needs "lib/ssrbool-compiled.hl";;
needs "lib/ssrnat-compiled.hl";;
needs "arith/interval_arith.hl";;
needs "Multivariate/realanalysis.ml";;
open Interval_arith;;
prioritize_real();;
let derivative = new_definition `derivative f = \y. @d. (f has_real_derivative d) (atreal y)`;;
let nth_derivative = new_definition `nth_derivative n f = iter n derivative f`;;
let nth_differentiable = define `(nth_differentiable 0 f x <=> f real_continuous atreal x) /\
	(nth_differentiable (SUC n) f x <=> nth_differentiable n f x /\ 
			nth_derivative n f real_differentiable atreal x)`;;
let nth_differentiable_on = new_definition `nth_differentiable_on n s f <=>
	!x. x IN s ==> nth_differentiable n f x`;;
let nth_differentiable_on_int = new_definition `nth_differentiable_on_int n int f <=>
	!x. interval_arith x int ==> nth_differentiable n f x`;;
let nth_diff_weak = new_definition `nth_diff_weak n f x <=> f real_continuous atreal x /\ 
	?F. F 0 = f /\ !i. i < n ==> (F i has_real_derivative F (SUC i) x) (atreal x)`;;
let nth_diff_strong = new_definition `nth_diff_strong n f x <=> 
	?s. real_open s /\ x IN s /\ nth_differentiable_on n s f`;;
let nth_diff_strong_int = new_definition `nth_diff_strong_int n int f <=>
	!x. interval_arith x int ==> nth_diff_strong n f x`;;
(* Section NthDerivatives *)
begin_section "NthDerivatives";;

(* Lemma has_derivative_cond *)
let has_derivative_cond = section_proof ["f";"x"]
`(?d. (f has_real_derivative d) (atreal x)) ==>
	(f has_real_derivative (derivative f x)) (atreal x)`
[
   (BETA_TAC THEN (case THEN (move ["d"])) THEN (move ["df"]));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`derivative f x = d`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) (exact_tac));
   (((((use_arg_then "derivative")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (((use_arg_then "SELECT_UNIQUE") (thm_tac apply_tac)) THEN (simp_tac) THEN (move ["y"])) THEN ((THENL) (split_tac) [(move ["df2"]); ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)))]));
   ((((fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_DERIVATIVE_UNIQUE_ATREAL") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)) THEN (done_tac));
];;

(* Lemma has_derivative_alt *)
let has_derivative_alt = section_proof ["f";"x"]
`f real_differentiable atreal x ==>
	(f has_real_derivative (derivative f x)) (atreal x)`
[
   (((((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "has_derivative_cond") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (done_tac));
];;

(* Lemma derivative_unique *)
let derivative_unique = section_proof ["f";"f'";"x"]
`(f has_real_derivative f') (atreal x) ==> derivative f x = f'`
[
   ((BETA_TAC THEN (move ["df"])) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_DERIVATIVE_UNIQUE_ATREAL") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_cond")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN ((use_arg_then "f'") (term_tac exists_tac)) THEN (done_tac));
];;

(* Lemma derivative_unique_on *)
let derivative_unique_on = section_proof ["s";"f";"f'"]
`(!x. x IN s ==> (f has_real_derivative f' x) (atreal x)) ==>
	(!x. x IN s ==> f' x = derivative f x)`
[
   (BETA_TAC THEN (move ["df"]) THEN (move ["x"]) THEN (move ["xs"]));
   (((((fun arg_tac -> (fun arg_tac -> (use_arg_then "derivative_unique") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`f' x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma has_derivative_lemma *)
let has_derivative_lemma = section_proof ["f";"f'";"x"]
`f real_differentiable atreal x /\
	derivative f x = f' ==>
	(f has_real_derivative f') (atreal x)`
[
   ((BETA_TAC THEN (case THEN ((move ["diff"]) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))))) THEN (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_derivative0 *)
let nth_derivative0 = section_proof ["f"]
`nth_derivative 0 f = f`
[
   (((((use_arg_then "nth_derivative")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL iter)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_derivativeS *)
let nth_derivativeS = section_proof ["n";"f"]
`nth_derivative (SUC n) f = derivative (nth_derivative n f)`
[
   (((repeat_tactic 1 9 (((use_arg_then "nth_derivative")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "iterS")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_Sderivative *)
let nth_Sderivative = section_proof ["n";"f"]
`nth_derivative (SUC n) f = nth_derivative n (derivative f)`
[
   (((((use_arg_then "nth_derivative")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iterSr")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_derivative")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma nth_derivative1 *)
let nth_derivative1 = section_proof ["f"]
`nth_derivative 1 f = derivative f`
[
   (((((use_arg_then "ONE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_derivative2 *)
let nth_derivative2 = section_proof ["f"]
`nth_derivative 2 f = derivative (derivative f)`
[
   (((((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `2 = SUC(SUC 0)`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "iterS")(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL iter)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_derivative_add *)
let nth_derivative_add = section_proof ["n";"m";"f"]
`nth_derivative n (nth_derivative m f) = nth_derivative (n + m) f`
[
   (((repeat_tactic 1 9 (((use_arg_then "nth_derivative")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "iter_add")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_diff_continuous *)
let nth_diff_continuous = section_proof ["n";"f";"x"]
`nth_differentiable n f x ==>
	f real_continuous atreal x`
[
   (((THENL) (((use_arg_then "n") (disch_tac [])) THEN (clear_assumption "n") THEN elim) [ALL_TAC; ((move ["n"]) THEN (move ["IHn"]))]) THEN ((((use_arg_then "nth_differentiable")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))));
   ((BETA_TAC THEN (case THEN (DISCH_THEN (fun snd_th -> (use_arg_then "IHn") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (simp_tac)) THEN (done_tac));
];;

(* Lemma nth_differentiable_cond *)
let nth_differentiable_cond = section_proof ["n";"f";"x"]
`nth_differentiable n f x ==>
   !i. i < n ==> (nth_derivative i f has_real_derivative (nth_derivative (SUC i) f x)) (atreal x)`
[
   ((THENL_FIRST) ((THENL) (((use_arg_then "n") (disch_tac [])) THEN (clear_assumption "n") THEN elim) [ALL_TAC; ((move ["n"]) THEN (move ["IHn"]))]) ((((use_arg_then "ltn0")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((((use_arg_then "nth_differentiable")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN ((DISCH_THEN (fun snd_th -> (use_arg_then "IHn") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["df_n"]) THEN (move ["dfn"]))) THEN (move ["i"]));
   (((((use_arg_then "ltE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "leqSS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "leq_eqVlt")(thm_tac (new_rewrite [] []))))) THEN ((THENL) case [(((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))); ((DISCH_THEN (fun snd_th -> (use_arg_then "df_n") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN ((TRY done_tac)))]));
   (((((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_differentiable_on_cond *)
let nth_differentiable_on_cond = section_proof ["n";"s";"f"]
`nth_differentiable_on n s f ==>
 !x. x IN s ==> 
  !i. i < n ==> (nth_derivative i f has_real_derivative (nth_derivative (SUC i) f x)) (atreal x)`
[
   ((((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (move ["cond"]) THEN (move ["x"]));
   (((DISCH_THEN (fun snd_th -> (use_arg_then "cond") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (DISCH_THEN (fun snd_th -> (use_arg_then "nth_differentiable_cond") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (done_tac));
];;

(* Lemma nth_differentiable_eq *)
let nth_differentiable_eq = section_proof ["n";"f";"x"]
`nth_differentiable n f x <=>
  f real_continuous atreal x /\
  !i. i < n ==> (nth_derivative i f has_real_derivative (nth_derivative (SUC i) f x)) (atreal x)`
[
   ((THENL) (split_tac) [(move ["dn_f"]); (case THEN (move ["f_cont"]))]);
   (((((fun arg_tac -> (use_arg_then "nth_diff_continuous") (fun fst_arg -> (use_arg_then "dn_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_differentiable_cond") (disch_tac [])) THEN (clear_assumption "nth_differentiable_cond") THEN (exact_tac)) THEN (done_tac));
   (((THENL) (((use_arg_then "n") (disch_tac [])) THEN (clear_assumption "n") THEN elim) [ALL_TAC; ((move ["n"]) THEN (move ["IHn"]))]) THEN (((((use_arg_then "nth_differentiable")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "ltE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "leqSS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "leq_eqVlt")(thm_tac (new_rewrite [] []))))) THEN (move ["cond"])));
   ((THENL_FIRST) ((((use_arg_then "IHn")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((BETA_TAC THEN (move ["i"]) THEN (move ["i_lt_n"])) THEN (((use_arg_then "cond") (disch_tac [])) THEN (clear_assumption "cond") THEN (DISCH_THEN apply_tac)) THEN (done_tac)));
   ((((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> arg_tac (Arg_term (`nth_derivative (SUC n) f x`))) (term_tac exists_tac)));
   (((use_arg_then "cond") (disch_tac [])) THEN (clear_assumption "cond") THEN (exact_tac));
];;

(* Lemma nth_differentiable_on_int2 *)
let nth_differentiable_on_int2 = section_proof ["f";"int"]
`nth_differentiable_on_int 2 int f ==>
	?f' f''. f' = derivative f /\ f'' = nth_derivative 2 f /\
		!x. interval_arith x int ==> (f has_real_derivative f' x) (atreal x) /\
					(f' has_real_derivative f'' x) (atreal x)`
[
   ((((use_arg_then "nth_differentiable_on_int")(thm_tac (new_rewrite [] [])))) THEN (move ["h"]));
   ((((fun arg_tac -> arg_tac (Arg_term (`derivative f`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`nth_derivative 2 f`))) (term_tac exists_tac))) THEN (simp_tac) THEN (move ["x"]) THEN (move ["ineq"]));
   ((((use_arg_then "nth_derivative1")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then "nth_derivative0") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (((use_arg_then "ONE")(thm_tac (new_rewrite [1] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `2 = SUC 1`)))(thm_tac (new_rewrite [] [])))));
   (((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then "nth_differentiable_cond") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`2`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "h")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma nth_mth_diff *)
let nth_mth_diff = section_proof ["n";"m";"f";"x"]
`n <= m ==> nth_differentiable m f x ==> nth_differentiable n f x`
[
   ((BETA_TAC THEN (move ["n_le_m"])) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (move ["cond"])))) THEN ((((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (move ["i"]) THEN (move ["i_lt"])));
   ((((use_arg_then "cond") (disch_tac [])) THEN (clear_assumption "cond") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "n_le_m") (disch_tac [])) THEN (clear_assumption "n_le_m") THEN ((use_arg_then "ltn_leq_trans") (disch_tac [])) THEN (clear_assumption "ltn_leq_trans") THEN (DISCH_THEN apply_tac)) THEN (done_tac));
];;

(* Lemma nth_differentiable1 *)
let nth_differentiable1 = section_proof ["f";"x"]
`nth_differentiable 1 f x <=> f real_differentiable atreal x`
[
   (((((use_arg_then "ONE")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL nth_differentiable)))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN ((THENL) (split_tac) [(((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))); (move ["df"])]));
   ((((use_arg_then "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_diff_imp_diff *)
let nth_diff_imp_diff = section_proof ["n";"f";"x"]
`0 < n ==> nth_differentiable n f x ==>
	f real_differentiable atreal x`
[
   ((((((use_arg_then "ltE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["ineq"]) THEN (move ["df"])) THEN (((use_arg_then "nth_differentiable1")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_mth_diff") (disch_tac [])) THEN (clear_assumption "nth_mth_diff") THEN (DISCH_THEN apply_tac)) THEN ((use_arg_then "n") (term_tac exists_tac)) THEN (done_tac));
];;

(* Lemma nth_derivative_continuous *)
let nth_derivative_continuous = section_proof ["n";"f";"x";"i"]
`nth_differentiable n f x ==> i < n ==>
	nth_derivative i f real_continuous atreal x`
[
   ((((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (move ["df"]));
   (((DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (move ["cond"])) THEN (((use_arg_then "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL") THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> arg_tac (Arg_term (`nth_derivative (SUC i) f x`))) (term_tac exists_tac)) THEN (done_tac));
];;

(* Lemma ith_derivative_differentiable *)
let ith_derivative_differentiable = section_proof ["i";"n";"f";"x"]
`nth_differentiable n f x ==> i < n ==>
	nth_differentiable (n - i) (nth_derivative i f) x`
[
   (BETA_TAC THEN (move ["dnf"]));
   ((((use_arg_then "dnf") (disch_tac [])) THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (move ["cond"]) THEN (move ["i_lt_n"])));
   (((((fun arg_tac -> (use_arg_then "nth_derivative_continuous") (fun fst_arg -> (use_arg_then "dnf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["j"]) THEN (move ["j_lt_ni"]));
   (((repeat_tactic 1 9 (((use_arg_then "nth_derivative_add")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "addSn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "cond")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "j_lt_ni") (disch_tac [])) THEN (clear_assumption "j_lt_ni") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma nth_diff_strong_imp_diff *)
let nth_diff_strong_imp_diff = section_proof ["n";"f";"x"]
`nth_diff_strong n f x ==>	nth_differentiable n f x`
[
   (((((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["xs"]))) THEN ((((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (move ["h"])));
   (((use_arg_then "h") (disch_tac [])) THEN (clear_assumption "h") THEN (exact_tac));
];;

(* Section DerivativeArith *)
begin_section "DerivativeArith";;

(* Section ElementaryDerivatives *)
begin_section "ElementaryDerivatives";;

(* Lemma derivative_x *)
let derivative_x = section_proof []
`derivative (\x. x) = (\x. &1)`
[
   ((((use_arg_then "eq_ext")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (simp_tac));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ID")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma derivative_const *)
let derivative_const = section_proof ["c"]
`derivative (\x. c) = (\x. &0)`
[
   ((((use_arg_then "eq_ext")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (simp_tac));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_CONST")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma derivative_inv *)
let derivative_inv = section_proof ["x"]
`~(x = &0) ==> derivative inv x = -- inv (x * x)`
[
   (BETA_TAC THEN (move ["xn0"]));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_INV_BASIC")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_atn *)
let derivative_atn = section_proof []
`derivative atn = (\x. inv (&1 + x * x))`
[
   (((((use_arg_then "eq_ext")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["x"]) THEN (simp_tac));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ATN")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma derivative_exp *)
let derivative_exp = section_proof []
`derivative exp = exp`
[
   ((((use_arg_then "eq_ext")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (simp_tac));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_EXP")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma derivative_acs *)
let derivative_acs = section_proof ["x"]
`abs x < &1 ==> derivative acs x = --inv(sqrt(&1 - x * x))`
[
   (BETA_TAC THEN (move ["x_ineq"]));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ACS")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_sqrt *)
let derivative_sqrt = section_proof ["x"]
`&0 < x ==> derivative sqrt x = inv (&2 * sqrt x)`
[
   (BETA_TAC THEN (move ["xg0"]));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_SQRT")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma derivative_composition *)
let derivative_composition = section_proof ["f";"g";"x"]
`f real_differentiable atreal x ==>
	g real_differentiable atreal (f x) ==>
	derivative (\x. g (f x)) x = derivative f x * derivative g (f x)`
[
   ((BETA_TAC THEN (DISCH_THEN (fun snd_th -> (use_arg_then "has_derivative_alt") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["df"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "has_derivative_alt") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["dg"])) THEN (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)));
   ((THENL_FIRST) (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL HAS_REAL_DERIVATIVE_CHAIN))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`derivative f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`derivative g`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`\y. y = f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "g") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN (ANTS_TAC)) ((BETA_TAC THEN (move ["y"]) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   ((BETA_TAC THEN (case THEN (move ["_"]))) THEN (DISCH_THEN apply_tac) THEN (done_tac));
];;

(* Section ElementaryCompose *)
begin_section "ElementaryCompose";;

(* Lemma REAL_DIFFERENTIABLE_AT_INV *)
let REAL_DIFFERENTIABLE_AT_INV = section_proof ["x"]
`~(x = &0) ==> inv real_differentiable atreal x`
[
   (BETA_TAC THEN (move ["xn0"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_DIFFERENTIABLE_INV_ATREAL") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`(\x. x:real)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (((simp_tac) THEN (((use_arg_then "REAL_DIFFERENTIABLE_ID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xn0")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (done_tac));
];;
(add_section_var (mk_var ("f", (`:real->real`))));;
(add_section_var (mk_var ("x", (`:real`))));;
(add_section_hyp "df" (`f real_differentiable atreal x`));;

(* Lemma derivative_compose_atn *)
let derivative_compose_atn = section_proof []
`(\x. atn (f x)) real_differentiable atreal x /\
	derivative (\x. atn (f x)) x = derivative f x / (&1 + f x * f x)`
[
   (split_tac);
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`atn`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "REAL_DIFFERENTIABLE_AT_ATN")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_AT_ATN")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((((use_arg_then "derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_compose_exp *)
let derivative_compose_exp = section_proof []
`(\x. exp (f x)) real_differentiable atreal x /\
	derivative (\x. exp (f x)) x = exp (f x) * derivative f x`
[
   (split_tac);
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`exp`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "REAL_DIFFERENTIABLE_AT_EXP")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_AT_EXP")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((((use_arg_then "derivative_exp")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_compose_inv *)
let derivative_compose_inv = section_proof []
`~(f x = &0) ==>
	(\x. inv (f x)) real_differentiable atreal x /\
	derivative (\x. inv (f x)) x = -- inv (f x * f x) * derivative f x`
[
   ((THENL_FIRST) ((BETA_TAC THEN (move ["fn0"])) THEN (split_tac)) ((((use_arg_then "REAL_DIFFERENTIABLE_INV_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_INV_ATREAL") THEN (DISCH_THEN apply_tac)) THEN (done_tac)));
   (((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "derivative_inv")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_AT_INV")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_compose_sqrt *)
let derivative_compose_sqrt = section_proof []
`&0 < f x ==>
	(\x. sqrt (f x)) real_differentiable atreal x /\
	derivative (\x. sqrt (f x)) x = derivative f x / (&2 * sqrt (f x))`
[
   ((BETA_TAC THEN (move ["f_pos"])) THEN (split_tac));
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`sqrt`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_AT_SQRT") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_AT_SQRT") THEN (DISCH_THEN apply_tac)) THEN (done_tac));
   ((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_AT_SQRT")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((((use_arg_then "derivative_sqrt")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_compose_acs *)
let derivative_compose_acs = section_proof []
`abs (f x) < &1 ==>
	(\x. acs (f x)) real_differentiable atreal x /\
	derivative (\x. acs (f x)) x = -- (derivative f x / sqrt (&1 - f x * f x))`
[
   ((BETA_TAC THEN (move ["f_abs"])) THEN (split_tac));
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`acs`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_AT_ACS")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_AT_ACS")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((((use_arg_then "derivative_acs")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_MUL_RNEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section ElementaryCompose *)
let REAL_DIFFERENTIABLE_AT_INV = finalize_theorem REAL_DIFFERENTIABLE_AT_INV;;
let derivative_compose_atn = finalize_theorem derivative_compose_atn;;
let derivative_compose_exp = finalize_theorem derivative_compose_exp;;
let derivative_compose_inv = finalize_theorem derivative_compose_inv;;
let derivative_compose_sqrt = finalize_theorem derivative_compose_sqrt;;
let derivative_compose_acs = finalize_theorem derivative_compose_acs;;
end_section "ElementaryCompose";;

(* Finalization of the section ElementaryDerivatives *)
let derivative_x = finalize_theorem derivative_x;;
let derivative_const = finalize_theorem derivative_const;;
let derivative_inv = finalize_theorem derivative_inv;;
let derivative_atn = finalize_theorem derivative_atn;;
let derivative_exp = finalize_theorem derivative_exp;;
let derivative_acs = finalize_theorem derivative_acs;;
let derivative_sqrt = finalize_theorem derivative_sqrt;;
let derivative_composition = finalize_theorem derivative_composition;;
let REAL_DIFFERENTIABLE_AT_INV = finalize_theorem REAL_DIFFERENTIABLE_AT_INV;;
let derivative_compose_atn = finalize_theorem derivative_compose_atn;;
let derivative_compose_exp = finalize_theorem derivative_compose_exp;;
let derivative_compose_inv = finalize_theorem derivative_compose_inv;;
let derivative_compose_sqrt = finalize_theorem derivative_compose_sqrt;;
let derivative_compose_acs = finalize_theorem derivative_compose_acs;;
end_section "ElementaryDerivatives";;
(add_section_var (mk_var ("f", (`:real -> real`))); add_section_var (mk_var ("g", (`:real -> real`))));;
(add_section_var (mk_var ("x", (`:real`))); add_section_var (mk_var ("c", (`:real`))));;
(add_section_hyp "df" (`f real_differentiable atreal x`));;

(* Lemma derivative_scale *)
let derivative_scale = section_proof []
`derivative (\x. c * f x) x = c * derivative f x`
[
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_LMUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_neg *)
let derivative_neg = section_proof []
`derivative (\x. -- f x) x = -- derivative f x`
[
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_NEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_pow *)
let derivative_pow = section_proof ["n"]
`derivative (\x. f x pow n) x = &n * f x pow (n - 1) * derivative f x`
[
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_POW_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;
(add_section_hyp "dg" (`g real_differentiable atreal x`));;

