(*
Experimental theory: long division of polynomials
$Id$
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: ab_semigroup_add_class.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_tac j = "deg r - 1" in le_less_trans)
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 clarify
apply (rule_tac a = "q2 * g + r1 - q2 * g" and b = "q2 * g + r2 - q2 * g" in box_equals)
apply simp
apply (simp (no_asm))
apply (simp (no_asm))
apply (rule long_div_quo_unique)
apply assumption+
done
end