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7998
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(*
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Experimental theory: long division of polynomials
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$Id$
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Author: Clemens Ballarin, started 23 June 1999
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*)
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16417
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theory LongDiv imports PolyHomo begin
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7998
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21423
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definition
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lcoeff :: "'a::ring up => 'a" where
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"lcoeff p = coeff p (deg p)"
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7998
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21423
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definition
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eucl_size :: "'a::zero up => nat" where
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"eucl_size p = (if p = 0 then 0 else deg p + 1)"
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14723
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lemma SUM_shrink_below_lemma:
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"!! f::(nat=>'a::ring). (ALL i. i < m --> f i = 0) -->
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setsum (%i. f (i+m)) {..d} = setsum f {..m+d}"
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apply (induct_tac d)
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15481
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apply (induct_tac m)
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21423
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apply simp
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apply force
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apply (simp add: ab_semigroup_add_class.add_commute [of m])
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done
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lemma SUM_extend_below:
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"!! f::(nat=>'a::ring).
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[| m <= n; !!i. i < m ==> f i = 0; P (setsum (%i. f (i+m)) {..n-m}) |]
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==> P (setsum f {..n})"
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by (simp add: SUM_shrink_below_lemma add_diff_inverse leD)
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lemma up_repr2D:
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"!! p::'a::ring up.
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[| deg p <= n; P (setsum (%i. monom (coeff p i) i) {..n}) |]
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==> P p"
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by (simp add: up_repr_le)
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(* Start of LongDiv *)
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lemma deg_lcoeff_cancel:
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"!!p::('a::ring up).
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[| deg p <= deg r; deg q <= deg r;
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coeff p (deg r) = - (coeff q (deg r)); deg r ~= 0 |] ==>
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deg (p + q) < deg r"
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apply (rule_tac j = "deg r - 1" in le_less_trans)
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prefer 2
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apply arith
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apply (rule deg_aboveI)
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apply (case_tac "deg r = m")
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apply clarify
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apply simp
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(* case "deg q ~= m" *)
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apply (subgoal_tac "deg p < m & deg q < m")
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apply (simp (no_asm_simp) add: deg_aboveD)
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apply arith
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done
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lemma deg_lcoeff_cancel2:
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"!!p::('a::ring up).
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[| deg p <= deg r; deg q <= deg r;
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p ~= -q; coeff p (deg r) = - (coeff q (deg r)) |] ==>
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deg (p + q) < deg r"
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apply (rule deg_lcoeff_cancel)
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apply assumption+
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apply (rule classical)
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apply clarify
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apply (erule notE)
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apply (rule_tac p = p in up_repr2D, assumption)
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apply (rule_tac p = q in up_repr2D, assumption)
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apply (rotate_tac -1)
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apply (simp add: smult_l_minus)
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done
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lemma long_div_eucl_size:
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"!!g::('a::ring up). g ~= 0 ==>
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Ex (% (q, r, k).
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(lcoeff g)^k *s f = q * g + r & (eucl_size r < eucl_size g))"
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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)
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(* TO DO: replace by measure_induct *)
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apply (rule_tac f = eucl_size in wf_measure)
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apply (case_tac "eucl_size x < eucl_size g")
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apply (rule_tac x = "(0, x, 0)" in exI)
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apply (simp (no_asm_simp))
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(* case "eucl_size x >= eucl_size g" *)
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apply (drule_tac x = "lcoeff g *s x - (monom (lcoeff x) (deg x - deg g)) * g" in spec)
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apply (erule impE)
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apply (simp (no_asm_use) add: inv_image_def measure_def lcoeff_def)
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apply (case_tac "x = 0")
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apply (rotate_tac -1)
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apply (simp add: eucl_size_def)
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(* case "x ~= 0 *)
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apply (rotate_tac -1)
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apply (simp add: eucl_size_def)
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apply (rule impI)
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apply (rule deg_lcoeff_cancel2)
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(* replace by linear arithmetic??? *)
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apply (rule_tac [2] le_trans)
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apply (rule_tac [2] deg_smult_ring)
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prefer 2
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apply simp
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apply (simp (no_asm))
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apply (rule le_trans)
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apply (rule deg_mult_ring)
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apply (rule le_trans)
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(**)
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apply (rule add_le_mono)
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apply (rule le_refl)
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(* term order forces to use this instead of add_le_mono1 *)
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apply (rule deg_monom_ring)
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apply (simp (no_asm_simp))
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apply force
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apply (simp (no_asm))
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(**)
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(* This change is probably caused by application of commutativity *)
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apply (rule_tac m = "deg g" and n = "deg x" in SUM_extend)
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apply (simp (no_asm))
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apply (simp (no_asm_simp))
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apply arith
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apply (rule_tac m = "deg g" and n = "deg g" in SUM_extend_below)
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apply (rule le_refl)
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apply (simp (no_asm_simp))
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apply arith
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apply (simp (no_asm))
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(**)
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(* end of subproof deg f1 < deg f *)
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apply (erule exE)
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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)
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apply clarify
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apply (drule sym)
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apply (tactic {* simp_tac (simpset() addsimps [thm "l_distr", thm "a_assoc"]
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delsimprocs [ring_simproc]) 1 *})
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apply (tactic {* asm_simp_tac (simpset() delsimprocs [ring_simproc]) 1 *})
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apply (tactic {* simp_tac (simpset () addsimps [thm "minus_def", thm "smult_r_distr",
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thm "smult_r_minus", thm "monom_mult_smult", thm "smult_assoc1", thm "smult_assoc2"]
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delsimprocs [ring_simproc]) 1 *})
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apply simp
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done
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ML {* simplify (simpset() addsimps [thm "eucl_size_def"]
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delsimprocs [ring_simproc]) (thm "long_div_eucl_size") *}
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thm long_div_eucl_size [simplified]
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lemma long_div_ring:
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"!!g::('a::ring up). g ~= 0 ==>
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Ex (% (q, r, k).
