src/HOL/Arith_Tools.thy

avoid Theory.rep_theory;

(*  Title:      HOL/Arith_Tools.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Author:     Amine Chaieb, TU Muenchen
*)

header {* Setup of arithmetic tools *}

theory Arith_Tools
imports Groebner_Basis Dense_Linear_Order
uses
  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
  "~~/src/Provers/Arith/extract_common_term.ML"
  "int_factor_simprocs.ML"
  "nat_simprocs.ML"
begin

subsection {* Simprocs for the Naturals *}

declaration {* K nat_simprocs_setup *}

subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}

text{*Where K above is a literal*}

lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)

text {*Now just instantiating @{text n} to @{text "number_of v"} does
  the right simplification, but with some redundant inequality
  tests.*}
lemma neg_number_of_pred_iff_0:
  "neg (number_of (Numeral.pred v)::int) = (number_of v = (0::nat))"
apply (subgoal_tac "neg (number_of (Numeral.pred v)) = (number_of v < Suc 0) ")
apply (simp only: less_Suc_eq_le le_0_eq)
apply (subst less_number_of_Suc, simp)
done

text{*No longer required as a simprule because of the @{text inverse_fold}
   simproc*}
lemma Suc_diff_number_of:
     "neg (number_of (uminus v)::int) ==>
      Suc m - (number_of v) = m - (number_of (Numeral.pred v))"
apply (subst Suc_diff_eq_diff_pred)
apply simp
apply (simp del: nat_numeral_1_eq_1)
apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
                        neg_number_of_pred_iff_0)
done

lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
by (simp add: numerals split add: nat_diff_split)


subsubsection{*For @{term nat_case} and @{term nat_rec}*}

lemma nat_case_number_of [simp]:
     "nat_case a f (number_of v) =
        (let pv = number_of (Numeral.pred v) in
         if neg pv then a else f (nat pv))"
by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)

lemma nat_case_add_eq_if [simp]:
     "nat_case a f ((number_of v) + n) =
       (let pv = number_of (Numeral.pred v) in
         if neg pv then nat_case a f n else f (nat pv + n))"
apply (subst add_eq_if)
apply (simp split add: nat.split
            del: nat_numeral_1_eq_1
            add: numeral_1_eq_Suc_0 [symmetric] Let_def
                 neg_imp_number_of_eq_0 neg_number_of_pred_iff_0)
done

lemma nat_rec_number_of [simp]:
     "nat_rec a f (number_of v) =
        (let pv = number_of (Numeral.pred v) in
         if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
apply (case_tac " (number_of v) ::nat")
apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
apply (simp split add: split_if_asm)
done

lemma nat_rec_add_eq_if [simp]:
     "nat_rec a f (number_of v + n) =
        (let pv = number_of (Numeral.pred v) in
         if neg pv then nat_rec a f n
                   else f (nat pv + n) (nat_rec a f (nat pv + n)))"
apply (subst add_eq_if)
apply (simp split add: nat.split
            del: nat_numeral_1_eq_1
            add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0
                 neg_number_of_pred_iff_0)
done


subsubsection{*Various Other Lemmas*}

text {*Evens and Odds, for Mutilated Chess Board*}

text{*Lemmas for specialist use, NOT as default simprules*}
lemma nat_mult_2: "2 * z = (z+z::nat)"
proof -
  have "2*z = (1 + 1)*z" by simp
  also have "... = z+z" by (simp add: left_distrib)
  finally show ?thesis .
qed

lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
by (subst mult_commute, rule nat_mult_2)

text{*Case analysis on @{term "n<2"}*}
lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
by arith

lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
by arith

lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
by (simp add: nat_mult_2 [symmetric])

lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
apply (subgoal_tac "m mod 2 < 2")
apply (erule less_2_cases [THEN disjE])
apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
done

lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
apply (subgoal_tac "m mod 2 < 2")
apply (force simp del: mod_less_divisor, simp)
done

text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}

lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
by simp

lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
by simp

text{*Can be used to eliminate long strings of Sucs, but not by default*}
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
by simp


