Quelle Approximation.thy Sprache: Isabelle

(* Author:     Johannes Hoelzl, TU Muenchen
   Coercions removed by Dmitriy Traytel *)

theory Approximation
imports
  Complex_Main
  Approximation_Bounds
  "HOL-Library.Code_Target_Numeral_Float"
keywords "approximate" :: diag
begin

section "Implement floatarith"

subsection "Define syntax and semantics"

datatype floatarith
  = Add floatarith floatarith
  | Minus floatarith
  | Mult floatarith floatarith
  | Inverse floatarith
  | Cos floatarith
  | Arctan floatarith
  | Abs floatarith
  | Max floatarith floatarith
  | Min floatarith floatarith
  | Pi
  | Sqrt floatarith
  | Exp floatarith
  | Powr floatarith floatarith
  | Ln floatarith
  | Power floatarith nat
  | Floor floatarith
  | Var nat
  | Num float

fun interpret_floatarith :: "floatarith \ real list \ real" where
"interpret_floatarith (Add a b) vs = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
"interpret_floatarith (Minus a) vs = - (interpret_floatarith a vs)" |
"interpret_floatarith (Mult a b) vs = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
"interpret_floatarith (Inverse a) vs = inverse (interpret_floatarith a vs)" |
"interpret_floatarith (Cos a) vs = cos (interpret_floatarith a vs)" |
"interpret_floatarith (Arctan a) vs = arctan (interpret_floatarith a vs)" |
"interpret_floatarith (Min a b) vs = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
"interpret_floatarith (Max a b) vs = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
"interpret_floatarith (Abs a) vs = \interpret_floatarith a vs\" |
"interpret_floatarith Pi vs = pi" |
"interpret_floatarith (Sqrt a) vs = sqrt (interpret_floatarith a vs)" |
"interpret_floatarith (Exp a) vs = exp (interpret_floatarith a vs)" |
"interpret_floatarith (Powr a b) vs = interpret_floatarith a vs powr interpret_floatarith b vs" |
"interpret_floatarith (Ln a) vs = ln (interpret_floatarith a vs)" |
"interpret_floatarith (Power a n) vs = (interpret_floatarith a vs)^n" |
"interpret_floatarith (Floor a) vs = floor (interpret_floatarith a vs)" |
"interpret_floatarith (Num f) vs = f" |
"interpret_floatarith (Var n) vs = vs ! n"

lemma interpret_floatarith_divide:
  "interpret_floatarith (Mult a (Inverse b)) vs =
    (interpret_floatarith a vs) / (interpret_floatarith b vs)"
  unfolding divide_inverse interpret_floatarith.simps ..

lemma interpret_floatarith_diff:
  "interpret_floatarith (Add a (Minus b)) vs =
    (interpret_floatarith a vs) - (interpret_floatarith b vs)"
  unfolding interpret_floatarith.simps by simp

lemma interpret_floatarith_sin:
  "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) vs =
    sin (interpret_floatarith a vs)"
  unfolding sin_cos_eq interpret_floatarith.simps
    interpret_floatarith_divide interpret_floatarith_diff
  by auto

subsection "Implement approximation function"

fun lift_bin :: "(float interval) option \ (float interval) option \ (float interval \ float interval \ (float interval) option) \ (float interval) option" where
"lift_bin (Some ivl1) (Some ivl2) f = f ivl1 ivl2" |
"lift_bin a b f = None"

fun lift_bin' :: "(float interval) option \ (float interval) option \ (float interval \ float interval \ float interval) \ (float interval) option" where
"lift_bin' (Some ivl1) (Some ivl2) f = Some (f ivl1 ivl2)" |
"lift_bin' a b f = None"

fun lift_un :: "float interval option \ (float interval \ float interval option) \ float interval option" where
"lift_un (Some ivl) f = f ivl" |
"lift_un b f = None"

lemma lift_un_eq:\<comment> \<open>TODO\<close> "lift_un x f = Option.bind x f"
  by (cases x) auto

fun lift_un' :: "(float interval) option \ (float interval \ (float interval)) \ (float interval) option" where
"lift_un' (Some ivl1) f = Some (f ivl1)" |
"lift_un' b f = None"

definition bounded_by :: "real list \ (float interval) option list \ bool" where
  "bounded_by xs vs \ (\ i < length vs. case vs ! i of None \ True | Some ivl \ xs ! i \\<^sub>r ivl)"

lemma bounded_byE:
  assumes "bounded_by xs vs"
  shows "\ i. i < length vs \ case vs ! i of None \ True
         | Some ivl \<Rightarrow> xs ! i \<in>\<^sub>r ivl"
  using assms
  by (auto simp: bounded_by_def)

lemma bounded_by_update:
  assumes "bounded_by xs vs"
    and bnd: "xs ! i \\<^sub>r ivl"
  shows "bounded_by xs (vs[i := Some ivl])"
  using assms
  by (cases "i < length vs") (auto simp: bounded_by_def nth_list_update split: option.splits)

lemma bounded_by_None: "bounded_by xs (replicate (length xs) None)"
  unfolding bounded_by_def by auto

fun approx approx' :: "nat \ floatarith \ (float interval) option list \ (float interval) option"
  where
    "approx' prec a bs = (case (approx prec a bs) of Some ivl \ Some (round_interval prec ivl) | None \ None)" |
    "approx prec (Add a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (plus_float_interval prec)" |
    "approx prec (Minus a) bs = lift_un' (approx' prec a bs) uminus" |
    "approx prec (Mult a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (mult_float_interval prec)" |
    "approx prec (Inverse a) bs = lift_un (approx' prec a bs) (inverse_float_interval prec)" |
    "approx prec (Cos a) bs = lift_un' (approx' prec a bs) (cos_float_interval prec)" |
    "approx prec Pi bs = Some (pi_float_interval prec)" |
    "approx prec (Min a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (min_interval)" |
    "approx prec (Max a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (max_interval)" |
    "approx prec (Abs a) bs = lift_un' (approx' prec a bs) (abs_interval)" |
    "approx prec (Arctan a) bs = lift_un' (approx' prec a bs) (arctan_float_interval prec)" |
    "approx prec (Sqrt a) bs = lift_un' (approx' prec a bs) (sqrt_float_interval prec)" |
    "approx prec (Exp a) bs = lift_un' (approx' prec a bs) (exp_float_interval prec)" |
    "approx prec (Powr a b) bs = lift_bin (approx' prec a bs) (approx' prec b bs) (powr_float_interval prec)" |
    "approx prec (Ln a) bs = lift_un (approx' prec a bs) (ln_float_interval prec)" |
    "approx prec (Power a n) bs = lift_un' (approx' prec a bs) (power_float_interval prec n)" |
    "approx prec (Floor a) bs = lift_un' (approx' prec a bs) (floor_float_interval)" |
    "approx prec (Num f) bs = Some (interval_of f)" |
    "approx prec (Var i) bs = (if i < length bs then bs ! i else None)"

