
wenzelm@23252: (*  Title:      HOL/Groebner_Basis.thy
wenzelm@23252:     Author:     Amine Chaieb, TU Muenchen
wenzelm@23252: *)
wenzelm@23252: 
wenzelm@23252: header {* Semiring normalization and Groebner Bases *}
haftmann@28402: 
wenzelm@23252: theory Groebner_Basis
haftmann@28402: imports Arith_Tools
wenzelm@23252: uses
wenzelm@23252:   "Tools/Groebner_Basis/misc.ML"
wenzelm@23252:   "Tools/Groebner_Basis/normalizer_data.ML"
wenzelm@23252:   ("Tools/Groebner_Basis/normalizer.ML")
chaieb@23312:   ("Tools/Groebner_Basis/groebner.ML")
wenzelm@23252: begin
wenzelm@23252: 
wenzelm@23252: subsection {* Semiring normalization *}
wenzelm@23252: 
wenzelm@23252: setup NormalizerData.setup
wenzelm@23252: 
wenzelm@23252: 
wenzelm@23258: locale gb_semiring =
wenzelm@23252:   fixes add mul pwr r0 r1
wenzelm@23252:   assumes add_a:"(add x (add y z) = add (add x y) z)"
wenzelm@23252:     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
wenzelm@23252:     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
wenzelm@23252:     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
wenzelm@23252:     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
wenzelm@23252:     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
wenzelm@23252: begin
wenzelm@23252: 
wenzelm@23252: lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
wenzelm@23252: proof (induct p)
wenzelm@23252:   case 0
wenzelm@23252:   then show ?case by (auto simp add: pwr_0 mul_1)
wenzelm@23252: next
wenzelm@23252:   case Suc
wenzelm@23252:   from this [symmetric] show ?case
wenzelm@23252:     by (auto simp add: pwr_Suc mul_1 mul_a)
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@23252: lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
wenzelm@23252: proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
wenzelm@23252:   fix q x y
wenzelm@23252:   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
wenzelm@23252:   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
wenzelm@23252:     by (simp add: mul_a)
wenzelm@23252:   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
wenzelm@23252:   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
wenzelm@23252:   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
wenzelm@23252:     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@23252: lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
wenzelm@23252: proof (induct p arbitrary: q)
wenzelm@23252:   case 0
wenzelm@23252:   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
wenzelm@23252: next
wenzelm@23252:   case Suc
wenzelm@23252:   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@23252: 
wenzelm@23252: subsubsection {* Declaring the abstract theory *}
wenzelm@23252: 
wenzelm@23252: lemma semiring_ops:
wenzelm@23252:   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
wenzelm@28856:     and "TERM r0" and "TERM r1" .
wenzelm@23252: 
wenzelm@23252: lemma semiring_rules:
wenzelm@23252:   "add (mul a m) (mul b m) = mul (add a b) m"
wenzelm@23252:   "add (mul a m) m = mul (add a r1) m"
wenzelm@23252:   "add m (mul a m) = mul (add a r1) m"
wenzelm@23252:   "add m m = mul (add r1 r1) m"
wenzelm@23252:   "add r0 a = a"
wenzelm@23252:   "add a r0 = a"
wenzelm@23252:   "mul a b = mul b a"
wenzelm@23252:   "mul (add a b) c = add (mul a c) (mul b c)"
wenzelm@23252:   "mul r0 a = r0"
wenzelm@23252:   "mul a r0 = r0"
wenzelm@23252:   "mul r1 a = a"
wenzelm@23252:   "mul a r1 = a"
wenzelm@23252:   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
wenzelm@23252:   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
wenzelm@23252:   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
wenzelm@23252:   "mul (mul lx ly) rx = mul (mul lx rx) ly"
wenzelm@23252:   "mul (mul lx ly) rx = mul lx (mul ly rx)"