text{*These lemmas collapse some needless occurrences of Suc:
    at least three Sucs, since two and fewer are rewritten back to Suc again!
    We already have some rules to simplify operands smaller than 3.*}

lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
by (simp add: Suc3_eq_add_3)

lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
by (simp add: Suc3_eq_add_3)

lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
by (simp add: Suc3_eq_add_3)

lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
by (simp add: Suc3_eq_add_3)

lemmas Suc_div_eq_add3_div_number_of =
    Suc_div_eq_add3_div [of _ "number_of v", standard]
declare Suc_div_eq_add3_div_number_of [simp]

lemmas Suc_mod_eq_add3_mod_number_of =
    Suc_mod_eq_add3_mod [of _ "number_of v", standard]
declare Suc_mod_eq_add3_mod_number_of [simp]


subsubsection{*Special Simplification for Constants*}

text{*These belong here, late in the development of HOL, to prevent their
interfering with proofs of abstract properties of instances of the function
@{term number_of}*}

text{*These distributive laws move literals inside sums and differences.*}
lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard]
declare left_distrib_number_of [simp]

lemmas right_distrib_number_of = right_distrib [of "number_of v", standard]
declare right_distrib_number_of [simp]


lemmas left_diff_distrib_number_of =
    left_diff_distrib [of _ _ "number_of v", standard]
declare left_diff_distrib_number_of [simp]

lemmas right_diff_distrib_number_of =
    right_diff_distrib [of "number_of v", standard]
declare right_diff_distrib_number_of [simp]


text{*These are actually for fields, like real: but where else to put them?*}
lemmas zero_less_divide_iff_number_of =
    zero_less_divide_iff [of "number_of w", standard]
declare zero_less_divide_iff_number_of [simp,noatp]

lemmas divide_less_0_iff_number_of =
    divide_less_0_iff [of "number_of w", standard]
declare divide_less_0_iff_number_of [simp,noatp]

lemmas zero_le_divide_iff_number_of =
    zero_le_divide_iff [of "number_of w", standard]
declare zero_le_divide_iff_number_of [simp,noatp]

lemmas divide_le_0_iff_number_of =
    divide_le_0_iff [of "number_of w", standard]
declare divide_le_0_iff_number_of [simp,noatp]


(****
IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
then these special-case declarations may be useful.

text{*These simprules move numerals into numerators and denominators.*}
lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
by (simp add: times_divide_eq)

lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
by (simp add: times_divide_eq)

lemmas times_divide_eq_right_number_of =
    times_divide_eq_right [of "number_of w", standard]
declare times_divide_eq_right_number_of [simp]

lemmas times_divide_eq_right_number_of =
    times_divide_eq_right [of _ _ "number_of w", standard]
declare times_divide_eq_right_number_of [simp]

lemmas times_divide_eq_left_number_of =
    times_divide_eq_left [of _ "number_of w", standard]
declare times_divide_eq_left_number_of [simp]

lemmas times_divide_eq_left_number_of =
    times_divide_eq_left [of _ _ "number_of w", standard]
declare times_divide_eq_left_number_of [simp]

****)

text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
  strange, but then other simprocs simplify the quotient.*}

lemmas inverse_eq_divide_number_of =
    inverse_eq_divide [of "number_of w", standard]
declare inverse_eq_divide_number_of [simp]


text {*These laws simplify inequalities, moving unary minus from a term
into the literal.*}
lemmas less_minus_iff_number_of =
    less_minus_iff [of "number_of v", standard]
declare less_minus_iff_number_of [simp,noatp]

lemmas le_minus_iff_number_of =
    le_minus_iff [of "number_of v", standard]
declare le_minus_iff_number_of [simp,noatp]

lemmas equation_minus_iff_number_of =
    equation_minus_iff [of "number_of v", standard]
declare equation_minus_iff_number_of [simp,noatp]


lemmas minus_less_iff_number_of =
    minus_less_iff [of _ "number_of v", standard]
declare minus_less_iff_number_of [simp,noatp]