lemma approx_approx':
  assumes Pa: "\ivl. approx prec a vs = Some ivl \ interpret_floatarith a xs \\<^sub>r ivl"
    and approx': "approx' prec a vs = Some ivl"
  shows "interpret_floatarith a xs \\<^sub>r ivl"
proof -
  obtain ivl' where S: "approx prec a vs = Some ivl'" and ivl_def: "ivl = round_interval prec ivl'"
    using approx' unfolding approx'.simps by (cases "approx prec a vs") auto
  show ?thesis
    by (auto simp: ivl_def intro!: in_round_intervalI Pa[OF S])
qed

lemma approx:
  assumes "bounded_by xs vs"
    and "approx prec arith vs = Some ivl"
  shows "interpret_floatarith arith xs \\<^sub>r ivl"
  using \<open>approx prec arith vs = Some ivl\<close>
  using  \<open>bounded_by xs vs\<close>[THEN bounded_byE]
  by (induct arith arbitrary: ivl)
    (force split: option.splits if_splits
      intro!: plus_float_intervalI mult_float_intervalI uminus_in_float_intervalI
      inverse_float_interval_eqI abs_in_float_intervalI
      min_intervalI max_intervalI
      cos_float_intervalI
      arctan_float_intervalI pi_float_interval sqrt_float_intervalI exp_float_intervalI
      powr_float_interval_eqI ln_float_interval_eqI power_float_intervalI floor_float_intervalI
      intro: in_round_intervalI)+

datatype form = Bound floatarith floatarith floatarith form
              | Assign floatarith floatarith form
              | Less floatarith floatarith
              | LessEqual floatarith floatarith
              | AtLeastAtMost floatarith floatarith floatarith
              | Conj form form
              | Disj form form

fun interpret_form :: "form \ real list \ bool" where
"interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \ { interpret_floatarith a vs .. interpret_floatarith b vs } \ interpret_form f vs)" |
"interpret_form (Assign x a f) vs = (interpret_floatarith x vs = interpret_floatarith a vs \ interpret_form f vs)" |
"interpret_form (Less a b) vs = (interpret_floatarith a vs < interpret_floatarith b vs)" |
"interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \ interpret_floatarith b vs)" |
"interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \ { interpret_floatarith a vs .. interpret_floatarith b vs })" |
"interpret_form (Conj f g) vs \ interpret_form f vs \ interpret_form g vs" |
"interpret_form (Disj f g) vs \ interpret_form f vs \ interpret_form g vs"

fun approx_form' and approx_form :: "nat \ form \ (float interval) option list \ nat list \ bool" where
"approx_form' prec f 0 n ivl bs ss = approx_form prec f (bs[n := Some ivl]) ss" |
"approx_form' prec f (Suc s) n ivl bs ss =
  (let (ivl1, ivl2) = split_float_interval ivl
   in (if approx_form' prec f s n ivl1 bs ss then approx_form' prec f s n ivl2 bs ss else False))" |
"approx_form prec (Bound (Var n) a b f) bs ss =
   (case (approx prec a bs, approx prec b bs)
   of (Some ivl1, Some ivl2) \<Rightarrow> approx_form' prec f (ss ! n) n (sup ivl1 ivl2) bs ss
    | _ \<Rightarrow> False)" |
"approx_form prec (Assign (Var n) a f) bs ss =
   (case (approx prec a bs)
   of (Some ivl) \<Rightarrow> approx_form' prec f (ss ! n) n ivl bs ss
    | _ \<Rightarrow> False)" |
"approx_form prec (Less a b) bs ss =
   (case (approx prec a bs, approx prec b bs)
   of (Some ivl, Some ivl') \ float_plus_up prec (upper ivl) (-lower ivl') < 0
    | _ \<Rightarrow> False)" |
"approx_form prec (LessEqual a b) bs ss =
   (case (approx prec a bs, approx prec b bs)
   of (Some ivl, Some ivl') \ float_plus_up prec (upper ivl) (-lower ivl') \ 0
    | _ \<Rightarrow> False)" |
"approx_form prec (AtLeastAtMost x a b) bs ss =
   (case (approx prec x bs, approx prec a bs, approx prec b bs)
   of (Some ivlx, Some ivl, Some ivl') \
      float_plus_up prec (upper ivl) (-lower ivlx) \<le> 0 \<and>
      float_plus_up prec (upper ivlx) (-lower ivl') \ 0
    | _ \<Rightarrow> False)" |
"approx_form prec (Conj a b) bs ss \ approx_form prec a bs ss \ approx_form prec b bs ss" |
"approx_form prec (Disj a b) bs ss \ approx_form prec a bs ss \ approx_form prec b bs ss" |
"approx_form _ _ _ _ = False"

lemma lazy_conj: "(if A then B else False) = (A \ B)" by simp

lemma approx_form_approx_form':
  assumes "approx_form' prec f s n ivl bs ss"
    and "(x::real) \\<^sub>r ivl"
  obtains ivl' where "x \\<^sub>r ivl'"
    and "approx_form prec f (bs[n := Some ivl']) ss"
using assms proof (induct s arbitrary: ivl)
  case 0
  from this(1)[of ivl] this(2,3)
  show thesis by auto
next
  case (Suc s)
  then obtain ivl1 ivl2 where ivl_split: "split_float_interval ivl = (ivl1, ivl2)"
    by (fastforce dest: split_float_intervalD)
  from split_float_interval_realD[OF this \<open>x \<in>\<^sub>r ivl\<close>]
  consider "x \\<^sub>r ivl1" | "x \\<^sub>r ivl2"
    by atomize_elim
  then show thesis
  proof cases
    case *: 1
    from Suc.hyps[OF _ _ *] Suc.prems
    show ?thesis
      by (simp add: lazy_conj ivl_split split: prod.splits)
  next
    case *: 2
    from Suc.hyps[OF _ _ *] Suc.prems
    show ?thesis
      by (simp add: lazy_conj ivl_split split: prod.splits)
  qed
qed

lemma approx_form_aux:
  assumes "approx_form prec f vs ss"
    and "bounded_by xs vs"
  shows "interpret_form f xs"
using assms proof (induct f arbitrary: vs)
  case (Bound x a b f)
  then obtain n
    where x_eq: "x = Var n" by (cases x) auto

  with Bound.prems obtain ivl1 ivl2
    where l_eq: "approx prec a vs = Some ivl1"
    and u_eq: "approx prec b vs = Some ivl2"
    and approx_form': "approx_form' prec f (ss ! n) n (sup ivl1 ivl2) vs ss"
    by (cases "approx prec a vs", simp) (cases "approx prec b vs", auto)

  have "interpret_form f xs"
    if "xs ! n \ { interpret_floatarith a xs .. interpret_floatarith b xs }"
  proof -
    from approx[OF Bound.prems(2) l_eq] and approx[OF Bound.prems(2) u_eq] that
    have "xs ! n \\<^sub>r (sup ivl1 ivl2)"
      by (auto simp: set_of_eq sup_float_def max_def inf_float_def min_def)

    from approx_form_approx_form'[OF approx_form' this]
    obtain ivlx where bnds: "xs ! n \\<^sub>r ivlx"
      and approx_form: "approx_form prec f (vs[n := Some ivlx]) ss" .

