signature TIMING_FUNCTIONS =
sig
type 'a wctxt = {
ctxt: local_theory,
origins: term list,
f: term -> 'a
}
type 'a converter = {
constc : 'a wctxt -> term -> 'a,
funcc : 'a wctxt -> term -> term list -> 'a,
ifc : 'a wctxt -> typ -> term -> term -> term -> 'a,
casec : 'a wctxt -> term -> term list -> 'a,
letc : 'a wctxt -> typ -> term -> string list -> typ list -> term -> 'a
}
val walk : local_theory -> term list -> 'a converter -> term -> 'a
type pfunc = { names : string list, terms : term list, typs : typ list }
val fun_pretty': Proof.context -> pfunc -> Pretty.T
val fun_pretty: Proof.context -> Function.info -> Pretty.T
val print_timing': Proof.context -> pfunc -> pfunc -> unit
val print_timing: Proof.context -> Function.info -> Function.info -> unit
val reg_and_proove_time_func: local_theory -> term list -> term list
-> bool -> Function.info * local_theory
val reg_time_func: local_theory -> term list -> term list
-> bool -> Function.info * local_theory
val time_dom_tac: Proof.context -> thm -> thm list -> int -> tactic
end
structure Timing_Functions : TIMING_FUNCTIONS =
struct
(* Configure config variable to adjust the prefix *)
val bprefix = Attrib.setup_config_string @{binding "time_prefix"} (K "T_")
(* Configure config variable to adjust the suffix *)
val bsuffix = Attrib.setup_config_string @{binding "time_suffix"} (K "")
(* some default values to build terms easier *)
val zero = Const (@{const_name "Groups.zero"}, HOLogic.natT)
val one = Const (@{const_name "Groups.one"}, HOLogic.natT)
(* Extracts terms from function info *)
fun terms_of_info (info: Function.info) =
map Thm.prop_of (case #simps info of SOME s => s
| NONE => error "No terms of function found in info")
type pfunc = {
names : string list,
terms : term list,
typs : typ list
}
fun info_pfunc (info: Function.info): pfunc =
let
val {defname, fs, ...} = info;
val T = case hd fs of (Const (_,T)) => T
| (Free (_,T)) => T
| _ => error "Internal error: Invalid info to print"
in
{ names=[Binding.name_of defname], terms=terms_of_info info, typs=[T] }
end
(* Auxiliary functions for printing functions *)
fun fun_pretty' ctxt (pfunc: pfunc) =
let
val {names, terms, typs} = pfunc;
val header_beg = Pretty.str "fun ";
fun prepHeadCont (nm,T) = [Pretty.str (nm ^ " :: "), (Pretty.quote (Syntax.pretty_typ ctxt T))]
val header_content =
List.concat (prepHeadCont (hd names,hd typs) :: map ((fn l => Pretty.str "\nand " :: l) o prepHeadCont) (ListPair.zip (tl names, tl typs)));
val header_end = Pretty.str " where\n ";
val header = [header_beg] @ header_content @ [header_end];
fun separate sep prts =
flat (Library.separate [Pretty.str sep] (map single prts));
val ptrms = (separate "\n| " (map (Syntax.pretty_term ctxt) terms));
in
Pretty.text_fold (header @ ptrms)
end
fun fun_pretty ctxt = fun_pretty' ctxt o info_pfunc
fun print_timing' ctxt (opfunc: pfunc) (tpfunc: pfunc) =
let
val {names, ...} = opfunc;
val poriginal = Pretty.item [Pretty.str "Original function:\n", fun_pretty' ctxt opfunc]
val ptiming = Pretty.item [Pretty.str ("Running time function:\n"), fun_pretty' ctxt tpfunc]
in
Pretty.writeln (Pretty.text_fold [Pretty.str ("Converting " ^ (hd names) ^ (String.concat (map (fn nm => ", " ^ nm) (tl names))) ^ "\n"), poriginal, Pretty.str "\n", ptiming])
end
fun print_timing ctxt (oinfo: Function.info) (tinfo: Function.info) =
print_timing' ctxt (info_pfunc oinfo) (info_pfunc tinfo)
fun contains l e = exists (fn e' => e' = e) l
fun contains' comp l e = exists (comp e) l
fun index [] _ = 0
| index (x::xs) el = (if x = el then 0 else 1 + index xs el)
fun used_for_const orig_used t i = orig_used (t,i)
(* Split name by . *)
val split_name = String.fields (fn s => s = #".")