lemmas minus_le_iff_number_of =
    minus_le_iff [of _ "number_of v", standard]
declare minus_le_iff_number_of [simp,noatp]

lemmas minus_equation_iff_number_of =
    minus_equation_iff [of _ "number_of v", standard]
declare minus_equation_iff_number_of [simp,noatp]


text{*To Simplify Inequalities Where One Side is the Constant 1*}

lemma less_minus_iff_1 [simp,noatp]:
  fixes b::"'b::{ordered_idom,number_ring}"
  shows "(1 < - b) = (b < -1)"
by auto

lemma le_minus_iff_1 [simp,noatp]:
  fixes b::"'b::{ordered_idom,number_ring}"
  shows "(1 \<le> - b) = (b \<le> -1)"
by auto

lemma equation_minus_iff_1 [simp,noatp]:
  fixes b::"'b::number_ring"
  shows "(1 = - b) = (b = -1)"
by (subst equation_minus_iff, auto)

lemma minus_less_iff_1 [simp,noatp]:
  fixes a::"'b::{ordered_idom,number_ring}"
  shows "(- a < 1) = (-1 < a)"
by auto

lemma minus_le_iff_1 [simp,noatp]:
  fixes a::"'b::{ordered_idom,number_ring}"
  shows "(- a \<le> 1) = (-1 \<le> a)"
by auto

lemma minus_equation_iff_1 [simp,noatp]:
  fixes a::"'b::number_ring"
  shows "(- a = 1) = (a = -1)"
by (subst minus_equation_iff, auto)


text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}

lemmas mult_less_cancel_left_number_of =
    mult_less_cancel_left [of "number_of v", standard]
declare mult_less_cancel_left_number_of [simp,noatp]

lemmas mult_less_cancel_right_number_of =
    mult_less_cancel_right [of _ "number_of v", standard]
declare mult_less_cancel_right_number_of [simp,noatp]

lemmas mult_le_cancel_left_number_of =
    mult_le_cancel_left [of "number_of v", standard]
declare mult_le_cancel_left_number_of [simp,noatp]

lemmas mult_le_cancel_right_number_of =
    mult_le_cancel_right [of _ "number_of v", standard]
declare mult_le_cancel_right_number_of [simp,noatp]


text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}

lemmas le_divide_eq_number_of = le_divide_eq [of _ _ "number_of w", standard]
declare le_divide_eq_number_of [simp]

lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard]
declare divide_le_eq_number_of [simp]

lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard]
declare less_divide_eq_number_of [simp]

lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard]
declare divide_less_eq_number_of [simp]

lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard]
declare eq_divide_eq_number_of [simp]

lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard]
declare divide_eq_eq_number_of [simp]



subsubsection{*Optional Simplification Rules Involving Constants*}

text{*Simplify quotients that are compared with a literal constant.*}

lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]


text{*Not good as automatic simprules because they cause case splits.*}
lemmas divide_const_simps =
  le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
  divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1

text{*Division By @{text "-1"}*}

lemma divide_minus1 [simp]:
     "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
by simp

lemma minus1_divide [simp]:
     "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
by (simp add: divide_inverse inverse_minus_eq)

lemma half_gt_zero_iff:
     "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
by auto

lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
declare half_gt_zero [simp]

(* The following lemma should appear in Divides.thy, but there the proof
   doesn't work. *)

lemma nat_dvd_not_less:
  "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
  by (unfold dvd_def) auto

ML {*
val divide_minus1 = @{thm divide_minus1};
val minus1_divide = @{thm minus1_divide};
*}


subsection{* Groebner Bases for fields *}

interpretation class_fieldgb:
  fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)

lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
  by simp
lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
  by simp
lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
  by simp
lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
  by simp

lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp

lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
  by (simp add: add_divide_distrib)
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
  by (simp add: add_divide_distrib)

declaration{*
let
 val zr = @{cpat "0"}
 val zT = ctyp_of_term zr
 val geq = @{cpat "op ="}
 val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
 val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
 val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
 val add_num_frac = mk_meta_eq @{thm "add_num_frac"}

