needs "../formal_lp/ineqs/constants_approx.hl";;

module Lp_approx_ineqs = struct

open Constants_approx;;

(* Integral approximation of inequalities and equalities *)
let mk_decimal =
  let decimal_const = `DECIMAL` and
      neg_op_real = `(--):real->real` in
    fun (n,m) ->
      let tm = mk_comb(mk_comb(decimal_const, mk_numeral (abs_num n)), mk_numeral m) in
	if (n </ Int 0) then mk_comb(neg_op_real, tm) else tm;;


(*
Given a rational number term `r` and a natural number k,
returns a pair of theorems:
|- low <= r, |- r <= high,
with decimal low and high such that
|low - r| <= 10^(-k) and |high - r| <= 10^(-k)
*)
let real_rat_approx =
  let le_op_real = `(<=):real->real->bool` in
    fun tm precision ->
      let n = rat_of_term tm in
      let m = Int 10 **/ (Int precision) in
      let n1 = n */ m in
      let low, high = floor_num n1, ceiling_num n1 in
      let l_tm, h_tm =
	if precision = 0 then
	  term_of_rat low, term_of_rat high
	else
	  mk_decimal (low, m), mk_decimal (high, m) in
      let l_th = EQT_ELIM (REAL_RAT_LE_CONV (mk_binop le_op_real l_tm tm)) in
      let h_th = EQT_ELIM (REAL_RAT_LE_CONV (mk_binop le_op_real tm h_tm)) in
	l_th, h_th;;


(* Splits a sum into two parts: with and without free variables *)
let split_sum_conv =
  let sum_th0 = REAL_ARITH `x = x + &0 + &0` and
      sum_th1 = REAL_ARITH `(x + y) + b + c = y + (x + b) + c:real` and
      sum_th2 = REAL_ARITH `(x + y) + b + c = y + b + (x + c):real` and
      sum_th1' = REAL_ARITH `x + b + c = (x + b) + c:real` and
      sum_th2' = REAL_ARITH `x + b + c = b + (x + c):real` in
  let x_var_real = `x:real` and
      y_var_real = `y:real` and
      b_var_real = `b:real` and
      c_var_real = `c:real` and
      add_op_real = `(+):real->real->real` in
  let rec split_sum_raw_conv tm =
    let xy_tm, bc_tm = dest_binop add_op_real tm in
    let b_tm, c_tm = dest_binop add_op_real bc_tm in
      if (is_binop add_op_real xy_tm) then
        let x_tm, y_tm = dest_binop add_op_real xy_tm in
        let inst_th = INST[x_tm, x_var_real; y_tm, y_var_real; b_tm, b_var_real; c_tm, c_var_real] in
        let th1 = if (frees x_tm <> []) then inst_th sum_th1 else inst_th sum_th2 in
        let th2 = split_sum_raw_conv (rand(concl th1)) in
          TRANS th1 th2
      else
        let inst_th = INST[xy_tm, x_var_real; b_tm, b_var_real; c_tm, c_var_real] in
          if (frees xy_tm <> []) then inst_th sum_th1' else inst_th sum_th2' in

    fun tm ->
      let th0 = INST[tm, x_var_real] sum_th0 in
      let th1 = split_sum_raw_conv (rand(concl th0)) in
	TRANS th0 th1;;



let rearrange_mul_conv =
  let mul_op_real = `( * ):real->real->real` in
  let rec dest_mul tm =
    if (is_binop mul_op_real tm) then
      let lhs, rhs = dest_binop mul_op_real tm in
      let cs, vars = dest_mul rhs in
        if (frees lhs = []) then
          lhs::cs, vars
        else
          cs, lhs::vars
    else
      if (frees tm = []) then
        [tm], []
      else
        [], [tm] in
  let mk_mul list =
    if (list = []) then failwith "rearrange_mul: empty list"
    else itlist (fun l r -> mk_binop mul_op_real l r) (tl list) (hd list) in

    fun tm ->
      let cs, vars = dest_mul tm in
      let cs_mul, vars_mul = mk_mul cs, mk_mul vars in
      let t = mk_eq(tm, mk_binop mul_op_real cs_mul vars_mul) in
	EQT_ELIM (REWRITE_CONV[REAL_MUL_AC] t);;