wenzelm@23252:   "mul lx (mul rx ry) = mul (mul lx rx) ry"
wenzelm@23252:   "mul lx (mul rx ry) = mul rx (mul lx ry)"
wenzelm@23252:   "add (add a b) (add c d) = add (add a c) (add b d)"
wenzelm@23252:   "add (add a b) c = add a (add b c)"
wenzelm@23252:   "add a (add c d) = add c (add a d)"
wenzelm@23252:   "add (add a b) c = add (add a c) b"
wenzelm@23252:   "add a c = add c a"
wenzelm@23252:   "add a (add c d) = add (add a c) d"
wenzelm@23252:   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
wenzelm@23252:   "mul x (pwr x q) = pwr x (Suc q)"
wenzelm@23252:   "mul (pwr x q) x = pwr x (Suc q)"
wenzelm@23252:   "mul x x = pwr x 2"
wenzelm@23252:   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
wenzelm@23252:   "pwr (pwr x p) q = pwr x (p * q)"
wenzelm@23252:   "pwr x 0 = r1"
wenzelm@23252:   "pwr x 1 = x"
wenzelm@23252:   "mul x (add y z) = add (mul x y) (mul x z)"
wenzelm@23252:   "pwr x (Suc q) = mul x (pwr x q)"
wenzelm@23252:   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
wenzelm@23252:   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
wenzelm@23252: proof -
wenzelm@23252:   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
wenzelm@23252: next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
wenzelm@23252: next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
wenzelm@23252: next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
wenzelm@23252: next show "add r0 a = a" using add_0 by simp
wenzelm@23252: next show "add a r0 = a" using add_0 add_c by simp
wenzelm@23252: next show "mul a b = mul b a" using mul_c by simp
wenzelm@23252: next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
wenzelm@23252: next show "mul r0 a = r0" using mul_0 by simp
wenzelm@23252: next show "mul a r0 = r0" using mul_0 mul_c by simp
wenzelm@23252: next show "mul r1 a = a" using mul_1 by simp
wenzelm@23252: next show "mul a r1 = a" using mul_1 mul_c by simp
wenzelm@23252: next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
wenzelm@23252:     using mul_c mul_a by simp
wenzelm@23252: next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
wenzelm@23252:     using mul_a by simp
wenzelm@23252: next
wenzelm@23252:   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
wenzelm@23252:   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
wenzelm@23252:   finally
wenzelm@23252:   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
wenzelm@23252:     using mul_c by simp
wenzelm@23252: next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
wenzelm@23252: next
wenzelm@23252:   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
wenzelm@23252: next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
wenzelm@23252: next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
wenzelm@23252: next show "add (add a b) (add c d) = add (add a c) (add b d)"
wenzelm@23252:     using add_c add_a by simp
wenzelm@23252: next show "add (add a b) c = add a (add b c)" using add_a by simp
wenzelm@23252: next show "add a (add c d) = add c (add a d)"
wenzelm@23252:     apply (simp add: add_a) by (simp only: add_c)
wenzelm@23252: next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
wenzelm@23252: next show "add a c = add c a" by (rule add_c)
wenzelm@23252: next show "add a (add c d) = add (add a c) d" using add_a by simp
wenzelm@23252: next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
wenzelm@23252: next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
wenzelm@23252: next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
wenzelm@23252: next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
wenzelm@23252: next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
wenzelm@23252: next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
wenzelm@23252: next show "pwr x 0 = r1" using pwr_0 .