 fun prove_nz ctxt =
  let val ss = local_simpset_of ctxt
  in fn T => fn t =>
    let
      val z = instantiate_cterm ([(zT,T)],[]) zr
      val eq = instantiate_cterm ([(eqT,T)],[]) geq
      val th = Simplifier.rewrite (ss addsimps simp_thms)
           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
                  (Thm.capply (Thm.capply eq t) z)))
    in equal_elim (symmetric th) TrueI
    end
  end

 fun proc ctxt phi ss ct =
  let
    val ((x,y),(w,z)) =
         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
    val T = ctyp_of_term x
    val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
  in SOME (implies_elim (implies_elim th y_nz) z_nz)
  end
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE

 fun proc2 ctxt phi ss ct =
  let
    val (l,r) = Thm.dest_binop ct
    val T = ctyp_of_term l
  in (case (term_of l, term_of r) of
      (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
        let val (x,y) = Thm.dest_binop l val z = r
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
            val ynz = prove_nz ctxt T y
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
        end
     | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
        let val (x,y) = Thm.dest_binop r val z = l
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
            val ynz = prove_nz ctxt T y
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
        end
     | _ => NONE)
  end
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE

 fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
   | is_number t = can HOLogic.dest_number t

 val is_number = is_number o term_of

 fun proc3 phi ss ct =
  (case term_of ct of
    Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
      in SOME (mk_meta_eq th) end
  | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
      in SOME (mk_meta_eq th) end
  | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
      in SOME (mk_meta_eq th) end
  | _ => NONE)
  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE

fun add_frac_frac_simproc ctxt =
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
                     name = "add_frac_frac_simproc",
                     proc = proc ctxt, identifier = []}

fun add_frac_num_simproc ctxt =
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
                     name = "add_frac_num_simproc",
                     proc = proc2 ctxt, identifier = []}

val ord_frac_simproc =
  make_simproc
    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
             @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
             name = "ord_frac_simproc", proc = proc3, identifier = []}

val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
               "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]

val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
                 "add_Suc", "add_number_of_left", "mult_number_of_left",
                 "Suc_eq_add_numeral_1"])@
                 (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
                 @ arith_simps@ nat_arith @ rel_simps
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
           @{thm "divide_Numeral1"},
           @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
           @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
           @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
           @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
           @{thm "diff_def"}, @{thm "minus_divide_left"},
           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]

local
open Conv
in
fun comp_conv ctxt = (Simplifier.rewrite
(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
              addsimps ths addsimps comp_arith addsimps simp_thms
              addsimprocs field_cancel_numeral_factors
               addsimprocs [add_frac_frac_simproc ctxt, add_frac_num_simproc ctxt,
                            ord_frac_simproc]
                addcongs [@{thm "if_weak_cong"}]))
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
end

fun numeral_is_const ct =
  case term_of ct of
   Const (@{const_name "HOL.divide"},_) $ a $ b =>
     numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
 | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
 | t => can HOLogic.dest_number t

fun dest_const ct = case term_of ct of
   Const (@{const_name "HOL.divide"},_) $ a $ b=>
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))

fun mk_const phi cT x =
 let val (a, b) = Rat.quotient_of_rat x
 in if b = 1 then Numeral.mk_cnumber cT a
    else Thm.capply
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
                     (Numeral.mk_cnumber cT a))
         (Numeral.mk_cnumber cT b)
  end

in
 NormalizerData.funs @{thm class_fieldgb.axioms}
   {is_const = K numeral_is_const,
    dest_const = K dest_const,
    mk_const = mk_const,
    conv = K comp_conv}
end