    from \<open>bounded_by xs vs\<close> bnds have "bounded_by xs (vs[n := Some ivlx])"
      by (rule bounded_by_update)
    with Bound.hyps[OF approx_form] show ?thesis
      by blast
  qed
  thus ?case
    using interpret_form.simps x_eq and interpret_floatarith.simps by simp
next
  case (Assign x a f)
  then obtain n where x_eq: "x = Var n"
    by (cases x) auto

  with Assign.prems obtain ivl
    where bnd_eq: "approx prec a vs = Some ivl"
    and x_eq: "x = Var n"
    and approx_form': "approx_form' prec f (ss ! n) n ivl vs ss"
    by (cases "approx prec a vs") auto

  have "interpret_form f xs"
    if bnds: "xs ! n = interpret_floatarith a xs"
  proof -
    from approx[OF Assign.prems(2) bnd_eq] bnds
    have "xs ! n \\<^sub>r ivl" by auto
    from approx_form_approx_form'[OF approx_form' this]
    obtain ivlx where bnds: "xs ! n \\<^sub>r ivlx"
      and approx_form: "approx_form prec f (vs[n := Some ivlx]) ss" .

    from \<open>bounded_by xs vs\<close> bnds have "bounded_by xs (vs[n := Some ivlx])"
      by (rule bounded_by_update)
    with Assign.hyps[OF approx_form] show ?thesis
      by blast
  qed
  thus ?case
    using interpret_form.simps x_eq and interpret_floatarith.simps by simp
next
  case (Less a b)
  then obtain ivl ivl'
    where l_eq: "approx prec a vs = Some ivl"
      and u_eq: "approx prec b vs = Some ivl'"
      and inequality: "real_of_float (float_plus_up prec (upper ivl) (-lower ivl')) < 0"
    by (cases "approx prec a vs", auto, cases "approx prec b vs", auto)
  from le_less_trans[OF float_plus_up inequality]
    approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq]
  show ?case by (auto simp: set_of_eq)
next
  case (LessEqual a b)
  then obtain ivl ivl'
    where l_eq: "approx prec a vs = Some ivl"
      and u_eq: "approx prec b vs = Some ivl'"
      and inequality: "real_of_float (float_plus_up prec (upper ivl) (-lower ivl')) \ 0"
    by (cases "approx prec a vs", auto, cases "approx prec b vs", auto)
  from order_trans[OF float_plus_up inequality]
    approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq]
  show ?case by (auto simp: set_of_eq)
next
  case (AtLeastAtMost x a b)
  then obtain ivlx ivl ivl'
    where x_eq: "approx prec x vs = Some ivlx"
      and l_eq: "approx prec a vs = Some ivl"
      and u_eq: "approx prec b vs = Some ivl'"
    and inequality: "real_of_float (float_plus_up prec (upper ivl) (-lower ivlx)) \ 0"
      "real_of_float (float_plus_up prec (upper ivlx) (-lower ivl')) \ 0"
    by (cases "approx prec x vs", auto,
      cases "approx prec a vs", auto,
      cases "approx prec b vs", auto)
  from order_trans[OF float_plus_up inequality(1)] order_trans[OF float_plus_up inequality(2)]
    approx[OF AtLeastAtMost.prems(2) l_eq] approx[OF AtLeastAtMost.prems(2) u_eq] approx[OF AtLeastAtMost.prems(2) x_eq]
  show ?case by (auto simp: set_of_eq)
qed auto

lemma approx_form:
  assumes "n = length xs"
    and "approx_form prec f (replicate n None) ss"
  shows "interpret_form f xs"
  using approx_form_aux[OF _ bounded_by_None] assms by auto

subsection \<open>Implementing Taylor series expansion\<close>

fun isDERIV :: "nat \ floatarith \ real list \ bool" where
"isDERIV x (Add a b) vs = (isDERIV x a vs \ isDERIV x b vs)" |
"isDERIV x (Mult a b) vs = (isDERIV x a vs \ isDERIV x b vs)" |
"isDERIV x (Minus a) vs = isDERIV x a vs" |
"isDERIV x (Inverse a) vs = (isDERIV x a vs \ interpret_floatarith a vs \ 0)" |
"isDERIV x (Cos a) vs = isDERIV x a vs" |
"isDERIV x (Arctan a) vs = isDERIV x a vs" |
"isDERIV x (Min a b) vs = False" |
"isDERIV x (Max a b) vs = False" |
"isDERIV x (Abs a) vs = False" |
"isDERIV x Pi vs = True" |
"isDERIV x (Sqrt a) vs = (isDERIV x a vs \ interpret_floatarith a vs > 0)" |
"isDERIV x (Exp a) vs = isDERIV x a vs" |
"isDERIV x (Powr a b) vs =
    (isDERIV x a vs \<and> isDERIV x b vs \<and> interpret_floatarith a vs > 0)" |
"isDERIV x (Ln a) vs = (isDERIV x a vs \ interpret_floatarith a vs > 0)" |
"isDERIV x (Floor a) vs = (isDERIV x a vs \ interpret_floatarith a vs \ \)" |
"isDERIV x (Power a 0) vs = True" |
"isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" |
"isDERIV x (Num f) vs = True" |
"isDERIV x (Var n) vs = True"

fun DERIV_floatarith :: "nat \ floatarith \ floatarith" where
"DERIV_floatarith x (Add a b) = Add (DERIV_floatarith x a) (DERIV_floatarith x b)" |
"DERIV_floatarith x (Mult a b) = Add (Mult a (DERIV_floatarith x b)) (Mult (DERIV_floatarith x a) b)" |
"DERIV_floatarith x (Minus a) = Minus (DERIV_floatarith x a)" |
"DERIV_floatarith x (Inverse a) = Minus (Mult (DERIV_floatarith x a) (Inverse (Power a 2)))" |
"DERIV_floatarith x (Cos a) = Minus (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Arctan a) = Mult (Inverse (Add (Num 1) (Power a 2))) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Min a b) = Num 0" |
"DERIV_floatarith x (Max a b) = Num 0" |
"DERIV_floatarith x (Abs a) = Num 0" |
"DERIV_floatarith x Pi = Num 0" |
"DERIV_floatarith x (Sqrt a) = (Mult (Inverse (Mult (Sqrt a) (Num 2))) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Exp a) = Mult (Exp a) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Powr a b) =
   Mult (Powr a b) (Add (Mult (DERIV_floatarith x b) (Ln a)) (Mult (Mult (DERIV_floatarith x a) b) (Inverse a)))" |
"DERIV_floatarith x (Ln a) = Mult (Inverse a) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Power a 0) = Num 0" |
"DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Floor a) = Num 0" |
"DERIV_floatarith x (Num f) = Num 0" |
"DERIV_floatarith x (Var n) = (if x = n then Num 1 else Num 0)"

