src/HOL/Arith_Tools.thy
 author chaieb Sun, 22 Jul 2007 17:53:42 +0200 changeset 23901 7392193f9ecf parent 23881 851c74f1bb69 child 24075 366d4d234814 permissions -rw-r--r--
Tunes Proof
```
(*  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 *}

setup 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)"

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"

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))"

"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))"
del: nat_numeral_1_eq_1
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)
done

"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)))"
del: nat_numeral_1_eq_1
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)"

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)"

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

lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"

lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"

Suc_div_eq_add3_div [of _ "number_of v", standard]

Suc_mod_eq_add3_mod [of _ "number_of v", standard]

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]

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

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

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

(****
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)"

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

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]

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

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

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

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

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

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

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

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

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

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

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

lemma minus_equation_iff_1 [simp]:
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]

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

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

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

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)"

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

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

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

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

make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
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",
(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
ord_frac_simproc]
[@{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
```