needs "../formal_lp/formal_interval/more_float.hl";;
needs "../formal_lp/list/list_conversions.hl";;

let MY_RULE_FLOAT = UNDISCH_ALL o NUMERALS_TO_NUM o 
  PURE_REWRITE_RULE[FLOAT_OF_NUM; min_exp_def; GSYM IMP_IMP] o SPEC_ALL;;


(*************************************)
(* list_sum2 evaluation *)

let list_sum2 = new_definition `list_sum2 f l1 l2 = ITLIST2 (\a b c. f a b + c) l1 l2 (&0)`;;

let LIST_SUM2_0_LE' = (MY_RULE_FLOAT o prove)(`list_sum2 (f:A->B->real) [] [] <= &0`, 
					      REWRITE_TAC[list_sum2; ITLIST2; REAL_LE_REFL]);;
let LIST_SUM2_1_LE' = (MY_RULE o prove)(`f h1 h2 <= x ==> list_sum2 (f:A->B->real) [h1] [h2] <= x`, 
					REWRITE_TAC[list_sum2; ITLIST2; REAL_ADD_RID]);;
let LIST_SUM2_LE' = (MY_RULE o prove)(`f h1 h2 <= x /\ list_sum2 f t1 t2 <= y /\ x + y <= z ==>
					list_sum2 (f:A->B->real) (CONS h1 t1) (CONS h2 t2) <= z`,
				      REWRITE_TAC[list_sum2; ITLIST2] THEN STRIP_TAC THEN
					MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `x + y:real` THEN
					ASM_SIMP_TAC[REAL_LE_ADD2]);;


let list_sum2_le_conv pp f_le_conv tm =
  let ltm, list2_tm = dest_comb tm in
  let ltm2, list1_tm = dest_comb ltm in
  let f_tm = rand ltm2 in
  let list1_ty = type_of list1_tm and
      list2_ty = type_of list2_tm and
      f_ty = type_of f_tm in
  let ty1 = (hd o snd o dest_type) list1_ty and
      ty2 = (hd o snd o dest_type) list2_ty in
  let f_var = mk_var ("f", f_ty) and
      h1_var, t1_var = mk_var ("h1", ty1), mk_var ("t1", list1_ty) and
      h2_var, t2_var = mk_var ("h2", ty2), mk_var ("t2", list2_ty) in
  let inst_t = INST[f_tm, f_var] o INST_TYPE[ty1, aty; ty2, bty] in
  let list2_0, list2_1, list2_le = inst_t LIST_SUM2_0_LE', inst_t LIST_SUM2_1_LE', inst_t LIST_SUM2_LE' in

  let rec rec_conv = fun list1_tm list2_tm ->
    if (is_comb list1_tm) then
      let h1_tm, t1_tm = dest_cons list1_tm and
	  h2_tm, t2_tm = dest_cons list2_tm in
      let f_le_th = f_le_conv pp h1_tm h2_tm in
      let x_tm = (rand o concl) f_le_th in
      let inst0 = INST[h1_tm, h1_var; h2_tm, h2_var; x_tm, x_var_real] in
	if is_comb t1_tm then
	  let sum2_t_th = rec_conv t1_tm t2_tm in
	  let y_tm = (rand o concl) sum2_t_th in
	  let xy_th = float_add_hi pp x_tm y_tm in
	  let z_tm = (rand o concl) xy_th in
	    (MY_PROVE_HYP xy_th o MY_PROVE_HYP sum2_t_th o MY_PROVE_HYP f_le_th o
	       INST[y_tm, y_var_real; z_tm, z_var_real; t1_tm, t1_var; t2_tm, t2_var] o 
	       inst0) list2_le
	else
	  if is_comb t2_tm then failwith ("sum2_le_conv: t1 = []; t2 = "^string_of_term t2_tm) else
	    (MY_PROVE_HYP f_le_th o inst0) list2_1
    else
      if is_comb list2_tm  then failwith ("sum2_le_conv: list1 = []; list2 = "^string_of_term list2_tm) else
	list2_0 in

    rec_conv list1_tm list2_tm;;
	    

(*
let pp = 5;;
let pi_approx = (fst o dest_pair o rand o concl) pi_approx_array.(pp) and
    pi2_approx = (fst o dest_pair o rand o concl) pi2_approx_array.(pp);;

let l1 = mk_list (replicate pi_approx 5, real_ty) and
    l2 = mk_list (replicate pi2_approx 5, real_ty);;

let tm = mk_binop `list_sum2 ( * )` l1 l2;;
(*let test_conv pp tm =
  let tm1, tm2 = dest_binop mul_op_real tm in float_mul_hi pp tm1 tm2;; *)
list_sum2_le_conv pp float_mul_hi tm;;
(* 10: 4.504 *)
test 1000 (list_sum2_le_conv pp float_mul_hi) tm;;
*)


(**************************)
(* \a b c. a * iabs b + c *)

let error_mul_f2 = new_definition `error_mul_f2 a int = a * iabs int`;;
let ERROR_MUL_F2' = (SYM o MY_RULE) error_mul_f2;;


