MAlpha accum = accum
| do_mtype_comp Eq xs (MFun (M11, a1, M12)) (MFun (M21, a2, M22))
(SOME accum) =
accum |> do_sign_atom_comp Eq xs a1 a2 |> do_mtype_comp Eq xs M11 M21
|> do_mtype_comp Eq xs M12 M22
| do_mtype_comp Leq xs (MFun (M11, a1, M12)) (MFun (M21, a2, M22))
(SOME accum) =
(if exists_alpha_sub_mtype M11 then
accum |> do_sign_atom_comp Leq xs a1 a2
|> do_mtype_comp Leq xs M21 M11
|> (case a2 of
S Minus => I
| S Plus => do_mtype_comp Leq xs M11 M21
| V x => do_mtype_comp Leq (x :: xs) M11 M21)
else
SOME accum)
|> do_mtype_comp Leq xs M12 M22
| do_mtype_comp cmp xs (M1 as MPair (M11, M12)) (M2 as MPair (M21, M22))
accum =
(accum |> fold (uncurry (do_mtype_comp cmp xs)) [(M11, M21), (M12, M22)]
handle Library.UnequalLengths =>
raise MTYPE ("Nitpick_Mono.do_mtype_comp", [M1, M2], []))
| do_mtype_comp _ _ (MType _) (MType _) accum =
accum (* no need to compare them thanks to the cache *)
| do_mtype_comp cmp _ M1 M2 _ =
raise MTYPE ("Nitpick_Mono.do_mtype_comp (" ^ string_for_comp_op cmp ^ ")",
[M1, M2], [])
fun add_mtype_comp cmp M1 M2 ((lits, comps, sexps) : constraint_set) =
(trace_msg (fn () => "*** Add " ^ string_for_mtype M1 ^ " " ^
string_for_comp_op cmp ^ " " ^ string_for_mtype M2);
case do_mtype_comp cmp [] M1 M2 (SOME (lits, comps)) of
NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
| SOME (lits, comps) => (lits, comps, sexps))
val add_mtypes_equal = add_mtype_comp Eq
val add_is_sub_mtype = add_mtype_comp Leq
fun do_notin_mtype_fv _ _ _ NONE = NONE
| do_notin_mtype_fv Minus _ MAlpha accum = accum
| do_notin_mtype_fv Plus [] MAlpha _ = NONE
| do_notin_mtype_fv Plus [(x, sn)] MAlpha (SOME (lits, sexps)) =
SOME lits |> do_literal (x, sn) |> Option.map (rpair sexps)
| do_notin_mtype_fv Plus sexp MAlpha (SOME (lits, sexps)) =
SOME (lits, insert (op =) sexp sexps)
| do_notin_mtype_fv sn sexp (MFun (M1, S sn', M2)) accum =
accum |> (if sn' = Plus andalso sn = Plus then
do_notin_mtype_fv Plus sexp M1
else
I)
|> (if sn' = Minus orelse sn = Plus then
do_notin_mtype_fv Minus sexp M1
else
I)
|> do_notin_mtype_fv sn sexp M2
| do_notin_mtype_fv Plus sexp (MFun (M1, V x, M2)) accum =
accum |> (case do_literal (x, Minus) (SOME sexp) of
NONE => I
| SOME sexp' => do_notin_mtype_fv Plus sexp' M1)
|> do_notin_mtype_fv Minus sexp M1
|> do_notin_mtype_fv Plus sexp M2
| do_notin_mtype_fv Minus sexp (MFun (M1, V x, M2)) accum =
accum |> (case do_literal (x, Plus) (SOME sexp) of
NONE => I
| SOME sexp' => do_notin_mtype_fv Plus sexp' M1)
|> do_notin_mtype_fv Minus sexp M2
| do_notin_mtype_fv sn sexp (MPair (M1, M2)) accum =
accum |> fold (do_notin_mtype_fv sn sexp) [M1, M2]
| do_notin_mtype_fv sn sexp (MType (_, Ms)) accum =
accum |> fold (do_notin_mtype_fv sn sexp) Ms
| do_notin_mtype_fv _ _ M _ =
raise MTYPE ("Nitpick_Mono.do_notin_mtype_fv", [M], [])
fun add_notin_mtype_fv sn M ((lits, comps, sexps) : constraint_set) =
(trace_msg (fn () => "*** Add " ^ string_for_mtype M ^ " is " ^
(case sn of Minus => "concrete" | Plus => "complete") ^
".");
case do_notin_mtype_fv sn [] M (SOME (lits, sexps)) of
NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
| SOME (lits, sexps) => (lits, comps, sexps))
