four changes to Nitpick:
1. avoid writing absolute paths in Kodkodi files for input/output files of external SAT solvers (e.g. MiniSat), to dodge Cygwin problems
2. do eta-contraction in the monotonicity check
3. improved quantifier massaging algorithms using ideas from Paradox
4. repaired "check_potential" and "check_genuine"
(* Title: HOL/Tools/Nitpick/nitpick_mono.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2009, 2010
Monotonicity predicate for higher-order logic.
*)
signature NITPICK_MONO =
sig
datatype sign = Plus | Minus
type extended_context = Nitpick_HOL.extended_context
val formulas_monotonic :
extended_context -> typ -> sign -> term list -> term list -> term -> bool
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 ctype =
CAlpha |
CFun of ctype * sign_atom * ctype |
CPair of ctype * ctype |
CType of string * ctype list |
CRec of string * typ list
type cdata =
{ext_ctxt: extended_context,
alpha_T: typ,
max_fresh: int Unsynchronized.ref,
datatype_cache: ((string * typ list) * ctype) list Unsynchronized.ref,
constr_cache: (styp * ctype) list Unsynchronized.ref}
exception CTYPE of string * ctype list
(* string -> unit *)
fun print_g (s : string) = ()
(* var -> string *)
val string_for_var = signed_string_of_int
(* string -> var list -> string *)
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>"
(* sign -> string *)
fun string_for_sign Plus = "+"
| string_for_sign Minus = "-"
(* sign -> sign -> sign *)
fun xor sn1 sn2 = if sn1 = sn2 then Plus else Minus
(* sign -> sign *)
val negate = xor Minus
(* sign_atom -> string *)
fun string_for_sign_atom (S sn) = string_for_sign sn
| string_for_sign_atom (V j) = string_for_var j
(* literal -> string *)
fun string_for_literal (x, sn) = string_for_var x ^ " = " ^ string_for_sign sn
val bool_C = CType (@{type_name bool}, [])
(* ctype -> bool *)
fun is_CRec (CRec _) = true
| is_CRec _ = false
val no_prec = 100
val prec_CFun = 1
val prec_CPair = 2
(* tuple_set -> int *)
fun precedence_of_ctype (CFun _) = prec_CFun
| precedence_of_ctype (CPair _) = prec_CPair
| precedence_of_ctype _ = no_prec
(* ctype -> string *)
val string_for_ctype =
let
(* int -> ctype -> string *)
fun aux outer_prec C =
let
val prec = precedence_of_ctype C
val need_parens = (prec < outer_prec)
in
(if need_parens then "(" else "") ^
(case C of
CAlpha => "\<alpha>"
| CFun (C1, a, C2) =>
aux (prec + 1) C1 ^ " \<Rightarrow>\<^bsup>" ^
string_for_sign_atom a ^ "\<^esup> " ^ aux prec C2
| CPair (C1, C2) => aux (prec + 1) C1 ^ " \<times> " ^ aux prec C2
| CType (s, []) =>
if s = @{type_name prop} orelse s = @{type_name bool} then "o" else s
| CType (s, Cs) => "(" ^ commas (map (aux 0) Cs) ^ ") " ^ s
| CRec (s, _) => "[" ^ s ^ "]") ^
(if need_parens then ")" else "")
end
in aux 0 end
(* ctype -> ctype list *)
fun flatten_ctype (CPair (C1, C2)) = maps flatten_ctype [C1, C2]
| flatten_ctype (CType (_, Cs)) = maps flatten_ctype Cs
| flatten_ctype C = [C]
(* extended_context -> typ -> cdata *)
fun initial_cdata ext_ctxt alpha_T =
({ext_ctxt = ext_ctxt, alpha_T = alpha_T, max_fresh = Unsynchronized.ref 0,
datatype_cache = Unsynchronized.ref [],
constr_cache = Unsynchronized.ref []} : cdata)
(* typ -> typ -> bool *)
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)
(* theory -> typ -> typ -> bool *)
fun could_exist_alpha_sub_ctype _ (alpha_T as TFree _) T =
could_exist_alpha_subtype alpha_T T
| could_exist_alpha_sub_ctype thy alpha_T T =
(T = alpha_T orelse is_datatype thy T)
(* ctype -> bool *)
fun exists_alpha_sub_ctype CAlpha = true
| exists_alpha_sub_ctype (CFun (C1, _, C2)) =
exists exists_alpha_sub_ctype [C1, C2]
| exists_alpha_sub_ctype (CPair (C1, C2)) =
exists exists_alpha_sub_ctype [C1, C2]
| exists_alpha_sub_ctype (CType (_, Cs)) = exists exists_alpha_sub_ctype Cs
| exists_alpha_sub_ctype (CRec _) = true
(* ctype -> bool *)