(* Lemma derivative_add *)
let derivative_add = section_proof []
`derivative (\x. f x + g x) x = derivative f x + derivative g x`
[
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_ADD")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma derivative_mul *)
let derivative_mul = section_proof []
`derivative (\x. f x * g x) x = f x * derivative g x + derivative f x * g x`
[
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma derivative_sub *)
let derivative_sub = section_proof []
`derivative (\x. f x - g x) x = derivative f x - derivative g x`
[
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_SUB")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma derivative_div *)
let derivative_div = section_proof []
`~(g x = &0) ==> 
  derivative (\x. f x / g x) x = (derivative f x * g x - f x * derivative g x) / (g x * g x)`
[
   ((BETA_TAC THEN (move ["gn0"])) THEN (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_DIV_ATREAL")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))));
   ((repeat_tactic 1 9 (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section DerivativeArith *)
let derivative_x = finalize_theorem derivative_x;;
let derivative_const = finalize_theorem derivative_const;;
let derivative_inv = finalize_theorem derivative_inv;;
let derivative_atn = finalize_theorem derivative_atn;;
let derivative_exp = finalize_theorem derivative_exp;;
let derivative_acs = finalize_theorem derivative_acs;;
let derivative_sqrt = finalize_theorem derivative_sqrt;;
let derivative_composition = finalize_theorem derivative_composition;;
let REAL_DIFFERENTIABLE_AT_INV = finalize_theorem REAL_DIFFERENTIABLE_AT_INV;;
let derivative_compose_atn = finalize_theorem derivative_compose_atn;;
let derivative_compose_exp = finalize_theorem derivative_compose_exp;;
let derivative_compose_inv = finalize_theorem derivative_compose_inv;;
let derivative_compose_sqrt = finalize_theorem derivative_compose_sqrt;;
let derivative_compose_acs = finalize_theorem derivative_compose_acs;;
let derivative_scale = finalize_theorem derivative_scale;;
let derivative_neg = finalize_theorem derivative_neg;;
let derivative_pow = finalize_theorem derivative_pow;;
let derivative_add = finalize_theorem derivative_add;;
let derivative_mul = finalize_theorem derivative_mul;;
let derivative_sub = finalize_theorem derivative_sub;;
let derivative_div = finalize_theorem derivative_div;;
end_section "DerivativeArith";;

(* Section MoreDerivativeArith *)
begin_section "MoreDerivativeArith";;

(* Lemma differentiable_sum_numseg *)
let differentiable_sum_numseg = section_proof ["G";"n";"m";"x"]
`(!i. i IN n..m ==> G i real_differentiable atreal x) ==>
	(\x. sum (n..m) (\i. G i x)) real_differentiable atreal x`
[
   ((((THENL) (((use_arg_then "m") (disch_tac [])) THEN (clear_assumption "m") THEN elim) [ALL_TAC; ((move ["m"]) THEN (move ["IHm"]))]) THEN (move ["dG"])) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL SUM_CLAUSES_NUMSEG)))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then "EXCLUDED_MIDDLE") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`n = 0`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN case THEN (simp_tac) THEN (move ["n_eq_0"])) THEN ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_CONST")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((TRY done_tac))));
   (((((use_arg_then "dG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "n_eq_0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "leqnn")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_ROT (-1)) (((fun arg_tac -> (use_arg_then "EXCLUDED_MIDDLE") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`n <= SUC m`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN case THEN (simp_tac) THEN (move ["n_le_Sm"])));
   ((((use_arg_then "IHm") (disch_tac [])) THEN (clear_assumption "IHm") THEN (DISCH_THEN apply_tac) THEN (move ["i"]) THEN (move ["i_in"])) THEN (((use_arg_then "dG") (disch_tac [])) THEN (clear_assumption "dG") THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then "i_in") (disch_tac [])) THEN (clear_assumption "i_in") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "REAL_DIFFERENTIABLE_ADD")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((use_arg_then "dG")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "IHm")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "n_le_Sm")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "leqnn")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["ineq"])) THEN (((use_arg_then "dG") (disch_tac [])) THEN (clear_assumption "dG") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma derivative_sum_numseg *)
let derivative_sum_numseg = section_proof ["G";"n";"m";"x"]
`(!i. i IN n..m ==> G i real_differentiable atreal x) ==>
	derivative (\x. sum (n..m) (\i. G i x)) x = sum (n..m) (\i. derivative (G i) x)`
[
   ((((THENL) (((use_arg_then "m") (disch_tac [])) THEN (clear_assumption "m") THEN elim) [ALL_TAC; ((move ["m"]) THEN (move ["IHm"]))]) THEN (move ["dG"])) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL SUM_CLAUSES_NUMSEG)))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> arg_tac (Arg_term (`n = 0`))) (disch_tac [])) THEN case THEN (simp_tac)) THEN ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((TRY done_tac)) THEN (((use_arg_then "derivative_const")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg])))) THEN (done_tac));
   ((THENL_ROT (-1)) (((fun arg_tac -> (use_arg_then "EXCLUDED_MIDDLE") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`n <= SUC m`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN case THEN (simp_tac) THEN (move ["n_le_Sm"])));
   ((((use_arg_then "IHm") (disch_tac [])) THEN (clear_assumption "IHm") THEN (DISCH_THEN apply_tac) THEN (move ["i"]) THEN (move ["i_in"])) THEN (((use_arg_then "dG") (disch_tac [])) THEN (clear_assumption "dG") THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then "i_in") (disch_tac [])) THEN (clear_assumption "i_in") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (((use_arg_then "IHm")(gsym_then (thm_tac (new_rewrite [] [])))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["i_in"])) THEN (((use_arg_then "dG") (disch_tac [])) THEN (clear_assumption "dG") THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then "i_in") (disch_tac [])) THEN (clear_assumption "i_in") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (arith_tac));
   ((((use_arg_then "derivative_add")(thm_tac (new_rewrite [] [])))) THEN ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((TRY done_tac))));
   (((((use_arg_then "dG")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "leqnn")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "andbT")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "differentiable_sum_numseg")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]) THEN (move ["i_in"]));
   ((((use_arg_then "dG") (disch_tac [])) THEN (clear_assumption "dG") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "i_in") (disch_tac [])) THEN (clear_assumption "i_in") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section MoreDerivativeArith *)
let differentiable_sum_numseg = finalize_theorem differentiable_sum_numseg;;
let derivative_sum_numseg = finalize_theorem derivative_sum_numseg;;
end_section "MoreDerivativeArith";;

(* Lemma HAS_REAL_DERIVATIVE_LOCAL *)
let HAS_REAL_DERIVATIVE_LOCAL = section_proof ["f";"g";"x";"g'"]
`(g has_real_derivative g') (atreal x) /\
	(?s. real_open s /\ x IN s /\ (!y. y IN s ==> f y = g y))
	==> (f has_real_derivative g') (atreal x)`
[
   (BETA_TAC THEN (case THEN (move ["dg"])) THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["f_eq_g"]));
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN BETA_TAC) THEN (((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then "HAS_REAL_DERIVATIVE_WITHIN_REAL_OPEN") (fun fst_arg -> (use_arg_then "open_s") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (move ["dg"])));
   (((fun arg_tac -> (use_arg_then "HAS_REAL_DERIVATIVE_TRANSFORM_WITHIN") (fun fst_arg -> (use_arg_then "dg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`&1`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_LT_01")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (case THEN ALL_TAC) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "f_eq_g") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma differentiable_local *)
let differentiable_local = section_proof ["f";"g";"x";"s"]
`g real_differentiable atreal x /\ real_open s /\ x IN s /\
	(!y. y IN s ==> f y = g y) ==> f real_differentiable atreal x`
[
   ((repeat_tactic 1 9 (((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN ALL_TAC) THEN (case THEN (move ["f'"])) THEN (move ["dg"]) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["eq"]));
   (((use_arg_then "f'") (term_tac exists_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac)));
   (((use_arg_then "g") (term_tac exists_tac)) THEN ((((use_arg_then "dg")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN ((use_arg_then "s") (term_tac exists_tac)) THEN (done_tac));
];;

(* Section NthDerivativeArith *)
begin_section "NthDerivativeArith";;
(add_section_var (mk_var ("f", (`:real->real`))); add_section_var (mk_var ("g", (`:real->real`))));;
(add_section_var (mk_var ("int", (`:real#real`))));;
(add_section_var (mk_var ("n", (`:num`))));;
(add_section_hyp "df" (`nth_diff_strong_int n int f`));;

(* Lemma nth_derivative_scale_strong *)
let nth_derivative_scale_strong = section_proof ["c";"i";"x"]
`interval_arith x int ==> i <= n ==>
	?s. real_open s /\ x IN s /\ 
	!y. y IN s ==> nth_derivative i (\y. c * f y) y = c * nth_derivative i f y`
[
   ((((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN (((((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] []))))) THEN (move ["df"])));
   (((DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"]))) THEN (((((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))) THEN (move ["diff"])));
   ((THENL_FIRST) ((THENL) (((use_arg_then "i") (disch_tac [])) THEN (clear_assumption "i") THEN elim) [(move ["_"]); ((move ["i"]) THEN (move ["IHi"]))]) ((repeat_tactic 1 9 (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN ((use_arg_then "s") (term_tac exists_tac)) THEN (done_tac)));
   (((((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["i_lt_n"]));
   ((((fun arg_tac -> (use_arg_then "IHi") (fun fst_arg -> (fun arg_tac -> (use_arg_then "ltnW") (fun fst_arg -> (use_arg_then "i_lt_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["xt"])) THEN (move ["eq"])) THEN (((use_arg_then "IHi") (disch_tac [])) THEN (clear_assumption "IHi") THEN BETA_TAC THEN (move ["_"])));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER t`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xt")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (case THEN ((move ["ys"]) THEN (move ["yt"])))));
   (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`\y. c * nth_derivative i f y`))) (term_tac exists_tac));
   ((((use_arg_then "HAS_REAL_DERIVATIVE_LMUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((fun arg_tac -> (use_arg_then "diff") (fun fst_arg -> (use_arg_then "ys") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER t`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "yt")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ys")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["z"]) THEN (move ["z_in"]) THEN (simp_tac)));
   ((((use_arg_then "eq")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_derivative_scale_strong_all *)
let nth_derivative_scale_strong_all = section_proof ["c";"x"]
`interval_arith x int ==>
	?s. real_open s /\ x IN s /\ 
	!i y. i <= n /\ y IN s ==> nth_derivative i (\y. c * f y) y = c * nth_derivative i f y`
[
   (BETA_TAC THEN (DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then "nth_derivative_scale_strong") (fun fst_arg -> (use_arg_then "c") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"]));
   ((fun arg_tac -> arg_tac (Arg_term (`\i s. real_open s /\ x IN s /\ 
	(!y. y IN s ==> nth_derivative i (\y. c * f y) y = c * nth_derivative i f y)`))) (term_tac (set_tac "P")));
   ((fun arg_tac -> arg_tac (Arg_term (`!i. i <= n:num ==> P i ((@) (P i))`))) (term_tac (have_gen_tac [](move ["sel_P"]))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["i_le_n"])) THEN ((use_arg_then "SELECT_AX") (thm_tac apply_tac)));
   (((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (move ["cond"]));
   (((use_arg_then "s") (term_tac exists_tac)) THEN ((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`INTERS (IMAGE (\i. (@) (P i)) (0..n))`))) (term_tac (set_tac "S")));
   ((use_arg_then "S") (term_tac exists_tac));
   ((((use_arg_then "S_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_OPEN_INTERS")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (((repeat_tactic 1 9 (((use_arg_then "IN_INTERS")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] [])))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (BETA_TAC THEN (move ["i"]) THEN (move ["y"]) THEN (case THEN (move ["i_le_n"])));
   ((((fun arg_tac -> (use_arg_then "sel_P") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["y_in"]) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`(@) (P i)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))));
   ((THENL_FIRST) (ANTS_TAC) (((use_arg_then "i") (term_tac exists_tac)) THEN (((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((use_arg_then "y_in") (disch_tac [])) THEN (clear_assumption "y_in") THEN (exact_tac));
];;

(* Lemma nth_derivative_scale *)
let nth_derivative_scale = section_proof ["c";"i";"x"]
`interval_arith x int ==> i <= n ==>
	nth_derivative i (\x. c * f x) x = c * nth_derivative i f x`
[
   (BETA_TAC THEN (move ["ineq"]) THEN (move ["i_le_n"]));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_scale_strong") (fun fst_arg -> (use_arg_then "c") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["xs"])) THEN (move ["h"]));
   (((use_arg_then "h") (disch_tac [])) THEN (clear_assumption "h") THEN (exact_tac));
];;

(* Lemma nth_diff_scale *)
let nth_diff_scale = section_proof ["c"]
`nth_diff_strong_int n int (\x. c * f x)`
[
   ((((use_arg_then "df") (disch_tac [])) THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] [])))))));
   (BETA_TAC THEN (move ["df"]) THEN (move ["x"]) THEN (move ["ineq"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_scale_strong_all") (fun fst_arg -> (use_arg_then "c") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["diff"]));
   (((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["xt"])) THEN (move ["diff2"]));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER t`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xt")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (case THEN ((move ["ys"]) THEN (move ["yt"])))));
   (((((use_arg_then "REAL_CONTINUOUS_LMUL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff2")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]) THEN (move ["i_lt_n"]));
   ((((use_arg_then "diff")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "ys")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`(\y. c * nth_derivative i f y)`))) (term_tac exists_tac));
   ((THENL_FIRST) ((((use_arg_then "HAS_REAL_DERIVATIVE_LMUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((fun arg_tac -> (use_arg_then "diff2") (fun fst_arg -> (use_arg_then "yt") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ys")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["z"]) THEN (move ["zs"]) THEN (simp_tac)));
   ((((use_arg_then "diff") (disch_tac [])) THEN (clear_assumption "diff") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "ltnW")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;
(add_section_hyp "dg" (`nth_diff_strong_int n int g`));;

(* Lemma nth_derivative_add_strong *)
let nth_derivative_add_strong = section_proof ["i";"x"]
`interval_arith x int ==> i <= n ==>
   ?s. real_open s /\ x IN s /\ 
      !y. y IN s ==> nth_derivative i (\y. f y + g y) y = nth_derivative i f y + nth_derivative i g y`
[
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN (((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))))) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["ineq"])));
   ((((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["sf"])) THEN (case THEN (move ["open_sf"])) THEN (case THEN (move ["xsf"]))) THEN (((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC THEN (move ["_"])));
   ((((fun arg_tac -> (use_arg_then "dg") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["sg"])) THEN (case THEN (move ["open_sg"])) THEN (case THEN (move ["xsg"]))) THEN (((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN BETA_TAC THEN (move ["_"])));
   (((repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] [])))))) THEN (move ["diff_g"]) THEN (move ["diff_f"]));
   ((THENL_FIRST) ((THENL) (((use_arg_then "i") (disch_tac [])) THEN (clear_assumption "i") THEN elim) [(move ["_"]); ((move ["i"]) THEN (move ["IHi"]))]) ((repeat_tactic 1 9 (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN ((use_arg_then "sf") (term_tac exists_tac)) THEN (done_tac)));
   (((((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["i_lt_n"]));
   ((((fun arg_tac -> (use_arg_then "IHi") (fun fst_arg -> (fun arg_tac -> (use_arg_then "ltnW") (fun fst_arg -> (use_arg_then "i_lt_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["xt"])) THEN (move ["eq"])) THEN (((use_arg_then "IHi") (disch_tac [])) THEN (clear_assumption "IHi") THEN BETA_TAC THEN (move ["_"])));
   ((fun arg_tac -> arg_tac (Arg_term (`sf INTER sg INTER t`))) (term_tac exists_tac));
   (((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xt")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsf")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsg")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (case THEN (move ["ysf"])) THEN (case THEN ((move ["ysg"]) THEN (move ["yt"]))));
   (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`(\y. nth_derivative i f y + nth_derivative i g y)`))) (term_tac exists_tac));
   ((((use_arg_then "HAS_REAL_DERIVATIVE_ADD")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((fun arg_tac -> (use_arg_then "diff_f") (fun fst_arg -> (use_arg_then "ysf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_g") (fun fst_arg -> (use_arg_then "ysg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`sf INTER sg INTER t`))) (term_tac exists_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "yt")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ysf")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ysg")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))));
   ((BETA_TAC THEN (move ["z"]) THEN (move ["z_in"]) THEN (simp_tac)) THEN (((use_arg_then "eq")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_derivative_add_strong_all *)
let nth_derivative_add_strong_all = section_proof ["x"]
`interval_arith x int ==>
	?s. real_open s /\ x IN s /\ 
	!i y. i <= n /\ y IN s ==> 
		nth_derivative i (\y. f y + g y) y = nth_derivative i f y + nth_derivative i g y`
[
   (BETA_TAC THEN (DISCH_THEN (fun snd_th -> (use_arg_then "nth_derivative_add_strong") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"]));
   ((fun arg_tac -> arg_tac (Arg_term (`\i s. real_open s /\ x IN s /\ 
   (!y. y IN s ==> nth_derivative i (\y. f y + g y) y = nth_derivative i f y + nth_derivative i g y)`))) (term_tac (set_tac "P")));
   ((fun arg_tac -> arg_tac (Arg_term (`!i. i <= n:num ==> P i ((@) (P i))`))) (term_tac (have_gen_tac [](move ["sel_P"]))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["i_le_n"])) THEN ((use_arg_then "SELECT_AX") (thm_tac apply_tac)));
   (((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (move ["cond"]));
   (((use_arg_then "s") (term_tac exists_tac)) THEN ((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`INTERS (IMAGE (\i. (@) (P i)) (0..n))`))) (term_tac (set_tac "S")));
   ((use_arg_then "S") (term_tac exists_tac));
   ((((use_arg_then "S_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_OPEN_INTERS")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (((repeat_tactic 1 9 (((use_arg_then "IN_INTERS")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] [])))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (BETA_TAC THEN (move ["i"]) THEN (move ["y"]) THEN (case THEN (move ["i_le_n"])));
   ((((fun arg_tac -> (use_arg_then "sel_P") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["y_in"]) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`(@) (P i)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))));
   ((THENL_FIRST) (ANTS_TAC) (((use_arg_then "i") (term_tac exists_tac)) THEN (((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((use_arg_then "y_in") (disch_tac [])) THEN (clear_assumption "y_in") THEN (exact_tac));
];;

(* Lemma nth_derivative_add *)
let nth_derivative_add = section_proof ["i";"x"]
`interval_arith x int ==> i <= n ==>
	nth_derivative i (\x. f x + g x) x = nth_derivative i f x + nth_derivative i g x`
[
   (BETA_TAC THEN (move ["ineq"]) THEN (move ["i_le_n"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_add_strong") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["xs"])) THEN (move ["h"]));
   (((use_arg_then "h") (disch_tac [])) THEN (clear_assumption "h") THEN (exact_tac));
];;

(* Lemma nth_diff_add *)
let nth_diff_add = section_proof []
`nth_diff_strong_int n int (\x. f x + g x)`
[
   (((use_arg_then "dg") (disch_tac [])) THEN ((use_arg_then "df") (disch_tac [])) THEN BETA_TAC);
   ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))));
   (BETA_TAC THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["x"]) THEN (move ["ineq"]));
   ((fun arg_tac -> (use_arg_then "nth_derivative_add_strong_all") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["diff"]));
   (((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["tf"])) THEN (case THEN (move ["open_tf"])) THEN (case THEN (move ["xtf"])) THEN (move ["diff_f"]));
   (((fun arg_tac -> (use_arg_then "dg") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["tg"])) THEN (case THEN (move ["open_tg"])) THEN (case THEN (move ["xtg"])) THEN (move ["diff_g"]));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER tf INTER tg`))) (term_tac exists_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xtf")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xtg")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))));
   (BETA_TAC THEN (move ["y"]) THEN (case THEN ((move ["ys"]) THEN (case THEN ((move ["ytf"]) THEN (move ["ytg"]))))));
   (((((use_arg_then "REAL_CONTINUOUS_ADD")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff_f")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "diff_g")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]) THEN (move ["i_lt_n"]));
   ((((use_arg_then "diff")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "ys")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`(\y. nth_derivative i f y + nth_derivative i g y)`))) (term_tac exists_tac));
   ((THENL_FIRST) ((((use_arg_then "HAS_REAL_DERIVATIVE_ADD")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((fun arg_tac -> (use_arg_then "diff_f") (fun fst_arg -> (use_arg_then "ytf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_g") (fun fst_arg -> (use_arg_then "ytg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ys")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["z"]) THEN (move ["zs"]) THEN (simp_tac)));
   ((((use_arg_then "diff") (disch_tac [])) THEN (clear_assumption "diff") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "ltnW")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_derivative_sub_strong *)
let nth_derivative_sub_strong = section_proof ["i";"x"]
`interval_arith x int ==> i <= n ==>
   ?s. real_open s /\ x IN s /\ 
      !y. y IN s ==> nth_derivative i (\y. f y - g y) y = nth_derivative i f y - nth_derivative i g y`
[
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN (((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))))) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["ineq"])));
   ((((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["sf"])) THEN (case THEN (move ["open_sf"])) THEN (case THEN (move ["xsf"]))) THEN (((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC THEN (move ["_"])));
   ((((fun arg_tac -> (use_arg_then "dg") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["sg"])) THEN (case THEN (move ["open_sg"])) THEN (case THEN (move ["xsg"]))) THEN (((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN BETA_TAC THEN (move ["_"])));
   (((repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] [])))))) THEN (move ["diff_g"]) THEN (move ["diff_f"]));
   ((THENL_FIRST) ((THENL) (((use_arg_then "i") (disch_tac [])) THEN (clear_assumption "i") THEN elim) [(move ["_"]); ((move ["i"]) THEN (move ["IHi"]))]) ((repeat_tactic 1 9 (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN ((use_arg_then "sf") (term_tac exists_tac)) THEN (done_tac)));
   (((((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["i_lt_n"]));
   ((((fun arg_tac -> (use_arg_then "IHi") (fun fst_arg -> (fun arg_tac -> (use_arg_then "ltnW") (fun fst_arg -> (use_arg_then "i_lt_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["xt"])) THEN (move ["eq"])) THEN (((use_arg_then "IHi") (disch_tac [])) THEN (clear_assumption "IHi") THEN BETA_TAC THEN (move ["_"])));
   ((fun arg_tac -> arg_tac (Arg_term (`sf INTER sg INTER t`))) (term_tac exists_tac));
   (((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xt")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsf")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsg")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (case THEN (move ["ysf"])) THEN (case THEN ((move ["ysg"]) THEN (move ["yt"]))));
   (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`(\y. nth_derivative i f y - nth_derivative i g y)`))) (term_tac exists_tac));
   ((((use_arg_then "HAS_REAL_DERIVATIVE_SUB")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((fun arg_tac -> (use_arg_then "diff_f") (fun fst_arg -> (use_arg_then "ysf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_g") (fun fst_arg -> (use_arg_then "ysg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`sf INTER sg INTER t`))) (term_tac exists_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "yt")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ysf")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ysg")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))));
   ((BETA_TAC THEN (move ["z"]) THEN (move ["z_in"]) THEN (simp_tac)) THEN (((use_arg_then "eq")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_derivative_sub_strong_all *)
let nth_derivative_sub_strong_all = section_proof ["x"]
`interval_arith x int ==>
	?s. real_open s /\ x IN s /\ 
	!i y. i <= n /\ y IN s ==> 
		nth_derivative i (\y. f y - g y) y = nth_derivative i f y - nth_derivative i g y`
[
   (BETA_TAC THEN (DISCH_THEN (fun snd_th -> (use_arg_then "nth_derivative_sub_strong") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"]));
   ((fun arg_tac -> arg_tac (Arg_term (`\i s. real_open s /\ x IN s /\ 
   (!y. y IN s ==> nth_derivative i (\y. f y - g y) y = nth_derivative i f y - nth_derivative i g y)`))) (term_tac (set_tac "P")));
   ((fun arg_tac -> arg_tac (Arg_term (`!i. i <= n:num ==> P i ((@) (P i))`))) (term_tac (have_gen_tac [](move ["sel_P"]))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["i_le_n"])) THEN ((use_arg_then "SELECT_AX") (thm_tac apply_tac)));
   (((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (move ["cond"]));
   (((use_arg_then "s") (term_tac exists_tac)) THEN ((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`INTERS (IMAGE (\i. (@) (P i)) (0..n))`))) (term_tac (set_tac "S")));
   ((use_arg_then "S") (term_tac exists_tac));
   ((((use_arg_then "S_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_OPEN_INTERS")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (((repeat_tactic 1 9 (((use_arg_then "IN_INTERS")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] [])))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (BETA_TAC THEN (move ["i"]) THEN (move ["y"]) THEN (case THEN (move ["i_le_n"])));
   ((((fun arg_tac -> (use_arg_then "sel_P") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["y_in"]) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`(@) (P i)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))));
   ((THENL_FIRST) (ANTS_TAC) (((use_arg_then "i") (term_tac exists_tac)) THEN (((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((use_arg_then "y_in") (disch_tac [])) THEN (clear_assumption "y_in") THEN (exact_tac));
];;