wenzelm@23252: next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
wenzelm@23252: next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
wenzelm@23252: next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
wenzelm@23252: next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
wenzelm@23252: next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
wenzelm@23252:     by (simp add: nat_number pwr_Suc mul_pwr)
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@23252: 
wenzelm@26462: lemmas gb_semiring_axioms' =
wenzelm@26314:   gb_semiring_axioms [normalizer
wenzelm@23252:     semiring ops: semiring_ops
wenzelm@26314:     semiring rules: semiring_rules]
wenzelm@23252: 
wenzelm@23252: end
wenzelm@23252: 
ballarin@29233: interpretation class_semiring!: gb_semiring
ballarin@29223:     "op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"
nipkow@29667:   proof qed (auto simp add: algebra_simps power_Suc)
wenzelm@23252: 
wenzelm@23252: lemmas nat_arith =
huffman@28987:   add_nat_number_of
huffman@28987:   diff_nat_number_of
huffman@28987:   mult_nat_number_of
huffman@28987:   eq_nat_number_of
huffman@28987:   less_nat_number_of
wenzelm@23252: 
wenzelm@23252: lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
wenzelm@23252:   by (simp add: numeral_1_eq_1)
huffman@28986: 
huffman@29039: lemmas comp_arith =
huffman@29039:   Let_def arith_simps nat_arith rel_simps neg_simps if_False
wenzelm@23252:   if_True add_0 add_Suc add_number_of_left mult_number_of_left
wenzelm@23252:   numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
huffman@28986:   numeral_0_eq_0[symmetric] numerals[symmetric]
huffman@28986:   iszero_simps not_iszero_Numeral1
wenzelm@23252: 
wenzelm@23252: lemmas semiring_norm = comp_arith
wenzelm@23252: 
wenzelm@23252: ML {*
wenzelm@23573: local
wenzelm@23252: 
wenzelm@23573: open Conv;
wenzelm@23573: 
wenzelm@23573: fun numeral_is_const ct =
wenzelm@23573:   can HOLogic.dest_number (Thm.term_of ct);
wenzelm@23252: 
wenzelm@23573: fun int_of_rat x =
wenzelm@23573:   (case Rat.quotient_of_rat x of (i, 1) => i
wenzelm@23573:   | _ => error "int_of_rat: bad int");
wenzelm@23252: 
wenzelm@23573: val numeral_conv =
wenzelm@23573:   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
wenzelm@23573:   Simplifier.rewrite (HOL_basic_ss addsimps
wenzelm@23573:     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
wenzelm@23573: 
wenzelm@23573: in
wenzelm@23573: 
wenzelm@23573: fun normalizer_funs key =
wenzelm@23573:   NormalizerData.funs key
wenzelm@23252:    {is_const = fn phi => numeral_is_const,
wenzelm@23252:     dest_const = fn phi => fn ct =>
wenzelm@23252:       Rat.rat_of_int (snd
wenzelm@23252:         (HOLogic.dest_number (Thm.term_of ct)
wenzelm@23252:           handle TERM _ => error "ring_dest_const")),
wenzelm@23573:     mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
chaieb@23330:     conv = fn phi => K numeral_conv}
wenzelm@23573: 
wenzelm@23573: end
wenzelm@23252: *}
wenzelm@23252: 
wenzelm@26462: declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
wenzelm@23573: 
wenzelm@23252: 
wenzelm@23258: locale gb_ring = gb_semiring +
wenzelm@23252:   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@23252:     and neg :: "'a \<Rightarrow> 'a"
wenzelm@23252:   assumes neg_mul: "neg x = mul (neg r1) x"
wenzelm@23252:     and sub_add: "sub x y = add x (neg y)"
wenzelm@23252: begin
wenzelm@23252: 
wenzelm@28856: lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
wenzelm@23252: 
wenzelm@23252: lemmas ring_rules = neg_mul sub_add
wenzelm@23252: 
wenzelm@26462: lemmas gb_ring_axioms' =
wenzelm@26314:   gb_ring_axioms [normalizer
wenzelm@26314:     semiring ops: semiring_ops
wenzelm@26314:     semiring rules: semiring_rules
wenzelm@26314:     ring ops: ring_ops
wenzelm@26314:     ring rules: ring_rules]
wenzelm@23252: 
wenzelm@23252: end
wenzelm@23252: 
wenzelm@23252: 
ballarin@29233: interpretation class_ring!: gb_ring "op +" "op *" "op ^"
ballarin@29223:     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"
haftmann@28823:   proof qed simp_all
wenzelm@23252: 
wenzelm@23252: 