fun exists_alpha_sub_ctype_fresh CAlpha = true
| exists_alpha_sub_ctype_fresh (CFun (_, V _, _)) = true
| exists_alpha_sub_ctype_fresh (CFun (_, _, C2)) =
exists_alpha_sub_ctype_fresh C2
| exists_alpha_sub_ctype_fresh (CPair (C1, C2)) =
exists exists_alpha_sub_ctype_fresh [C1, C2]
| exists_alpha_sub_ctype_fresh (CType (_, Cs)) =
exists exists_alpha_sub_ctype_fresh Cs
| exists_alpha_sub_ctype_fresh (CRec _) = true
(* string * typ list -> ctype list -> ctype *)
fun constr_ctype_for_binders z Cs =
fold_rev (fn C => curry3 CFun C (S Minus)) Cs (CRec z)
(* ((string * typ list) * ctype) list -> ctype list -> ctype -> ctype *)
fun repair_ctype _ _ CAlpha = CAlpha
| repair_ctype cache seen (CFun (C1, a, C2)) =
CFun (repair_ctype cache seen C1, a, repair_ctype cache seen C2)
| repair_ctype cache seen (CPair Cp) =
CPair (pairself (repair_ctype cache seen) Cp)
| repair_ctype cache seen (CType (s, Cs)) =
CType (s, maps (flatten_ctype o repair_ctype cache seen) Cs)
| repair_ctype cache seen (CRec (z as (s, _))) =
case AList.lookup (op =) cache z |> the of
CRec _ => CType (s, [])
| C => if member (op =) seen C then CType (s, [])
else repair_ctype cache (C :: seen) C
(* ((string * typ list) * ctype) list Unsynchronized.ref -> unit *)
fun repair_datatype_cache cache =
let
(* (string * typ list) * ctype -> unit *)
fun repair_one (z, C) =
Unsynchronized.change cache
(AList.update (op =) (z, repair_ctype (!cache) [] C))
in List.app repair_one (rev (!cache)) end
(* (typ * ctype) list -> (styp * ctype) list Unsynchronized.ref -> unit *)
fun repair_constr_cache dtype_cache constr_cache =
let
(* styp * ctype -> unit *)
fun repair_one (x, C) =
Unsynchronized.change constr_cache
(AList.update (op =) (x, repair_ctype dtype_cache [] C))
in List.app repair_one (!constr_cache) end
(* cdata -> typ -> ctype *)
fun fresh_ctype_for_type ({ext_ctxt as {thy, ...}, alpha_T, max_fresh,
datatype_cache, constr_cache, ...} : cdata) =
let
(* typ -> typ -> ctype *)
fun do_fun T1 T2 =
let
val C1 = do_type T1
val C2 = do_type T2
val a = if is_boolean_type (body_type T2) andalso
exists_alpha_sub_ctype_fresh C1 then
V (Unsynchronized.inc max_fresh)
else
S Minus
in CFun (C1, a, C2) end
(* typ -> ctype *)
and do_type T =
if T = alpha_T then
CAlpha
else case T of
Type ("fun", [T1, T2]) => do_fun T1 T2
| Type (@{type_name fun_box}, [T1, T2]) => do_fun T1 T2
| Type ("*", [T1, T2]) => CPair (pairself do_type (T1, T2))
| Type (z as (s, _)) =>
if could_exist_alpha_sub_ctype thy alpha_T T then
case AList.lookup (op =) (!datatype_cache) z of
SOME C => C
| NONE =>
let
val _ = Unsynchronized.change datatype_cache (cons (z, CRec z))
val xs = datatype_constrs ext_ctxt T
val (all_Cs, constr_Cs) =
fold_rev (fn (_, T') => fn (all_Cs, constr_Cs) =>
let
val binder_Cs = map do_type (binder_types T')
val new_Cs = filter exists_alpha_sub_ctype_fresh
binder_Cs
val constr_C = constr_ctype_for_binders z
binder_Cs
in
(union (op =) new_Cs all_Cs,
constr_C :: constr_Cs)
end)
xs ([], [])
val C = CType (s, all_Cs)
val _ = Unsynchronized.change datatype_cache
(AList.update (op =) (z, C))
val _ = Unsynchronized.change constr_cache
(append (xs ~~ constr_Cs))
in
if forall (not o is_CRec o snd) (!datatype_cache) then
(repair_datatype_cache datatype_cache;
repair_constr_cache (!datatype_cache) constr_cache;
AList.lookup (op =) (!datatype_cache) z |> the)
else
C
end
else
CType (s, [])
| _ => CType (Refute.string_of_typ T, [])
in do_type end
(* ctype -> ctype list *)
fun prodC_factors (CPair (C1, C2)) = maps prodC_factors [C1, C2]
| prodC_factors C = [C]
(* ctype -> ctype list * ctype *)
fun curried_strip_ctype (CFun (C1, S Minus, C2)) =
curried_strip_ctype C2 |>> append (prodC_factors C1)
| curried_strip_ctype C = ([], C)
(* string -> ctype -> ctype *)
fun sel_ctype_from_constr_ctype s C =
let val (arg_Cs, dataC) = curried_strip_ctype C in