(* |- x = a, |- P x y -> P a y *)
let rewrite_lhs eq_th th =
  let ltm, rhs = dest_comb (concl th) in
  let th0 = AP_THM (AP_TERM (rator ltm) eq_th) rhs in
    EQ_MP th0 th;;

let a_var_real = `a:real`;;
let le_op_real = `(<=):real->real->bool`;;

let error_mul_f2_le_conv pp tm1 tm2 =
  let eq_th = INST[tm1, a_var_real; tm2, int_var] ERROR_MUL_F2' in
  let iabs_th = float_iabs tm2 in
  let iabs_tm = (rand o concl) iabs_th in
  let mul_th = float_mul_hi pp tm1 iabs_tm in
  let th0 = AP_TERM (mk_comb (mul_op_real, tm1)) iabs_th in
  let th1 = AP_THM (AP_TERM le_op_real th0) (rand (concl mul_th)) in
  let le_th = EQ_MP (SYM th1) mul_th in
    rewrite_lhs eq_th le_th;;

let ERROR_MUL_F2_LEMMA' = (MY_RULE o prove)(`iabs int = x /\ a * x <= y ==> error_mul_f2 a int <= y`,
					  REWRITE_TAC[error_mul_f2] THEN STRIP_TAC THEN ASM_REWRITE_TAC[]);;

let error_mul_f2_le_conv2 pp tm1 tm2 =
  let iabs_th = float_iabs tm2 in
  let x_tm = (rand o concl) iabs_th in
  let mul_th = float_mul_hi pp tm1 x_tm in
  let y_tm = (rand o concl) mul_th in
    (MY_PROVE_HYP iabs_th o MY_PROVE_HYP mul_th o
       INST[tm2, int_var; tm1, a_var_real; x_tm, x_var_real; y_tm, y_var_real]) ERROR_MUL_F2_LEMMA';;


(*
let pp = 5;;
let tm1 = pi_approx and tm2 = (rand o concl) pi2_approx_array.(pp);;
error_mul_f2_le_conv pp tm1 tm2;;
error_mul_f2_le_conv2 pp tm1 tm2;;
(* 10: 0.772  *)
test 1000 (error_mul_f2_le_conv pp tm1) tm2;;
(* 10: 0.768 *)
test 1000 (error_mul_f2_le_conv2 pp tm1) tm2;;

let n = 6;;
let l1 = mk_list (replicate pi_approx n, real_ty) and
    l2 = mk_list (replicate (mk_pair (pi2_approx, pi2_approx)) n, `:real#real`);;

let tm = mk_comb (mk_comb (`list_sum2 error_mul_f2`, l1), l2);;

list_sum2_le_conv pp error_mul_f2_le_conv2 tm;;
(* 10: 5.920 *)
test 1000 (list_sum2_le_conv pp error_mul_f2_le_conv2) tm;;
*)


(**************************)
(* \a b c. a * iabs b + c *)

let error_mul_f1 = new_definition `error_mul_f1 w x list = x * list_sum2 error_mul_f2 w list`;;

let ERROR_MUL_F1_LEMMA' = (MY_RULE o prove)(`x * list_sum2 error_mul_f2 w list <= z ==>
					      error_mul_f1 w x list <= z`, REWRITE_TAC[error_mul_f1]);;

let list_sum2_error2_const = `list_sum2 error_mul_f2` and
    w_var_list = `w:(real)list` and
    list_var = `list:(real#real)list`;;

let error_mul_f1_le_conv w_tm pp x_tm list_tm =
  (* TODO: if x = 0 then do not need to compute the sum *)
  let sum2_tm = mk_binop list_sum2_error2_const w_tm list_tm in
  let sum2_le_th = list_sum2_le_conv pp error_mul_f2_le_conv2 sum2_tm in
  let ineq_th = mul_ineq_pos_const_hi pp x_tm sum2_le_th in
  let z_tm = (rand o concl) ineq_th in
    (MY_PROVE_HYP ineq_th o
       INST[x_tm, x_var_real; z_tm, z_var_real; w_tm, w_var_list; list_tm, list_var]) ERROR_MUL_F1_LEMMA';;

(*
let pp = 5;;
let w_tm = mk_list (replicate pi5 6, real_ty) and
    x_tm = pi5;;
let list_tm = mk_list (replicate (mk_pair (two_float, two_float)) 6, `:real#real`);;

error_mul_f1_le_conv w_tm pp x_tm list_tm;;

(* 10: 3.672  *)
test 1000 (error_mul_f1_le_conv w_tm pp x_tm) list_tm;;
*)

(***********************)

(*
let n = 6;;
let pp = 5;;

let xx = mk_vector_list (replicate two_float n) and
    zz = mk_vector_list (replicate pi5 n);;

let eval_poly = eval_second_bounded_poly pp delta_x_poly;;
let th = eval_poly xx zz;;
let domain6_th = mk_m_center_domain n pp (rand xx) (rand zz);;
let w_tm = (rand o rand o concl) domain6_th;;
let dd_tm = (rand o concl) th;;

let op = mk_icomb (`list_sum2`, mk_comb (`error_mul_f1`, w_tm));;
let tm = mk_binop op w_tm dd_tm;;

list_sum2_le_conv pp (error_mul_f1_le_conv w_tm) tm;;
(* 10: 4.236 *)
test 100 (list_sum2_le_conv pp (error_mul_f1_le_conv w_tm)) tm;;
*)