(* returns true if it's an if term *)
fun is_if (Const (@{const_name "HOL.If"},_)) = true
| is_if _ = false
(* returns true if it's a case term *)
fun is_case (Const (n,_)) = n |> split_name |> List.last |> String.isPrefix "case_"
| is_case _ = false
(* returns true if it's a let term *)
fun is_let (Const (@{const_name "HOL.Let"},_)) = true
| is_let _ = false
(* change type of original function to new type (_ \<Rightarrow> ... \<Rightarrow> _ to _ \<Rightarrow> ... \<Rightarrow> nat)
and replace all function arguments f with (t*T_f) if used *)
fun change_typ' used (Type ("fun", [T1, T2])) =
Type ("fun", [check_for_fun' (used 0) T1, change_typ' (fn i => used (i+1)) T2])
| change_typ' _ _ = HOLogic.natT
and check_for_fun' true (f as Type ("fun", [_,_])) = HOLogic.mk_prodT (f, change_typ' (K false) f)
| check_for_fun' false (f as Type ("fun", [_,_])) = change_typ' (K false) f
| check_for_fun' _ t = t
val change_typ = change_typ' (K false)
(* Convert string name of function to its timing equivalent *)
fun fun_name_to_time ctxt s name =
let
val prefix = Config.get ctxt bprefix
val suffix = (if s then Config.get ctxt bsuffix else "")
fun replace_last_name [n] = [prefix ^ n ^ suffix]
| replace_last_name (n::ns) = n :: (replace_last_name ns)
| replace_last_name _ = error "Internal error: Invalid function name to convert"
val parts = split_name name
in
String.concatWith "." (replace_last_name parts)
end
(* Count number of arguments of a function *)
fun count_args (Type (n, [_,res])) = (if n = "fun" then 1 + count_args res else 0)
| count_args _ = 0
(* Check if number of arguments matches function *)
val _ = dest_Const
fun check_args s (t, args) =
(if length args = count_args (type_of t) then ()
else error ("Partial applications/Lambdas not allowed (" ^ s ^ ")"))
(* Removes Abs *)
fun rem_abs f (Abs (_,_,t)) = rem_abs f t
| rem_abs f t = f t
(* Map right side of equation *)
fun map_r f (pT $ (eq $ l $ r)) = (pT $ (eq $ l $ f r))
| map_r _ _ = error "Internal error: No right side of equation found"
(* Get left side of equation *)
fun get_l (_ $ (_ $ l $ _)) = l
| get_l _ = error "Internal error: No left side of equation found"
(* Get right side of equation *)
fun get_r (_ $ (_ $ _ $ r)) = r
| get_r _ = error "Internal error: No right side of equation found"
(* Return name of Const *)
fun Const_name (Const (nm,_)) = SOME nm
| Const_name _ = NONE
fun is_Used (Type ("Product_Type.prod", _)) = true
| is_Used _ = false
(* Custom compare function for types ignoring variable names *)
fun typ_comp (Type (A,a)) (Type (B,b)) = (A = B) andalso List.foldl (fn ((c,i),s) => typ_comp c i andalso s) true (ListPair.zip (a, b))
| typ_comp (Type _) _ = false
| typ_comp _ (Type _) = false
| typ_comp _ _ = true
fun const_comp (Const (nm,T)) (Const (nm',T')) = nm = nm' andalso typ_comp T T'
| const_comp _ _ = false
fun time_term ctxt s (Const (nm,T)) =
let
val T_nm = fun_name_to_time ctxt s nm
val T_T = change_typ T
in
(SOME (Syntax.check_term ctxt (Const (T_nm,T_T))))
handle (ERROR _) =>
case Syntax.read_term ctxt (Long_Name.base_name T_nm)
of (Const (T_nm,T_T)) =>
let
fun col_Used i (Type ("fun", [Type ("fun", _), Ts])) (Type ("fun", [T', Ts'])) =
(if is_Used T' then [i] else []) @ col_Used (i+1) Ts Ts'
| col_Used i (Type ("fun", [_, Ts])) (Type ("fun", [_, Ts'])) = col_Used (i+1) Ts Ts'
| col_Used _ _ _ = []
in
SOME (Const (T_nm,change_typ' (contains (col_Used 0 T T_T)) T))