CFun (dataC, S Minus,
case sel_no_from_name s of ~1 => bool_C | n => nth arg_Cs n)
end
(* cdata -> styp -> ctype *)
fun ctype_for_constr (cdata as {ext_ctxt as {thy, ...}, alpha_T, constr_cache,
...}) (x as (_, T)) =
if could_exist_alpha_sub_ctype thy alpha_T T then
case AList.lookup (op =) (!constr_cache) x of
SOME C => C
| NONE => if T = alpha_T then
let val C = fresh_ctype_for_type cdata T in
(Unsynchronized.change constr_cache (cons (x, C)); C)
end
else
(fresh_ctype_for_type cdata (body_type T);
AList.lookup (op =) (!constr_cache) x |> the)
else
fresh_ctype_for_type cdata T
fun ctype_for_sel (cdata as {ext_ctxt, ...}) (x as (s, _)) =
x |> boxed_constr_for_sel ext_ctxt |> ctype_for_constr cdata
|> sel_ctype_from_constr_ctype s
(* literal list -> ctype -> ctype *)
fun instantiate_ctype lits =
let
(* ctype -> ctype *)
fun aux CAlpha = CAlpha
| aux (CFun (C1, V x, C2)) =
let
val a = case AList.lookup (op =) lits x of
SOME sn => S sn
| NONE => V x
in CFun (aux C1, a, aux C2) end
| aux (CFun (C1, a, C2)) = CFun (aux C1, a, aux C2)
| aux (CPair Cp) = CPair (pairself aux Cp)
| aux (CType (s, Cs)) = CType (s, map aux Cs)
| aux (CRec z) = CRec z
in aux end
datatype comp_op = Eq | Leq
type comp = sign_atom * sign_atom * comp_op * var list
type sign_expr = literal list
datatype constraint_set =
UnsolvableCSet |
CSet of literal list * comp list * sign_expr list
(* comp_op -> string *)
fun string_for_comp_op Eq = "="
| string_for_comp_op Leq = "\<le>"
(* sign_expr -> string *)
fun string_for_sign_expr [] = "\<bot>"
| string_for_sign_expr lits =
space_implode " \<or> " (map string_for_literal lits)
(* constraint_set *)
val slack = CSet ([], [], [])
(* literal -> literal list option -> literal list option *)
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)
(* comp_op -> var list -> sign_atom -> sign_atom -> literal list * comp list
-> (literal list * comp list) option *)
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 (accum as (lits, comps)) =
SOME (lits, insert (op =) (a1, a2, cmp, xs) comps)
(* comp -> var list -> ctype -> ctype -> (literal list * comp list) option
-> (literal list * comp list) option *)
fun do_ctype_comp _ _ _ _ NONE = NONE
| do_ctype_comp _ _ CAlpha CAlpha accum = accum
| do_ctype_comp Eq xs (CFun (C11, a1, C12)) (CFun (C21, a2, C22))
(SOME accum) =
accum |> do_sign_atom_comp Eq xs a1 a2 |> do_ctype_comp Eq xs C11 C21
|> do_ctype_comp Eq xs C12 C22
| do_ctype_comp Leq xs (CFun (C11, a1, C12)) (CFun (C21, a2, C22))
(SOME accum) =
(if exists_alpha_sub_ctype C11 then
accum |> do_sign_atom_comp Leq xs a1 a2
|> do_ctype_comp Leq xs C21 C11
|> (case a2 of
S Minus => I
| S Plus => do_ctype_comp Leq xs C11 C21
| V x => do_ctype_comp Leq (x :: xs) C11 C21)
else
SOME accum)
|> do_ctype_comp Leq xs C12 C22
| do_ctype_comp cmp xs (C1 as CPair (C11, C12)) (C2 as CPair (C21, C22))
accum =
(accum |> fold (uncurry (do_ctype_comp cmp xs)) [(C11, C21), (C12, C22)]
handle Library.UnequalLengths =>
raise CTYPE ("Nitpick_Mono.do_ctype_comp", [C1, C2]))
| do_ctype_comp cmp xs (CType _) (CType _) accum =
accum (* no need to compare them thanks to the cache *)
| do_ctype_comp _ _ C1 C2 _ =
raise CTYPE ("Nitpick_Mono.do_ctype_comp", [C1, C2])
(* comp_op -> ctype -> ctype -> constraint_set -> constraint_set *)
fun add_ctype_comp _ _ _ UnsolvableCSet = UnsolvableCSet
| add_ctype_comp cmp C1 C2 (CSet (lits, comps, sexps)) =
(print_g ("*** Add " ^ string_for_ctype C1 ^ " " ^ string_for_comp_op cmp ^
" " ^ string_for_ctype C2);
case do_ctype_comp cmp [] C1 C2 (SOME (lits, comps)) of
NONE => (print_g "**** Unsolvable"; UnsolvableCSet)
| SOME (lits, comps) => CSet (lits, comps, sexps))
(* ctype -> ctype -> constraint_set -> constraint_set *)
val add_ctypes_equal = add_ctype_comp Eq
val add_is_sub_ctype = add_ctype_comp Leq
