(* Experimental theory: long division of polynomials $Id: LongDiv.thy,v 1.11 2007/09/27 15:55:29 paulson Exp $ Author: Clemens Ballarin, started 23 June 1999 *) theory LongDiv imports PolyHomo begin definition lcoeff :: "'a::ring up => 'a" where "lcoeff p = coeff p (deg p)" definition eucl_size :: "'a::zero up => nat" where "eucl_size p = (if p = 0 then 0 else deg p + 1)" lemma SUM_shrink_below_lemma: "!! f::(nat=>'a::ring). (ALL i. i < m --> f i = 0) --> setsum (%i. f (i+m)) {..d} = setsum f {..m+d}" apply (induct_tac d) apply (induct_tac m) apply simp apply force apply (simp add: add_commute [of m]) done lemma SUM_extend_below: "!! 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 (simp add: SUM_shrink_below_lemma add_diff_inverse leD) lemma up_repr2D: "!! p::'a::ring up. [| deg p <= n; P (setsum (%i. monom (coeff p i) i) {..n}) |] ==> P p" by (simp add: up_repr_le) (* Start of LongDiv *) lemma deg_lcoeff_cancel: "!!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" apply (rule le_less_trans [of _ "deg r - 1"]) prefer 2 apply arith apply (rule deg_aboveI) apply (case_tac "deg r = m") apply clarify apply simp (* case "deg q ~= m" *) apply (subgoal_tac "deg p < m & deg q < m") apply (simp (no_asm_simp) add: deg_aboveD) apply arith done lemma deg_lcoeff_cancel2: "!!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" apply (rule deg_lcoeff_cancel) apply assumption+ apply (rule classical) apply clarify apply (erule notE) apply (rule_tac p = p in up_repr2D, assumption) apply (rule_tac p = q in up_repr2D, assumption) apply (rotate_tac -1) apply (simp add: smult_l_minus) done lemma long_div_eucl_size: "!!g::('a::ring up). g ~= 0 ==> Ex (% (q, r, k). (lcoeff g)^k *s f = q * g + r & (eucl_size r < eucl_size g))" apply (rule_tac P = "%f. Ex (% (q, r, k) . (lcoeff g) ^k *s f = q * g + r & (eucl_size r < eucl_size g))" in wf_induct) (* TO DO: replace by measure_induct *) apply (rule_tac f = eucl_size in wf_measure) apply (case_tac "eucl_size x < eucl_size g") apply (rule_tac x = "(0, x, 0)" in exI) apply (simp (no_asm_simp)) (* case "eucl_size x >= eucl_size g" *) apply (drule_tac x = "lcoeff g *s x - (monom (lcoeff x) (deg x - deg g)) * g" in spec) apply (erule impE) apply (simp (no_asm_use) add: inv_image_def measure_def lcoeff_def) apply (case_tac "x = 0") apply (rotate_tac -1) apply (simp add: eucl_size_def) (* case "x ~= 0 *) apply (rotate_tac -1) apply (simp add: eucl_size_def) apply (rule impI) apply (rule deg_lcoeff_cancel2) (* replace by linear arithmetic??? *) apply (rule_tac [2] le_trans) apply (rule_tac [2] deg_smult_ring) prefer 2 apply simp apply (simp (no_asm)) apply (rule le_trans) apply (rule deg_mult_ring) apply (rule le_trans) (**) apply (rule add_le_mono) apply (rule le_refl) (* term order forces to use this instead of add_le_mono1 *) apply (rule deg_monom_ring) apply (simp (no_asm_simp)) apply force apply (simp (no_asm)) (**) (* This change is probably caused by