(* 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") 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" . end interpretation class_semiring: gb_semiring ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"] by unfold_locales (auto simp add: ring_eq_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 {* fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct); val numeral_conv = Conv.then_conv (Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}), Simplifier.rewrite (HOL_basic_ss addsimps [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)})); *} ML {* fun int_of_rat x = (case Rat.quotient_of_rat x of (i, 1) => i | _ => error "int_of_rat: bad int") *} declaration {* NormalizerData.funs @{thm class_semiring.axioms} {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 => Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)), conv = fn phi => numeral_conv} *} 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" . 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 {* NormalizerData.funs @{thm class_ring.axioms} {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 => Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)), conv = fn phi => numeral_conv} *} 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" subsection {* Gröbner 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 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" . 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" . 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_eq_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_eq_simps) hence "(y - z) * (w - x) = 0" by (simp add: ring_eq_simps) with no_zero_divirors_neq0 [OF ynz'] have "w - x = 0" by blast thus "w = x" by simp qed declaration {* NormalizerData.funs @{thm class_ringb.axioms} {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 => Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)), conv = fn phi => numeral_conv} *} interpretation natgb: semiringb ["op +" "op *" "op ^" "0::nat" "1"] proof (unfold_locales, simp add: ring_eq_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 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_eq_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_eq_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 {* NormalizerData.funs @{thm natgb.axioms} {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 => Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)), conv = fn phi => numeral_conv} *} 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" ML {* fun algebra_tac ctxt i = ObjectLogic.full_atomize_tac i THEN (fn st => rtac (Groebner.ring_conv ctxt (Thm.dest_arg (nth (cprems_of st) (i - 1)))) i st); *} method_setup algebra = {* Method.ctxt_args (Method.SIMPLE_METHOD' o algebra_tac) *} "" end