end
| _ => error ("Timing function of " ^ nm ^ " is not defined")
end
| time_term _ _ _ = error "Internal error: No valid function given"
type 'a wctxt = {
ctxt: local_theory,
origins: term list,
f: term -> 'a
}
type 'a converter = {
constc : 'a wctxt -> term -> 'a,
funcc : 'a wctxt -> term -> term list -> 'a,
ifc : 'a wctxt -> typ -> term -> term -> term -> 'a,
casec : 'a wctxt -> term -> term list -> 'a,
letc : 'a wctxt -> typ -> term -> string list -> typ list -> term -> 'a
}
(* Walks over term and calls given converter *)
fun walk_func (t1 $ t2) ts = walk_func t1 (t2::ts)
| walk_func t ts = (t, ts)
fun walk_func' t = walk_func t []
fun build_func (f, []) = f
| build_func (f, (t::ts)) = build_func (f$t, ts)
fun walk_abs (Abs (nm,T,t)) nms Ts = walk_abs t (nm::nms) (T::Ts)
| walk_abs t nms Ts = (t, nms, Ts)
fun build_abs t (nm::nms) (T::Ts) = build_abs (Abs (nm,T,t)) nms Ts
| build_abs t [] [] = t
| build_abs _ _ _ = error "Internal error: Invalid terms to build abs"
fun walk ctxt (origin: term list) (conv as {ifc, casec, funcc, letc, ...} : 'a converter) (t as _ $ _) =
let
val (f, args) = walk_func t []
val this = (walk ctxt origin conv)
val _ = (case f of Abs _ => error "Lambdas not supported" | _ => ())
val wctxt = {ctxt = ctxt, origins = origin, f = this}
in
(if is_if f then
(case f of (Const (_,T)) =>
(case args of [cond, t, f] => ifc wctxt T cond t f
| _ => error "Partial applications not supported (if)")
| _ => error "Internal error: invalid if term")
else if is_case f then casec wctxt f args
else if is_let f then
(case f of (Const (_,lT)) =>
(case args of [exp, t] =>
let val (t,nms,Ts) = walk_abs t [] [] in letc wctxt lT exp nms Ts t end
| _ => error "Partial applications not allowed (let)")
| _ => error "Internal error: invalid let term")
else funcc wctxt f args)
end
| walk ctxt origin (conv as {constc, ...}) c =
constc {ctxt = ctxt, origins = origin, f = walk ctxt origin conv} c
(* 1. Fix all terms *)
(* Exchange Var in types and terms to Free *)
fun fixTerms (Var(ixn,T)) = Free (fst ixn, T)
| fixTerms t = t
fun fixTypes (TVar ((t, _), T)) = TFree (t, T)
| fixTypes t = t
fun noFun (Type ("fun", _)) = error "Functions in datatypes are not allowed in case constructions"
| noFun T = T
fun casecBuildBounds n t = if n > 0 then casecBuildBounds (n-1) (t $ (Bound (n-1))) else t
fun casecAbs wctxt n (Type ("fun",[T,Tr])) (Abs (v,Ta,t)) = (map_atyps noFun T; Abs (v,Ta,casecAbs wctxt n Tr t))
| casecAbs wctxt n (Type ("fun",[T,Tr])) t =
(map_atyps noFun T; Abs ("uu",T,casecAbs wctxt (n + 1) Tr t))
| casecAbs wctxt n _ t = (#f wctxt) (casecBuildBounds n (Term.incr_bv n 0 t))
fun fixCasecCases _ _ [t] = [t]
| fixCasecCases wctxt (Type (_,[T,Tr])) (t::ts) = casecAbs wctxt 0 T t :: fixCasecCases wctxt Tr ts
| fixCasecCases _ _ _ = error "Internal error: invalid case types/terms"
fun fixCasec wctxt (t as Const (_,T)) args =
(check_args "cases" (t,args); build_func (t,fixCasecCases wctxt T args))
| fixCasec _ _ _ = error "Internal error: invalid case term"
fun fixPartTerms ctxt (term: term list) t =
let
val _ = check_args "args" (walk_func (get_l t) [])
in
map_r (walk ctxt term {
funcc = (fn wctxt => fn t => fn args =>
(check_args "func" (t,args); build_func (t, map (#f wctxt) args))),
constc = (fn _ => fn c => (case c of Abs _ => error "Lambdas not supported" | _ => c)),
ifc = (fn wctxt => fn T => fn cond => fn tt => fn tf =>