lemma DERIV_floatarith:
  assumes "n < length vs"
  assumes isDERIV: "isDERIV n f (vs[n := x])"
  shows "DERIV (\ x'. interpret_floatarith f (vs[n := x'])) x :>
               interpret_floatarith (DERIV_floatarith n f) (vs[n := x])"
   (is "DERIV (?i f) x :> _")
using isDERIV
proof (induct f arbitrary: x)
  case (Inverse a)
  thus ?case
    by (auto intro!: derivative_eq_intros simp add: algebra_simps power2_eq_square)
next
  case (Cos a)
  thus ?case
    by (auto intro!: derivative_eq_intros
           simp del: interpret_floatarith.simps(5)
           simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a])
next
  case (Power a n)
  thus ?case
    by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc)
next
  case (Floor a)
  thus ?case
    by (auto intro!: derivative_eq_intros DERIV_isCont floor_has_real_derivative)
next
  case (Ln a)
  thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse)
next
  case (Var i)
  thus ?case using \<open>n < length vs\<close> by auto
next
  case (Powr a b)
  note [derivative_intros] = has_real_derivative_powr'
  from Powr show ?case
    by (auto intro!: derivative_eq_intros simp: field_simps)
qed (auto intro!: derivative_eq_intros)

declare approx.simps[simp del]

fun isDERIV_approx :: "nat \ nat \ floatarith \ (float interval) option list \ bool" where
"isDERIV_approx prec x (Add a b) vs = (isDERIV_approx prec x a vs \ isDERIV_approx prec x b vs)" |
"isDERIV_approx prec x (Mult a b) vs = (isDERIV_approx prec x a vs \ isDERIV_approx prec x b vs)" |
"isDERIV_approx prec x (Minus a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Inverse a) vs =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some ivl \<Rightarrow> 0 < lower ivl \<or> upper ivl < 0 | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Cos a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Arctan a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Min a b) vs = False" |
"isDERIV_approx prec x (Max a b) vs = False" |
"isDERIV_approx prec x (Abs a) vs = False" |
"isDERIV_approx prec x Pi vs = True" |
"isDERIV_approx prec x (Sqrt a) vs =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some ivl \<Rightarrow> 0 < lower ivl | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Exp a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Powr a b) vs =
  (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs \<and> (case approx prec a vs of Some ivl \<Rightarrow> 0 < lower ivl | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Ln a) vs =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some ivl \<Rightarrow> 0 < lower ivl | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Power a 0) vs = True" |
"isDERIV_approx prec x (Floor a) vs =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some ivl \<Rightarrow> lower ivl > floor (upper ivl) \<and> upper ivl < ceiling (lower ivl) | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Num f) vs = True" |
"isDERIV_approx prec x (Var n) vs = True"

lemma isDERIV_approx:
  assumes "bounded_by xs vs"
    and isDERIV_approx: "isDERIV_approx prec x f vs"
  shows "isDERIV x f xs"
  using isDERIV_approx
proof (induct f)
  case (Inverse a)
  then obtain ivl where approx_Some: "approx prec a vs = Some ivl"
    and *: "0 < lower ivl \ upper ivl < 0"
    by (cases "approx prec a vs") auto
  with approx[OF \<open>bounded_by xs vs\<close> approx_Some]
  have "interpret_floatarith a xs \ 0" by (auto simp: set_of_eq)
  thus ?case using Inverse by auto
next
  case (Ln a)
  then obtain ivl where approx_Some: "approx prec a vs = Some ivl"
    and *: "0 < lower ivl"
    by (cases "approx prec a vs") auto
  with approx[OF \<open>bounded_by xs vs\<close> approx_Some]
  have "0 < interpret_floatarith a xs" by (auto simp: set_of_eq)
  thus ?case using Ln by auto
next
  case (Sqrt a)
  then obtain ivl where approx_Some: "approx prec a vs = Some ivl"
    and *: "0 < lower ivl"
    by (cases "approx prec a vs") auto
  with approx[OF \<open>bounded_by xs vs\<close> approx_Some]
  have "0 < interpret_floatarith a xs" by (auto simp: set_of_eq)
  thus ?case using Sqrt by auto
next
  case (Power a n)
  thus ?case by (cases n) auto
next
  case (Powr a b)
  from Powr obtain ivl1 where a: "approx prec a vs = Some ivl1"
    and pos: "0 < lower ivl1"
    by (cases "approx prec a vs") auto
  with approx[OF \<open>bounded_by xs vs\<close> a]
  have "0 < interpret_floatarith a xs" by (auto simp: set_of_eq)
  with Powr show ?case by auto
next
  case (Floor a)
  then obtain ivl where approx_Some: "approx prec a vs = Some ivl"
    and "real_of_int \real_of_float (upper ivl)\ < real_of_float (lower ivl)"
      "real_of_float (upper ivl) < real_of_int \real_of_float (lower ivl)\"
    and "isDERIV x a xs"
    by (cases "approx prec a vs") auto
  with approx[OF \<open>bounded_by xs vs\<close> approx_Some] le_floor_iff
  show ?case
    by (force elim!: Ints_cases simp: set_of_eq)
qed auto

lemma bounded_by_update_var:
  assumes "bounded_by xs vs"
    and "vs ! i = Some ivl"
    and bnd: "x \\<^sub>r ivl"
  shows "bounded_by (xs[i := x]) vs"
  using assms
  by (cases "i < length xs")
    (force simp: bounded_by_def nth_list_update list_update_beyond split: option.splits)+

lemma isDERIV_approx':
  assumes "bounded_by xs vs"
    and vs_x: "vs ! x = Some ivl"
    and X_in: "X \\<^sub>r ivl"
    and approx: "isDERIV_approx prec x f vs"
  shows "isDERIV x f (xs[x := X])"
proof -
  from bounded_by_update_var[OF \<open>bounded_by xs vs\<close> vs_x X_in] approx
  show ?thesis by (rule isDERIV_approx)
qed

lemma DERIV_approx:
  assumes "n < length xs"
    and bnd: "bounded_by xs vs"
    and isD: "isDERIV_approx prec n f vs"
    and app: "approx prec (DERIV_floatarith n f) vs = Some ivl" (is "approx _ ?D _ = _")
  shows "\(x::real). x \\<^sub>r ivl \
             DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x"
         (is "\ x. _ \ DERIV (?i f) _ :> _")
proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI)
  let "?i f" = "\x. interpret_floatarith f (xs[n := x])"
  from approx[OF bnd app]
  show "?i ?D (xs!n) \\<^sub>r ivl"
    using \<open>n < length xs\<close> by auto
  from DERIV_floatarith[OF \<open>n < length xs\<close>, of f "xs!n"] isDERIV_approx[OF bnd isD]
  show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))"
    by simp
qed

lemma lift_bin_aux:
  assumes lift_bin_Some: "lift_bin a b f = Some ivl"
  obtains ivl1 ivl2
  where "a = Some ivl1"
    and "b = Some ivl2"
    and "f ivl1 ivl2 = Some ivl"
  using assms by (cases a, simp, cases b, simp, auto)

fun approx_tse where
"approx_tse prec n 0 c k f bs = approx prec f bs" |
"approx_tse prec n (Suc s) c k f bs =
  (if isDERIV_approx prec n f bs then
    lift_bin (approx prec f (bs[n := Some (interval_of c)]))
             (approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs)
             (\<lambda>ivl1 ivl2. approx prec
                 (Add (Var 0)
                      (Mult (Inverse (Num (Float (int k) 0)))
                                 (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
                                       (Var (Suc 0))))) [Some ivl1, Some ivl2, bs!n])
  else approx prec f bs)"