(* Moves everything with free variables on the left and performs basic reductions *)
let ineq_rewrite_conv = 
  let le_add_th = REAL_ARITH `x + y <= b + c <=> x - b <= c - y:real` in
    REWRITE_CONV[real_ge; real_div; DECIMAL] THENC
      LAND_CONV REAL_POLY_CONV THENC RAND_CONV REAL_POLY_CONV THENC
      LAND_CONV split_sum_conv THENC RAND_CONV split_sum_conv THENC
      ONCE_REWRITE_CONV[le_add_th] THENC
      LAND_CONV REAL_POLY_CONV THENC RAND_CONV REAL_POLY_CONV THENC
      REWRITE_CONV[GSYM real_div] THENC
      ONCE_DEPTH_CONV rearrange_mul_conv;;


(* Approximation *)

let le_mul1_th = REAL_ARITH `!x. &1 * x <= x`;;
let ge_mul1_th = REAL_ARITH `!x. x <= &1 * x`;;
let INTERVAL_LO = prove(`interval_arith x (a,b) ==> a <= x`, SIMP_TAC[interval_arith]);;
let INTERVAL_HI = prove(`interval_arith x (a,b) ==> x <= b`, SIMP_TAC[interval_arith]);;

let add_op_real = `(+):real->real->real`;;

let rec low_approx tm precision =
  let low_approx1 tm precision =
    if (is_binop mul_op_real tm && frees tm <> []) then
      let c, var = dest_binop mul_op_real tm in
      let interval_th = approx_interval (create_interval c) precision in
      let l_th = MATCH_MP INTERVAL_LO interval_th in
      let a, b = dest_binop le_op_real (concl l_th) in
        (PROVE_HYP l_th o UNDISCH_ALL o REWRITE_RULE[GSYM IMP_IMP] o SPECL[a; b; var]) REAL_LE_RMUL
    else
      if (frees tm = []) then
        MATCH_MP INTERVAL_LO (approx_interval (create_interval tm) precision)
      else
        SPEC tm le_mul1_th in

  if (is_binop add_op_real tm && frees tm <> []) then
    let lhs, rhs = dest_binop add_op_real tm in
    let l_th = low_approx1 lhs precision in
    let r_th = low_approx rhs precision in
      MATCH_MP REAL_LE_ADD2 (CONJ l_th r_th)
  else
    low_approx1 tm precision;;


let rec hi_approx tm precision =
  let hi_approx1 tm precision =
    if (is_binop mul_op_real tm && frees tm <> []) then
      let c, var = dest_binop mul_op_real tm in
      let interval_th = approx_interval (create_interval c) precision in
      let h_th = MATCH_MP INTERVAL_HI interval_th in
      let a, b = dest_binop le_op_real (concl h_th) in
        (PROVE_HYP h_th o UNDISCH_ALL o REWRITE_RULE[GSYM IMP_IMP] o SPECL[a; b; var]) REAL_LE_RMUL
    else
      if (frees tm = []) then
        MATCH_MP INTERVAL_HI (approx_interval (create_interval tm) precision)
      else
        SPEC tm ge_mul1_th in

  if (is_binop add_op_real tm && frees tm <> []) then
    let lhs, rhs = dest_binop add_op_real tm in
    let l_th = hi_approx1 lhs precision in
    let r_th = hi_approx rhs precision in
      MATCH_MP REAL_LE_ADD2 (CONJ l_th r_th)
  else
    hi_approx1 tm precision;;



let approx_le_ineq precision tm =
  let lhs, rhs = dest_binop le_op_real tm in
  let lhs_th = low_approx lhs precision in
  let rhs_th = hi_approx rhs precision in
  let ll, lr = dest_binop le_op_real (concl lhs_th) in
  let rl, rr = dest_binop le_op_real (concl rhs_th) in

  let th0 = ASSUME tm in
  let th1 = SPECL[ll; lr; rhs] REAL_LE_TRANS in
  let th2 = SPECL[ll; rhs; rr] REAL_LE_TRANS in
  let s1 = MATCH_MP th1 (CONJ lhs_th th0) in
    MATCH_MP th2 (CONJ s1 rhs_th);;


let integer_approx_le_ineq precision ineq =
  let lhs, rhs = dest_binop le_op_real ineq in
  let m = (Int 10 **/ Int precision) in
  let m_num, m_real = mk_numeral m, term_of_rat m in
  let m_pos = SPEC m_num REAL_POS in
  let mul_th = SPECL[m_real; lhs; rhs] REAL_LE_LMUL in
  let th0 = MATCH_MP mul_th (CONJ m_pos (ASSUME ineq)) in
  let th1 = (CONV_RULE ineq_rewrite_conv) th0 in
  let approx = approx_le_ineq 0 (concl th1) in
    PROVE_HYP th1 approx;;


let create_approximations precision_list ineq =
  let ineq_th = ineq_rewrite_conv ineq in
  let rhs = rand(concl ineq_th) in
  let th0 = CONV_RULE (LAND_CONV (REWRITE_CONV[DECIMAL] THENC REAL_POLY_CONV))
    (approx_le_ineq 8 rhs) in

  let int_approx p =
    let th1 = integer_approx_le_ineq p (concl th0) in
    let th2 = PROVE_HYP th0 th1 in
    let th3 = DISCH rhs th2 in
      REWRITE_RULE[GSYM ineq_th] th3 in

    map int_approx precision_list;;