val add_mtype_is_concrete = add_notin_mtype_fv Minus
val add_mtype_is_complete = add_notin_mtype_fv Plus
val bool_from_minus = true
fun bool_from_sign Plus = not bool_from_minus
| bool_from_sign Minus = bool_from_minus
fun sign_from_bool b = if b = bool_from_minus then Minus else Plus
fun prop_for_literal (x, sn) =
(not (bool_from_sign sn) ? PropLogic.Not) (PropLogic.BoolVar x)
fun prop_for_sign_atom_eq (S sn', sn) =
if sn = sn' then PropLogic.True else PropLogic.False
| prop_for_sign_atom_eq (V x, sn) = prop_for_literal (x, sn)
fun prop_for_sign_expr xs = PropLogic.exists (map prop_for_literal xs)
fun prop_for_exists_eq xs sn =
PropLogic.exists (map (fn x => prop_for_literal (x, sn)) xs)
fun prop_for_comp (a1, a2, Eq, []) =
PropLogic.SAnd (prop_for_comp (a1, a2, Leq, []),
prop_for_comp (a2, a1, Leq, []))
| prop_for_comp (a1, a2, Leq, []) =
PropLogic.SOr (prop_for_sign_atom_eq (a1, Plus),
prop_for_sign_atom_eq (a2, Minus))
| prop_for_comp (a1, a2, cmp, xs) =
PropLogic.SOr (prop_for_exists_eq xs Minus, prop_for_comp (a1, a2, cmp, []))
fun literals_from_assignments max_var assigns lits =
fold (fn x => fn accum =>
if AList.defined (op =) lits x then
accum
else case assigns x of
SOME b => (x, sign_from_bool b) :: accum
| NONE => accum) (max_var downto 1) lits
fun string_for_comp (a1, a2, cmp, xs) =
string_for_sign_atom a1 ^ " " ^ string_for_comp_op cmp ^
subscript_string_for_vars " \<and> " xs ^ " " ^ string_for_sign_atom a2
fun print_problem lits comps sexps =
trace_msg (fn () => "*** Problem:\n" ^
cat_lines (map string_for_literal lits @
map string_for_comp comps @
map string_for_sign_expr sexps))
fun print_solution lits =
let val (pos, neg) = List.partition (curry (op =) Plus o snd) lits in
trace_msg (fn () => "*** Solution:\n" ^
"+: " ^ commas (map (string_for_var o fst) pos) ^ "\n" ^
"-: " ^ commas (map (string_for_var o fst) neg))
end
fun solve max_var (lits, comps, sexps) =
let
fun do_assigns assigns =
SOME (literals_from_assignments max_var assigns lits
|> tap print_solution)
val _ = print_problem lits comps sexps
val prop = PropLogic.all (map prop_for_literal lits @
map prop_for_comp comps @
map prop_for_sign_expr sexps)
val default_val = bool_from_sign Minus
in
if PropLogic.eval (K default_val) prop then
do_assigns (K (SOME default_val))
else
let
(* use the first ML solver (to avoid startup overhead) *)
val solvers = !SatSolver.solvers
|> filter (member (op =) ["dptsat", "dpll"] o fst)
in
case snd (hd solvers) prop of
SatSolver.SATISFIABLE assigns => do_assigns assigns
| _ => NONE
end
end
type mtype_schema = mtyp * constraint_set
type mtype_context =
{bound_Ts: typ list,
bound_Ms: mtyp list,
frees: (styp * mtyp) list,
consts: (styp * mtyp) list}
type accumulator = mtype_context * constraint_set
val initial_gamma = {bound_Ts = [], bound_Ms = [], frees = [], consts = []}
fun push_bound T M {bound_Ts, bound_Ms, frees, consts} =
{bound_Ts = T :: bound_Ts, bound_Ms = M :: bound_Ms, frees = frees,
consts = consts}
fun pop_bound {bound_Ts, bound_Ms, frees, consts} =
{bound_Ts = tl bound_Ts, bound_Ms = tl bound_Ms, frees = frees,
consts = consts}