wenzelm@26462: declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
wenzelm@23252: 
wenzelm@23252: use "Tools/Groebner_Basis/normalizer.ML"
wenzelm@23252: 
chaieb@27666: 
wenzelm@23252: method_setup sring_norm = {*
wenzelm@23252:   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
wenzelm@23458: *} "semiring normalizer"
wenzelm@23252: 
wenzelm@23252: 
chaieb@23327: locale gb_field = gb_ring +
chaieb@23327:   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
chaieb@23327:     and inverse:: "'a \<Rightarrow> 'a"
chaieb@23327:   assumes divide: "divide x y = mul x (inverse y)"
chaieb@23327:      and inverse: "inverse x = divide r1 x"
chaieb@23327: begin
chaieb@23327: 
wenzelm@26462: lemmas gb_field_axioms' =
wenzelm@26314:   gb_field_axioms [normalizer
wenzelm@26314:     semiring ops: semiring_ops
wenzelm@26314:     semiring rules: semiring_rules
wenzelm@26314:     ring ops: ring_ops
wenzelm@26314:     ring rules: ring_rules]
chaieb@23327: 
chaieb@23327: end
chaieb@23327: 
wenzelm@23458: 
wenzelm@23266: subsection {* Groebner Bases *}
wenzelm@23252: 
wenzelm@23258: locale semiringb = gb_semiring +
wenzelm@23252:   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
wenzelm@23252:   and add_mul_solve: "add (mul w y) (mul x z) =
wenzelm@23252:     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
wenzelm@23252: begin
wenzelm@23252: 
wenzelm@23252: 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)"
wenzelm@23252: proof-
wenzelm@23252:   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
wenzelm@23252:   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
wenzelm@23252:     using add_mul_solve by blast
wenzelm@23252:   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)"
wenzelm@23252:     by simp
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@23252: lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
wenzelm@23252:   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
wenzelm@23252: proof(clarify)
wenzelm@23252:   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
wenzelm@23252:     and eq: "add b (mul r c) = add b (mul r d)"
wenzelm@23252:   hence "mul r c = mul r d" using cnd add_cancel by simp
wenzelm@23252:   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
wenzelm@23252:     using mul_0 add_cancel by simp
wenzelm@23252:   thus "False" using add_mul_solve nz cnd by simp
wenzelm@23252: qed
wenzelm@23252: 
chaieb@25250: lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
chaieb@25250: proof-
chaieb@25250:   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
chaieb@25250:   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
chaieb@25250: qed
chaieb@25250: 
wenzelm@26462: declare gb_semiring_axioms' [normalizer del]
wenzelm@23252: 
wenzelm@26462: lemmas semiringb_axioms' = semiringb_axioms [normalizer
wenzelm@23252:   semiring ops: semiring_ops
wenzelm@23252:   semiring rules: semiring_rules
wenzelm@26314:   idom rules: noteq_reduce add_scale_eq_noteq]
wenzelm@23252: 
wenzelm@23252: end
wenzelm@23252: 
chaieb@25250: locale ringb = semiringb + gb_ring + 
chaieb@25250:   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
wenzelm@23252: begin
wenzelm@23252: 
wenzelm@26462: declare gb_ring_axioms' [normalizer del]
wenzelm@23252: 
wenzelm@26462: lemmas ringb_axioms' = ringb_axioms [normalizer
wenzelm@23252:   semiring ops: semiring_ops
wenzelm@23252:   semiring rules: semiring_rules
wenzelm@23252:   ring ops: ring_ops
wenzelm@23252:   ring rules: ring_rules
chaieb@25250:   idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314:   ideal rules: subr0_iff add_r0_iff]
wenzelm@23252: 
wenzelm@23252: end
wenzelm@23252: 
chaieb@25250: 
wenzelm@23252: lemma no_zero_divirors_neq0:
wenzelm@23252:   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
wenzelm@23252:     and ab: "a*b = 0" shows "b = 0"
wenzelm@23252: proof -
wenzelm@23252:   { assume bz: "b \<noteq> 0"
wenzelm@23252:     from no_zero_divisors [OF az bz] ab have False by blast }
wenzelm@23252:   thus "b = 0" by blast
wenzelm@23252: qed
wenzelm@23252: 
ballarin@29233: interpretation class_ringb!: ringb