(* sign -> sign_expr -> ctype -> (literal list * sign_expr list) option
-> (literal list * sign_expr list) option *)
fun do_notin_ctype_fv _ _ _ NONE = NONE
| do_notin_ctype_fv Minus _ CAlpha accum = accum
| do_notin_ctype_fv Plus [] CAlpha _ = NONE
| do_notin_ctype_fv Plus [(x, sn)] CAlpha (SOME (lits, sexps)) =
SOME lits |> do_literal (x, sn) |> Option.map (rpair sexps)
| do_notin_ctype_fv Plus sexp CAlpha (SOME (lits, sexps)) =
SOME (lits, insert (op =) sexp sexps)
| do_notin_ctype_fv sn sexp (CFun (C1, S sn', C2)) accum =
accum |> (if sn' = Plus andalso sn = Plus then
do_notin_ctype_fv Plus sexp C1
else
I)
|> (if sn' = Minus orelse sn = Plus then
do_notin_ctype_fv Minus sexp C1
else
I)
|> do_notin_ctype_fv sn sexp C2
| do_notin_ctype_fv Plus sexp (CFun (C1, V x, C2)) accum =
accum |> (case do_literal (x, Minus) (SOME sexp) of
NONE => I
| SOME sexp' => do_notin_ctype_fv Plus sexp' C1)
|> do_notin_ctype_fv Minus sexp C1
|> do_notin_ctype_fv Plus sexp C2
| do_notin_ctype_fv Minus sexp (CFun (C1, V x, C2)) accum =
accum |> (case do_literal (x, Plus) (SOME sexp) of
NONE => I
| SOME sexp' => do_notin_ctype_fv Plus sexp' C1)
|> do_notin_ctype_fv Minus sexp C2
| do_notin_ctype_fv sn sexp (CPair (C1, C2)) accum =
accum |> fold (do_notin_ctype_fv sn sexp) [C1, C2]
| do_notin_ctype_fv sn sexp (CType (_, Cs)) accum =
accum |> fold (do_notin_ctype_fv sn sexp) Cs
| do_notin_ctype_fv _ _ C _ =
raise CTYPE ("Nitpick_Mono.do_notin_ctype_fv", [C])
(* sign -> ctype -> constraint_set -> constraint_set *)
fun add_notin_ctype_fv _ _ UnsolvableCSet = UnsolvableCSet
| add_notin_ctype_fv sn C (CSet (lits, comps, sexps)) =
(print_g ("*** Add " ^ string_for_ctype C ^ " is right-" ^
(case sn of Minus => "unique" | Plus => "total") ^ ".");
case do_notin_ctype_fv sn [] C (SOME (lits, sexps)) of
NONE => (print_g "**** Unsolvable"; UnsolvableCSet)
| SOME (lits, sexps) => CSet (lits, comps, sexps))
(* ctype -> constraint_set -> constraint_set *)
val add_ctype_is_right_unique = add_notin_ctype_fv Minus
val add_ctype_is_right_total = add_notin_ctype_fv Plus
(* constraint_set -> constraint_set -> constraint_set *)
fun unite (CSet (lits1, comps1, sexps1)) (CSet (lits2, comps2, sexps2)) =
(case SOME lits1 |> fold do_literal lits2 of
NONE => (print_g "**** Unsolvable"; UnsolvableCSet)
| SOME lits => CSet (lits, comps1 @ comps2, sexps1 @ sexps2))
| unite _ _ = UnsolvableCSet
(* sign -> bool *)
fun bool_from_sign Plus = false
| bool_from_sign Minus = true
(* bool -> sign *)
fun sign_from_bool false = Plus
| sign_from_bool true = Minus
(* literal -> PropLogic.prop_formula *)
fun prop_for_literal (x, sn) =
(not (bool_from_sign sn) ? PropLogic.Not) (PropLogic.BoolVar x)
(* sign_atom -> PropLogic.prop_formula *)
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)
(* sign_expr -> PropLogic.prop_formula *)
fun prop_for_sign_expr xs = PropLogic.exists (map prop_for_literal xs)
(* var list -> sign -> PropLogic.prop_formula *)
fun prop_for_exists_eq xs sn =
PropLogic.exists (map (fn x => prop_for_literal (x, sn)) xs)
(* comp -> PropLogic.prop_formula *)
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, []))
(* var -> (int -> bool option) -> literal list -> literal list *)
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
(* literal list -> sign_atom -> sign option *)
fun lookup_sign_atom _ (S sn) = SOME sn
| lookup_sign_atom lit (V x) = AList.lookup (op =) lit x
(* comp -> string *)
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
(* literal list -> comp list -> sign_expr list -> unit *)
fun print_problem lits comps sexps =
print_g ("*** Problem:\n" ^ cat_lines (map string_for_literal lits @
map string_for_comp comps @
map string_for_sign_expr sexps))
(* literal list -> unit *)
fun print_solution lits =
let val (pos, neg) = List.partition (curry (op =) Plus o snd) lits in
print_g ("*** Solution:\n" ^
