boehmes@36898: (*  Title:      HOL/Tools/SMT/z3_model.ML
boehmes@36898:     Author:     Sascha Boehme and Philipp Meyer, TU Muenchen
boehmes@36898: 
boehmes@36898: Parser for counterexamples generated by Z3.
boehmes@36898: *)
boehmes@36898: 
boehmes@36898: signature Z3_MODEL =
boehmes@36898: sig
boehmes@39536:   val parse_counterex: Proof.context -> SMT_Translate.recon -> string list ->
boehmes@39536:     term list
boehmes@36898: end
boehmes@36898: 
boehmes@36898: structure Z3_Model: Z3_MODEL =
boehmes@36898: struct
boehmes@36898: 
boehmes@36898: (* counterexample expressions *)
boehmes@36898: 
boehmes@36898: datatype expr = True | False | Number of int * int option | Value of int |
boehmes@39536:   Array of array | App of string * expr list
boehmes@36898: and array = Fresh of expr | Store of (array * expr) * expr
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* parsing *)
boehmes@36898: 
boehmes@36898: val space = Scan.many Symbol.is_ascii_blank
boehmes@39536: fun spaced p = p --| space
boehmes@39536: fun in_parens p = spaced (Scan.$$ "(") |-- p --| spaced (Scan.$$ ")")
boehmes@39536: fun in_braces p = spaced (Scan.$$ "{") |-- p --| spaced (Scan.$$ "}")
boehmes@36898: 
boehmes@36898: val digit = (fn
boehmes@36898:   "0" => SOME 0 | "1" => SOME 1 | "2" => SOME 2 | "3" => SOME 3 |
boehmes@36898:   "4" => SOME 4 | "5" => SOME 5 | "6" => SOME 6 | "7" => SOME 7 |
boehmes@36898:   "8" => SOME 8 | "9" => SOME 9 | _ => NONE)
boehmes@36898: 
boehmes@39536: val nat_num = spaced (Scan.repeat1 (Scan.some digit) >>
boehmes@39536:   (fn ds => fold (fn d => fn i => i * 10 + d) ds 0))
boehmes@39536: val int_num = spaced (Scan.optional ($$ "-" >> K (fn i => ~i)) I :|--
boehmes@39536:   (fn sign => nat_num >> sign))
boehmes@36898: 
boehmes@36898: val is_char = Symbol.is_ascii_letter orf Symbol.is_ascii_digit orf
wenzelm@40627:   member (op =) (raw_explode "_+*-/%~=<>$&|?!.@^#")
boehmes@39536: val name = spaced (Scan.many1 is_char >> implode)
boehmes@39536: 
boehmes@39536: fun $$$ s = spaced (Scan.this_string s)
boehmes@36898: 
boehmes@39536: fun array_expr st = st |> in_parens (
boehmes@39536:   $$$ "const" |-- expr >> Fresh ||
boehmes@39536:   $$$ "store" |-- array_expr -- expr -- expr >> Store)
boehmes@36898: 
boehmes@39536: and expr st = st |> (
boehmes@39536:   $$$ "true" >> K True ||
boehmes@39536:   $$$ "false" >> K False ||
boehmes@39536:   int_num -- Scan.option ($$$ "/" |-- int_num) >> Number ||
boehmes@39536:   $$$ "val!" |-- nat_num >> Value ||
boehmes@39536:   name >> (App o rpair []) ||
boehmes@39536:   array_expr >> Array ||
boehmes@39536:   in_parens (name -- Scan.repeat1 expr) >> App)
boehmes@36898: 
boehmes@39536: fun args st = ($$$ "->" >> K [] || expr ::: args) st
boehmes@39536: val args_case = args -- expr
boehmes@39536: val else_case = $$$ "else" -- $$$ "->" |-- expr >> pair ([] : expr list)
boehmes@36898: 
boehmes@36898: val func =
boehmes@36898:   let fun cases st = (else_case >> single || args_case ::: cases) st
boehmes@36898:   in in_braces cases end
boehmes@36898: 
boehmes@39536: val cex = space |--
boehmes@39536:   Scan.repeat (name --| $$$ "->" -- (func || expr >> (single o pair [])))