handle List.Empty => initial_gamma (* FIXME: needed? *)
fun consider_term (mdata as {hol_ctxt as {thy, ctxt, stds, fast_descrs,
def_table, ...},
alpha_T, max_fresh, ...}) =
let
fun is_enough_eta_expanded t =
case strip_comb t of
(Const x, ts) =>
the_default 0 (arity_of_built_in_const thy stds fast_descrs x)
<= length ts
| _ => true
val mtype_for = fresh_mtype_for_type mdata false
fun plus_set_mtype_for_dom M =
MFun (M, S (if exists_alpha_sub_mtype M then Plus else Minus), bool_M)
fun do_all T (gamma, cset) =
let
val abs_M = mtype_for (domain_type (domain_type T))
val body_M = mtype_for (body_type T)
in
(MFun (MFun (abs_M, S Minus, body_M), S Minus, body_M),
(gamma, cset |> add_mtype_is_complete abs_M))
end
fun do_equals T (gamma, cset) =
let val M = mtype_for (domain_type T) in
(MFun (M, S Minus, MFun (M, V (Unsynchronized.inc max_fresh),
mtype_for (nth_range_type 2 T))),
(gamma, cset |> add_mtype_is_concrete M))
end
fun do_robust_set_operation T (gamma, cset) =
let
val set_T = domain_type T
val M1 = mtype_for set_T
val M2 = mtype_for set_T
val M3 = mtype_for set_T
in
(MFun (M1, S Minus, MFun (M2, S Minus, M3)),
(gamma, cset |> add_is_sub_mtype M1 M3 |> add_is_sub_mtype M2 M3))
end
fun do_fragile_set_operation T (gamma, cset) =
let
val set_T = domain_type T
val set_M = mtype_for set_T
fun custom_mtype_for (T as Type (@{type_name fun}, [T1, T2])) =
if T = set_T then set_M
else MFun (custom_mtype_for T1, S Minus, custom_mtype_for T2)
| custom_mtype_for T = mtype_for T
in
(custom_mtype_for T, (gamma, cset |> add_mtype_is_concrete set_M))
end
fun do_pair_constr T accum =
case mtype_for (nth_range_type 2 T) of
M as MPair (a_M, b_M) =>
(MFun (a_M, S Minus, MFun (b_M, S Minus, M)), accum)
| M => raise MTYPE ("Nitpick_Mono.consider_term.do_pair_constr", [M], [])
fun do_nth_pair_sel n T =
case mtype_for (domain_type T) of
M as MPair (a_M, b_M) =>
pair (MFun (M, S Minus, if n = 0 then a_M else b_M))
| M => raise MTYPE ("Nitpick_Mono.consider_term.do_nth_pair_sel", [M], [])
fun do_bounded_quantifier t0 abs_s abs_T connective_t bound_t body_t accum =
let
val abs_M = mtype_for abs_T
val (bound_m, accum) =
accum |>> push_bound abs_T abs_M |> do_term bound_t
val expected_bound_M = plus_set_mtype_for_dom abs_M
val (body_m, accum) =
accum ||> add_mtypes_equal expected_bound_M (mtype_of_mterm bound_m)
|> do_term body_t ||> apfst pop_bound
val bound_M = mtype_of_mterm bound_m
val (M1, a, M2) = dest_MFun bound_M
in
(MApp (MRaw (t0, MFun (bound_M, S Minus, bool_M)),
MAbs (abs_s, abs_T, M1, a,
MApp (MApp (MRaw (connective_t,
mtype_for (fastype_of connective_t)),
MApp (bound_m, MRaw (Bound 0, M1))),
body_m))), accum)
end
and do_term t (accum as (gamma as {bound_Ts, bound_Ms, frees, consts},
cset)) =
(trace_msg (fn () => " \<Gamma> \<turnstile> " ^
Syntax.string_of_term ctxt t ^ " : _?");
case t of
Const (x as (s, T)) =>
(case AList.lookup (op =) consts x of
SOME M => (M, accum)
| NONE =>
if not (could_exist_alpha_subtype alpha_T T) then
(mtype_for T, accum)
else case s of
@{const_name all} => do_all T accum
| @{const_name "=="} => do_equals T accum
| @{const_name All} => do_all T accum
| @{const_name Ex} =>