lemma bounded_by_Cons:
  assumes bnd: "bounded_by xs vs"
    and x: "x \\<^sub>r ivl"
  shows "bounded_by (x#xs) ((Some ivl)#vs)"
proof -
  have "case ((Some ivl)#vs) ! i of Some ivl \ (x#xs)!i \\<^sub>r ivl | None \ True"
    if *: "i < length ((Some ivl)#vs)" for i
  proof (cases i)
    case 0
    with x show ?thesis by auto
  next
    case (Suc i)
    with * have "i < length vs" by auto
    from bnd[THEN bounded_byE, OF this]
    show ?thesis unfolding Suc nth_Cons_Suc .
  qed
  thus ?thesis
    by (auto simp add: bounded_by_def)
qed

lemma approx_tse_generic:
  assumes "bounded_by xs vs"
    and bnd_c: "bounded_by (xs[x := c]) vs"
    and "x < length vs" and "x < length xs"
    and bnd_x: "vs ! x = Some ivlx"
    and ate: "approx_tse prec x s c k f vs = Some ivl"
  shows "\ n. (\ m < n. \ (z::real) \ set_of (real_interval ivlx).
      DERIV (\<lambda> y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) z :>
            (interpret_floatarith ((DERIV_floatarith x ^^ (Suc m)) f) (xs[x := z])))
   \<and> (\<forall> (t::real) \<in> set_of (real_interval ivlx).  (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) *
                  interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := c]) *
                  (xs!x - c)^i) +
      inverse (real (\<Prod> j \<in> {k..<k+n}. j)) *
      interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) *
      (xs!x - c)^n \<in>\<^sub>r ivl)" (is "\<exists> n. ?taylor f k ivl n")
  using ate
proof (induct s arbitrary: k f ivl)
  case 0
  {
    fix t::real assume "t \\<^sub>r ivlx"
    note bounded_by_update_var[OF \<open>bounded_by xs vs\<close> bnd_x this]
    from approx[OF this 0[unfolded approx_tse.simps]]
    have "(interpret_floatarith f (xs[x := t])) \\<^sub>r ivl"
      by (auto simp add: algebra_simps)
  }
  thus ?case by (auto intro!: exI[of _ 0])
next
  case (Suc s)
  show ?case
  proof (cases "isDERIV_approx prec x f vs")
    case False
    note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]]
    {
      fix t::real assume "t \\<^sub>r ivlx"
      note bounded_by_update_var[OF \<open>bounded_by xs vs\<close> bnd_x this]
      from approx[OF this ap]
      have "(interpret_floatarith f (xs[x := t])) \\<^sub>r ivl"
        by (auto simp add: algebra_simps)
    }
    thus ?thesis by (auto intro!: exI[of _ 0])
  next
    case True
    with Suc.prems
    obtain ivl1 ivl2
      where a: "approx prec f (vs[x := Some (interval_of c)]) = Some ivl1"
        and ate: "approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs = Some ivl2"
        and final: "approx prec
          (Add (Var 0)
               (Mult (Inverse (Num (Float (int k) 0)))
                     (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
                           (Var (Suc 0))))) [Some ivl1, Some ivl2, vs!x] = Some ivl"
      by (auto elim!: lift_bin_aux)

    from bnd_c \<open>x < length xs\<close>
    have bnd: "bounded_by (xs[x:=c]) (vs[x:= Some (interval_of c)])"
      by (auto intro!: bounded_by_update)

    from approx[OF this a]
    have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := c]) \\<^sub>r ivl1"
              (is "?f 0 (real_of_float c) \ _")
      by auto

    have funpow_Suc[symmetric]: "(f ^^ Suc n) x = (f ^^ n) (f x)"
      for f :: "'a \ 'a" and n :: nat and x :: 'a
      by (induct n) auto
    from Suc.hyps[OF ate, unfolded this] obtain n
      where DERIV_hyp: "\m z. \ m < n ; (z::real) \\<^sub>r ivlx \ \
        DERIV (?f (Suc m)) z :> ?f (Suc (Suc m)) z"
      and hyp: "\t \ set_of (real_interval ivlx).
        (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) c * (xs!x - c)^i) +
          inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - c)^n \<in>\<^sub>r ivl2"
          (is "\ t \ _. ?X (Suc k) f n t \ _")
      by blast

    have DERIV: "DERIV (?f m) z :> ?f (Suc m) z"
      if "m < Suc n" and bnd_z: "z \\<^sub>r ivlx" for m and z::real
    proof (cases m)
      case 0
      with DERIV_floatarith[OF \<open>x < length xs\<close>
        isDERIV_approx'[OF \bounded_by xs vs\ bnd_x bnd_z True]]
      show ?thesis by simp
    next
      case (Suc m')
      hence "m' < n"
        using \<open>m < Suc n\<close> by auto
      from DERIV_hyp[OF this bnd_z] show ?thesis
        using Suc by simp
    qed

    have "\k i. k < i \ {k ..< i} = insert k {Suc k ..< i}" by auto
    hence prod_head_Suc: "\k i. \{k ..< k + Suc i} = k * \{Suc k ..< Suc k + i}"
      by auto
    have sum_move0: "\k F. sum F {0.. k. F (Suc k)) {0..
      unfolding sum.shift_bounds_Suc_ivl[symmetric]
      unfolding sum.atLeast_Suc_lessThan[OF zero_less_Suc] ..
    define C where "C = xs!x - c"

    {
      fix t::real assume t: "t \\<^sub>r ivlx"
      hence "bounded_by [xs!x] [vs!x]"
        using \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>]
        by (cases "vs!x", auto simp add: bounded_by_def)

      with hyp[THEN bspec, OF t] f_c
      have "bounded_by [?f 0 c, ?X (Suc k) f n t, xs!x] [Some ivl1, Some ivl2, vs!x]"
        by (auto intro!: bounded_by_Cons)
      from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]]
      have "?X (Suc k) f n t * (xs!x - real_of_float c) * inverse k + ?f 0 c \\<^sub>r ivl"
        by (auto simp add: algebra_simps)
      also have "?X (Suc k) f n t * (xs!x - real_of_float c) * inverse (real k) + ?f 0 c =
               (\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i c * (xs!x - c)^i) +
               inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
        unfolding funpow_Suc C_def[symmetric] sum_move0 prod_head_Suc
        by (auto simp add: algebra_simps)
          (simp only: mult.left_commute [of _ "inverse (real k)"] sum_distrib_left [symmetric])
      finally have "?T \\<^sub>r ivl" .
    }
    thus ?thesis using DERIV by blast
  qed
qed

lemma prod_fact: "real (\ {1..<1 + k}) = fact (k :: nat)"
  by (simp add: fact_prod atLeastLessThanSuc_atLeastAtMost)