(* Lemma nth_derivative_sub *)
let nth_derivative_sub = section_proof ["i";"x"]
`interval_arith x int ==> i <= n ==>
	nth_derivative i (\x. f x - g x) x = nth_derivative i f x - nth_derivative i g x`
[
   (BETA_TAC THEN (move ["ineq"]) THEN (move ["i_le_n"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_sub_strong") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["xs"])) THEN (move ["h"]));
   (((use_arg_then "h") (disch_tac [])) THEN (clear_assumption "h") THEN (exact_tac));
];;

(* Lemma nth_diff_sub *)
let nth_diff_sub = section_proof []
`nth_diff_strong_int n int (\x. f x - g x)`
[
   (((use_arg_then "dg") (disch_tac [])) THEN ((use_arg_then "df") (disch_tac [])) THEN BETA_TAC);
   ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))));
   (BETA_TAC THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["x"]) THEN (move ["ineq"]));
   ((fun arg_tac -> (use_arg_then "nth_derivative_sub_strong_all") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["diff"]));
   (((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["tf"])) THEN (case THEN (move ["open_tf"])) THEN (case THEN (move ["xtf"])) THEN (move ["diff_f"]));
   (((fun arg_tac -> (use_arg_then "dg") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["tg"])) THEN (case THEN (move ["open_tg"])) THEN (case THEN (move ["xtg"])) THEN (move ["diff_g"]));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER tf INTER tg`))) (term_tac exists_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xtf")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xtg")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))));
   (BETA_TAC THEN (move ["y"]) THEN (case THEN ((move ["ys"]) THEN (case THEN ((move ["ytf"]) THEN (move ["ytg"]))))));
   (((((use_arg_then "REAL_CONTINUOUS_SUB")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff_f")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "diff_g")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]) THEN (move ["i_lt_n"]));
   ((((use_arg_then "diff")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "ys")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`(\y. nth_derivative i f y - nth_derivative i g y)`))) (term_tac exists_tac));
   ((THENL_FIRST) ((((use_arg_then "HAS_REAL_DERIVATIVE_SUB")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((fun arg_tac -> (use_arg_then "diff_f") (fun fst_arg -> (use_arg_then "ytf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_g") (fun fst_arg -> (use_arg_then "ytg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ys")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["z"]) THEN (move ["zs"]) THEN (simp_tac)));
   ((((use_arg_then "diff") (disch_tac [])) THEN (clear_assumption "diff") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "ltnW")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma nth_derivative_mul_strong *)
let nth_derivative_mul_strong = section_proof ["i";"x"]
`interval_arith x int ==> i <= n ==>
   ?s. real_open s /\ x IN s /\ 
      !y. y IN s ==> nth_derivative i (\y. f y * g y) y = 
	sum (0..i) (\k. &(binom (i, k)) * nth_derivative k f y * nth_derivative (i - k) g y)`
[
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN (((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))))) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["ineq"])));
   ((((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["sf"])) THEN (case THEN (move ["open_sf"])) THEN (case THEN (move ["xsf"]))) THEN (((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC THEN (move ["_"])));
   ((((fun arg_tac -> (use_arg_then "dg") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["sg"])) THEN (case THEN (move ["open_sg"])) THEN (case THEN (move ["xsg"]))) THEN (((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN BETA_TAC THEN (move ["_"])));
   (((repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] [])))))) THEN (move ["diff_g"]) THEN (move ["diff_f"]));
   ((THENL) (((use_arg_then "i") (disch_tac [])) THEN (clear_assumption "i") THEN elim) [(move ["_"]); ((move ["i"]) THEN (move ["IHi"]))]);
   (((use_arg_then "sf") (term_tac exists_tac)) THEN (((((use_arg_then "open_sf")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsf")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "SUM_SING_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (move ["y"]) THEN (move ["_"]) THEN (simp_tac)));
   (((((use_arg_then "subn0")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "binom")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ltE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["i_lt_n"]));
   ((((fun arg_tac -> (use_arg_then "IHi") (fun fst_arg -> (fun arg_tac -> (use_arg_then "ltnW") (fun fst_arg -> (use_arg_then "i_lt_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["xt"])) THEN (move ["eq"])) THEN (((use_arg_then "IHi") (disch_tac [])) THEN (clear_assumption "IHi") THEN BETA_TAC THEN (move ["_"])));
   ((fun arg_tac -> arg_tac (Arg_term (`sf INTER sg INTER t`))) (term_tac exists_tac));
   (((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xt")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsf")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xsg")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (case THEN (move ["ysf"])) THEN (case THEN ((move ["ysg"]) THEN (move ["yt"]))));
   (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac));
   (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`\y. sum (0..i) (\k. &(binom (i, k)) * nth_derivative k f y * nth_derivative (i - k) g y)`))) (term_tac exists_tac));
   ((THENL_ROT (-1)) (split_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`sf INTER sg INTER t`))) (term_tac exists_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "yt")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ysf")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ysg")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))));
   ((BETA_TAC THEN (move ["z"]) THEN (move ["z_in"]) THEN (simp_tac)) THEN (((use_arg_then "eq")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   (((use_arg_then "has_derivative_lemma") (disch_tac [])) THEN (clear_assumption "has_derivative_lemma") THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`!k. k IN 0..i ==>
	nth_derivative k f real_differentiable atreal y /\
	nth_derivative (i - k) g real_differentiable atreal y`))) (term_tac (have_gen_tac [](move ["diff_cond"]))));
   (((((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] [])))) THEN (move ["k"]) THEN (move ["ineq"]) THEN (simp_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`nth_derivative (SUC k) f y`))) (term_tac exists_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_f") (fun fst_arg -> (use_arg_then "ysf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "i_lt_n") (disch_tac [])) THEN (clear_assumption "i_lt_n") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`nth_derivative (SUC (i - k)) g y`))) (term_tac exists_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_g") (fun fst_arg -> (use_arg_then "ysg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "i_lt_n") (disch_tac [])) THEN (clear_assumption "i_lt_n") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`!k. k IN 0..i ==> 
	(\y. &(binom (i, k)) * nth_derivative k f y * nth_derivative (i - k) g y)
		real_differentiable atreal y`))) (term_tac (have_gen_tac [](move ["diff_cond2"]))));
   (BETA_TAC THEN (move ["k"]) THEN (move ["k_in"]) THEN (simp_tac));
   (((repeat_tactic 1 9 (((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_CONST")(thm_tac (new_rewrite [] [])))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((repeat_tactic 1 9 (((use_arg_then "diff_cond")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))) THEN (done_tac));
   (((((use_arg_then "differentiable_sum_numseg")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "derivative_sum_numseg")(thm_tac (new_rewrite [] [])))))) THEN ((simp_tac) THEN (((use_arg_then "diff_cond2")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((TRY done_tac))));
   ((fun arg_tac -> arg_tac (Arg_term (`sum (0..i) _`))) (term_tac (set_tac "lhs")));
   ((fun arg_tac -> arg_tac (Arg_term (`sum (0 + 1..i + 1) (\k. &(binom (i, k - 1)) * nth_derivative k f y * nth_derivative (SUC i - k) g y)`))) (term_tac (set_tac "sum1")));
   ((fun arg_tac -> arg_tac (Arg_term (`sum (0..i + 1) (\k. &(binom (i, k)) * nth_derivative k f y * nth_derivative (SUC i - k) g y)`))) (term_tac (set_tac "sum2")));
   ((fun arg_tac -> arg_tac (Arg_term (`lhs = sum1 + sum2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then "sum1_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "SUM_OFFSET")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "addn1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "succnK")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "subSS")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "sum2_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "addn1")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL SUM_CLAUSES_NUMSEG)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `0 <= SUC i`)))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`binom(i, SUC i) = 0`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((((use_arg_then "BINOM_EQ_0")(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "REAL_MUL_LZERO")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_ADD_RID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SUM_ADD_NUMSEG")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "lhs_def")(gsym_then (thm_tac (new_rewrite [] []))))));
   (((use_arg_then "SUM_EQ") (thm_tac apply_tac)) THEN (move ["k"]) THEN (move ["k_in"]) THEN (simp_tac));
   (((((use_arg_then "derivative_scale")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((repeat_tactic 0 10 (((use_arg_then "diff_cond")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))));
   (((((use_arg_then "REAL_ADD_LDISTRIB")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_EQ_MUL_LCANCEL")(thm_tac (new_rewrite [] []))))) THEN (DISJ2_TAC));
   ((((use_arg_then "derivative_mul")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((repeat_tactic 0 10 (((use_arg_then "diff_cond")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))));
   ((((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_derivativeS")(gsym_then (thm_tac (new_rewrite [] [])))))));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`SUC (i - k) = SUC i - k`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) ((done_tac) THEN (done_tac)));
   ((((use_arg_then "k_in") (disch_tac [])) THEN (clear_assumption "k_in") THEN BETA_TAC) THEN (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SUM_CLAUSES_LEFT")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "sum2_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 1 (((use_arg_then "SUM_CLAUSES_LEFT")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (TRY ((arith_tac))));
   ((repeat_tactic 1 9 (((use_arg_then "binom")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_ADD_ASSOC")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_EQ_ADD_LCANCEL")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "sum1_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "SUM_ADD_NUMSEG")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then "addn1") (fun fst_arg -> (use_arg_then "i") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   (((use_arg_then "SUM_EQ") (thm_tac apply_tac)) THEN (move ["k"]) THEN (move ["k_in"]) THEN (simp_tac));
   (((((use_arg_then "REAL_ADD_RDISTRIB")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_EQ_MUL_RCANCEL")(thm_tac (new_rewrite [] []))))) THEN (DISJ1_TAC));
   ((THENL_FIRST) (((THENL) (((use_arg_then "k_in") (disch_tac [])) THEN (clear_assumption "k_in") THEN ((use_arg_then "k") (disch_tac [])) THEN (clear_assumption "k") THEN case) [ALL_TAC; (move ["k"])]) THEN (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then "binom")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ONE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "subSS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "subn0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_OF_NUM_ADD")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma nth_derivative_mul_strong_all *)
let nth_derivative_mul_strong_all = section_proof ["x"]
`interval_arith x int ==>
	?s. real_open s /\ x IN s /\ 
	!i y. i <= n /\ y IN s ==> 
	  nth_derivative i (\y. f y * g y) y = 
		sum (0..i) (\k. &(binom (i, k)) * nth_derivative k f y * nth_derivative (i - k) g y)`
[
   (BETA_TAC THEN (DISCH_THEN (fun snd_th -> (use_arg_then "nth_derivative_mul_strong") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"]));
   ((fun arg_tac -> arg_tac (Arg_term (`\i s. real_open s /\ x IN s /\ 
   (!y. y IN s ==> nth_derivative i (\y. f y * g y) y = 
	sum (0..i) (\k. &(binom (i, k)) * nth_derivative k f y * nth_derivative (i - k) g y))`))) (term_tac (set_tac "P")));
   ((fun arg_tac -> arg_tac (Arg_term (`!i. i <= n:num ==> P i ((@) (P i))`))) (term_tac (have_gen_tac [](move ["sel_P"]))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["i_le_n"])) THEN ((use_arg_then "SELECT_AX") (thm_tac apply_tac)));
   (((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (move ["cond"]));
   (((use_arg_then "s") (term_tac exists_tac)) THEN ((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`INTERS (IMAGE (\i. (@) (P i)) (0..n))`))) (term_tac (set_tac "S")));
   ((use_arg_then "S") (term_tac exists_tac));
   ((((use_arg_then "S_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_OPEN_INTERS")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "FINITE_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (((repeat_tactic 1 9 (((use_arg_then "IN_INTERS")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_IMAGE")(thm_tac (new_rewrite [] [])))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["t"]) THEN (case THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "sel_P") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN (done_tac));
   (BETA_TAC THEN (move ["i"]) THEN (move ["y"]) THEN (case THEN (move ["i_le_n"])));
   ((((fun arg_tac -> (use_arg_then "sel_P") (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then "P_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (simp_tac)) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["y_in"]) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`(@) (P i)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))));
   ((THENL_FIRST) (ANTS_TAC) (((use_arg_then "i") (term_tac exists_tac)) THEN (((use_arg_then "IN_NUMSEG_0")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((use_arg_then "y_in") (disch_tac [])) THEN (clear_assumption "y_in") THEN (exact_tac));
];;

(* Lemma nth_derivative_mul *)
let nth_derivative_mul = section_proof ["i";"x"]
`interval_arith x int ==> i <= n ==>
	nth_derivative i (\x. f x * g x) x = 
  	  sum (0..i) (\k. &(binom (i, k)) * nth_derivative k f x * nth_derivative (i - k) g x)`
[
   (BETA_TAC THEN (move ["ineq"]) THEN (move ["i_le_n"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_mul_strong") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "i_le_n") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["xs"])) THEN (move ["h"]));
   (((use_arg_then "h") (disch_tac [])) THEN (clear_assumption "h") THEN (exact_tac));
];;

(* Lemma nth_diff_mul *)
let nth_diff_mul = section_proof []
`nth_diff_strong_int n int (\x. f x * g x)`
[
   (((use_arg_then "dg") (disch_tac [])) THEN ((use_arg_then "df") (disch_tac [])) THEN BETA_TAC);
   ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))));
   (BETA_TAC THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["x"]) THEN (move ["ineq"]));
   ((fun arg_tac -> (use_arg_then "nth_derivative_mul_strong_all") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["diff"]));
   (((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["tf"])) THEN (case THEN (move ["open_tf"])) THEN (case THEN (move ["xtf"])) THEN (move ["diff_f"]));
   (((fun arg_tac -> (use_arg_then "dg") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["tg"])) THEN (case THEN (move ["open_tg"])) THEN (case THEN (move ["xtg"])) THEN (move ["diff_g"]));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER tf INTER tg`))) (term_tac exists_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xtf")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "xtg")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))));
   (BETA_TAC THEN (move ["y"]) THEN (case THEN ((move ["ys"]) THEN (case THEN ((move ["ytf"]) THEN (move ["ytg"]))))));
   (((((use_arg_then "REAL_CONTINUOUS_MUL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff_f")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "diff_g")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]) THEN (move ["i_lt_n"]));
   ((((use_arg_then "nth_derivativeS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_alt")(thm_tac (new_rewrite [] [])))));
   (((use_arg_then "differentiable_local") (disch_tac [])) THEN (clear_assumption "differentiable_local") THEN (DISCH_THEN apply_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`\y. sum (0..i) (\k. &(binom(i, k)) * nth_derivative k f y * nth_derivative (i - k) g y)`))) (term_tac exists_tac)) THEN ((use_arg_then "s") (term_tac exists_tac)));
   ((THENL_LAST) (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ys")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (split_tac)) ((BETA_TAC THEN (move ["z"]) THEN (move ["zs"])) THEN ((((use_arg_then "diff")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "ltnW")(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   (((((use_arg_then "differentiable_sum_numseg")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "IN_NUMSEG")(thm_tac (new_rewrite [] []))))) THEN (move ["k"]) THEN (move ["k_in"]) THEN (simp_tac));
   (((repeat_tactic 1 9 (((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_CONST")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))));
   ((repeat_tactic 1 9 (((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`nth_derivative (SUC k) f y`))) (term_tac exists_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_f") (fun fst_arg -> (use_arg_then "ytf") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "i_lt_n") (disch_tac [])) THEN (clear_assumption "i_lt_n") THEN ((use_arg_then "k_in") (disch_tac [])) THEN (clear_assumption "k_in") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`nth_derivative (SUC (i - k)) g y`))) (term_tac exists_tac)) THEN (((fun arg_tac -> (use_arg_then "diff_g") (fun fst_arg -> (use_arg_then "ytg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "i_lt_n") (disch_tac [])) THEN (clear_assumption "i_lt_n") THEN ((use_arg_then "k_in") (disch_tac [])) THEN (clear_assumption "k_in") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section NthDerivativeArith *)
let nth_derivative_scale_strong = finalize_theorem nth_derivative_scale_strong;;
let nth_derivative_scale_strong_all = finalize_theorem nth_derivative_scale_strong_all;;
let nth_derivative_scale = finalize_theorem nth_derivative_scale;;
let nth_diff_scale = finalize_theorem nth_diff_scale;;
let nth_derivative_add_strong = finalize_theorem nth_derivative_add_strong;;
let nth_derivative_add_strong_all = finalize_theorem nth_derivative_add_strong_all;;
let nth_derivative_add = finalize_theorem nth_derivative_add;;
let nth_diff_add = finalize_theorem nth_diff_add;;
let nth_derivative_sub_strong = finalize_theorem nth_derivative_sub_strong;;
let nth_derivative_sub_strong_all = finalize_theorem nth_derivative_sub_strong_all;;
let nth_derivative_sub = finalize_theorem nth_derivative_sub;;
let nth_diff_sub = finalize_theorem nth_diff_sub;;
let nth_derivative_mul_strong = finalize_theorem nth_derivative_mul_strong;;
let nth_derivative_mul_strong_all = finalize_theorem nth_derivative_mul_strong_all;;
let nth_derivative_mul = finalize_theorem nth_derivative_mul;;
let nth_diff_mul = finalize_theorem nth_diff_mul;;
end_section "NthDerivativeArith";;