((Const (@{const_name "HOL.If"}, T)) $ (#f wctxt) cond $ ((#f wctxt) tt) $ ((#f wctxt) tf))),
casec = fixCasec,
letc = (fn wctxt => fn expT => fn exp => fn nms => fn Ts => fn t =>
let
val f' = if length nms = 0 then
(case (#f wctxt) (t$exp) of t$_ => t | _ => error "Internal error: case could not be fixed (let)")
else (#f wctxt) t
in (Const (@{const_name "HOL.Let"},expT) $ ((#f wctxt) exp) $ build_abs f' nms Ts) end)
}) t
end
(* 2. Check for properties about the function *)
(* 2.1 Check if function is recursive *)
fun or f (a,b) = f a orelse b
fun find_rec ctxt term = (walk ctxt term {
funcc = (fn wctxt => fn t => fn args =>
List.exists (fn term => (Const_name t) = (Const_name term)) term
orelse List.foldr (or (#f wctxt)) false args),
constc = (K o K) false,
ifc = (fn wctxt => fn _ => fn cond => fn tt => fn tf =>
(#f wctxt) cond orelse (#f wctxt) tt orelse (#f wctxt) tf),
casec = (fn wctxt => fn t => fn cs =>
(#f wctxt) t orelse List.foldr (or (rem_abs (#f wctxt))) false cs),
letc = (fn wctxt => fn _ => fn exp => fn _ => fn _ => fn t =>
(#f wctxt) exp orelse (#f wctxt) t)
}) o get_r
fun is_rec ctxt (term: term list) = List.foldr (or (find_rec ctxt term)) false
(* 2.2 Check for higher-order function if original function is used *)
fun find_used' ctxt term t T_t =
let
val (ident, _) = walk_func (get_l t) []
val (T_ident, T_args) = walk_func (get_l T_t) []
fun filter_passed [] = []
| filter_passed ((f as Free (_, Type ("Product_Type.prod",[Type ("fun",_), Type ("fun", _)])))::args) =
f :: filter_passed args
| filter_passed (_::args) = filter_passed args
val frees' = (walk ctxt term {
funcc = (fn wctxt => fn t => fn args =>
(case t of (Const ("Product_Type.prod.snd", _)) => []
| _ => (if t = T_ident then [] else filter_passed args)
@ List.foldr (fn (l,r) => (#f wctxt) l @ r) [] args)),
constc = (K o K) [],
ifc = (fn wctxt => fn _ => fn cond => fn tt => fn tf => (#f wctxt) cond @ (#f wctxt) tt @ (#f wctxt) tf),
casec = (fn wctxt => fn _ => fn cs => List.foldr (fn (l,r) => (#f wctxt) l @ r) [] cs),
letc = (fn wctxt => fn _ => fn exp => fn _ => fn _ => fn t => (#f wctxt) exp @ (#f wctxt) t)
}) (get_r T_t)
fun build _ [] _ = false
| build i (a::args) item =
(if item = (ident,i) then contains frees' a else build (i+1) args item)
in
build 0 T_args
end
fun find_used ctxt term terms T_terms =
ListPair.zip (terms, T_terms)
|> List.map (fn (t, T_t) => find_used' ctxt term t T_t)
|> List.foldr (fn (f,g) => fn item => f item orelse g item) (K false)
(* 3. Convert equations *)
(* Some Helper *)
val plusTyp = @{typ "nat => nat => nat"}
fun plus (SOME a) (SOME b) = SOME (Const (@{const_name "Groups.plus"}, plusTyp) $ a $ b)
| plus (SOME a) NONE = SOME a
| plus NONE (SOME b) = SOME b
| plus NONE NONE = NONE
fun opt_term NONE = HOLogic.zero
| opt_term (SOME t) = t
fun use_origin (Free (nm, T as Type ("fun",_))) = HOLogic.mk_fst (Free (nm,HOLogic.mk_prodT (T, change_typ T)))
| use_origin t = t
(* Conversion of function term *)
fun fun_to_time ctxt orig_used _ (origin: term list) (func as Const (nm,T)) =
let
val used' = used_for_const orig_used func
in
if contains' const_comp origin func then SOME (Free (func |> Term.term_name |> fun_name_to_time ctxt true, change_typ' used' T)) else
if Zero_Funcs.is_zero (Proof_Context.theory_of ctxt) (nm,T) then NONE else
time_term ctxt false func
end
| fun_to_time ctxt _ used _ (f as Free (nm,T)) = SOME (
