(* Title: HOL/Decision_Procs/approximation_generator.ML
Author: Fabian Immler, TU Muenchen
*)
signature APPROXIMATION_GENERATOR =
sig
val custom_seed: int Config.T
val precision: int Config.T
val epsilon: real Config.T
val approximation_generator:
Proof.context ->
(term * term list) list ->
bool -> int list -> (bool * term list) option * Quickcheck.report option
val setup: theory -> theory
end;
structure Approximation_Generator : APPROXIMATION_GENERATOR =
struct
val custom_seed = Attrib.setup_config_int \<^binding>\<open>quickcheck_approximation_custom_seed\<close> (K ~1)
val precision = Attrib.setup_config_int \<^binding>\<open>quickcheck_approximation_precision\<close> (K 30)
val epsilon = Attrib.setup_config_real \<^binding>\<open>quickcheck_approximation_epsilon\<close> (K 0.0)
val random_float = @{code "random_class.random::_ \<Rightarrow> _ \<Rightarrow> (float \<times> (unit \<Rightarrow> term)) \<times> _"}
fun nat_of_term t =
(HOLogic.dest_nat t handle TERM _ => snd (HOLogic.dest_number t)
handle TERM _ => raise TERM ("nat_of_term", [t]));
fun int_of_term t = snd (HOLogic.dest_number t) handle TERM _ => raise TERM ("int_of_term", [t]);
fun real_of_man_exp m e = Real.fromManExp {man = Real.fromInt m, exp = e}
fun mapprox_float (\<^term>\<open>Float\<close> $ m $ e) = real_of_man_exp (int_of_term m) (int_of_term e)
| mapprox_float t = Real.fromInt (snd (HOLogic.dest_number t))
handle TERM _ => raise TERM ("mapprox_float", [t]);
(* TODO: define using compiled terms? *)
fun mapprox_floatarith (\<^term>\<open>Add\<close> $ a $ b) xs = mapprox_floatarith a xs + mapprox_floatarith b xs
| mapprox_floatarith (\<^term>\<open>Minus\<close> $ a) xs = ~ (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Mult\<close> $ a $ b) xs = mapprox_floatarith a xs * mapprox_floatarith b xs
| mapprox_floatarith (\<^term>\<open>Inverse\<close> $ a) xs = 1.0 / mapprox_floatarith a xs
| mapprox_floatarith (\<^term>\<open>Cos\<close> $ a) xs = Math.cos (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Arctan\<close> $ a) xs = Math.atan (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Abs\<close> $ a) xs = abs (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Max\<close> $ a $ b) xs =
Real.max (mapprox_floatarith a xs, mapprox_floatarith b xs)
| mapprox_floatarith (\<^term>\<open>Min\<close> $ a $ b) xs =
Real.min (mapprox_floatarith a xs, mapprox_floatarith b xs)
| mapprox_floatarith \<^term>\<open>Pi\<close> _ = Math.pi
| mapprox_floatarith (\<^term>\<open>Sqrt\<close> $ a) xs = Math.sqrt (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Exp\<close> $ a) xs = Math.exp (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Powr\<close> $ a $ b) xs =
Math.pow (mapprox_floatarith a xs, mapprox_floatarith b xs)
| mapprox_floatarith (\<^term>\<open>Ln\<close> $ a) xs = Math.ln (mapprox_floatarith a xs)
| mapprox_floatarith (\<^term>\<open>Power\<close> $ a $ n) xs =
Math.pow (mapprox_floatarith a xs, Real.fromInt (nat_of_term n))
| mapprox_floatarith (\<^term>\<open>Floor\<close> $ a) xs = Real.fromInt (floor (mapprox_floatarith a xs))
| mapprox_floatarith (\<^term>\<open>Var\<close> $ n) xs = nth xs (nat_of_term n)
| mapprox_floatarith (\<^term>\<open>Num\<close> $ m) _ = mapprox_float m
| mapprox_floatarith t _ = raise TERM ("mapprox_floatarith", [t])
fun mapprox_atLeastAtMost eps x a b xs =
let
val x' = mapprox_floatarith x xs
in
mapprox_floatarith a xs + eps <= x' andalso x' + eps <= mapprox_floatarith b xs
end
fun mapprox_form eps (\<^term>\<open>Bound\<close> $ x $ a $ b $ f) xs =
(not (mapprox_atLeastAtMost eps x a b xs)) orelse mapprox_form eps f xs
| mapprox_form eps (\<^term>\<open>Assign\<close> $ x $ a $ f) xs =
(Real.!= (mapprox_floatarith x xs, mapprox_floatarith a xs)) orelse mapprox_form eps f xs
| mapprox_form eps (\<^term>\<open>Less\<close> $ a $ b) xs = mapprox_floatarith a xs + eps < mapprox_floatarith b xs
| mapprox_form eps (\<^term>\<open>LessEqual\<close> $ a $ b) xs = mapprox_floatarith a xs + eps <= mapprox_floatarith b xs
| mapprox_form eps (\<^term>\<open>AtLeastAtMost\<close> $ x $ a $ b) xs = mapprox_atLeastAtMost eps x a b xs
| mapprox_form eps (\<^term>\<open>Conj\<close> $ f $ g) xs = mapprox_form eps f xs andalso mapprox_form eps g xs
| mapprox_form eps (\<^term>\<open>Disj\<close> $ f $ g) xs = mapprox_form eps f xs orelse mapprox_form eps g xs
| mapprox_form _ t _ = raise TERM ("mapprox_form", [t])
fun dest_interpret_form (\<^const>\<open>interpret_form\<close> $ b $ xs) = (b, xs)