*}


subsection {* Ferrante and Rackoff algorithm over ordered fields *}

lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
proof-
  assume H: "c < 0"
  have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
  also have "\<dots> = (0 < x)" by simp
  finally show  "(c*x < 0) == (x > 0)" by simp
qed

lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
proof-
  assume H: "c > 0"
  hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
  also have "\<dots> = (0 > x)" by simp
  finally show  "(c*x < 0) == (x < 0)" by simp
qed

lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
proof-
  assume H: "c < 0"
  have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
  also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
  also have "\<dots> = ((- 1/c)*t < x)" by simp
  finally show  "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
qed

lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
proof-
  assume H: "c > 0"
  have "c*x + t< 0 = (c*x < -t)"  by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
  also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
  also have "\<dots> = ((- 1/c)*t > x)" by simp
  finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
qed

lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
  using less_diff_eq[where a= x and b=t and c=0] by simp

lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
proof-
  assume H: "c < 0"
  have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
  also have "\<dots> = (0 <= x)" by simp
  finally show  "(c*x <= 0) == (x >= 0)" by simp
qed

lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
proof-
  assume H: "c > 0"
  hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
  also have "\<dots> = (0 >= x)" by simp
  finally show  "(c*x <= 0) == (x <= 0)" by simp
qed

lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
proof-
  assume H: "c < 0"
  have "c*x + t <= 0 = (c*x <= -t)"  by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
  also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
  also have "\<dots> = ((- 1/c)*t <= x)" by simp
  finally show  "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
qed

lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
proof-
  assume H: "c > 0"
  have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
  also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
  also have "\<dots> = ((- 1/c)*t >= x)" by simp
  finally show  "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
qed

lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
  using le_diff_eq[where a= x and b=t and c=0] by simp

lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
proof-
  assume H: "c \<noteq> 0"
  have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
  also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_simps)
  finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
qed
lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
  using eq_diff_eq[where a= x and b=t and c=0] by simp


interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order
 ["op <=" "op <"
   "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
proof (unfold_locales, dlo, dlo, auto)
  fix x y::'a assume lt: "x < y"
  from  less_half_sum[OF lt] show "x < (x + y) /2" by simp
next
  fix x y::'a assume lt: "x < y"
  from  gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
qed

declaration{*
let
fun earlier [] x y = false
        | earlier (h::t) x y =
    if h aconvc y then false else if h aconvc x then true else earlier t x y;

fun dest_frac ct = case term_of ct of
   Const (@{const_name "HOL.divide"},_) $ a $ b=>
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))

fun mk_frac phi cT x =
 let val (a, b) = Rat.quotient_of_rat x
 in if b = 1 then Numeral.mk_cnumber cT a
    else Thm.capply
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
                     (Numeral.mk_cnumber cT a))
         (Numeral.mk_cnumber cT b)
 end

fun whatis x ct = case term_of ct of
  Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
     if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
     else ("Nox",[])
| Const(@{const_name "HOL.plus"}, _)$y$_ =>
     if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
     else ("Nox",[])
| Const(@{const_name "HOL.times"}, _)$_$y =>
     if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
     else ("Nox",[])
| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);

fun xnormalize_conv ctxt [] ct = reflexive ct
| xnormalize_conv ctxt (vs as (x::_)) ct =
   case term_of ct of
   Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) =>
    (case whatis x (Thm.dest_arg1 ct) of
    ("c*x+t",[c,t]) =>
       let
        val cr = dest_frac c
        val clt = Thm.dest_fun2 ct
        val cz = Thm.dest_arg ct
        val neg = cr </ Rat.zero
        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
               (Thm.capply @{cterm "Trueprop"}
                  (if neg then Thm.capply (Thm.capply clt c) cz
                    else Thm.capply (Thm.capply clt cz) c))
        val cth = equal_elim (symmetric cthp) TrueI
        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
             (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
      in rth end
    | ("x+t",[t]) =>
       let
        val T = ctyp_of_term x
        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
       in  rth end
    | ("c*x",[c]) =>
       let
        val cr = dest_frac c
        val clt = Thm.dest_fun2 ct
        val cz = Thm.dest_arg ct
        val neg = cr </ Rat.zero
        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
               (Thm.capply @{cterm "Trueprop"}
                  (if neg then Thm.capply (Thm.capply clt c) cz
                    else Thm.capply (Thm.capply clt cz) c))
        val cth = equal_elim (symmetric cthp) TrueI
        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
             (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
        val rth = th
      in rth end
    | _ => reflexive ct)