lemma approx_tse:
  assumes "bounded_by xs vs"
    and bnd_x: "vs ! x = Some ivlx"
    and bnd_c: "real_of_float c \\<^sub>r ivlx"
    and "x < length vs" and "x < length xs"
    and ate: "approx_tse prec x s c 1 f vs = Some ivl"
  shows "interpret_floatarith f xs \\<^sub>r ivl"
proof -
  define F where [abs_def]: "F n z =
    interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])" for n z
  hence F0: "F 0 = (\ z. interpret_floatarith f (xs[x := z]))" by auto

  hence "bounded_by (xs[x := c]) vs" and "x < length vs" "x < length xs"
    using \<open>bounded_by xs vs\<close> bnd_x bnd_c \<open>x < length vs\<close> \<open>x < length xs\<close>
    by (auto intro!: bounded_by_update_var)

  from approx_tse_generic[OF \<open>bounded_by xs vs\<close> this bnd_x ate]
  obtain n
    where DERIV: "\ m z. m < n \ real_of_float (lower ivlx) \ z \ z \ real_of_float (upper ivlx) \ DERIV (F m) z :> F (Suc m) z"
    and hyp: "\ (t::real). t \\<^sub>r ivlx \
           (\<Sum> j = 0..<n. inverse(fact j) * F j c * (xs!x - c)^j) +
             inverse ((fact n)) * F n t * (xs!x - c)^n
             \<in>\<^sub>r ivl" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _")
    unfolding F_def atLeastAtMost_iff[symmetric] prod_fact
    by (auto simp: set_of_eq Ball_def)

  have bnd_xs: "xs ! x \\<^sub>r ivlx"
    using \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>] bnd_x by auto

  show ?thesis
  proof (cases n)
    case 0
    thus ?thesis
      using hyp[OF bnd_xs] unfolding F_def by auto
  next
    case (Suc n')
    show ?thesis
    proof (cases "xs ! x = c")
      case True
      from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis
        unfolding F_def Suc sum.atLeast_Suc_lessThan[OF zero_less_Suc] sum.shift_bounds_Suc_ivl
        by auto
    next
      case False
      have "lower ivlx \ real_of_float c" "real_of_float c \ upper ivlx" "lower ivlx \ xs!x" "xs!x \ upper ivlx"
        using Suc bnd_c \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>] bnd_x
        by (auto simp: set_of_eq)
      from Taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False]
      obtain t::real where t_bnd: "if xs ! x < c then xs ! x < t \ t < c else c < t \ t < xs ! x"
        and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) =
           (\<Sum>m = 0..<Suc n'. F m c / (fact m) * (xs ! x - c) ^ m) +
           F (Suc n') t / (fact (Suc n')) * (xs ! x - c) ^ Suc n'"
        unfolding atLeast0LessThan by blast

      from t_bnd bnd_xs bnd_c have *: "t \\<^sub>r ivlx"
        by (cases "xs ! x < c") (auto simp: set_of_eq)

      have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t"
        unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse)
      also have "\ \\<^sub>r ivl"
        using * by (rule hyp)
      finally show ?thesis
        by simp
    qed
  qed
qed

fun approx_tse_form' where
"approx_tse_form' prec t f 0 ivl cmp =
  (case approx_tse prec 0 t (mid ivl) 1 f [Some ivl]
     of Some ivl \<Rightarrow> cmp ivl | None \<Rightarrow> False)" |
"approx_tse_form' prec t f (Suc s) ivl cmp =
  (let (ivl1, ivl2) = split_float_interval ivl
   in (if approx_tse_form' prec t f s ivl1 cmp then
      approx_tse_form' prec t f s ivl2 cmp else False))"

lemma approx_tse_form':
  fixes x :: real
  assumes "approx_tse_form' prec t f s ivl cmp"
    and "x \\<^sub>r ivl"
  shows "\ivl' ivly. x \\<^sub>r ivl' \ ivl' \ ivl \ cmp ivly \
    approx_tse prec 0 t (mid ivl') 1 f [Some ivl'] = Some ivly"
  using assms
proof (induct s arbitrary: ivl)
  case 0
  then obtain ivly
    where *: "approx_tse prec 0 t (mid ivl) 1 f [Some ivl] = Some ivly"
    and **: "cmp ivly" by (auto elim!: case_optionE)
  with 0 show ?case by auto
next
  case (Suc s)
  let ?ivl1 = "fst (split_float_interval ivl)"
  let ?ivl2 = "snd (split_float_interval ivl)"
  from Suc.prems
  have l: "approx_tse_form' prec t f s ?ivl1 cmp"
    and u: "approx_tse_form' prec t f s ?ivl2 cmp"
    by (auto simp add: Let_def lazy_conj)

  have subintervals: "?ivl1 \ ivl" "?ivl2 \ ivl"
    using mid_le
    by (auto simp: less_eq_interval_def split_float_interval_bounds)

  from split_float_interval_realD[OF _ \<open>x \<in>\<^sub>r ivl\<close>] consider "x \<in>\<^sub>r ?ivl1" | "x \<in>\<^sub>r ?ivl2"
    by (auto simp: prod_eq_iff)
  then show ?case
  proof cases
    case 1
    from Suc.hyps[OF l this]
    obtain ivl' ivly where
      "x \\<^sub>r ivl' \ ivl' \ fst (split_float_interval ivl) \ cmp ivly \
        approx_tse prec 0 t (mid ivl') 1 f [Some ivl'] = Some ivly"
      by blast
    then show ?thesis
      using subintervals
      by (auto)
  next
    case 2
    from Suc.hyps[OF u this]
    obtain ivl' ivly where
      "x \\<^sub>r ivl' \ ivl' \ snd (split_float_interval ivl) \ cmp ivly \
        approx_tse prec 0 t (mid ivl') 1 f [Some ivl'] = Some ivly"
      by blast
    then show ?thesis
      using subintervals
      by (auto)
  qed
qed

lemma approx_tse_form'_less:
  fixes x :: real
  assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s ivl (\ ivl. 0 < lower ivl)"
    and x: "x \\<^sub>r ivl"
  shows "interpret_floatarith b [x] < interpret_floatarith a [x]"
proof -
  from approx_tse_form'[OF tse x]
  obtain ivl' ivly
    where x': "x \\<^sub>r ivl'"
    and "ivl' \ ivl" and "0 < lower ivly"
    and tse: "approx_tse prec 0 t (mid ivl') 1 (Add a (Minus b)) [Some ivl'] = Some ivly"
    by blast

  hence "bounded_by [x] [Some ivl']"
    by (auto simp add: bounded_by_def)
  from approx_tse[OF this _ _ _ _ tse, of ivl'] x' mid_le
  have "lower ivly \ interpret_floatarith a [x] - interpret_floatarith b [x]"
    by (auto simp: set_of_eq)
  from order_less_le_trans[OF _ this, of 0] \<open>0 < lower ivly\<close> show ?thesis
    by auto
qed