boehmes@36898: 
boehmes@36898: fun read_cex ls =
wenzelm@40627:   maps (cons "\n" o raw_explode) ls
boehmes@36898:   |> try (fst o Scan.finite Symbol.stopper cex)
boehmes@36898:   |> the_default []
boehmes@36898: 
boehmes@36898: 
boehmes@39536: (* normalization *)
boehmes@39536: 
boehmes@39536: local
boehmes@39536:   fun matches terms f n =
boehmes@39536:     (case Symtab.lookup terms n of
boehmes@39536:       NONE => false
boehmes@39536:     | SOME t => f t)
boehmes@39536: 
boehmes@39536:   fun subst f (n, cases) = (n, map (fn (args, v) => (map f args, f v)) cases)
boehmes@39536: in
boehmes@39536: 
boehmes@39536: fun reduce_function (n, [c]) = SOME ((n, 0), [c])
boehmes@39536:   | reduce_function (n, cases) =
boehmes@39536:       let val (patterns, else_case as (_, e)) = split_last cases
boehmes@39536:       in
boehmes@39536:         (case patterns of
boehmes@39536:           [] => NONE
boehmes@39536:         | (args, _) :: _ => SOME ((n, length args),
boehmes@39536:             filter_out (equal e o snd) patterns @ [else_case]))
boehmes@39536:       end
boehmes@39536: 
boehmes@39536: fun drop_skolem_constants terms = filter (Symtab.defined terms o fst o fst)
boehmes@39536: 
boehmes@39536: fun substitute_constants terms =
boehmes@39536:   let
boehmes@39536:     fun check vs1 [] = rev vs1
boehmes@39536:       | check vs1 ((v as ((n, k), [([], Value i)])) :: vs2) =
boehmes@39536:           if matches terms (fn Free _ => true | _ => false) n orelse k > 0
boehmes@39536:           then check (v :: vs1) vs2
boehmes@39536:           else
boehmes@39536:             let
boehmes@39536:               fun sub (e as Value j) = if i = j then App (n, []) else e
boehmes@39536:                 | sub e = e
boehmes@39536:             in check (map (subst sub) vs1) (map (subst sub) vs2) end
boehmes@39536:       | check vs1 (v :: vs2) = check (v :: vs1) vs2
boehmes@39536:   in check [] end
boehmes@39536: 
boehmes@39536: fun remove_int_nat_coercions terms vs =
boehmes@39536:   let
boehmes@39536:     fun match ts ((n, _), _) = matches terms (member (op aconv) ts) n
boehmes@39536: 
boehmes@40551:     val (default_int, ints) =
boehmes@40579:       (case find_first (match [@{const of_nat (int)}]) vs of
boehmes@40551:         NONE => (NONE, [])
boehmes@40551:       | SOME (_, cases) =>
boehmes@40551:           let val (cs, (_, e)) = split_last cases
boehmes@40551:           in (SOME e, map (apfst hd) cs) end)
boehmes@40551: 
boehmes@40551:     fun nat_of @{typ nat} (v as Value _) =
boehmes@40551:           AList.lookup (op =) ints v |> the_default (the_default v default_int)
boehmes@40551:       | nat_of _ e = e
boehmes@40551: 
boehmes@40551:     fun subst_nat T k ([], e) =
boehmes@40551:           let fun app f i = if i <= 0 then I else app f (i-1) o f
boehmes@40551:           in ([], nat_of (app Term.range_type k T) e) end
boehmes@40551:       | subst_nat T k (arg :: args, e) =
boehmes@40551:           subst_nat (Term.range_type T) (k-1) (args, e)
boehmes@40551:           |> apfst (cons (nat_of (Term.domain_type T) arg))
boehmes@40551: 
boehmes@40551:     fun subst_nats (v as ((n, k), cases)) =