if used f then HOLogic.mk_snd (Free (nm,HOLogic.mk_prodT (T,change_typ T)))
else Free (fun_name_to_time ctxt false nm, change_typ T)
)
| fun_to_time _ _ _ _ _ = error "Internal error: invalid function to convert"
(* Convert arguments of left side of a term *)
fun conv_arg ctxt used _ (f as Free (nm,T as Type("fun",_))) =
if used f then Free (nm, HOLogic.mk_prodT (T, change_typ' (K false) T))
else Free (fun_name_to_time ctxt false nm, change_typ' (K false) T)
| conv_arg _ _ _ x = x
fun conv_args ctxt used origin = map (conv_arg ctxt used origin)
(* Handle function calls *)
fun build_zero (Type ("fun", [T, R])) = Abs ("uu", T, build_zero R)
| build_zero _ = zero
fun funcc_use_origin used (f as Free (nm, T as Type ("fun",_))) =
if used f then HOLogic.mk_fst (Free (nm,HOLogic.mk_prodT (T, change_typ T)))
else error "Internal error: Error in used detection"
| funcc_use_origin _ t = t
fun funcc_conv_arg _ used _ (t as (_ $ _)) = map_aterms (funcc_use_origin used) t
| funcc_conv_arg wctxt used u (f as Free (nm, T as Type ("fun",_))) =
if used f then
if u then Free (nm, HOLogic.mk_prodT (T, change_typ T))
else HOLogic.mk_snd (Free (nm,HOLogic.mk_prodT (T,change_typ T)))
else Free (fun_name_to_time (#ctxt wctxt) false nm, change_typ T)
| funcc_conv_arg wctxt _ true (f as Const (_,T as Type ("fun",_))) =
(Const (@{const_name "Product_Type.Pair"},
Type ("fun", [T,Type ("fun", [change_typ T, HOLogic.mk_prodT (T,change_typ T)])]))
$ f $ (Option.getOpt (fun_to_time (#ctxt wctxt) (K false) (K false) (#origins wctxt) f, build_zero T)))
| funcc_conv_arg wctxt _ false (f as Const (_,T as Type ("fun",_))) =
Option.getOpt (fun_to_time (#ctxt wctxt) (K false) (K false) (#origins wctxt) f, build_zero T)
| funcc_conv_arg _ _ _ t = t
fun funcc_conv_args _ _ _ [] = []
| funcc_conv_args wctxt used (Type ("fun", [t, ts])) (a::args) =
funcc_conv_arg wctxt used (is_Used t) a :: funcc_conv_args wctxt used ts args
| funcc_conv_args _ _ _ _ = error "Internal error: Non matching type"
fun funcc orig_used used wctxt func args =
let
fun get_T (Free (_,T)) = T
| get_T (Const (_,T)) = T
| get_T (_ $ (Free (_,Type (_, [_, T])))) = T (* Case of snd was constructed *)
| get_T _ = error "Internal error: Forgotten type"
in
List.foldr (I #-> plus)
(case fun_to_time (#ctxt wctxt) orig_used used (#origins wctxt) func
of SOME t => SOME (build_func (t,funcc_conv_args wctxt used (get_T t) args))
| NONE => NONE)
(map (#f wctxt) args)
end
(* Handle case terms *)
fun casecIsCase (Type (n1, [_,Type (n2, _)])) = (n1 = "fun" andalso n2 = "fun")
| casecIsCase _ = false
fun casecLastTyp (Type (n, [T1,T2])) = Type (n, [T1, change_typ T2])
| casecLastTyp _ = error "Internal error: Invalid case type"
fun casecTyp (Type (n, [T1, T2])) =
Type (n, [change_typ T1, (if casecIsCase T2 then casecTyp else casecLastTyp) T2])
| casecTyp _ = error "Internal error: Invalid case type"
fun casecAbs f (Abs (v,Ta,t)) = (case casecAbs f t of (nconst,t) => (nconst,Abs (v,Ta,t)))
| casecAbs f t = (case f t of NONE => (false,HOLogic.zero) | SOME t => (true,t))
fun casecArgs _ [t] = (false, [map_aterms use_origin t])
| casecArgs f (t::ar) =
(case casecAbs f t of (nconst, tt) =>
casecArgs f ar ||> (fn ar => tt :: ar) |>> (if nconst then K true else I))
| casecArgs _ _ = error "Internal error: Invalid case term"
fun casec wctxt (Const (t,T)) args =
if not (casecIsCase T) then error "Internal error: Invalid case type" else
