(* Title: HOL/Groebner_Basis.thy
ID: $Id$
Author: Amine Chaieb, TU Muenchen
*)
header {* Semiring normalization and Groebner Bases *}
theory Groebner_Basis
imports NatBin
uses
"Tools/Groebner_Basis/misc.ML"
"Tools/Groebner_Basis/normalizer_data.ML"
("Tools/Groebner_Basis/normalizer.ML")
("Tools/Groebner_Basis/groebner.ML")
begin
subsection {* Semiring normalization *}
setup NormalizerData.setup
locale gb_semiring =
fixes add mul pwr r0 r1
assumes add_a:"(add x (add y z) = add (add x y) z)"
and add_c: "add x y = add y x" and add_0:"add r0 x = x"
and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0"
and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
begin
lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
proof (induct p)
case 0
then show ?case by (auto simp add: pwr_0 mul_1)
next
case Suc
from this [symmetric] show ?case
by (auto simp add: pwr_Suc mul_1 mul_a)
qed
lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
fix q x y
assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
by (simp add: mul_a)
also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
qed
lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
proof (induct p arbitrary: q)
case 0
show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
next
case Suc
thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
qed
subsubsection {* Declaring the abstract theory *}
lemma semiring_ops:
includes meta_term_syntax
shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
and "TERM r0" and "TERM r1"
by rule+
lemma semiring_rules:
"add (mul a m) (mul b m) = mul (add a b) m"
"add (mul a m) m = mul (add a r1) m"
"add m (mul a m) = mul (add a r1) m"
"add m m = mul (add r1 r1) m"
"add r0 a = a"
"add a r0 = a"
"mul a b = mul b a"
"mul (add a b) c = add (mul a c) (mul b c)"
"mul r0 a = r0"
"mul a r0 = r0"
"mul r1 a = a"
"mul a r1 = a"
"mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
"mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
"mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
"mul (mul lx ly) rx = mul (mul lx rx) ly"
"mul (mul lx ly) rx = mul lx (mul ly rx)"
"mul lx (mul rx ry) = mul (mul lx rx) ry"
"mul lx (mul rx ry) = mul rx (mul lx ry)"
"add (add a b) (add c d) = add (add a c) (add b d)"
"add (add a b) c = add a (add b c)"
"add a (add c d) = add c (add a d)"
"add (add a b) c = add (add a c) b"
"add a c = add c a"
"add a (add c d) = add (add a c) d"
"mul (pwr x p) (pwr x q) = pwr x (p + q)"
"mul x (pwr x q) = pwr x (Suc q)"
"mul (pwr x q) x = pwr x (Suc q)"
"mul x x = pwr x 2"
"pwr (mul x y) q = mul (pwr x q) (pwr y q)"
"pwr (pwr x p) q = pwr x (p * q)"
"pwr x 0 = r1"
"pwr x 1 = x"
"mul x (add y z) = add (mul x y) (mul x z)"
"pwr x (Suc q) = mul x (pwr x q)"
"pwr x (2*n) = mul (pwr x n) (pwr x n)"
"pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
proof -
show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
next show "add r0 a = a" using add_0 by simp
next show "add a r0 = a" using add_0 add_c by simp
next show "mul a b = mul b a" using mul_c by simp
next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
next show "mul r0 a = r0" using mul_0 by simp
next show "mul a r0 = r0" using mul_0 mul_c by simp
next show "mul r1 a = a" using mul_1 by simp
next show "mul a r1 = a" using mul_1 mul_c by simp
next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
using mul_c mul_a by simp
next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
using mul_a by simp
next
have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
finally
show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
using mul_c by simp
next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
next
show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
next show "add (add a b) (add c d) = add (add a c) (add b d)"
using add_c add_a by simp
next show "add (add a b) c = add a (add b c)" using add_a by simp
next show "add a (add c d) = add c (add a d)"
apply (simp add: add_a) by (simp only: add_c)
next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
next show "add a c = add c a" by (rule add_c)
next show "add a (add c d) = add (add a c) d" using add_a by simp
next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
next show "pwr x 0 = r1" using pwr_0 .
next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
by (simp add: nat_number pwr_Suc mul_pwr)
qed
lemmas gb_semiring_axioms' =
gb_semiring_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules]
end
interpretation class_semiring: gb_semiring
["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
by unfold_locales (auto simp add: ring_simps power_Suc)
lemmas nat_arith =
add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
by (simp add: numeral_1_eq_1)
lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
if_True add_0 add_Suc add_number_of_left mult_number_of_left
numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
iszero_number_of_Bit1 iszero_number_of_Bit0 nonzero_number_of_Min
iszero_number_of_Pls iszero_0 not_iszero_Numeral1
lemmas semiring_norm = comp_arith
ML {*
local
open Conv;
fun numeral_is_const ct =
can HOLogic.dest_number (Thm.term_of ct);
fun int_of_rat x =
(case Rat.quotient_of_rat x of (i, 1) => i
| _ => error "int_of_rat: bad int");
val numeral_conv =
Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
Simplifier.rewrite (HOL_basic_ss addsimps
(@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
in
fun normalizer_funs key =
NormalizerData.funs key
{is_const = fn phi => numeral_is_const,
dest_const = fn phi => fn ct =>
Rat.rat_of_int (snd
(HOLogic.dest_number (Thm.term_of ct)
handle TERM _ => error "ring_dest_const")),
mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
conv = fn phi => K numeral_conv}
end
*}
declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
locale gb_ring = gb_semiring +
fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and neg :: "'a \<Rightarrow> 'a"
assumes neg_mul: "neg x = mul (neg r1) x"
and sub_add: "sub x y = add x (neg y)"
begin
lemma ring_ops:
includes meta_term_syntax
shows "TERM (sub x y)" and "TERM (neg x)" .