application of commutativity *) apply (rule_tac m = "deg g" and n = "deg x" in SUM_extend) apply (simp (no_asm)) apply (simp (no_asm_simp)) apply arith apply (rule_tac m = "deg g" and n = "deg g" in SUM_extend_below) apply (rule le_refl) apply (simp (no_asm_simp)) apply arith apply (simp (no_asm)) (**) (* end of subproof deg f1 < deg f *) apply (erule exE) apply (rule_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) " in exI) apply clarify apply (drule sym) apply (tactic {* simp_tac (simpset() addsimps [thm "l_distr", thm "a_assoc"] delsimprocs [ring_simproc]) 1 *}) apply (tactic {* asm_simp_tac (simpset() delsimprocs [ring_simproc]) 1 *}) apply (tactic {* simp_tac (simpset () addsimps [thm "minus_def", thm "smult_r_distr", thm "smult_r_minus", thm "monom_mult_smult", thm "smult_assoc1", thm "smult_assoc2"] delsimprocs [ring_simproc]) 1 *}) apply simp done ML {* simplify (simpset() addsimps [thm "eucl_size_def"] delsimprocs [ring_simproc]) (thm "long_div_eucl_size") *} thm long_div_eucl_size [simplified] lemma long_div_ring: "!!g::('a::ring up). g ~= 0 ==> Ex (% (q, r, k). (lcoeff g)^k *s f = q * g + r & (r = 0 | deg r < deg g))" apply (tactic {* forw_inst_tac [("f", "f")] (simplify (simpset() addsimps [thm "eucl_size_def"] delsimprocs [ring_simproc]) (thm "long_div_eucl_size")) 1 *}) apply (tactic {* auto_tac (claset(), simpset() delsimprocs [ring_simproc]) *}) apply (case_tac "aa = 0") apply blast (* case "aa ~= 0 *) apply (rotate_tac -1) apply auto done (* Next one fails *) lemma long_div_unit: "!!g::('a::ring up). [| g ~= 0; (lcoeff g) dvd 1 |] ==> Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))" apply (frule_tac f = "f" in long_div_ring) apply (erule exE) apply (rule_tac x = "((% (q,r,k) . (inverse (lcoeff g ^k) *s q)) x, (% (q,r,k) . inverse (lcoeff g ^k) *s r) x) " in exI) apply clarify apply (rule conjI) apply (drule sym) apply (tactic {* asm_simp_tac (simpset() addsimps [thm "smult_r_distr" RS sym, thm "smult_assoc2"] delsimprocs [ring_simproc]) 1 *}) apply (simp (no_asm_simp) add: l_inverse_ring unit_power smult_assoc1 [symmetric]) (* degree property *) apply (erule disjE) apply (simp (no_asm_simp)) apply (rule disjI2) apply (rule le_less_trans) apply (rule deg_smult_ring) apply (simp (no_asm_simp)) done lemma long_div_theorem: "!!g::('a::field up). g ~= 0 ==> Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))" apply (rule long_div_unit) apply assumption apply (simp (no_asm_simp) add: lcoeff_def lcoeff_nonzero field_ax) done lemma uminus_zero: "- (0::'a::ring) = 0" by simp lemma diff_zero_imp_eq: "!!a::'a::ring. a - b = 0 ==> a = b" apply (rule_tac s = "a - (a - b) " in trans) apply (tactic {* asm_simp_tac (simpset() delsimprocs [ring_simproc]) 1 *}) apply simp apply (simp (no_asm)) done lemma eq_imp_diff_zero: "!!a::'a::ring. a = b ==> a + (-b) = 0" by simp lemma long_div_quo_unique: "!!