let val (nconst, args') = casecArgs (#f wctxt) args in
plus
((#f wctxt) (List.last args))
(if nconst then
SOME (build_func (Const (t,casecTyp T), args'))
else NONE)
end
| casec _ _ _ = error "Internal error: Invalid case term"
(* Handle if terms -> drop the term if true and false terms are zero *)
fun ifc wctxt _ cond tt ft =
let
val f = #f wctxt
val rcond = map_aterms use_origin cond
val tt = f tt
val ft = f ft
in
plus (f cond) (case (tt,ft) of (NONE, NONE) => NONE | _ =>
if tt = ft then tt else
(SOME ((Const (@{const_name "HOL.If"}, @{typ "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"})) $ rcond $ (opt_term tt) $ (opt_term ft))))
end
fun letc_change_typ (Type ("fun", [T1, Type ("fun", [T2, _])])) = (Type ("fun", [T1, Type ("fun", [change_typ T2, HOLogic.natT])]))
| letc_change_typ _ = error "Internal error: invalid let type"
fun letc wctxt expT exp nms Ts t =
plus (#f wctxt exp)
(if length nms = 0 (* In case of "length nms = 0" the expression got reducted
Here we need Bound 0 to gain non-partial application *)
then (case #f wctxt (t $ Bound 0) of SOME (t' $ Bound 0) =>
(SOME (Const (@{const_name "HOL.Let"}, letc_change_typ expT) $ (map_aterms use_origin exp) $ t'))
(* Expression is not used and can therefore let be dropped *)
| SOME t' => SOME t'
| NONE => NONE)
else (case #f wctxt t of SOME t' =>
SOME (if Term.is_dependent t' then Const (@{const_name "HOL.Let"}, letc_change_typ expT) $ (map_aterms use_origin exp) $ build_abs t' nms Ts
else Term.subst_bounds([exp],t'))
| NONE => NONE))
(* The converter for timing functions given to the walker *)
fun converter orig_used used : term option converter = {
constc = fn _ => fn t =>
(case t of Const ("HOL.undefined", _) => SOME (Const ("HOL.undefined", @{typ "nat"}))
| _ => NONE),
funcc = (funcc orig_used used),
ifc = ifc,
casec = casec,
letc = letc
}
fun top_converter is_rec _ _ = opt_term o (fn exp => plus exp (if is_rec then SOME one else NONE))
(* Use converter to convert right side of a term *)
fun to_time ctxt origin is_rec orig_used used term =
top_converter is_rec ctxt origin (walk ctxt origin (converter orig_used used) term)
(* Converts a term to its running time version *)
fun convert_term ctxt (origin: term list) is_rec orig_used (pT $ (Const (eqN, _) $ l $ r)) =
let
val (l' as (l_const, l_params)) = walk_func l []
val used =
l_const
|> used_for_const orig_used
|> (fn f => fn n => f (index l_params n))
in
pT
$ (Const (eqN, @{typ "nat \<Rightarrow> nat \<Rightarrow> bool"})
$ (build_func (l' |>> (fun_to_time ctxt orig_used used origin) |>> Option.valOf ||> conv_args ctxt used origin))
$ (to_time ctxt origin is_rec orig_used used r))
end
| convert_term _ _ _ _ _ = error "Internal error: invalid term to convert"
(* 4. Tactic to prove "f_dom n" *)
fun time_dom_tac ctxt induct_rule domintros =
(Induction.induction_tac ctxt true [] [[]] [] (SOME [induct_rule]) []
THEN_ALL_NEW ((K (auto_tac ctxt)) THEN' (fn i => FIRST' (
(if i <= length domintros then [Metis_Tactic.metis_tac [] ATP_Problem_Generate.combsN ctxt [List.nth (domintros, i-1)]] else []) @
[Metis_Tactic.metis_tac [] ATP_Problem_Generate.combsN ctxt domintros]) i)))
fun get_terms theory (term: term) =
let
val equations = Spec_Rules.retrieve theory term
|> map #rules
|> map (map Thm.prop_of)
handle Empty => error "Function or terms of function not found"
in
equations
|> filter (fn ts => typ_comp (ts |> hd |> get_l |> walk_func' |> fst |> dest_Const |> snd) (term |> dest_Const |> snd))
