(* 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
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 (Int.pred v)::int) = (number_of v = (0::nat))"
apply (subgoal_tac "neg (number_of (Int.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 (Int.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 (Int.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 (Int.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 (Int.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 (Int.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_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
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)
ML{*
local
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 ss T 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
fun proc 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 ss 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 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 ss 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 ss 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
val add_frac_frac_simproc =
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
name = "add_frac_frac_simproc",
proc = proc, identifier = []}
val add_frac_num_simproc =
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
name = "add_frac_num_simproc",
proc = proc2, 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"])
@ @{thms arith_simps} @ nat_arith @ @{thms 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
val comp_conv = (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, add_frac_num_simproc,
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)))
handle TERM _ => error "ring_dest_const")
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
val field_comp_conv = comp_conv;
val fieldgb_declaration =
NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
{is_const = K numeral_is_const,
dest_const = K dest_const,
mk_const = mk_const,
conv = K (K comp_conv)}
end;
*}
declaration{* fieldgb_declaration *}
end