(* Title: HOL/Decision_Procs/approximation.ML
Author: Johannes Hoelzl, TU Muenchen
*)
signature APPROXIMATION =
sig
val reify_form: Proof.context -> term -> term
val approx: int -> Proof.context -> term -> term
val approximate: Proof.context -> term -> term
val approximation_tac : int -> (string * int) list -> int option -> Proof.context -> int -> tactic
end
structure Approximation =
struct
fun reorder_bounds_tac ctxt prems i =
let
fun variable_of_bound \<^Const_>\<open>Trueprop for \<^Const_>\<open>Set.member _ for \<open>Free (name, _)\<close> _\<close>\<close> = name
| variable_of_bound \<^Const_>\<open>Trueprop for \<^Const_>\<open>HOL.eq _ for \<open>Free (name, _)\<close> _\<close>\<close> = name
| variable_of_bound t = raise TERM ("variable_of_bound", [t])
val variable_bounds
= map (`(variable_of_bound o Thm.prop_of)) prems
fun add_deps (name, bnds)
= Graph.add_deps_acyclic (name,
remove (op =) name (Term.add_free_names (Thm.prop_of bnds) []))
val order = Graph.empty
|> fold Graph.new_node variable_bounds
|> fold add_deps variable_bounds
|> Graph.strong_conn |> map the_single |> rev
|> map_filter (AList.lookup (op =) variable_bounds)
fun prepend_prem th tac =
tac THEN resolve_tac ctxt [th RSN (2, @{thm mp})] i
in
fold prepend_prem order all_tac
end
fun approximate ctxt t = case fastype_of t
of \<^Type>\<open>bool\<close> =>
Approximation_Computation.approx_bool ctxt t
| \<^typ>\<open>float interval option\<close> =>
Approximation_Computation.approx_arith ctxt t
| \<^typ>\<open>float interval option list\<close> =>
Approximation_Computation.approx_form_eval ctxt t
| _ => error ("Bad term: " ^ Syntax.string_of_term ctxt t);
fun rewrite_interpret_form_tac ctxt prec splitting taylor i st = let
fun lookup_splitting (Free (name, _)) =
(case AList.lookup (op =) splitting name
of SOME s => HOLogic.mk_number \<^Type>\<open>nat\<close> s
| NONE => \<^term>\<open>0 :: nat\<close>)
| lookup_splitting t = raise TERM ("lookup_splitting", [t])
val vs = nth (Thm.prems_of st) (i - 1)
|> Logic.strip_imp_concl
|> HOLogic.dest_Trueprop
|> Term.strip_comb |> snd |> List.last
|> HOLogic.dest_list
val p = prec
|> HOLogic.mk_number \<^Type>\<open>nat\<close>
|> Thm.cterm_of ctxt
in case taylor
of NONE => let
val n = vs |> length
|> HOLogic.mk_number \<^Type>\<open>nat\<close>
|> Thm.cterm_of ctxt
val ss = vs
|> map lookup_splitting
|> HOLogic.mk_list \<^Type>\<open>nat\<close>
|> Thm.cterm_of ctxt
in
(resolve_tac ctxt [
\<^instantiate>\<open>n and prec = p and ss in
lemma (schematic)
\<open>n = length xs \<Longrightarrow> approx_form prec f (replicate n None) ss \<Longrightarrow> interpret_form f xs\<close>
by (rule approx_form)\<close>] i
THEN simp_tac (put_simpset (simpset_of \<^context>) ctxt) i) st
end
| SOME t =>
if length vs <> 1
then raise (TERM ("More than one variable used for taylor series expansion", [Thm.prop_of st]))
else let
val t = t
|> HOLogic.mk_number \<^Type>\<open>nat\<close>
|> Thm.cterm_of ctxt
val s = vs |> map lookup_splitting |> hd
|> Thm.cterm_of ctxt
in
resolve_tac ctxt [
\<^instantiate>\<open>s and t and prec = p in
lemma (schematic) "approx_tse_form prec t s f \<Longrightarrow> interpret_form f [x]"
by (rule approx_tse_form)\<close>] i st
end
end
fun calculated_subterms \<^Const_>\<open>Trueprop for t\<close> = calculated_subterms t
| calculated_subterms \<^Const_>\<open>implies for _ t\<close> = calculated_subterms t
| calculated_subterms \<^Const_>\<open>less_eq \<^Type>\<open>real\<close> for t1 t2\<close> = [t1, t2]
| calculated_subterms \<^Const_>\<open>less \<^Type>\<open>real\<close> for t1 t2\<close> = [t1, t2]