let val set_T = domain_type T in
do_term (Abs (Name.uu, set_T,
@{const Not} $ (HOLogic.mk_eq
(Abs (Name.uu, domain_type set_T,
@{const False}),
Bound 0)))) accum
|>> mtype_of_mterm
end
| @{const_name HOL.eq} => do_equals T accum
| @{const_name The} =>
(trace_msg (K "*** The"); raise UNSOLVABLE ())
| @{const_name Eps} =>
(trace_msg (K "*** Eps"); raise UNSOLVABLE ())
| @{const_name If} =>
do_robust_set_operation (range_type T) accum
|>> curry3 MFun bool_M (S Minus)
| @{const_name Pair} => do_pair_constr T accum
| @{const_name fst} => do_nth_pair_sel 0 T accum
| @{const_name snd} => do_nth_pair_sel 1 T accum
| @{const_name Id} =>
(MFun (mtype_for (domain_type T), S Minus, bool_M), accum)
| @{const_name converse} =>
let
val x = Unsynchronized.inc max_fresh
fun mtype_for_set T =
MFun (mtype_for (domain_type T), V x, bool_M)
val ab_set_M = domain_type T |> mtype_for_set
val ba_set_M = range_type T |> mtype_for_set
in (MFun (ab_set_M, S Minus, ba_set_M), accum) end
| @{const_name trancl} => do_fragile_set_operation T accum
| @{const_name rel_comp} =>
let
val x = Unsynchronized.inc max_fresh
fun mtype_for_set T =
MFun (mtype_for (domain_type T), V x, bool_M)
val bc_set_M = domain_type T |> mtype_for_set
val ab_set_M = domain_type (range_type T) |> mtype_for_set
val ac_set_M = nth_range_type 2 T |> mtype_for_set
in
(MFun (bc_set_M, S Minus, MFun (ab_set_M, S Minus, ac_set_M)),
accum)
end
| @{const_name image} =>
let
val a_M = mtype_for (domain_type (domain_type T))
val b_M = mtype_for (range_type (domain_type T))
in
(MFun (MFun (a_M, S Minus, b_M), S Minus,
MFun (plus_set_mtype_for_dom a_M, S Minus,
plus_set_mtype_for_dom b_M)), accum)
end
| @{const_name finite} =>
let val M1 = mtype_for (domain_type (domain_type T)) in
(MFun (plus_set_mtype_for_dom M1, S Minus, bool_M), accum)
end
| @{const_name Sigma} =>
let
val x = Unsynchronized.inc max_fresh
fun mtype_for_set T =
MFun (mtype_for (domain_type T), V x, bool_M)
val a_set_T = domain_type T
val a_M = mtype_for (domain_type a_set_T)
val b_set_M = mtype_for_set (range_type (domain_type
(range_type T)))
val a_set_M = mtype_for_set a_set_T
val a_to_b_set_M = MFun (a_M, S Minus, b_set_M)
val ab_set_M = mtype_for_set (nth_range_type 2 T)
in
(MFun (a_set_M, S Minus,
MFun (a_to_b_set_M, S Minus, ab_set_M)), accum)
end
| _ =>
if s = @{const_name safe_The} orelse
s = @{const_name safe_Eps} then
let
val a_set_M = mtype_for (domain_type T)
val a_M = dest_MFun a_set_M |> #1
in (MFun (a_set_M, S Minus, a_M), accum) end
else if s = @{const_name ord_class.less_eq} andalso
is_set_type (domain_type T) then
do_fragile_set_operation T accum
else if is_sel s then
(mtype_for_sel mdata x, accum)
else if is_constr ctxt stds x then
(mtype_for_constr mdata x, accum)
else if is_built_in_const thy stds fast_descrs x then
(fresh_mtype_for_type mdata true T, accum)
else
let val M = mtype_for T in
(M, ({bound_Ts = bound_Ts, bound_Ms = bound_Ms,
frees = frees, consts = (x, M) :: consts}, cset))
end) |>> curry MRaw t
| Free (x as (_, T)) =>
(case AList.lookup (op =) frees x of
SOME M => (M, accum)
| NONE =>
let val M = mtype_for T in
(M, ({bound_Ts = bound_Ts, bound_Ms = bound_Ms,