(* Finalization of the section NthDerivatives *)
let has_derivative_cond = finalize_theorem has_derivative_cond;;
let has_derivative_alt = finalize_theorem has_derivative_alt;;
let derivative_unique = finalize_theorem derivative_unique;;
let derivative_unique_on = finalize_theorem derivative_unique_on;;
let has_derivative_lemma = finalize_theorem has_derivative_lemma;;
let nth_derivative0 = finalize_theorem nth_derivative0;;
let nth_derivativeS = finalize_theorem nth_derivativeS;;
let nth_Sderivative = finalize_theorem nth_Sderivative;;
let nth_derivative1 = finalize_theorem nth_derivative1;;
let nth_derivative2 = finalize_theorem nth_derivative2;;
let nth_derivative_add = finalize_theorem nth_derivative_add;;
let nth_diff_continuous = finalize_theorem nth_diff_continuous;;
let nth_differentiable_cond = finalize_theorem nth_differentiable_cond;;
let nth_differentiable_on_cond = finalize_theorem nth_differentiable_on_cond;;
let nth_differentiable_eq = finalize_theorem nth_differentiable_eq;;
let nth_differentiable_on_int2 = finalize_theorem nth_differentiable_on_int2;;
let nth_mth_diff = finalize_theorem nth_mth_diff;;
let nth_differentiable1 = finalize_theorem nth_differentiable1;;
let nth_diff_imp_diff = finalize_theorem nth_diff_imp_diff;;
let nth_derivative_continuous = finalize_theorem nth_derivative_continuous;;
let ith_derivative_differentiable = finalize_theorem ith_derivative_differentiable;;
let nth_diff_strong_imp_diff = finalize_theorem nth_diff_strong_imp_diff;;
let derivative_x = finalize_theorem derivative_x;;
let derivative_const = finalize_theorem derivative_const;;
let derivative_inv = finalize_theorem derivative_inv;;
let derivative_atn = finalize_theorem derivative_atn;;
let derivative_exp = finalize_theorem derivative_exp;;
let derivative_acs = finalize_theorem derivative_acs;;
let derivative_sqrt = finalize_theorem derivative_sqrt;;
let derivative_composition = finalize_theorem derivative_composition;;
let REAL_DIFFERENTIABLE_AT_INV = finalize_theorem REAL_DIFFERENTIABLE_AT_INV;;
let derivative_compose_atn = finalize_theorem derivative_compose_atn;;
let derivative_compose_exp = finalize_theorem derivative_compose_exp;;
let derivative_compose_inv = finalize_theorem derivative_compose_inv;;
let derivative_compose_sqrt = finalize_theorem derivative_compose_sqrt;;
let derivative_compose_acs = finalize_theorem derivative_compose_acs;;
let derivative_scale = finalize_theorem derivative_scale;;
let derivative_neg = finalize_theorem derivative_neg;;
let derivative_pow = finalize_theorem derivative_pow;;
let derivative_add = finalize_theorem derivative_add;;
let derivative_mul = finalize_theorem derivative_mul;;
let derivative_sub = finalize_theorem derivative_sub;;
let derivative_div = finalize_theorem derivative_div;;
let differentiable_sum_numseg = finalize_theorem differentiable_sum_numseg;;
let derivative_sum_numseg = finalize_theorem derivative_sum_numseg;;
let HAS_REAL_DERIVATIVE_LOCAL = finalize_theorem HAS_REAL_DERIVATIVE_LOCAL;;
let differentiable_local = finalize_theorem differentiable_local;;
let nth_derivative_scale_strong = finalize_theorem nth_derivative_scale_strong;;
let nth_derivative_scale_strong_all = finalize_theorem nth_derivative_scale_strong_all;;
let nth_derivative_scale = finalize_theorem nth_derivative_scale;;
let nth_diff_scale = finalize_theorem nth_diff_scale;;
let nth_derivative_add_strong = finalize_theorem nth_derivative_add_strong;;
let nth_derivative_add_strong_all = finalize_theorem nth_derivative_add_strong_all;;
let nth_derivative_add = finalize_theorem nth_derivative_add;;
let nth_diff_add = finalize_theorem nth_diff_add;;
let nth_derivative_sub_strong = finalize_theorem nth_derivative_sub_strong;;
let nth_derivative_sub_strong_all = finalize_theorem nth_derivative_sub_strong_all;;
let nth_derivative_sub = finalize_theorem nth_derivative_sub;;
let nth_diff_sub = finalize_theorem nth_diff_sub;;
let nth_derivative_mul_strong = finalize_theorem nth_derivative_mul_strong;;
let nth_derivative_mul_strong_all = finalize_theorem nth_derivative_mul_strong_all;;
let nth_derivative_mul = finalize_theorem nth_derivative_mul;;
let nth_diff_mul = finalize_theorem nth_diff_mul;;
end_section "NthDerivatives";;
let lin_approx = new_definition `lin_approx f x f_bounds df_bounds <=> 
	interval_arith (f x) f_bounds /\
	(?f'. (f has_real_derivative f') (atreal x) /\ interval_arith f' df_bounds)`;;
let has_bounded_second_derivative = new_definition `has_bounded_second_derivative f int dd_bounds <=>
	nth_diff_strong_int 2 int f /\ 
	bounded_on_int (nth_derivative 2 f) int dd_bounds`;;
let taylor_interval = new_definition 
  `taylor_interval f x y z w f_bounds df_bounds ddf_bounds <=>
	x <= y /\ y <= z /\ y - x <= w /\ z - y <= w /\
	lin_approx f y f_bounds df_bounds /\
	has_bounded_second_derivative f (x, z) ddf_bounds`;;
(* Lemma nth_diff_strong_imp_diff_int *)
let nth_diff_strong_imp_diff_int = section_proof ["n";"int";"f"]
`nth_diff_strong_int n int f ==>
	nth_differentiable_on_int n int f`
[
   ((((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_on_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))));
   (BETA_TAC THEN (move ["h"]) THEN (move ["x"]) THEN (move ["ineq"]));
   ((((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["xs"]))) THEN (exact_tac) THEN (done_tac));
];;

(* Lemma has_bounded_second_derivative_old *)
let has_bounded_second_derivative_old = section_proof ["f";"int";"dd_bounds"]
`has_bounded_second_derivative f int dd_bounds ==>
	?f' f''. (!x. interval_arith x int ==> (f has_real_derivative f' x) (atreal x) /\
       		(f' has_real_derivative f'' x) (atreal x) /\	interval_arith (f'' x) dd_bounds)`
[
   ((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (DISCH_THEN (fun snd_th -> (use_arg_then "nth_diff_strong_imp_diff_int") (thm_tac (match_mp_then snd_th MP_TAC))))));
   ((DISCH_THEN (fun snd_th -> (use_arg_then "nth_differentiable_on_int2") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["f'"])) THEN (case THEN (move ["f''"])));
   ((BETA_TAC THEN (case THEN (move ["eq1"])) THEN (case THEN (move ["eq2"])) THEN (move ["h"]) THEN (move ["b"])) THEN ((((use_arg_then "f'") (term_tac exists_tac)) THEN ((use_arg_then "f''") (term_tac exists_tac))) THEN (move ["x"]) THEN (move ["ineq"])));
   ((((use_arg_then "b") (disch_tac [])) THEN (clear_assumption "b") THEN BETA_TAC) THEN (((((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "eq2")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["b"])));
   (((repeat_tactic 1 9 (((use_arg_then "h")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "b")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma exists_and_left *)
let exists_and_left = section_proof ["P";"Q"]
`(?x. P x /\ Q x) ==> (?x. P x)`
[
   (BETA_TAC THEN (case THEN (move ["x"])) THEN (case THEN ((move ["Px"]) THEN (move ["_"]))));
   (((use_arg_then "x") (term_tac exists_tac)) THEN (done_tac));
];;

(* Lemma lim_ineq *)
let lim_ineq = section_proof ["a";"b"]
`(!e. &0 < e ==> a <= b + e) <=> (a <= b)`
[
   ((THENL_ROT (-1)) ((THENL) (split_tac) [ALL_TAC; ((move ["ineq"]) THEN (move ["e"]) THEN (move ["e0"]))]));
   (((((fun arg_tac -> (use_arg_then "REAL_ADD_RID") (fun fst_arg -> (use_arg_then "a") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_LE_ADD2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ineq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "contraLR") (disch_tac [])) THEN (clear_assumption "contraLR") THEN (DISCH_THEN apply_tac)) THEN (((((use_arg_then "NOT_FORALL_THM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "NOT_IMP")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_NOT_LE")(thm_tac (new_rewrite [] [])))))) THEN (move ["ba"])));
   (((fun arg_tac -> arg_tac (Arg_term (`(a - b) / &2`))) (term_tac exists_tac)) THEN (((use_arg_then "ba") (disch_tac [])) THEN (clear_assumption "ba") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma continuous_leq *)
let continuous_leq = section_proof ["f";"c";"a"]
`(?b. a < b /\ !x. x IN real_interval (a, b) ==> f x <= c) /\
	f real_continuous atreal a ==> f a <= c`
[
   (BETA_TAC THEN (case THEN ((case THEN (move ["b"])) THEN (case THEN (move ["ab"])) THEN (move ["ineq"]))));
   (((((use_arg_then "REAL_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REALLIM_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (move ["f_cont"]));
   ((((use_arg_then "lim_ineq")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["e"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "f_cont") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (case THEN (move ["d"])) THEN (case THEN (move ["d0"])) THEN (move ["cond"]));
   ((fun arg_tac -> arg_tac (Arg_term (`a + min (b - a) d / &2`))) (term_tac (set_tac "r")));
   ((THENL_FIRST) ((((fun arg_tac -> (use_arg_then "cond") (fun fst_arg -> (use_arg_then "r") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (ANTS_TAC)) ((((use_arg_then "ab") (disch_tac [])) THEN (clear_assumption "ab") THEN ((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN ((use_arg_then "r_def") (disch_tac [])) THEN (clear_assumption "r_def") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((THENL_LAST) ((((fun arg_tac -> (use_arg_then "ineq") (fun fst_arg -> (use_arg_then "r") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (ANTS_TAC)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ab") (disch_tac [])) THEN (clear_assumption "ab") THEN ((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN ((use_arg_then "r_def") (disch_tac [])) THEN (clear_assumption "r_def") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma continuous_reflection *)
let continuous_reflection = section_proof ["f";"x"]
`f real_continuous atreal x <=>
	(\x. f (--x)) real_continuous atreal (--x)`
[
   (((repeat_tactic 1 9 (((use_arg_then "REAL_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then "REALLIM_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_NEG_NEG")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN ((split_tac) THEN (move ["h"]) THEN (move ["e"]) THEN (move ["e0"])));
   (((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "e0") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d0"])) THEN (move ["cond"]));
   (((use_arg_then "d") (term_tac exists_tac)) THEN (((((use_arg_then "d0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["y"]) THEN (move ["ineqs"])) THEN (((use_arg_then "cond") (disch_tac [])) THEN (clear_assumption "cond") THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then "ineqs") (disch_tac [])) THEN (clear_assumption "ineqs") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   (((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (use_arg_then "e0") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d0"])) THEN (move ["cond"]));
   (((use_arg_then "d") (term_tac exists_tac)) THEN (((((use_arg_then "d0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["y"]) THEN (move ["ineqs"])));
   ((((fun arg_tac -> (use_arg_then "cond") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`--y`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (((use_arg_then "REAL_NEG_NEG")(thm_tac (new_rewrite [] [])))) THEN (DISCH_THEN apply_tac) THEN (((use_arg_then "ineqs") (disch_tac [])) THEN (clear_assumption "ineqs") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma continuous_leq_segment *)
let continuous_leq_segment = section_proof ["f";"c";"a";"b"]
`a < b ==> f real_continuous atreal a ==> f real_continuous atreal b ==>
	(!x. x IN real_interval (a, b) ==> f x <= c) ==>
	(!x. x IN real_interval [a, b] ==> f x <= c)`
[
   (BETA_TAC THEN (move ["ab"]) THEN (move ["f_cont_a"]) THEN (move ["f_cont_b"]) THEN (move ["ineq"]) THEN (move ["x"]));
   ((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 2 0 (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b. a <= b <=> (a = b \/ a < b)`)))(thm_tac (new_rewrite [] []))))));
   (case THEN ((THENL) case [((((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["_"])); (move ["ax"])]));
   ((((use_arg_then "continuous_leq") (disch_tac [])) THEN (clear_assumption "continuous_leq") THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "f_cont_a")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andbT")(thm_tac (new_rewrite [] []))))) THEN ((use_arg_then "b") (term_tac exists_tac)) THEN (done_tac));
   ((THENL_LAST) ((THENL) case [(((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))); (move ["xb"])]) ((((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`\x. f (--x)`))) (term_tac (set_tac "g")));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`!x. f x = g (--x)`))) (term_tac (have_gen_tac [](move ["fg"])))) (((((use_arg_then "g_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "REAL_NEG_NEG")(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   ((((use_arg_then "fg")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "continuous_leq") (disch_tac [])) THEN (clear_assumption "continuous_leq") THEN (DISCH_THEN apply_tac)) THEN (split_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`--a`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_LT_NEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ab")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (move ["y"])));
   ((((fun arg_tac -> (use_arg_then "ineq") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`--y`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "fg")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_NEG_NEG")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then "g_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "continuous_reflection")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma pair_eq *)
let pair_eq = section_proof ["p"]
`p = (FST p, SND p)`
[
   (done_tac);
];;

(* Section Taylor *)
begin_section "Taylor";;

(* Lemma iabs_alt *)
let iabs_alt = section_proof ["lo";"hi";"a"]
`interval_arith a (lo, hi) ==> iabs (lo, hi) = max (abs lo) (abs hi)`
[
   (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma iabs_pos *)
let iabs_pos = section_proof ["lo";"hi";"a"]
`interval_arith a (lo, hi) ==> &0 <= iabs (lo, hi)`
[
   (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;
(add_section_var (mk_var ("f", (`:real -> real`))));;
(add_section_var (mk_var ("x", (`:real`))); add_section_var (mk_var ("y", (`:real`))); add_section_var (mk_var ("z", (`:real`))); add_section_var (mk_var ("w", (`:real`))));;
(add_section_var (mk_var ("f_bounds", (`:real#real`))); add_section_var (mk_var ("df_bounds", (`:real#real`))); add_section_var (mk_var ("ddf_bounds", (`:real#real`))));;
(add_section_var (mk_var ("dd_bound", (`:real`))));;
(add_section_hyp "dd_bound_eq" (`dd_bound = iabs ddf_bounds`));;
(add_section_hyp "tif" (`taylor_interval f x y z w f_bounds df_bounds ddf_bounds`));;

(* Lemma f_continuous *)
let f_continuous = section_proof []
`!t. t IN real_interval [x, z] ==> f real_continuous atreal t`
[
   (((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["t"]) THEN (move ["t_in"]));
   (((use_arg_then "HAS_REAL_DERIVATIVE_IMP_CONTINUOUS_ATREAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_IMP_CONTINUOUS_ATREAL") THEN (DISCH_THEN apply_tac));
   ((((use_arg_then "tif") (disch_tac [])) THEN (clear_assumption "tif") THEN BETA_TAC) THEN (((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "has_bounded_second_derivative_old") (thm_tac (match_mp_then snd_th MP_TAC))))));
   (BETA_TAC THEN (case THEN (move ["f'"])) THEN (case THEN (move ["f''"])) THEN (move ["df"]));
   (((fun arg_tac -> arg_tac (Arg_term (`f' t`))) (term_tac exists_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma taylor_error *)
let taylor_error = section_proof ["t"]
`x <= t /\ t <= z ==> 
	abs (f t - f y) <= w * iabs df_bounds + w * w * dd_bound / &2`
[
   (BETA_TAC THEN (move ["t_ineqs"]));
   (((use_arg_then "tif") (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN (((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN ALL_TAC) THEN (case THEN (move ["domain_ineqs"])) THEN (move ["lin_app"])));
   ((DISCH_THEN (fun snd_th -> (use_arg_then "has_bounded_second_derivative_old") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["f'"])) THEN (case THEN (move ["f''"])) THEN (move ["df"]));
   ((fun arg_tac -> arg_tac (Arg_term (`abs (t - y) <= w`))) (term_tac (have_gen_tac [](move ["abs_ty"]))));
   ((((use_arg_then "t_ineqs") (disch_tac [])) THEN (clear_assumption "t_ineqs") THEN ((use_arg_then "domain_ineqs") (disch_tac [])) THEN (clear_assumption "domain_ineqs") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`interval_arith y (x, z)`))) (term_tac (have_gen_tac [](move ["y_in"])))) ((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`&0 <= dd_bound /\ !p. p IN real_interval [x, z] ==> abs (f'' p) <= dd_bound`))) (term_tac (have_gen_tac [](move ["dd_prop"]))));
   ((((use_arg_then "dd_bound_eq")(thm_tac (new_rewrite [] [])))) THEN (split_tac));
   ((((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "y_in") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (BETA_TAC THEN (move ["p"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN ((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`abs (f t - (f y + f' y * (t - y) pow 1)) <= dd_bound * abs (t - y) pow (1 + 1) / &2`))) (term_tac (have_gen_tac []ALL_TAC))));
   (((fun arg_tac -> arg_tac (Arg_term (`f' y * _`))) (term_tac (set_tac "b1"))) THEN (((fun arg_tac -> arg_tac (Arg_term (`dd_bound * _`))) (term_tac (set_tac "b2"))) THEN (move ["ineq1"])));
   ((((use_arg_then "REAL_LE_TRANS") (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`abs (f t - (f y + b1)) + abs b1`))) (term_tac exists_tac)) THEN (split_tac));
   (((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b. a - b = (a - (b + b1)) + b1`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_ABS_TRIANGLE")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ADD2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "b1_def")(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (((use_arg_then "REAL_POW_1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_ABS_MUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_MUL2")(thm_tac (new_rewrite [] [])))));
   ((repeat_tactic 1 9 (((use_arg_then "REAL_ABS_POS")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "abs_ty")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "lin_app") (disch_tac [])) THEN (clear_assumption "lin_app") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["ff"])) THEN (case THEN (move ["df'"]))));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`ff = f' y`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) (((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac)));
   ((((fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_DERIVATIVE_UNIQUE_ATREAL") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)) THEN ((((use_arg_then "df'")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_TRANS") (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((use_arg_then "b2") (term_tac exists_tac)));
   ((((use_arg_then "ineq1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "b2_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_ASSOC")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_LE_MUL2")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `1 + 1 = 2`)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_LE_REFL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_INV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LE_MUL")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "dd_prop")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LE_POW_2")(thm_tac (new_rewrite [] [])))))) THEN (TRY ((arith_tac))));
   ((((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbT")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_LE_MUL2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_REFL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "dd_prop")(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_LE_POW_2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbT")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_LE_SQUARE_ABS")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_ABS_ABS")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "abs_ty") (disch_tac [])) THEN (clear_assumption "abs_ty") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`\i. if i = 0 then f else if i = 1 then f' else if i = 2 then f'' else I`))) (term_tac (set_tac "Df")));
   ((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `0 + 1 = 1 /\ ~(1 = 0) /\ 1 + 1 = 2 /\ 
	~(2 = 1) /\ ~(2 = 1) /\ ~(2 = 0)`))) (fun arg -> thm_tac MP_TAC arg THEN (move ["arith"])));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_TAYLOR") (fun fst_arg -> (use_arg_then "Df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`1`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`real_interval [x, z]`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "dd_bound") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (ANTS_TAC);
   (((((use_arg_then "IS_REALINTERVAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["p"]) THEN (case THEN (move ["p_in"]))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `i <= 1 <=> i = 0 \/ i = 1`)))(thm_tac (new_rewrite [] [])))));
   ((case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN ((((use_arg_then "Df_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN ((use_arg_then "HAS_REAL_DERIVATIVE_ATREAL_WITHIN") (thm_tac apply_tac)) THEN ((repeat_tactic 1 9 (((use_arg_then "arith")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((BETA_TAC THEN (move ["p"]) THEN (move ["p_in"])) THEN ((((use_arg_then "Df_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)));
   (((repeat_tactic 1 9 (((use_arg_then "arith")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "dd_prop")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((BETA_TAC THEN ((fun arg_tac -> (fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "t") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))) THEN ((repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "domain_ineqs")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "t_ineqs")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)));
   ((((use_arg_then "ONE")(thm_tac (new_rewrite [1] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL SUM_CLAUSES_NUMSEG)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SUM_SING_NUMSEG")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "Df_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `0 <= 1 /\ ~(1 = 0)`)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL real_pow)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "FACT")(thm_tac (new_rewrite [] [])))));
   (((repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `FACT 1 = 1 /\ FACT (1 + 1) = 2`)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_DIV_1")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_MUL_RID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma taylor_upper_bound *)
let taylor_upper_bound = section_proof []
`!t. x <= t /\ t <= z ==>
	f t <= SND f_bounds + (w * iabs df_bounds + w * w * dd_bound / &2)`
[
   (((fun arg_tac -> (use_arg_then "pair_eq") (fun fst_arg -> (use_arg_then "f_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (move ["eq"]));
   (BETA_TAC THEN (move ["t"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "taylor_error") (thm_tac (match_mp_then snd_th MP_TAC)))));
   ((((use_arg_then "tif") (disch_tac [])) THEN (clear_assumption "tif") THEN BETA_TAC) THEN (((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN ALL_TAC) THEN (case THEN (move ["_"])) THEN (case THEN (move ["f_int"])) THEN (move ["_"]) THEN (move ["_"])));
   ((((use_arg_then "f_int") (disch_tac [])) THEN (clear_assumption "f_int") THEN BETA_TAC) THEN ((((use_arg_then "eq")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma taylor_lower_bound *)
let taylor_lower_bound = section_proof []
`!t. x <= t /\ t <= z ==>
	FST f_bounds - (w * iabs df_bounds + w * w * dd_bound / &2) <= f t`
[
   ((fun arg_tac -> (use_arg_then "pair_eq") (fun fst_arg -> (use_arg_then "f_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["eq"])));
   (BETA_TAC THEN (move ["t"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "taylor_error") (thm_tac (match_mp_then snd_th MP_TAC)))));
   ((((use_arg_then "tif") (disch_tac [])) THEN (clear_assumption "tif") THEN BETA_TAC) THEN (((((use_arg_then "eq")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN ALL_TAC) THEN (case THEN (move ["_"])) THEN (case THEN (move ["f_int"])) THEN (move ["_"]) THEN (move ["_"])));
   ((((use_arg_then "f_int") (disch_tac [])) THEN (clear_assumption "f_int") THEN BETA_TAC) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (arith_tac));
];;

