(* Title: HOL/Decision_Procs/approximation.ML
Author: Johannes Hoelzl, TU Muenchen
*)
signature APPROXIMATION =
sig
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: APPROXIMATION =
struct
fun reorder_bounds_tac ctxt prems i =
let
fun variable_of_bound (Const (@{const_name Trueprop}, _) $
(Const (@{const_name Set.member}, _) $
Free (name, _) $ _)) = name
| variable_of_bound (Const (@{const_name Trueprop}, _) $
(Const (@{const_name HOL.eq}, _) $
Free (name, _) $ _)) = 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 approximation_conv ctxt ct =
approximation_oracle (Proof_Context.theory_of ctxt, Thm.term_of ct |> tap (tracing o Syntax.string_of_term ctxt));
fun approximate ctxt t =
approximation_oracle (Proof_Context.theory_of ctxt, t)
|> Thm.prop_of |> Logic.dest_equals |> snd;
(* Should be in HOL.thy ? *)
fun gen_eval_tac conv ctxt =
CONVERSION
(Object_Logic.judgment_conv ctxt (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt))
THEN' resolve_tac ctxt [TrueI]
val form_equations = @{thms interpret_form_equations};
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 @{typ nat} s
| NONE => @{term "0 :: nat"}
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 @{typ nat}
|> Thm.cterm_of ctxt
in case taylor
of NONE => let
val n = vs |> length
|> HOLogic.mk_number @{typ nat}
|> Thm.cterm_of ctxt
val s = vs
|> map lookup_splitting
|> HOLogic.mk_list @{typ nat}
|> Thm.cterm_of ctxt
in
(resolve_tac ctxt [Thm.instantiate ([], [((("n", 0), @{typ nat}), n),
((("prec", 0), @{typ nat}), p),
((("ss", 0), @{typ "nat list"}), s)])
@{thm approx_form}] 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 @{typ nat}
|> Thm.cterm_of ctxt
val s = vs |> map lookup_splitting |> hd
|> Thm.cterm_of ctxt
in
resolve_tac ctxt [Thm.instantiate ([], [((("s", 0), @{typ nat}), s),
((("t", 0), @{typ nat}), t),
((("prec", 0), @{typ nat}), p)])
@{thm approx_tse_form}] i st
end
end
fun calculated_subterms (@{const Trueprop} $ t) = calculated_subterms t
| calculated_subterms (@{const HOL.implies} $ _ $ t) = calculated_subterms t
| calculated_subterms (@{term "op <= :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
| calculated_subterms (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
| calculated_subterms (@{term "op : :: real \<Rightarrow> real set \<Rightarrow> bool"} $ t1 $
(@{term "atLeastAtMost :: real \<Rightarrow> real \<Rightarrow> real set"} $ t2 $ t3)) = [t1, t2, t3]
| calculated_subterms t = raise TERM ("calculated_subterms", [t])
fun dest_interpret_form (@{const "interpret_form"} $ b $ xs) = (b, xs)
| dest_interpret_form t = raise TERM ("dest_interpret_form", [t])
fun dest_interpret (@{const "interpret_floatarith"} $ b $ xs) = (b, xs)
| dest_interpret t = raise TERM ("dest_interpret", [t])
fun dest_float (@{const "Float"} $ m $ e) =
(snd (HOLogic.dest_number m), snd (HOLogic.dest_number e))
fun dest_ivl (Const (@{const_name "Some"}, _) $
(Const (@{const_name Pair}, _) $ u $ l)) = SOME (dest_float u, dest_float l)
| dest_ivl (Const (@{const_name "None"}, _)) = NONE
| dest_ivl t = raise TERM ("dest_result", [t])
fun mk_approx' prec t = (@{const "approx'"}
$ HOLogic.mk_number @{typ nat} prec
$ t $ @{term "[] :: (float * float) option list"})
fun mk_approx_form_eval prec t xs = (@{const "approx_form_eval"}
$ HOLogic.mk_number @{typ nat} prec
$ t $ xs)
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 c p 0 digits cnt = (digits, cnt, 0)
| frac c 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 - (IntInf.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 @{typ real} m
else if e = 1 then @{term "divide :: real \<Rightarrow> real \<Rightarrow> real"} $
HOLogic.mk_number @{typ real} m $
@{term "10"}
else @{term "divide :: real \<Rightarrow> real \<Rightarrow> real"} $
HOLogic.mk_number @{typ real} m $
(@{term "power 10 :: nat \<Rightarrow> real"} $
HOLogic.mk_number @{typ nat} (~e)) end)
in @{term "atLeastAtMost :: real \<Rightarrow> real \<Rightarrow> real set"} $ mk_float10 true l $ mk_float10 false u end)
| mk_result _ NONE = @{term "UNIV :: real set"}
fun realify t =
let
val t = Logic.varify_global t
val m = map (fn (name, _) => (name, @{typ real})) (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 prepare_form ctxt term = apply_tactic ctxt term (
REPEAT (FIRST' [eresolve_tac ctxt @{thms intervalE},
eresolve_tac ctxt @{thms meta_eqE},
resolve_tac ctxt @{thms impI}] 1)
THEN Subgoal.FOCUS (fn {prems, context = ctxt', ...} => reorder_bounds_tac ctxt' prems 1) ctxt 1
THEN DETERM (TRY (filter_prems_tac ctxt (K false) 1)))
fun reify_form ctxt term = apply_tactic ctxt term
(Reification.tac ctxt form_equations NONE 1)
fun approx_form prec ctxt t =
realify t
|> prepare_form ctxt
|> (fn arith_term => reify_form ctxt arith_term
|> HOLogic.dest_Trueprop |> dest_interpret_form
|> (fn (data, xs) =>
mk_approx_form_eval prec data (HOLogic.mk_list @{typ "(float * float) option"}
(map (fn _ => @{term "None :: (float * float) option"}) (HOLogic.dest_list xs)))
|> approximate ctxt
|> HOLogic.dest_list
|> curry ListPair.zip (HOLogic.dest_list xs @ calculated_subterms arith_term)
|> map (fn (elem, s) => @{term "op : :: real \<Rightarrow> real set \<Rightarrow> bool"} $ elem $ mk_result prec (dest_ivl s))
|> foldr1 HOLogic.mk_conj))
fun approx_arith prec ctxt t = realify t
|> Thm.cterm_of ctxt
|> Reification.conv ctxt form_equations
|> 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 = @{typ prop} then approx_form prec ctxt t
else if type_of t = @{typ bool} then approx_form prec ctxt (@{const Trueprop} $ t)
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' = Variable.auto_fixes 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 "("} |-- Parse.!!! (Scan.repeat1 Parse.xname --| @{keyword ")"})) [];
val _ =
Outer_Syntax.command @{command_keyword approximate} "print approximation of term"
(opt_modes -- Parse.term
>> (fn (modes, t) => Toplevel.keep (approximate_cmd modes t)));
fun approximation_tac prec splitting taylor 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 DETERM (Reification.tac ctxt form_equations NONE i)
THEN rewrite_interpret_form_tac ctxt prec splitting taylor i
THEN gen_eval_tac (approximation_conv ctxt) ctxt i
end;