ballarin@29223:   "op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"
nipkow@29667: proof(unfold_locales, simp add: algebra_simps power_Suc, auto)
wenzelm@23252:   fix w x y z ::"'a::{idom,recpower,number_ring}"
wenzelm@23252:   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
wenzelm@23252:   hence ynz': "y - z \<noteq> 0" by simp
wenzelm@23252:   from p have "w * y + x* z - w*z - x*y = 0" by simp
nipkow@29667:   hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
nipkow@29667:   hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
wenzelm@23252:   with  no_zero_divirors_neq0 [OF ynz']
wenzelm@23252:   have "w - x = 0" by blast
wenzelm@23252:   thus "w = x"  by simp
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@26462: declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
wenzelm@23252: 
ballarin@29233: interpretation natgb!: semiringb
ballarin@29223:   "op +" "op *" "op ^" "0::nat" "1"
nipkow@29667: proof (unfold_locales, simp add: algebra_simps power_Suc)
wenzelm@23252:   fix w x y z ::"nat"
wenzelm@23252:   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
wenzelm@23252:     hence "y < z \<or> y > z" by arith
wenzelm@23252:     moreover {
wenzelm@23252:       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
wenzelm@23252:       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
nipkow@29667:       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
wenzelm@23252:       hence "x*k = w*k" by simp
wenzelm@23252:       hence "w = x" using kp by (simp add: mult_cancel2) }
wenzelm@23252:     moreover {
wenzelm@23252:       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
wenzelm@23252:       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
nipkow@29667:       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
wenzelm@23252:       hence "w*k = x*k" by simp
wenzelm@23252:       hence "w = x" using kp by (simp add: mult_cancel2)}
wenzelm@23252:     ultimately have "w=x" by blast }
wenzelm@23252:   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
wenzelm@23252: qed
wenzelm@23252: 
wenzelm@26462: declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
wenzelm@23252: 
chaieb@23327: locale fieldgb = ringb + gb_field
chaieb@23327: begin
chaieb@23327: 
wenzelm@26462: declare gb_field_axioms' [normalizer del]
chaieb@23327: 
wenzelm@26462: lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
chaieb@23327:   semiring ops: semiring_ops
chaieb@23327:   semiring rules: semiring_rules
chaieb@23327:   ring ops: ring_ops
chaieb@23327:   ring rules: ring_rules
chaieb@25250:   idom rules: noteq_reduce add_scale_eq_noteq
wenzelm@26314:   ideal rules: subr0_iff add_r0_iff]
wenzelm@26314: 
chaieb@23327: end
chaieb@23327: 
chaieb@23327: 
wenzelm@23258: lemmas bool_simps = simp_thms(1-34)
wenzelm@23252: lemma dnf:
wenzelm@23252:     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
wenzelm@23252:     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
wenzelm@23252:   by blast+
wenzelm@23252: 
wenzelm@23252: lemmas weak_dnf_simps = dnf bool_simps
wenzelm@23252: 
wenzelm@23252: lemma nnf_simps:
wenzelm@23252:     "(\<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)"
wenzelm@23252:     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
wenzelm@23252:   by blast+
wenzelm@23252: 
wenzelm@23252: lemma PFalse:
wenzelm@23252:     "P \<equiv> False \<Longrightarrow> \<not> P"
wenzelm@23252:     "\<not> P \<Longrightarrow> (P \<equiv> False)"
wenzelm@23252:   by auto
wenzelm@23252: use "Tools/Groebner_Basis/groebner.ML"
wenzelm@23252: 
chaieb@23332: method_setup algebra =
wenzelm@23458: {*
chaieb@23332: let
chaieb@23332:  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
chaieb@23332:  val addN = "add"
chaieb@23332:  val delN = "del"
chaieb@23332:  val any_keyword = keyword addN || keyword delN
chaieb@23332:  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
chaieb@23332: in
chaieb@23332: fn src => Method.syntax 
chaieb@23332:     ((Scan.optional (keyword addN |-- thms) []) -- 
chaieb@23332:     (Scan.optional (keyword delN |-- thms) [])) src 
chaieb@23332:  #> (fn ((add_ths, del_ths), ctxt) => 