(* Lemma taylor_derivative_error *)
let taylor_derivative_error = section_proof []
`!t. x <= t /\ t <= z ==> 
	abs (derivative f t - derivative f y) <= w * dd_bound`
[
   (BETA_TAC THEN (move ["t"]) THEN (move ["t_ineqs"]));
   (((use_arg_then "tif") (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN (((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN ALL_TAC) THEN (case THEN (move ["domain_ineqs"])) THEN (move ["_"])));
   ((DISCH_THEN (fun snd_th -> (use_arg_then "has_bounded_second_derivative_old") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["f'"])) THEN (case THEN (move ["f''"])) THEN (move ["df"]));
   ((fun arg_tac -> arg_tac (Arg_term (`abs (t - y) <= w`))) (term_tac (have_gen_tac [](move ["abs_ty"]))));
   ((((use_arg_then "t_ineqs") (disch_tac [])) THEN (clear_assumption "t_ineqs") THEN ((use_arg_then "domain_ineqs") (disch_tac [])) THEN (clear_assumption "domain_ineqs") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`!p. x <= p /\ p <= z ==> derivative f p = f' p`))) (term_tac (have_gen_tac [](move ["der_eq"]))));
   (((((use_arg_then "IN_REAL_INTERVAL")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "EQ_SYM_EQ")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "derivative_unique_on") (disch_tac [])) THEN (clear_assumption "derivative_unique_on") THEN (DISCH_THEN apply_tac)));
   ((((((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["p"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (simp_tac)) THEN (done_tac));
   ((repeat_tactic 1 9 (((use_arg_then "der_eq")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`&0 <= dd_bound /\ !p. p IN real_interval [x, z] ==> abs (f'' p) <= dd_bound`))) (term_tac (have_gen_tac [](move ["dd_prop"]))));
   (((((use_arg_then "dd_bound_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (split_tac));
   ((((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> (use_arg_then "pair_eq") (fun fst_arg -> (use_arg_then "ddf_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "domain_ineqs")(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   ((BETA_TAC THEN (move ["p"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN ((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`\i. if i = 0 then f' else if i = 1 then f'' else I`))) (term_tac (set_tac "Df")));
   ((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `0 + 1 = 1 /\ ~(1 = 0) /\ 1 + 1 = 2 /\ 
	~(2 = 1) /\ ~(2 = 1) /\ ~(2 = 0)`))) (fun arg -> thm_tac MP_TAC arg THEN (move ["arith"])));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_TAYLOR") (fun fst_arg -> (use_arg_then "Df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`0`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`real_interval [x, z]`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "dd_bound") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (ANTS_TAC);
   (((((use_arg_then "IS_REALINTERVAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["i"]) THEN (move ["p"]) THEN (case THEN (move ["p_in"]))) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `i <= 0 <=> i = 0`)))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then "Df_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then "arith")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ATREAL_WITHIN") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_ATREAL_WITHIN") THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "IN_REAL_INTERVAL")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((BETA_TAC THEN (move ["p"]) THEN (move ["p_in"])) THEN ((((use_arg_then "Df_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac)));
   (((repeat_tactic 1 9 (((use_arg_then "arith")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "dd_prop")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((BETA_TAC THEN ((fun arg_tac -> (fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "t") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))) THEN ((repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "domain_ineqs")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "t_ineqs")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)));
   ((((use_arg_then "ONE")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "SUM_SING_NUMSEG")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "Df_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL real_pow)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "arith")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "FACT")(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `FACT 1 = 1`)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_DIV_1")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_MUL_RID")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "REAL_POW_1")(thm_tac (new_rewrite [] [])))) THEN (move ["ineq"]));
   ((((use_arg_then "REAL_LE_TRANS") (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`dd_bound * abs (t - y)`))) (term_tac exists_tac)));
   (((((use_arg_then "ineq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_RMUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "dd_prop")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "abs_ty")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma derivative_interval *)
let derivative_interval = section_proof []
`FST df_bounds <= derivative f y /\ derivative f y <= SND df_bounds`
[
   ((fun arg_tac -> (use_arg_then "pair_eq") (fun fst_arg -> (use_arg_then "df_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["eq"])));
   ((((use_arg_then "tif") (disch_tac [])) THEN (clear_assumption "tif") THEN BETA_TAC) THEN (((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN ALL_TAC) THEN (case THEN (move ["ineqs"])) THEN (case THEN (move ["g"])) THEN (case THEN ((move ["fg"]) THEN (move ["g_int"]))) THEN (move ["_"])));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`derivative f y = g`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) ((((use_arg_then "g_int") (disch_tac [])) THEN (clear_assumption "g_int") THEN BETA_TAC) THEN ((((use_arg_then "eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then "REAL_DERIVATIVE_UNIQUE_ATREAL") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac));
   (((((use_arg_then "fg")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_derivative_cond")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN ((use_arg_then "g") (term_tac exists_tac)) THEN (done_tac));
];;

(* Lemma taylor_derivative_upper_bound *)
let taylor_derivative_upper_bound = section_proof []
`!t. x <= t /\ t <= z ==>
	derivative f t <= SND df_bounds + w * dd_bound`
[
   ((BETA_TAC THEN (move ["t"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "taylor_derivative_error") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (((use_arg_then "derivative_interval") (disch_tac [])) THEN (clear_assumption "derivative_interval") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma taylor_derivative_lower_bound *)
let taylor_derivative_lower_bound = section_proof []
`!t. x <= t /\ t <= z ==>
	FST df_bounds - w * dd_bound <= derivative f t`
[
   ((BETA_TAC THEN (move ["t"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "taylor_derivative_error") (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (((use_arg_then "derivative_interval") (disch_tac [])) THEN (clear_assumption "derivative_interval") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section Taylor *)
let iabs_alt = finalize_theorem iabs_alt;;
let iabs_pos = finalize_theorem iabs_pos;;
let f_continuous = finalize_theorem f_continuous;;
let taylor_error = finalize_theorem taylor_error;;
let taylor_upper_bound = finalize_theorem taylor_upper_bound;;
let taylor_lower_bound = finalize_theorem taylor_lower_bound;;
let taylor_derivative_error = finalize_theorem taylor_derivative_error;;
let derivative_interval = finalize_theorem derivative_interval;;
let taylor_derivative_upper_bound = finalize_theorem taylor_derivative_upper_bound;;
let taylor_derivative_lower_bound = finalize_theorem taylor_derivative_lower_bound;;
end_section "Taylor";;

(* Section LinearApproximation *)
begin_section "LinearApproximation";;
(add_section_var (mk_var ("f", (`:real->real`))));;
(add_section_var (mk_var ("f_bounds", (`:real#real`))); add_section_var (mk_var ("df_bounds", (`:real#real`))));;
(add_section_var (mk_var ("x", (`:real`))));;

(* Lemma lin_approx_eq *)
let lin_approx_eq = section_proof []
`lin_approx f x f_bounds df_bounds <=>
	(f real_differentiable atreal x /\ interval_arith (f x) f_bounds /\
	interval_arith (derivative f x) df_bounds)`
[
   (((((use_arg_then "lin_approx")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] []))))) THEN ((THENL) (split_tac) [(case THEN ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (case THEN (move ["f'"])) THEN (case THEN ((move ["df'"]) THEN (move ["int_f'"]))))); ((case THEN ((case THEN (move ["f'"])) THEN (move ["df"]))) THEN (case THEN ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (move ["df_int"]))))]));
   ((THENL_FIRST) (split_tac) (((use_arg_then "f'") (term_tac exists_tac)) THEN (done_tac)));
   ((((fun arg_tac -> (fun arg_tac -> (use_arg_then "derivative_unique") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f'") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then "f'") (term_tac exists_tac)) THEN ((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "derivative_unique") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f'") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;
(add_section_hyp "approx_f" (`lin_approx f x f_bounds df_bounds`));;

(* Lemma lin_approx_imp_f_interval *)
let lin_approx_imp_f_interval = section_proof []
`interval_arith (f x) f_bounds`
[
   ((((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
];;

(* Lemma lin_approx_imp_df_interval *)
let lin_approx_imp_df_interval = section_proof []
`interval_arith (derivative f x) df_bounds`
[
   ((((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
];;

(* Lemma lin_approx_imp_f_diff *)
let lin_approx_imp_f_diff = section_proof []
`f real_differentiable atreal x`
[
   ((((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
];;

(* Finalization of the section LinearApproximation *)
let lin_approx_eq = finalize_theorem lin_approx_eq;;
let lin_approx_imp_f_interval = finalize_theorem lin_approx_imp_f_interval;;
let lin_approx_imp_df_interval = finalize_theorem lin_approx_imp_df_interval;;
let lin_approx_imp_f_diff = finalize_theorem lin_approx_imp_f_diff;;
end_section "LinearApproximation";;

(* Section MoreLinearApproximation *)
begin_section "MoreLinearApproximation";;
(add_section_var (mk_var ("f", (`:real->real`))); add_section_var (mk_var ("g", (`:real->real`))));;
(add_section_var (mk_var ("x", (`:real`))));;
(add_section_var (mk_var ("f_bounds", (`:real#real`))); add_section_var (mk_var ("df_bounds", (`:real#real`))));;
(add_section_var (mk_var ("g_bounds", (`:real#real`))); add_section_var (mk_var ("dg_bounds", (`:real#real`))));;

(* Lemma interval_arith_not_zero *)
let interval_arith_not_zero = section_proof ["x";"int"]
`interval_arith x int ==> interval_not_zero int ==>
	~(x = &0)`
[
   (((((fun arg_tac -> (use_arg_then "PAIR") (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_not_zero")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma interval_arith_pos *)
let interval_arith_pos = section_proof ["x";"int"]
`interval_arith x int ==> interval_pos int ==> &0 < x`
[
   ((((use_arg_then "int") (disch_tac [])) THEN (clear_assumption "int") THEN case) THEN ((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_pos")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma interval_arith_abs *)
let interval_arith_abs = section_proof ["x";"int";"y"]
`interval_arith x int ==> iabs int < y ==> abs x < y`
[
   (((((fun arg_tac -> (use_arg_then "PAIR") (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "iabs")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma lin_approx_x *)
let lin_approx_x = section_proof []
`lin_approx (\x. x) x (x, x) (&1, &1)`
[
   (((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_x")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "REAL_DIFFERENTIABLE_ID")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_LE_REFL")(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma lin_approx_const *)
let lin_approx_const = section_proof ["c"]
`lin_approx (\x. c) x (c, c) (&0, &0)`
[
   ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_const")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "REAL_DIFFERENTIABLE_CONST")(thm_tac (new_rewrite [] [])))));
   ((repeat_tactic 1 9 (((use_arg_then "REAL_LE_REFL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Section LinearApproxUnivariateComposition *)
begin_section "LinearApproxUnivariateComposition";;
(add_section_hyp "approx_f" (`lin_approx f x f_bounds df_bounds`));;

(* Lemma lin_approx_compose_atn *)
let lin_approx_compose_atn = section_proof []
`(\x. atn (f x)) real_differentiable atreal x /\
	derivative (\x. atn (f x)) x = derivative f x / (&1 + f x * f x)`
[
   ((((use_arg_then "derivative_compose_atn") (disch_tac [])) THEN (clear_assumption "derivative_compose_atn") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lin_approx_compose_exp *)
let lin_approx_compose_exp = section_proof []
`(\x. exp (f x)) real_differentiable atreal x /\
	derivative (\x. exp (f x)) x = exp (f x) * derivative f x`
[
   ((((use_arg_then "derivative_compose_exp") (disch_tac [])) THEN (clear_assumption "derivative_compose_exp") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lin_approx_compose_inv *)
let lin_approx_compose_inv = section_proof []
`interval_not_zero f_bounds ==>
	(\x. inv (f x)) real_differentiable atreal x /\
	derivative (\x. inv (f x)) x = -- inv (f x * f x) * derivative f x`
[
   ((((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN ((move ["df"]) THEN (move ["h"]))) THEN (move ["f0"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`~(f x = &0)`))) (term_tac (have_gen_tac [](move ["fn0"])))) ((((use_arg_then "interval_arith_not_zero") (disch_tac [])) THEN (clear_assumption "interval_arith_not_zero") THEN (DISCH_THEN apply_tac)) THEN ((use_arg_then "f_bounds") (term_tac exists_tac)) THEN (done_tac)));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then "derivative_compose_inv") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "fn0") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (exact_tac));
];;

(* Lemma lin_approx_compose_sqrt *)
let lin_approx_compose_sqrt = section_proof []
`interval_pos f_bounds ==>
	(\x. sqrt (f x)) real_differentiable atreal x /\
	derivative (\x. sqrt (f x)) x = derivative f x / (&2 * sqrt (f x))`
[
   ((((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN ((move ["df"]) THEN (move ["h"]))) THEN (move ["f_ineq"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`&0 < f x`))) (term_tac (have_gen_tac [](move ["f_pos"])))) ((((use_arg_then "interval_arith_pos") (disch_tac [])) THEN (clear_assumption "interval_arith_pos") THEN (DISCH_THEN apply_tac)) THEN ((use_arg_then "f_bounds") (term_tac exists_tac)) THEN (done_tac)));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then "derivative_compose_sqrt") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f_pos") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (exact_tac));
];;

(* Lemma lin_approx_compose_acs *)
let lin_approx_compose_acs = section_proof []
`iabs f_bounds < &1 ==>
	(\x. acs (f x)) real_differentiable atreal x /\
	derivative (\x. acs (f x)) x = -- (derivative f x / sqrt (&1 - f x * f x))`
[
   ((((use_arg_then "approx_f") (disch_tac [])) THEN (clear_assumption "approx_f") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN ((move ["df"]) THEN (move ["h"]))) THEN (move ["f_ineq"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`abs (f x) < &1`))) (term_tac (have_gen_tac [](move ["f_abs"])))) ((((use_arg_then "interval_arith_abs") (disch_tac [])) THEN (clear_assumption "interval_arith_abs") THEN (DISCH_THEN apply_tac)) THEN ((use_arg_then "f_bounds") (term_tac exists_tac)) THEN (done_tac)));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then "derivative_compose_acs") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f_abs") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (exact_tac));
];;

(* Finalization of the section LinearApproxUnivariateComposition *)
let lin_approx_compose_atn = finalize_theorem lin_approx_compose_atn;;
let lin_approx_compose_exp = finalize_theorem lin_approx_compose_exp;;
let lin_approx_compose_inv = finalize_theorem lin_approx_compose_inv;;
let lin_approx_compose_sqrt = finalize_theorem lin_approx_compose_sqrt;;
let lin_approx_compose_acs = finalize_theorem lin_approx_compose_acs;;
end_section "LinearApproxUnivariateComposition";;
(add_section_hyp "approx_f" (`lin_approx f x f_bounds df_bounds`));;
(add_section_hyp "approx_g" (`lin_approx g x g_bounds dg_bounds`));;

(* Lemma lin_approx_imp_add_diff *)
let lin_approx_imp_add_diff = section_proof []
`(\x. f x + g x) real_differentiable atreal x`
[
   (((use_arg_then "REAL_DIFFERENTIABLE_ADD") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_ADD") THEN (DISCH_THEN apply_tac));
   (((((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_g") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lin_approx_imp_sub_diff *)
let lin_approx_imp_sub_diff = section_proof []
`(\x. f x - g x) real_differentiable atreal x`
[
   (((use_arg_then "REAL_DIFFERENTIABLE_SUB") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_SUB") THEN (DISCH_THEN apply_tac));
   (((((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_g") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lin_approx_imp_mul_diff *)
let lin_approx_imp_mul_diff = section_proof []
`(\x. f x * g x) real_differentiable atreal x`
[
   (((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_MUL_ATREAL") THEN (DISCH_THEN apply_tac));
   (((((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_g") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lin_approx_imp_div_diff *)
let lin_approx_imp_div_diff = section_proof []
`interval_not_zero g_bounds ==> (\x. f x / g x) real_differentiable atreal x`
[
   ((BETA_TAC THEN (move ["gn0"])) THEN (((use_arg_then "REAL_DIFFERENTIABLE_DIV_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_DIV_ATREAL") THEN (DISCH_THEN apply_tac)));
   ((((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then "lin_approx_imp_f_diff") (fun fst_arg -> (use_arg_then "approx_g") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "interval_arith_not_zero") (disch_tac [])) THEN (clear_assumption "interval_arith_not_zero") THEN (DISCH_THEN apply_tac)) THEN ((use_arg_then "g_bounds") (term_tac exists_tac)));
   ((((use_arg_then "approx_g") (disch_tac [])) THEN (clear_assumption "approx_g") THEN BETA_TAC) THEN ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section MoreLinearApproximation *)
let interval_arith_not_zero = finalize_theorem interval_arith_not_zero;;
let interval_arith_pos = finalize_theorem interval_arith_pos;;
let interval_arith_abs = finalize_theorem interval_arith_abs;;
let lin_approx_x = finalize_theorem lin_approx_x;;
let lin_approx_const = finalize_theorem lin_approx_const;;
let lin_approx_compose_atn = finalize_theorem lin_approx_compose_atn;;
let lin_approx_compose_exp = finalize_theorem lin_approx_compose_exp;;
let lin_approx_compose_inv = finalize_theorem lin_approx_compose_inv;;
let lin_approx_compose_sqrt = finalize_theorem lin_approx_compose_sqrt;;
let lin_approx_compose_acs = finalize_theorem lin_approx_compose_acs;;
let lin_approx_imp_add_diff = finalize_theorem lin_approx_imp_add_diff;;
let lin_approx_imp_sub_diff = finalize_theorem lin_approx_imp_sub_diff;;
let lin_approx_imp_mul_diff = finalize_theorem lin_approx_imp_mul_diff;;
let lin_approx_imp_div_diff = finalize_theorem lin_approx_imp_div_diff;;
end_section "MoreLinearApproximation";;

(* Section LinearApproxArith *)
begin_section "LinearApproxArith";;
(add_section_var (mk_var ("f1", (`:real->real`))); add_section_var (mk_var ("f2", (`:real->real`))));;
(add_section_var (mk_var ("f1_bounds", (`:real#real`))); add_section_var (mk_var ("f2_bounds", (`:real#real`))));;
(add_section_var (mk_var ("df1_lo", (`:real`))); add_section_var (mk_var ("df1_hi", (`:real`))); add_section_var (mk_var ("df2_lo", (`:real`))); add_section_var (mk_var ("df2_hi", (`:real`))));;
(add_section_var (mk_var ("f_bounds", (`:real#real`))));;
(add_section_var (mk_var ("df_lo", (`:real`))); add_section_var (mk_var ("df_hi", (`:real`))));;
(add_section_var (mk_var ("x", (`:real`))));;
(add_section_hyp "approx1" (`lin_approx f1 x f1_bounds (df1_lo, df1_hi)`));;

(* Lemma lin_approx_scale *)
let lin_approx_scale = section_proof ["c"]
`&0 <= c ==> interval_arith (c * f1 x) f_bounds /\
	df_lo <= c * df1_lo /\ c * df1_hi <= df_hi ==>
	lin_approx (\x. c * f1 x) x f_bounds (df_lo, df_hi)`
[
   (((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (move ["c0"]) THEN (move ["ineqs"]));
   (((((use_arg_then "lin_approx")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (split_tac));
   ((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ineqs") (disch_tac [])) THEN (clear_assumption "ineqs") THEN ((use_arg_then "approx1") (disch_tac [])) THEN (clear_assumption "approx1") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "approx1") (disch_tac [])) THEN (clear_assumption "approx1") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["int_f1"])) THEN (case THEN (move ["f1'"])) THEN (case THEN ((move ["df1"]) THEN (move ["int_f1'"])))));
   (((fun arg_tac -> arg_tac (Arg_term (`c * f1'`))) (term_tac exists_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_LMUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "int_f1'") (disch_tac [])) THEN (clear_assumption "int_f1'") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (move ["ineqs2"])) THEN (split_tac) THEN (((use_arg_then "REAL_LE_TRANS") (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)));
   (((fun arg_tac -> arg_tac (Arg_term (`c * df1_lo`))) (term_tac exists_tac)) THEN ((((use_arg_then "ineqs")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_LMUL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`c * df1_hi`))) (term_tac exists_tac)) THEN ((((use_arg_then "ineqs")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_LMUL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;
(add_section_hyp "approx2" (`lin_approx f2 x f2_bounds (df2_lo, df2_hi)`));;

(* Lemma lin_approx_add *)
let lin_approx_add = section_proof []
`interval_arith (f1 x + f2 x) f_bounds /\
	df_lo <= df1_lo + df2_lo /\ df1_hi + df2_hi <= df_hi ==>
	lin_approx (\x. f1 x + f2 x) x f_bounds (df_lo, df_hi)`
[
   (((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (move ["ineqs"]));
   (((((use_arg_then "lin_approx")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (split_tac));
   ((((use_arg_then "pair_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ineqs") (disch_tac [])) THEN (clear_assumption "ineqs") THEN ((use_arg_then "approx2") (disch_tac [])) THEN (clear_assumption "approx2") THEN ((use_arg_then "approx1") (disch_tac [])) THEN (clear_assumption "approx1") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "approx2") (disch_tac [])) THEN (clear_assumption "approx2") THEN ((use_arg_then "approx1") (disch_tac [])) THEN (clear_assumption "approx1") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "lin_approx")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["int_f1"])) THEN (case THEN (move ["f1'"])) THEN (case THEN ((move ["df1"]) THEN (move ["int_f1'"])))));
   (BETA_TAC THEN (case THEN (move ["int_f2"])) THEN (case THEN (move ["f2'"])) THEN (case THEN ((move ["df2"]) THEN (move ["int_f2'"]))));
   (((fun arg_tac -> arg_tac (Arg_term (`f1' + f2':real`))) (term_tac exists_tac)) THEN ((((use_arg_then "HAS_REAL_DERIVATIVE_ADD")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "int_f2'") (disch_tac [])) THEN (clear_assumption "int_f2'") THEN ((use_arg_then "int_f1'") (disch_tac [])) THEN (clear_assumption "int_f1'") THEN ((use_arg_then "ineqs") (disch_tac [])) THEN (clear_assumption "ineqs") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section LinearApproxArith *)
let lin_approx_scale = finalize_theorem lin_approx_scale;;
let lin_approx_add = finalize_theorem lin_approx_add;;
end_section "LinearApproxArith";;

(* Section SecondDerivativeBound *)
begin_section "SecondDerivativeBound";;
(add_section_var (mk_var ("f1", (`:real->real`))); add_section_var (mk_var ("f2", (`:real->real`))));;
(add_section_var (mk_var ("int", (`:real#real`))));;
(add_section_var (mk_var ("dd1", (`:real#real`))); add_section_var (mk_var ("dd2", (`:real#real`))));;

(* Lemma nth_diff_strong2_eq_alt *)
let nth_diff_strong2_eq_alt = section_proof ["f";"x"]
`nth_diff_strong 2 f x <=> ?s. real_open s /\ x IN s /\
	!y. y IN s ==> (f has_real_derivative derivative f y) (atreal y) /\
	(derivative f has_real_derivative nth_derivative 2 f y) (atreal y)`
[
   ((((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `i < 2 <=> i = 0 \/ i = 1`)))(thm_tac (new_rewrite [] [])))));
   ((split_tac) THEN ALL_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["df"]));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (case THEN (move ["_"])) THEN (move ["h"])));
   ((((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`1`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN ((fun arg_tac -> (use_arg_then "h") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`0`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (simp_tac)) THEN (((((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "TWO")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"])));
   ((((use_arg_then "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "real_differentiable")(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> arg_tac (Arg_term (`derivative f y`))) (term_tac exists_tac)) THEN (done_tac));
   ((BETA_TAC THEN (move ["i"])) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN ((repeat_tactic 0 10 (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then "TWO")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then "nth_derivative0")(fun tmp_arg1 -> (use_arg_then "nth_derivative1")(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma nth_diff_strong2_eq *)
let nth_diff_strong2_eq = section_proof ["f";"x"]
`nth_diff_strong 2 f x <=> ?s. real_open s /\ x IN s /\
	!y. y IN s ==> f real_differentiable atreal y /\ derivative f real_differentiable atreal y`
[
   ((((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "TWO")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ONE")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL nth_differentiable)))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] [])))));
   ((split_tac) THEN ALL_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["df"]));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"])) THEN (done_tac));
   (((use_arg_then "s") (term_tac exists_tac)) THEN (((((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["y"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then "df") (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (move ["h"])));
   (((repeat_tactic 1 9 (((use_arg_then "h")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lin_approx_compose *)
let lin_approx_compose = section_proof ["f";"g";"y";"g_bounds";"f_bounds";"d_bounds"]
`nth_diff_strong_int 2 g_bounds f ==>
	  g real_differentiable atreal y ==>
	  interval_arith (g y) g_bounds ==>
	  bounded_on_int f g_bounds f_bounds ==>
	  interval_arith (derivative g y * derivative f (g y)) d_bounds ==>
	  lin_approx (\x. f (g x)) y f_bounds d_bounds`
[
   (((((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["g_int"]) THEN (move ["f_int"]) THEN (move ["dfg_int"]));
   ((fun arg_tac -> arg_tac (Arg_term (`f real_differentiable atreal (g y)`))) (term_tac (have_gen_tac [](move ["dfgy"]))));
   ((((fun arg_tac -> (use_arg_then "df") (fun fst_arg -> (use_arg_then "g_int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["ys"])) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "lin_approx_eq")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "dfg_int")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "f_int")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andbT")(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") (disch_tac [])) THEN (clear_assumption "REAL_DIFFERENTIABLE_COMPOSE_ATREAL") THEN (exact_tac)) THEN (done_tac));
];;

