Theory Groebner_Basis

Up to index of Isabelle/HOL

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

(*  Title:      HOL/Groebner_Basis.thy
    ID:         $Id: Groebner_Basis.thy,v 1.12 2007/10/31 11:19:35 chaieb Exp $
    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 "!!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 "… = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
  also have "… = (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 "… = 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: "¬ 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 => 'a => 'a"
    and neg :: "'a => '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 => 'a => 'a"
    and inverse:: "'a => '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 <-> y = z"
  and add_mul_solve: "add (mul w y) (mul x z) =
    add (mul w z) (mul x y) <-> w = x ∨ y = z"
begin

lemma noteq_reduce: "a ≠ b ∧ c ≠ d <-> add (mul a c) (mul b d) ≠ add (mul a d) (mul b c)"
proof-
  have "a ≠ b ∧ c ≠ d <-> ¬ (a = b ∨ c = d)" by simp
  also have "… <-> add (mul a c) (mul b d) ≠ add (mul a d) (mul b c)"
    using add_mul_solve by blast
  finally show "a ≠ b ∧ c ≠ d <-> add (mul a c) (mul b d) ≠ add (mul a d) (mul b c)"
    by simp
qed

lemma add_scale_eq_noteq: "[|r ≠ r0 ; (a = b) ∧ ~(c = d)|]
  ==> add a (mul r c) ≠ add b (mul r d)"
proof(clarify)
  assume nz: "r≠ r0" and cnd: "c≠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 <-> a = r0"
proof-
  have "a = r0 <-> add x a = add x r0" by (simp add: add_cancel)
  thus "x = add x a <-> a = r0" by (auto simp add: add_c add_0)
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 + 
  assumes subr0_iff: "sub x y = r0 <-> x = y"
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
  ideal rules: subr0_iff add_r0_iff]:
  "ringb add mul pwr r0 r1 sub neg" by fact

end


lemma no_zero_divirors_neq0:
  assumes az: "(a::'a::no_zero_divisors) ≠ 0"
    and ab: "a*b = 0" shows "b = 0"
proof -
  { assume bz: "b ≠ 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 ≠ z"
  hence ynz': "y - z ≠ 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 ≠ z"
    hence "y < z ∨ y > z" by arith
    moreover {
      assume lt:"y <z" hence "∃k. z = y + k ∧ 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 "∃k. y = z + k ∧ 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 ∨ 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
  ideal rules: subr0_iff add_r0_iff]:
  "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 ∧ Q) = (Q ∧ P)" "(P ∨ Q) = (Q ∨ P)"
  by blast+

lemmas weak_dnf_simps = dnf bool_simps

lemma nnf_simps:
    "(¬(P ∧ Q)) = (¬P ∨ ¬Q)" "(¬(P ∨ Q)) = (¬P ∧ ¬Q)" "(P --> Q) = (¬P ∨ Q)"
    "(P = Q) = ((P ∧ Q) ∨ (¬P ∧ ¬ Q))" "(¬ ¬(P)) = P"
  by blast+

lemma PFalse:
    "P ≡ False ==> ¬ P"
    "¬ P ==> (P ≡ 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"

end

Semiring normalization

lemma mul_pwr:

  mul (pwr x p) (pwr x q) = pwr x (p + q)

lemma pwr_mul:

  pwr (mul x y) q = mul (pwr x q) (pwr y q)

lemma pwr_pwr:

  pwr (pwr x p) q = pwr x (p * q)

Declaring the abstract theory

lemma semiring_ops(1):

  TERM add x y

and semiring_ops(2):

  TERM mul x y

and semiring_ops(3):

  TERM pwr x n

and semiring_ops(4):

  TERM r0

and semiring_ops(5):

  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))

lemma axioms:

  gb_semiring add mul pwr r0 r1

lemma nat_arith:

  number_of v + number_of v' =
  (if neg (number_of v) then number_of v'
   else if neg (number_of v') then number_of v else number_of (v + v'))
  number_of v - number_of v' =
  (if neg (number_of v') then number_of v
   else let d = number_of (v + - v') in if neg d then 0 else nat d)
  number_of v * number_of v' =
  (if neg (number_of v) then 0 else number_of (v * v'))
  (number_of v = number_of v') =
  (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
   else if neg (number_of v') then iszero (number_of v)
        else iszero (number_of (v + - v')))
  (number_of v < number_of v') =
  (if neg (number_of v) then neg (number_of (- v'))
   else neg (number_of (v + - v')))

lemma not_iszero_Numeral1:

  ¬ iszero Numeral1

lemma comp_arith:

  Let s f == f s
  bit.B0  bit.B1
  bit.B1  bit.B0
  Numeral.Pls BIT bit.B0 = Numeral.Pls
  Numeral.Min BIT bit.B1 = Numeral.Min
  Numeral.pred Numeral.Pls = Numeral.Min
  Numeral.pred Numeral.Min = Numeral.Min BIT bit.B0
  Numeral.pred (k BIT bit.B1) = k BIT bit.B0
  Numeral.pred (k BIT bit.B0) = Numeral.pred k BIT bit.B1
  Numeral.succ Numeral.Pls = Numeral.Pls BIT bit.B1
  Numeral.succ Numeral.Min = Numeral.Pls
  Numeral.succ (k BIT bit.B1) = Numeral.succ k BIT bit.B0
  Numeral.succ (k BIT bit.B0) = k BIT bit.B1
  Numeral.Pls + k = k
  Numeral.Min + k = Numeral.pred k
  k BIT bit.B0 + l BIT b = (k + l) BIT b
  k BIT bit.B1 + l BIT bit.B0 = (k + l) BIT bit.B1
  k BIT bit.B1 + l BIT bit.B1 = (k + Numeral.succ l) BIT bit.B0
  - Numeral.Pls = Numeral.Pls
  - Numeral.Min = Numeral.Pls BIT bit.B1
  - k BIT bit.B1 = Numeral.pred (- k) BIT bit.B1
  - k BIT bit.B0 = (- k) BIT bit.B0
  Numeral.Pls * w = Numeral.Pls
  Numeral.Min * k = - k
  k BIT bit.B1 * l = (k * l) BIT bit.B0 + l
  k BIT bit.B0 * l = (k * l) BIT bit.B0
  k + Numeral.Pls = k
  k + Numeral.Min = Numeral.pred k
  ¦0::'a¦ = (0::'a)
  ¦1::'a¦ = (1::'a)
  number_of v + number_of w = number_of (v + w)
  - number_of w = number_of (- w)
  - (1::'a) = (-1::'a)
  number_of v * number_of w = number_of (v * w)
  number_of v - number_of w = number_of (v + - w)
  ¦number_of x¦ = (if number_of x < (0::'a) then - number_of x else number_of x)
  number_of v + number_of v' =
  (if neg (number_of v) then number_of v'
   else if neg (number_of v') then number_of v else number_of (v + v'))
  number_of v - number_of v' =
  (if neg (number_of v') then number_of v
   else let d = number_of (v + - v') in if neg d then 0 else nat d)
  number_of v * number_of v' =
  (if neg (number_of v) then 0 else number_of (v * v'))
  (number_of v = number_of v') =
  (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
   else if neg (number_of v') then iszero (number_of v)
        else iszero (number_of (v + - v')))
  (number_of v < number_of v') =
  (if neg (number_of v) then neg (number_of (- v'))
   else neg (number_of (v + - v')))
  (number_of x = number_of y) = iszero (number_of (x + - y))
  iszero (0::'a)
  ¬ iszero (-1::'a)
  iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
  ¬ iszero (number_of (w BIT bit.B1))
  (number_of x < number_of y) = neg (number_of (x + - y))
  ¬ neg Numeral0
  ¬ neg (0::'a)
  ¬ neg (1::'a)
  ¬ iszero (1::'a)
  neg (-1::'a)
  neg (number_of (w BIT x)) = neg (number_of w)
  (number_of x  number_of y) = (¬ neg (number_of (y + - x)))
  (if False then x else y) = y
  (if True then x else y) = x
  0 + n = n
  Suc m + n = Suc (m + n)
  number_of v + (number_of w + z) = number_of (v + w) + z
  number_of v * (number_of w * z) = number_of (v * w) * z
  (1::'a) = Numeral1
  Suc n = n + 1
  (0::'a) = Numeral0
  0 = Numeral0
  1 = Numeral1
  Suc (Suc 0) = 2
  ¬ iszero (1::'a)
  ¬ iszero (number_of (w BIT bit.B1))
  iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
  ¬ iszero (-1::'a)
  iszero Numeral0
  iszero (0::'a)
  ¬ iszero Numeral1

lemma semiring_norm:

  Let s f == f s
  bit.B0  bit.B1
  bit.B1  bit.B0
  Numeral.Pls BIT bit.B0 = Numeral.Pls
  Numeral.Min BIT bit.B1 = Numeral.Min
  Numeral.pred Numeral.Pls = Numeral.Min
  Numeral.pred Numeral.Min = Numeral.Min BIT bit.B0
  Numeral.pred (k BIT bit.B1) = k BIT bit.B0
  Numeral.pred (k BIT bit.B0) = Numeral.pred k BIT bit.B1
  Numeral.succ Numeral.Pls = Numeral.Pls BIT bit.B1
  Numeral.succ Numeral.Min = Numeral.Pls
  Numeral.succ (k BIT bit.B1) = Numeral.succ k BIT bit.B0
  Numeral.succ (k BIT bit.B0) = k BIT bit.B1
  Numeral.Pls + k = k
  Numeral.Min + k = Numeral.pred k
  k BIT bit.B0 + l BIT b = (k + l) BIT b
  k BIT bit.B1 + l BIT bit.B0 = (k + l) BIT bit.B1
  k BIT bit.B1 + l BIT bit.B1 = (k + Numeral.succ l) BIT bit.B0
  - Numeral.Pls = Numeral.Pls
  - Numeral.Min = Numeral.Pls BIT bit.B1
  - k BIT bit.B1 = Numeral.pred (- k) BIT bit.B1
  - k BIT bit.B0 = (- k) BIT bit.B0
  Numeral.Pls * w = Numeral.Pls
  Numeral.Min * k = - k
  k BIT bit.B1 * l = (k * l) BIT bit.B0 + l
  k BIT bit.B0 * l = (k * l) BIT bit.B0
  k + Numeral.Pls = k
  k + Numeral.Min = Numeral.pred k
  ¦0::'a¦ = (0::'a)
  ¦1::'a¦ = (1::'a)
  number_of v + number_of w = number_of (v + w)
  - number_of w = number_of (- w)
  - (1::'a) = (-1::'a)
  number_of v * number_of w = number_of (v * w)
  number_of v - number_of w = number_of (v + - w)
  ¦number_of x¦ = (if number_of x < (0::'a) then - number_of x else number_of x)
  number_of v + number_of v' =
  (if neg (number_of v) then number_of v'
   else if neg (number_of v') then number_of v else number_of (v + v'))
  number_of v - number_of v' =
  (if neg (number_of v') then number_of v
   else let d = number_of (v + - v') in if neg d then 0 else nat d)
  number_of v * number_of v' =
  (if neg (number_of v) then 0 else number_of (v * v'))
  (number_of v = number_of v') =
  (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
   else if neg (number_of v') then iszero (number_of v)
        else iszero (number_of (v + - v')))
  (number_of v < number_of v') =
  (if neg (number_of v) then neg (number_of (- v'))
   else neg (number_of (v + - v')))
  (number_of x = number_of y) = iszero (number_of (x + - y))
  iszero (0::'a)
  ¬ iszero (-1::'a)
  iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
  ¬ iszero (number_of (w BIT bit.B1))
  (number_of x < number_of y) = neg (number_of (x + - y))
  ¬ neg Numeral0
  ¬ neg (0::'a)
  ¬ neg (1::'a)
  ¬ iszero (1::'a)
  neg (-1::'a)
  neg (number_of (w BIT x)) = neg (number_of w)
  (number_of x  number_of y) = (¬ neg (number_of (y + - x)))
  (if False then x else y) = y
  (if True then x else y) = x
  0 + n = n
  Suc m + n = Suc (m + n)
  number_of v + (number_of w + z) = number_of (v + w) + z
  number_of v * (number_of w * z) = number_of (v * w) * z
  (1::'a) = Numeral1
  Suc n = n + 1
  (0::'a) = Numeral0
  0 = Numeral0
  1 = Numeral1
  Suc (Suc 0) = 2
  ¬ iszero (1::'a)
  ¬ iszero (number_of (w BIT bit.B1))
  iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
  ¬ iszero (-1::'a)
  iszero Numeral0
  iszero (0::'a)
  ¬ iszero Numeral1

lemma ring_ops(1):