| calculated_subterms \<^Const_>\<open>Set.member \<^Type>\<open>real\<close> for
t1 \<^Const_>\<open>atLeastAtMost \<^Type>\<open>real\<close> for t2 t3\<close>\<close> = [t1, t2, t3]
| calculated_subterms t = raise TERM ("calculated_subterms", [t])
fun dest_interpret_form \<^Const_>\<open>interpret_form for b xs\<close> = (b, xs)
| dest_interpret_form t = raise TERM ("dest_interpret_form", [t])
fun dest_interpret \<^Const_>\<open>interpret_floatarith for b xs\<close> = (b, xs)
| dest_interpret t = raise TERM ("dest_interpret", [t])
fun dest_interpret_env \<^Const_>\<open>interpret_form for _ xs\<close> = xs
| dest_interpret_env \<^Const_>\<open>interpret_floatarith for _ xs\<close> = xs
| dest_interpret_env t = raise TERM ("dest_interpret_env", [t])
fun dest_float \<^Const_>\<open>Float for m e\<close> = (snd (HOLogic.dest_number m), snd (HOLogic.dest_number e))
| dest_float t = raise TERM ("dest_float", [t])
fun dest_ivl \<^Const_>\<open>Some _ for \<^Const_>\<open>Interval _ for \<^Const_>\<open>Pair _ _ for u l\<close>\<close>\<close> =
SOME (dest_float u, dest_float l)
| dest_ivl \<^Const_>\<open>None _\<close> = NONE
| dest_ivl t = raise TERM ("dest_result", [t])
fun mk_approx' prec t =
\<^Const>\<open>approx' for \<open>HOLogic.mk_number \<^Type>\<open>nat\<close> prec\<close> t \<^Const>\<open>Nil \<^typ>\<open>float interval option\<close>\<close>\<close>
fun mk_approx_form_eval prec t xs =
\<^Const>\<open>approx_form_eval for \<open>HOLogic.mk_number \<^Type>\<open>nat\<close> prec\<close> t xs\<close>
fun float2_float10 prec round_down (m, e) = (
let
val (m, e) = (if e < 0 then (m,e) else (m * Integer.pow e 2, 0))
fun frac _ _ 0 digits cnt = (digits, cnt, 0)
| frac _ 0 r digits cnt = (digits, cnt, r)
| frac c p r digits cnt = (let
val (d, r) = Integer.div_mod (r * 10) (Integer.pow (~e) 2)
in frac (c orelse d <> 0) (if d <> 0 orelse c then p - 1 else p) r
(digits * 10 + d) (cnt + 1)
end)
val sgn = Int.sign m
val m = abs m
val round_down = (sgn = 1 andalso round_down) orelse
(sgn = ~1 andalso not round_down)
val (x, r) = Integer.div_mod m (Integer.pow (~e) 2)
val p = ((if x = 0 then prec else prec - (Integer.log2 x + 1)) * 3) div 10 + 1
val (digits, e10, r) = if p > 0 then frac (x <> 0) p r 0 0 else (0,0,0)
val digits = if round_down orelse r = 0 then digits else digits + 1
in (sgn * (digits + x * (Integer.pow e10 10)), ~e10)
end)
fun mk_result prec (SOME (l, u)) =
(let
fun mk_float10 rnd x = (let val (m, e) = float2_float10 prec rnd x
in if e = 0 then HOLogic.mk_number \<^Type>\<open>real\<close> m
else if e = 1 then \<^Const>\<open>divide \<^Type>\<open>real\<close>\<close> $
HOLogic.mk_number \<^Type>\<open>real\<close> m $
\<^term>\<open>10\<close>
else \<^Const>\<open>divide \<^Type>\<open>real\<close>\<close> $
HOLogic.mk_number \<^Type>\<open>real\<close> m $
(\<^term>\<open>power 10 :: nat \<Rightarrow> real\<close> $
HOLogic.mk_number \<^Type>\<open>nat\<close> (~e)) end)
in \<^Const>\<open>atLeastAtMost \<^Type>\<open>real\<close> for \<open>mk_float10 true l\<close> \<open>mk_float10 false u\<close>\<close> end)
| mk_result _ NONE = \<^term>\<open>UNIV :: real set\<close>
fun realify t =
let
val t = Logic.varify_global t
val m = map (fn (name, _) => (name, \<^Type>\<open>real\<close>)) (Term.add_tvars t [])
val t = Term.subst_TVars m t
in t end
fun apply_tactic ctxt term tactic =
Thm.cterm_of ctxt term
|> Goal.init
|> SINGLE tactic
|> the |> Thm.prems_of |> hd
fun preproc_form_conv ctxt =
Simplifier.rewrite
