+
+static unsigned clifford_check6(const matrix & A)
+{
+ varidx v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4),
+ psi(symbol("psi"),4), lam(symbol("lambda"), 4),
+ xi(symbol("xi"), 4), rho(symbol("rho"),4);
+
+ matrix A_symm(4,4), A2(4, 4);
+ A_symm = A.add(A.transpose()).mul(half);
+ A2 = A_symm.mul(A_symm);
+
+ ex e, e1;
+ int result = 0;
+
+ // checks general identities and contractions for clifford_unit
+ e = dirac_ONE(2) * clifford_unit(mu, A, 2) * dirac_ONE(2);
+ result += check_equal(e, clifford_unit(mu, A, 2));
+
+ e = clifford_unit(idx(2, 4), A) * clifford_unit(idx(1, 4), A)
+ * clifford_unit(idx(1, 4), A) * clifford_unit(idx(2, 4), A);
+ result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
+
+ e = clifford_unit(varidx(2, 4), A) * clifford_unit(varidx(1, 4), A)
+ * clifford_unit(varidx(1, 4), A) * clifford_unit(varidx(2, 4), A);
+ result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
+
+ e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A);
+ result += check_equal_simplify(e, A.trace() * dirac_ONE());
+
+ e = clifford_unit(nu, A) * clifford_unit(nu, A);
+ result += check_equal_simplify(e, indexed(A_symm, sy_symm(), nu, nu) * dirac_ONE());
+
+ e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A) * clifford_unit(mu, A);
+ result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A));
+
+ e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu.toggle_variance(), A);
+
+ result += check_equal_simplify_term(e, 2 * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu) *clifford_unit(nu, A)-A.trace()*clifford_unit(mu, A), mu);
+
+ e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A)
+ * clifford_unit(mu, A) * clifford_unit(mu.toggle_variance(), A);
+ result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
+
+ e = clifford_unit(mu, A) * clifford_unit(nu, A)
+ * clifford_unit(nu.toggle_variance(), A) * clifford_unit(mu.toggle_variance(), A);
+ result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
+
+ e = clifford_unit(mu, A) * clifford_unit(nu, A)
+ * clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu.toggle_variance(), A);
+
+ result += check_equal_simplify_term2(e, 2*indexed(A_symm, sy_symm(), nu.toggle_variance(), mu.toggle_variance()) * clifford_unit(mu, A) * clifford_unit(nu, A) - pow(A.trace(), 2)*dirac_ONE());
+
+ e = clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu, A)
+ * clifford_unit(mu, A) * clifford_unit(nu.toggle_variance(), A);
+
+ result += check_equal_simplify_term2(e, 2*indexed(A_symm, nu, mu) * clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu.toggle_variance(), A) - pow(A.trace(), 2)*dirac_ONE());
+
+ e = clifford_unit(nu.toggle_variance(), A) * clifford_unit(rho.toggle_variance(), A)
+ * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A);
+ e = e.simplify_indexed().collect(clifford_unit(mu, A));
+
+ result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu.toggle_variance(), rho)*indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) *clifford_unit(nu, A)
+ - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho.toggle_variance(), mu)
+ + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
+
+ e = clifford_unit(nu.toggle_variance(), A) * clifford_unit(rho, A)
+ * clifford_unit(mu, A) * clifford_unit(rho.toggle_variance(), A) * clifford_unit(nu, A);
+ e = e.simplify_indexed().collect(clifford_unit(mu, A));
+
+ result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu.toggle_variance(), rho)*indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) *clifford_unit(nu, A)
+ - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho.toggle_variance(), mu)
+ + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
+
+ e = clifford_unit(mu, A) * clifford_unit(nu, A) + clifford_unit(nu, A) * clifford_unit(mu, A);
+ result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(A_symm, sy_symm(), mu, nu));
+
+ e = (clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A)
+ + clifford_unit(nu, A) * clifford_unit(lam, A) * clifford_unit(mu, A)
+ + clifford_unit(lam, A) * clifford_unit(mu, A) * clifford_unit(nu, A)
+ - clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(lam, A)
+ - clifford_unit(lam, A) * clifford_unit(nu, A) * clifford_unit(mu, A)
+ - clifford_unit(mu, A) * clifford_unit(lam, A) * clifford_unit(nu, A)) / 6
+ + indexed(A_symm, sy_symm(), mu, nu) * clifford_unit(lam, A)
+ - indexed(A_symm, sy_symm(), mu, lam) * clifford_unit(nu, A)
+ + indexed(A_symm, sy_symm(), nu, lam) * clifford_unit(mu, A)
+ - clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A);
+ result += check_equal(canonicalize_clifford(e), 0);