"+: " ^ commas (map (string_for_var o fst) pos) ^ "\n" ^
"-: " ^ commas (map (string_for_var o fst) neg))
end
(* var -> constraint_set -> literal list list option *)
fun solve _ UnsolvableCSet = (print_g "*** Problem: Unsolvable"; NONE)
| solve max_var (CSet (lits, comps, sexps)) =
let
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)
(* 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 =>
SOME (literals_from_assignments max_var assigns lits
|> tap print_solution)
| _ => NONE
end
(* var -> constraint_set -> bool *)
val is_solvable = is_some oo solve
type ctype_schema = ctype * constraint_set
type ctype_context =
{bounds: ctype list,
frees: (styp * ctype) list,
consts: (styp * ctype) list}
type accumulator = ctype_context * constraint_set
val initial_gamma = {bounds = [], frees = [], consts = []}
val unsolvable_accum = (initial_gamma, UnsolvableCSet)
(* ctype -> ctype_context -> ctype_context *)
fun push_bound C {bounds, frees, consts} =
{bounds = C :: bounds, frees = frees, consts = consts}
(* ctype_context -> ctype_context *)
fun pop_bound {bounds, frees, consts} =
{bounds = tl bounds, frees = frees, consts = consts}
handle List.Empty => initial_gamma
(* cdata -> term -> accumulator -> ctype * accumulator *)
fun consider_term (cdata as {ext_ctxt as {ctxt, thy, def_table, ...}, alpha_T,
max_fresh, ...}) =
let
(* typ -> ctype *)
val ctype_for = fresh_ctype_for_type cdata
(* ctype -> ctype *)
fun pos_set_ctype_for_dom C =
CFun (C, S (if exists_alpha_sub_ctype C then Plus else Minus), bool_C)
(* typ -> accumulator -> ctype * accumulator *)
fun do_quantifier T (gamma, cset) =
let
val abs_C = ctype_for (domain_type (domain_type T))
val body_C = ctype_for (range_type T)
in
(CFun (CFun (abs_C, S Minus, body_C), S Minus, body_C),
(gamma, cset |> add_ctype_is_right_total abs_C))
end
fun do_equals T (gamma, cset) =
let val C = ctype_for (domain_type T) in
(CFun (C, S Minus, CFun (C, V (Unsynchronized.inc max_fresh),
ctype_for (nth_range_type 2 T))),
(gamma, cset |> add_ctype_is_right_unique C))
end
fun do_robust_set_operation T (gamma, cset) =
let
val set_T = domain_type T
val C1 = ctype_for set_T
val C2 = ctype_for set_T
val C3 = ctype_for set_T
in
(CFun (C1, S Minus, CFun (C2, S Minus, C3)),
(gamma, cset |> add_is_sub_ctype C1 C3 |> add_is_sub_ctype C2 C3))
end
fun do_fragile_set_operation T (gamma, cset) =
let
val set_T = domain_type T
val set_C = ctype_for set_T
(* typ -> ctype *)
fun custom_ctype_for (T as Type ("fun", [T1, T2])) =
if T = set_T then set_C
else CFun (custom_ctype_for T1, S Minus, custom_ctype_for T2)
| custom_ctype_for T = ctype_for T
in
(custom_ctype_for T, (gamma, cset |> add_ctype_is_right_unique set_C))
end
(* typ -> accumulator -> ctype * accumulator *)
fun do_pair_constr T accum =
case ctype_for (nth_range_type 2 T) of
C as CPair (a_C, b_C) =>
(CFun (a_C, S Minus, CFun (b_C, S Minus, C)), accum)
| C => raise CTYPE ("Nitpick_Mono.consider_term.do_pair_constr", [C])
(* int -> typ -> accumulator -> ctype * accumulator *)
fun do_nth_pair_sel n T =
case ctype_for (domain_type T) of
C as CPair (a_C, b_C) =>
pair (CFun (C, S Minus, if n = 0 then a_C else b_C))
| C => raise CTYPE ("Nitpick_Mono.consider_term.do_nth_pair_sel", [C])
val unsolvable = (CType ("unsolvable", []), unsolvable_accum)
(* typ -> term -> accumulator -> ctype * accumulator *)
fun do_bounded_quantifier abs_T bound_t body_t accum =
let
val abs_C = ctype_for abs_T
val (bound_C, accum) = accum |>> push_bound abs_C |> do_term bound_t
val expected_bound_C = pos_set_ctype_for_dom abs_C
in
accum ||> add_ctypes_equal expected_bound_C bound_C |> do_term body_t
||> apfst pop_bound
end
(* term -> accumulator -> ctype * accumulator *)
and do_term _ (_, UnsolvableCSet) = unsolvable
| do_term t (accum as (gamma as {bounds, frees, consts}, cset)) =
(case t of