frees = (x, M) :: frees, consts = consts}, cset))
end) |>> curry MRaw t
| Var _ => (trace_msg (K "*** Var"); raise UNSOLVABLE ())
| Bound j => (MRaw (t, nth bound_Ms j), accum)
| Abs (s, T, t') =>
(case fin_fun_body T (fastype_of1 (T :: bound_Ts, t')) t' of
SOME t' =>
let
val M = mtype_for T
val a = V (Unsynchronized.inc max_fresh)
val (m', accum) = do_term t' (accum |>> push_bound T M)
in (MAbs (s, T, M, a, m'), accum |>> pop_bound) end
| NONE =>
((case t' of
t1' $ Bound 0 =>
if not (loose_bvar1 (t1', 0)) andalso
is_enough_eta_expanded t1' then
do_term (incr_boundvars ~1 t1') accum
else
raise SAME ()
| (t11 as Const (@{const_name HOL.eq}, _)) $ Bound 0 $ t13 =>
if not (loose_bvar1 (t13, 0)) then
do_term (incr_boundvars ~1 (t11 $ t13)) accum
else
raise SAME ()
| _ => raise SAME ())
handle SAME () =>
let
val M = mtype_for T
val (m', accum) = do_term t' (accum |>> push_bound T M)
in
(MAbs (s, T, M, S Minus, m'), accum |>> pop_bound)
end))
| (t0 as Const (@{const_name All}, _))
$ Abs (s', T', (t10 as @{const HOL.implies}) $ (t11 $ Bound 0) $ t12) =>
do_bounded_quantifier t0 s' T' t10 t11 t12 accum
| (t0 as Const (@{const_name Ex}, _))
$ Abs (s', T', (t10 as @{const HOL.conj}) $ (t11 $ Bound 0) $ t12) =>
do_bounded_quantifier t0 s' T' t10 t11 t12 accum
| Const (@{const_name Let}, _) $ t1 $ t2 =>
do_term (betapply (t2, t1)) accum
| t1 $ t2 =>
let
val (m1, accum) = do_term t1 accum
val (m2, accum) = do_term t2 accum
in
let
val T11 = domain_type (fastype_of1 (bound_Ts, t1))
val T2 = fastype_of1 (bound_Ts, t2)
val M11 = mtype_of_mterm m1 |> dest_MFun |> #1
val M2 = mtype_of_mterm m2
in (MApp (m1, m2), accum ||> add_is_sub_mtype M2 M11) end
end)
|> tap (fn (m, _) => trace_msg (fn () => " \<Gamma> \<turnstile> " ^
string_for_mterm ctxt m))
in do_term end
fun force_minus_funs 0 _ = I
| force_minus_funs n (M as MFun (M1, _, M2)) =
add_mtypes_equal M (MFun (M1, S Minus, M2))
#> force_minus_funs (n - 1) M2
| force_minus_funs _ M =
raise MTYPE ("Nitpick_Mono.force_minus_funs", [M], [])
fun consider_general_equals mdata def (x as (_, T)) t1 t2 accum =
let
val (m1, accum) = consider_term mdata t1 accum
val (m2, accum) = consider_term mdata t2 accum
val M1 = mtype_of_mterm m1
val M2 = mtype_of_mterm m2
val accum = accum ||> add_mtypes_equal M1 M2
val body_M = fresh_mtype_for_type mdata false (nth_range_type 2 T)
val m = MApp (MApp (MRaw (Const x,
MFun (M1, S Minus, MFun (M2, S Minus, body_M))), m1), m2)
in
(m, if def then
let val (head_m, arg_ms) = strip_mcomb m1 in
accum ||> force_minus_funs (length arg_ms) (mtype_of_mterm head_m)
end
else
accum)
end
fun consider_general_formula (mdata as {hol_ctxt = {ctxt, ...}, ...}) =
let
val mtype_for = fresh_mtype_for_type mdata false
val do_term = consider_term mdata
fun do_formula sn t accum =
let
fun do_quantifier (quant_x as (quant_s, _)) abs_s abs_T body_t =
let
val abs_M = mtype_for abs_T
val side_cond = ((sn = Minus) = (quant_s = @{const_name Ex}))
val (body_m, accum) =
accum ||> side_cond ? add_mtype_is_complete abs_M
|>> push_bound abs_T abs_M |> do_formula sn body_t
val body_M = mtype_of_mterm body_m
in
(MApp (MRaw (Const quant_x,