lemma approx_tse_form'_le:
  fixes x :: real
  assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s ivl (\ ivl. 0 \ lower ivl)"
    and x: "x \\<^sub>r ivl"
  shows "interpret_floatarith b [x] \ interpret_floatarith a [x]"
proof -
  from approx_tse_form'[OF tse x]
  obtain ivl' ivly
    where x': "x \\<^sub>r ivl'"
    and "ivl' \ ivl" and "0 \ lower ivly"
    and tse: "approx_tse prec 0 t (mid ivl') 1 (Add a (Minus b)) [Some (ivl')] = Some ivly"
    by blast

  hence "bounded_by [x] [Some ivl']" by (auto simp add: bounded_by_def)
  from approx_tse[OF this _ _ _ _ tse, of ivl'] x' mid_le
  have "lower ivly \ interpret_floatarith a [x] - interpret_floatarith b [x]"
    by (auto simp: set_of_eq)
  from order_trans[OF _ this, of 0] \<open>0 \<le> lower ivly\<close> show ?thesis
    by auto
qed

fun approx_tse_concl where
"approx_tse_concl prec t (Less lf rt) s ivl ivl' \
    approx_tse_form' prec t (Add rt (Minus lf)) s (sup ivl ivl') (\<lambda> ivl. 0 < lower ivl)" |
"approx_tse_concl prec t (LessEqual lf rt) s ivl ivl' \
    approx_tse_form' prec t (Add rt (Minus lf)) s (sup ivl ivl') (\<lambda> ivl. 0 \<le> lower ivl)" |
"approx_tse_concl prec t (AtLeastAtMost x lf rt) s ivl ivl' \
    (if approx_tse_form' prec t (Add x (Minus lf)) s (sup ivl ivl') (\<lambda> ivl. 0 \<le> lower ivl) then
      approx_tse_form' prec t (Add rt (Minus x)) s (sup ivl ivl') (\<lambda> ivl. 0 \<le> lower ivl) else False)" |
"approx_tse_concl prec t (Conj f g) s ivl ivl' \
    approx_tse_concl prec t f s ivl ivl' \ approx_tse_concl prec t g s ivl ivl'" |
"approx_tse_concl prec t (Disj f g) s ivl ivl' \
    approx_tse_concl prec t f s ivl ivl' \ approx_tse_concl prec t g s ivl ivl'" |
"approx_tse_concl _ _ _ _ _ _ \ False"

definition
  "approx_tse_form prec t s f =
    (case f of
      Bound x a b f \<Rightarrow>
        x = Var 0 \<and>
        (case (approx prec a [None], approx prec b [None]) of
          (Some ivl, Some ivl') \ approx_tse_concl prec t f s ivl ivl'
        | _ \<Rightarrow> False)
    | _ \<Rightarrow> False)"

lemma approx_tse_form:
  assumes "approx_tse_form prec t s f"
  shows "interpret_form f [x]"
proof (cases f)
  case f_def: (Bound i a b f')
  with assms obtain ivl ivl'
    where a: "approx prec a [None] = Some ivl"
    and b: "approx prec b [None] = Some ivl'"
    unfolding approx_tse_form_def by (auto elim!: case_optionE)

  from f_def assms have "i = Var 0"
    unfolding approx_tse_form_def by auto
  hence i: "interpret_floatarith i [x] = x" by auto

  {
    let ?f = "\z. interpret_floatarith z [x]"
    assume "?f i \ { ?f a .. ?f b }"
    with approx[OF _ a, of "[x]"] approx[OF _ b, of "[x]"]
    have bnd: "x \\<^sub>r sup ivl ivl'" unfolding bounded_by_def i
      by (auto simp: set_of_eq sup_float_def inf_float_def min_def max_def)

    have "interpret_form f' [x]"
      using assms[unfolded f_def]
    proof (induct f')
      case (Less lf rt)
      with a b
      have "approx_tse_form' prec t (Add rt (Minus lf)) s (sup ivl ivl') (\ ivl. 0 < lower ivl)"
        unfolding approx_tse_form_def by auto
      from approx_tse_form'_less[OF this bnd]
      show ?case using Less by auto
    next
      case (LessEqual lf rt)
      with f_def a b assms
      have "approx_tse_form' prec t (Add rt (Minus lf)) s (sup ivl ivl') (\ ivl. 0 \ lower ivl)"
        unfolding approx_tse_form_def by auto
      from approx_tse_form'_le[OF this bnd]
      show ?case using LessEqual by auto
    next
      case (AtLeastAtMost x lf rt)
      with f_def a b assms
      have "approx_tse_form' prec t (Add rt (Minus x)) s (sup ivl ivl') (\ ivl. 0 \ lower ivl)"
        and "approx_tse_form' prec t (Add x (Minus lf)) s (sup ivl ivl') (\ ivl. 0 \ lower ivl)"
        unfolding approx_tse_form_def lazy_conj by (auto split: if_split_asm)
      from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd]
      show ?case using AtLeastAtMost by auto
    qed (auto simp: f_def approx_tse_form_def elim!: case_optionE)
  }
  thus ?thesis unfolding f_def by auto
qed (insert assms, auto simp add: approx_tse_form_def)

text \<open>\<^term>\<open>approx_form_eval\<close> is only used for the {\tt value}-command.\<close>

fun approx_form_eval :: "nat \ form \ (float interval) option list \ (float interval) option list" where
"approx_form_eval prec (Bound (Var n) a b f) bs =
   (case (approx prec a bs, approx prec b bs)
   of (Some ivl1, Some ivl2) \<Rightarrow> approx_form_eval prec f (bs[n := Some (sup ivl1 ivl2)])
    | _ \<Rightarrow> bs)" |
"approx_form_eval prec (Assign (Var n) a f) bs =
   (case (approx prec a bs)
   of (Some ivl) \<Rightarrow> approx_form_eval prec f (bs[n := Some ivl])
    | _ \<Rightarrow> bs)" |
"approx_form_eval prec (Less a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
"approx_form_eval prec (LessEqual a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
"approx_form_eval prec (AtLeastAtMost x a b) bs =
   bs @ [approx prec x bs, approx prec a bs, approx prec b bs]" |
"approx_form_eval _ _ bs = bs"

subsection \<open>Implement proof method \texttt{approximation}\<close>

ML \<open>
val _ = \<comment> \<open>Trusting the oracle \<close>@{oracle_name "holds_by_evaluation"}
signature APPROXIMATION_COMPUTATION = sig
val approx_bool: Proof.context -> term -> term
val approx_arith: Proof.context -> term -> term
val approx_form_eval: Proof.context -> term -> term
val approx_conv: Proof.context -> conv
end

structure Approximation_Computation : APPROXIMATION_COMPUTATION = struct

  fun approx_conv ctxt ct =
    @{computation_check
      terms: Trueprop
        "0 :: nat" "1 :: nat" "2 :: nat" "3 :: nat" Suc
          "(+)::nat\nat\nat" "(-)::nat\nat\nat" "(*)::nat\nat\nat"
        "0 :: int" "1 :: int" "2 :: int" "3 :: int" "-1 :: int"
          "(+)::int\int\int" "(-)::int\int\int" "(*)::int\int\int" "uminus::int\int"
        "replicate :: _ \ (float interval) option \ _"
        "interval_of::float\float interval"
        approx'
        approx_form
        approx_tse_form
        approx_form_eval
      datatypes: bool
        int
        nat
        integer
        "nat list"
        float
        "(float interval) option list"
        floatarith
        form
    } ctxt ct