Const (x as (s, T)) =>
(case AList.lookup (op =) consts x of
SOME C => (C, accum)
| NONE =>
if not (could_exist_alpha_subtype alpha_T T) then
(ctype_for T, accum)
else case s of
@{const_name all} => do_quantifier T accum
| @{const_name "=="} => do_equals T accum
| @{const_name All} => do_quantifier T accum
| @{const_name Ex} => do_quantifier T accum
| @{const_name "op ="} => do_equals T accum
| @{const_name The} => (print_g "*** The"; unsolvable)
| @{const_name Eps} => (print_g "*** Eps"; unsolvable)
| @{const_name If} =>
do_robust_set_operation (range_type T) accum
|>> curry3 CFun bool_C (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} =>
(CFun (ctype_for (domain_type T), S Minus, bool_C), accum)
| @{const_name insert} =>
let
val set_T = domain_type (range_type T)
val C1 = ctype_for (domain_type set_T)
val C1' = pos_set_ctype_for_dom C1
val C2 = ctype_for set_T
val C3 = ctype_for set_T
in
(CFun (C1, S Minus, CFun (C2, S Minus, C3)),
(gamma, cset |> add_ctype_is_right_unique C1
|> add_is_sub_ctype C1' C3
|> add_is_sub_ctype C2 C3))
end
| @{const_name converse} =>
let
val x = Unsynchronized.inc max_fresh
(* typ -> ctype *)
fun ctype_for_set T =
CFun (ctype_for (domain_type T), V x, bool_C)
val ab_set_C = domain_type T |> ctype_for_set
val ba_set_C = range_type T |> ctype_for_set
in (CFun (ab_set_C, S Minus, ba_set_C), accum) end
| @{const_name trancl} => do_fragile_set_operation T accum
| @{const_name rtrancl} => (print_g "*** rtrancl"; unsolvable)
| @{const_name lower_semilattice_fun_inst.inf_fun} =>
do_robust_set_operation T accum
| @{const_name upper_semilattice_fun_inst.sup_fun} =>
do_robust_set_operation T accum
| @{const_name finite} =>
let val C1 = ctype_for (domain_type (domain_type T)) in
(CFun (pos_set_ctype_for_dom C1, S Minus, bool_C), accum)
end
| @{const_name rel_comp} =>
let
val x = Unsynchronized.inc max_fresh
(* typ -> ctype *)
fun ctype_for_set T =
CFun (ctype_for (domain_type T), V x, bool_C)
val bc_set_C = domain_type T |> ctype_for_set
val ab_set_C = domain_type (range_type T) |> ctype_for_set
val ac_set_C = nth_range_type 2 T |> ctype_for_set
in
(CFun (bc_set_C, S Minus, CFun (ab_set_C, S Minus, ac_set_C)),
accum)
end
| @{const_name image} =>
let
val a_C = ctype_for (domain_type (domain_type T))
val b_C = ctype_for (range_type (domain_type T))
in
(CFun (CFun (a_C, S Minus, b_C), S Minus,
CFun (pos_set_ctype_for_dom a_C, S Minus,
pos_set_ctype_for_dom b_C)), accum)
end
| @{const_name Sigma} =>
let
val x = Unsynchronized.inc max_fresh
(* typ -> ctype *)
fun ctype_for_set T =
CFun (ctype_for (domain_type T), V x, bool_C)
val a_set_T = domain_type T
val a_C = ctype_for (domain_type a_set_T)
val b_set_C = ctype_for_set (range_type (domain_type
(range_type T)))
val a_set_C = ctype_for_set a_set_T
val a_to_b_set_C = CFun (a_C, S Minus, b_set_C)
val ab_set_C = ctype_for_set (nth_range_type 2 T)
in
(CFun (a_set_C, S Minus,
CFun (a_to_b_set_C, S Minus, ab_set_C)), accum)
end
| @{const_name minus_fun_inst.minus_fun} =>
let
val set_T = domain_type T
val left_set_C = ctype_for set_T
val right_set_C = ctype_for set_T
in
(CFun (left_set_C, S Minus,
CFun (right_set_C, S Minus, left_set_C)),
(gamma, cset |> add_ctype_is_right_unique right_set_C
|> add_is_sub_ctype right_set_C left_set_C))
end
| @{const_name ord_fun_inst.less_eq_fun} =>
do_fragile_set_operation T accum
| @{const_name Tha} =>
let
val a_C = ctype_for (domain_type (domain_type T))
val a_set_C = pos_set_ctype_for_dom a_C
in (CFun (a_set_C, S Minus, a_C), accum) end
| @{const_name FunBox} =>
let val dom_C = ctype_for (domain_type T) in
(CFun (dom_C, S Minus, dom_C), accum)
end
| _ => if is_sel s then
if constr_name_for_sel_like s = @{const_name FunBox} then
let val dom_C = ctype_for (domain_type T) in
(CFun (dom_C, S Minus, dom_C), accum)
end
else
(ctype_for_sel cdata x, accum)