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


(*
let tm = `(&34/ &13 + pi/sol0)* x + &2 + -- &14 / &3 * z <= pi + rho218 + z * sol0 - u / &1000`;;
create_approximations [3;4;5] tm;;
*)


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

(* Additional step for generalizing hypotheses *)

let LIST_SUM_LMUL = prove(`!(f:A->real) c n. list_sum n (\x. c * f x) = c * list_sum n f`,
			  REWRITE_TAC[Seq2.list_sum_lmul]);;
let le_hyp_gen = prove(`!f y. (!x. &0 <= f x) ==> &0 <= f y`, SIMP_TAC[]);;
let le_list_sum_hyp_gen = prove(`!(f:A->real) n. (!x. &0 <= f x) ==> &0 <= list_sum n f`,
				REPEAT STRIP_TAC THEN MATCH_MP_TAC Seq2.list_sum_ge0 THEN ASM_REWRITE_TAC[]);;


let generalize_hyp th =
  let gen_hyp tm =
    let fn, arg = dest_comb (rand tm) in
      if (is_comb fn && is_const (rator fn) && (fst o dest_const o rator) fn = "list_sum") then
        let f, n = arg, rand fn in
          UNDISCH_ALL (ISPECL[f; n] le_list_sum_hyp_gen)
      else
        UNDISCH_ALL (ISPECL[fn; arg] le_hyp_gen) in

  let hyp_ths = map gen_hyp (hyp th) in
    itlist PROVE_HYP hyp_ths th;;


(*
let tm = `list_sum n (\x. s x * r) + &1 / &3 * (azim_dart (V,E) o g) x <= pi`;;
let ths = create_approximations [3] tm;;
map generalize_hyp ths;;
*)


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


let generate_ineqs =
  let imp_th = TAUT `(A ==> B) ==> ((P ==> A) ==> (P ==> B))` in
  let and_imp_th = TAUT `(A ==> B /\ C) <=> ((A ==> B) /\ (A ==> C))` in
  let p_bool = `P:bool` in
  let strip_imp = splitlist dest_imp in
  let list_mk_imp = itlist (curry mk_imp) in

  let create_approxs pos_ths original_th precision_list =
    let var, ineq' = (dest_abs o rand o rator o concl) original_th in
    let cond_tms, ineq = strip_imp ineq' in
    let final_step = fun approx_th ->
      let p_tm', q_tm' = dest_imp (concl approx_th) in
      let p_tm, q_tm = mk_abs(var, list_mk_imp cond_tms p_tm'), mk_abs(var, list_mk_imp cond_tms q_tm') in
      let list_tm = (rand o concl) original_th in
      let mono_th = BETA_RULE (ISPECL[p_tm; q_tm; list_tm] (GEN_ALL MONO_ALL)) in
      let approx_th2 = itlist (fun p_tm th -> MATCH_MP (INST [p_tm, p_bool] imp_th) th) cond_tms approx_th in
      let s1 = MATCH_MP mono_th (GEN var approx_th2) in
	MATCH_MP s1 original_th in
    
    let approx_ths0 = map generalize_hyp (create_approximations precision_list ineq) in
    let approx_ths1 = map (itlist PROVE_HYP pos_ths) approx_ths0 in
    let approx_ths = map (REWRITE_RULE[GSYM LIST_SUM_LMUL]) approx_ths1 in
      map final_step approx_ths in

    fun pos_ths precision_list ineq_th ->
      let _, ineq = (strip_imp o snd o dest_abs o rand o rator o concl) ineq_th in
	if (is_eq ineq) then
	  let eq_th = PURE_REWRITE_RULE[GSYM REAL_LE_ANTISYM; and_imp_th; GSYM AND_ALL] ineq_th in
	  let ths1 = create_approxs pos_ths (CONJUNCT1 eq_th) precision_list in
	  let ths2 = create_approxs pos_ths (CONJUNCT2 eq_th) precision_list in
            ths1, ths2
	else
	  create_approxs pos_ths ineq_th precision_list, [];;



(*
generate_ineqs [] [3;4] (ASSUME `ALL (\n. list_sum n (\x. f x) - &1 <= pi) s`);;
*)

end;;