(* Lemma second_derivative_atn_eq *)
let second_derivative_atn_eq = section_proof ["x"]
`((\x. inv (&1 + x pow 2)) has_real_derivative 
	(-- &2 * x) * inv (&1 + x pow 2) pow 2) (atreal x)`
[
   (((((use_arg_then "REAL_POW_INV")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_div")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_NEG_LMUL")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_INV_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((((fun arg_tac -> (use_arg_then "REAL_ADD_LID") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`&2 * x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ADD")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_CONST")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "REAL_POW_2")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `&2 * x = x * &1 + &1 * x`)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "HAS_REAL_DERIVATIVE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "REAL_POS_NZ")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LT_ADD1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_POW_2")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_atn *)
let second_derivative_atn = section_proof []
`derivative (derivative atn) = (\x. (-- &2 * x) * inv (&1 + x pow 2) pow 2)`
[
   ((((((use_arg_then "derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "eq_ext")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["x"]) THEN (simp_tac)) THEN (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)));
   (((((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "second_derivative_atn_eq")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_atn *)
let diff2_atn = section_proof ["x"]
`nth_diff_strong 2 atn x`
[
   ((((use_arg_then "nth_diff_strong2_eq_alt")(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> arg_tac (Arg_term (`(:real)`))) (term_tac exists_tac)));
   (((((use_arg_then "IN_UNIV")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_OPEN_UNIV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "IN_UNIV")(thm_tac (new_rewrite [] []))))) THEN (move ["y"]) THEN (simp_tac));
   ((((use_arg_then "derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ATN")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "second_derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "second_derivative_atn_eq")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_inv *)
let second_derivative_inv = section_proof ["x"]
`~(x = &0) ==> nth_derivative 2 inv x = &2 * inv (x pow 3)`
[
   ((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (move ["xn0"]));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac)));
   (((fun arg_tac -> arg_tac (Arg_term (`(\x. -- inv (x * x))`))) (term_tac exists_tac)) THEN (split_tac));
   (((fun arg_tac -> arg_tac (Arg_theorem ((DISCH_ALL o REAL_DIFF_CONV) `((\x. -- inv(x * x)) has_real_derivative f) (atreal x)`))) (disch_tac [])) THEN BETA_TAC);
   ((((use_arg_then "REAL_ENTIRE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xn0")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_NEG_NEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_RID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_2")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `(x * x) pow 2 = x * x pow 3`)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] []))))));
   (((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a. (&2 * x) * inv x * a = &2 * (x * inv x) * a`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - abs x, x + abs x)`))) (term_tac exists_tac));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))))) THEN ((THENL) (split_tac) [ALL_TAC; ((move ["y"]) THEN (move ["ineq"]))]));
   ((((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "derivative_inv")(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac)));
   ((((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma diff2_inv *)
let diff2_inv = section_proof ["x"]
`~(x = &0) ==> nth_diff_strong 2 inv x`
[
   ((((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] [])))) THEN (move ["xn0"]));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - abs x, x + abs x)`))) (term_tac exists_tac));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))))) THEN ((THENL) (split_tac) [ALL_TAC; ((move ["y"]) THEN (move ["ineq"]))]));
   ((((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then "REAL_DIFFERENTIABLE_AT_INV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "differentiable_local") (disch_tac [])) THEN (clear_assumption "differentiable_local") THEN (DISCH_THEN apply_tac)) THEN (((fun arg_tac -> arg_tac (Arg_term (`\x. --inv (x * x)`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`real_interval (y - abs y, y + abs y)`))) (term_tac exists_tac))));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((((use_arg_then "REAL_DIFFERENTIABLE_NEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_INV_ATREAL")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_DIFFERENTIABLE_ID")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_ENTIRE")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL) (split_tac) [ALL_TAC; ((move ["z"]) THEN (move ["ineq2"]))]);
   ((((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then "derivative_inv")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then "xn0") (disch_tac [])) THEN (clear_assumption "xn0") THEN ((use_arg_then "ineq2") (disch_tac [])) THEN (clear_assumption "ineq2") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma SQRT_POW *)
let SQRT_POW = section_proof ["x";"n"]
`&0 <= x ==> sqrt x pow n = sqrt (x pow n)`
[
   ((BETA_TAC THEN (move ["ineq"])) THEN ((THENL) (((use_arg_then "n") (disch_tac [])) THEN (clear_assumption "n") THEN elim) [ALL_TAC; ((move ["n"]) THEN (move ["IHn"]))]) THEN ((repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL real_pow)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "SQRT_1")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))));
   (((((use_arg_then "IHn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SQRT_MUL")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_POW_LE")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_sqrt *)
let second_derivative_sqrt = section_proof ["x"]
`&0 < x ==> nth_derivative 2 sqrt x = -- inv(&4 * sqrt (x pow 3))`
[
   ((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (move ["x_pos"]));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac)));
   (((fun arg_tac -> arg_tac (Arg_term (`(\x. inv (&2 * sqrt x))`))) (term_tac exists_tac)) THEN (split_tac));
   (((fun arg_tac -> arg_tac (Arg_theorem ((DISCH_ALL o REAL_DIFF_CONV) `((\x. inv(&2 * sqrt x)) has_real_derivative f) (atreal x)`))) (disch_tac [])) THEN BETA_TAC);
   ((fun arg_tac -> arg_tac (Arg_term (`~(&2 * sqrt x = &0)`))) (term_tac (have_gen_tac [](move ["ineq"]))));
   (((((use_arg_then "REAL_ENTIRE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "negb_or")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SQRT_EQ_0")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))) THEN (((use_arg_then "x_pos") (disch_tac [])) THEN (clear_assumption "x_pos") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "x_pos")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ineq")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_INV_POW")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_LNEG")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_MUL_ASSOC")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL real_pow)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_INV_POW")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_POW_MUL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "SQRT_POW")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then "REAL_MUL_ASSOC")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL real_pow)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_ASSOC")(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `&2 pow 2 = &4`)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `SUC 2 = 3`)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (&0, x + &1)`))) (term_tac exists_tac));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))))) THEN ((THENL) (split_tac) [ALL_TAC; ((move ["y"]) THEN (move ["ineq"]))]));
   ((((use_arg_then "x_pos") (disch_tac [])) THEN (clear_assumption "x_pos") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "derivative_sqrt")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma diff2_sqrt *)
let diff2_sqrt = section_proof ["x"]
`&0 < x ==> nth_diff_strong 2 sqrt x`
[
   ((((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] [])))) THEN (move ["x_pos"]));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (&0, x + &1)`))) (term_tac exists_tac));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))))) THEN ((THENL) (split_tac) [ALL_TAC; ((move ["y"]) THEN (move ["ineq"]))]));
   ((((use_arg_then "x_pos") (disch_tac [])) THEN (clear_assumption "x_pos") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then "REAL_DIFFERENTIABLE_AT_SQRT")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((((use_arg_then "x_pos") (disch_tac [])) THEN (clear_assumption "x_pos") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "differentiable_local") (disch_tac [])) THEN (clear_assumption "differentiable_local") THEN (DISCH_THEN apply_tac)) THEN (((fun arg_tac -> arg_tac (Arg_term (`\x. inv (&2 * sqrt x)`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`real_interval (&0, y + &1)`))) (term_tac exists_tac))));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((((use_arg_then "REAL_DIFFERENTIABLE_INV_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_DIFFERENTIABLE_CONST")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_AT_SQRT")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "REAL_ENTIRE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SQRT_EQ_0")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))) THEN (((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((THENL) (split_tac) [ALL_TAC; ((move ["z"]) THEN (move ["ineq2"]))]) ((((use_arg_then "x_pos") (disch_tac [])) THEN (clear_assumption "x_pos") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "derivative_sqrt")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma real_powS *)
let real_powS = section_proof ["x";"n"]
`x pow (SUC n) = x * x pow n`
[
   ((((use_arg_then "real_pow")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma second_derivative_acs *)
let second_derivative_acs = section_proof ["x"]
`abs x < &1 ==> 
	nth_derivative 2 acs x = --(x / sqrt ((&1 - x * x) pow 3))`
[
   ((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (move ["x_ineq"]));
   ((((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac)));
   (((fun arg_tac -> arg_tac (Arg_term (`\x. --inv (sqrt (&1 - x * x))`))) (term_tac exists_tac)) THEN (split_tac));
   (((fun arg_tac -> arg_tac (Arg_theorem ((DISCH_ALL o REAL_DIFF_CONV) `((\x. --inv (sqrt (&1 - x * x))) has_real_derivative f) (atreal x)`))) (disch_tac [])) THEN BETA_TAC);
   ((fun arg_tac -> arg_tac (Arg_term (`&0 < &1 - x * x /\ ~(sqrt(&1 - x * x) = &0)`))) (term_tac (have_gen_tac [](move ["ineqs"]))));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`&0 < &1 - x * x`))) (term_tac (have_gen_tac [](move ["h"])))));
   (((((use_arg_then "SQRT_EQ_0")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))) THEN (((use_arg_then "h") (disch_tac [])) THEN (clear_assumption "h") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `&1 - x * x = (&1 - x) * (&1 + x)`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LT_MUL")(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((repeat_tactic 1 9 (((use_arg_then "ineqs")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_SUB_LZERO")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_NEG_NEG")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_ASSOC")(gsym_then (thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then "REAL_INV_MUL")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "real_powS")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "SQRT_POW")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_RID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_2")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a. (&2 * x) * inv (&2) * a = (&2 * inv (&2)) * x * a`)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `SUC 2 = 3`)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`&1 - abs x`))) (term_tac (set_tac "e")));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - e, x + e)`))) (term_tac exists_tac));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))))) THEN ((THENL) (split_tac) [ALL_TAC; ((move ["y"]) THEN (move ["ineq"]))]));
   ((((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "derivative_acs")(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac)));
   ((((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma diff2_acs *)
let diff2_acs = section_proof ["x"]
`abs x < &1 ==> nth_diff_strong 2 acs x`
[
   ((((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] [])))) THEN (move ["x_ineq"]));
   ((fun arg_tac -> arg_tac (Arg_term (`&1 - abs x`))) (term_tac (set_tac "e")));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - e, x + e)`))) (term_tac exists_tac));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))))) THEN ((THENL) (split_tac) [ALL_TAC; ((move ["y"]) THEN (move ["ineq"]))]));
   ((((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then "REAL_DIFFERENTIABLE_AT_ACS")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) ((((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`&1 - abs y`))) (term_tac (set_tac "e2")));
   ((((use_arg_then "differentiable_local") (disch_tac [])) THEN (clear_assumption "differentiable_local") THEN (DISCH_THEN apply_tac)) THEN (((fun arg_tac -> arg_tac (Arg_term (`\x. --inv (sqrt (&1 - x * x))`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`real_interval (y - e2, y + e2)`))) (term_tac exists_tac))));
   (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((((use_arg_then "REAL_DIFFERENTIABLE_NEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_INV_ATREAL")(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> arg_tac (Arg_term (`&0 < &1 - y * y`))) (term_tac (have_gen_tac [](move ["gt0"]))));
   (((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `&1 - y * y = (&1 - y) * (&1 + y)`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LT_MUL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "SQRT_EQ_0")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_LT_IMP_NZ")(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`(\x. sqrt (&1 - x * x)) = sqrt o (\x. &1 - x * x)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   (((((use_arg_then "eq_ext")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "o_THM")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
   ((((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_AT_SQRT")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then "REAL_DIFFERENTIABLE_SUB")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_DIFFERENTIABLE_CONST")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then "REAL_DIFFERENTIABLE_ID")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((THENL) (split_tac) [ALL_TAC; ((move ["z"]) THEN (move ["ineq2"]))]);
   ((((use_arg_then "e2_def") (disch_tac [])) THEN (clear_assumption "e2_def") THEN ((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "derivative_acs")(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac)));
   ((((use_arg_then "ineq2") (disch_tac [])) THEN (clear_assumption "ineq2") THEN ((use_arg_then "e2_def") (disch_tac [])) THEN (clear_assumption "e2_def") THEN ((use_arg_then "e_def") (disch_tac [])) THEN (clear_assumption "e_def") THEN ((use_arg_then "ineq") (disch_tac [])) THEN (clear_assumption "ineq") THEN ((use_arg_then "x_ineq") (disch_tac [])) THEN (clear_assumption "x_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Section SecondDerivativeCompose *)
begin_section "SecondDerivativeCompose";;

(* Lemma REAL_CONTINUOUS_OPEN_PREIMAGE *)
let REAL_CONTINUOUS_OPEN_PREIMAGE = section_proof ["f";"s";"t"]
`f real_continuous_on s ==> real_open s ==> real_open t ==>
	real_open {x | x IN s /\ f x IN t}`
[
   (BETA_TAC THEN (move ["f_cont"]) THEN (move ["open_s"]) THEN (move ["open_t"]));
   (((use_arg_then "REAL_OPEN")(thm_tac (new_rewrite [] []))));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`IMAGE lift {x | x IN s /\ f x IN t} = 
	{x | x IN (IMAGE lift s) /\ (lift o f o drop) x IN (IMAGE lift t)}`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))));
   (((((use_arg_then "CONTINUOUS_OPEN_PREIMAGE")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_OPEN")(gsym_then (thm_tac (new_rewrite [] []))))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_CONTINUOUS_ON")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((((use_arg_then "EXTENSION")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "IN_IMAGE_LIFT_DROP")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_IMAGE_LIFT_DROP")(thm_tac (new_rewrite [] [])))))) THEN (move ["z"]) THEN (simp_tac)) THEN (split_tac));
   ((BETA_TAC THEN (case THEN (move ["x"])) THEN (case THEN (move ["x_in"])) THEN (move ["x_eq"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`lift x`))) (term_tac exists_tac)));
   (((repeat_tactic 1 9 (((use_arg_then "o_THM")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "LIFT_DROP")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "x_in")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "x_eq")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "LIFT_DROP")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((repeat_tactic 1 9 (((use_arg_then "o_THM")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "LIFT_DROP")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["x"])) THEN (case THEN (move ["x_in"])) THEN (move ["x_eq"]));
   (((fun arg_tac -> arg_tac (Arg_term (`drop x`))) (term_tac exists_tac)) THEN (done_tac));
];;

(* Lemma second_derivative_compose *)
let second_derivative_compose = section_proof ["f";"g";"x"]
`nth_diff_strong 2 g x ==> nth_diff_strong 2 f (g x) ==>
	nth_derivative 2 (\x. f (g x)) x = 
	nth_derivative 2 f (g x) * (derivative g x) pow 2 + derivative f (g x) * nth_derivative 2 g x`
[
   (BETA_TAC THEN (move ["dg"]) THEN (move ["df"]));
   ((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_unique") (disch_tac [])) THEN (clear_assumption "derivative_unique") THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_LOCAL") (disch_tac [])) THEN (clear_assumption "HAS_REAL_DERIVATIVE_LOCAL") THEN (DISCH_THEN apply_tac)));
   ((THENL_ROT (-1)) (((fun arg_tac -> arg_tac (Arg_term (`\x. derivative f (g x) * derivative g x`))) (term_tac exists_tac)) THEN (split_tac)));
   ((((use_arg_then "dg") (disch_tac [])) THEN ((use_arg_then "df") (disch_tac [])) THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["gxs"])) THEN (move ["d_f"]) THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["xt"])) THEN (move ["d_g"])));
   ((fun arg_tac -> arg_tac (Arg_term (`{z | z IN t /\ g z IN s}`))) (term_tac (set_tac "s'")));
   ((fun arg_tac -> arg_tac (Arg_term (`real_open s'`))) (term_tac (have_gen_tac [](move ["open_s'"]))));
   ((((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_CONTINUOUS_OPEN_PREIMAGE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "open_t")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbT")(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then "REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (move ["y"]) THEN (move ["yt"]));
   (((((use_arg_then "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "d_g")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`t INTER s'`))) (term_tac exists_tac)) THEN ((((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   (((((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xt")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN ((use_arg_then "x") (term_tac exists_tac)) THEN (done_tac));
   ((BETA_TAC THEN (move ["y"]) THEN (case THEN ((move ["yt"]) THEN (move ["ys'"])))) THEN ((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))));
   (((((use_arg_then "d_g")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "d_f")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then "ys'") (disch_tac [])) THEN (clear_assumption "ys'") THEN BETA_TAC) THEN (((((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["z"])) THEN (simp_tac)) THEN (done_tac));
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong2_eq_alt")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (move ["d_f"]) THEN (case THEN (move ["_"])) THEN (move ["d_g"])));
   ((THENL_ROT (-1)) (((((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_POW_2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_ASSOC")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_MUL_ATREAL")(thm_tac (new_rewrite [] []))))) THEN (split_tac)));
   ((((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "d_g")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`derivative f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN (((use_arg_then "REAL_DIFF_CHAIN_ATREAL")(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "d_g")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "d_f")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_compose *)
let diff2_compose = section_proof ["f";"g";"x"]
`nth_diff_strong 2 g x ==> nth_diff_strong 2 f (g x) ==>
	nth_diff_strong 2 (\x. f (g x)) x`
[
   ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (move ["dg"]) THEN (case THEN (move ["t"])) THEN (case THEN (move ["open_t"])) THEN (case THEN (move ["gxt"])) THEN (move ["df"]));
   ((fun arg_tac -> arg_tac (Arg_term (`{z | z IN s /\ g z IN t}`))) (term_tac (set_tac "s'")));
   ((fun arg_tac -> arg_tac (Arg_term (`real_open s'`))) (term_tac (have_gen_tac [](move ["open_s'"]))));
   ((((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_CONTINUOUS_OPEN_PREIMAGE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "open_t")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "open_s")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbT")(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then "REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (move ["y"]) THEN (move ["yt"]));
   (((((use_arg_then "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "dg")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`s INTER s'`))) (term_tac exists_tac)) THEN ((((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   (((((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "xs")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "x") (term_tac exists_tac)) THEN (simp_tac)) THEN (done_tac));
   (BETA_TAC THEN (move ["y"]) THEN (case THEN ((move ["ys"]) THEN (move ["ys'"]))));
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "dg")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac))));
   ((((use_arg_then "ys'") (disch_tac [])) THEN (clear_assumption "ys'") THEN BETA_TAC) THEN (((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["z"])) THEN (simp_tac)) THEN (done_tac));
   (((use_arg_then "differentiable_local") (disch_tac [])) THEN (clear_assumption "differentiable_local") THEN (DISCH_THEN apply_tac));
   ((THENL_ROT (-1)) ((((fun arg_tac -> arg_tac (Arg_term (`\x. derivative g x * derivative f (g x)`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`s INTER s'`))) (term_tac exists_tac))) THEN (split_tac)));
   (((((use_arg_then "REAL_OPEN_INTER")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then "IN_INTER")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ys")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ys'")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["z"]) THEN (move ["z_in"]));
   ((((use_arg_then "derivative_composition")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "dg")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then "z_in") (disch_tac [])) THEN (clear_assumption "z_in") THEN BETA_TAC) THEN (((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (simp_tac)) THEN (done_tac));
   ((((use_arg_then "REAL_DIFFERENTIABLE_MUL_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((use_arg_then "dg")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then "o_THM") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`derivative f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "ETA_AX")(fun arg -> ONCE_REWRITE_TAC[get_arg_thm arg]))) THEN ((((use_arg_then "REAL_DIFFERENTIABLE_COMPOSE_ATREAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "dg")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))));
   ((((use_arg_then "ys'") (disch_tac [])) THEN (clear_assumption "ys'") THEN BETA_TAC) THEN (((use_arg_then "s'_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((((use_arg_then "IN_ELIM_THM")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["z"])) THEN (simp_tac)) THEN (done_tac));
];;