else if is_constr thy x then
(ctype_for_constr cdata x, accum)
else if is_built_in_const true x then
case def_of_const thy def_table x of
SOME t' => do_term t' accum
| NONE => (print_g ("*** built-in " ^ s); unsolvable)
else
let val C = ctype_for T in
(C, ({bounds = bounds, frees = frees,
consts = (x, C) :: consts}, cset))
end)
| Free (x as (_, T)) =>
(case AList.lookup (op =) frees x of
SOME C => (C, accum)
| NONE =>
let val C = ctype_for T in
(C, ({bounds = bounds, frees = (x, C) :: frees,
consts = consts}, cset))
end)
| Var _ => (print_g "*** Var"; unsolvable)
| Bound j => (nth bounds j, accum)
| Abs (_, T, @{const False}) => (ctype_for (T --> bool_T), accum)
| Abs (s, T, t') =>
((case t' of
t1' $ Bound 0 =>
if not (loose_bvar1 (t1', 0)) then
do_term (incr_boundvars ~1 t1') accum
else
raise SAME ()
| _ => raise SAME ())
handle SAME () =>
let
val C = ctype_for T
val (C', accum) = do_term t' (accum |>> push_bound C)
in (CFun (C, S Minus, C'), accum |>> pop_bound) end)
| Const (@{const_name All}, _)
$ Abs (_, T', @{const "op -->"} $ (t1 $ Bound 0) $ t2) =>
do_bounded_quantifier T' t1 t2 accum
| Const (@{const_name Ex}, _)
$ Abs (_, T', @{const "op &"} $ (t1 $ Bound 0) $ t2) =>
do_bounded_quantifier T' t1 t2 accum
| Const (@{const_name Let}, _) $ t1 $ t2 =>
do_term (betapply (t2, t1)) accum
| t1 $ t2 =>
let
val (C1, accum) = do_term t1 accum
val (C2, accum) = do_term t2 accum
in
case accum of
(_, UnsolvableCSet) => unsolvable
| _ => case C1 of
CFun (C11, _, C12) =>
(C12, accum ||> add_is_sub_ctype C2 C11)
| _ => raise CTYPE ("Nitpick_Mono.consider_term.do_term \
\(op $)", [C1])
end)
|> tap (fn (C, _) =>
print_g (" \<Gamma> \<turnstile> " ^
Syntax.string_of_term ctxt t ^ " : " ^
string_for_ctype C))
in do_term end
(* cdata -> sign -> term -> accumulator -> accumulator *)
fun consider_general_formula (cdata as {ext_ctxt as {ctxt, ...}, ...}) =
let
(* typ -> ctype *)
val ctype_for = fresh_ctype_for_type cdata
(* term -> accumulator -> ctype * accumulator *)
val do_term = consider_term cdata
(* sign -> term -> accumulator -> accumulator *)
fun do_formula _ _ (_, UnsolvableCSet) = unsolvable_accum
| do_formula sn t (accum as (gamma as {bounds, frees, consts}, cset)) =
let
(* term -> accumulator -> accumulator *)
val do_co_formula = do_formula sn
val do_contra_formula = do_formula (negate sn)
(* string -> typ -> term -> accumulator *)
fun do_quantifier quant_s abs_T body_t =
let
val abs_C = ctype_for abs_T
val side_cond = ((sn = Minus) = (quant_s = @{const_name Ex}))
val cset = cset |> side_cond ? add_ctype_is_right_total abs_C
in
(gamma |> push_bound abs_C, cset) |> do_co_formula body_t
|>> pop_bound
end
(* typ -> term -> accumulator *)
fun do_bounded_quantifier abs_T body_t =
accum |>> push_bound (ctype_for abs_T) |> do_co_formula body_t
|>> pop_bound
(* term -> term -> accumulator *)
fun do_equals t1 t2 =
case sn of
Plus => do_term t accum |> snd
| Minus => let
val (C1, accum) = do_term t1 accum
val (C2, accum) = do_term t2 accum
in accum ||> add_ctypes_equal C1 C2 end
in
case t of
Const (s0 as @{const_name all}, _) $ Abs (_, T1, t1) =>
do_quantifier s0 T1 t1
| Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2
| @{const "==>"} $ t1 $ t2 =>
accum |> do_contra_formula t1 |> do_co_formula t2
| @{const Trueprop} $ t1 => do_co_formula t1 accum
| @{const Not} $ t1 => do_contra_formula t1 accum
| Const (@{const_name All}, _)
$ Abs (_, T1, t1 as @{const "op -->"} $ (_ $ Bound 0) $ _) =>
do_bounded_quantifier T1 t1
| Const (s0 as @{const_name All}, _) $ Abs (_, T1, t1) =>
do_quantifier s0 T1 t1
| Const (@{const_name Ex}, _)
$ Abs (_, T1, t1 as @{const "op &"} $ (_ $ Bound 0) $ _) =>
do_bounded_quantifier T1 t1
| Const (s0 as @{const_name Ex}, _) $ Abs (_, T1, t1) =>
do_quantifier s0 T1 t1
| Const (@{const_name "op ="}, _) $ t1 $ t2 => do_equals t1 t2
