(* Long division of polynomials $Id: LongDiv.ML,v 1.12 2004/05/10 14:40:54 obua Exp $ Author: Clemens Ballarin, started 23 June 1999 *) (* legacy bindings and theorems *) val deg_aboveI = thm "deg_aboveI"; val smult_l_minus = thm "smult_l_minus"; val deg_monom_ring = thm "deg_monom_ring"; val deg_smult_ring = thm "deg_smult_ring"; val smult_l_distr = thm "smult_l_distr"; val smult_r_distr = thm "smult_r_distr"; val smult_r_minus = thm "smult_r_minus"; val smult_assoc2 = thm "smult_assoc2"; val smult_assoc1 = thm "smult_assoc1"; val monom_mult_smult = thm "monom_mult_smult"; val field_ax = thm "field_ax"; val lcoeff_nonzero = thm "lcoeff_nonzero"; val lcoeff_def = thm "lcoeff_def"; val eucl_size_def = thm "eucl_size_def"; val SUM_shrink_below_lemma = thm "SUM_shrink_below_lemma"; Goal "!! f::(nat=>'a::ring). \ \ [| m <= n; !!i. i < m ==> f i = 0; P (setsum (%i. f (i+m)) {..n-m}) |] \ \ ==> P (setsum f {..n})"; by (asm_full_simp_tac (simpset() addsimps [SUM_shrink_below_lemma, add_diff_inverse, leD]) 1); qed "SUM_extend_below"; Goal "!! p::'a::ring up. \ \ [| deg p <= n; P (setsum (%i. monom (coeff p i) i) {..n}) |] \ \ ==> P p"; by (asm_full_simp_tac (simpset() addsimps [thm "up_repr_le"]) 1); qed "up_repr2D"; (* Start of LongDiv *) Goal "!!p::('a::ring up). \ \ [| deg p <= deg r; deg q <= deg r; \ \ coeff p (deg r) = - (coeff q (deg r)); deg r ~= 0 |] ==> \ \ deg (p + q) < deg r"; by (res_inst_tac [("j", "deg r - 1")] le_less_trans 1); by (arith_tac 2); by (rtac deg_aboveI 1); by (strip_tac 1); by (case_tac "deg r = m" 1); by (Clarify_tac 1); by (Asm_full_simp_tac 1); (* case "deg q ~= m" *) by (subgoal_tac "deg p < m & deg q < m" 1); by (asm_simp_tac (simpset() addsimps [deg_aboveD]) 1); by (arith_tac 1); qed "deg_lcoeff_cancel"; Goal "!!p::('a::ring up). \ \ [| deg p <= deg r; deg q <= deg r; \ \ p ~= -q; coeff p (deg r) = - (coeff q (deg r)) |] ==> \ \ deg (p + q) < deg r"; by (rtac deg_lcoeff_cancel 1); by (REPEAT (atac 1)); by (rtac classical 1); by (Clarify_tac 1); by (etac notE 1); by (res_inst_tac [("p", "p")] up_repr2D 1 THEN atac 1); by (res_inst_tac [("p", "q")] up_repr2D 1 THEN atac 1); by (rotate_tac ~1 1); by (asm_full_simp_tac (simpset() addsimps [smult_l_minus]) 1); qed "deg_lcoeff_cancel2"; Goal "!!g::('a::ring up). g ~= 0 ==> \ \ Ex (% (q, r, k). \ \ (lcoeff g)^k *s f = q * g + r & (eucl_size r < eucl_size g))"; by (res_inst_tac [("P", "%f. Ex (% (q, r, k). \ \ (lcoeff g)^k *s f = q * g + r & (eucl_size r < eucl_size g))")] wf_induct 1); (* TO DO: replace by measure_induct *) by (res_inst_tac [("f", "eucl_size")] wf_measure 1); by (case_tac "eucl_size x < eucl_size g" 1); by (res_inst_tac [("x", "(0, x, 0)")] exI 1); by (Asm_simp_tac 1); (* case "eucl_size x >= eucl_size g" *) by (dres_inst_tac [("x", "lcoeff g *s x - (monom (lcoeff x) (deg x - deg g)) * g")] spec 1); by (etac impE 1); by (full_simp_tac (simpset() addsimps [inv_image_def, measure_def, lcoeff_def]) 1); by (case_tac "x = 0" 1); by (rotate_tac ~1 1); by (asm_full_simp_tac (simpset() addsimps [eucl_size_def]) 1); (* case "x ~= 0 *) by (rotate_tac ~1 1); by (asm_full_simp_tac (simpset() addsimps [eucl_size_def]) 1); (* by (Simp_tac 1); *) by (rtac impI 1); by (rtac deg_lcoeff_cancel2 1); (* replace by linear arithmetic??? *) by (rtac le_trans 2); by (rtac deg_smult_ring 2); by (Simp_tac 2); by (Simp_tac 1); by (rtac le_trans 1); by (rtac deg_mult_ring 1); by (rtac le_trans 1); (**) by (rtac add_le_mono 1); by (rtac le_refl 1); (* term order forces to use this instead of add_le_mono1 *) by (rtac deg_monom_ring 1); by (Asm_simp_tac 1); (**) (* by (rtac add_le_mono1 1); by (rtac deg_smult_ring 1); (* by (asm_simp_tac (simpset() addsimps [leI]) 1); *) by (Asm_simp_tac 1); by (cut_inst_tac [("m", "deg x - deg g"), ("'a", "'a")] deg_monom_ring 1); by (arith_tac 1); *) by (Force_tac 1); by (Simp_tac 1); (**) (* This change is probably caused by application of commutativity *) by (res_inst_tac [("m", "deg g"), ("n", "deg x")] SUM_extend 1); by (Simp_tac 1); by (Asm_simp_tac 1); by (arith_tac 1); by (res_inst_tac [("m", "deg g"), ("n", "deg g")] SUM_extend_below 1); by (rtac le_refl 1); by (Asm_simp_tac 1); by (arith_tac 1); by (Simp_tac 1); (**) (* by (res_inst_tac [("m", "deg x - deg g"), ("n", "deg x")] SUM_extend 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps [less_not_refl2 RS not_sym]) 1); by (res_inst_tac [("m", "deg x - deg g"), ("n", "deg x - deg g")] SUM_extend_below 1); by (rtac le_refl 1); by (asm_simp_tac (simpset() addsimps [less_not_refl2]) 1); by (asm_simp_tac (simpset() addsimps [diff_diff_right, leI, m_comm]) 1); *) (* end of subproof deg f1 < deg f *) by (etac exE 1); by (res_inst_tac [("x", "((%(q,r,k). (monom (lcoeff g ^ k * lcoeff x) (deg x - deg g) + q)) xa, (%(q,r,k). r) xa, (%(q,r,k). Suc k) xa)")] exI 1); by (Clarify_tac 1); by (dtac sym 1); by (simp_tac (simpset() addsimps [l_distr, a_assoc] delsimprocs [ring_simproc]) 1); by (asm_simp_tac (simpset() delsimprocs [ring_simproc]) 1); by (simp_tac (simpset() addsimps [minus_def, smult_r_distr, smult_r_minus, monom_mult_smult, smult_assoc1, smult_assoc2] delsimprocs [ring_simproc]) 1); by (Simp_tac 1); qed "long_div_eucl_size"; Goal "!!g::('a::ring up). g ~= 0 ==> \ \ Ex (% (q, r, k). \ \ (lcoeff g)^k *s f = q * g + r & (r = 0 | deg r < deg g))"; by (forw_inst_tac [("f", "f")] (simplify (simpset() addsimps [eucl_size_def] delsimprocs [ring_simproc]) long_div_eucl_size) 1); by (auto_tac (claset(), simpset() delsimprocs [ring_simproc])); by (case_tac "aa = 0" 1); by (Blast_tac 1); (* case "aa ~= 0 *) by (rotate_tac ~1 1); by Auto_tac; qed "long_div_ring"; (* Next one fails *) Goal "!!g::('a::ring up). [| g ~= 0; (lcoeff g) dvd 1 |] ==> \ \ Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))"; by (forw_inst_tac [("f", "f")] long_div_ring 1); by (etac exE 1); by (res_inst_tac [("x", "((%(q,r,k). (inverse(lcoeff g ^k) *s q)) x, \ \ (%(q,r,k). inverse(lcoeff g ^k) *s r) x)")] exI 1); by (Clarify_tac 1); (* by (Simp_tac 1); *) by (rtac conjI 1); by (dtac sym 1); by (asm_simp_tac (simpset() addsimps [smult_r_distr RS sym, smult_assoc2] delsimprocs [ring_simproc]) 1); by (asm_simp_tac (simpset() addsimps [l_inverse_ring, unit_power, smult_assoc1 RS sym]) 1); (* degree property *) by (etac disjE 1); by (Asm_simp_tac 1); by (rtac disjI2 1); by (rtac le_less_trans 1); by (rtac deg_smult_ring 1); by (Asm_simp_tac 1); qed "long_div_unit"; Goal "!!g::('a::field up). g ~= 0 ==> \ \ Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))"; by (rtac long_div_unit 1); by (assume_tac 1); by (asm_simp_tac (simpset() addsimps [lcoeff_def, lcoeff_nonzero, field_ax]) 1); qed "long_div_theorem"; Goal "- (0::'a::ring) = 0"; by (Simp_tac 1); val uminus_zero = result(); Goal "!!a::'a::ring. a - b = 0 ==> a = b"; by (res_inst_tac [("s", "a - (a - b)")] trans 1); by (asm_simp_tac (simpset() delsimprocs [ring_simproc]) 1); by (Simp_tac 1); by (Simp_tac 1); val diff_zero_imp_eq = result(); Goal "!!a::'a::ring. a = b ==> a + (-b) = 0"; by (Asm_simp_tac 1); val eq_imp_diff_zero = result(); Goal "!!g::('a::field up). [| g ~= 0; \ \ f = q1 * g + r1; (r1 = 0 | deg r1 < deg g); \ \ f = q2 * g + r2; (r2 = 0 | deg r2 < deg g) |] ==> q1 = q2"; by (subgoal_tac "(q1 - q2) * g = r2 - r1" 1); (* 1 *) by (thin_tac "f = ?x" 1); by (thin_tac "f = ?x" 1); by (rtac diff_zero_imp_eq 1); by (rtac classical 1); by (etac disjE 1); (* r1 = 0 *) by (etac disjE 1); (* r2 = 0 *) by (asm_full_simp_tac (simpset() addsimps [thm "integral_iff", minus_def, l_zero, uminus_zero] delsimprocs [ring_simproc]) 1); (* r2 ~= 0 *) by (dres_inst_tac [("f", "deg"), ("y", "r2 - r1")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [minus_def, l_zero, uminus_zero] delsimprocs [ring_simproc]) 1); (* r1 ~=0 *) by (etac disjE 1); (* r2 = 0 *) by (dres_inst_tac [("f", "deg"), ("y", "r2 - r1")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [minus_def, l_zero, uminus_zero] delsimprocs [ring_simproc]) 1); (* r2 ~= 0 *) by (dres_inst_tac [("f", "deg"), ("y", "r2 - r1")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [minus_def] delsimprocs [ring_simproc]) 1); by (dtac (order_eq_refl RS add_leD2) 1); by (dtac leD 1); by (etac notE 1 THEN rtac (deg_add RS le_less_trans) 1); by (Asm_simp_tac 1); (* proof of 1 *) by (rtac diff_zero_imp_eq 1); by (hyp_subst_tac 1); by (dres_inst_tac [("a", "?x+?y")] eq_imp_diff_zero 1); by (Asm_full_simp_tac 1); qed "long_div_quo_unique"; Goal "!!g::('a::field up). [| g ~= 0; \ \ f = q1 * g + r1; (r1 = 0 | deg r1 < deg g); \ \ f = q2 * g + r2; (r2 = 0 | deg r2 < deg g) |] ==> r1 = r2"; by (subgoal_tac "q1 = q2" 1); by (Clarify_tac 1); by (res_inst_tac [("a", "q2 * g + r1 - q2 * g"), ("b", "q2 * g + r2 - q2 * g")] box_equals 1); by (Asm_full_simp_tac 1); by (Simp_tac 1); by (Simp_tac 1); by (rtac long_div_quo_unique 1); by (REPEAT (atac 1)); qed "long_div_rem_unique";