(* Lemma continuous_not0_exists_open *)
let continuous_not0_exists_open = section_proof ["f";"x"]
`~(f x = &0) ==> f real_continuous atreal x ==>
	?s. real_open s /\ x IN s /\ !y. y IN s ==> ~(f y = &0)`
[
   ((((use_arg_then "real_continuous_atreal")(thm_tac (new_rewrite [] [])))) THEN (move ["fn0"]) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`abs (f x)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)));
   ((THENL_FIRST) (ANTS_TAC) ((((use_arg_then "fn0") (disch_tac [])) THEN (clear_assumption "fn0") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   (BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d0"])) THEN (move ["ineq"]));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - d, x + d)`))) (term_tac exists_tac));
   ((THENL_FIRST) (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN ((THENL) (split_tac) [ALL_TAC; (move ["y"])])) ((((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "fn0") (disch_tac [])) THEN (clear_assumption "fn0") THEN ((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN ((fun arg_tac -> (use_arg_then "ineq") (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma continuous_gt_exists_open *)
let continuous_gt_exists_open = section_proof ["a";"f";"x"]
`a < f x ==> f real_continuous atreal x ==>
	?s. real_open s /\ x IN s /\ !y. y IN s ==> a < f y`
[
   ((BETA_TAC THEN (move ["f_ineq"])) THEN ((((use_arg_then "real_continuous_atreal")(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`f x - a`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))));
   ((THENL_FIRST) (ANTS_TAC) ((((use_arg_then "f_ineq") (disch_tac [])) THEN (clear_assumption "f_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   (BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d0"])) THEN (move ["ineq"]));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - d, x + d)`))) (term_tac exists_tac));
   ((THENL_FIRST) (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN ((THENL) (split_tac) [ALL_TAC; (move ["y"])])) ((((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "f_ineq") (disch_tac [])) THEN (clear_assumption "f_ineq") THEN ((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN ((fun arg_tac -> (use_arg_then "ineq") (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma continuous_lt_exists_open *)
let continuous_lt_exists_open = section_proof ["a";"f";"x"]
`f x < a ==> f real_continuous atreal x ==>
	?s. real_open s /\ x IN s /\ !y. y IN s ==> f y < a`
[
   ((BETA_TAC THEN (move ["f_ineq"])) THEN ((((use_arg_then "real_continuous_atreal")(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`a - f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))));
   ((THENL_FIRST) (ANTS_TAC) ((((use_arg_then "f_ineq") (disch_tac [])) THEN (clear_assumption "f_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   (BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d0"])) THEN (move ["ineq"]));
   ((fun arg_tac -> arg_tac (Arg_term (`real_interval (x - d, x + d)`))) (term_tac exists_tac));
   ((THENL_FIRST) (((((use_arg_then "REAL_OPEN_REAL_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_REAL_INTERVAL")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN ((THENL) (split_tac) [ALL_TAC; (move ["y"])])) ((((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "f_ineq") (disch_tac [])) THEN (clear_assumption "f_ineq") THEN ((use_arg_then "d0") (disch_tac [])) THEN (clear_assumption "d0") THEN ((fun arg_tac -> (use_arg_then "ineq") (fun fst_arg -> (use_arg_then "y") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;
(add_section_var (mk_var ("f", (`:real->real`))));;
(add_section_var (mk_var ("x", (`:real`))));;
(add_section_hyp "df" (`nth_diff_strong 2 f x`));;

(* Lemma second_derivative_compose_atn *)
let second_derivative_compose_atn = section_proof []
`nth_derivative 2 (\x. atn (f x)) x = 
	(nth_derivative 2 f x * (&1 + f x * f x) - &2 * f x * derivative f x pow 2) / (&1 + f x * f x) pow 2`
[
   ((((use_arg_then "second_derivative_compose")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff2_atn")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "second_derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "real_sub")(gsym_then (thm_tac (new_rewrite [] []))))));
   (((fun arg_tac -> arg_tac (Arg_term (`_1 * _2`))) (term_tac (set_tac "lhs1"))) THEN ((fun arg_tac -> arg_tac (Arg_term (`_1 * _2`))) (term_tac (set_tac "lhs2"))));
   ((((use_arg_then "real_div")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_INV_POW")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_SUB_RDISTRIB")(thm_tac (new_rewrite [] [])))));
   (((fun arg_tac -> arg_tac (Arg_term (`_1 * _2`))) (term_tac (set_tac "rhs1"))) THEN ((fun arg_tac -> arg_tac (Arg_term (`_1 * _2`))) (term_tac (set_tac "rhs2"))));
   (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `lhs1 = rhs1 /\ lhs2 = rhs2 ==> lhs1 - lhs2 = rhs1 - rhs2`))) (disch_tac [])) THEN (DISCH_THEN apply_tac));
   (((((use_arg_then "lhs2_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "rhs2_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_ASSOC")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] [(`_1 * _2 pow 2`)])))) THEN (((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((split_tac) THEN ((TRY done_tac))));
   ((((use_arg_then "rhs1_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_POW_2")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_ASSOC")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. a * b * c * d = a * (b * c) * d`)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "lhs1_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((fun arg_tac -> (use_arg_then "REAL_LE_SQUARE") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma second_derivative_compose_inv *)
let second_derivative_compose_inv = section_proof []
`~(f x = &0) ==>
	nth_derivative 2 (\x. inv (f x)) x =
	(&2 * derivative f x pow 2 - nth_derivative 2 f x * f x) / (f x pow 3)`
[
   (BETA_TAC THEN (move ["fn0"]));
   ((((use_arg_then "second_derivative_compose")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff2_inv")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "second_derivative_inv")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "derivative_inv")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_sub")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_SUB_RDISTRIB")(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. a = c /\ b = d ==> a - b = c - d`))) (disch_tac [])) THEN (DISCH_THEN apply_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then "REAL_INV_POW")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `3 = SUC 2`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_powS")(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. (a * b) * c * d = a * (b * c) * d`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_POW_2")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_MUL_SYM")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_sqrt *)
let second_derivative_compose_sqrt = section_proof []
`&0 < f x ==>
	nth_derivative 2 (\x. sqrt (f x)) x =
	(&2 * nth_derivative 2 f x * f x - derivative f x pow 2) / (&4 * sqrt (f x pow 3))`
[
   (BETA_TAC THEN (move ["f_pos"]));
   ((((use_arg_then "second_derivative_compose")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff2_sqrt")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "second_derivative_sqrt")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "derivative_sqrt")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_sub")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_SUB_RDISTRIB")(thm_tac (new_rewrite [] [])))));
   ((THENL_LAST) ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. a = c /\ b = d ==> a - b = c - d`))) (disch_tac [])) THEN (DISCH_THEN apply_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`sqrt (f x pow 3) = sqrt (f x) * f x`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   (((((use_arg_then "SQRT_POW")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `3 = SUC 2`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_powS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SQRT_POW_2")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_IMP_LE")(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (repeat_tactic 1 9 (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))));
   (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. (&2 * a * b) * inv (&4) * c * d = (inv (&2) * c) * a * (b * d)`)))(thm_tac (new_rewrite [] []))));
   (((((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_POS_NZ")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_MUL_RID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_acs *)
let second_derivative_compose_acs = section_proof []
`abs (f x) < &1 ==>
	nth_derivative 2 (\x. acs (f x)) x =
	-- ((nth_derivative 2 f x * (&1 - f x * f x) + f x * derivative f x pow 2) / 
	sqrt ((&1 - f x * f x) pow 3))`
[
   (BETA_TAC THEN (move ["f_ineq"]));
   ((((use_arg_then "second_derivative_compose")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "diff2_acs")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "second_derivative_acs")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "derivative_acs")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((repeat_tactic 1 9 (((use_arg_then "REAL_MUL_LNEG")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_NEG_ADD")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_EQ_NEG2")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_ADD_RDISTRIB")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_ADD_SYM")(thm_tac (new_rewrite [] [])))));
   ((THENL_LAST) ((((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. a = c /\ b = d ==> a + b = c + d`))) (disch_tac [])) THEN (DISCH_THEN apply_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`&1 - f x * f x`))) (term_tac (set_tac "y")));
   ((fun arg_tac -> arg_tac (Arg_term (`&0 <= y /\ &0 < y /\ ~(y = &0)`))) (term_tac (have_gen_tac [](move ["y_ineq"]))));
   ((((use_arg_then "y_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a. &1 - a * a = (&1 - a) * (&1 + a)`)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_ENTIRE")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_MUL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "REAL_LT_MUL")(thm_tac (new_rewrite [] [])))))) THEN (((use_arg_then "f_ineq") (disch_tac [])) THEN (clear_assumption "f_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`sqrt (y pow 3) = sqrt y * y`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   (((((use_arg_then "SQRT_POW")(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `3 = SUC 2`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_powS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "SQRT_POW_2")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (REAL_ARITH `!a b c d. (a * b) * c * d = c * a * (b * d)`)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_MUL_RID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_compose_atn *)
let diff2_compose_atn = section_proof []
`nth_diff_strong 2 (\x. atn (f x)) x`
[
   (((((use_arg_then "diff2_compose")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "diff2_atn")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_compose_inv *)
let diff2_compose_inv = section_proof []
`~(f x = &0) ==> nth_diff_strong 2 (\x. inv (f x)) x`
[
   ((BETA_TAC THEN (move ["fn0"])) THEN ((((use_arg_then "diff2_compose")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "diff2_inv")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_compose_sqrt *)
let diff2_compose_sqrt = section_proof []
`&0 < f x ==> nth_diff_strong 2 (\x. sqrt (f x)) x`
[
   ((BETA_TAC THEN (move ["f_pos"])) THEN ((((use_arg_then "diff2_compose")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "diff2_sqrt")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_compose_acs *)
let diff2_compose_acs = section_proof []
`abs (f x) < &1 ==> nth_diff_strong 2 (\x. acs (f x)) x`
[
   ((BETA_TAC THEN (move ["f_abs"])) THEN ((((use_arg_then "diff2_compose")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "diff2_acs")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section SecondDerivativeCompose *)
let REAL_CONTINUOUS_OPEN_PREIMAGE = finalize_theorem REAL_CONTINUOUS_OPEN_PREIMAGE;;
let second_derivative_compose = finalize_theorem second_derivative_compose;;
let diff2_compose = finalize_theorem diff2_compose;;
let continuous_not0_exists_open = finalize_theorem continuous_not0_exists_open;;
let continuous_gt_exists_open = finalize_theorem continuous_gt_exists_open;;
let continuous_lt_exists_open = finalize_theorem continuous_lt_exists_open;;
let second_derivative_compose_atn = finalize_theorem second_derivative_compose_atn;;
let second_derivative_compose_inv = finalize_theorem second_derivative_compose_inv;;
let second_derivative_compose_sqrt = finalize_theorem second_derivative_compose_sqrt;;
let second_derivative_compose_acs = finalize_theorem second_derivative_compose_acs;;
let diff2_compose_atn = finalize_theorem diff2_compose_atn;;
let diff2_compose_inv = finalize_theorem diff2_compose_inv;;
let diff2_compose_sqrt = finalize_theorem diff2_compose_sqrt;;
let diff2_compose_acs = finalize_theorem diff2_compose_acs;;
end_section "SecondDerivativeCompose";;

(* Lemma second_derivative_mul *)
let second_derivative_mul = section_proof ["f";"g";"x"]
`nth_diff_strong 2 f x ==>
  nth_diff_strong 2 g x ==>
  nth_derivative 2 (\x. f x * g x) x =
  f x * nth_derivative 2 g x + &2 * derivative f x * derivative g x + nth_derivative 2 f x * g x`
[
   ((BETA_TAC THEN (move ["df0"]) THEN (move ["dg0"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(x,x)`))) (term_tac (set_tac "int"))));
   ((fun arg_tac -> arg_tac (Arg_term (`interval_arith x int /\ nth_diff_strong_int 2 int f /\ nth_diff_strong_int 2 int g`))) (term_tac (have_gen_tac [](case THEN ((move ["ineq"]) THEN (case THEN ((move ["df"]) THEN (move ["dg"]))))))));
   ((repeat_tactic 1 9 (((use_arg_then "int_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "CONST_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   (((split_tac) THEN (move ["y"])) THEN (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ANTISYM")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_mul") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "dg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`2`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN ((((use_arg_then "leqnn")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))));
   ((((use_arg_then "TWO")(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then "ONE")(thm_tac (new_rewrite [1] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL SUM_CLAUSES_NUMSEG)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `!n. 0 <= SUC n`)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "TWO")(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "subnn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "subn0")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `2 - 1 = 1`)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then "binom")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "BINOM_1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "BINOM_REFL")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_ADD_ASSOC")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_div *)
let second_derivative_div = section_proof ["f";"g";"x"]
`~(g x = &0) ==>
	nth_diff_strong 2 f x ==>
	nth_diff_strong 2 g x ==>
	nth_derivative 2 (\x. f x / g x) x = 
	((nth_derivative 2 f x * g x - f x * nth_derivative 2 g x) * g x -
	&2 * derivative g x * (derivative f x * g x - f x * derivative g x)) / (g x pow 3)`
[
   (BETA_TAC THEN (move ["gn0"]) THEN (move ["diff_f"]) THEN (move ["diff_g"]));
   (((fun arg_tac -> arg_tac (Arg_term (`derivative g x`))) (term_tac (set_tac "dg"))) THEN ((fun arg_tac -> arg_tac (Arg_term (`derivative f x`))) (term_tac (set_tac "df"))));
   (((fun arg_tac -> arg_tac (Arg_term (`nth_derivative 2 g x`))) (term_tac (set_tac "ddg"))) THEN ((fun arg_tac -> arg_tac (Arg_term (`nth_derivative 2 f x`))) (term_tac (set_tac "ddf"))));
   ((repeat_tactic 1 9 (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "second_derivative_mul")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "second_derivative_compose_inv")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "diff2_compose_inv")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((use_arg_then "derivative_compose_inv")(thm_tac (new_rewrite [] []))));
   ((((use_arg_then "diff_g") (disch_tac [])) THEN (clear_assumption "diff_g") THEN BETA_TAC) THEN ((((use_arg_then "nth_diff_strong2_eq")(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["s"])) THEN (case THEN (move ["open_s"])) THEN (case THEN (move ["xs"])) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (done_tac));
   ((((use_arg_then "ddf_def")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ddg_def")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df_def")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "dg_def")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))));
   ((fun arg_tac -> arg_tac (Arg_term (`_1 + _2`))) (term_tac (set_tac "lhs")));
   (((fun arg_tac -> arg_tac (Arg_theorem (REAL_RING `!f g x. ((ddf * g x - f x * ddg) * g x - &2 * dg * (df * g x - f x * dg)) * 
						inv (g x pow 3) =
			f x * (&2 * dg pow 2 - ddg * g x) * inv (g x pow 3) +
			&2 * df * --(g x * inv (g x pow 3)) * dg +
			ddf * (g x * g x * inv (g x pow 3))`)))(thm_tac (new_rewrite [] []))));
   ((fun arg_tac -> arg_tac (Arg_term (`g x * inv (g x pow 3) = inv (g x pow 2)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   (((((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `3 = SUC 2`)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_powS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_ASSOC")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`g x * inv (g x pow 2) = inv (g x)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   (((((use_arg_then "TWO")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "real_powS")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_INV_MUL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_ASSOC")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_MUL_RINV")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "REAL_MUL_LID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_POW_1")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((fun arg_tac -> (use_arg_then "REAL_POW_2") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`g x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "lhs_def")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma diff2_div *)
let diff2_div = section_proof ["f";"g";"x"]
`~(g x = &0) ==> nth_diff_strong 2 f x ==> nth_diff_strong 2 g x ==>
	nth_diff_strong 2 (\x. f x / g x) x`
[
   ((BETA_TAC THEN (move ["gn0"]) THEN (move ["df0"]) THEN (move ["dg0"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(x,x)`))) (term_tac (set_tac "int"))));
   ((fun arg_tac -> arg_tac (Arg_term (`interval_arith x int /\ nth_diff_strong_int 2 int f /\ nth_diff_strong_int 2 int g`))) (term_tac (have_gen_tac [](case THEN ((move ["ineq"]) THEN (case THEN ((move ["df"]) THEN (move ["dg"]))))))));
   ((repeat_tactic 1 9 (((use_arg_then "int_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "CONST_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   (((split_tac) THEN (move ["y"])) THEN (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ANTISYM")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_diff_mul") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`2`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`\x. inv (g x)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN ((split_tac) THEN ((TRY done_tac)) THEN (move ["y"])));
   (((((use_arg_then "int_def")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ANTISYM")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((use_arg_then "diff2_compose_inv")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma second_derivative_scale *)
let second_derivative_scale = section_proof ["f";"c";"x"]
`nth_diff_strong 2 f x ==>
	nth_derivative 2 (\x. c * f x) x = c * nth_derivative 2 f x`
[
   ((BETA_TAC THEN (move ["df0"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(x,x)`))) (term_tac (set_tac "int"))));
   ((fun arg_tac -> arg_tac (Arg_term (`interval_arith x int /\ nth_diff_strong_int 2 int f`))) (term_tac (have_gen_tac [](case THEN ((move ["ineq"]) THEN (move ["df"]))))));
   ((repeat_tactic 1 9 (((use_arg_then "int_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "CONST_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   ((BETA_TAC THEN (move ["y"])) THEN (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ANTISYM")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_scale") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "c") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`2`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN ((((use_arg_then "leqnn")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_add *)
let second_derivative_add = section_proof ["f";"g";"x"]
`nth_diff_strong 2 f x ==>
	nth_diff_strong 2 g x ==>
	nth_derivative 2 (\x. f x + g x) x = nth_derivative 2 f x + nth_derivative 2 g x`
[
   ((BETA_TAC THEN (move ["df0"]) THEN (move ["dg0"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(x,x)`))) (term_tac (set_tac "int"))));
   ((fun arg_tac -> arg_tac (Arg_term (`interval_arith x int /\ nth_diff_strong_int 2 int f /\ nth_diff_strong_int 2 int g`))) (term_tac (have_gen_tac [](case THEN ((move ["ineq"]) THEN (case THEN ((move ["df"]) THEN (move ["dg"]))))))));
   ((repeat_tactic 1 9 (((use_arg_then "int_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "CONST_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   (((split_tac) THEN (move ["y"])) THEN (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ANTISYM")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_add") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "dg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`2`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN ((((use_arg_then "leqnn")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_sub *)
let second_derivative_sub = section_proof ["f";"g";"x"]
`nth_diff_strong 2 f x ==>
	nth_diff_strong 2 g x ==>
	nth_derivative 2 (\x. f x - g x) x = nth_derivative 2 f x - nth_derivative 2 g x`
[
   ((BETA_TAC THEN (move ["df0"]) THEN (move ["dg0"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(x,x)`))) (term_tac (set_tac "int"))));
   ((fun arg_tac -> arg_tac (Arg_term (`interval_arith x int /\ nth_diff_strong_int 2 int f /\ nth_diff_strong_int 2 int g`))) (term_tac (have_gen_tac [](case THEN ((move ["ineq"]) THEN (case THEN ((move ["df"]) THEN (move ["dg"]))))))));
   ((repeat_tactic 1 9 (((use_arg_then "int_def")(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "CONST_INTERVAL")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))));
   (((split_tac) THEN (move ["y"])) THEN (((((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "REAL_LE_ANTISYM")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "nth_derivative_sub") (fun fst_arg -> (use_arg_then "df") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "dg") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`2`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN ((((use_arg_then "leqnn")(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_bounds *)
let second_derivative_compose_bounds = section_proof ["f";"g";"int";"g_bounds";"dd_bounds"]
`nth_diff_strong_int 2 int g ==>
	bounded_on_int g int g_bounds ==>
	nth_diff_strong_int 2 g_bounds f ==>
	bounded_on_int (\x. nth_derivative 2 f (g x) * derivative g x pow 2 + 
			derivative f (g x) * nth_derivative 2 g x) int dd_bounds ==>
	has_bounded_second_derivative (\x. f (g x)) int dd_bounds`
[
   ((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac));
   ((BETA_TAC THEN (move ["dg"]) THEN (move ["g_bounded"]) THEN (move ["df"]) THEN (move ["bounded"])) THEN ((split_tac) THEN (move ["x"]) THEN (move ["ineq"])));
   (((((use_arg_then "diff2_compose")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "dg")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "g_bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_compose")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "dg")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "g_bounded")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_atn_bounds *)
let second_derivative_atn_bounds = section_proof ["dd_bounds"]
`bounded_on_int (\x. (-- &2 * x) * inv(&1 + x pow 2) pow 2) int dd_bounds ==>
	has_bounded_second_derivative atn int dd_bounds`
[
   (((((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] []))))) THEN (move ["ineq"]));
   ((THENL_LAST) (split_tac) (((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "second_derivative_atn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   (((((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "diff2_atn")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_atn_bounds *)
let second_derivative_compose_atn_bounds = section_proof ["f";"dd_bounds"]
`nth_diff_strong_int 2 int f ==>
	bounded_on_int (\x. (nth_derivative 2 f x * (&1 + f x * f x) - &2 * f x * derivative f x pow 2) /
			(&1 + f x * f x) pow 2) int dd_bounds ==>
	has_bounded_second_derivative (\x. atn (f x)) int dd_bounds`
[
   ((((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["df"]) THEN (move ["bounded"])) THEN ((split_tac) THEN (move ["x"]) THEN (move ["ineq"])));
   (((((use_arg_then "diff2_compose_atn")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_compose_atn")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_inv_bounds *)
let second_derivative_inv_bounds = section_proof ["dd_bounds"]
`interval_not_zero int ==>
	bounded_on_int (\x. &2 * inv (x pow 3)) int dd_bounds ==>
	has_bounded_second_derivative inv int dd_bounds`
[
   ((((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["int0"]) THEN (move ["bounded"])) THEN ((split_tac) THEN (move ["x"]) THEN (move ["ineq"])));
   (((((use_arg_then "diff2_inv")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_not_zero") (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_inv")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_not_zero") (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_inv_bounds *)
let second_derivative_compose_inv_bounds = section_proof ["f";"f_bounds";"dd_bounds"]
`bounded_on_int f int f_bounds ==> interval_not_zero f_bounds ==>
	nth_diff_strong_int 2 int f ==>
	bounded_on_int (\x. (&2 * derivative f x pow 2 - nth_derivative 2 f x * f x) / f x pow 3) 
			int dd_bounds ==>
	has_bounded_second_derivative (\x. inv (f x)) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["f_bounded"]) THEN (move ["f0"]) THEN (move ["df"]) THEN (move ["bounded"]));
   ((fun arg_tac -> arg_tac (Arg_term (`!x. interval_arith x int ==> ~(f x = &0)`))) (term_tac (have_gen_tac [](move ["fn0"]))));
   ((BETA_TAC THEN (move ["x"]) THEN (move ["ineq"])) THEN ((((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_not_zero") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "f_bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((split_tac) THEN (move ["x"]) THEN (move ["ineq"]));
   (((((use_arg_then "diff2_compose_inv")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "fn0")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_compose_inv")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "fn0")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_sqrt_bounds *)
let second_derivative_sqrt_bounds = section_proof ["dd_bounds"]
`interval_pos int ==>
	bounded_on_int (\x. --inv (&4 * sqrt (x pow 3))) int dd_bounds ==>
	has_bounded_second_derivative sqrt int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["int_pos"]) THEN (move ["bounded"]));
   ((split_tac) THEN (move ["x"]) THEN (move ["ineq"]));
   (((((use_arg_then "diff2_sqrt")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_pos") (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_sqrt")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_pos") (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_sqrt_bounds *)
let second_derivative_compose_sqrt_bounds = section_proof ["f";"f_bounds";"dd_bounds"]
`bounded_on_int f int f_bounds ==> interval_pos f_bounds ==>
	nth_diff_strong_int 2 int f ==>
	bounded_on_int (\x. (&2 * nth_derivative 2 f x * f x - derivative f x pow 2) /
			(&4 * sqrt (f x pow 3))) 
			int dd_bounds ==>
	has_bounded_second_derivative (\x. sqrt (f x)) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["f_bounded"]) THEN (move ["f_int"]) THEN (move ["df"]) THEN (move ["bounded"]));
   ((fun arg_tac -> arg_tac (Arg_term (`!x. interval_arith x int ==> &0 < f x`))) (term_tac (have_gen_tac [](move ["f_pos"]))));
   ((BETA_TAC THEN (move ["x"]) THEN (move ["ineq"])) THEN ((((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_pos") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "f_bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((split_tac) THEN (move ["x"]) THEN (move ["ineq"]));
   (((((use_arg_then "diff2_compose_sqrt")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "f_pos")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_compose_sqrt")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "f_pos")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_acs_bounds *)
let second_derivative_acs_bounds = section_proof ["dd_bounds"]
`iabs int < &1 ==>
	bounded_on_int (\x. --(x / sqrt ((&1 - x * x) pow 3))) int dd_bounds ==>
	has_bounded_second_derivative acs int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["int_abs"]) THEN (move ["bounded"]));
   ((split_tac) THEN (move ["x"]) THEN (move ["ineq"]));
   (((((use_arg_then "diff2_acs")(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_abs") (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_acs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_abs") (fun fst_arg -> (use_arg_then "x") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "int") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_compose_acs_bounds *)
let second_derivative_compose_acs_bounds = section_proof ["f";"f_bounds";"dd_bounds"]
`bounded_on_int f int f_bounds ==> iabs f_bounds < &1 ==>
	nth_diff_strong_int 2 int f ==>
	bounded_on_int (\x. --((nth_derivative 2 f x * (&1 - f x * f x) + f x * derivative f x pow 2) /
				sqrt ((&1 - f x * f x) pow 3))) int dd_bounds ==>
	has_bounded_second_derivative (\x. acs (f x)) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (simp_tac)) THEN (move ["f_bounded"]) THEN (move ["f_abs"]) THEN (move ["df"]) THEN (move ["bounded"]));
   ((fun arg_tac -> arg_tac (Arg_term (`!x. interval_arith x int ==> abs (f x) < &1`))) (term_tac (have_gen_tac [](move ["fabs"]))));
   ((BETA_TAC THEN (move ["x"]) THEN (move ["ineq"])) THEN ((((fun arg_tac -> (fun arg_tac -> (use_arg_then "interval_arith_abs") (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`f x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "f_bounds") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "f_bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((split_tac) THEN (move ["x"]) THEN (move ["ineq"]));
   (((((use_arg_then "diff2_compose_acs")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "fabs")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "second_derivative_compose_acs")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "fabs")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_scale_bounds *)
let second_derivative_scale_bounds = section_proof ["c";"f";"dd_bounds"]
`nth_diff_strong_int 2 int f ==>
	bounded_on_int (\x. c * nth_derivative 2 f x) int dd_bounds ==>
	has_bounded_second_derivative (\x. c * f x) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))))) THEN (simp_tac) THEN (move ["df"]) THEN (move ["b"]));
   (((((use_arg_then "nth_diff_scale")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (move ["ineq"]));
   ((((use_arg_then "second_derivative_scale")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "b")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (exact_tac) THEN (done_tac));
];;