(put_simpset HOL_basic_ss ctxt addsimps
(Named_Theorems.get ctxt \<^named_theorems>\<open>approximation_preproc\<close>))
fun reify_form_conv ctxt ct =
let
val thm =
Reification.conv ctxt @{thms interpret_form.simps interpret_floatarith.simps} ct
handle ERROR msg =>
cat_error ("Reification failed: " ^ msg)
("Approximation does not support " ^
quote (Syntax.string_of_term ctxt (Thm.term_of ct)))
fun check_env (Free _) = ()
| check_env (Var _) = ()
| check_env t =
cat_error "Term not supported by approximation:" (Syntax.string_of_term ctxt t)
val _ = Thm.rhs_of thm |> Thm.term_of |> dest_interpret_env |> HOLogic.dest_list |> map check_env
in thm end
fun reify_form_tac ctxt i = CONVERSION (Conv.arg_conv (reify_form_conv ctxt)) i
fun prepare_form_tac ctxt i =
REPEAT (FIRST' [eresolve_tac ctxt @{thms intervalE},
eresolve_tac ctxt @{thms meta_eqE},
resolve_tac ctxt @{thms impI}] i)
THEN Subgoal.FOCUS (fn {prems, context = ctxt', ...} => reorder_bounds_tac ctxt' prems i) ctxt i
THEN DETERM (TRY (filter_prems_tac ctxt (K false) i))
THEN CONVERSION (Conv.arg_conv (preproc_form_conv ctxt)) i
fun prepare_form ctxt term = apply_tactic ctxt term (prepare_form_tac ctxt 1)
fun apply_reify_form ctxt t = apply_tactic ctxt t (reify_form_tac ctxt 1)
fun reify_form ctxt t = HOLogic.mk_Trueprop t
|> prepare_form ctxt
|> apply_reify_form ctxt
|> HOLogic.dest_Trueprop
fun approx_form prec ctxt t =
realify t
|> prepare_form ctxt
|> (fn arith_term => apply_reify_form ctxt arith_term
|> HOLogic.dest_Trueprop
|> dest_interpret_form
|> (fn (data, xs) =>
mk_approx_form_eval prec data (HOLogic.mk_list \<^typ>\<open>float interval option\<close>
(map (fn _ => \<^Const>\<open>None \<^typ>\<open>float interval option\<close>\<close>) (HOLogic.dest_list xs)))
|> approximate ctxt
|> HOLogic.dest_list
|> curry ListPair.zip (HOLogic.dest_list xs @ calculated_subterms arith_term)
|> map (fn (elem, s) => \<^Const>\<open>Set.member \<^Type>\<open>real\<close> for elem \<open>mk_result prec (dest_ivl s)\<close>\<close>)
|> foldr1 HOLogic.mk_conj))
fun approx_arith prec ctxt t = realify t
|> Thm.cterm_of ctxt
|> (preproc_form_conv ctxt then_conv reify_form_conv ctxt)
|> Thm.prop_of
|> Logic.dest_equals |> snd
|> dest_interpret |> fst
|> mk_approx' prec
|> approximate ctxt
|> dest_ivl
|> mk_result prec
fun approx prec ctxt t =
if type_of t = \<^Type>\<open>prop\<close> then approx_form prec ctxt t
else if type_of t = \<^Type>\<open>bool\<close> then approx_form prec ctxt \<^Const>\<open>Trueprop for t\<close>
else approx_arith prec ctxt t
fun approximate_cmd modes raw_t state =
let
val ctxt = Toplevel.context_of state;
val t = Syntax.read_term ctxt raw_t;
val t' = approx 30 ctxt t;
val ty' = Term.type_of t';
val ctxt' = Proof_Context.augment t' ctxt;
in
Print_Mode.with_modes modes (fn () =>
Pretty.block [Pretty.quote (Syntax.pretty_term ctxt' t'), Pretty.fbrk,
Pretty.str "::", Pretty.brk 1, Pretty.quote (Syntax.pretty_typ ctxt' ty')]) ()
end |> Pretty.writeln;
val opt_modes =
Scan.optional (\<^keyword>\<open>(\<close> |-- Parse.!!! (Scan.repeat1 Parse.name --| \<^keyword>\<open>)\<close>)) [];
val _ =
Outer_Syntax.command \<^command_keyword>\<open>approximate\<close> "print approximation of term"
(opt_modes -- Parse.term
>> (fn (modes, t) => Toplevel.keep (approximate_cmd modes t)));
fun approximation_tac prec splitting taylor ctxt =
prepare_form_tac ctxt
THEN' reify_form_tac ctxt
THEN' rewrite_interpret_form_tac ctxt prec splitting taylor
THEN' CONVERSION (Approximation_Computation.approx_conv ctxt)
THEN' resolve_tac ctxt [TrueI]
end;