MFun (MFun (abs_M, S Minus, body_M), S Minus,
body_M)),
MAbs (abs_s, abs_T, abs_M, S Minus, body_m)),
accum |>> pop_bound)
end
fun do_equals x t1 t2 =
case sn of
Plus => do_term t accum
| Minus => consider_general_equals mdata false x t1 t2 accum
in
(trace_msg (fn () => " \<Gamma> \<turnstile> " ^
Syntax.string_of_term ctxt t ^ " : o\<^sup>" ^
string_for_sign sn ^ "?");
case t of
Const (x as (@{const_name all}, _)) $ Abs (s1, T1, t1) =>
do_quantifier x s1 T1 t1
| Const (x as (@{const_name "=="}, _)) $ t1 $ t2 => do_equals x t1 t2
| @{const Trueprop} $ t1 =>
let val (m1, accum) = do_formula sn t1 accum in
(MApp (MRaw (@{const Trueprop}, mtype_for (bool_T --> prop_T)),
m1), accum)
end
| @{const Not} $ t1 =>
let val (m1, accum) = do_formula (negate sn) t1 accum in
(MApp (MRaw (@{const Not}, mtype_for (bool_T --> bool_T)), m1),
accum)
end
| Const (x as (@{const_name All}, _)) $ Abs (s1, T1, t1) =>
do_quantifier x s1 T1 t1
| Const (x0 as (s0 as @{const_name Ex}, T0))
$ (t1 as Abs (s1, T1, t1')) =>
(case sn of
Plus => do_quantifier x0 s1 T1 t1'
| Minus =>
(* FIXME: Move elsewhere *)
do_term (@{const Not}
$ (HOLogic.eq_const (domain_type T0) $ t1
$ Abs (Name.uu, T1, @{const False}))) accum)
| Const (x as (@{const_name HOL.eq}, _)) $ t1 $ t2 =>
do_equals x t1 t2
| Const (@{const_name Let}, _) $ t1 $ t2 =>
do_formula sn (betapply (t2, t1)) accum
| (t0 as Const (s0, _)) $ t1 $ t2 =>
if s0 = @{const_name "==>"} orelse
s0 = @{const_name Pure.conjunction} orelse
s0 = @{const_name HOL.conj} orelse
s0 = @{const_name HOL.disj} orelse
s0 = @{const_name HOL.implies} then
let
val impl = (s0 = @{const_name "==>"} orelse
s0 = @{const_name HOL.implies})
val (m1, accum) = do_formula (sn |> impl ? negate) t1 accum
val (m2, accum) = do_formula sn t2 accum
in
(MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2),
accum)
end
else
do_term t accum
| _ => do_term t accum)
end
|> tap (fn (m, _) =>
trace_msg (fn () => "\<Gamma> \<turnstile> " ^
string_for_mterm ctxt m ^ " : o\<^sup>" ^
string_for_sign sn))
in do_formula end
(* The harmless axiom optimization below is somewhat too aggressive in the face
of (rather peculiar) user-defined axioms. *)
val harmless_consts =
[@{const_name ord_class.less}, @{const_name ord_class.less_eq}]
val bounteous_consts = [@{const_name bisim}]
fun is_harmless_axiom ({no_harmless = true, ...} : mdata) _ = false
| is_harmless_axiom {hol_ctxt = {thy, stds, fast_descrs, ...}, ...} t =
Term.add_consts t []
|> filter_out (is_built_in_const thy stds fast_descrs)
|> (forall (member (op =) harmless_consts o original_name o fst) orf
exists (member (op =) bounteous_consts o fst))
fun consider_nondefinitional_axiom mdata t =
if is_harmless_axiom mdata t then pair (MRaw (t, dummy_M))
else consider_general_formula mdata Plus t
fun consider_definitional_axiom (mdata as {hol_ctxt = {ctxt, ...}, ...}) t =
if not (is_constr_pattern_formula ctxt t) then
consider_nondefinitional_axiom mdata t
else if is_harmless_axiom mdata t then
pair (MRaw (t, dummy_M))
else
let
val mtype_for = fresh_mtype_for_type mdata false
val do_term = consider_term mdata
fun do_all quant_t abs_s abs_T body_t accum =
let