| dest_interpret_form t = raise TERM ("dest_interpret_form", [t])
fun optionT t = Type (\<^type_name>\<open>option\<close>, [t])
fun mk_Some t = Const (\<^const_name>\<open>Some\<close>, t --> optionT t)
fun random_float_list size xs seed =
fold (K (apsnd (random_float size) #-> (fn c => apfst (fn b => b::c)))) xs ([],seed)
fun real_of_Float (@{code Float} (m, e)) =
real_of_man_exp (@{code integer_of_int} m) (@{code integer_of_int} e)
fun is_True \<^term>\<open>True\<close> = true
| is_True _ = false
val postproc_form_eqs =
@{lemma
"real_of_float (Float 0 a) = 0"
"real_of_float (Float (numeral m) 0) = numeral m"
"real_of_float (Float 1 0) = 1"
"real_of_float (Float (- 1) 0) = - 1"
"real_of_float (Float 1 (numeral e)) = 2 ^ numeral e"
"real_of_float (Float 1 (- numeral e)) = 1 / 2 ^ numeral e"
"real_of_float (Float a 1) = a * 2"
"real_of_float (Float a (-1)) = a / 2"
"real_of_float (Float (- a) b) = - real_of_float (Float a b)"
"real_of_float (Float (numeral m) (numeral e)) = numeral m * 2 ^ (numeral e)"
"real_of_float (Float (numeral m) (- numeral e)) = numeral m / 2 ^ (numeral e)"
"- (c * d::real) = -c * d"
"- (c / d::real) = -c / d"
"- (0::real) = 0"
"int_of_integer (numeral k) = numeral k"
"int_of_integer (- numeral k) = - numeral k"
"int_of_integer 0 = 0"
"int_of_integer 1 = 1"
"int_of_integer (- 1) = - 1"
by auto
}
fun rewrite_with ctxt thms = Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps thms)
fun conv_term ctxt conv r = Thm.cterm_of ctxt r |> conv |> Thm.prop_of |> Logic.dest_equals |> snd
fun approx_random ctxt prec eps frees e xs genuine_only size seed =
let
val (rs, seed') = random_float_list size xs seed
fun mk_approx_form e ts =
\<^const>\<open>approx_form\<close> $
HOLogic.mk_number \<^typ>\<open>nat\<close> prec $
e $
(HOLogic.mk_list \<^typ>\<open>(float * float) option\<close>
(map (fn t => mk_Some \<^typ>\<open>float * float\<close> $ HOLogic.mk_prod (t, t)) ts)) $
\<^term>\<open>[] :: nat list\<close>
in
(if
mapprox_form eps e (map (real_of_Float o fst) rs)
handle
General.Overflow => false
| General.Domain => false
| General.Div => false
| IEEEReal.Unordered => false
then
let
val ts = map (fn x => snd x ()) rs
val ts' = map
(AList.lookup op = (map dest_Free xs ~~ ts)
#> the_default Term.dummy
#> curry op $ \<^term>\<open>real_of_float::float\<Rightarrow>_\<close>
#> conv_term ctxt (rewrite_with ctxt postproc_form_eqs))
frees
in
if Approximation.approximate ctxt (mk_approx_form e ts) |> is_True
then SOME (true, ts')
else (if genuine_only then NONE else SOME (false, ts'))
end
else NONE, seed')
end
val preproc_form_eqs =
@{lemma
"(a::real) \<in> {b .. c} \<longleftrightarrow> b \<le> a \<and> a \<le> c"
"a = b \<longleftrightarrow> a \<le> b \<and> b \<le> a"
"(p \<longrightarrow> q) \<longleftrightarrow> \<not>p \<or> q"
"(p \<longleftrightarrow> q) \<longleftrightarrow> (p \<longrightarrow> q) \<and> (q \<longrightarrow> p)"
by auto}
fun reify_goal ctxt t =
HOLogic.mk_not t
|> conv_term ctxt (rewrite_with ctxt preproc_form_eqs)
|> Approximation.reify_form ctxt
|> dest_interpret_form
||> HOLogic.dest_list
fun approximation_generator_raw ctxt t =
let
val iterations = Config.get ctxt Quickcheck.iterations
val prec = Config.get ctxt precision
val eps = Config.get ctxt epsilon
val cs = Config.get ctxt custom_seed
val seed = (Code_Numeral.natural_of_integer (cs + 1), Code_Numeral.natural_of_integer 1)
val run = if cs < 0
then (fn f => fn seed => (Random_Engine.run f, seed))
else (fn f => fn seed => f seed)
val frees = Term.add_frees t []
val (e, xs) = reify_goal ctxt t
fun single_tester b s =
approx_random ctxt prec eps frees e xs b s |> run
fun iterate _ _ 0 _ = NONE
| iterate genuine_only size j seed =
case single_tester genuine_only size seed of
(NONE, seed') => iterate genuine_only size (j - 1) seed'
| (SOME q, _) => SOME q
in
fn genuine_only => fn size => (iterate genuine_only size iterations seed, NONE)
end
fun approximation_generator ctxt [(t, _)] =
(fn genuine_only =>
fn [_, size] =>
approximation_generator_raw ctxt t genuine_only
(Code_Numeral.natural_of_integer size))
| approximation_generator _ _ =
error "Quickcheck-approximation does not support type variables (or finite instantiations)"
val test_goals =
Quickcheck_Common.generator_test_goal_terms
("approximation", (fn _ => fn _ => false, approximation_generator))
val active = Attrib.setup_config_bool \<^binding>\<open>quickcheck_approximation_active\<close> (K false)
val setup = Context.theory_map (Quickcheck.add_tester ("approximation", (active, test_goals)))
end