boehmes@40551:       (case Symtab.lookup terms n of
boehmes@40551:         NONE => v
boehmes@40551:       | SOME t => ((n, k), map (subst_nat (Term.fastype_of t) k) cases))
boehmes@39536:   in
boehmes@40551:     map subst_nats vs
boehmes@40579:     |> filter_out (match [@{const of_nat (int)}, @{const nat}])
boehmes@39536:   end
boehmes@39536: 
boehmes@39536: fun filter_valid_valuations terms = map_filter (fn
boehmes@39536:     (_, []) => NONE
boehmes@39536:   | ((n, i), cases) =>
boehmes@39536:       let
boehmes@39536:         fun valid_expr (Array a) = valid_array a
boehmes@39536:           | valid_expr (App (n, es)) =
boehmes@39536:               Symtab.defined terms n andalso forall valid_expr es
boehmes@39536:           | valid_expr _ = true
boehmes@39536:         and valid_array (Fresh e) = valid_expr e
boehmes@39536:           | valid_array (Store ((a, e1), e2)) =
boehmes@39536:               valid_array a andalso valid_expr e1 andalso valid_expr e2
boehmes@39536:         fun valid_case (es, e) = forall valid_expr (e :: es)
boehmes@39536:       in
boehmes@39536:         if not (forall valid_case cases) then NONE
boehmes@39536:         else Option.map (rpair cases o rpair i) (Symtab.lookup terms n)
boehmes@39536:       end)
boehmes@39536: 
boehmes@39536: end
boehmes@39536: 
boehmes@39536: 
boehmes@36898: (* translation into terms *)
boehmes@36898: 
boehmes@39536: fun with_context ctxt terms f vs =
boehmes@39536:   fst (fold_map f vs (ctxt, terms, Inttab.empty))
boehmes@36898: 
boehmes@39536: fun fresh_term T (ctxt, terms, values) =
boehmes@39536:   let val (n, ctxt') = yield_singleton Variable.variant_fixes "" ctxt
boehmes@39536:   in (Free (n, T), (ctxt', terms, values)) end
boehmes@36898: 
boehmes@39536: fun term_of_value T i (cx as (_, _, values)) =
boehmes@39536:   (case Inttab.lookup values i of
boehmes@36898:     SOME t => (t, cx)
boehmes@36898:   | NONE =>
boehmes@39536:       let val (t, (ctxt', terms', values')) = fresh_term T cx
boehmes@39536:       in (t, (ctxt', terms', Inttab.update (i, t) values')) end)
boehmes@39536: 
boehmes@39536: fun get_term n (cx as (_, terms, _)) = (the (Symtab.lookup terms n), cx)
boehmes@36898: 
boehmes@40579: fun trans_expr _ True = pair @{const True}
boehmes@40579:   | trans_expr _ False = pair @{const False}
boehmes@36898:   | trans_expr T (Number (i, NONE)) = pair (HOLogic.mk_number T i)
boehmes@36898:   | trans_expr T (Number (i, SOME j)) =
boehmes@36898:       pair (Const (@{const_name divide}, [T, T] ---> T) $
boehmes@36898:         HOLogic.mk_number T i $ HOLogic.mk_number T j)
boehmes@36898:   | trans_expr T (Value i) = term_of_value T i
boehmes@36898:   | trans_expr T (Array a) = trans_array T a
boehmes@39536:   | trans_expr _ (App (n, es)) =
boehmes@39536:       let val get_Ts = take (length es) o Term.binder_types o Term.fastype_of
boehmes@39536:       in
boehmes@39536:         get_term n #-> (fn t =>
boehmes@39536:         fold_map (uncurry trans_expr) (get_Ts t ~~ es) #>>
boehmes@39536:         Term.list_comb o pair t)
boehmes@39536:       end
boehmes@36898: 
boehmes@36898: and trans_array T a =