val abs_M = mtype_for abs_T
val (body_m, accum) =
accum |>> push_bound abs_T abs_M |> do_formula body_t
val body_M = mtype_of_mterm body_m
in
(MApp (MRaw (quant_t,
MFun (MFun (abs_M, S Minus, body_M), S Minus, body_M)),
MAbs (abs_s, abs_T, abs_M, S Minus, body_m)),
accum |>> pop_bound)
end
and do_conjunction t0 t1 t2 accum =
let
val (m1, accum) = do_formula t1 accum
val (m2, accum) = do_formula t2 accum
in
(MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2), accum)
end
and do_implies t0 t1 t2 accum =
let
val (m1, accum) = do_term t1 accum
val (m2, accum) = do_formula t2 accum
in
(MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2), accum)
end
and do_formula t accum =
case t of
(t0 as Const (@{const_name all}, _)) $ Abs (s1, T1, t1) =>
do_all t0 s1 T1 t1 accum
| @{const Trueprop} $ t1 =>
let val (m1, accum) = do_formula t1 accum in
(MApp (MRaw (@{const Trueprop}, mtype_for (bool_T --> prop_T)),
m1), accum)
end
| Const (x as (@{const_name "=="}, _)) $ t1 $ t2 =>
consider_general_equals mdata true x t1 t2 accum
| (t0 as @{const "==>"}) $ t1 $ t2 => do_implies t0 t1 t2 accum
| (t0 as @{const Pure.conjunction}) $ t1 $ t2 =>
do_conjunction t0 t1 t2 accum
| (t0 as Const (@{const_name All}, _)) $ Abs (s0, T1, t1) =>
do_all t0 s0 T1 t1 accum
| Const (x as (@{const_name HOL.eq}, _)) $ t1 $ t2 =>
consider_general_equals mdata true x t1 t2 accum
| (t0 as @{const HOL.conj}) $ t1 $ t2 => do_conjunction t0 t1 t2 accum
| (t0 as @{const HOL.implies}) $ t1 $ t2 => do_implies t0 t1 t2 accum
| _ => raise TERM ("Nitpick_Mono.consider_definitional_axiom.\
\do_formula", [t])
in do_formula t end
fun string_for_mtype_of_term ctxt lits t M =
Syntax.string_of_term ctxt t ^ " : " ^ string_for_mtype (resolve_mtype lits M)
fun print_mtype_context ctxt lits ({frees, consts, ...} : mtype_context) =
trace_msg (fn () =>
map (fn (x, M) => string_for_mtype_of_term ctxt lits (Free x) M) frees @
map (fn (x, M) => string_for_mtype_of_term ctxt lits (Const x) M) consts
|> cat_lines)
fun amass f t (ms, accum) =
let val (m, accum) = f t accum in (m :: ms, accum) end
fun infer which no_harmless (hol_ctxt as {ctxt, ...}) binarize alpha_T
(nondef_ts, def_ts) =
let
val _ = trace_msg (fn () => "****** " ^ which ^ " analysis: " ^
string_for_mtype MAlpha ^ " is " ^
Syntax.string_of_typ ctxt alpha_T)
val mdata as {max_fresh, constr_mcache, ...} =
initial_mdata hol_ctxt binarize no_harmless alpha_T
val accum = (initial_gamma, ([], [], []))
val (nondef_ms, accum) =
([], accum) |> amass (consider_general_formula mdata Plus) (hd nondef_ts)
|> fold (amass (consider_nondefinitional_axiom mdata))
(tl nondef_ts)
val (def_ms, (gamma, cset)) =
([], accum) |> fold (amass (consider_definitional_axiom mdata)) def_ts
in
case solve (!max_fresh) cset of
SOME lits => (print_mtype_context ctxt lits gamma;
SOME (lits, (nondef_ms, def_ms), !constr_mcache))
| _ => NONE
end
handle UNSOLVABLE () => NONE
| MTYPE (loc, Ms, Ts) =>
raise BAD (loc, commas (map string_for_mtype Ms @
map (Syntax.string_of_typ ctxt) Ts))
| MTERM (loc, ms) =>
raise BAD (loc, commas (map (string_for_mterm ctxt) ms))
val formulas_monotonic = is_some oooo infer "Monotonicity" false