chaieb@25250:        Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
chaieb@23332: end
chaieb@25250: *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
chaieb@27666: declare dvd_def[algebra]
chaieb@27666: declare dvd_eq_mod_eq_0[symmetric, algebra]
nipkow@30027: declare mod_div_trivial[algebra]
nipkow@30027: declare mod_mod_trivial[algebra]
chaieb@27666: declare conjunct1[OF DIVISION_BY_ZERO, algebra]
chaieb@27666: declare conjunct2[OF DIVISION_BY_ZERO, algebra]
chaieb@27666: declare zmod_zdiv_equality[symmetric,algebra]
chaieb@27666: declare zdiv_zmod_equality[symmetric, algebra]
chaieb@27666: declare zdiv_zminus_zminus[algebra]
chaieb@27666: declare zmod_zminus_zminus[algebra]
chaieb@27666: declare zdiv_zminus2[algebra]
chaieb@27666: declare zmod_zminus2[algebra]
chaieb@27666: declare zdiv_zero[algebra]
chaieb@27666: declare zmod_zero[algebra]
nipkow@30031: declare mod_by_1[algebra]
nipkow@30031: declare div_by_1[algebra]
chaieb@27666: declare zmod_minus1_right[algebra]
chaieb@27666: declare zdiv_minus1_right[algebra]
chaieb@27666: declare mod_div_trivial[algebra]
chaieb@27666: declare mod_mod_trivial[algebra]
chaieb@27666: declare zmod_zmult_self1[algebra]
chaieb@27666: declare zmod_zmult_self2[algebra]
chaieb@27666: declare zmod_eq_0_iff[algebra]
chaieb@27666: declare zdvd_0_left[algebra]
chaieb@27666: declare zdvd1_eq[algebra]
chaieb@27666: declare zmod_eq_dvd_iff[algebra]
chaieb@27666: declare nat_mod_eq_iff[algebra]
wenzelm@23252: 
haftmann@28402: subsection{* Groebner Bases for fields *}
haftmann@28402: 
ballarin@29233: interpretation class_fieldgb!:
ballarin@29223:   fieldgb "op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
haftmann@28402: 
haftmann@28402: lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
haftmann@28402: lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
haftmann@28402:   by simp
haftmann@28402: lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
haftmann@28402:   by simp
haftmann@28402: lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
haftmann@28402:   by simp
haftmann@28402: lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
haftmann@28402:   by simp
haftmann@28402: 
haftmann@28402: lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
haftmann@28402: 
haftmann@28402: lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
haftmann@28402:   by (simp add: add_divide_distrib)
haftmann@28402: lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
haftmann@28402:   by (simp add: add_divide_distrib)
haftmann@28402: 
haftmann@28402: 
haftmann@28402: ML{* 
haftmann@28402: local
haftmann@28402:  val zr = @{cpat "0"}
haftmann@28402:  val zT = ctyp_of_term zr
haftmann@28402:  val geq = @{cpat "op ="}
haftmann@28402:  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
haftmann@28402:  val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
haftmann@28402:  val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
haftmann@28402:  val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
haftmann@28402: 
haftmann@28402:  fun prove_nz ss T t =
haftmann@28402:     let
haftmann@28402:       val z = instantiate_cterm ([(zT,T)],[]) zr
haftmann@28402:       val eq = instantiate_cterm ([(eqT,T)],[]) geq
haftmann@28402:       val th = Simplifier.rewrite (ss addsimps simp_thms)
haftmann@28402:            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
haftmann@28402:                   (Thm.capply (Thm.capply eq t) z)))
haftmann@28402:     in equal_elim (symmetric th) TrueI
haftmann@28402:     end
haftmann@28402: 
haftmann@28402:  fun proc phi ss ct =
haftmann@28402:   let
haftmann@28402:     val ((x,y),(w,z)) =
haftmann@28402:          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
haftmann@28402:     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
haftmann@28402:     val T = ctyp_of_term x
haftmann@28402:     val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
haftmann@28402:     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
haftmann@28402:   in SOME (implies_elim (implies_elim th y_nz) z_nz)
haftmann@28402:   end
haftmann@28402:   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