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(lcoeff g)^k *s f = q * g + r & (r = 0 | deg r < deg g))"
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apply (tactic {* forw_inst_tac [("f", "f")]
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(simplify (simpset() addsimps [thm "eucl_size_def"]
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delsimprocs [ring_simproc]) (thm "long_div_eucl_size")) 1 *})
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apply (tactic {* auto_tac (claset(), simpset() delsimprocs [ring_simproc]) *})
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apply (case_tac "aa = 0")
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apply blast
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(* case "aa ~= 0 *)
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apply (rotate_tac -1)
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apply auto
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done
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(* Next one fails *)
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lemma long_div_unit:
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"!!g::('a::ring up). [| g ~= 0; (lcoeff g) dvd 1 |] ==>
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Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))"
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apply (frule_tac f = "f" in long_div_ring)
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apply (erule exE)
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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)
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apply clarify
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apply (rule conjI)
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apply (drule sym)
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apply (tactic {* asm_simp_tac
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(simpset() addsimps [thm "smult_r_distr" RS sym, thm "smult_assoc2"]
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delsimprocs [ring_simproc]) 1 *})
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apply (simp (no_asm_simp) add: l_inverse_ring unit_power smult_assoc1 [symmetric])
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(* degree property *)
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apply (erule disjE)
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apply (simp (no_asm_simp))
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apply (rule disjI2)
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apply (rule le_less_trans)
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apply (rule deg_smult_ring)
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apply (simp (no_asm_simp))
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done
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lemma long_div_theorem:
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"!!g::('a::field up). g ~= 0 ==>
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Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))"
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apply (rule long_div_unit)
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apply assumption
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apply (simp (no_asm_simp) add: lcoeff_def lcoeff_nonzero field_ax)
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done
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lemma uminus_zero: "- (0::'a::ring) = 0"
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by simp
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lemma diff_zero_imp_eq: "!!a::'a::ring. a - b = 0 ==> a = b"
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apply (rule_tac s = "a - (a - b) " in trans)
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apply (tactic {* asm_simp_tac (simpset() delsimprocs [ring_simproc]) 1 *})
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apply simp
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apply (simp (no_asm))
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done
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lemma eq_imp_diff_zero: "!!a::'a::ring. a = b ==> a + (-b) = 0"
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by simp
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lemma long_div_quo_unique:
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"!!g::('a::field up). [| g ~= 0;
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f = q1 * g + r1; (r1 = 0 | deg r1 < deg g);
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f = q2 * g + r2; (r2 = 0 | deg r2 < deg g) |] ==> q1 = q2"
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apply (subgoal_tac "(q1 - q2) * g = r2 - r1") (* 1 *)
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apply (erule_tac V = "f = ?x" in thin_rl)
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apply (erule_tac V = "f = ?x" in thin_rl)
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apply (rule diff_zero_imp_eq)
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apply (rule classical)
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apply (erule disjE)
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(* r1 = 0 *)
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apply (erule disjE)
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(* r2 = 0 *)
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apply (tactic {* asm_full_simp_tac (simpset()
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addsimps [thm "integral_iff", thm "minus_def", thm "l_zero", thm "uminus_zero"]
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delsimprocs [ring_simproc]) 1 *})
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(* r2 ~= 0 *)
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apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong)
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apply (tactic {* asm_full_simp_tac (simpset() addsimps
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[thm "minus_def", thm "l_zero", thm "uminus_zero"] delsimprocs [ring_simproc]) 1 *})
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(* r1 ~=0 *)
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apply (erule disjE)
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(* r2 = 0 *)
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apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong)
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apply (tactic {* asm_full_simp_tac (simpset() addsimps
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[thm "minus_def", thm "l_zero", thm "uminus_zero"] delsimprocs [ring_simproc]) 1 *})
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(* r2 ~= 0 *)
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apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong)
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apply (tactic {* asm_full_simp_tac (simpset() addsimps [thm "minus_def"]
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delsimprocs [ring_simproc]) 1 *})
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apply (drule order_eq_refl [THEN add_leD2])
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apply (drule leD)
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apply (erule notE, rule deg_add [THEN le_less_trans])
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apply (simp (no_asm_simp))
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(* proof of 1 *)
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apply (rule diff_zero_imp_eq)
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apply hypsubst
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apply (drule_tac a = "?x+?y" in eq_imp_diff_zero)
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apply simp
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done
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lemma long_div_rem_unique:
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"!!g::('a::field up). [| g ~= 0;
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f = q1 * g + r1; (r1 = 0 | deg r1 < deg g);
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f = q2 * g + r2; (r2 = 0 | deg r2 < deg g) |] ==> r1 = r2"
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apply (subgoal_tac "q1 = q2")
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apply clarify
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apply (rule_tac a = "q2 * g + r1 - q2 * g" and b = "q2 * g + r2 - q2 * g" in box_equals)
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apply simp
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apply (simp (no_asm))
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apply (simp (no_asm))
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apply (rule long_div_quo_unique)
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apply assumption+
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14723
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done
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7998
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end
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