fun fin_fun_constr T1 T2 =
(@{const_name FinFun}, (T1 --> T2) --> Type (@{type_name fin_fun}, [T1, T2]))
fun finitize_funs (hol_ctxt as {thy, ctxt, stds, fast_descrs, constr_cache,
...})
binarize finitizes alpha_T tsp =
case infer "Finiteness" true hol_ctxt binarize alpha_T tsp of
SOME (lits, msp, constr_mtypes) =>
if forall (curry (op =) Minus o snd) lits then
tsp
else
let
fun should_finitize T a =
case triple_lookup (type_match thy) finitizes T of
SOME (SOME false) => false
| _ => resolve_sign_atom lits a = S Plus
fun type_from_mtype T M =
case (M, T) of
(MAlpha, _) => T
| (MFun (M1, a, M2), Type (@{type_name fun}, Ts)) =>
Type (if should_finitize T a then @{type_name fin_fun}
else @{type_name fun}, map2 type_from_mtype Ts [M1, M2])
| (MPair (M1, M2), Type (@{type_name prod}, Ts)) =>
Type (@{type_name prod}, map2 type_from_mtype Ts [M1, M2])
| (MType _, _) => T
| _ => raise MTYPE ("Nitpick_Mono.finitize_funs.type_from_mtype",
[M], [T])
fun finitize_constr (x as (s, T)) =
(s, case AList.lookup (op =) constr_mtypes x of
SOME M => type_from_mtype T M
| NONE => T)
fun term_from_mterm new_Ts old_Ts m =
case m of
MRaw (t, M) =>
let
val T = fastype_of1 (old_Ts, t)
val T' = type_from_mtype T M
in
case t of
Const (x as (s, _)) =>
if s = @{const_name finite} then
case domain_type T' of
set_T' as Type (@{type_name fin_fun}, _) =>
Abs (Name.uu, set_T', @{const True})
| _ => Const (s, T')
else if s = @{const_name "=="} orelse
s = @{const_name HOL.eq} then
let
val T =
case T' of
Type (_, [T1, Type (_, [T2, T3])]) =>
T1 --> T2 --> T3
| _ => raise TYPE ("Nitpick_Mono.finitize_funs.\
\term_from_mterm", [T, T'], [])
in coerce_term hol_ctxt new_Ts T' T (Const (s, T)) end
else if is_built_in_const thy stds fast_descrs x then
coerce_term hol_ctxt new_Ts T' T t
else if is_constr ctxt stds x then
Const (finitize_constr x)
else if is_sel s then
let
val n = sel_no_from_name s
val x' =
x |> binarized_and_boxed_constr_for_sel hol_ctxt binarize
|> finitize_constr
val x'' =
binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize
x' n
in Const x'' end
else
Const (s, T')
| Free (s, T) => Free (s, type_from_mtype T M)
| Bound _ => t
| _ => raise MTERM ("Nitpick_Mono.finitize_funs.term_from_mterm",
[m])
end
| MApp (m1, m2) =>
let
val (t1, t2) = pairself (term_from_mterm new_Ts old_Ts) (m1, m2)
val (T1, T2) = pairself (curry fastype_of1 new_Ts) (t1, t2)
val (t1', T2') =
case T1 of
Type (s, [T11, T12]) =>
(if s = @{type_name fin_fun} then
select_nth_constr_arg ctxt stds (fin_fun_constr T11 T12) t1
0 (T11 --> T12)
else
t1, T11)
| _ => raise TYPE ("Nitpick_Mono.finitize_funs.term_from_mterm",
[T1], [])
in betapply (t1', coerce_term hol_ctxt new_Ts T2' T2 t2) end
| MAbs (s, old_T, M, a, m') =>
let
val new_T = type_from_mtype old_T M
val t' = term_from_mterm (new_T :: new_Ts) (old_T :: old_Ts) m'
val T' = fastype_of1 (new_T :: new_Ts, t')
in
Abs (s, new_T, t')
|> should_finitize (new_T --> T') a
? construct_value ctxt stds (fin_fun_constr new_T T') o single
end
in
Unsynchronized.change constr_cache (map (apsnd (map finitize_constr)));
pairself (map (term_from_mterm [] [])) msp
end
| NONE => tsp
end;