(*  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

 lemma "axioms" [normalizer

     semiring ops: semiring_ops

     semiring rules: semiring_rules]:

   "gb_semiring add mul pwr r0 r1" by fact

 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_1 iszero_number_of_0 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.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

 lemma "axioms" [normalizer

   semiring ops: semiring_ops

   semiring rules: semiring_rules

   ring ops: ring_ops

   ring rules: ring_rules]:

   "gb_ring add mul pwr r0 r1 sub neg" by fact

 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.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

 lemma "axioms" [normalizer

   semiring ops: semiring_ops

   semiring rules: semiring_rules

   ring ops: ring_ops

   ring rules: ring_rules]:

   "gb_field add mul pwr r0 r1 sub neg divide inverse" by fact

 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

 declare "axioms" [normalizer del]

 lemma "axioms" [normalizer

   semiring ops: semiring_ops

   semiring rules: semiring_rules

   idom rules: noteq_reduce add_scale_eq_noteq]:

   "semiringb add mul pwr r0 r1" by fact

 end

 locale ringb = semiringb + gb_ring

 begin

 declare "axioms" [normalizer del]

 lemma "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]:

   "ringb add mul pwr r0 r1 sub neg" by fact

 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.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.axioms} *}

 locale fieldgb = ringb + gb_field

 begin

 declare "axioms" [normalizer del]

 lemma "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]:

   "fieldgb add mul pwr r0 r1 sub neg divide inverse" by unfold_locales

 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.ring_tac add_ths del_ths ctxt))

 end

 *} "solve polynomial equations over (semi)rings using Groebner bases"

 end