--- a/src/HOL/Groebner_Basis.thy Mon Sep 29 12:31:58 2008 +0200
+++ b/src/HOL/Groebner_Basis.thy Mon Sep 29 12:31:59 2008 +0200
@@ -4,8 +4,9 @@
*)
header {* Semiring normalization and Groebner Bases *}
+
theory Groebner_Basis
-imports NatBin
+imports Arith_Tools
uses
"Tools/Groebner_Basis/misc.ML"
"Tools/Groebner_Basis/normalizer_data.ML"
@@ -461,4 +462,220 @@
declare zmod_eq_dvd_iff[algebra]
declare nat_mod_eq_iff[algebra]
+
+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