
diff -r 237190475d79 -r 7957d26c3334 src/HOL/Algebra/poly/LongDiv.thy
--- a/src/HOL/Algebra/poly/LongDiv.thy	Mon Mar 25 19:53:44 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,257 +0,0 @@
-(*  Author: Clemens Ballarin, started 23 June 1999
-
-Experimental theory: long division of polynomials.
-*)
-
-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)
-  using [[simproc del: ring]]
-  apply (simp (no_asm_use) add: l_distr a_assoc)
-  apply (simp (no_asm_simp))
-  apply (simp (no_asm_use) add: minus_def smult_r_distr smult_r_minus
-    monom_mult_smult smult_assoc2)
-  using [[simproc ring]]
-  apply (simp add: smult_assoc1 [symmetric])
-  done
-
-lemma long_div_ring_aux:
-  "(g :: 'a::ring up) ~= 0 ==>
-    Ex (\<lambda>(q, r, k). lcoeff g ^ k *s f = q * g + r \<and>
-      (if r = 0 then 0 else deg r + 1) < (if g = 0 then 0 else deg g + 1))"
-proof -
-  note [[simproc del: ring]]
-  assume "g ~= 0"
-  then show ?thesis
-    by (rule long_div_eucl_size [simplified eucl_size_def])
-qed
-
-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 (frule_tac f = f in long_div_ring_aux)
-  using [[simproc del: ring]]
-  apply auto
-  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)
-   using [[simproc del: ring]]
-   apply (simp (no_asm_simp) add: smult_r_distr [symmetric] smult_assoc2)
-   using [[simproc ring]]
-   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 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 *)
-     using [[simproc del: ring]]
-     apply (simp add: integral_iff minus_def l_zero uminus_zero)
-  (* r2 ~= 0 *)
-    apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong)
-    apply (simp add: minus_def l_zero uminus_zero)
-  (* r1 ~=0 *)
-   apply (erule disjE)
-  (* r2 = 0 *)
-    apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong)
-    apply (simp add: minus_def l_zero uminus_zero)
-  (* r2 ~= 0 *)
-   apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong)
-   apply (simp add: minus_def)
-   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)
-  using [[simproc ring]]
-  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