boehmes@36898:   let val dT = Term.domain_type T and rT = Term.range_type T
boehmes@36898:   in
boehmes@36898:     (case a of
boehmes@36898:       Fresh e => trans_expr rT e #>> (fn t => Abs ("x", dT, t))
boehmes@36898:     | Store ((a', e1), e2) =>
boehmes@36898:         trans_array T a' ##>> trans_expr dT e1 ##>> trans_expr rT e2 #>>
boehmes@36898:         (fn ((m, k), v) =>
boehmes@36898:           Const (@{const_name fun_upd}, [T, dT, rT] ---> T) $ m $ k $ v))
boehmes@36898:   end
boehmes@36898: 
boehmes@39536: fun trans_pattern T ([], e) = trans_expr T e #>> pair []
boehmes@39536:   | trans_pattern T (arg :: args, e) =
boehmes@39536:       trans_expr (Term.domain_type T) arg ##>>
boehmes@39536:       trans_pattern (Term.range_type T) (args, e) #>>
boehmes@39536:       (fn (arg', (args', e')) => (arg' :: args', e'))
boehmes@36898: 
boehmes@39536: fun mk_fun_upd T U = Const (@{const_name fun_upd}, [T --> U, T, U, T] ---> U)
boehmes@39536: 
boehmes@39536: fun split_type T = (Term.domain_type T, Term.range_type T)
boehmes@36898: 
boehmes@39536: fun mk_update ([], u) _ = u
boehmes@39536:   | mk_update ([t], u) f =
boehmes@39536:       uncurry mk_fun_upd (split_type (Term.fastype_of f)) $ f $ t $ u
boehmes@39536:   | mk_update (t :: ts, u) f =
boehmes@39536:       let
boehmes@39536:         val (dT, rT) = split_type (Term.fastype_of f)
boehmes@39536:         val (dT', rT') = split_type rT
boehmes@39536:       in
boehmes@39536:         mk_fun_upd dT rT $ f $ t $
boehmes@39536:           mk_update (ts, u) (Term.absdummy (dT', Const ("_", rT')))
boehmes@39536:       end
boehmes@39536: 
boehmes@39536: fun mk_lambda Ts (t, pats) =
boehmes@39536:   fold_rev (curry Term.absdummy) Ts t |> fold mk_update pats
boehmes@36898: 
boehmes@39536: fun translate' T i [([], e)] =
boehmes@39536:       if i = 0 then trans_expr T e
boehmes@39536:       else 
boehmes@39536:         let val ((Us1, Us2), U) = Term.strip_type T |>> chop i
boehmes@39536:         in trans_expr (Us2 ---> U) e #>> mk_lambda Us1 o rpair [] end
boehmes@39536:   | translate' T i cases =
boehmes@39536:       let
boehmes@39536:         val (pat_cases, def) = split_last cases |> apsnd snd
boehmes@39536:         val ((Us1, Us2), U) = Term.strip_type T |>> chop i
boehmes@39536:       in
boehmes@39536:         trans_expr (Us2 ---> U) def ##>>
boehmes@39536:         fold_map (trans_pattern T) pat_cases #>>
boehmes@39536:         mk_lambda Us1
boehmes@39536:       end
boehmes@39536: 
boehmes@39536: fun translate ((t, i), cases) =
boehmes@39536:   translate' (Term.fastype_of t) i cases #>> HOLogic.mk_eq o pair t
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* overall procedure *)
boehmes@36898: 
boehmes@39536: fun parse_counterex ctxt ({terms, ...} : SMT_Translate.recon) ls =
boehmes@36898:   read_cex ls
boehmes@39536:   |> map_filter reduce_function
boehmes@39536:   |> drop_skolem_constants terms
boehmes@39536:   |> substitute_constants terms
boehmes@39536:   |> remove_int_nat_coercions terms
boehmes@39536:   |> filter_valid_valuations terms
boehmes@39536:   |> with_context ctxt terms translate
boehmes@36898: 
boehmes@36898: end
boehmes@39536: