(*  Title:      HOL/Groebner_Basis.thy

    Author:     Amine Chaieb, TU Muenchen

*)

header {* Semiring normalization and Groebner Bases *}

theory Groebner_Basis

imports Numeral_Simprocs Nat_Transfer

uses

  "Tools/Groebner_Basis/normalizer.ML"

  ("Tools/Groebner_Basis/groebner.ML")

begin

subsection {* Semiring normalization *}

setup Normalizer.setup

locale normalizing_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

lemma semiring_ops:

  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"

    and "TERM r0" and "TERM r1" .

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" unfolding One_nat_def 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 normalizing_semiring_axioms' =

  normalizing_semiring_axioms [normalizer

    semiring ops: semiring_ops

    semiring rules: semiring_rules]

end

sublocale comm_semiring_1

  < normalizing!: normalizing_semiring plus times power zero one

proof

qed (simp_all add: algebra_simps)

declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}

locale normalizing_ring = normalizing_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: shows "TERM (sub x y)" and "TERM (neg x)" .

lemmas ring_rules = neg_mul sub_add

lemmas normalizing_ring_axioms' =

  normalizing_ring_axioms [normalizer

    semiring ops: semiring_ops

    semiring rules: semiring_rules

    ring ops: ring_ops

    ring rules: ring_rules]

end

sublocale comm_ring_1

  < normalizing!: normalizing_ring plus times power zero one minus uminus

proof

qed (simp_all add: diff_minus)

declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}

locale normalizing_field = normalizing_ring +

  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

    and inverse:: "'a \<Rightarrow> 'a"

  assumes divide_inverse: "divide x y = mul x (inverse y)"

     and inverse_divide: "inverse x = divide r1 x"

begin

lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .

lemmas field_rules = divide_inverse inverse_divide

lemmas normalizing_field_axioms' =

  normalizing_field_axioms [normalizer

    semiring ops: semiring_ops

    semiring rules: semiring_rules

    ring ops: ring_ops

    ring rules: ring_rules

    field ops: field_ops

    field rules: field_rules]

end

locale normalizing_semiring_cancel = normalizing_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 normalizing_semiring_axioms' [normalizer del]

lemmas normalizing_semiring_cancel_axioms' =

  normalizing_semiring_cancel_axioms [normalizer

    semiring ops: semiring_ops

    semiring rules: semiring_rules

    idom rules: noteq_reduce add_scale_eq_noteq]

end

locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 

  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"

begin

declare normalizing_ring_axioms' [normalizer del]

lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_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

sublocale idom

  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus

proof

  fix w x y z

  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"

  proof

    assume "w * y + x * z = w * z + x * y"

    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)

    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)

    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)

    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)

    then show "w = x \<or> y = z" by auto

  qed (auto simp add: add_ac)

qed (simp_all add: algebra_simps)

declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}

interpretation normalizing_nat!: normalizing_semiring_cancel

  "op +" "op *" "op ^" "0::nat" "1"

proof (unfold_locales, simp add: algebra_simps)

  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 algebra_simps)

      hence "x*k = w*k" by simp

      hence "w = x" using kp by simp }

    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 algebra_simps)

      hence "w*k = x*k" by simp

      hence "w = x" using kp by simp }

    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.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}

locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field

begin

declare normalizing_field_axioms' [normalizer del]

lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer

  semiring ops: semiring_ops

  semiring rules: semiring_rules

  ring ops: ring_ops

  ring rules: ring_rules

  field ops: field_ops

  field rules: field_rules

  idom rules: noteq_reduce add_scale_eq_noteq

  ideal rules: subr0_iff add_r0_iff]

end

sublocale field 

  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse

proof

qed (simp_all add: divide_inverse)

declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}

subsection {* Groebner Bases *}

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

ML {*

structure Algebra_Simplification = Named_Thms(

  val name = "algebra"

  val description = "pre-simplification rules for algebraic methods"

)

*}

setup Algebra_Simplification.setup

declare dvd_def[algebra]

declare dvd_eq_mod_eq_0[symmetric, algebra]

declare mod_div_trivial[algebra]

declare 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 mod_by_1[algebra]

declare div_by_1[algebra]

declare zmod_minus1_right[algebra]

declare zdiv_minus1_right[algebra]

declare mod_div_trivial[algebra]

declare mod_mod_trivial[algebra]

declare mod_mult_self2_is_0[algebra]

declare mod_mult_self1_is_0[algebra]

declare zmod_eq_0_iff[algebra]

declare dvd_0_left_iff[algebra]

declare zdvd1_eq[algebra]

declare zmod_eq_dvd_iff[algebra]

declare nat_mod_eq_iff[algebra]

use "Tools/Groebner_Basis/groebner.ML"

method_setup algebra = Groebner.algebra_method

  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"

end