(* Lemma second_derivative_add_bounds *)
let second_derivative_add_bounds = section_proof ["f";"g";"dd_bounds"]
`nth_diff_strong_int 2 int f ==>
	nth_diff_strong_int 2 int g ==>
	bounded_on_int (\x. nth_derivative 2 f x + nth_derivative 2 g x) int dd_bounds ==>
	has_bounded_second_derivative (\x. f x + g x) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))))) THEN (simp_tac) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["b"]));
   (((((use_arg_then "nth_diff_add")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (move ["ineq"]));
   ((((use_arg_then "second_derivative_add")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "b")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_sub_bounds *)
let second_derivative_sub_bounds = section_proof ["f";"g";"dd_bounds"]
`nth_diff_strong_int 2 int f ==>
	nth_diff_strong_int 2 int g ==>
	bounded_on_int (\x. nth_derivative 2 f x - nth_derivative 2 g x) int dd_bounds ==>
	has_bounded_second_derivative (\x. f x - g x) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))))) THEN (simp_tac) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["b"]));
   (((((use_arg_then "nth_diff_sub")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (move ["ineq"]));
   ((((use_arg_then "second_derivative_sub")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "b")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_mul_bounds *)
let second_derivative_mul_bounds = section_proof ["f";"g";"dd_bounds"]
`nth_diff_strong_int 2 int f ==>
	nth_diff_strong_int 2 int g ==>
	bounded_on_int (\x. f x * nth_derivative 2 g x + &2 * derivative f x * derivative g x +
			nth_derivative 2 f x * g x) int dd_bounds ==>
	has_bounded_second_derivative (\x. f x * g x) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))))) THEN (simp_tac) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["b"]));
   (((((use_arg_then "nth_diff_mul")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["x"]) THEN (move ["ineq"]));
   ((((use_arg_then "second_derivative_mul")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "b")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then "dg") (disch_tac [])) THEN (clear_assumption "dg") THEN ((use_arg_then "df") (disch_tac [])) THEN (clear_assumption "df") THEN BETA_TAC) THEN ((repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma second_derivative_div_bounds *)
let second_derivative_div_bounds = section_proof ["f";"g";"g_bounds";"dd_bounds"]
`bounded_on_int g int g_bounds ==> interval_not_zero g_bounds ==>
	nth_diff_strong_int 2 int f ==>
	nth_diff_strong_int 2 int g ==>
	bounded_on_int (\x. ((nth_derivative 2 f x * g x - f x * nth_derivative 2 g x) * g x -
	  &2 * derivative g x * (derivative f x * g x - f x * derivative g x)) / g x pow 3) int dd_bounds ==>
	has_bounded_second_derivative (\x. f x / g x) int dd_bounds`
[
   (((((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))))) THEN (move ["bg"]) THEN (move ["gn0"]) THEN (simp_tac) THEN (move ["df"]) THEN (move ["dg"]) THEN (move ["b"]));
   ((fun arg_tac -> arg_tac (Arg_term (`!x. interval_arith x int ==> ~(g x = &0)`))) (term_tac (have_gen_tac [](move ["g0"]))));
   ((BETA_TAC THEN (move ["x"]) THEN (move ["ineq"])) THEN (((use_arg_then "interval_arith_not_zero") (disch_tac [])) THEN (clear_assumption "interval_arith_not_zero") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`g_bounds`))) (term_tac exists_tac)) THEN (((use_arg_then "bg")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((THENL_FIRST) ((split_tac) THEN (move ["x"]) THEN (move ["ineq"])) (((((use_arg_then "diff2_div")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "g0")(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "dg")(thm_tac (new_rewrite [] [])))))) THEN (done_tac)));
   (((((use_arg_then "second_derivative_div")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then "g0")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "df")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then "dg")(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then "b")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section SecondDerivativeBound *)
let nth_diff_strong2_eq_alt = finalize_theorem nth_diff_strong2_eq_alt;;
let nth_diff_strong2_eq = finalize_theorem nth_diff_strong2_eq;;
let lin_approx_compose = finalize_theorem lin_approx_compose;;
let second_derivative_atn_eq = finalize_theorem second_derivative_atn_eq;;
let second_derivative_atn = finalize_theorem second_derivative_atn;;
let diff2_atn = finalize_theorem diff2_atn;;
let second_derivative_inv = finalize_theorem second_derivative_inv;;
let diff2_inv = finalize_theorem diff2_inv;;
let SQRT_POW = finalize_theorem SQRT_POW;;
let second_derivative_sqrt = finalize_theorem second_derivative_sqrt;;
let diff2_sqrt = finalize_theorem diff2_sqrt;;
let real_powS = finalize_theorem real_powS;;
let second_derivative_acs = finalize_theorem second_derivative_acs;;
let diff2_acs = finalize_theorem diff2_acs;;
let REAL_CONTINUOUS_OPEN_PREIMAGE = finalize_theorem REAL_CONTINUOUS_OPEN_PREIMAGE;;
let second_derivative_compose = finalize_theorem second_derivative_compose;;
let diff2_compose = finalize_theorem diff2_compose;;
let continuous_not0_exists_open = finalize_theorem continuous_not0_exists_open;;
let continuous_gt_exists_open = finalize_theorem continuous_gt_exists_open;;
let continuous_lt_exists_open = finalize_theorem continuous_lt_exists_open;;
let second_derivative_compose_atn = finalize_theorem second_derivative_compose_atn;;
let second_derivative_compose_inv = finalize_theorem second_derivative_compose_inv;;
let second_derivative_compose_sqrt = finalize_theorem second_derivative_compose_sqrt;;
let second_derivative_compose_acs = finalize_theorem second_derivative_compose_acs;;
let diff2_compose_atn = finalize_theorem diff2_compose_atn;;
let diff2_compose_inv = finalize_theorem diff2_compose_inv;;
let diff2_compose_sqrt = finalize_theorem diff2_compose_sqrt;;
let diff2_compose_acs = finalize_theorem diff2_compose_acs;;
let second_derivative_mul = finalize_theorem second_derivative_mul;;
let second_derivative_div = finalize_theorem second_derivative_div;;
let diff2_div = finalize_theorem diff2_div;;
let second_derivative_scale = finalize_theorem second_derivative_scale;;
let second_derivative_add = finalize_theorem second_derivative_add;;
let second_derivative_sub = finalize_theorem second_derivative_sub;;
let second_derivative_compose_bounds = finalize_theorem second_derivative_compose_bounds;;
let second_derivative_atn_bounds = finalize_theorem second_derivative_atn_bounds;;
let second_derivative_compose_atn_bounds = finalize_theorem second_derivative_compose_atn_bounds;;
let second_derivative_inv_bounds = finalize_theorem second_derivative_inv_bounds;;
let second_derivative_compose_inv_bounds = finalize_theorem second_derivative_compose_inv_bounds;;
let second_derivative_sqrt_bounds = finalize_theorem second_derivative_sqrt_bounds;;
let second_derivative_compose_sqrt_bounds = finalize_theorem second_derivative_compose_sqrt_bounds;;
let second_derivative_acs_bounds = finalize_theorem second_derivative_acs_bounds;;
let second_derivative_compose_acs_bounds = finalize_theorem second_derivative_compose_acs_bounds;;
let second_derivative_scale_bounds = finalize_theorem second_derivative_scale_bounds;;
let second_derivative_add_bounds = finalize_theorem second_derivative_add_bounds;;
let second_derivative_sub_bounds = finalize_theorem second_derivative_sub_bounds;;
let second_derivative_mul_bounds = finalize_theorem second_derivative_mul_bounds;;
let second_derivative_div_bounds = finalize_theorem second_derivative_div_bounds;;
end_section "SecondDerivativeBound";;

(* Section TaylorArith *)
begin_section "TaylorArith";;
let cell_domain = new_definition `cell_domain x y z w <=> 
	x <= y /\ y <= z /\ y - x <= w /\ z - y <= w`;;
(* Lemma taylor_x *)
let taylor_x = section_proof ["x";"y";"z";"w"]
`cell_domain x y z w ==>
	taylor_interval (\x. x) x y z w (y, y) (&1, &1) (&0, &0)`
[
   ((((use_arg_then "cell_domain")(thm_tac (new_rewrite [] [])))) THEN (move ["ineqs"]));
   ((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "ineqs")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "lin_approx_x")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   (((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] []))));
   (((((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["p"]) THEN (move ["_"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(:real)`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_OPEN_UNIV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_UNIV")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["q"]) THEN (simp_tac)));
   (((((use_arg_then "REAL_CONTINUOUS_AT_ID")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]));
   ((((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `i < 2 <=> i = 0 \/ i = 1`)))(thm_tac (new_rewrite [] [])))) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_x")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_ID")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "TWO")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_x")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "derivative_const")(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then "HAS_REAL_DERIVATIVE_CONST")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_x")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "derivative_const")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_LE_REFL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma taylor_const *)
let taylor_const = section_proof ["c";"x";"y";"z";"w"]
`cell_domain x y z w ==>
	taylor_interval (\x. c) x y z w (c, c) (&0, &0) (&0, &0)`
[
   ((((use_arg_then "cell_domain")(thm_tac (new_rewrite [] [])))) THEN (move ["ineqs"]));
   ((((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "ineqs")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "lin_approx_const")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))));
   (((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] []))));
   (((((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_diff_strong")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_on")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "nth_differentiable_eq")(thm_tac (new_rewrite [] []))))) THEN (split_tac));
   ((BETA_TAC THEN (move ["p"]) THEN (move ["_"])) THEN ((fun arg_tac -> arg_tac (Arg_term (`(:real)`))) (term_tac exists_tac)) THEN (((((use_arg_then "REAL_OPEN_UNIV")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "IN_UNIV")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "andTb")(thm_tac (new_rewrite [] [])))))) THEN (move ["q"]) THEN (simp_tac)));
   (((((use_arg_then "REAL_CONTINUOUS_CONST")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andTb")(thm_tac (new_rewrite [] []))))) THEN (move ["i"]));
   ((((fun arg_tac -> arg_tac (Arg_theorem (ARITH_RULE `i < 2 <=> i = 0 \/ i = 1`)))(thm_tac (new_rewrite [] [])))) THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then "nth_derivative0")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "ONE")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "derivative_const")(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_CONST")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then "nth_derivative1")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "TWO")(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "derivative_const")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "HAS_REAL_DERIVATIVE_CONST")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then "nth_derivative2")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "derivative_const")(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "REAL_LE_REFL")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;
(add_section_var (mk_var ("f", (`:real->real`))));;
(add_section_var (mk_var ("x", (`:real`))); add_section_var (mk_var ("y", (`:real`))); add_section_var (mk_var ("z", (`:real`))); add_section_var (mk_var ("w", (`:real`))));;
(add_section_var (mk_var ("f_bounds", (`:real#real`))); add_section_var (mk_var ("df_bounds", (`:real#real`))); add_section_var (mk_var ("dd_bounds", (`:real#real`))));;
(add_section_var (mk_var ("f_lo", (`:real`))); add_section_var (mk_var ("f_hi", (`:real`))); add_section_var (mk_var ("df_lo", (`:real`))); add_section_var (mk_var ("df_hi", (`:real`))));;

(* Lemma taylor_f_bounds *)
let taylor_f_bounds = section_proof ["df";"dd";"lo";"hi";"t"]
`taylor_interval f x y z w (f_lo, f_hi) df_bounds dd_bounds ==>
	iabs df_bounds = df ==>
	iabs dd_bounds = dd ==>
	w * (df + w * dd * inv(&2)) <= t ==>
	f_hi + t <= hi ==> lo <= f_lo - t ==>
	bounded_on_int f (x, z) (lo, hi)`
[
   (BETA_TAC THEN (move ["taylor_f"]) THEN (move ["iabs_df"]) THEN (move ["iabs_dd"]) THEN (move ["t_ineq"]) THEN (move ["hi_ineq"]) THEN (move ["lo_ineq"]));
   (((((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))))) THEN (move ["p"]) THEN (move ["ineq"]));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then "taylor_upper_bound") (fun fst_arg -> (fun arg_tac -> (use_arg_then "EQ_SYM") (fun fst_arg -> (use_arg_then "iabs_dd") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "taylor_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN ((fun arg_tac -> (fun arg_tac -> (use_arg_then "taylor_lower_bound") (fun fst_arg -> (fun arg_tac -> (use_arg_then "EQ_SYM") (fun fst_arg -> (use_arg_then "iabs_dd") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "taylor_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   (((simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then "real_div")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "iabs_df")(thm_tac (new_rewrite [] []))))) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)) THEN (move ["ineq1"]) THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)) THEN (move ["ineq2"]));
   ((((use_arg_then "t_ineq") (disch_tac [])) THEN (clear_assumption "t_ineq") THEN ((use_arg_then "hi_ineq") (disch_tac [])) THEN (clear_assumption "hi_ineq") THEN ((use_arg_then "lo_ineq") (disch_tac [])) THEN (clear_assumption "lo_ineq") THEN ((use_arg_then "ineq2") (disch_tac [])) THEN (clear_assumption "ineq2") THEN ((use_arg_then "ineq1") (disch_tac [])) THEN (clear_assumption "ineq1") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma taylor_df_bounds *)
let taylor_df_bounds = section_proof ["dd";"lo";"hi"]
`taylor_interval f x y z w f_bounds (df_lo, df_hi) dd_bounds ==>
	iabs dd_bounds = dd ==>
	df_hi + w * dd <= hi ==>
	lo <= df_lo - w * dd ==>
	bounded_on_int (derivative f) (x, z) (lo, hi)`
[
   (BETA_TAC THEN (move ["taylor_f"]) THEN (move ["iabs_dd"]) THEN (move ["hi_ineq"]) THEN (move ["lo_ineq"]));
   (((((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "interval_arith")(thm_tac (new_rewrite [] [])))))) THEN (move ["p"]) THEN (move ["ineq"]));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "taylor_derivative_lower_bound") (fun fst_arg -> (fun arg_tac -> (use_arg_then "EQ_SYM") (fun fst_arg -> (use_arg_then "iabs_dd") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "taylor_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then "taylor_derivative_upper_bound") (fun fst_arg -> (fun arg_tac -> (use_arg_then "EQ_SYM") (fun fst_arg -> (use_arg_then "iabs_dd") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "taylor_f") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then "ineq") (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((use_arg_then "hi_ineq") (disch_tac [])) THEN (clear_assumption "hi_ineq") THEN ((use_arg_then "lo_ineq") (disch_tac [])) THEN (clear_assumption "lo_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma bounded_on_int_compose *)
let bounded_on_int_compose = section_proof ["g";"int";"g_bounds"]
`bounded_on_int g int g_bounds ==>
	bounded_on_int f g_bounds f_bounds ==>
	bounded_on_int (\x. f (g x)) int f_bounds`
[
   ((repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] []))))) THEN (move ["g_bounded"]) THEN (move ["f_bounded"]) THEN (move ["x"]) THEN (move ["ineq"]) THEN (simp_tac));
   (((((use_arg_then "f_bounded")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "g_bounded")(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma bounded_on_int_imp_interval_arith *)
let bounded_on_int_imp_interval_arith = section_proof ["int"]
`bounded_on_int f int f_bounds ==>
	interval_arith y int ==> interval_arith (f y) f_bounds`
[
   ((((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))) THEN (move ["bounded_f"]) THEN (move ["int_y"]));
   ((((use_arg_then "bounded_f")(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma taylor_interval_narrow *)
let taylor_interval_narrow = section_proof ["x0";"z0";"w0"]
`taylor_interval f x y z w f_bounds df_bounds dd_bounds ==>
	cell_domain x0 y z0 w0 ==>
	x <= x0 ==> z0 <= z ==>
	taylor_interval f x0 y z0 w0 f_bounds df_bounds dd_bounds`
[
   (((repeat_tactic 1 9 (((use_arg_then "taylor_interval")(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then "cell_domain")(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then "andbA")(thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN ((case THEN ((move ["ineq"]) THEN (move ["lin"]))) THEN (move ["dd"]))) THEN (move ["c"]) THEN (move ["i1"]) THEN (move ["i2"]));
   ((THENL_FIRST) (((((use_arg_then "lin")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "andbT")(thm_tac (new_rewrite [] []))))) THEN (split_tac)) ((((use_arg_then "i2") (disch_tac [])) THEN (clear_assumption "i2") THEN ((use_arg_then "i1") (disch_tac [])) THEN (clear_assumption "i1") THEN ((use_arg_then "c") (disch_tac [])) THEN (clear_assumption "c") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then "dd") (disch_tac [])) THEN (clear_assumption "dd") THEN BETA_TAC) THEN (((repeat_tactic 1 9 (((use_arg_then "has_bounded_second_derivative")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "nth_diff_strong_int")(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then "bounded_on_int")(thm_tac (new_rewrite [] [])))))) THEN (move ["df"])));
   ((split_tac) THEN (move ["p"]) THEN (move ["int_p"]));
   ((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "i2") (disch_tac [])) THEN (clear_assumption "i2") THEN ((use_arg_then "i1") (disch_tac [])) THEN (clear_assumption "i1") THEN ((use_arg_then "int_p") (disch_tac [])) THEN (clear_assumption "int_p") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL interval_arith)))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then "df")(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then "i2") (disch_tac [])) THEN (clear_assumption "i2") THEN ((use_arg_then "i1") (disch_tac [])) THEN (clear_assumption "i1") THEN ((use_arg_then "int_p") (disch_tac [])) THEN (clear_assumption "int_p") THEN BETA_TAC) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL interval_arith)))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section TaylorArith *)
let taylor_x = finalize_theorem taylor_x;;
let taylor_const = finalize_theorem taylor_const;;
let taylor_f_bounds = finalize_theorem taylor_f_bounds;;
let taylor_df_bounds = finalize_theorem taylor_df_bounds;;
let bounded_on_int_compose = finalize_theorem bounded_on_int_compose;;
let bounded_on_int_imp_interval_arith = finalize_theorem bounded_on_int_imp_interval_arith;;
let taylor_interval_narrow = finalize_theorem taylor_interval_narrow;;
end_section "TaylorArith";;