lemmas ring_rules = neg_mul sub_add
lemmas gb_ring_axioms' =
gb_ring_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules]
end
interpretation class_ring: gb_ring ["op +" "op *" "op ^"
"0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
by unfold_locales simp_all
declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
use "Tools/Groebner_Basis/normalizer.ML"
method_setup sring_norm = {*
Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
*} "semiring normalizer"
locale gb_field = gb_ring +
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and inverse:: "'a \<Rightarrow> 'a"
assumes divide: "divide x y = mul x (inverse y)"
and inverse: "inverse x = divide r1 x"
begin
lemmas gb_field_axioms' =
gb_field_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules]
end
subsection {* Groebner Bases *}
locale semiringb = gb_semiring +
assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
and add_mul_solve: "add (mul w y) (mul x z) =
add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
begin
lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
proof-
have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
using add_mul_solve by blast
finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
by simp
qed
lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
\<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
proof(clarify)
assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
and eq: "add b (mul r c) = add b (mul r d)"
hence "mul r c = mul r d" using cnd add_cancel by simp
hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
using mul_0 add_cancel by simp
thus "False" using add_mul_solve nz cnd by simp
qed
lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
proof-
have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
qed
declare gb_semiring_axioms' [normalizer del]
lemmas semiringb_axioms' = semiringb_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
idom rules: noteq_reduce add_scale_eq_noteq]
end
locale ringb = semiringb + gb_ring +
assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
begin
declare gb_ring_axioms' [normalizer del]
lemmas ringb_axioms' = ringb_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules
idom rules: noteq_reduce add_scale_eq_noteq
ideal rules: subr0_iff add_r0_iff]
end
lemma no_zero_divirors_neq0:
assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
and ab: "a*b = 0" shows "b = 0"
proof -
{ assume bz: "b \<noteq> 0"
from no_zero_divisors [OF az bz] ab have False by blast }
thus "b = 0" by blast
qed
interpretation class_ringb: ringb
["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
proof(unfold_locales, simp add: ring_simps power_Suc, auto)
fix w x y z ::"'a::{idom,recpower,number_ring}"
assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
hence ynz': "y - z \<noteq> 0" by simp
from p have "w * y + x* z - w*z - x*y = 0" by simp
hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
with no_zero_divirors_neq0 [OF ynz']
have "w - x = 0" by blast
thus "w = x" by simp
qed
declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
interpretation natgb: semiringb
["op +" "op *" "op ^" "0::nat" "1"]
proof (unfold_locales, simp add: ring_simps power_Suc)
fix w x y z ::"nat"
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
hence "y < z \<or> y > z" by arith
moreover {
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
then obtain k where kp: "k>0" and yz:"z = y + k" by blast
from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
hence "x*k = w*k" by simp
hence "w = x" using kp by (simp add: mult_cancel2) }
moreover {
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
then obtain k where kp: "k>0" and yz:"y = z + k" by blast
from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
hence "w*k = x*k" by simp
hence "w = x" using kp by (simp add: mult_cancel2)}
ultimately have "w=x" by blast }
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
qed
declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
locale fieldgb = ringb + gb_field
begin
declare gb_field_axioms' [normalizer del]
lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules
idom rules: noteq_reduce add_scale_eq_noteq
ideal rules: subr0_iff add_r0_iff]
end
lemmas bool_simps = simp_thms(1-34)
lemma dnf:
"(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
"(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
by blast+
lemmas weak_dnf_simps = dnf bool_simps
lemma nnf_simps:
"(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
by blast+
lemma PFalse:
"P \<equiv> False \<Longrightarrow> \<not> P"
"\<not> P \<Longrightarrow> (P \<equiv> False)"
by auto
use "Tools/Groebner_Basis/groebner.ML"
method_setup algebra =
{*
let
fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
val addN = "add"
val delN = "del"
val any_keyword = keyword addN || keyword delN
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
in
fn src => Method.syntax
((Scan.optional (keyword addN |-- thms) []) --
(Scan.optional (keyword delN |-- thms) [])) src
#> (fn ((add_ths, del_ths), ctxt) =>
Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
end
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
declare dvd_def[algebra]
declare dvd_eq_mod_eq_0[symmetric, algebra]
declare nat_mod_div_trivial[algebra]
declare nat_mod_mod_trivial[algebra]
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
declare zmod_zdiv_equality[symmetric,algebra]
declare zdiv_zmod_equality[symmetric, algebra]
declare zdiv_zminus_zminus[algebra]
declare zmod_zminus_zminus[algebra]
declare zdiv_zminus2[algebra]
declare zmod_zminus2[algebra]
declare zdiv_zero[algebra]
declare zmod_zero[algebra]
declare zmod_1[algebra]
declare zdiv_1[algebra]
declare zmod_minus1_right[algebra]
declare zdiv_minus1_right[algebra]
declare mod_div_trivial[algebra]
declare mod_mod_trivial[algebra]
declare zmod_zmult_self1[algebra]
declare zmod_zmult_self2[algebra]
declare zmod_eq_0_iff[algebra]
declare zdvd_0_left[algebra]
declare zdvd1_eq[algebra]
declare zmod_eq_dvd_iff[algebra]
declare nat_mod_eq_iff[algebra]
end