  fun term_of_bool true = \<^Const>\<open>True\<close>
    | term_of_bool false = \<^Const>\<open>False\<close>;

  val mk_int = HOLogic.mk_number \<^Type>\<open>int\<close> o @{code integer_of_int};

  fun term_of_float (@{code Float} (k, l)) =
    \<^Const>\<open>Float for \<open>mk_int k\<close> \<open>mk_int l\<close>\<close>;

  fun term_of_float_interval x = @{term "Interval::_\float interval"} $
    HOLogic.mk_prod
      (apply2 term_of_float (@{code lowerF} x, @{code upperF} x))

  fun term_of_float_interval_option NONE = \<^Const>\<open>None \<^typ>\<open>float interval option\<close>\<close>
    | term_of_float_interval_option (SOME ff) =
        \<^Const>\<open>Some \<^typ>\<open>float interval\<close> for \<open>term_of_float_interval ff\<close>\<close>;

  val term_of_float_interval_option_list =
    HOLogic.mk_list \<^typ>\<open>float interval option\<close> o map term_of_float_interval_option;

  val approx_bool = @{computation bool}
    (fn _ => fn x => case x of SOME b => term_of_bool b
      | NONE => error "Computation approx_bool failed.")
  val approx_arith = @{computation "float interval option"}
    (fn _ => fn x => case x of SOME ffo => term_of_float_interval_option ffo
      | NONE => error "Computation approx_arith failed.")
  val approx_form_eval = @{computation "float interval option list"}
    (fn _ => fn x => case x of SOME ffos => term_of_float_interval_option_list ffos
      | NONE => error "Computation approx_form_eval failed.")

end
\<close>

lemma intervalE: "a \ x \ x \ b \ \ x \ { a .. b } \ P\ \ P"
  by auto

lemma meta_eqE: "x \ a \ \ x = a \ P\ \ P"
  by auto

named_theorems approximation_preproc

lemma approximation_preproc_floatarith[approximation_preproc]:
  "0 = real_of_float 0"
  "1 = real_of_float 1"
  "0 = Float 0 0"
  "1 = Float 1 0"
  "numeral a = Float (numeral a) 0"
  "numeral a = real_of_float (numeral a)"
  "x - y = x + - y"
  "x / y = x * inverse y"
  "ceiling x = - floor (- x)"
  "log x y = ln y * inverse (ln x)"
  "sin x = cos (pi / 2 - x)"
  "tan x = sin x / cos x"
  by (simp_all add: inverse_eq_divide ceiling_def log_def sin_cos_eq tan_def real_of_float_eq)

lemma approximation_preproc_int[approximation_preproc]:
  "real_of_int 0 = real_of_float 0"
  "real_of_int 1 = real_of_float 1"
  "real_of_int (i + j) = real_of_int i + real_of_int j"
  "real_of_int (- i) = - real_of_int i"
  "real_of_int (i - j) = real_of_int i - real_of_int j"
  "real_of_int (i * j) = real_of_int i * real_of_int j"
  "real_of_int (i div j) = real_of_int (floor (real_of_int i / real_of_int j))"
  "real_of_int (min i j) = min (real_of_int i) (real_of_int j)"
  "real_of_int (max i j) = max (real_of_int i) (real_of_int j)"
  "real_of_int (abs i) = abs (real_of_int i)"
  "real_of_int (i ^ n) = (real_of_int i) ^ n"
  "real_of_int (numeral a) = real_of_float (numeral a)"
  "i mod j = i - i div j * j"
  "i = j \ real_of_int i = real_of_int j"
  "i \ j \ real_of_int i \ real_of_int j"
  "i < j \ real_of_int i < real_of_int j"
  "i \ {j .. k} \ real_of_int i \ {real_of_int j .. real_of_int k}"
  by (simp_all add: floor_divide_of_int_eq minus_div_mult_eq_mod [symmetric])

lemma approximation_preproc_nat[approximation_preproc]:
  "real 0 = real_of_float 0"
  "real 1 = real_of_float 1"
  "real (i + j) = real i + real j"
  "real (i - j) = max (real i - real j) 0"
  "real (i * j) = real i * real j"
  "real (i div j) = real_of_int (floor (real i / real j))"
  "real (min i j) = min (real i) (real j)"
  "real (max i j) = max (real i) (real j)"
  "real (i ^ n) = (real i) ^ n"
  "real (numeral a) = real_of_float (numeral a)"
  "i mod j = i - i div j * j"
  "n = m \ real n = real m"
  "n \ m \ real n \ real m"
  "n < m \ real n < real m"
  "n \ {m .. l} \ real n \ {real m .. real l}"
  by (simp_all add: real_div_nat_eq_floor_of_divide minus_div_mult_eq_mod [symmetric])

ML_file \<open>approximation.ML\<close>

method_setup approximation = \<open>
  let
    val free =
      Args.context -- Args.term >> (fn (_, Free (n, _)) => n | (ctxt, t) =>
        error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
  in
    Scan.lift Parse.nat --
    Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon)
      |-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) [] --
    Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon) |--
    (free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat)) >>
    (fn ((prec, splitting), taylor) => fn ctxt =>
      SIMPLE_METHOD' (Approximation.approximation_tac prec splitting taylor ctxt))
  end
\<close> "real number approximation"

section "Quickcheck Generator"

lemma approximation_preproc_push_neg[approximation_preproc]:
  fixes a b::real
  shows
    "\ (a < b) \ b \ a"
    "\ (a \ b) \ b < a"
    "\ (a = b) \ b < a \ a < b"
    "\ (p \ q) \ \ p \ \ q"
    "\ (p \ q) \ \ p \ \ q"
    "\ \ q \ q"
  by auto

ML_file \<open>approximation_generator.ML\<close>
setup "Approximation_Generator.setup"

section "Avoid pollution of name space"

bundle floatarith_syntax
begin
notation Add (\<open>Add\<close>)
notation Minus (\<open>Minus\<close>)
notation Mult (\<open>Mult\<close>)
notation Inverse (\<open>Inverse\<close>)
notation Cos (\<open>Cos\<close>)
notation Arctan (\<open>Arctan\<close>)
notation Abs (\<open>Abs\<close>)
notation Max (\<open>Max\<close>)
notation Min (\<open>Min\<close>)
notation Pi (\<open>Pi\<close>)
notation Sqrt (\<open>Sqrt\<close>)
notation Exp (\<open>Exp\<close>)
notation Powr (\<open>Powr\<close>)
notation Ln (\<open>Ln\<close>)
notation Power (\<open>Power\<close>)
notation Floor (\<open>Floor\<close>)
notation Var (\<open>Var\<close>)
notation Num (\<open>Num\<close>)
end

hide_const (open)
  Add
  Minus
  Mult
  Inverse
  Cos
  Arctan
  Abs
  Max
  Min
  Pi
  Sqrt
  Exp
  Powr
  Ln
  Power
  Floor
  Var
  Num

end

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