|  Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) =>
   (case whatis x (Thm.dest_arg1 ct) of
    ("c*x+t",[c,t]) =>
       let
        val T = ctyp_of_term x
        val cr = dest_frac c
        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
        val cz = Thm.dest_arg ct
        val neg = cr </ Rat.zero
        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
               (Thm.capply @{cterm "Trueprop"}
                  (if neg then Thm.capply (Thm.capply clt c) cz
                    else Thm.capply (Thm.capply clt cz) c))
        val cth = equal_elim (symmetric cthp) TrueI
        val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
             (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
      in rth end
    | ("x+t",[t]) =>
       let
        val T = ctyp_of_term x
        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
       in  rth end
    | ("c*x",[c]) =>
       let
        val T = ctyp_of_term x
        val cr = dest_frac c
        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
        val cz = Thm.dest_arg ct
        val neg = cr </ Rat.zero
        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
               (Thm.capply @{cterm "Trueprop"}
                  (if neg then Thm.capply (Thm.capply clt c) cz
                    else Thm.capply (Thm.capply clt cz) c))
        val cth = equal_elim (symmetric cthp) TrueI
        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
             (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
        val rth = th
      in rth end
    | _ => reflexive ct)

|  Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
   (case whatis x (Thm.dest_arg1 ct) of
    ("c*x+t",[c,t]) =>
       let
        val T = ctyp_of_term x
        val cr = dest_frac c
        val ceq = Thm.dest_fun2 ct
        val cz = Thm.dest_arg ct
        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
            (Thm.capply @{cterm "Trueprop"}
             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
        val cth = equal_elim (symmetric cthp) TrueI
        val th = implies_elim
                 (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
      in rth end
    | ("x+t",[t]) =>
       let
        val T = ctyp_of_term x
        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
       in  rth end
    | ("c*x",[c]) =>
       let
        val T = ctyp_of_term x
        val cr = dest_frac c
        val ceq = Thm.dest_fun2 ct
        val cz = Thm.dest_arg ct
        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
            (Thm.capply @{cterm "Trueprop"}
             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
        val cth = equal_elim (symmetric cthp) TrueI
        val rth = implies_elim
                 (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
      in rth end
    | _ => reflexive ct);

local
  val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
  val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
  val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
in
fun field_isolate_conv phi ctxt vs ct = case term_of ct of
  Const(@{const_name HOL.less},_)$a$b =>
   let val (ca,cb) = Thm.dest_binop ct
       val T = ctyp_of_term ca
       val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
       val nth = Conv.fconv_rule
         (Conv.arg_conv (Conv.arg1_conv
              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   in rth end
| Const(@{const_name HOL.less_eq},_)$a$b =>
   let val (ca,cb) = Thm.dest_binop ct
       val T = ctyp_of_term ca
       val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
       val nth = Conv.fconv_rule
         (Conv.arg_conv (Conv.arg1_conv
              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   in rth end

| Const("op =",_)$a$b =>
   let val (ca,cb) = Thm.dest_binop ct
       val T = ctyp_of_term ca
       val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
       val nth = Conv.fconv_rule
         (Conv.arg_conv (Conv.arg1_conv
              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
   in rth end
| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
| _ => reflexive ct
end;

fun classfield_whatis phi =
 let
  fun h x t =
   case term_of t of
     Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
                            else Ferrante_Rackoff_Data.Nox
   | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
                            else Ferrante_Rackoff_Data.Nox
   | Const(@{const_name HOL.less},_)$y$z =>
       if term_of x aconv y then Ferrante_Rackoff_Data.Lt
        else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
        else Ferrante_Rackoff_Data.Nox
   | Const (@{const_name HOL.less_eq},_)$y$z =>
         if term_of x aconv y then Ferrante_Rackoff_Data.Le
         else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
         else Ferrante_Rackoff_Data.Nox
   | _ => Ferrante_Rackoff_Data.Nox
 in h end;
fun class_field_ss phi =
   HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
   addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]

in
Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
  {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
end
*}

end