|> hd
end
(* Register timing function of a given function *)
fun reg_time_func (lthy: local_theory) (term: term list) (terms: term list) print =
let
val _ =
case time_term lthy true (hd term)
handle (ERROR _) => NONE
of SOME _ => error ("Timing function already declared: " ^ (Term.term_name (hd term)))
| NONE => ()
(* 1. Fix all terms *)
(* Exchange Var in types and terms to Free and check constraints *)
val terms = map
(map_aterms fixTerms
#> map_types (map_atyps fixTypes)
#> fixPartTerms lthy term)
terms
(* 2. Find properties about the function *)
(* 2.1 Check if function is recursive *)
val is_rec = is_rec lthy term terms
(* 3. Convert every equation
- Change type of toplevel equation from _ \<Rightarrow> _ \<Rightarrow> bool to nat \<Rightarrow> nat \<Rightarrow> bool
- On left side change name of function to timing function
- Convert right side of equation with conversion schema
*)
fun convert used = map (convert_term lthy term is_rec used)
fun repeat T_terms =
let
val orig_used = find_used lthy term terms T_terms
val T_terms' = convert orig_used terms
in
if T_terms' <> T_terms then repeat T_terms' else T_terms'
end
val T_terms = repeat (convert (K true) terms)
val orig_used = find_used lthy term terms T_terms
(* 4. Register function and prove termination *)
val names = map Term.term_name term
val timing_names = map (fun_name_to_time lthy true) names
val bindings = map (fn nm => (Binding.name nm, NONE, NoSyn)) timing_names
fun pat_completeness_auto ctxt =
Pat_Completeness.pat_completeness_tac ctxt 1 THEN auto_tac ctxt
val specs = map (fn eq => (((Binding.empty, []), eq), [], [])) T_terms
(* For partial functions sequential=true is needed in order to support them
We need sequential=false to support the automatic proof of termination over dom
*)
fun register seq =
let
val _ = (if seq then warning "Falling back on sequential function..." else ())
val fun_config = Function_Common.FunctionConfig
{sequential=seq, default=NONE, domintros=true, partials=false}
in
Function.add_function bindings specs fun_config pat_completeness_auto lthy
end
(* Context for printing without showing question marks *)
val print_ctxt = lthy
|> Config.put show_question_marks false
|> Config.put show_sorts false (* Change it for debugging *)
val print_ctxt = List.foldl (fn (term, ctxt) => Variable.add_fixes_implicit term ctxt) print_ctxt (term @ T_terms)
(* Print result if print *)
val _ = if not print then () else
let
val nms = map (fst o dest_Const) term
val used = map (used_for_const orig_used) term
val typs = map (snd o dest_Const) term
in
print_timing' print_ctxt { names=nms, terms=terms, typs=typs }
{ names=timing_names, terms=T_terms, typs=map (fn (used, typ) => change_typ' used typ) (ListPair.zip (used, typs)) }
end
in
register false
handle (ERROR _) =>
register true
| Match =>
register true
end
fun proove_termination (term: term list) terms print (T_info: Function.info, lthy: local_theory) =
let
(* Start proving the termination *)
val infos = SOME (map (Function.get_info lthy) term) handle Empty => NONE
val timing_names = map (fun_name_to_time lthy true o Term.term_name) term
(* Proof by lexicographic_order_tac *)
val (time_info, lthy') =
(Function.prove_termination NONE
(Lexicographic_Order.lexicographic_order_tac false lthy) lthy)
handle (ERROR _) =>
let
val _ = warning "Falling back on proof over dom..."