haftmann@28402: 
haftmann@28402:  fun proc2 phi ss ct =
haftmann@28402:   let
haftmann@28402:     val (l,r) = Thm.dest_binop ct
haftmann@28402:     val T = ctyp_of_term l
haftmann@28402:   in (case (term_of l, term_of r) of
haftmann@28402:       (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
haftmann@28402:         let val (x,y) = Thm.dest_binop l val z = r
haftmann@28402:             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
haftmann@28402:             val ynz = prove_nz ss T y
haftmann@28402:         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
haftmann@28402:         end
haftmann@28402:      | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402:         let val (x,y) = Thm.dest_binop r val z = l
haftmann@28402:             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
haftmann@28402:             val ynz = prove_nz ss T y
haftmann@28402:         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
haftmann@28402:         end
haftmann@28402:      | _ => NONE)
haftmann@28402:   end
haftmann@28402:   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
haftmann@28402: 
haftmann@28402:  fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
haftmann@28402:    | is_number t = can HOLogic.dest_number t
haftmann@28402: 
haftmann@28402:  val is_number = is_number o term_of
haftmann@28402: 
haftmann@28402:  fun proc3 phi ss ct =
haftmann@28402:   (case term_of ct of
haftmann@28402:     Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
haftmann@28402:       let
haftmann@28402:         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402:         val _ = map is_number [a,b,c]
haftmann@28402:         val T = ctyp_of_term c
haftmann@28402:         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
haftmann@28402:       in SOME (mk_meta_eq th) end
haftmann@28402:   | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
haftmann@28402:       let
haftmann@28402:         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402:         val _ = map is_number [a,b,c]
haftmann@28402:         val T = ctyp_of_term c
haftmann@28402:         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
haftmann@28402:       in SOME (mk_meta_eq th) end
haftmann@28402:   | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
haftmann@28402:       let
haftmann@28402:         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
haftmann@28402:         val _ = map is_number [a,b,c]
haftmann@28402:         val T = ctyp_of_term c
haftmann@28402:         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
haftmann@28402:       in SOME (mk_meta_eq th) end
haftmann@28402:   | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402:     let
haftmann@28402:       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402:         val _ = map is_number [a,b,c]
haftmann@28402:         val T = ctyp_of_term c
haftmann@28402:         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
haftmann@28402:       in SOME (mk_meta_eq th) end
haftmann@28402:   | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402:     let
haftmann@28402:       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402:         val _ = map is_number [a,b,c]
haftmann@28402:         val T = ctyp_of_term c
haftmann@28402:         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
haftmann@28402:       in SOME (mk_meta_eq th) end
haftmann@28402:   | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
haftmann@28402:     let
haftmann@28402:       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
haftmann@28402:         val _ = map is_number [a,b,c]
haftmann@28402:         val T = ctyp_of_term c
haftmann@28402:         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
haftmann@28402:       in SOME (mk_meta_eq th) end
haftmann@28402:   | _ => NONE)
haftmann@28402:   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
haftmann@28402: 
haftmann@28402: val add_frac_frac_simproc =
haftmann@28402:        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
haftmann@28402:                      name = "add_frac_frac_simproc",