  TERM sub x y

and ring_ops(2):

  TERM neg x

lemma ring_rules:

  neg x = mul (neg r1) x
  sub x y = add x (neg y)

lemma axioms:

  gb_ring add mul pwr r0 r1 sub neg

lemma axioms:

  gb_field add mul pwr r0 r1 sub neg divide inverse

Groebner Bases

lemma noteq_reduce:

  (a  bc  d) = (add (mul a c) (mul b d)  add (mul a d) (mul b c))

lemma add_scale_eq_noteq:

  [| r  r0; a = bc  d |] ==> add a (mul r c)  add b (mul r d)

lemma add_r0_iff:

  (x = add x a) = (a = r0)

lemma axioms:

  semiringb add mul pwr r0 r1

lemma axioms:

  ringb add mul pwr r0 r1 sub neg

lemma no_zero_divirors_neq0:

  [| a  (0::'a); a * b = (0::'a) |] ==> b = (0::'a)

lemma axioms:

  fieldgb add mul pwr r0 r1 sub neg divide inverse

lemma bool_simps:

  (¬ ¬ P) = P
  ((¬ P) = (¬ Q)) = (P = Q)
  (P  Q) = (P = (¬ Q))
  (P ∨ ¬ P) = True
  PP) = True
  (x = x) = True
  (¬ True) = False
  (¬ False) = True
  P)  P
  P P)
  (True = P) = P
  (P = True) = P
  (False = P) = (¬ P)
  (P = False) = (¬ P)
  (True --> P) = P
  (False --> P) = True
  (P --> True) = True
  (P --> P) = True
  (P --> False) = (¬ P)
  (P --> ¬ P) = (¬ P)
  (P ∧ True) = P
  (True ∧ P) = P
  (P ∧ False) = False
  (False ∧ P) = False
  (PP) = P
  (PPQ) = (PQ)
  (P ∧ ¬ P) = False
  PP) = False
  (P ∨ True) = True
  (True ∨ P) = True
  (P ∨ False) = P
  (False ∨ P) = P
  (PP) = P
  (PPQ) = (PQ)

lemma dnf:

  (P ∧ (QR)) = (PQPR)
  ((QR) ∧ P) = (QPRP)
  (PQ) = (QP)
  (PQ) = (QP)

lemma weak_dnf_simps:

  (P ∧ (QR)) = (PQPR)
  ((QR) ∧ P) = (QPRP)
  (PQ) = (QP)
  (PQ) = (QP)
  (¬ ¬ P) = P
  ((¬ P) = (¬ Q)) = (P = Q)
  (P  Q) = (P = (¬ Q))
  (P ∨ ¬ P) = True
  PP) = True
  (x = x) = True
  (¬ True) = False
  (¬ False) = True
  P)  P
  P P)
  (True = P) = P
  (P = True) = P
  (False = P) = (¬ P)
  (P = False) = (¬ P)
  (True --> P) = P
  (False --> P) = True
  (P --> True) = True
  (P --> P) = True
  (P --> False) = (¬ P)
  (P --> ¬ P) = (¬ P)
  (P ∧ True) = P
  (True ∧ P) = P
  (P ∧ False) = False
  (False ∧ P) = False
  (PP) = P
  (PPQ) = (PQ)
  (P ∧ ¬ P) = False
  PP) = False
  (P ∨ True) = True
  (True ∨ P) = True
  (P ∨ False) = P
  (False ∨ P) = P
  (PP) = P
  (PPQ) = (PQ)

lemma nnf_simps:

  (¬ (PQ)) = (¬ P ∨ ¬ Q)
  (¬ (PQ)) = (¬ P ∧ ¬ Q)
  (P --> Q) = (¬ PQ)
  (P = Q) = (PQ ∨ ¬ P ∧ ¬ Q)
  (¬ ¬ P) = P

lemma PFalse:

  P == False ==> ¬ P
  ¬ P ==> P == False