val _ = (if length term > 1 then error "Proof over dom not supported for mutual recursive functions" else ())
fun args (a$(Var ((nm,_),T))) = args a |> (fn ar => (nm,T)::ar)
| args (a$(Const (_,T))) = args a |> (fn ar => ("uu",T)::ar)
| args _ = []
val dom_vars =
terms |> hd |> get_l |> map_types (map_atyps fixTypes)
|> args |> Variable.variant_frees lthy []
val dom_args =
List.foldl (fn (t,p) => HOLogic.mk_prod ((Free t),p)) (Free (hd dom_vars)) (tl dom_vars)
val {inducts, ...} = case infos of SOME [i] => i | _ => error "Proof over dom failed as no induct rule was found"
val induct = (Option.valOf inducts |> hd)
val domintros = Proof_Context.get_fact lthy (Facts.named (hd timing_names ^ ".domintros"))
val prop = HOLogic.mk_Trueprop (#dom T_info $ dom_args)
(* Prove a helper lemma *)
val dom_lemma = Goal.prove lthy (map fst dom_vars) [] prop
(fn {context, ...} => HEADGOAL (time_dom_tac context induct domintros))
(* Add dom_lemma to simplification set *)
val simp_lthy = Simplifier.add_simp dom_lemma lthy
in
(* Use lemma to prove termination *)
Function.prove_termination NONE
(auto_tac simp_lthy) lthy
end
(* Context for printing without showing question marks *)
val print_ctxt = lthy'
|> Config.put show_question_marks false
|> Config.put show_sorts false (* Change it for debugging *)
(* Print result if print *)
val _ = if not print then () else
let
val nms = map (fst o dest_Const) term
val typs = map (snd o dest_Const) term
in
print_timing' print_ctxt { names=nms, terms=terms, typs=typs } (info_pfunc time_info)
end
in
(time_info, lthy')
end
fun reg_and_proove_time_func (lthy: local_theory) (term: term list) (terms: term list) print =
reg_time_func lthy term terms false
|> proove_termination term terms print
fun fix_definition (Const ("Pure.eq", _) $ l $ r) = Const ("HOL.Trueprop", @{typ "bool \<Rightarrow> prop"})
$ (Const ("HOL.eq", @{typ "bool \<Rightarrow> bool \<Rightarrow> bool"}) $ l $ r)
| fix_definition t = t
fun check_definition [t] = [t]
| check_definition _ = error "Only a single definition is allowed"
fun isTypeClass' (Const (nm,_)) =
(case split_name nm |> rev
of (_::nm::_) => String.isSuffix "_class" nm
| _ => false)
| isTypeClass' _ = false
val isTypeClass =
(List.foldr (fn (a,b) => a orelse b) false) o (map isTypeClass')
fun detect_typ (ctxt: local_theory) (term: term) =
let
val class_term = (case term of Const (nm,_) => Syntax.read_term ctxt nm
| _ => error "Could not find term of class")
fun find_free (Type (_,class)) (Type (_,inst)) =
List.foldl (fn ((c,i),s) => (case s of NONE => find_free c i | t => t)) (NONE) (ListPair.zip (class, inst))
| find_free (TFree _) (TFree _) = NONE
| find_free (TFree _) (Type (nm,_)) = SOME nm
| find_free _ _ = error "Unhandled case in detecting type"
in
find_free (type_of class_term) (type_of term)
|> Option.map (hd o rev o split_name)
end
fun set_suffix (fterms: term list) ctxt =
let
val isTypeClass = isTypeClass fterms
val _ = (if length fterms > 1 andalso isTypeClass then error "No mutual recursion inside instantiation allowed" else ())
val suffix = (if isTypeClass then detect_typ ctxt (hd fterms) else NONE)
in
(case suffix of NONE => I | SOME s => Config.put bsuffix ("_" ^ s)) ctxt
end
(* Convert function into its timing function (called by command) *)
fun reg_time_fun_cmd (funcs, thms) (ctxt: local_theory) =
let
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val (_, ctxt') = reg_and_proove_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms)
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
true
in ctxt'
end
(* Convert function into its timing function (called by command) with termination proof provided by user*)
fun reg_time_function_cmd (funcs, thms) (ctxt: local_theory) =
let
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val ctxt' = reg_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms)
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
true
|> snd
in ctxt'
end
(* Convert function into its timing function (called by command) *)
fun reg_time_definition_cmd (funcs, thms) (ctxt: local_theory) =
let
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val (_, ctxt') = reg_and_proove_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms) |> check_definition |> map fix_definition
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
true
in ctxt'
end
val parser = (Scan.repeat1 Parse.prop) -- (Scan.option (Parse.keyword_improper "equations" -- Parse.thms1 >> snd))
val _ = Outer_Syntax.local_theory @{command_keyword "time_fun"}
"Defines runtime function of a function"
(parser >> reg_time_fun_cmd)
val _ = Outer_Syntax.local_theory @{command_keyword "time_function"}
"Defines runtime function of a function"
(parser >> reg_time_function_cmd)
val _ = Outer_Syntax.local_theory @{command_keyword "time_definition"}
"Defines runtime function of a definition"
(parser >> reg_time_definition_cmd)
end