haftmann@28402:                      proc = proc, identifier = []}
haftmann@28402: 
haftmann@28402: val add_frac_num_simproc =
haftmann@28402:        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
haftmann@28402:                      name = "add_frac_num_simproc",
haftmann@28402:                      proc = proc2, identifier = []}
haftmann@28402: 
haftmann@28402: val ord_frac_simproc =
haftmann@28402:   make_simproc
haftmann@28402:     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
haftmann@28402:              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
haftmann@28402:              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
haftmann@28402:              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
haftmann@28402:              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
haftmann@28402:              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
haftmann@28402:              name = "ord_frac_simproc", proc = proc3, identifier = []}
haftmann@28402: 
haftmann@28402: val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
haftmann@28402:            @{thm "divide_Numeral1"},
haftmann@28402:            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
haftmann@28402:            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
haftmann@28402:            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
haftmann@28402:            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
haftmann@28402:            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
haftmann@28402:            @{thm "diff_def"}, @{thm "minus_divide_left"},
haftmann@28402:            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
haftmann@28402: 
haftmann@28402: local
haftmann@28402: open Conv
haftmann@28402: in
haftmann@28402: val comp_conv = (Simplifier.rewrite
haftmann@28402: (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
huffman@28987:               addsimps ths addsimps simp_thms
haftmann@28402:               addsimprocs field_cancel_numeral_factors
haftmann@28402:                addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
haftmann@28402:                             ord_frac_simproc]
haftmann@28402:                 addcongs [@{thm "if_weak_cong"}]))
haftmann@28402: then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
haftmann@28402:   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
wenzelm@23252: end
haftmann@28402: 
haftmann@28402: fun numeral_is_const ct =
haftmann@28402:   case term_of ct of
haftmann@28402:    Const (@{const_name "HOL.divide"},_) $ a $ b =>
haftmann@28402:      numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
haftmann@28402:  | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
haftmann@28402:  | t => can HOLogic.dest_number t
haftmann@28402: 
haftmann@28402: fun dest_const ct = ((case term_of ct of
haftmann@28402:    Const (@{const_name "HOL.divide"},_) $ a $ b=>
haftmann@28402:     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
haftmann@28402:  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
haftmann@28402:    handle TERM _ => error "ring_dest_const")
haftmann@28402: 
haftmann@28402: fun mk_const phi cT x =
haftmann@28402:  let val (a, b) = Rat.quotient_of_rat x
haftmann@28402:  in if b = 1 then Numeral.mk_cnumber cT a
haftmann@28402:     else Thm.capply
haftmann@28402:          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
haftmann@28402:                      (Numeral.mk_cnumber cT a))
haftmann@28402:          (Numeral.mk_cnumber cT b)
haftmann@28402:   end
haftmann@28402: 
haftmann@28402: in
haftmann@28402:  val field_comp_conv = comp_conv;
haftmann@28402:  val fieldgb_declaration = 
haftmann@28402:   NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
haftmann@28402:    {is_const = K numeral_is_const,
haftmann@28402:     dest_const = K dest_const,
haftmann@28402:     mk_const = mk_const,
haftmann@28402:     conv = K (K comp_conv)}
haftmann@28402: end;
haftmann@28402: *}
haftmann@28402: 
haftmann@28402: declaration fieldgb_declaration
haftmann@28402: 
haftmann@28402: end