+
+ // lst_to_clifford() and clifford_inverse() check
+ realsymbol x("x"), y("y"), t("t"), z("z");
+
+ ex c = clifford_unit(nu, A, 1);
+ e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);
+ e1 = clifford_inverse(e);
+ result += check_equal((e*e1).simplify_indexed(), dirac_ONE(1));
+
+ // Moebius map (both forms) checks for symmetric metrics only
+ matrix M1(2, 2), M2(2, 2);
+ c = clifford_unit(nu, A);
+
+ e = clifford_moebius_map(0, dirac_ONE(),
+ dirac_ONE(), 0, lst(t, x, y, z), A); // this is just the inversion
+ M1 = 0, dirac_ONE(),
+ dirac_ONE(), 0;
+ e1 = clifford_moebius_map(M1, lst(t, x, y, z), A); // the inversion again
+ result += check_equal_lst(e, e1);
+
+ e1 = clifford_to_lst(clifford_inverse(lst_to_clifford(lst(t, x, y, z), mu, A)), c);
+ result += check_equal_lst(e, e1);
+
+ e = clifford_moebius_map(dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), nu, A),
+ 0, dirac_ONE(), lst(t, x, y, z), A); //this is just a shift
+ M2 = dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), c),
+ 0, dirac_ONE();
+ e1 = clifford_moebius_map(M2, lst(t, x, y, z), c); // the same shift
+ result += check_equal_lst(e, e1);
+
+ result += check_equal(e, lst(t+1, x+2, y+3, z+4));
+
+ // Check the group law for Moebius maps
+ e = clifford_moebius_map(M1, ex_to<lst>(e1), c); //composition of M1 and M2
+ e1 = clifford_moebius_map(M1.mul(M2), lst(t, x, y, z), c); // the product M1*M2
+ result += check_equal_lst(e, e1);
+
+ return result;
+}
+
+
+static unsigned clifford_check7(const ex & G, const symbol & dim)
+{
+ // checks general identities and contractions
+
+ unsigned result = 0;
+
+ varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
+ psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
+
+ ex e;
+ clifford unit = ex_to<clifford>(clifford_unit(mu, G));
+ ex scalar = unit.get_metric(varidx(0, dim), varidx(0, dim));
+
+ e = dirac_ONE() * dirac_ONE();
+ result += check_equal(e, dirac_ONE());
+
+ e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
+ result += check_equal(e, clifford_unit(mu, G));
+
+ e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
+ * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
+ result += check_equal(e, dirac_ONE()*pow(scalar, 2));
+
+ e = clifford_unit(mu, G) * clifford_unit(nu, G)
+ * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
+ result += check_equal_simplify(e, pow(dim*scalar, 2) * dirac_ONE());
+
+ e = clifford_unit(mu, G) * clifford_unit(nu, G)
+ * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
+ result += check_equal_simplify(e, (2*dim - pow(dim, 2))*pow(scalar,2)*dirac_ONE());
+
+ e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
+ * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
+ e = e.simplify_indexed().collect(clifford_unit(mu, G));
+ result += check_equal(e, pow(scalar*(dim-2), 2).expand() * clifford_unit(mu, G));
+
+ // canonicalize_clifford() checks, only for symmetric metrics
+ if (ex_to<symmetry>(ex_to<indexed>(ex_to<clifford>(clifford_unit(mu, G)).get_metric()).get_symmetry()).has_symmetry()) {
+ e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
+ result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*unit.get_metric(nu, mu));
+
+ e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
+ + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
+ + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
+ - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
+ - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
+ + unit.get_metric(mu, nu) * clifford_unit(lam, G)
+ - unit.get_metric(mu, lam) * clifford_unit(nu, G)
+ + unit.get_metric(nu, lam) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
+ result += check_equal(canonicalize_clifford(e), 0);
+ } else {
+ e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
+ result += check_equal(canonicalize_clifford(e), dirac_ONE()*(unit.get_metric(mu, nu) + unit.get_metric(nu, mu)));
+
+ e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
+ + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
+ + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
+ - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
+ - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
+ + half * (unit.get_metric(mu, nu) + unit.get_metric(nu, mu)) * clifford_unit(lam, G)
+ - half * (unit.get_metric(mu, lam) + unit.get_metric(lam, mu)) * clifford_unit(nu, G)
+ + half * (unit.get_metric(nu, lam) + unit.get_metric(lam, nu)) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
+ result += check_equal(canonicalize_clifford(e), 0);
+ }
+ return result;
+}
+
+unsigned exam_clifford()