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" apply (subgoal_tac "(q1 - q2) * g = r2 - r1") (* 1 *) apply (erule_tac V = "f = ?x" in thin_rl) apply (erule_tac V = "f = ?x" in thin_rl) apply (rule diff_zero_imp_eq) apply (rule classical) apply (erule disjE) (* r1 = 0 *) apply (erule disjE) (* r2 = 0 *) apply (tactic {* asm_full_simp_tac (simpset() addsimps [thm "integral_iff", thm "minus_def", thm "l_zero", thm "uminus_zero"] delsimprocs [ring_simproc]) 1 *}) (* r2 ~= 0 *) apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong) apply (tactic {* asm_full_simp_tac (simpset() addsimps [thm "minus_def", thm "l_zero", thm "uminus_zero"] delsimprocs [ring_simproc]) 1 *}) (* r1 ~=0 *) apply (erule disjE) (* r2 = 0 *) apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong) apply (tactic {* asm_full_simp_tac (simpset() addsimps [thm "minus_def", thm "l_zero", thm "uminus_zero"] delsimprocs [ring_simproc]) 1 *}) (* r2 ~= 0 *) apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong) apply (tactic {* asm_full_simp_tac (simpset() addsimps [thm "minus_def"] delsimprocs [ring_simproc]) 1 *}) apply (drule order_eq_refl [THEN add_leD2]) apply (drule leD) apply (erule notE, rule deg_add [THEN le_less_trans]) apply (simp (no_asm_simp)) (* proof of 1 *) apply (rule diff_zero_imp_eq) apply hypsubst apply (drule_tac a = "?x+?y" in eq_imp_diff_zero) apply simp done lemma long_div_rem_unique: "!!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" apply (subgoal_tac "q1 = q2") apply (metis a_comm a_lcancel m_comm) apply (metis a_comm l_zero long_div_quo_unique m_comm conc) done end
lemma SUM_shrink_below_lemma:
(∀i<m. f i = (0::'a)) --> (∑i≤d. f (i + m)) = setsum f {..m + d}
lemma SUM_extend_below:
[| m ≤ n; !!i. i < m ==> f i = (0::'a); P (∑i≤n - m. f (i + m)) |]
==> P (setsum f {..n})
lemma up_repr2D:
[| deg p ≤ n; P (∑i≤n. coeff p i*X^i) |] ==> P p
lemma deg_lcoeff_cancel:
[| deg p ≤ deg r; deg q ≤ deg r; coeff p (deg r) = - coeff q (deg r);
deg r ≠ 0 |]
==> deg (p + q) < deg r
lemma deg_lcoeff_cancel2:
[| deg p ≤ deg r; deg q ≤ deg r; p ≠ - q; coeff p (deg r) = - coeff q (deg r) |]
==> deg (p + q) < deg r
lemma long_div_eucl_size:
g ≠ 0
==> Ex (λ(q, r, k). lcoeff g ^ k *s f = q * g + r ∧ eucl_size r < eucl_size g)
lemma long_div_ring:
g ≠ 0
==> Ex (λ(q, r, k). lcoeff g ^ k *s f = q * g + r ∧ (r = 0 ∨ deg r < deg g))
lemma long_div_unit:
[| g ≠ 0; lcoeff g dvd (1::'a) |]
==> Ex (λ(q, r). f = q * g + r ∧ (r = 0 ∨ deg r < deg g))
lemma long_div_theorem:
g ≠ 0 ==> Ex (λ(q, r). f = q * g + r ∧ (r = 0 ∨ deg r < deg g))
lemma uminus_zero:
- (0::'a) = (0::'a)
lemma diff_zero_imp_eq:
a - b = (0::'a) ==> a = b
lemma eq_imp_diff_zero:
a = b ==> a + - b = (0::'a)
lemma long_div_quo_unique:
[| g ≠ 0; f = q1.0 * g + r1.0; r1.0 = 0 ∨ deg r1.0 < deg g; f = q2.0 * g + r2.0;
r2.0 = 0 ∨ deg r2.0 < deg g |]
==> q1.0 = q2.0
lemma long_div_rem_unique:
[| g ≠ 0; f = q1.0 * g + r1.0; r1.0 = 0 ∨ deg r1.0 < deg g; f = q2.0 * g + r2.0;
r2.0 = 0 ∨ deg r2.0 < deg g |]
==> r1.0 = r2.0