| @{const "op &"} $ t1 $ t2 =>
accum |> do_co_formula t1 |> do_co_formula t2
| @{const "op |"} $ t1 $ t2 =>
accum |> do_co_formula t1 |> do_co_formula t2
| @{const "op -->"} $ t1 $ t2 =>
accum |> do_contra_formula t1 |> do_co_formula t2
| Const (@{const_name If}, _) $ t1 $ t2 $ t3 =>
accum |> do_term t1 |> snd |> fold do_co_formula [t2, t3]
| Const (@{const_name Let}, _) $ t1 $ t2 =>
do_co_formula (betapply (t2, t1)) accum
| _ => do_term t accum |> snd
end
|> tap (fn _ => print_g ("\<Gamma> \<turnstile> " ^
Syntax.string_of_term ctxt t ^
" : 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}]
(* term -> bool *)
fun is_harmless_axiom t =
Term.add_consts t [] |> filter_out (is_built_in_const true)
|> (forall (member (op =) harmless_consts o original_name o fst)
orf exists (member (op =) bounteous_consts o fst))
(* cdata -> sign -> term -> accumulator -> accumulator *)
fun consider_nondefinitional_axiom cdata sn t =
not (is_harmless_axiom t) ? consider_general_formula cdata sn t
(* cdata -> term -> accumulator -> accumulator *)
fun consider_definitional_axiom (cdata as {ext_ctxt as {thy, ...}, ...}) t =
if not (is_constr_pattern_formula thy t) then
consider_nondefinitional_axiom cdata Plus t
else if is_harmless_axiom t then
I
else
let
(* term -> accumulator -> ctype * accumulator *)
val do_term = consider_term cdata
(* typ -> term -> accumulator -> accumulator *)
fun do_all abs_T body_t accum =
let val abs_C = fresh_ctype_for_type cdata abs_T in
accum |>> push_bound abs_C |> do_formula body_t |>> pop_bound
end
(* term -> term -> accumulator -> accumulator *)
and do_implies t1 t2 = do_term t1 #> snd #> do_formula t2
and do_equals t1 t2 accum =
let
val (C1, accum) = do_term t1 accum
val (C2, accum) = do_term t2 accum
in accum ||> add_ctypes_equal C1 C2 end
(* term -> accumulator -> accumulator *)
and do_formula _ (_, UnsolvableCSet) = unsolvable_accum
| do_formula t accum =
case t of
Const (@{const_name all}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum
| @{const Trueprop} $ t1 => do_formula t1 accum
| Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2 accum
| @{const "==>"} $ t1 $ t2 => do_implies t1 t2 accum
| @{const Pure.conjunction} $ t1 $ t2 =>
accum |> do_formula t1 |> do_formula t2
| Const (@{const_name All}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum
| Const (@{const_name "op ="}, _) $ t1 $ t2 => do_equals t1 t2 accum
| @{const "op &"} $ t1 $ t2 => accum |> do_formula t1 |> do_formula t2
| @{const "op -->"} $ t1 $ t2 => do_implies t1 t2 accum
| _ => raise TERM ("Nitpick_Mono.consider_definitional_axiom.\
\do_formula", [t])
in do_formula t end
(* Proof.context -> literal list -> term -> ctype -> string *)
fun string_for_ctype_of_term ctxt lits t C =
Syntax.string_of_term ctxt t ^ " : " ^
string_for_ctype (instantiate_ctype lits C)
(* theory -> literal list -> ctype_context -> unit *)
fun print_ctype_context ctxt lits ({frees, consts, ...} : ctype_context) =
map (fn (x, C) => string_for_ctype_of_term ctxt lits (Free x) C) frees @
map (fn (x, C) => string_for_ctype_of_term ctxt lits (Const x) C) consts
|> cat_lines |> print_g
(* extended_context -> typ -> sign -> term list -> term list -> term -> bool *)
fun formulas_monotonic (ext_ctxt as {ctxt, ...}) alpha_T sn def_ts nondef_ts
core_t =
let
val _ = print_g ("****** " ^ string_for_ctype CAlpha ^ " is " ^
Syntax.string_of_typ ctxt alpha_T)
val cdata as {max_fresh, ...} = initial_cdata ext_ctxt alpha_T
val (gamma, cset) =
(initial_gamma, slack)
|> fold (consider_definitional_axiom cdata) def_ts
|> fold (consider_nondefinitional_axiom cdata Plus) nondef_ts
|> consider_general_formula cdata sn core_t
in
case solve (!max_fresh) cset of
SOME lits => (print_ctype_context ctxt lits gamma; true)
| _ => false
end
handle CTYPE (loc, Cs) => raise BAD (loc, commas (map string_for_ctype Cs))
end;